ANSYS, Inc. TheoryReferenceANSYS Release 9.0

002114November 2004

ANSYS, Inc. is aUL registeredISO 9001: 2000Company.

ANSYS, Inc. Theory Reference

ANSYS Release 9.0

ANSYS, Inc.Southpointe275 Technology DriveCanonsburg, PA 15317ansysinfo@ansys.comhttp://www.ansys.com(T) 724-746-3304(F) 724-514-9494

Copyright and Trademark InformationCopyright © 2004 SAS IP, Inc. All rights reserved. Unauthorized use, distribution or duplication is prohibited.

ANSYS, DesignSpace, CFX, DesignModeler, DesignXplorer, ANSYS Workbench environment, AI*Environment, CADOE and any and all ANSYS, Inc. productnames referenced on any media, manual or the like, are registered trademarks or trademarks of subsidiaries of ANSYS, Inc. located in the United States orother countries. ICEM CFD is a trademark licensed by ANSYS, Inc. All other trademarks and registered trademarks are property of their respective owners.

ANSYS, Inc. is a UL registered ISO 9001: 2000 Company.

ANSYS Inc. products may contain U.S. Patent No. 6,055,541.

Microsoft, Windows, Windows 2000 and Windows XP are registered trademarks of Microsoft Corporation.Inventor and Mechanical Desktop are registered trademarks of Autodesk, Inc.SolidWorks is a registered trademark of SolidWorks Corporation.Pro/ENGINEER is a registered trademark of Parametric Technology Corporation.Unigraphics, Solid Edge and Parasolid are registered trademarks of Electronic Data Systems Corporation (EDS).ACIS and ACIS Geometric Modeler are registered trademarks of Spatial Technology, Inc.

FLEXlm License Manager is a trademark of Macrovision Corporation.

This ANSYS, Inc. software product and program documentation is ANSYS Confidential Information and are furnished by ANSYS, Inc. under an ANSYSsoftware license agreement that contains provisions concerning non-disclosure, copying, length and nature of use, warranties, disclaimers and remedies,and other provisions. The Program and Documentation may be used or copied only in accordance with the terms of that license agreement.

See the ANSYS, Inc. online documentation or the ANSYS, Inc. documentation CD for the complete Legal Notice.

If this is a copy of a document published by and reproduced with the permission of ANSYS, Inc., it might not reflect the organization or physical appearanceof the original. ANSYS, Inc. is not liable for any errors or omissions introduced by the copying process. Such errors are the responsibility of the partyproviding the copy.

Edited by: Peter Kohnke, Ph.D.

Table of Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–11.1. Purpose of the ANSYS Theory Reference ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–11.2. Notation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–11.3. Applicable Products ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–3

1.3.1. ANSYS Products ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–31.3.2. ANSYS Workbench Products ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–4

1.4. Using the ANSYS, Inc. Theory Reference for the ANSYS Workbench Product ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–41.4.1. Elements Used by the ANSYS Workbench Product ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–41.4.2. Solvers Used by the ANSYS Workbench Product ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–41.4.3. Other Features ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–5

2. Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–12.1. Structural Fundamentals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–1

2.1.1. Stress-Strain Relationships ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–12.1.2. Orthotropic Material Transformation for Axisymmetric Models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–52.1.3. Temperature-Dependent Coefficient of Thermal Expansion ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–6

2.2. Derivation of Structural Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–82.3. Structural Strain and Stress Evaluations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–12

2.3.1. Integration Point Strains and Stresses ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–122.3.2. Surface Stresses ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–122.3.3. Shell Element Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–13

2.4. Combined Stresses and Strains ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–152.4.1. Combined Strains ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–152.4.2. Combined Stresses ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–162.4.3. Failure Criteria .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–172.4.4. Maximum Strain Failure Criteria .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–172.4.5. Maximum Stress Failure Criteria .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–182.4.6. Tsai-Wu Failure Criteria .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–182.4.7. Safety Tools in the ANSYS Workbench Product ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–19

3. Structures with Geometric Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–13.1. Large Strain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–1

3.1.1. Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–13.1.2. Implementation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–33.1.3. Definition of Thermal Strains ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–53.1.4. Element Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–63.1.5. Applicable Input ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–73.1.6. Applicable Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–7

3.2. Large Rotation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–73.2.1. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–83.2.2. Implementation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–83.2.3. Element Transformation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–93.2.4. Deformational Displacements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–103.2.5. Updating Rotations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–113.2.6. Applicable Input ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–113.2.7. Applicable Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–113.2.8. Consistent Tangent Stiffness Matrix and Finite Rotation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–11

3.3. Stress Stiffening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–133.3.1. Overview and Usage ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–133.3.2. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–133.3.3. Implementation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–163.3.4. Pressure Load Stiffness ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–18

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

3.3.5. Applicable Input ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–193.3.6. Applicable Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–19

3.4. Spin Softening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–193.5. General Element Formulations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–22

3.5.1. Fundamental Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–233.5.2. Classical Pure Displacement Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–243.5.3. Mixed u-P Formulations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–263.5.4. u-P Formulation I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–273.5.5. u-P Formulation II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–293.5.6. u-P Formulation III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–303.5.7. Volumetric Constraint Equations in u-P Formulations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–30

3.6. Constraints and Lagrange Multiplier Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–314. Structures with Material Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–1

4.1. Rate-Independent Plasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–24.1.1. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–34.1.2. Yield Criterion ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–34.1.3. Flow Rule ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–64.1.4. Hardening Rule ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–64.1.5. Plastic Strain Increment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–84.1.6. Implementation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–94.1.7. Elastoplastic Stress-Strain Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–114.1.8. Specialization for Hardening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–114.1.9. Specification for Nonlinear Isotropic Hardening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–124.1.10. Specialization for Bilinear Kinematic Hardening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–134.1.11. Specialization for Multilinear Kinematic Hardening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–154.1.12. Specialization for Nonlinear Kinematic Hardening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–174.1.13. Specialization for Anisotropic Plasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–194.1.14. Hill Potential Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–194.1.15. Generalized Hill Potential Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–214.1.16. Specialization for Drucker-Prager ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–254.1.17. Cast Iron Material Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–27

4.2. Rate-Dependent Plasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–314.2.1. Creep Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–31

4.2.1.1. Definition and Limitations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–314.2.1.2. Calculation of Creep ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–324.2.1.3. Time Step Size ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–34

4.2.2. Rate-Dependent Plasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–344.2.2.1. Perzyna Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–344.2.2.2. Peirce Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–34

4.2.3. Anand Viscoplasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–354.2.3.1. Overview ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–354.2.3.2. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–354.2.3.3. Implementation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–36

4.3. Gasket Material .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–374.3.1. Stress and Deformation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–384.3.2. Material Definition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–384.3.3. Thermal Deformation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–39

4.4. Nonlinear Elasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–394.4.1. Overview and Guidelines for Use ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–39

4.5. Shape Memory Alloy Material Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–404.5.1. Background ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–404.5.2. The Continuum Mechanics Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–41

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.vi

4.6. Hyperelasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–444.6.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–444.6.2. Finite Strain Elasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–444.6.3. Deviatoric-Volumetric Multiplicative Split .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–454.6.4. Strain Energy Potentials .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–46

4.6.4.1. Neo-Hookean ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–464.6.4.2. Mooney-Rivlin ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–474.6.4.3. Polynomial Form .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–484.6.4.4. Ogden Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–484.6.4.5. Arruda-Boyce Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–494.6.4.6. Gent Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–494.6.4.7. Yeoh Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–504.6.4.8. Ogden Compressible Foam Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–504.6.4.9. Blatz-Ko Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–51

4.6.5. USER Subroutine ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–514.6.6. Mooney-Rivlin (Using TB,MOONEY Command) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–514.6.7. Output Quantities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–534.6.8. Determining Mooney-Rivlin Material Constants ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–534.6.9. Uniaxial Tension (Equivalently, Equibiaxial Compression) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–564.6.10. Equibiaxial Tension (Equivalently, Uniaxial Compression) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–574.6.11. Pure Shear ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–574.6.12. Least Squares Fit Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–584.6.13. Material Stability Check ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–59

4.7. Viscoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–604.7.1. Small Strain Viscoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–604.7.2. Constitutive Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–604.7.3. Numerical Integration ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–624.7.4. Thermorheological Simplicity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–634.7.5. Large Deformation Viscoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–644.7.6. Visco-Hypoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–644.7.7. Large Strain Viscoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–654.7.8. Shift Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–67

4.8. Concrete ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–694.8.1. The Domain (Compression - Compression - Compression) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–714.8.2. The Domain (Tension - Compression - Compression) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–744.8.3. The Domain (Tension - Tension - Compression) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–744.8.4. The Domain (Tension - Tension - Tension) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–75

4.9. Swelling ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–765. Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–1

5.1. Electromagnetic Field Fundamentals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–15.1.1. Magnetic Scalar Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–45.1.2. Solution Strategies ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–4

5.1.2.1. RSP Strategy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–55.1.2.2. DSP Strategy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–65.1.2.3. GSP Strategy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–7

5.1.3. Magnetic Vector Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–85.1.4. Edge Flux Degrees of Freedom .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–95.1.5. Limitation of the Nodal Vector Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–105.1.6. Harmonic Analysis Using Complex Formalism .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–115.1.7. Nonlinear Time-Harmonic Magnetic Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–135.1.8. Electric Scalar Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–14

5.1.8.1. Quasistatic Electric Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–15

viiANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

5.1.8.2. Electrostatic Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–165.2. Derivation of Electromagnetic Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–17

5.2.1. Magnetic Scalar Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–175.2.1.1. Degrees of freedom .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–175.2.1.2. Coefficient Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–175.2.1.3. Applied Loads ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–17

5.2.2. Magnetic Vector Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–185.2.2.1. Degrees of Freedom .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–185.2.2.2. Coefficient Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–185.2.2.3. Applied Loads ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–19

5.2.3. Electric Scalar Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–215.2.3.1. Quasistatic Electric Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–215.2.3.2. Electrostatic Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–22

5.3. Electromagnetic Field Evaluations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–225.3.1. Magnetic Scalar Potential Results .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–225.3.2. Magnetic Vector Potential Results .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–235.3.3. Magnetic Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–24

5.3.3.1. Lorentz forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–245.3.3.2. Maxwell Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–255.3.3.3. Virtual Work Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–26

5.3.3.3.1. Element Shape Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–265.3.3.3.2. Nodal Perturbation Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–27

5.3.4. Joule Heat in a Magnetic Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–275.3.5. Electric Scalar Potential Results .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–28

5.3.5.1. Quasistatic Electric Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–285.3.5.2. Electrostatic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–29

5.3.6. Electrostatic Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–295.3.7. Electric Constitutive Error .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–30

5.4. Voltage Forced and Circuit-Coupled Magnetic Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–315.4.1. Voltage Forced Magnetic Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–315.4.2. Circuit-Coupled Magnetic Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–32

5.5. High-Frequency Electromagnetic Field Simulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–335.5.1. High-Frequency Electromagnetic Field FEA Principle ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–335.5.2. Boundary Conditions and Perfectly Matched Layers (PML) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–38

5.5.2.1. PEC Boundary Condition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–385.5.2.2. PMC Boundary Condition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–385.5.2.3. Impedance Boundary Condition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–395.5.2.4. Perfectly Matched Layers ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–405.5.2.5. Periodic Boundary Condition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–42

5.5.3. Excitation Sources ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–425.5.3.1. Waveguide Modal Sources ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–425.5.3.2. Current Excitation Source ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–425.5.3.3. Plane Wave Source ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–435.5.3.4. Surface Magnetic Field Source ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–435.5.3.5. Electric Field Source ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–44

5.5.4. High-Frequency Parameters Evaluations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–445.5.4.1. Electric Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–445.5.4.2. Magnetic Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–445.5.4.3. Poynting Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–445.5.4.4. Power Flow ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–455.5.4.5. Stored Energy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–455.5.4.6. Dielectric Loss ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–45

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.viii

5.5.4.7. Surface Loss ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–455.5.4.8. Quality Factor ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–455.5.4.9. Voltage ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–465.5.4.10. Current ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–465.5.4.11. Characteristic Impedance ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–465.5.4.12. Scattering Matrix (S-Parameter) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–465.5.4.13. Surface Equivalence Principle ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–485.5.4.14. Radar Cross Section (RCS) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–495.5.4.15. Antenna Pattern ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–505.5.4.16. Antenna Radiation Power ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–515.5.4.17. Antenna Directive Gain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–515.5.4.18. Antenna Power Gain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–515.5.4.19. Antenna Radiation Efficiency ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–525.5.4.20. Electromagnetic Field of Phased Array Antenna ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–525.5.4.21. Specific Absorption Rate (SAR) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–525.5.4.22. Power Reflection and Transmission Coefficient ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–525.5.4.23. Reflection and Transmission Coefficient in Periodic Structure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–535.5.4.24. The Smith Chart .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–545.5.4.25. Conversion Among Scattering Matrix (S-parameter), Admittance Matrix (Y-parameter),and Impedance Matrix (Z-parameter) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–54

5.6. Inductance, Flux and Energy Computation by LMATRIX and SENERGY Macros ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–555.6.1. Differential Inductance Definition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–565.6.2. Review of Inductance Computation Methods ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–575.6.3. Inductance Computation Method Used ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–575.6.4. Transformer and Motion Induced Voltages ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–575.6.5. Absolute Flux Computation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–585.6.6. Inductance Computations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–595.6.7. Absolute Energy Computation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–59

5.7. Electromagnetic Particle Tracing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–605.8. Maxwell Stress Tensor ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–61

5.8.1. Notation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–615.8.2. Fundamental Relations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–625.8.3. Derived Relations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–635.8.4. Maxwell Tensor From Maxwell's Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–64

5.9. Electromechanical Transducers ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–655.10. Capacitance Computation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–655.11. Open Boundary Analysis with a Trefftz Domain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–685.12. Circuit Analysis, Reduced Order Modeling ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–69

5.12.1. Mechanical Circuit Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–695.12.2. Electrical Circuit Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–705.12.3. Coupled Field Circuit Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–70

5.13. Conductance Computation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–706. Heat Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–1

6.1. Heat Flow Fundamentals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–16.1.1. Conduction and Convection ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–16.1.2. Radiation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–3

6.1.2.1. View Factors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–46.1.2.2. Radiation Usage ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–4

6.2. Derivation of Heat Flow Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–56.3. Heat Flow Evaluations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–7

6.3.1. Integration Point Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–76.3.2. Surface Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–7

ixANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

6.4. Radiation Matrix Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–86.4.1. Non-Hidden Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–96.4.2. Hidden Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–96.4.3. View Factors of Axisymmetric Bodies ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–106.4.4. Space Node ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–12

6.5. Radiosity Solution Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–126.5.1. View Factor Calculation - Hemicube Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–13

7. Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–17.1. Fluid Flow Fundamentals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–1

7.1.1. Continuity Equation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–17.1.2. Momentum Equation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–27.1.3. Compressible Energy Equation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–37.1.4. Incompressible Energy Equation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–47.1.5. Turbulence ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–5

7.1.5.1. Standard k-ε Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–77.1.5.2. RNG Turbulence Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–97.1.5.3. NKE Turbulence Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–107.1.5.4. GIR Turbulence Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–117.1.5.5. SZL Turbulence Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–127.1.5.6. Standard k-ω Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–127.1.5.7. SST Turbulence Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–137.1.5.8. Near-Wall Treatment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–14

7.1.6. Pressure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–167.1.7. Multiple Species Transport .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–177.1.8. Arbitrary Lagrangian-Eulerian (ALE) Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–18

7.2. Derivation of Fluid Flow Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–197.2.1. Discretization of Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–197.2.2. Transient Term .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–207.2.3. Advection Term .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–227.2.4. Monotone Streamline Upwind Approach (MSU) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–227.2.5. Streamline Upwind/Petro-Galerkin Approach (SUPG) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–237.2.6. Collocated Galerkin Approach (COLG) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–247.2.7. Diffusion Terms ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–247.2.8. Source Terms ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–257.2.9. Segregated Solution Algorithm .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–25

7.3. Volume of Fluid Method for Free Surface Flows ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–317.3.1. Overview ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–317.3.2. CLEAR-VOF Advection ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–317.3.3. CLEAR-VOF Reconstruction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–337.3.4. Treatment of Finite Element Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–347.3.5. Treatment of Volume Fraction Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–357.3.6. Treatment of Surface Tension Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–37

7.4. Fluid Solvers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–387.5. Overall Convergence and Stability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–39

7.5.1. Convergence ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–397.5.2. Stability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–40

7.5.2.1. Relaxation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–407.5.2.2. Inertial Relaxation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–407.5.2.3. Artificial Viscosity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–40

7.5.3. Residual File .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–417.5.4. Modified Inertial Relaxation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–41

7.6. Fluid Properties ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–42

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.x

7.6.1. Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–427.6.2. Viscosity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–437.6.3. Thermal Conductivity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–467.6.4. Specific Heat ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–477.6.5. Surface Tension Coefficient ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–477.6.6. Wall Static Contact Angle ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–487.6.7. Multiple Species Property Options ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–48

7.7. Derived Quantities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–497.7.1. Mach Number ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–497.7.2. Total Pressure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–497.7.3. Y-Plus and Wall Shear Stress ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–507.7.4. Stream Function ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–51

7.7.4.1. Cartesian Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–517.7.4.2. Axisymmetric Geometry (about x) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–517.7.4.3. Axisymmetric Geometry (about y) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–517.7.4.4. Polar Coordinates ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–51

7.7.5. Heat Transfer Film Coefficient ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–527.7.5.1. Matrix Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–527.7.5.2. Thermal Gradient Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–527.7.5.3. Film Coefficient Evaluation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–52

7.8. Squeeze Film Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–537.8.1. Flow Between Flat Surfaces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–537.8.2. Flow in Channels .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–54

7.9. Slide Film Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–558. Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–1

8.1. Acoustic Fluid Fundamentals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–18.1.1. Governing Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–18.1.2. Discretization of the Lossless Wave Equation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–2

8.2. Derivation of Acoustics Fluid Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–38.3. Absorption of Acoustical Pressure Wave ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–4

8.3.1. Addition of Dissipation due to Damping at the Boundary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–48.4. Acoustics Fluid-Structure Coupling ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–68.5. Acoustics Output Quantities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8–7

9. This chapter intentionally omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–110. This chapter intentionally omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–111. Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1

11.1. Coupled Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–111.1.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1

11.1.1.1. Advantages ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–211.1.1.2. Disadvantages ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–2

11.1.2. Coupling ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–211.1.2.1. Thermal-Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–411.1.2.2. Magneto-Structural Analysis (Vector Potential) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–411.1.2.3. Magneto-Structural Analysis (Scalar Potential) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–411.1.2.4. Electromagnetic Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–511.1.2.5. Electro-Magneto-Thermo-Structural Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–511.1.2.6. Electro-Magneto-Thermal Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–611.1.2.7. Piezoelectric Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–611.1.2.8. Piezoresistive Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–611.1.2.9. Thermo-Pressure Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–711.1.2.10. Velocity-Thermo-Pressure Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–711.1.2.11. Pressure-Structural (Acoustic) Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–8

xiANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

11.1.2.12. Thermo-Electric Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–911.1.2.13. Magnetic-Thermal Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–911.1.2.14. Circuit-Magnetic Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–10

11.2. Piezoelectrics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1311.2.1. Structural Mass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1611.2.2. Structural Damping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.3. Structural Stiffness ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.4. Dielectric Conductivity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.5. Piezoelectric Coupling Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.6. Structural Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.7. Electrical Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.8. Elastic Energy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.9. Dielectric Energy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1711.2.10. Electromechanical Coupling Energy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–17

11.3. Piezoresistivity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1811.4. Thermoelectrics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1911.5. Review of Coupled Electromechanical Methods ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–21

12. Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–112.1. 2-D Lines ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2

12.1.1. 2-D Lines without RDOF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–212.1.2. 2-D Lines with RDOF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2

12.2. 3-D Lines ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–312.2.1. 3-D 2 Node Lines without RDOF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–312.2.2. 3-D 2 Node Lines with RDOF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–412.2.3. 3-D 3 Node Lines ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4

12.3. Axisymmetric Shells .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–512.3.1. Axisymmetric Shell without ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–512.3.2. Axisymmetric Shell with ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–5

12.4. Axisymmetric Harmonic Shells .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–612.4.1. Axisymmetric Harmonic Shells without ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–612.4.2. Axisymmetric Harmonic Shells with ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–6

12.5. 3-D Shells .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–712.5.1. 3-D 3-Node Triangular Shells without RDOF (CST) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–812.5.2. 3-D 6-Node Triangular Shells without RDOF (LST) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–812.5.3. 3-D 3-Node Triangular Shells with RDOF but without SD ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–812.5.4. 3-D 3-Node Triangular Shells with RDOF and with SD ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–912.5.5. 3-D 6-Node Triangular Shells with RDOF and with SD ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1012.5.6. 3-D 4-Node Quadrilateral Shells without RDOF and without ESF (Q4) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1112.5.7. 3-D 4-Node Quadrilateral Shells without RDOF but with ESF (QM6) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1112.5.8. 3-D 8-Node Quadrilateral Shells without RDOF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1212.5.9. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and without ESF ... . . . . . . . . . . . . . . . . 12–1212.5.10. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and with ESF ... . . . . . . . . . . . . . . . . . . . . 12–1312.5.11. 3-D 4-Node Quadrilateral Shells with RDOF and with SD ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1312.5.12. 3-D 8-Node Quadrilateral Shells with RDOF and with SD ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–15

12.6. 2-D and Axisymmetric Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1512.6.1. 2-D and Axisymmetric 3 Node Triangular Solids (CST) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1612.6.2. 2-D and Axisymmetric 6 Node Triangular Solids (LST) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1712.6.3. 2-D and Axisymmetric 4 Node Quadrilateral Solid without ESF (Q4) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1712.6.4. 2-D and Axisymmetric 4 Node Quadrilateral Solids with ESF (QM6) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1812.6.5. 2-D and Axisymmetric 8 Node Quadrilateral Solids (Q8) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1812.6.6. 2-D and Axisymmetric 4 Node Quadrilateral Infinite Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–19

12.6.6.1. Lagrangian Isoparametric Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–19

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xii

12.6.6.2. Mapping Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2012.6.7. 2-D and Axisymmetric 8 Node Quadrilateral Infinite Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–20

12.6.7.1. Lagrangian Isoparametric Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2012.6.7.2. Mapping Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–21

12.7. Axisymmetric Harmonic Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2112.7.1. Axisymmetric Harmonic 3 Node Triangular Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2112.7.2. Axisymmetric Harmonic 6 Node Triangular Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2212.7.3. Axisymmetric Harmonic 4 Node Quadrilateral Solids without ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2212.7.4. Axisymmetric Harmonic 4 Node Quadrilateral Solids with ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2212.7.5. Axisymmetric Harmonic 8 Node Quadrilateral Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–23

12.8. 3-D Solids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2312.8.1. 4 Node Tetrahedra ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2412.8.2. 10 Node Tetrahedra ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2412.8.3. 5 Node Pyramids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2512.8.4. 13 Node Pyramids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2612.8.5. 6 Node Wedges without ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2712.8.6. 6 Node Wedges with ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2812.8.7. 15 Node Wedges as a Condensation of 20 Node Brick ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2812.8.8. 15 Node Wedges Based on Wedge Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2912.8.9. 8 Node Bricks without ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3012.8.10. 8 Node Bricks with ESF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3112.8.11. 20 Node Bricks ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3212.8.12. 8 Node Infinite Bricks ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–33

12.8.12.1. Lagrangian Isoparametric Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3312.8.12.2. Mapping Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–34

12.8.13. 3-D 20 Node Infinite Bricks ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3412.8.13.1. Lagrangian Isoparametric Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3512.8.13.2. Mapping Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–36

12.9. Electromagnetic Edge Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3612.9.1. 2-D 8 Node Quad Geometry and DOFs ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3712.9.2. 3-D 20 Node Brick Geometry and DOFs ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–39

12.10. High Frequency Electromagnetic Tangential Vector Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4112.10.1. Tetrahedral Elements (HF119) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4112.10.2. Hexahedral Elements (HF120) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4312.10.3. Triangular Elements (HF118) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4512.10.4. Quadrilateral Elements (HF118) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–47

13. Element Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–113.1. Integration Point Locations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1

13.1.1. Lines (1, 2, or 3 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–113.1.2. Quadrilaterals (2 x 2 or 3 x 3 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–113.1.3. Bricks and Pyramids (2 x 2 x 2 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–213.1.4. Triangles (1, 3, or 6 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–313.1.5. Tetrahedra (1, 4, 5, or 11 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–313.1.6. Triangles and Tetrahedra (2 x 2 or 2 x 2 x 2 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–513.1.7. Wedges (3 x 2 or 3 x 3 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–513.1.8. Wedges (2 x 2 x 2 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–613.1.9. Bricks (14 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–613.1.10. Nonlinear Bending (5 Points) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–7

13.2. Lumped Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–813.2.1. Diagonalization Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–813.2.2. Limitations of Lumped Mass Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–9

13.3. Reuse of Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–9

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13.3.1. Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–913.3.2. Structure Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1013.3.3. Override Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–10

13.4. Temperature-Dependent Material Properties ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1013.5. Positive Definite Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–11

13.5.1. Matrices Representing the Complete Structure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1113.5.2. Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–12

13.6. Nodal and Centroidal Data Evaluation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1213.7. Element Shape Testing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–13

13.7.1. Overview ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1313.7.2. 3-D Solid Element Faces and Cross-Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1313.7.3. Aspect Ratio ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1613.7.4. Aspect Ratio Calculation for Triangles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1613.7.5. Aspect Ratio Calculation for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1713.7.6. Angle Deviation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1813.7.7. Angle Deviation Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1813.7.8. Parallel Deviation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1913.7.9. Parallel Deviation Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1913.7.10. Maximum Corner Angle ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2013.7.11. Maximum Corner Angle Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2013.7.12. Jacobian Ratio ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–22

13.7.12.1. Jacobian Ratio Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2213.7.13. Warping Factor ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–24

13.7.13.1. Warping Factor Calculation for Quadrilateral Shell Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2413.7.13.2. Warping Factor Calculation for 3-D Solid Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–26

14. Element Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–114.1. LINK1 - 2-D Spar (or Truss) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1

14.1.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–114.1.2. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1

14.2. PLANE2 - 2-D 6-Node Triangular Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–214.2.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2

14.3. BEAM3 - 2-D Elastic Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–214.3.1. Element Matrices and Load Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–314.3.2. Stress Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5

14.4. BEAM4 - 3-D Elastic Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–614.4.1. Stiffness and Mass Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–614.4.2. Gyroscopic Damping Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1014.4.3. Pressure and Temperature Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1014.4.4. Local to Global Conversion ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1014.4.5. Stress Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12

14.5. SOLID5 - 3-D Coupled-Field Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1314.5.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14

14.6. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1414.7. COMBIN7 - Revolute Joint .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14

14.7.1. Element Description ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1414.7.2. Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1614.7.3. Modification of Real Constants ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18

14.8. LINK8 - 3-D Spar (or Truss) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1914.8.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1914.8.2. Element Matrices and Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1914.8.3. Force and Stress ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22

14.9. INFIN9 - 2-D Infinite Boundary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23

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ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xiv

14.9.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2314.9.2. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23

14.10. LINK10 - Tension-only or Compression-only Spar ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2614.10.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2614.10.2. Element Matrices and Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26

14.11. LINK11 - Linear Actuator ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2814.11.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2914.11.2. Element Matrices and Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2914.11.3. Force, Stroke, and Length ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30

14.12. CONTAC12 - 2-D Point-to-Point Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3114.12.1. Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3114.12.2. Orientation of the Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3314.12.3. Rigid Coulomb Friction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33

14.13. PLANE13 - 2-D Coupled-Field Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3414.13.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35

14.14. COMBIN14 - Spring-Damper ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3514.14.1. Types of Input ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3614.14.2. Stiffness Pass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3614.14.3. Output Quantities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–37

14.15. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3814.16. PIPE16 - Elastic Straight Pipe ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–38

14.16.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3914.16.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3914.16.3. Stiffness Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3914.16.4. Mass Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4014.16.5. Gyroscopic Damping Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4014.16.6. Stress Stiffness Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4114.16.7. Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4114.16.8. Stress Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–43

14.17. PIPE17 - Elastic Pipe Tee ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4914.17.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–49

14.18. PIPE18 - Elastic Curved Pipe (Elbow) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4914.18.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5014.18.2. Stiffness Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5014.18.3. Mass Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5314.18.4. Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5314.18.5. Stress Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–54

14.19. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5414.20. PIPE20 - Plastic Straight Pipe ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–54

14.20.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5514.20.2. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5514.20.3. Stress and Strain Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–55

14.21. MASS21 - Structural Mass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5914.22. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6014.23. BEAM23 - 2-D Plastic Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–60

14.23.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6114.23.2. Integration Points ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6114.23.3. Tangent Stiffness Matrix for Plasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6514.23.4. Newton-Raphson Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6714.23.5. Stress and Strain Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–70

14.24. BEAM24 - 3-D Thin-walled Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–7114.24.1. Assumptions and Restrictions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–72

xvANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

14.24.2. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–7214.24.3. Temperature Distribution Across Cross-Section ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–7214.24.4. Calculation of Cross-Section Section Properties ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–7314.24.5. Offset Transformation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–78

14.25. PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8114.25.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8214.25.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8214.25.3. Use of Temperature ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–82

14.26. CONTAC26 - 2-D Point-to-Ground Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8214.27. MATRIX27 - Stiffness, Damping, or Mass Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–82

14.27.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8214.28. SHELL28 - Shear/Twist Panel .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–83

14.28.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8314.28.2. Commentary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8314.28.3. Output Terms ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–84

14.29. FLUID29 - 2-D Acoustic Fluid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8514.29.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–85

14.30. FLUID30 - 3-D Acoustic Fluid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8614.30.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–86

14.31. LINK31 - Radiation Link ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8614.31.1. Standard Radiation (KEYOPT(3) = 0) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8614.31.2. Empirical Radiation (KEYOPT(3) = 1) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8714.31.3. Solution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–87

14.32. LINK32 - 2-D Conduction Bar ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8814.32.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8814.32.2. Matrices and Load Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–88

14.33. LINK33 - 3-D Conduction Bar ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8914.33.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8914.33.2. Matrices and Load Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8914.33.3. Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–90

14.34. LINK34 - Convection Link ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9014.34.1. Conductivity Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9014.34.2. Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–91

14.35. PLANE35 - 2-D 6-Node Triangular Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9214.35.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–92

14.36. SOURC36 - Current Source ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9314.36.1. Description ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–93

14.37. COMBIN37 - Control .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9314.37.1. Element Characteristics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9414.37.2. Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9514.37.3. Adjustment of Real Constants ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9514.37.4. Evaluation of Control Parameter ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–96

14.38. FLUID38 - Dynamic Fluid Coupling ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9714.38.1. Description ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9714.38.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9814.38.3. Mass Matrix Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9814.38.4. Damping Matrix Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–99

14.39. COMBIN39 - Nonlinear Spring ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10014.39.1. Input ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10014.39.2. Element Stiffness Matrix and Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10114.39.3. Choices for Element Behavior .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–102

14.40. COMBIN40 - Combination ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–105

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xvi

14.40.1. Characteristics of the Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10514.40.2. Element Matrices for Structural Applications ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10614.40.3. Determination of F1 and F2 for Structural Applications ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10714.40.4. Thermal Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–108

14.41. SHELL41 - Membrane Shell .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10814.41.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10914.41.2. Wrinkle Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–109

14.42. PLANE42 - 2-D Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11014.42.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–110

14.43. SHELL43 - 4-Node Plastic Large Strain Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11114.43.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11214.43.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11214.43.3. Assumed Displacement Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11214.43.4. Stress-Strain Relationships ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11214.43.5. In-Plane Rotational DOF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11314.43.6. Spurious Mode Control with Allman Rotation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11314.43.7. Natural Space Extra Shape Functions with Allman Rotation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11514.43.8. Warping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11514.43.9. Stress Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–116

14.44. BEAM44 - 3-D Elastic Tapered Unsymmetric Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11614.44.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11614.44.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11714.44.3. Tapered Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11714.44.4. Shear Center Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11714.44.5. Offset at the Ends of the Member ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11914.44.6. End Moment Release ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12214.44.7. Local to Global Conversion ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12214.44.8. Stress Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–123

14.45. SOLID45 - 3-D Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12414.45.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–124

14.46. SOLID46 - 3-D 8-Node Layered Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12514.46.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12514.46.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12514.46.3. Stress-Strain Relationships ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12614.46.4. General Strain and Stress Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12814.46.5. Interlaminar Shear Stress Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–129

14.47. INFIN47 - 3-D Infinite Boundary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13114.47.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13114.47.2. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13114.47.3. Reduced Scalar Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13514.47.4. Difference Scalar Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13514.47.5. Generalized Scalar Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–136

14.48. CONTAC48 - 2-D Point-to-Surface Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13614.49. CONTAC49 - 3-D Point-to-Surface Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13614.50. MATRIX50 - Superelement (or Substructure) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–137

14.50.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13714.51. SHELL51 - Axisymmetric Structural Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–137

14.51.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13814.51.2. Integration Point Locations for Nonlinear Material Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13814.51.3. Large Deflections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–138

14.52. CONTAC52 - 3-D Point-to-Point Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13814.52.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–139

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14.52.2. Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13914.52.3. Orientation of Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–140

14.53. PLANE53 - 2-D 8-Node Magnetic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14014.53.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14014.53.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14014.53.3. VOLT DOF in 2-D and Axisymmetric Skin Effect Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–141

14.54. BEAM54 - 2-D Elastic Tapered Unsymmetric Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14214.54.1. Derivation of Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–142

14.55. PLANE55 - 2-D Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14314.55.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14314.55.2. Mass Transport Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–143

14.56. HYPER56 - 2-D 4-Node Mixed u-P Hyperelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14414.56.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–145

14.57. SHELL57 - Thermal Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14514.57.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–145

14.58. HYPER58 - 3-D 8-Node Mixed u-P Hyperelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14614.58.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14614.58.2. Mixed Hyperelastic Element Derivation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14614.58.3. Modified Strain Energy Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14614.58.4. Finite Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14714.58.5. Incompressibility .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14814.58.6. Instabilities in the Material Constitutive Law .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14914.58.7. Existence of Multiple Solutions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–149

14.59. PIPE59 - Immersed Pipe or Cable ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14914.59.1. Overview of the Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15014.59.2. Location of the Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15014.59.3. Stiffness Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15114.59.4. Mass Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15214.59.5. Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15214.59.6. Hydrostatic Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15314.59.7. Hydrodynamic Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15514.59.8. Stress Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–160

14.60. PIPE60 - Plastic Curved Pipe (Elbow) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16114.60.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16214.60.2. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16214.60.3. Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16214.60.4. Stress Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–165

14.61. SHELL61 - Axisymmetric-Harmonic Structural Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16814.61.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16914.61.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16914.61.3. Stress, Force, and Moment Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–169

14.62. SOLID62 - 3-D Magneto-Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17214.62.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–173

14.63. SHELL63 - Elastic Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17314.63.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17414.63.2. Foundation Stiffness ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17514.63.3. In-Plane Rotational Stiffness ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17514.63.4. Warping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17514.63.5. Options for Non-Uniform Material .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17614.63.6. Extrapolation of Results to the Nodes ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–178

14.64. SOLID64 - 3-D Anisotropic Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17814.64.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–179

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14.64.2. Stress-Strain Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17914.65. SOLID65 - 3-D Reinforced Concrete Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–179

14.65.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18014.65.2. Description ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18014.65.3. Linear Behavior - General .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18014.65.4. Linear Behavior - Concrete ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18014.65.5. Linear Behavior - Reinforcement ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18114.65.6. Nonlinear Behavior - Concrete ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18314.65.7. Modeling of a Crack ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18314.65.8. Modeling of Crushing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18714.65.9. Nonlinear Behavior - Reinforcement ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–187

14.66. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18714.67. PLANE67 - 2-D Coupled Thermal-Electric Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–187

14.67.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18814.68. LINK68 - Coupled Thermal-Electric Line ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–188

14.68.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18814.69. SOLID69 - 3-D Coupled Thermal-Electric Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–189

14.69.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18914.70. SOLID70 - 3-D Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–189

14.70.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19014.70.2. Fluid Flow in a Porous Medium .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–190

14.71. MASS71 - Thermal Mass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19214.71.1. Specific Heat Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19214.71.2. Heat Generation Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–192

14.72. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19314.73. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19314.74. HYPER74 - 2-D 8-Node Mixed u-P Hyperelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–193

14.74.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19314.74.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–194

14.75. PLANE75 - Axisymmetric-Harmonic 4-Node Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19414.75.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–194

14.76. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19414.77. PLANE77 - 2-D 8-Node Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–195

14.77.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19514.77.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–195

14.78. PLANE78 - Axisymmetric-Harmonic 8-Node Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19614.78.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19614.78.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–196

14.79. FLUID79 - 2-D Contained Fluid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19614.79.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–197

14.80. FLUID80 - 3-D Contained Fluid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19714.80.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19814.80.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19814.80.3. Material Properties ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19814.80.4. Free Surface Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–19914.80.5. Other Assumptions and Limitations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–200

14.81. FLUID81 - Axisymmetric-Harmonic Contained Fluid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20214.81.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20214.81.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20214.81.3. Load Vector Correction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–202

14.82. PLANE82 - 2-D 8-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20314.82.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–203

xixANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

14.82.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20314.83. PLANE83 - Axisymmetric-Harmonic 8-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–204

14.83.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20414.83.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–204

14.84. HYPER84 - 2-D Hyperelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20514.84.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20514.84.2. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–205

14.85. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20514.86. HYPER86 - 3-D Hyperelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–206

14.86.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20614.86.2. Virtual Work Statement ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20614.86.3. Element Matrix Derivation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20714.86.4. Reduced Integration on Volumetric Term in Stiffness Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20914.86.5. Description of Additional Output Strain Measures ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–210

14.87. SOLID87 - 3-D 10-Node Tetrahedral Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21214.87.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–212

14.88. VISCO88 - 2-D 8-Node Viscoelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21214.88.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–213

14.89. VISCO89 - 3-D 20-Node Viscoelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21314.89.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–214

14.90. SOLID90 - 3-D 20-Node Thermal Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21414.90.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–214

14.91. SHELL91 - Nonlinear Layered Structural Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21514.91.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21514.91.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21614.91.3. Stress-Strain Relationship ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21614.91.4. Stress, Force and Moment Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21614.91.5. Force and Moment Summations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21714.91.6. Interlaminar Shear Stress Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21814.91.7. Sandwich Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–220

14.92. SOLID92 - 3-D 10-Node Tetrahedral Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22014.92.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–221

14.93. SHELL93 - 8-Node Structural Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22114.93.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22214.93.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22214.93.3. Stress-Strain Relationships ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22214.93.4. Stress Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–223

14.94. CIRCU94 - Piezoelectric Circuit .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22314.94.1. Electric Circuit Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22314.94.2. Piezoelectric Circuit Element Matrices and Load Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–223

14.95. SOLID95 - 3-D 20-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22614.95.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–227

14.96. SOLID96 - 3-D Magnetic Scalar Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22714.96.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–228

14.97. SOLID97 - 3-D Magnetic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22814.97.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–228

14.98. SOLID98 - Tetrahedral Coupled-Field Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–22914.98.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–229

14.99. SHELL99 - Linear Layered Structural Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23014.99.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23014.99.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23014.99.3. Direct Matrix Input ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–231

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xx

14.99.4. Stress Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23314.99.5. Force and Moment Summations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23314.99.6. Shear Strain Adjustment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23514.99.7. Interlaminar Shear Stress Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–235

14.100. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23714.101. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23714.102. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23714.103. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23714.104. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23714.105. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23714.106. VISCO106 - 2-D 4-Node Viscoplastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–238

14.106.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23814.107. VISCO107 - 3-D 8-Node Viscoplastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–239

14.107.1. Basic Assumptions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23914.107.2. Element Tangent Matrices and Newton-Raphson Restoring Force ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–23914.107.3. Plastic Energy Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–241

14.108. VISCO108 - 2-D 8-Node Viscoplastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24214.108.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24214.108.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–242

14.109. TRANS109 - 2-D Electromechanical Transducer ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24314.110. INFIN110 - 2-D Infinite Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–244

14.110.1. Mapping Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24414.110.2. Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–246

14.111. INFIN111 - 3-D Infinite Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24814.111.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–248

14.112. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24914.113. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24914.114. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24914.115. INTER115 - 3-D Magnetic Interface ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–249

14.115.1. Element Matrix Derivation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24914.115.2. Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–250

14.116. FLUID116 - Coupled Thermal-Fluid Pipe ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–25414.116.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–25414.116.2. Combined Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–25414.116.3. Thermal Matrix Definitions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–25514.116.4. Fluid Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–257

14.117. SOLID117 - 3-D 20-Node Magnetic Edge ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26014.117.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26014.117.2. Matrix Formulation of Low Frequency Edge Element and Tree Gauging ... . . . . . . . . . . . . . . . . . . 14–261

14.118. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26214.119. HF119 - 3-D High-Frequency Magnetic Tetrahedral Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–262

14.119.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26214.119.2. Solution Shape Functions - H (curl) Conforming Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–262

14.120. HF120 - High-Frequency Magnetic Brick Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26414.120.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26414.120.2. Solution Shape Functions - H(curl) Conforming Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–264

14.121. PLANE121 - 2-D 8-Node Electrostatic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26714.121.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26714.121.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–267

14.122. SOLID122 - 3-D 20-Node Electrostatic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26714.122.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–268

14.123. SOLID123 - 3-D 10-Node Tetrahedral Electrostatic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–268

xxiANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

14.123.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26814.124. CIRCU124 - Electric Circuit .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–269

14.124.1. Electric Circuit Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–26914.124.2. Electric Circuit Element Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–270

14.125. CIRCU125 - Diode ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27114.125.1. Diode Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27214.125.2. Norton Equivalents ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27214.125.3. Element Matrix and Load Vector ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–273

14.126. TRANS126 - Electromechanical Transducer ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27414.127. SOLID127 - 3-D Tetrahedral Electrostatic Solid p-Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–277

14.127.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27714.128. SOLID128 - 3-D Brick Electrostatic Solid p-Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–278

14.128.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27814.129. FLUID129 - 2-D Infinite Acoustic .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–279

14.129.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27914.130. FLUID130 - 3-D Infinite Acoustic .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–279

14.130.1. Mathematical Formulation and F.E. Discretization ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28014.130.2. Finite Element Discretization ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–281

14.131. SHELL131 - 4-Node Layered Thermal Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28314.131.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–284

14.132. SHELL132 - 8-Node Layered Thermal Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28414.132.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–285

14.133. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28514.134. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28514.135. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28514.136. FLUID136 - 3-D Squeeze Film Fluid Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–285

14.136.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28514.136.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–285

14.137. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28614.138. FLUID138 - 3-D Viscous Fluid Link Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–286

14.138.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28614.139. FLUID139 - 3-D Slide Film Fluid Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–287

14.139.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28714.140. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28714.141. FLUID141 - 2-D Fluid-Thermal .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–288

14.141.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28914.142. FLUID142 - 3-D Fluid-Thermal .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–289

14.142.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29014.142.2. Distributed Resistance Main Diagonal Modification ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29014.142.3. Turbulent Kinetic Energy Source Term Linearization ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29114.142.4. Turbulent Kinetic Energy Dissipation Rate ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–292

14.143. SHELL143 - 4-Node Plastic Small Strain Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29314.143.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29414.143.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29414.143.3. Assumed Displacement Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29414.143.4. Stress-Strain Relationships ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29414.143.5. In-Plane Rotational DOF ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29414.143.6. Spurious Mode Control with Allman Rotation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29514.143.7. Natural Space Extra Shape Functions with Allman Rotation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29514.143.8. Warping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29514.143.9. Consistent Tangent ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29514.143.10. Stress Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–295

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xxii

14.144. ROM144 - Reduced Order Electrostatic-Structural .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29614.144.1. Element Matrices and Load Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–29614.144.2. Combination of Modal Coordinates and Nodal Displacement at Master Nodes ... . . . . . . . . 14–29814.144.3. Element Loads ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–300

14.145. PLANE145 - 2-D Quadrilateral Structural Solid p-Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30014.145.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–301

14.146. PLANE146 - 2-D Triangular Structural Solid p-Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30114.146.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–302

14.147. SOLID147 - 3-D Brick Structural Solid p-Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30214.147.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–302

14.148. SOLID148 - 3-D Tetrahedral Structural Solid p-Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30314.148.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–303

14.149. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30314.150. SHELL150 - 8-Node Structural Shell p-Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–304

14.150.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30414.150.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30414.150.3. Stress-Strain Relationships ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–305

14.151. SURF151 - 2-D Thermal Surface Effect .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30514.152. SURF152 - 3-D Thermal Surface Effect .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–306

14.152.1. Matrices and Load Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30614.152.2. Adiabatic Wall Temperature as Bulk Temperature ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30814.152.3. Film Coefficient Adjustment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–30914.152.4. Radiation Form Factor Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–310

14.153. SURF153 - 2-D Structural Surface Effect .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31114.154. SURF154 - 3-D Structural Surface Effect .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31214.155. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31514.156. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31514.157. SHELL157 - Thermal-Electric Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–315

14.157.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31514.158. HYPER158 - 3-D 10-Node Tetrahedral Mixed u-P Hyperelastic Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–316

14.158.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31614.159. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31614.160. LINK160 - Explicit 3-D Spar (or Truss) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31714.161. BEAM161 - Explicit 3-D Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31714.162. PLANE162 - Explicit 2-D Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31814.163. SHELL163 - Explicit Thin Structural Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31814.164. SOLID164 - Explicit 3-D Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31914.165. COMBI165 - Explicit Spring-Damper ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31914.166. MASS166 - Explicit 3-D Structural Mass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32014.167. LINK167 - Explicit Tension-Only Spar ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32014.168. SOLID168 - Explicit 3-D 10-Node Tetrahedral Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32014.169. TARGE169 - 2-D Target Segment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–321

14.169.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32114.169.2. Segment Types ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–321

14.170. TARGE170 - 3-D Target Segment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32214.170.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32214.170.2. Segment Types ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32214.170.3. Reaction Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–323

14.171. CONTA171 - 2-D 2-Node Surface-to-Surface Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32314.171.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–323

14.172. CONTA172 - 2-D 3-Node Surface-to-Surface Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32414.172.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–324

xxiiiANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

14.173. CONTA173 - 3-D 4-Node Surface-to-Surface Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32414.173.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–324

14.174. CONTA174 - 3-D 8-Node Surface-to-Surface Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32514.174.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32514.174.2. Contact Kinematics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32514.174.3. Frictional Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32714.174.4. Contact Algorithm .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32914.174.5. Thermal/Structural Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33114.174.6. Electric Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33214.174.7. Magnetic Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–333

14.175. CONTA175 - 2-D/3-D Node-to-Surface Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33314.175.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33414.175.2. Contact Models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33414.175.3. Contact Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–334

14.176. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33414.177. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33414.178. CONTA178 - 3-D Node-to-Node Contact .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–335

14.178.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33514.178.2. Contact Algorithms ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33514.178.3. Element Damper ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–337

14.179. PRETS179 - Pretension ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33714.179.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33814.179.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–338

14.180. LINK180 - 3-D Finite Strain Spar (or Truss) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33814.180.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33814.180.2. Element Mass Matrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–339

14.181. SHELL181 - 4-Node Finite Strain Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–33914.181.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34014.181.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34014.181.3. Assumed Displacement Shape Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34014.181.4. Membrane Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34014.181.5. Warping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–341

14.182. PLANE182 - 2-D 4-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34114.182.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34114.182.2. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–342

14.183. PLANE183 - 2-D 8-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34214.183.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34214.183.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–343

14.184. MPC184 - Multipoint Constraint Rigid Link and Rigid Beam Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34314.184.1. Slider Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34314.184.2. Spherical Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34414.184.3. Revolute Joint Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34514.184.4. Universal Joint Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–346

14.185. SOLID185 - 3-D 8-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34714.185.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34814.185.2. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–348

14.186. SOLID186 - 3-D 20-Node Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34814.186.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–349

14.187. SOLID187 - 3-D 10-Node Tetrahedral Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34914.187.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–350

14.188. BEAM188 - 3-D Linear Finite Strain Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35014.189. BEAM189 - 3-D Quadratic Finite Strain Beam .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–351

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14.189.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35114.189.2. Stress Evaluation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–352

14.190. SOLSH190 - 3-D 8-Node Solid Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35314.190.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35314.190.2. Theory ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–353

14.191. SOLID191 - 3-D 20-Node Layered Structural Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35414.191.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–354

14.192. INTER192 - 2-D 4-Node Gasket ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35514.192.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–355

14.193. INTER193 - 2-D 6-Node Gasket ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35514.193.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–356

14.194. INTER194 - 3-D 16-Node Gasket ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35614.194.1. Element Technology ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–356

14.195. INTER195 - 3-D 8-Node Gasket ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35714.195.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–357

14.196. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35714.197. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.198. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.199. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.200. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.201. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.202. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.203. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.204. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.205. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.206. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.207. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35814.208. SHELL208 - 2-Node Finite Strain Axisymmetric Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–359

14.208.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35914.208.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–359

14.209. SHELL209 - 2-Node Finite Strain Axisymmetric Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36014.209.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36014.209.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–360

14.210. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36014.211. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.212. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.213. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.214. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.215. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.216. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.217. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.218. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.219. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.220. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.221. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.222. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36114.223. PLANE223 - 2-D 8-Node Coupled-Field Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–362

14.223.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36214.224. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36314.225. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36314.226. SOLID226 - 3-D 20-Node Coupled-Field Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–363

14.226.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–364

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ANSYS, Inc. Theory Reference

14.227. SOLID227 - 3-D 10-Node Coupled-Field Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36414.227.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–365

14.228. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36514.229. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36514.230. PLANE230 - 2-D 8-Node Electric Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–365

14.230.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36514.230.2. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–366

14.231. SOLID231 - 3-D 20-Node Electric Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36614.231.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–366

14.232. SOLID232 - 3-D 10-Node Tetrahedral Electric Solid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36614.232.1. Other Applicable Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–367

14.233. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.234. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.235. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.236. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.237. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.238. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.239. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.240. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.241. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.242. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36714.243. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.244. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.245. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.246. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.247. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.248. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.249. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.250. Not Documented ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.251. SURF251 - 2-D Radiosity Surface ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–36814.252. SURF252 - 3-D Thermal Radiosity Surface ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–369

15. Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–115.1. Acceleration Effect .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–115.2. Inertia Relief .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–515.3. Damping Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–915.4. Element Reordering ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–11

15.4.1. Reordering Based on Topology with a Program-Defined Starting Surface ... . . . . . . . . . . . . . . . . . . . . . 15–1115.4.2. Reordering Based on Topology with a User- Defined Starting Surface ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1115.4.3. Reordering Based on Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1215.4.4. Automatic Reordering ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–12

15.5. Automatic Master DOF Selection ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1215.6. Automatic Time Stepping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–13

15.6.1. Time Step Prediction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1315.6.2. Time Step Bisection ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1415.6.3. The Response Eigenvalue for 1st Order Transients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1415.6.4. The Response Frequency for Structural Dynamics ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1515.6.5. Creep Time Increment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1615.6.6. Plasticity Time Increment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–16

15.7. Solving for Unknowns and Reactions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1615.7.1. Reaction Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–1715.7.2. Disequilibrium .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–19

15.8. Equation Solvers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–20

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xxvi

15.8.1. Direct Solvers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2015.8.2. Sparse Direct Solver .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2015.8.3. Frontal Solver .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2215.8.4. Iterative Solver .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–26

15.9. Mode Superposition Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2915.9.1. Modal Damping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–32

15.10. Reduced Order Modeling of Coupled Domains ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3315.10.1. Selection of Modal Basis Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3415.10.2. Element Loads ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3515.10.3. Mode Combinations for Finite Element Data Acquisition and Energy Computation ... . . . . . 15–3615.10.4. Function Fit Methods for Strain Energy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3615.10.5. Coupled Electrostatic-Structural Systems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3715.10.6. Computation of Capacitance Data and Function Fit .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–38

15.11. Newton-Raphson Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3815.11.1. Overview ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3815.11.2. Convergence ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4315.11.3. Predictor .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4415.11.4. Adaptive Descent ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4515.11.5. Line Search ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4615.11.6. Arc-Length Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–47

15.12. Constraint Equations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5015.12.1. Derivation of Matrix and Load Vector Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–50

15.13. This section intentionally omitted ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5115.14. Eigenvalue and Eigenvector Extraction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–52

15.14.1. Reduced Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5215.14.1.1. Transformation of the Generalized Eigenproblem to a Standard Eigenproblem .... . 15–5315.14.1.2. Reduce [A] to Tridiagonal Form .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5415.14.1.3. Eigenvalue Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5415.14.1.4. Eigenvector Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5415.14.1.5. Eigenvector Transformation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–54

15.14.2. Subspace Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5515.14.2.1. Convergence ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5615.14.2.2. Starting Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5615.14.2.3. Sturm Sequence Check ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5715.14.2.4. Shifting Strategy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5715.14.2.5. Sliding Window .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–58

15.14.3. Block Lanczos ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5815.14.4. Unsymmetric Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5815.14.5. Damped Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5915.14.6. QR Damped Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6015.14.7. Shifting ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6215.14.8. Repeated Eigenvalues ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–63

15.15. Analysis of Cyclic Symmetric Structures ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6315.15.1. Modal Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6315.15.2. Complete Mode Shape Derivation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6415.15.3. Cyclic Symmetry Transformations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–65

15.16. Mass Moments of Inertia .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6615.16.1. Accuracy of the Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6815.16.2. Effect of KSUM, LSUM, ASUM, and VSUM Commands ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–68

15.17. Energies ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6915.18. ANSYS Workbench Product Adaptive Solutions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–7115.19. Modal Projection Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–72

xxviiANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

15.19.1. Extraction of Modal Damping Parameter for Squeeze Film Problems ... . . . . . . . . . . . . . . . . . . . . . . . . . . 15–7216. This chapter intentionally omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–117. Analysis Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1

17.1. Static Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–117.1.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–117.1.2. Description of Structural Systems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–117.1.3. Description of Thermal, Magnetic and Other First Order Systems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2

17.2. Transient Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–317.2.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–317.2.2. Description of Structural and Other Second Order Systems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3

17.2.2.1. Solution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–817.2.3. Description of Thermal, Magnetic and Other First Order Systems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–13

17.3. Mode-Frequency Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1517.3.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1517.3.2. Description of Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–16

17.4. Harmonic Response Analyses ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1717.4.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1717.4.2. Description of Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1717.4.3. Complex Displacement Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1917.4.4. Nodal and Reaction Load Computation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1917.4.5. Solution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–20

17.4.5.1. Full Solution Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2017.4.5.2. Reduced Solution Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–20

17.4.5.2.1. Expansion Pass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2117.4.5.3. Mode Superposition Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–21

17.4.5.3.1. Expansion Pass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2317.4.6. Variational Technology Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–23

17.4.6.1. Viscous or Hysteretic Damping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2417.4.7. Automatic Frequency Spacing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–25

17.5. Buckling Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2617.5.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2617.5.2. Description of Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–27

17.6. Substructuring Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2817.6.1. Assumptions and Restrictions (within Superelement) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2817.6.2. Description of Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2817.6.3. Statics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2817.6.4. Transients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3017.6.5. Component Mode Synthesis (CMS) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–31

17.7. Spectrum Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3217.7.1. Assumptions and Restrictions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3217.7.2. Description of Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3317.7.3. Single-Point Response Spectrum .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3317.7.4. Damping ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3317.7.5. Participation Factors and Mode Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3417.7.6. Combination of Modes ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–37

17.7.6.1. Complete Quadratic Combination Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3817.7.6.2. Grouping Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3917.7.6.3. Double Sum Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3917.7.6.4. SRSS Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4017.7.6.5. NRL-SUM Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–40

17.7.7. Reduced Mass Summary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4017.7.8. Effective Mass ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–41

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17.7.9. Dynamic Design Analysis Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4117.7.10. Random Vibration Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4217.7.11. Description of Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4217.7.12. Response Power Spectral Densities and Mean Square Response ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–43

17.7.12.1. Dynamic Part .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4417.7.12.2. Pseudo-Static Part .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4417.7.12.3. Covariance Part .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4417.7.12.4. Equivalent Stress Mean Square Response ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–46

17.7.13. Cross Spectral Terms for Partially Correlated Input PSDs ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4717.7.14. Spatial Correlation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4717.7.15. Wave Propagation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4817.7.16. Multi-Point Response Spectrum Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–49

18. Pre and Postprocessing Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–118.1. Integration and Differentiation Procedures ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–1

18.1.1. Single Integration Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–118.1.2. Double Integration Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–118.1.3. Differentiation Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–218.1.4. Double Differentiation Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–2

18.2. Fourier Coefficient Evaluation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–318.3. Statistical Procedures ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–4

18.3.1. Mean, Covariance, Correlation Coefficient ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–418.3.2. Random Samples of a Uniform Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–518.3.3. Random Samples of a Gaussian Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–618.3.4. Random Samples of a Triangular Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–718.3.5. Random Samples of a Beta Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–818.3.6. Random Samples of a Gamma Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–9

19. Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–119.1. POST1 - Derived Nodal Data Processing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1

19.1.1. Derived Nodal Data Computation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–119.2. POST1 - Vector and Surface Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2

19.2.1. Vector Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–219.2.2. Surface Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2

19.3. POST1 - Path Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–319.3.1. Defining the Path ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–319.3.2. Defining Orientation Vectors of the Path ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–319.3.3. Mapping Nodal and Element Data onto the Path ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–519.3.4. Operating on Path Data ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–6

19.4. POST1 - Stress Linearization ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–719.4.1. Cartesian Case ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–719.4.2. Axisymmetric Case (General) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–919.4.3. Axisymmetric Case ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–15

19.5. POST1 - Fatigue Module ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1619.6. POST1 - Electromagnetic Macros ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–18

19.6.1. Flux Passing Thru a Closed Contour ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1819.6.2. Force on a Body ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1919.6.3. Magnetomotive Forces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1919.6.4. Power Loss ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1919.6.5. Terminal Parameters for a Stranded Coil .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2019.6.6. Energy Supplied ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2019.6.7. Terminal Inductance ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2119.6.8. Flux Linkage ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2119.6.9. Terminal Voltage ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–21

xxixANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

ANSYS, Inc. Theory Reference

19.6.10. Torque on a Body ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2219.6.11. Energy in a Magnetic Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2219.6.12. Relative Error in Electrostatic or Electromagnetic Field Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–23

19.6.12.1. Electrostatics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2319.6.12.1.1. Electric Field ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2319.6.12.1.2. Electric Flux Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–23

19.6.12.2. Electromagnetics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2419.6.12.2.1. Magnetic Field Intensity ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2419.6.12.2.2. Magnetic Flux Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–24

19.6.13. SPARM Macro-Parameters ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2519.6.14. Electromotive Force ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2519.6.15. Impedance of a Device ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2619.6.16. Computation of Equivalent Transmission-line Parameters ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2619.6.17. Quality Factor ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–29

19.7. POST1 - Error Approximation Technique ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3119.7.1. Error Approximation Technique for Displacement-Based Problems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3119.7.2. Error Approximation Technique for Temperature-Based Problems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–33

19.8. POST1 - Crack Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3519.9. POST1 - Harmonic Solid and Shell Element Postprocessing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–38

19.9.1. Thermal Solid Elements (PLANE75, PLANE78) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3819.9.2. Structural Solid Elements (PLANE25, PLANE83) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3919.9.3. Structural Shell Element (SHELL61) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–40

19.10. POST26 - Data Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4119.11. POST26 - Response Spectrum Generator (RESP) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–42

19.11.1. Time Step Size ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4419.12. POST1 and POST26 - Interpretation of Equivalent Strains ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–44

19.12.1. Physical Interpretation of Equivalent Strain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4519.12.2. Elastic Strain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4519.12.3. Plastic Strain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4519.12.4. Creep Strain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4619.12.5. Total Strain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–46

19.13. POST26 - Response Power Spectral Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4619.14. POST26 - Computation of Covariance ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–47

20. Design Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–120.1. Introduction to Design Optimization ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1

20.1.1. Feasible Versus Infeasible Design Sets ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–220.1.2. The Best Design Set ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–320.1.3. Optimization Methods and Design Tools .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–3

20.1.3.1. Single-Loop Analysis Tool .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–320.1.3.2. Random Tool .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–320.1.3.3. Sweep Tool .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–420.1.3.4. Factorial Tool .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–520.1.3.5. Gradient Tool .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–5

20.2. Subproblem Approximation Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–620.2.1. Function Approximations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–620.2.2. Minimizing the Subproblem Approximation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–720.2.3. Convergence ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–10

20.3. First Order Optimization Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1020.3.1. The Unconstrained Objective Function ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1120.3.2. The Search Direction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1220.3.3. Convergence ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–13

20.4. Topological Optimization ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–14

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xxx

20.4.1. General Optimization Problem Statement ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1420.4.2. Maximum Static Stiffness Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1520.4.3. Minimum Volume Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1520.4.4. Maximum Dynamic Stiffness Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–16

20.4.4.1. Weighted Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1720.4.4.2. Reciprocal Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1720.4.4.3. Euclidean Norm Formulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–18

20.4.5. Element Calculations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1821. Probabilistic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1

21.1. Probabilistic Modeling and Preprocessing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–221.1.1. Statistical Distributions for Random Input Variables ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2

21.1.1.1. Gaussian (Normal) Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–221.1.1.2. Truncated Gaussian Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–321.1.1.3. Lognormal Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–521.1.1.4. Triangular Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–621.1.1.5. Uniform Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–821.1.1.6. Exponential Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–921.1.1.7. Beta Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1021.1.1.8. Gamma Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1121.1.1.9. Weibull Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–12

21.2. Probabilistic Methods ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1321.2.1. Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1321.2.2. Common Features for all Probabilistic Methods ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–13

21.2.2.1. Random Numbers with Standard Uniform Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1321.2.2.2. Non-correlated Random Numbers with an Arbitrary Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . 21–1421.2.2.3. Correlated Random Numbers with an Arbitrary Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–14

21.2.3. Monte Carlo Simulation Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1421.2.3.1. Direct Monte Carlo Simulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1421.2.3.2. Latin Hypercube Sampling ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–15

21.2.4. The Response Surface Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1621.2.4.1. Central Composite Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1721.2.4.2. Box-Behnken Matrix Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–19

21.3. Regression Analysis for Building Response Surface Models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2121.3.1. General Definitions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2121.3.2. Linear Regression Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2221.3.3. F-Test for the Forward-Stepwise-Regression ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2321.3.4. Transformation of Random Output Parameter Values for Regression Fitting ... . . . . . . . . . . . . . . . . . 21–2421.3.5. Goodness-of-Fit Measures ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–25

21.3.5.1. Error Sum of Squares SSE ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2521.3.5.2. Coefficient of Determination R2 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2521.3.5.3. Maximum Absolute Residual .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–26

21.4. Probabilistic Postprocessing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2621.4.1. Statistical Procedures ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–27

21.4.1.1. Mean Value ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2721.4.1.2. Standard Deviation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2721.4.1.3. Minimum and Maximum Values ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–28

21.4.2. Correlation Coefficient Between Sampled Data ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2821.4.2.1. Pearson Linear Correlation Coefficient ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2821.4.2.2. Spearman Rank-Order Correlation Coefficient ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–29

21.4.3. Cumulative Distribution Function ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–3021.4.4. Evaluation of Probabilities From the Cumulative Distribution Function ... . . . . . . . . . . . . . . . . . . . . . . . . . 21–3021.4.5. Inverse Cumulative Distribution Function ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–30

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22. Reference Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–1Index ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index–1

List of Figures2.1. Stress Vector Definition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–22.2. Material Coordinate Systems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–52.3. Effects of Consistent Pressure Loading ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–123.1. Position Vectors and Motion of a Deforming Body ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–23.2. Polar Decomposition of a Shearing Deformation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–33.3. Element Transformation Definitions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–93.4. Definition of Deformational Rotations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–113.5. General Motion of a Fiber ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–143.6. Motion of a Fiber with Rigid Body Motion Removed ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–153.7. Spinning Spring-Mass System .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–203.8. Effects of Spin Softening and Stress Stiffening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–224.1. Stress-Strain Behavior of Each of the Plasticity Options ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–54.2. Various Yield Surfaces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–64.3. Types of Hardening Rules ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–74.4. Uniaxial Behavior .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–124.5. Uniaxial Behavior for Multilinear Kinematic Hardening ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–164.6. Plastic Work for a Uniaxial Case ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–234.7. Drucker-Prager and Mohr-Coulomb Yield Surfaces ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–264.8. Idealized Response of Gray Cast Iron in Tension and Compression ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–274.9. Cross-Section of Yield Surface ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–284.10. Meridian Section of Yield Surface ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–284.11. Flow Potential for Cast Iron ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–304.12. Pressure vs. Deflection Behavior of a Gasket Material .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–384.13. Stress-Strain Behavior for Nonlinear Elasticity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–404.14. Typical Superelasticity Behavior .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–414.15. Idealized Stress-Strain Diagram of Superelastic Behavior .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–434.16. Illustration of Deformation Modes ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–544.17. Equivalent Deformation Modes ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–554.18. Pure Shear from Direct Components ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–584.19. 3-D Failure Surface in Principal Stress Space ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–724.20. A Profile of the Failure Surface ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–734.21. Failure Surface in Principal Stress Space with Nearly Biaxial Stress ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–755.1. Electromagnetic Field Regions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–35.2. Patch Test Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–105.3. A Typical FEA Configuration for Electromagnetic Field Simulation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–345.4. Impedance Boundary Condition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–395.5. PML Configuration ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–415.6. Arbitrary Infinite Periodic Structure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–425.7. Soft Excitation Source ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–445.8. Two Ports Network ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–475.9. Surface Equivalent Currents ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–495.10. Input, Reflection, and Transmission Power in the System .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–535.11. Periodic Structure Under Plane Wave Excitation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–545.12. Energy and Co-energy for Non-Permanent Magnets ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–595.13. Energy and Co-energy for Permanent Magnets ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–605.14. Lumped Capacitor Model of Two Conductors and Ground ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–67

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5.15. Trefftz and Multiple Finite Element Domains ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–675.16. Typical Hybrid FEM-Trefftz Domain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–685.17. Multiple FE Domains Connected by One Trefftz Domain ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–695.18. Lumped Capacitor Model of Two Conductors and Ground ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–726.1. View Factor Calculation Terms ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–46.2. Receiving Surface Projection ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–106.3. Axisymmetric Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–116.4. End View of Showing n = 8 Segments ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–116.5. The Hemicube ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–146.6. Derivation of Delta-View Factors for Hemicube Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–147.1. Streamline Upwind Approach ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–227.2. Typical Advection Step in CLEAR-VOF Algorithm .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–327.3. Types of VFRC Boundary Conditions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–357.4. Stress vs. Strain Rate Relationship for “Ideal” Bingham Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–457.5. Stress vs. Strain Rate Relationship for “Biviscosity” Bingham Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–467.6. Flow Theory, Cut-off, and Maximum Frequency Interrelation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–5712.1. 2–D Line Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–212.2. 3–D Line Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–312.3. Axisymmetric Harmonic Shell Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–612.4. 3-D Shell Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–712.5. Interpolation Functions for Transverse Strains for Triangles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1012.6. Interpolation Functions for the Transverse Strains for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1412.7. 2-D and Axisymmetric Solid Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1612.8. 4 Node Quadrilateral Infinite Solid Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–1912.9. 8 Node Quadrilateral Infinite Solid Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2012.10. Axisymmetric Harmonic Solid Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2112.11. 3-D Solid Elements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2412.12. 10 Node Tetrahedra Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2512.13. 8 Node Brick Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2512.14. 13 Node Pyramid Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2612.15. 6 Node Wedge Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2712.16. 15 Node Wedge Element (SOLID90) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2812.17. 15 Node Wedge Element (SOLID95) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2912.18. 8 Node Brick Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3012.19. 20 Node Brick Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3212.20. 3-D 8 Node Brick Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3312.21. 20 Node Solid Brick Infinite Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3412.22. 2-D 8 Node Quad Edge Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3712.23. 3-D 20 Node Brick Edge Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3912.24. 1st-Order Tetrahedral Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4212.25. 2nd-Order Tetrahedral Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4312.26. 1st-Order Brick Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4412.27. 2nd-Order Brick Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4512.28. Mixed 1st-Order Triangular Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4612.29. Mixed 2nd-Order Triangular Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4712.30. Mixed 1st-Order Quadrilateral Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4812.31. Mixed 2nd-Order Quadrilateral Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–4913.1. Integration Point Locations for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–213.2. Integration Point Locations for Bricks and Pyramids ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–213.3. Integration Point Locations for Triangles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–313.4. Integration Point Locations for Tetrahedra ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–513.5. Integration Point Locations for Triangles and Tetrahedra ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–5

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ANSYS, Inc. Theory Reference

13.6. 6 and 9 Integration Point Locations for Wedges ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–613.7. 8 Integration Point Locations for Wedges ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–613.8. Integration Point Locations for 14 Point Rule ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–713.9. Nonlinear Bending Integration Point Locations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–713.10. Brick Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1313.11. Pyramid Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1413.12. Pyramid Element Cross-Section Construction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1413.13. Wedge Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1513.14. Tetrahedron Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1513.15. Tetrahedron Element Cross-Section Construction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1613.16. Triangle Aspect Ratio Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1613.17. Aspect Ratios for Triangles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1713.18. Quadrilateral Aspect Ratio Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1713.19. Aspect Ratios for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1813.20. Angle Deviations for SHELL28 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1913.21. Parallel Deviation Unit Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1913.22. Parallel Deviations for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2013.23. Maximum Corner Angles for Triangles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2113.24. Maximum Corner Angles for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2113.25. Jacobian Ratios for Triangles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2313.26. Jacobian Ratios for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2313.27. Jacobian Ratios for Quadrilaterals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2413.28. Shell Average Normal Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2513.29. Shell Element Projected onto a Plane ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2513.30. Quadrilateral Shell Having Warping Factor ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2613.31. Warping Factor for Bricks ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2714.1. Order of Degrees of Freedom .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–714.2. Joint Element Dynamic Behavior About the Revolute Axis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–1514.3. Definition of BE Subdomain and the Characteristics of the IBE ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2414.4. Force-Deflection Relations for Standard Case ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3314.5. Force-Deflection Relations for Rigid Coulomb Option ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3414.6. Thermal and Pressure Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4314.7. Elastic Pipe Direct Stress Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4514.8. Elastic Pipe Shear Stress Output ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4514.9. Stress Point Locations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4714.10. Mohr Circles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4814.11. Plane Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5014.12. Integration Points for End J .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5614.13. Integration Point Locations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6114.14. Beam Widths ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6314.15. Cross-Section Input and Principal Axes ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–7514.16. Definition of Sectorial Coordinate ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–7714.17. Reference Coordinate System .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8014.18. Uniform Shear on Rectangular Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8314.19. Uniform Shear on Separated Rectangular Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8414.20. Element Behavior .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–9414.21. Input Force-Deflection Curve ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10114.22. Stiffness Computation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10214.23. Input Force-Deflection Curve Reflected Through Origin ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10214.24. Force-Deflection Curve with KEYOPT(2) = 1 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10314.25. Nonconservative Unloading (KEYOPT(1) = 1) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10314.26. No Origin Shift on Reversed Loading (KEYOPT(1) = 1) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–104

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xxxiv

14.27. Origin Shift on Reversed Loading (KEYOPT(1) = 1) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10414.28. Crush Option (KEYOPT(2) = 2) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10514.29. Force-Deflection Relationship ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10614.30. Shape Functions for the Transverse Strains ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11314.31. Constant In-Plane Rotation Spurious Mode ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11414.32. Hourglass Mode ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11414.33. Offset Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–11914.34. Translation of Axes ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12114.35. Offset Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–12614.36. A Semi-infinite Boundary Element Zone and the Corresponding Boundary Element IJK ... . . . . . . . . . . . 14–13214.37. Infinite Element IJML and the Local Coordinate System .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–13314.38. Velocity Profiles for Wave-Current Interactions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15914.39. 3-D Plastic Curved Pipe Element Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16514.40. Integration Point Locations at End J .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–16514.41. Stress Locations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17014.42. Element Orientations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–17214.43. Reinforcement Orientation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18214.44. Strength of Cracked Condition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–18414.45. U-Tube with Fluid ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20014.46. Bending Without Resistance ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–20114.47. Integration Point Locations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–21914.48. Global to Local Mapping of a 1-D Infinite Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24514.49. Mapping of 2-D Solid Infinite Element ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–24514.50. A General Electromagnetics Analysis Field and Its Component Regions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–25014.51. I-V (Current-Voltage) Characteristics of CIRCU125 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27214.52. Norton Current Definition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27314.53. Electromechanical Transducer ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–27414.54. Absorbing Boundary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–28014.55. Form Factor Calculation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–31014.56. 2-D Segment Types ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32114.57. 3-D Segment Types ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32314.58. Contact Detection Point Location at Gauss Point .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32614.59. Penetration Distance ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32614.60. Smoothing Convex Corner ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32714.61. Friction Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–32814.62. 184.2 Slider Constraint Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34314.63. 184.3 Spherical Constraint Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34514.64. 184.4 Revolute Joint Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34614.65. 184.5 Universal Joint Geometry ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–34714.66. Section Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–35215.1. Rotational Coordinate System (Rotations 1 and 3) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–315.2. Rotational Coordinate System (Rotations 1 and 2) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–415.3. Rotational Coordinate System (Rotations 2 and 3) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–515.4. Ranges of Pivot Values ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2315.5. Wavefront Flow Chart .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2515.6. Sample Mesh ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2615.7. Single Degree of Freedom Oscillator .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3115.8. Set for Lagrange and Pascal Polynomials .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3715.9. Newton-Raphson Solution - One Iteration ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4015.10. Newton-Raphson Solution - Next Iteration ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4115.11. Incremental Newton-Raphson Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4215.12. Initial-Stiffness Newton-Raphson ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–43

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ANSYS, Inc. Theory Reference

15.13. Arc-Length Approach with Full Newton-Raphson Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4815.14. Typical Cyclic Symmetric Structure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6315.15. Basic Sector Definition ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–6415.16. Damping and Amplitude Ratio vs. Frequency ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–7215.17. Fluid Pressure From Modal Excitation Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–7317.1. Applied and Reaction Load Vectors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–217.2. Frequency Spacing ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2617.3. Types of Buckling Problems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–2717.4. Sphere of Influence Relating Spatially Correlated PSD Excitation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–4818.1. Integration Procedure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–118.2. Uniform Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–618.3. Cumulative Probability Function ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–618.4. Gaussian Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–718.5. Triangular Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–818.6. Beta Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–918.7. Gamma Density ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–1019.1. Typical Path Segment ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–319.2. Position and Unit Vectors of a Path ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–419.3. Mapping Data ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–619.4. Coordinates of Cross Section ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–719.5. Typical Stress Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–819.6. Axisymmetric Cross-Section ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1019.7. Geometry Used for Axisymmetric Evaluations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1019.8. Centerline Sections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1519.9. Non-Perpendicular Intersections ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–1619.10. Equivalent Two-Wire Transmission Line ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2719.11. Coaxial Cable Diagram .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2819.12. Local Coordinates Measured From a 3-D Crack Front ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3619.13. The Three Basic Modes of Fracture ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3619.14. Nodes Used for the Approximate Crack-Tip Displacements ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3819.15. Single Mass Oscillators .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4320.1. Extended Interior Penalty Function ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–921.1. Gaussian Distribution Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–321.2. Truncated Gaussian Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–421.3. Lognormal Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–621.4. Triangular Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–721.5. Uniform Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–821.6. Exponential Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–921.7. Beta Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1021.8. Gamma Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1121.9. Weibull Distribution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1221.10. Sample Set Generated with Direct Monte Carlo Simulation Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1521.11. Sample Set Generated with Latin Hypercube Sampling Method ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1621.12. Sample Set Based on a Central Composite Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1721.13. Sample Set Based on Box-Behnken Matrix Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–20

List of Tables1.1. General Terms ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–21.2. Superscripts and Subscripts .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–33.1. Interpolation Functions of Hydrostatic Pressure ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–27

ANSYS, Inc. Theory Reference

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.xxxvi

4.1. Notation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–34.2. Summary of Plasticity Options ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–74.3. Material Parameter Units for Anand Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–364.4. Concrete Material Table ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–697.1. Standard Model Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–87.2. RNG Model Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–97.3. NKE Turbulence Model Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–107.4. GIR Turbulence Model Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–117.5. SZL Turbulence Model Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–127.6. The k-ω Model Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–137.7. The SST Model Coefficients ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–147.8. Transport Equation Representation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–2011.1. Elements Used for Coupled Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–111.2. Coupling Methods ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–211.3. Nomenclature of Coefficient Matrices ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–1012.1. Shape Function Labels .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–113.1. Gauss Numerical Integration Constants ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–113.2. Numerical Integration for Triangles ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–313.3. Numerical Integration for Tetrahedra ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–313.4. Numerical Integration for 20-Node Brick ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–613.5. Thru-Thickness Numerical Integration ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–713.6. Assumed Data Variation of Stresses ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1213.7. Aspect Ratio Limits .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1813.8. Angle Deviation Limits .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–1913.9. Parallel Deviation Limits .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2013.10. Maximum Corner Angle Limits .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2113.11. Jacobian Ratio Limits .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2413.12. Applicability of Warping Tests .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2713.13. Warping Factor Limits .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2814.1. Value of Stiffness Coefficient (C1) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2714.2. Value of Stiffness Coefficient (C2) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2714.3. Stress Intensification Factors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4514.4. Cross-Sectional Computation Factors ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–6414.5. Number of Pressure DOFs and Interpolation Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–14714.6. Wave Theory Table ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–15515.1. Procedures Used for Eigenvalue and Eigenvector Extraction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–5215.2. Exceptions for Element Energies ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–7015.3. ANSYS Workbench Product Adaptivity Methods ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–7117.1. Nomenclature ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–317.2. Nomenclature ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1317.3. Types of Spectrum Loading ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–3319.1. POST26 Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–4121.1. Probability Matrix for Samples of Central Composite Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–1821.2. Probability Matrix for Samples of Box-Behnken Matrix Design ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–20

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Chapter 1: IntroductionWelcome to the ANSYS, Inc. Theory Reference. This manual presents theoretical descriptions of all elements, aswell as of many procedures and commands used in ANSYS, Inc. products. It is available to any of our productusers who need to understand how the program uses the input data to calculate the output. In addition, thismanual is indispensable for its explanations of how to interpret certain element and command results. In addition,the ANSYS, Inc. Theory Reference describes the relationship between input data and output results produced bythe programs, and is essential for a thorough understanding of how the programs' function.

1.1. Purpose of the ANSYS Theory Reference

The purpose of the ANSYS, Inc. Theory Reference is to inform you of the theoretical basis of ANSYS, Inc. products,including the ANSYS and ANSYS Workbench products. By understanding the underlying theory, you can use theANSYS, Inc. products more intelligently and with greater confidence, making better use of their capabilities whilebeing aware of their limitations. Of course, you are not expected to study the entire volume; you need only torefer to sections of it as required for specific elements and procedures. This manual does not, and cannot, presentall theory relating to finite element analysis. If you need the theory behind the basic finite element method, youshould obtain one of the many references available on the topic. If you need theory or information that goesbeyond that presented here, you should (as applicable) consult the indicated reference, run a simple test problemto try the feature of interest, or contact your ANSYS Support Distributor for more information.

The theory behind the basic analysis disciplines is presented in Chapter 2, “Structures” through Chapter 11,“Coupling”. Chapter 2, “Structures” covers structural theory, with Chapter 3, “Structures with Geometric Nonlin-earities” and Chapter 4, “Structures with Material Nonlinearities” adding geometric and structural material non-linearities. Chapter 5, “Electromagnetics” discusses electromagnetics, Chapter 6, “Heat Flow” deals with heatflow, Chapter 7, “Fluid Flow” handles fluid flow and Chapter 8, “Acoustics” deals with acoustics. Chapters 9 and10 are reserved for future topics. Coupled effects are treated in Chapter 11, “Coupling”.

Element theory is examined in Chapter 12, “Shape Functions”, Chapter 13, “Element Tools”, and Chapter 14,“Element Library”. Shape functions are presented in Chapter 12, “Shape Functions”, information about elementtools (integration point locations, matrix information, and other topics) is discussed in Chapter 13, “ElementTools”, and theoretical details of each ANSYS element are presented in Chapter 14, “Element Library”.

Chapter 15, “Analysis Tools” examines a number of analysis tools (acceleration effect, damping, element reordering,and many other features). Chapter 16 is reserved for a future topic. Chapter 17, “Analysis Procedures” discussesthe theory behind the different analysis types used in the ANSYS program.

Numerical processors used in preprocessing and postprocessing are covered in Chapter 18, “Pre and Postpro-cessing Tools”. Chapter 19, “Postprocessing” goes into a number of features from the general postprocessor(POST1) and the time-history postprocessor (POST26). Chapter 20, “Design Optimization” and Chapter 21,“Probabilistic Design” deal with design optimization and probabilistic design.

An index of keywords and commands has been compiled to give you handy access to the topic or command ofinterest.

1.2. Notation

The notation defined below is a partial list of the notation used throughout the manual. There are also sometables of definitions given in following sections:

• Chapter 11, “Coupling”

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

• Section 4.1: Rate-Independent Plasticity

Due to the wide variety of topics covered in this manual, some exceptions will exist.

Table 1.1 General Terms

MeaningTerm

strain-displacement matrix[B]

damping matrix[C]

specific heat matrix[Ct]

elasticity matrix[D]

Young's modulusE

force vectorF

identity matrix[I]

current vector, associated with electrical potential DOFsI

current vector, associated with magnetic potential DOFsJ

stiffness matrix[K]

conductivity matrix[Kt]

mass matrix[M]

null matrix[O]

pressure (vector)P, P

heat flow vectorQ

stress stiffness matrix[S]

temperature vectorT

time, thicknesst

local to global conversion matrix[TR]

displacement, displacement vectoru, v, w, u

electric potential vectorV

virtual internal workδ U

virtual external workδ V

fluid flow vectorW

element coordinatex, y, z

nodal coordinates (usually global Cartesian)X, Y, Z

coefficient of thermal expansionα

strainε

Poisson's ratioν

stressσ

Below is a partial list of superscripts and subscripts used on [K], [M], [C], [S], u, T, and/or F. See also Chapter 11,“Coupling”. The absence of a subscript on the above terms implies the total matrix in final form, ready for solution.

Chapter 1: Introduction

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Table 1.2 Superscripts and Subscripts

MeaningTerm

nodal effects caused by an acceleration fieldac

convection surfacec

creepcr

based on element in global coordinatese

elasticel

internal heat generationg

equilibrium iteration numberi

based on element in element coordinatesl

masterm

substep number (time step)n

effects applied directly to nodend

plasticitypl

pressurepr

slaves

swellingsw

thermalt, th

(flex over term) reduced matrices and vectors^

(dot over term) time derivative.

1.3. Applicable Products

This manual applies to the following ANSYS and ANSYS Workbench products:

1.3.1. ANSYS Products

ANSYS MultiphysicsANSYS MechanicalANSYS StructuralANSYS Mechanical with the electromagnetics add-onANSYS Mechanical with the FLOTRAN CFD add-onANSYS ProfessionalANSYS EmagANSYS FLOTRANANSYS PrepPostANSYS ED

Some command arguments and element KEYOPT settings have defaults in the derived products that are differentfrom those in the full ANSYS product. These cases are clearly documented under the “Product Restrictions” sectionof the affected commands and elements. If you plan to use your derived product input file in the ANSYS Mul-tiphysics product, you should explicitly input these settings in the derived product, rather than letting themdefault; otherwise, behavior in the full ANSYS product will be different.

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Section 1.3: Applicable Products

1.3.2. ANSYS Workbench Products

ANSYS DesignSpace (Simulation)ANSYS DesignSpace StructuralANSYS DesignSpace AdvansiaANSYS DesignSpace Entra

1.4. Using the ANSYS, Inc. Theory Reference for the ANSYS WorkbenchProduct

Many of the basic concepts and principles that are described in the ANSYS, Inc. Theory Reference apply to boththe ANSYS and ANSYS Workbench families of products; for instance, element formulations, number of integrationpoints per element, stress evaluation techniques, solve algorithms, contact mechanics. Items that will be ofparticular interest to ANSYS Workbench users include the elements and solvers. They are listed below; for moreinformation on these items, see the appropriate sections later in this manual.

1.4.1. Elements Used by the ANSYS Workbench Product

SOLID87 (Tetrahedral thermal solid)SOLID90 (Thermal solid)SOLID92 (Tetrahedral structural solid)SOLID95 (Structural solid)SHELL57 (Thermal shell)SHELL181 (Finite strain shell, full integration option)TARGE170 (Target segment)CONTA174 (Surface-to-surface contact)PRETS179 (Pretension)SOLID186 (Structural solid)SOLID187 (Tetrahedral structural solid)BEAM188 (Finite strain beam)

1.4.2. Solvers Used by the ANSYS Workbench Product

Sparse

The ANSYS Workbench product uses this solver for most structural and all thermal analyses.

PCG

The ANSYS Workbench product often uses this solver for some structural analyses, especially those with thickmodels; i.e., models that have more than one solid element through the thickness.

Boeing Block Lanczos

The ANSYS Workbench product uses this solver for modal analyses.

Chapter 1: Introduction

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1.4.3. Other Features

Shape Tool

The shape tool used by the ANSYS Workbench product is based on the same topological optimization capabilitiesas discussed in Section 20.4: Topological Optimization. Note that the shape tool is only available for stress shapeoptimization with solid models; no surface or thermal models are supported. Frequency shape optimization isnot available. In the ANSYS Workbench product, the maximum number of iteration loops to achieve a shapesolution is 40; in the ANSYS environment, you can control the number of iterations. In the ANSYS Workbenchproduct, only a single load case is considered in shape optimization.

Solution Convergence

This is discussed in Section 15.18: ANSYS Workbench Product Adaptive Solutions.

Safety Tool

The ANSYS Workbench product safety tool capability is described in Section 2.4.7: Safety Tools in the ANSYSWorkbench Product.

Fatigue Tool

The ANSYS Workbench product fatigue capabilities are described by Hancq, et al.(316).

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Section 1.4: Using the ANSYS, Inc. Theory Reference for the ANSYS Workbench Product

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Chapter 2: Structures

2.1. Structural Fundamentals

2.1.1. Stress-Strain Relationships

This section discusses material relationships for linear materials. Nonlinear materials are discussed in Chapter 4,“Structures with Material Nonlinearities”. The stress is related to the strains by:

(2–1) [ ] σ ε= D el

where:

σ = stress vector = σ σ σ σ σ σx y z xy yz xz

T (output as S)[D] = elasticity or elastic stiffness matrix or stress-strain matrix (defined in Equation 2–14 through Equa-tion 2–19) or inverse defined in Equation 2–4 or, for a few anisotropic elements, defined by full matrixdefinition (input with TB,ANEL.)

εel = ε - εth = elastic strain vector (output as EPEL)

ε = total strain vector = ε ε ε ε ε εx y z xy yz xz

T

εth = thermal strain vector (defined in Equation 2–3) (output as EPTH)

Note — εel (output as EPEL) are the strains that cause stresses.

The shear strains (εxy, εyz, and εxz) are the engineering shear strains, which are twice the tensor shear

strains. The ε notation is commonly used for tensor shear strains, but is used here as engineering shearstrains for simplicity of output.

A related quantity used in POST1 labeled “component total strain” (output as EPTO) is described inChapter 4, “Structures with Material Nonlinearities”.

The stress vector is shown in the figure below. The sign convention for direct stresses and strains usedthroughout the ANSYS program is that tension is positive and compression is negative. For shears, positive iswhen the two applicable positive axes rotate toward each other.

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Figure 2.1 Stress Vector Definition

Equation 2–1 may also be inverted to:

(2–2) [ ] ε ε σ= +th D 1

For the 3-D case, the thermal strain vector is:

(2–3) ε α α αthT

T xse

yse

zse=

∆ 0 0 0

where:

αxse

= secant coefficient of thermal expansion in the x direction (see Section 2.1.3: Temperature-DependentCoefficient of Thermal Expansion)∆T = T - Tref

T = current temperature at the point in questionTref = reference (strain-free) temperature (input on TREF command or as REFT on MP command)

The flexibility or compliance matrix, [D]-1 is:

(2–4)[ ]D

E E E

E E E

E E E

x xy x xz x

yx y y yz y

zx z zy z z− =

− −

− −

− −1

1 0 0 0

1 0 0 0

1 0

ν ν

ν ν

ν ν 00 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

G

G

G

xy

yz

xz

where typical terms are:

Chapter 2: Structures

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Ex = Young's modulus in the x direction (input as EX on MP command)

νxy = major Poisson's ratio (input as PRXY on MP command)

νyx = minor Poisson's ratio (input as NUXY on MP command)

Gxy = shear modulus in the xy plane (input as GXY on MP command)

Also, the [D]-1 matrix is presumed to be symmetric, so that:

(2–5)

ν νyx

y

xy

xE E=

(2–6)

ν νzx

z

xz

xE E=

(2–7)

ν νzy

z

yz

yE E=

Because of the above three relationships, νxy, νyz, νxz, νyx, νzy, and νzx are not independent quantities and therefore

the user should input either νxy, νyz, and νxz (input as PRXY, PRYZ, and PRXZ), or νyx, νzy, and νzx (input as NUXY,

NUYZ, and NUXZ). The use of Poisson's ratios for orthotropic materials sometimes causes confusion, so that careshould be taken in their use. Assuming that Ex is larger than Ey, νxy (PRXY) is larger than νyx (NUXY). Hence, νxy is

commonly referred to as the “major Poisson's ratio”, because it is larger than νyx, which is commonly referred to

as the “minor” Poisson's ratio. For orthotropic materials, the user needs to inquire of the source of the materialproperty data as to which type of input is appropriate. In practice, orthotropic material data are most oftensupplied in the major (PR-notation) form. For isotropic materials (Ex = Ey = Ez and νxy = νyz = νxz), so it makes no

difference which type of input is used.

Expanding Equation 2–2 with Equation 2–3 thru Equation 2–7 and writing out the six equations explicitly,

(2–8)ε α σ ν σ ν σ

x xx

x

xy y

x

xz z

xT

E E E= + − −∆

(2–9)ε α

ν σ σ ν σy y

xy x

x

y

y

yz z

yT

E E E= − + −∆

(2–10)ε α ν σ ν σ σ

z zxz x

x

yz y

y

z

zT

E E E= − − +∆

(2–11)ε

σxy

xy

xyG=

(2–12)ε

σyz

yz

yzG=

(2–13)ε σ

xzxz

xzG=

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Section 2.1: Structural Fundamentals

where typical terms are:

εx = direct strain in the x direction

σx = direct stress in the x direction

εxy = shear strain in the x-y plane

σxy = shear stress on the x-y plane

Alternatively, Equation 2–1 may be expanded by first inverting Equation 2–4 and then combining that resultwith Equation 2–3 and Equation 2–5 thru Equation 2–7 to give six explicit equations:

(2–14)

σ ν ε α ν ν νxx

yzz

yx x

yxy xz yz

z

y

Eh

EE

TE

hEE

AB= −

− + +1 2( ) ( ) ( )∆

u rruu

( ) ( )( )ε α ν ν ν ε αy yz

xz yz xy z zTEh

T− + + −∆ ∆

(2–15)

σ ν ν ν ε α νyy

xy xz yzz

yx x

yxz

z

x

E

hEE

TE

hEE

= +

− + −

( ) ( )∆ 1 2

− + +

−( ) ( )ε α ν ν ν ε αy y

zyz xz xy

y

xz zT

Eh

E

ET∆ ∆

(2–16)

σ ν ν ν ε α ν ν ν

ε

zz

xz yz xy x xz

yz xz xyy

x =

Eh

TEh

E

E( )( )

(

+ − + +

∆

y −− + −

−α ν ε αy

zxy

y

xz zT

Eh

E

ET∆ ∆) ( ) ( )1 2

(2–17)σ εxy xyG xy=

(2–18)σ εyz yz yzG=

(2–19)σ εxz xz xzG=

where:

(2–20)h

E

EEE

EE

EExy

y

xyz

z

yxz

z

xxy yz xz

z

x= − − − −1 22 2 2( ) ( ) ( )ν ν ν ν ν ν

If the shear moduli Gxy, Gyz, and Gxz are not input for isotropic materials, they are computed as:

(2–21)G G G

Exy yz xz

x

xy= = =

+2 1( )ν

For orthotropic materials, the user needs to inquire of the source of the material property data as to the correctvalues of the shear moduli, as there are no defaults provided by the program.

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The [D] matrix must be positive definite. The program checks each material property as used by each activeelement type to ensure that [D] is indeed positive definite. Positive definite matrices are defined in Section 13.5:Positive Definite Matrices. In the case of temperature dependent material properties, the evaluation is done atthe uniform temperature (input as BFUNIF,TEMP) for the first load step. The material is always positive definiteif the material is isotropic or if νxy, νyz, and νxz are all zero. When using the major Poisson's ratios (PRXY, PRYZ,

PRXZ), h as defined in Equation 2–20 must be positive for the material to be positive definite.

2.1.2. Orthotropic Material Transformation for Axisymmetric Models

The transformation of material property data from the R-θ-Z cylindrical system to the x-y-z system used for theinput requires special care. The conversion of the Young's moduli is fairly direct, whereas the correct method ofconversion of the Poisson's ratios is not obvious. Consider first how the Young's moduli transform from theglobal cylindrical system to the global Cartesian as used by the axisymmetric elements for a disc:

Figure 2.2 Material Coordinate Systems

EZ

ER

Eθ

Ex Ey

As needed by 3-D elements,using a polar coordinate system

As needed byaxisymmetric elements

(and hoop value = E )z

y

x

Thus, ER → Ex, Eθ → Ez, EZ → Ey. Starting with the global Cartesian system, the input for x-y-z coordinates gives the

following stress-strain matrix for the non-shear terms (from Equation 2–4):

(2–22)D

E E E

E E E

E E E

x y z

x x x

y y y

z z z

xy xz

yx yz

zx zy

− −− =

− −

− −

− −

11

1

1

ν ν

ν ν

ν ν

Rearranging so that the R-θ-Z axes match the x-y-z axes (i.e., x → R, y → Z, z → θ):

(2–23)[ ]D

E E E

E E E

E E ER Z

R RZ R R R

ZR Z Z Z Z

R Z

− −− =

− −− −− −

θ

θ

θ

θ θ θ θ θ

ν νν νν ν

11

1

1

If one coordinate system uses the major Poisson's ratios, and the other uses the minor Poisson's ratio, an addi-tional adjustment will need to be made.

Comparing Equation 2–22 and Equation 2–23 gives:

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Section 2.1: Structural Fundamentals

(2–24)E Ex R=

(2–25)E Ey Z=

(2–26)E Ez = θ

(2–27)ν νxy RZ

=

(2–28)ν ν θyz Z=

(2–29)ν ν θxz R=

This assumes that: νxy, νyz, νxz and νRZ, νRθ, νZθ are all major Poisson's ratios (i.e., Ex ≥ EY ≥ Ez and ER ≥ EZ ≥ Eθ).

If this is not the case (e.g., Eθ > EZ):

(2–30)ν νθ θ

θz z

z

EE

= = major Poisson ratio (input as PRYZ)

2.1.3. Temperature-Dependent Coefficient of Thermal Expansion

Considering a typical component, the thermal strain from Equation 2–3 is:

(2–31)ε αth serefT T T= −( )( )

where:

αse(T) = temperature-dependent secant coefficient of thermal expansion (SCTE)

αse(T) is input in one of three ways:

1. Input αse(T) directly (input as ALPX, ALPY, or ALPZ on MP command)

2. Computed using Equation 2–34 from αin(T), the instantaneous coefficients of thermal expansion (inputas CTEX, CTEY, or CTEZ on MP command)

3. Computed using Equation 2–32 from εith(T), the input thermal strains (input as THSX, THSY, or THSZ onMP command)

αse(T) is computed from εith(T) by rearranging Equation 2–31:

(2–32)α εse

ith

refT

TT T

( )( )=

−

Equation 2–32 assumes that when T = Tref, εith = 0. If this is not the case, the εith data is shifted automatically by

a constant so that it is true.

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εth(T) (thermal strain) is related to αin(T) by:

(2–33)ε αth in

T

TT T T

ref

( ) ( )= ∫

Combining this with equation Equation 2–32,

(2–34)α

αse

in

T

T

refT

T dT

T Tref( )

( )

=−

∫

No adjustment is needed for αin(T) as αse(T) is defined to be αin(T) when T = Tref.

As seen above, αse(T) is dependent on what was used for Tref. If αse(T) was defined using Tref as one value but

then the thermal strain was zero at another value, an adjustment needs to be made (using the MPAMOD com-mand). Consider:

(2–35)ε α αo

those

oT

TT T T dTin

o

= − = ∫( )( )

(2–36)ε α αr

thrse

refin

T

TT T T dT

ref

= − = ∫( )( )

Equation 2–35 and Equation 2–36 represent the thermal strain at a temperature T for two different startingpoints, To and Tref. Now let To be the temperature about which the data has been generated (definition temper-

ature), and Tref be the temperature at which all strains are zero (reference temperature). Thus, αose

is the supplied

data, and αrse

is what is needed as program input.

The right-hand side of Equation 2–35 may be expanded as:

(2–37)α α αin

T

Tin

T

Tin

T

TdT dT dT

o o

ref

ref∫ ∫ ∫= +

also,

(2–38)α αin

T

T

ose

ref ref odT T T To

ref

∫ = −( )( )

or

(2–39)α αin

T

T

rse

o ref odT T T To

ref

∫ = −( )( )

Combining Equation 2–35 through Equation 2–38,

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Section 2.1: Structural Fundamentals

(2–40)α α α αr

seose ref o

refose

ose

refT TT TT T

T T( ) ( ) ( ( ) )( )= + −−

−

Thus, Equation 2–40 must be accounted for when making an adjustment for the definition temperature beingdifferent from the strain-free temperature. This adjustment may be made (using the MPAMOD command).

Note that:

Equation 2–40 is nonlinear. Segments that were straight before may be no longer straight. Hence, extratemperatures may need to be specified initially (using the MPTEMP command).If Tref = To, Equation 2–40 is trivial.

If T = Tref, Equation 2–40 is undefined.

The values of T as used here are the temperatures used to define αse (input on MPTEMP command). Thus, when

using the αse adjustment procedure, it is recommended to avoid defining a T value to be the same as T = Tref (to

a tolerance of one degree). If a T value is the same as Tref, and:

• the T value is at either end of the input range, then the new αse value is simply the same as the new αvalue of the nearest adjacent point.

• the T value is not at either end of the input range, then the new αse value is the average of the two adjacentnew α values.

2.2. Derivation of Structural Matrices

The principle of virtual work states that a virtual (very small) change of the internal strain energy must be offsetby an identical change in external work due to the applied loads, or:

(2–41)δ δU V=

where:

U = strain energy (internal work) = U1 + U2

V = external work = V1 + V2 + V3

δ = virtual operator

The virtual strain energy is:

(2–42)δ δε σU1 = ∫ ( )d vol T

vol

where:

ε = strain vectorσ = stress vectorvol = volume of element

Continuing the derivation assuming linear materials and geometry, Equation 2–41 and Equation 2–42 are com-bined to give:

(2–43)δ δε ε δε εU1 = −∫ ( [ ] [ ] ) ( )T T th

vol D D d vol

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The strains may be related to the nodal displacements by:

(2–44) [ ] ε = B u

where:

[B] = strain-displacement matrix, based on the element shape functionsu = nodal displacement vector

It will be assumed that all effects are in the global Cartesian system. Combining Equation 2–44 with Equation 2–43,and noting that u does not vary over the volume:

(2–45)

δ δ

δ ε

U u B D B d vol u

u B D d vol

T Tvol

T T thvo

1 =

−

∫ [ ] [ ][ ] ( )

[ ] [ ] ( )ll∫

Another form of virtual strain energy is when a surface moves against a distributed resistance, as in a foundationstiffness. This may be written as:

(2–46)δ δ σU w d arean

Tfareaf2 = ∫ ( )

where:

wn = motion normal to the surface

σ = stress carried by the surfaceareaf = area of the distributed resistance

Both wn and σ will usually have only one nonzero component. The point-wise normal displacement is related

to the nodal displacements by:

(2–47) [ ] w N un n=

where:

[Nn] = matrix of shape functions for normal motions at the surface

The stress, σ, is

(2–48) σ = k wn

where:

k = the foundation stiffness in units of force per length per unit area

Combining Equation 2–46 thru Equation 2–48, and assuming that k is constant over the area,

(2–49)δ δU2 = ∫ [ ] [ ] ( ) u N N d area uT

nT

n fareak

f

Next, the external virtual work will be considered. The inertial effects will be studied first:

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Section 2.2: Derivation of Structural Matrices

(2–50)δ δV wFvol

d volTa

vol1 = −∫

( )

where:

w = vector of displacements of a general point

Fa = acceleration (D'Alembert) force vector

According to Newton's second law:

(2–51)

Fvol t

wa

= ∂∂

ρ2

2

where:

ρ = density (input as DENS on MP command)t = time

The displacements within the element are related to the nodal displacements by:

(2–52) [ ] w N u=

where [N] = matrix of shape functions. Combining Equation 2–50, Equation 2–51, and Equation 2–52 and assumingthat ρ is constant over the volume,

(2–53)δ δ ρ δ

δV1 = − ∫ [ ] [ ] ( ) u N N d vol

tuT T

vol

2

2

The pressure force vector formulation starts with:

(2–54)δ δV w P d arean

Tpareap2 = ∫ ( )

where:

P = the applied pressure vector (normally contains only one nonzero component)areap = area over which pressure acts

Combining equations Equation 2–52 and Equation 2–54,

(2–55)δ δV = u N 2

Tn [ ] ( )P d areapareap

∫

Unless otherwise noted, pressures are applied to the outside surface of each element and are normal to curvedsurfaces, if applicable.

Nodal forces applied to the element can be accounted for by:

(2–56)δ δV u FTend

3 =

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where:

Fend = nodal forces applied to the element

Finally, Equation 2–41, Equation 2–45, Equation 2–49, Equation 2–53, Equation 2–55, and Equation 2–56 may becombined to give:

(2–57)

[ ] [ ][ ] ( ) [ ] [ ] ( )

δ δ εu T Tvol

T T thvolB D B d vol u u B D d vol∫ ∫−

+ δδ

δ ρ δ

u N N d area u

u N N d vol

Tn

Tn farea

T Tvol

kf

[ ] [ ] ( )

[ ] [ ] ( )

∫

∫= −22

2δδ δ

tu u N P d area u FT

nT

pareaT

end

p [ ] ( ) + +∫

Noting that the δuT vector is a set of arbitrary virtual displacements common in all of the above terms, thecondition required to satisfy equation Equation 2–57 reduces to:

(2–58)([ ] [ ]) [ ] K K u F M u F Fe ef

eth

e epr

end+ − = + +&&

where:

[ ] [ ] [ ][ ] ( )K B D B d voleT

vol= =∫ element stiffness matrix

[ ] [ ] [ ] ( )K N d areaef T

n f= =k Nn element foundation stiffness matriixareaf∫

[ ] [ ] ( )F B D d voleth T th

vol= =∫ ε element thermal load vector

[ ] [ ] [ ] ( )M N N d voleT

vol= =∫ρ element mass matrix

&&ut

u= ∂∂

=2

2acceleration vector (such as gravity effectss)

[ ] ( )F P d areaepr T

pareap= =∫ Nn element pressure vector

Equation 2–58 represents the equilibrium equation on a one element basis.

The above matrices and load vectors were developed as “consistent”. Other formulations are possible. For example,if only diagonal terms for the mass matrix are requested (LUMPM,ON), the matrix is called “lumped” (see Sec-tion 13.2: Lumped Matrices). For most lumped mass matrices, the rotational degrees of freedom (DOFs) are re-moved. If the rotational DOFs are requested to be removed (KEYOPT commands with certain elements), thematrix or load vector is called “reduced”. Thus, use of the reduced pressure load vector does not generate momentsas part of the pressure load vector. Use of the consistent pressure load vector can cause erroneous internal mo-ments in a structure. An example of this would be a thin circular cylinder under internal pressure modelled withirregular shaped shell elements. As suggested by Figure 2.3: “Effects of Consistent Pressure Loading”, the consistentpressure loading generates an erroneous moment for two adjacent elements of dissimilar size.

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Section 2.2: Derivation of Structural Matrices

Figure 2.3 Effects of Consistent Pressure Loading

2.3. Structural Strain and Stress Evaluations

2.3.1. Integration Point Strains and Stresses

The element integration point strains and stresses are computed by combining equations Equation 2–1 andEquation 2–44 to get:

(2–59) [ ] ε εel thB u= −

(2–60) [ ] σ ε= D el

where:

εel = strains that cause stresses (output as EPEL)[B] = strain-displacement matrix evaluated at integration pointu = nodal displacement vector

εth = thermal strain vectorσ = stress vector (output as S)[D] = elasticity matrix

Nodal and centroidal stresses are computed from the integration point stresses as described in Section 13.6:Nodal and Centroidal Data Evaluation.

2.3.2. Surface Stresses

Surface stress output may be requested on “free” faces of 2-D and 3-D elements. “Free” means not connectedto other elements as well as not having any imposed displacements or nodal forces normal to the surface. Thefollowing steps are executed at each surface Gauss point to evaluate the surface stresses. The integration pointsused are the same as for an applied pressure to that surface.

1. Compute the in-plane strains of the surface at an integration point using:

[ ] ( )ε ε′ ′ ′ ′= −B u th(2–61)

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Hence, εx’

, εy’

and εxy’

are known. The prime (') represents the surface coordinate system, with z beingnormal to the surface.

2. A each point, set:

σz P’ = − (2–62)

(2–63)σxz’ = 0

(2–64)σyz

’ = 0

where P is the applied pressure. Equation 2–63 and Equation 2–64 are valid, as the surface for whichstresses are computed is presumed to be a free surface.

3. At each point, use the six material property equations represented by:

[ ] ’ ’ ’σ ε= D (2–65)

to compute the remaining strain and stress components ( εz’

, εxz’

, εyz’

, σx’

, σy

’ and

σxy’

.

4. Repeat and average the results across all integration points.

2.3.3. Shell Element Output

For elastic shell elements, the forces and moments per unit length (using shell nomenclature) are computed as:

(2–66)T dx x z

t

t=

−∫ σ/

/

2

2

(2–67)T dy y z

t

t=

−∫ σ/

/

2

2

(2–68)T dxy xy z

t

t=

−∫ σ/

/

2

2

(2–69)M z dx x z

t

t=

−∫ σ/

/

2

2

(2–70)M z dy y z

t

t=

−∫ σ/

/

2

2

(2–71)M z dxy xy z

t

t=

−∫ σ/

/

2

2

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Section 2.3: Structural Strain and Stress Evaluations

(2–72)N dx xz z

t

t=

−∫ σ/

/

2

2

(2–73)N dy yz z

t

t=

−∫ σ/

/

2

2

where:

Tx, Ty, Txy = in-plane forces per unit length (output as TX, TY, and TXY)

Mx, My, Mxy = bending moments per unit length (output as MX, MY, and MXY)

Nx, Ny = transverse shear forces per unit length (output as NX and NY)

t = thickness at midpoint of element, computed normal to center planeσx, etc. = direct stress (output as SX, etc.)

σxy, etc. = shear stress (output as SXY, etc.)

For shell elements with linearly elastic material, Equation 2–66 to Equation 2–73 reduce to:

(2–74)Tt

xx top x mid x bot=

+ +( ), , ,σ σ σ4

6

(2–75)Tt

yy top y mid y bot=

+ +( ), , ,σ σ σ4

6

(2–76)Tt

xyxy top xy mid xy bot=

+ +( ), , ,σ σ σ4

6

(2–77)Mt

xx top x bot=

−2

12

( ), ,σ σ

(2–78)Mt

yy top y bot=

−2

12

( ), ,σ σ

(2–79)Mt

xyxy top xy bot=

−2

12

( ), ,σ σ

(2–80)Nt

xxz top xz mid xz bot=

+ +( ), , ,σ σ σ4

6

(2–81)Nt

yyz top yz mid yz bot=

+ +( ), , ,σ σ σ4

6

For shell elements with nonlinear materials, Equation 2–66 to Equation 2–73 are numerically integrated.

It should be noted that the shell nomenclature and the nodal moment conventions are in apparent conflict witheach other. For example, a cantilever beam located along the x axis and consisting of shell elements in the x-y

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plane that deforms in the z direction under a pure bending load with coupled nodes at the free end, has thefollowing relationship:

(2–82)M b Fx MY=

where:

b = width of beamFMY = nodal moment applied to the free end (input as VALUE on F command with Lab = MY (not MX))

The shape functions of the shell element result in constant transverse strains and stresses through the thickness.Some shell elements adjust these values so that they will peak at the midsurface with 3/2 of the constant valueand be zero at both surfaces, as noted in the element discussions in Chapter 14, “Element Library”.

The thru-thickness stress (σz) is set equal to the negative of the applied pressure at the surfaces of the shell ele-

ments, and linearly interpolated in between.

2.4. Combined Stresses and Strains

When a model has only one functional direction of strains and stress (e.g., LINK8), comparison with an allowablevalue is straightforward. But when there is more than one component, the components are normally combinedinto one number to allow a comparison with an allowable. This section discusses different ways of doing thatcombination, representing different materials and/or technologies.

2.4.1. Combined Strains

The principal strains are calculated from the strain components by the cubic equation:

(2–83)

ε ε ε ε

ε ε ε ε

ε ε ε ε

x o xy xz

xy y o yz

xz yz z o

−

−

−

=

1

2

1

2

1

2

1

2

1

2

1

2

0

where:

εo = principal strain (3 values)

The three principal strains are labeled ε1, ε2, and ε3 (output as 1, 2, and 3 with strain items such as EPEL). The

principal strains are ordered so that ε1 is the most positive and ε3 is the most negative.

The strain intensity εI (output as INT with strain items such as EPEL) is the largest of the absolute values of ε1 - ε2,

ε2 - ε3, or ε3 - ε1. That is:

(2–84)ε ε ε ε ε ε εI MAX= − − −( , , )1 2 2 3 3 1

The von Mises or equivalent strain εe (output as EQV with strain items such as EPEL) is computed as:

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Section 2.4: Combined Stresses and Strains

(2–85)εν

ε ε ε ε ε εe =+

− + − + −

′

1

1

12 1 2

22 3

23 1

212( ) ( ) ( )

where:

′ = =ν effective Poisson’s ratio

materia Poisson’s ratio for l eelastic and thermal strains

0.5 for plastic, creep, and hypperelastic strains

2.4.2. Combined Stresses

The principal stresses (σ1, σ2, σ3) are calculated from the stress components by the cubic equation:

(2–86)

σ σ σ σ

σ σ σ σ

σ σ σ σ

x o xy xz

xy y o yz

xz yz z o

−

−

−

= 0

where:

σo = principal stress (3 values)

The three principal stresses are labeled σ1, σ2, and σ3 (output quantities S1, S2, and S3). The principal stresses are

ordered so that σ1 is the most positive (tensile) and σ3 is the most negative (compressive).

The stress intensity σI (output as SINT) is the largest of the absolute values of σ1 - σ2, σ2 - σ3, or σ3 - σ1. That is:

(2–87)σ σ σ σ σ σ σI = − − −MAX( )1 2 2 3 3 1

The von Mises or equivalent stress σe (output as SEQV) is computed as:

(2–88)σ σ σ σ σ σ σe = − + − + −

12 1 2

22 3

23 1

212( ) ( ) ( )

or

(2–89)σ σ σ σ σ σ σ σ σ σe x y y z z x xy yz xz= − + − + − + + +

12

62 2 2 2 2 2( ) ( ) ( ) ( )

12

When ν' = ν (input as PRXY or NUXY on MP command), the equivalent stress is related to the equivalent strainthrough

(2–90)σ εe eE=

where:

Chapter 2: Structures

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E = Young's modulus (input as EX on MP command)

2.4.3. Failure Criteria

Failure criteria are used to assess the possibility of failure of a material. This allows the consideration of orthotropicmaterials, which might be much weaker in one direction than another. Failure criteria are available in POST1 forall plane, shell, and solid structural elements (using the FC commands) and during solution for SHELL91, SHELL99,SOLID46, and SOLID191 (using TB,FAIL) only for composite elements.

Possible failure of a material can be evaluated by up to six different criteria, of which three are predefined. Theyare evaluated at the top and bottom (or middle) of each layer at each of the in-plane integration points. Thefailure criteria are:

2.4.4. Maximum Strain Failure Criteria

(2–91)ξ

εε

εε

ε

ε

1 = maximum of

whichever is applicablext

xtf

or

yt

yt

xc

xcf

ffor

zt

ztf

or

whichever is applicable

whicheve

ε

ε

εε

εε

yc

ycf

zc

zcf

rr is applicable

ε

ε

ε

ε

ε

ε

xy

xyf

yx

yzf

xz

xzf

where:

ξ1 = value of maximum strain failure criterion

εεxt

x=

0whichever is greater

εx = strain in layer x-direction

εε

xcx=

0

whichever is lesser

εxtf = failure strain in layer x-direction in tension

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Section 2.4: Combined Stresses and Strains

2.4.5. Maximum Stress Failure Criteria

(2–92)ξ

σσ

σσ

σ

σ

2 = maximum of

whichever is applicablext

xtf

or

yt

yt

xc

xcf

ffor

zt

ztf

or

whichever is applicable

whicheve

σ

σ

σσ

σσ

yc

ycf

zc

zcf

rr is applicable

σ

σ

σ

σ

σ

σ

xy

xyf

yx

yzf

xz

xzf

where:

ξ2 = value of maximum stress failure criterion

σσxt

x=

0whichever is greater

σx = stress in layer x-direction

σσ

xcx=

0

whichever is lesser

σxtf = failure stress in layer x-direction in tension

2.4.6. Tsai-Wu Failure Criteria

If the criterion used is the “strength index”:

(2–93)ξ3 = +A B

and if the criterion used is the inverse of the “strength ratio”:

(2–94)ξ3

21 02

2 1 0= − + +

. / ( / ) . /BA

B A A

where:

ξ3 = value of Tsai-Wu failure criterion

Chapter 2: Structures

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A x

xtf

xcf

y

ytf

ycf

z

ztf

zcf

xy

xyf

= − − − +( ) ( ) ( ) ( )

( )

σσ σ

σ

σ σσ

σ σ

σ

σ

2 2 2 2

2++ +

+ +

( )

( )

( )

( )

σ

σσσ

σ σ

σ σ σ σ

σ

yz

yzf

xz

xzf

xy x y

xtf

xcf

ytf

tcf

yzC C

2

2

2

2

yy z

ytf

ycf

ztf

zcf

xz x z

xtf

xcf

ztf

zcf

Cσ

σ σ σ σ

σ σ

σ σ σ σ+

Bxtf

xcf x

ytf

ycf y

ztf

zcf

= +

+ +

+ +1 1 1 1 1 1

σ σσ

σ σσ

σ σ

σz

Cxy, Cyz, Cxz = x-y, y-z, x-z, respectively, coupling coefficient for Tsai-Wu theory

The Tsai-Wu failure criteria used here are 3-D versions of the failure criterion reported in of Tsai and Hahn(190)for the 'strength index' and of Tsai(93) for the 'strength ratio'. Apparent differences are:

1. The program input used negative values for compression limits, whereas Tsai uses positive values for alllimits.

2.The program uses Cxy instead of the

Fxy*

used by Tsai and Hahn with Cxy being twice the value of Fxy

*.

2.4.7. Safety Tools in the ANSYS Workbench Product

The ANSYS Workbench product uses safety tools that are based on four different stress quantities:

1. Equivalent stress (σe).

This is the same as given in Equation 2–88.

2. Maximum tensile stress (σ1).

This is the same as given in Equation 2–86.

3. Maximum shear stress (τMAX)

This uses Mohr's circle:

(2–95)τ σ σMAX = −1 3

2

where:

σ1 and σ3 = principal stresses, defined in Equation 2–86.

4. Mohr-Coulomb stress

This theory uses a stress limit based on

(2–96)

σσ

σσ

1 3

tf

cf

+

where:

σtf = input tensile stress limit

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Section 2.4: Combined Stresses and Strains

σcf = input compression stress limit

Chapter 2: Structures

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Chapter 3: Structures with GeometricNonlinearitiesThis chapter discusses the different geometrically nonlinear options within the ANSYS program including largestrain, large deflection, stress stiffening, pressure load stiffness, and spin softening. Only elements with displace-ments degrees of freedom (DOFs) are applicable. Not included in this section are the multi-status elements (suchas LINK10, CONTAC12, COMBIN40, and CONTAC52, discussed in Chapter 14, “Element Library”) and the eigenvaluebuckling capability (discussed in Section 17.5: Buckling Analysis).

Geometric nonlinearities refer to the nonlinearities in the structure or component due to the changing geometryas it deflects. That is, the stiffness [K] is a function of the displacements u. The stiffness changes because theshape changes and/or the material rotates. The program can account for four types of geometric nonlinearities:

1. Large strain assumes that the strains are no longer infinitesimal (they are finite). Shape changes (e.g. area,thickness, etc.) are also accounted for. Deflections and rotations may be arbitrarily large.

2. Large rotation assumes that the rotations are large but the mechanical strains (those that cause stresses)are evaluated using linearized expressions. The structure is assumed not to change shape except for rigidbody motions. The elements of this class refer to the original configuration.

3. Stress stiffening assumes that both strains and rotations are small. A 1st order approximation to the rota-tions is used to capture some nonlinear rotation effects.

4. Spin softening also assumes that both strains and rotations are small. This option accounts for the radialmotion of a body's structural mass as it is subjected to an angular velocity. Hence it is a type of largedeflection but small rotation approximation.

All elements support the spin softening capability, while only some of the elements support the other options.Please refer to the ANSYS Elements Reference for details.

3.1. Large Strain

When the strains in a material exceed more than a few percent, the changing geometry due to this deformationcan no longer be neglected. Analyses which include this effect are called large strain, or finite strain, analyses. Alarge strain analysis is performed in a static (ANTYPE,STATIC) or transient (ANTYPE,TRANS) analysis while flagginglarge deformations (NLGEOM,ON) when the appropriate element type(s) is used.

The remainder of this section addresses the large strain formulation for elastic-plastic elements. These elementsuse a hypoelastic formulation so that they are restricted to small elastic strains (but allow for arbitrarily largeplastic strains). Section 4.6: Hyperelasticity addresses the large strain formulation for hyperelastic elements, whichallow arbitrarily large elastic strains.

3.1.1. Theory

The theory of large strain computations can be addressed by defining a few basic physical quantities (motionand deformation) and the corresponding mathematical relationship. The applied loads acting on a body makeit move from one position to another. This motion can be defined by studying a position vector in the “deformed”and “undeformed” configuration. Say the position vectors in the “deformed” and “undeformed” state are repres-ented by x and X respectively, then the motion (displacement) vector u is computed by (see Figure 3.1: “Po-sition Vectors and Motion of a Deforming Body”):

(3–1) u x X= −

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Figure 3.1 Position Vectors and Motion of a Deforming Body

The deformation gradient is defined as:

(3–2)[ ]

FxX

= ∂∂

which can be written in terms of the displacement of the point via Equation 3–1 as:

(3–3)[ ] [ ]

F IuX

= + ∂∂

where:

[I] = identity matrix

The information contained in the deformation gradient [F] includes the volume change, the rotation and theshape change of the deforming body. The volume change at a point is

(3–4)dVdV

det Fo

= [ ]

where:

Vo = original volume

V = current volume

det [⋅] = determinant of the matrix

The deformation gradient can be separated into a rotation and a shape change using the right polar decompos-ition theorem:

(3–5)[ ] [ ][ ]F R U=

where:

[R] = rotation matrix ([R]T[R] = [I])[U] = right stretch (shape change) matrix

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Once the stretch matrix is known, a logarithmic or Hencky strain measure is defined as:

(3–6)[ ] [ ]ε = ln U

([ε] is in tensor (matrix) form here, as opposed to the usual vector form ε). Since [U] is a 2nd order tensor (matrix),Equation 3–6 is determined through the spectral decomposition of [U]:

(3–7)[ ] ε λ= ∑

=ln e ei i i

T

i 1

3

where:

λi = eigenvalues of [U] (principal stretches)

ei = eigenvectors of [U] (principal directions)

The polar decomposition theorem (Equation 3–5) extracts a rotation [R] that represents the average rotation ofthe material at a point. Material lines initially orthogonal will not, in general, be orthogonal after deformation(because of shearing), see Figure 3.2: “Polar Decomposition of a Shearing Deformation”. The polar decompositionof this deformation, however, will indicate that they will remain orthogonal (lines x-y' in Figure 3.2: “Polar Decom-position of a Shearing Deformation”). For this reason, non-isotropic behavior (e.g. orthotropic elasticity or kin-ematic hardening plasticity) should be used with care with large strains, especially if large shearing deformationoccurs.

Figure 3.2 Polar Decomposition of a Shearing Deformation

3.1.2. Implementation

Computationally, the evaluation of Equation 3–6 is performed by one of two methods using the incrementalapproximation (since, in an elastic-plastic analysis, we are using an incremental solution procedure):

(3–8)[ ] [ ] [ ]ε ε= ≈ ∑∫d e D n

with

(3–9)[ ] [ ]∆ ∆εn nn U= l

where [∆Un] is the increment of the stretch matrix computed from the incremental deformation gradient:

(3–10)[ ] [ ][ ]∆ ∆ ∆F R Un n n=

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Section 3.1: Large Strain

where [∆Fn] is:

(3–11)[ ] [ ][ ]∆F F Fn n n= −−

11

[Fn] is the deformation gradient at the current time step and [Fn-1] is at the previous time step. Two methods are

employed for evaluating Equation 3–9.

Method 1 (Weber, et al.(127)) uses the idea of Equation 3–7:

(3–12)[ ] ∆ε λn i i i

T

in e e= ∑

=l

1

3

where λi and ei are the eigenvalue and eigenvector for the ith principal stretch increment of the incremental

stretch matrix [∆Un], Equation 3–10. This is the method employed by the large strain solids VISCO106, VISCO107

and VISCO108.

Method 2 (Hughes(156)) uses the approximate 2nd order accurate calculation:

(3–13)[ ] [ ] [ ][ ]/ /∆ ∆ε εnT

nR R= 1 2 1 2

where [R1/2] is the rotation matrix computed from the polar decomposition of the deformation gradient evaluated

at the midpoint configuration:

(3–14)[ ] [ ][ ]/ / /F R U1 2 1 2 1 2=

where [F1/2] is (using Equation 3–3):

(3–15)[ ] [ ]

/

/F Iu

X1 21 2= + ∂

∂

and the midpoint displacement is:

(3–16) ( )/u u un n1 2 1

12

= + −

un is the current displacement and un-1 is the displacement at the previous time step. [∆εn] is the “rotation-

neutralized” strain increment over the time step. The strain increment ∆%εn[ ]

is also computed from the midpointconfiguration:

(3–17) [ ]/∆ ∆%εn nB u= 1 2

∆un is the displacement increment over the time step and [B1/2] is the strain-displacement relationship evaluated

at the midpoint geometry:

(3–18) / ( )X X Xn n1 2 1

12

= + −

Chapter 3: Structures with Geometric Nonlinearities

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This method is an excellent approximation to the logarithmic strain if the strain steps are less than ~10%. Thismethod is used by the standard 2-D and 3-D solid and shell elements.

The computed strain increment [∆εn] (or equivalently ∆εn) can then be added to the previous strain εn-1 to

obtain the current total Hencky strain:

(3–19) ε ε εn n n= +−1 ∆

This strain can then be used in the stress updating procedures, see Section 4.1: Rate-Independent Plasticity andSection 4.2: Rate-Dependent Plasticity for discussions of the rate-independent and rate-dependent proceduresrespectively.

3.1.3. Definition of Thermal Strains

According to Callen(243), the coefficient of thermal expansion is defined as the fractional increase in the lengthper unit increase in the temperature. Mathematically,

(3–20)α = 1

lld

dT

where:

α = coefficient of thermal expansionl = current lengthT = temperature

Rearranging Equation 3–20 gives:

(3–21)d

dTll

= α

On the other hand, the logarithmic strain is defined as:

(3–22)εl l

ll

=

n

o

where:

εl = logarithmic strain

lo = initial length

Differential of Equation 3–22 yields:

(3–23)d

dεl ll

=

Comparison of Equation 3–21 and Equation 3–23 gives:

(3–24)d dTε αl =

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Section 3.1: Large Strain

Integration of Equation 3–24 yields:

(3–25)ε ε αl l− = −o oT T( )

where:

εol

= initial (reference) strain at temperature To

To = reference temperature

In the absence of initial strain ( εol = 0 ), then Equation 3–25 reduces to:

(3–26)ε αl = −( )T To

The thermal strain corresponds to the logarithmic strain. As an example problem, consider a line element of a

material with a constant coefficient of thermal expansion α. If the length of the line is l o at temperature To, then

the length after the temperature increases to T is:

(3–27)l l ll= = −o o oexp exp T Tε α[ ( )]

Now if one interpreted the thermal strain as the engineering (or nominal) strain, then the final length would bedifferent.

(3–28)ε αeoT T= −( )

where:

εe = engineering strain

The final length is then:

(3–29)l l l= + = + −oe

o oT T( ) [ ( )]1 1ε α

However, the difference should be very small as long as:

(3–30)α T To− = 1

because

(3–31)exp T T T To o[ ]( ) ( )α α− ≈ + −1

3.1.4. Element Formulation

The element matrices and load vectors are derived using an updated Lagrangian formulation. This producesequations of the form:

(3–32)[ ] K u F Fi iapp

inr∆ = −

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where the tangent matrix [ ]Ki has the form:

(3–33)[ ] [ ] [ ]K K Si i i= +

[Ki] is the usual stiffness matrix:

(3–34)[ ] [ ] [ ][ ] ( )K B D B d voli i

Ti i= ∫

[Bi] is the strain-displacement matrix in terms of the current geometry Xn and [Di] is the current stress-strain

matrix.

[Si] is the stress stiffness (or geometric stiffness) contribution, written symbolically as:

(3–35)[ ] [ ] [ ][ ] ( )S G G d voli i

Ti i= ∫ τ

where [Gi] is a matrix of shape function derivatives and [τi] is a matrix of the current Cauchy (true) stresses σi in

the global Cartesian system. The Newton-Raphson restoring force is:

(3–36)[ ] [ ] ( )F B d voli

nri

Ti= ∫ σ

All of the plane stress and shell elements account for the thickness changes due to the out-of-plane strain εz

(Hughes and Carnoy(157)). Shells, however, do not update their reference plane (as might be required in a largestrain out-of-plane bending deformation); the thickness change is assumed to be constant through the thickness.General element formulations using finite deformation are developed in Section 3.5: General Element Formulationsand are applicable to the 18x solid elements.

3.1.5. Applicable Input

NLGEOM,ON activates large strain computations in those elements which support it. SSTIF,ON activates thestress-stiffening contribution to the tangent matrix.

3.1.6. Applicable Output

For elements which have large strain capability, stresses (output as S) are true (Cauchy) stresses in the rotatedelement coordinate system (the element coordinate system follows the material as it rotates). Strains (output asEPEL, EPPL, etc.) are the logarithmic or Hencky strains, also in the rotated element coordinate system.

An exception is for the hyperelastic elements. For these elements, stress and strain components maintain theiroriginal orientations and some of these elements use other strain measures.

3.2. Large Rotation

If the rotations are large but the mechanical strains (those that cause stresses) are small, then a large rotationprocedure can be used. A large rotation analysis is performed in a static (ANTYPE,STATIC) or transient (AN-TYPE,TRANS) analysis while flagging large deformations (NLGEOM,ON) when the appropriate element type isused. Note that all large strain elements also support this capability, since both options account for the largerotations and for small strains, the logarithmic strain measure and the engineering strain measure coincide.

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Section 3.2: Large Rotation

3.2.1. Theory

Section 3.1: Large Strain presented the theory for general motion of a material point. Large rotation theory followsa similar development, except that the logarithmic strain measure (Equation 3–6) is replaced by the Biot, or small(engineering) strain measure:

(3–37)[ ] [ ] [ ]ε = −U I

where:

[U] = stretch matrix[I] = 3 x 3 identity matrix

3.2.2. Implementation

A corotational (or convected coordinate) approach is used in solving large rotation/small strain problems (Rankinand Brogan(66)). “Corotational” may be thought of as “rotated with”. The nonlinearities are contained in thestrain-displacement relationship which for this algorithm takes on the special form:

(3–38)[ ] [ ][ ]B B Tn v n=

where:

[Bv] = usual small strain-displacement relationship in the original (virgin) element coordinate system

[Tn] = orthogonal transformation relating the original element coordinates to the convected (or rotated)

element coordinates

The convected element coordinate frame differs from the original element coordinate frame by the amount ofrigid body rotation. Hence [Tn] is computed by separating the rigid body rotation from the total deformation

un using the polar decomposition theorem, Equation 3–5. From Equation 3–38, the element tangent stiffness

matrix has the form:

(3–39)[ ] [ ] [ ] [ ][ ][ ] ( )K T B D B T d vole n

Tv

Tv nvol

= ∫

and the element restoring force is:

(3–40) [ ] [ ] [ ] ( )F T B D d vole

nrn

Tv

Tnel

vol= ∫ ε

where the elastic strain is computed from:

(3–41) [ ] εnel

v dnB u=

und

is the element deformation which causes straining as described in a subsequent subsection.

The large rotation process can be summarized as a three step process for each element:

1. Determine the updated transformation matrix [Tn] for the element.

Chapter 3: Structures with Geometric Nonlinearities

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2.Extract the deformational displacement un

d from the total element displacement un for computing

the stresses as well as the restoring force Fe

nr .

3. After the rotational increments in ∆u are computed, update the node rotations appropriately. All threesteps require the concept of a rotational pseudovector in order to be efficiently implemented (Rankinand Brogan(66), Argyris(67)).

3.2.3. Element Transformation

The updated transformation matrix [Tn] relates the current element coordinate system to the global Cartesian

coordinate system as shown in Figure 3.3: “Element Transformation Definitions”.

Figure 3.3 Element Transformation Definitions

"!

$#%!

&!

' (

(

'

' (

)

)

)) *

*

*

[Tn] can be computed directly or the rotation of the element coordinate system [Rn] can be computed and related

to [Tn] by

(3–42)[ ] [ ][ ]T T Rn v n=

where [Tv] is the original transformation matrix. The determination of [Tn] is unique to the type of element involved,

whether it is a solid element, shell element, beam element, or spar element.

Solid Elements. The rotation matrix [Rn] for these elements is extracted from the displacement field using the

deformation gradient coupled with the polar decomposition theorem (see Malvern(87)).Shell Elements. The updated normal direction (element z direction) is computed directly from the updatedcoordinates. The computation of the element normal is given in Chapter 14, “Element Library” for each par-ticular shell element. The extraction procedure outlined for solid elements is used coupled with the inform-ation on the normal direction to compute the rotation matrix [Rn].

Beam Elements. The nodal rotation increments from ∆u are averaged to determine the average rotation ofthe element. The updated average element rotation and then the rotation matrix [Rn] is computed using

Rankin and Brogan(66). In special cases where the average rotation of the element computed in the above

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Section 3.2: Large Rotation

way differs significantly from the average rotation of the element computed from nodal translations, thequality of the results will be degraded.Link Elements. The updated transformation [Tn] is computed directly from the updated coordinates.

Generalized Mass Element (MASS21). The nodal rotation increment from ∆u is used to update the elementrotation which then yields the rotation matrix [Rn].

3.2.4. Deformational Displacements

The displacement field can be decomposed into a rigid body translation, a rigid body rotation, and a componentwhich causes strains:

(3–43) u u ur d= +

where:

ur = rigid body motion

ud = deformational displacements which cause strains

ud contains both translational as well as rotational DOF.

The translational component of the deformational displacement can be extracted from the displacement fieldby

(3–44) [ ]( ) u R x u xtd

n v v= + −

where:

utd

= translational component of the deformational displacement[Rn] = current element rotation matrix

xv = original element coordinates in the global coordinate system

u = element displacement vector in global coordinates

ud is in the global coordinate system.

For elements with rotational DOFs, the rotational components of the deformational displacement must becomputed. The rotational components are extracted by essentially “subtracting” the nodal rotations u from

the element rotation given by ur. In terms of the pseudovectors this operation is performed as follows for eachnode:

1. Compute a transformation matrix from the nodal pseudovector θn yielding [Tn].

2. Compute the relative rotation [Td] between [Rn] and [Tn]:

(3–45)[ ] [ ][ ]T R Tdn n

T=

This relative rotation contains the rotational deformations of that node as shown in Figure 3.4: “Definitionof Deformational Rotations”.

3. Extract the nodal rotational deformations ud from [Td].

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Because of the definition of the pseudovector, the deformational rotations extracted in step 3 are limited to lessthan 30°, since 2sin(θ /2) no longer approximates θ itself above 30°. This limitation only applies to the rotationaldistortion (i.e., bending) within a single element.

Figure 3.4 Definition of Deformational Rotations

3.2.5. Updating Rotations

Once the transformation [T] and deformational displacements ud are determined, the element matrices Equa-tion 3–39 and restoring force Equation 3–40 can be determined. The solution of the system of equations yieldsa displacement increment ∆u. The nodal rotations at the element level are updated with the rotational com-ponents of ∆u. The global rotations (in the output and on the results file) are not updated with the pseudovectorapproach, but are simply added to the previous rotation in un-1.

3.2.6. Applicable Input

The large rotation computations in those elements which support it are activated by the large deformation key(NLGEOM,ON). Stress-stiffening (SSTIF,ON) contributes to the tangent stiffness matrix (which may be requiredfor structures weak in bending resistance).

3.2.7. Applicable Output

Stresses (output as S) are engineering stresses in the rotated element coordinate system (the element coordinatesystem follows the material as it rotates). Strains (output as EPEL, EPPL, etc.) are engineering strains, also in therotated element coordinate system. This applies to element types that do not have large strain capability. Forelement types that have large strain capability, see Section 3.1: Large Strain.

3.2.8. Consistent Tangent Stiffness Matrix and Finite Rotation

It has been found in many situations that the use of consistent tangent stiffness in a nonlinear analysis can speedup the rate of convergence greatly. It normally results in a quadratic rate of convergence. A consistent tangentstiffness matrix is derived from the discretized finite element equilibrium equations without the introduction ofvarious approximations. The terminology of finite rotation in the context of geometrical nonlinearity impliesthat rotations can be arbitrarily large and can be updated accurately. A consistent tangent stiffness accountingfor finite rotations derived by Nour-Omid and Rankin(175) for beam/shell elements is used. The technology ofconsistent tangent matrix and finite rotation makes the buckling and postbuckling analysis a relatively easy task.

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Section 3.2: Large Rotation

KEYOPT(2) = 1 implemented in BEAM4 and SHELL63 uses this technology. The theory of finite rotation represent-ation and update has been described in Section 3.2: Large Rotation using a pseudovector representation. Thefollowing will outline the derivations of a consistent tangent stiffness matrix used for the corotational approach.

The nonlinear static finite element equations solved can be characterized by at the element level by:

(3–46)([ ] )intT F Fn

Te e

a

e

N− =

=∑ 0

1

where:

N = number of total elements

intFe = element internal force vector in the element coordinate system, generally see Equation 3–47

[Tn]T = transform matrix transferring the local internal force vector into the global coordinate system

Fea

= applied load vector at the element level in the global coordinate system

(3–47) [ ] ( )intF B d vole v

Te= ∫ σ

Hereafter, we shall focus on the derivation of the consistent tangent matrix at the element level without introdu-cing an approximation. The consistent tangent matrix is obtained by differentiating Equation 3–46 with respectto displacement variables ue:

(3–48)

[ ] [ ]

[ ]

intint

[ ]

K TFu

Tu

F

T

eT

consistent nT e

e

nT

ee

n

= +

=

∂∂

∂∂

TTv

T

een

T

ee

e

B eu

d vol

I

T vT

ud vol

II

B[ ] ( ) [ ] ( )

[ ]

∂∂

∂∂∫ ∫+

+

σ σ

∂∂∂[ ]

intTu

F

III

vT

ee

It can be seen that Part I is the main tangent matrix (Equation 3–39) and Part II is the stress stiffening matrix(Equation 3–35, Equation 3–62 or Equation 3–65). Part III is another part of the stress stiffening matrix (see Nour-Omid and Rankin(175)) traditionally neglected in the past. However, many numerical experiments have shown

that Part III of [ ]Ke

T is essential to the faster rate of convergence. KEYOPT(2) = 1 implemented in BEAM4 and

SHELL63 allows the use of [ ]Ke

T as shown in Equation 3–48. In some cases, Part III of

[ ]KeT

is unsymmetric; when

this occurs, a procedure of symmetrizing [ ]KeT

is invoked.

As Part III of the consistent tangent matrix utilizes the internal force vector intFe to form the matrix, it is required

that the internal vector intFe not be so large as to dominate the main tangent matrix (Part I). This can normally

Chapter 3: Structures with Geometric Nonlinearities

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be guaranteed if the realistic material and geometry are used, that is, the element is not used as a rigid link andthe actual thicknesses are input.

It is also noted that the consistent tangent matrix (Equation 3–48) is very suitable for use with the arc-lengthsolution method.

3.3. Stress Stiffening

3.3.1. Overview and Usage

Stress stiffening (also called geometric stiffening, incremental stiffening, initial stress stiffening, or differentialstiffening by other authors) is the stiffening (or weakening) of a structure due to its stress state. This stiffeningeffect normally needs to be considered for thin structures with bending stiffness very small compared to axialstiffness, such as cables, thin beams, and shells and couples the in-plane and transverse displacements. This effectalso augments the regular nonlinear stiffness matrix produced by large strain or large deflection effects(NLGEOM,ON). The effect of stress stiffening is accounted for by generating and then using an additional stiffnessmatrix, hereinafter called the “stress stiffness matrix”. The stress stiffness matrix is added to the regular stiffnessmatrix in order to give the total stiffness (SSTIF,ON command). Stress stiffening may be used for static (AN-TYPE,STATIC) or transient (ANTYPE,TRANS) analyses. Working with the stress stiffness matrix is the pressureload stiffness, discussed in Section 3.3.4: Pressure Load Stiffness.

The stress stiffness matrix is computed based on the stress state of the previous equilibrium iteration. Thus, togenerate a valid stress-stiffened problem, at least two iterations are normally required, with the first iterationbeing used to determine the stress state that will be used to generate the stress stiffness matrix of the seconditeration. If this additional stiffness affects the stresses, more iterations need to be done to obtain a convergedsolution.

In some linear analyses, the static (or initial) stress state may be large enough that the additional stiffness effectsmust be included for accuracy. Modal (ANTYPE,MODAL), reduced harmonic (ANTYPE,HARMIC with Method =FULL or REDUC on the HROPT command), reduced transient (ANTYPE,TRANS with Method = REDUC on theTRNOPT command) and substructure (ANTYPE,SUBSTR) analyses are linear analyses for which the prestressingeffects can be requested to be included (PSTRES,ON command). Note that in these cases the stress stiffnessmatrix is constant, so that the stresses computed in the analysis (e.g. the transient or harmonic stresses) are as-sumed small compared to the prestress stress.

If membrane stresses should become compressive rather than tensile, then terms in the stress stiffness matrixmay “cancel” the positive terms in the regular stiffness matrix and therefore yield a nonpositive-definite totalstiffness matrix, which indicates the onset of buckling. If this happens, it is indicated with the message: “Largenegative pivot value ___, at node ___ may be because buckling load has been exceeded”. It must be noted that astress stiffened model with insufficient boundary conditions to prevent rigid body motion may yield the samemessage.

The linear buckling load can be calculated directly by adding an unknown multiplier of the stress stiffness matrixto the regular stiffness matrix and performing an eigenvalue buckling problem (ANTYPE,BUCKLE) to calculatethe value of the unknown multiplier. This is discussed in more detail in Section 17.5: Buckling Analysis.

3.3.2. Theory

The strain-displacement equations for the general motion of a differential length fiber are derived below. Twodifferent results have been obtained and these are both discussed below. Consider the motion of a differentialfiber, originally at dS, and then at ds after deformation.

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Section 3.3: Stress Stiffening

Figure 3.5 General Motion of a Fiber

One end moves u, and the other end moves u + du, as shown in Figure 3.5: “General Motion of a Fiber”. Themotion of one end with the rigid body translation removed is u + du - u = du. du may be expanded as

(3–49) d

du

dv

dw

u =

where u is the displacement parallel to the original orientation of the fiber. This is shown in Figure 3.6: “Motionof a Fiber with Rigid Body Motion Removed”. Note that X, Y, and Z represent global Cartesian axes, and x, y, andz represent axes based on the original orientation of the fiber. By the Pythagorean theorem,

(3–50)ds dS du dv dw= + + +( ) ( ) ( )2 2 2

The stretch, Γ, is given by dividing ds by the original length dS:

(3–51)Λ = = +

+

+

dsdS

dudS

dvdS

dwdS

12 2 2

Chapter 3: Structures with Geometric Nonlinearities

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Figure 3.6 Motion of a Fiber with Rigid Body Motion Removed

As dS is along the local x axis,

(3–52)Λ = +

+

+

12 2 2

dudx

dvdx

dwdx

Next, Γ is expanded and converted to partial notation:

(3–53)Λ = + ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

1 22 2 2u

xux

vx

wx

The binominal theorem states that:

(3–54)1 12 8 16

2 3+ = + − +A

A A A...

when A2 < 1. One should be aware that using a limited number of terms of this series may restrict its applicabilityto small rotations and small strains. If the first two terms of the series in Equation 3–54 are used to expandEquation 3–53,

(3–55)Λ = + ∂

∂+ ∂

∂

+ ∂∂

+ ∂∂

112

2 2 2ux

ux

vx

wx

The resultant strain (same as extension since strains are assumed to be small) is then

(3–56)εx

ux

ux

vx

wx

= − = ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

Λ 112

2 2 2

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Section 3.3: Stress Stiffening

If, more accurately, the first three terms of Equation 3–54 are used and displacement derivatives of the third orderand above are dropped, Equation 3–54 reduces to:

(3–57)Λ = + ∂

∂+ ∂

∂

+ ∂∂

112

2 2ux

vx

wx

The resultant strain is:

(3–58)εx

ux

vx

wx

= − = ∂∂

+ ∂∂

+ ∂∂

Λ 112

2 2

For most 2-D and 3-D elements, Equation 3–56 is more convenient to use as no account of the loaded directionhas to be considered. The error associated with this is small as the strains were assumed to be small. For 1-Dstructures, and some 2-D elements, Equation 3–58 is used for its greater accuracy and causes no difficulty in itsimplementation.

3.3.3. Implementation

The stress-stiffness matrices are derived based on Equation 3–35, but using the nonlinear strain-displacementrelationships given in Equation 3–56 or Equation 3–58 (Cook(5)).

For a spar such as LINK8 the stress-stiffness matrix is given as:

(3–59)S

FLl[ ]=

−−

−−

0 0 0 0 0 00 1 0 0 1 00 0 1 0 0 10 0 0 0 0 00 1 0 0 1 00 0 1 0 0 1

The stress stiffness matrix for a 2-D beam (BEAM3) is given in Equation 3–60, which is the same as reported byPrzemieniecki(28). All beam and straight pipe elements use the same type of matrix. The 3-D beam and straightpipe elements (except BEAM188 and BEAM189) do not account for twist buckling. Forces used by straight pipeelements are based on not only the effect of axial stress with pipe wall, but also internal and external pressureson the “end-caps” of each element. This force is sometimes referred to as effective tension.

(3–60)[ ]S

FL

L L

L

L L

l =

− −

−

0

065

01

102

150 0 0 0

065

110

065

01

101

300

2

2

Symmetric

−−

110

215

2L L

where:

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F = force in memberL = length of member

The stress stiffness matrix for 2-D and 3-D solid elements is generated by the use of numerical integration. A 3-D solid element (SOLID45) is used here as an example:

(3–61)[ ]

[ ][ ]

[ ]S

SS

S

oo

ol =

0 00 00 0

where the matrices shown in Equation 3–61 have been reordered so that first all x-direction DOF are given, theny, and then z. [So] is an 8 by 8 matrix given by:

(3–62)[ ] [ ] [ ][ ] ( )S S S S d volo g

Tm gvol

= ∫

The matrices used by this equation are:

(3–63)[ ]Sm

x xy xz

xy y yz

xz yz x

=

σ σ σσ σ σσ σ σ

where σx, σxy etc. are stress based on the displacements of the previous iteration, and,

(3–64)[ ]

....

....

....

S

Nx

Nx

Nx

Ny

Ny

Ny

Nz

Nz

g =

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂

1 2 8

1 2 8

1 2 NNz8

∂

where Ni represents the ith shape function. This is the stress stiffness matrix for small strain analyses. For large

strain elements in a large strain analysis (NLGEOM,ON), the stress stiffening contribution is computed using theactual strain-displacement relationship (Equation 3–6).

One further case requires some explanation: axisymmetric structures with nonaxisymmetric deformations. Asany stiffening effects may only be axisymmetric, only axisymmetric cases are used for the prestress case.

Axisymmetric cases are defined as l (input as MODE on MODE command) = 0. Then, any subsequent load steps

with any value of l (including 0 itself) uses that same stress state, until another, more recent, l = 0 case isavailable. Also, torsional stresses are not incorporated into any stress stiffening effects.

Specializing this to SHELL61 (Axisymmetric-Harmonic Structural Shell), only two stresses are used for prestressing:σs, σθ, the meridional and hoop stresses, respectively. The element stress stiffness matrix is:

(3–65)[ ] [ ] [ ][ ] ( )S S S S d volg

Tm gvoll = ∫

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Section 3.3: Stress Stiffening

(3–66)

[ ]

[ ] [ ][ ]

S

S A N

m

s

s

g s

=

=

σσ

σσ

θ

θ

0 0 0

0 0 0

0 0 0

0 0 0

where [As] is defined below and [N] is defined by the element shape functions. [As] is an operator matrix and its

terms are:

(3–67)[ ]

sin

cosA

s

Cs R

CR

s

R

R

=− −

−

∂∂

∂∂

∂∂

∂∂

0 0

0 0

0

0 0

θ

θθ

θ

where:

C ==>

0 0 0

0

. if

1.0 if

ll

The three columns of the [As] matrix refer to u, v, and w motions, respectively. As suggested by the definition

for [Sm], the first two rows of [As] relate to σs and the second two rows relate to σθ. The first row of [As] is for motion

normal to the shell varying in the s direction and the second row is for hoop motions varying in the s direction.Similarly, the third row is for normal motions varying in the hoop direction. Thus Equation 3–58, rather thanEquation 3–56, is the type of nonlinear strain-displacement expression that has been used to develop Equa-tion 3–67.

3.3.4. Pressure Load Stiffness

Quite often concentrated forces are treated numerically by equivalent pressure over a known area. This is especiallycommon in the context of a linear static analysis. However, it is possible that different buckling loads may bepredicted from seemingly equivalent pressure and force loads in a eigenvalue buckling analysis. The differencecan be attributed to the fact that pressure is considered as a “follower” load. The force on the surface dependson the prescribed pressure magnitude and also on the surface orientation. Concentrated loads are not consideredas follower loads. The follower effects is a preload stiffness and plays a significant role in nonlinear and eigenvaluebuckling analysis. The follower effects manifest in the form of a “load stiffness matrix” in addition to the normalstress stiffening effects. As with any numerical analysis, it is recommended to use the type of loading which bestmodels the in-service component.

The effect of change of direction and/or area of an applied pressure is responsible for the pressure load stiffness

matrix ([Spr]) (see section 6.5.2 of Bonet and Wood(236)). It is used either for a large deflection analysis(NLGEOM,ON), regardless of the request for stress stiffening (SSTIF command), for an eigenvalue bucklinganalysis, or for a dynamic analysis that has prestressing flagged (PSTRES,ON command).

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The need of [Spr] is most dramatically seen when modelling the collapse of a ring due to external pressure usingeigenvalue buckling. The expected answer is:

(3–68)P

CEI

Rcr =

3

where:

Pcr = critical buckling load

E = Young's modulusI = moment of inertiaR = radius of the ringC = 3.0

This value of C = 3.0 is achieved when using [Spr], but when it is missing, C = 4.0, a 33% error.

[Spr] is available only for those elements identified as such in the ANSYS Elements Reference.

For static and transient analyses, its use is controlled by KEY3 on the SOLCONTROL command. For eigenvaluebuckling analyses, all elements with pressure load stiffness capability use that capability.

[Spr] is derived as an unsymmetric matrix. Symmetricizing is done, unless the command NROPT,UNSYM is used.Processing unsymmetric matrices takes more running time and storage, but may be more convergent.

3.3.5. Applicable Input

In a nonlinear analysis (ANTYPE,STATIC or ANTYPE,TRANS), the stress stiffness contribution is activated (SSTIF,ON)and then added to the stiffness matrix. When not using large deformations (NLGEOM,OFF), the rotations arepresumed to be small and the additional stiffness induced by the stress state is included. When using large de-formations (NLGEOM,ON), the stress stiffness augments the tangent matrix, affecting the rate of convergencebut not the final converged solution.

The stress stiffness contribution in the prestressed analysis is activated by the prestress flag (PSTRES,ON) anddirects the preceding analysis to save the stress state.

3.3.6. Applicable Output

In a small deflection/small strain analysis (NLGEOM,OFF), the 2-D and 3-D elements compute their strains using

Equation 3–56. The strains (output as EPEL, EPPL, etc.) therefore include the higher-order terms (e.g.

12

2∂∂

ux

in the strain computation. Also, nodal and reaction loads (output quantities F and M) will reflect the stress stiffnesscontribution, so that moment and force equilibrium include the higher order (small rotation) effects.

3.4. Spin Softening

The vibration of a spinning body will cause relative circumferential motions, which will change the direction ofthe centrifugal load which, in turn, will tend to destabilize the structure. As a small deflection analysis cannotdirectly account for changes in geometry, the effect can be accounted for by an adjustment of the stiffnessmatrix, called spin softening. Spin softening (input with KSPIN on the OMEGA command) is intended for useonly with modal (ANTYPE,MODAL), harmonic response (ANTYPE,HARMIC), reduced transient (ANTYPE,TRANS,with TRNOPT,REDUC) or substructure (ANTYPE,SUBSTR) analyses. When doing a static (ANTYPE,STATIC) or a

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Section 3.4: Spin Softening

full transient (ANTYPE,TRANS with TRNOPT,FULL) analysis, this effect is more accurately accounted for by largedeflections (NLGEOM,ON).

Consider a simple spring-mass system, with the spring oriented radially with respect to the axis of rotation, asshown in Figure 3.7: “Spinning Spring-Mass System”. Equilibrium of the spring and centrifugal forces on the massusing small deflection logic requires:

(3–69)Ku Mrs= ω2

where:

u = radial displacement of the mass from the rest positionr = radial rest position of the mass with respect to the axis of rotationωs = angular velocity of rotation

Figure 3.7 Spinning Spring-Mass System

However, to account for large deflection effects, Equation 3–69 must be expanded to:

(3–70)Ku M r us= +ω2 ( )

Rearranging terms,

(3–71)( )K M u Mrs s− =ω ω2 2

Defining:

(3–72)K K Ms= − ω2

and

(3–73)F Mrs= ω2

Equation 3–71 becomes simply,

(3–74)Ku F=

K is the stiffness needed in a small deflection solution to account for large deflection effects. F is the same asthat derived from small deflection logic. Thus, the large deflection effects are included in a small deflection

Chapter 3: Structures with Geometric Nonlinearities

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solution. This decrease in the effective stiffness matrix is called spin (or centrifugal) softening. See also Carne-gie(104) for additional development.

Extension of Equation 3–72 into three dimensions gives:

(3–75)K K Mxx xx y z xx= − +( )ω ω2 2

(3–76)K K Myy yy x z yy= − +( )ω ω2 2

(3–77)K K Mzz zz x y zz= − +( )ω ω2 2

where:

Kxx, Kyy, Kzz = x, y, and z components of stiffness as computed by the element

K K Kxx yy zz, , = x, y, and z components of stiffness adjusted foor spin softening

Mxx, Myy, Mzz = x, y, and z components of mass

wx, wy, wz = angular velocities of rotation about the x, y, and z axes

There are no modifications to the cross terms:

(3–78)K Kxy xy=

(3–79)K Kyz yz=

(3–80)K Kzx zx=

From Equation 3–75 thru Equation 3–77, it may be seen that there are spin softening effects only in the planeof rotation, not normal to the plane of rotation. Using the example of a modal analysis, Equation 3–72 can becombined with Equation 17–40 to give:

(3–81)[ ] [ ]K M− =ω2 0

or

(3–82)([ ] [ ]) [ ]K M Ms− − =ω ω2 2 0

where:

ω = the natural circular frequencies of the rotating body.

If stress stiffening is added to Equation 3–82, the resulting equation is:

(3–83)([ ] [ ] [ ]) [ ]K S M Ms+ − − =ω ω2 2 0

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Section 3.4: Spin Softening

Stress stiffening is normally applied whenever spin softening is activated, even though they are independenttheoretically. The modal analysis of a thin fan blade is shown in Figure 3.8: “Effects of Spin Softening and StressStiffening”.

Figure 3.8 Effects of Spin Softening and Stress Stiffening

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On Fan Blade Natural Frequencies

3.5. General Element Formulations

Element formulations developed in this section are applicable for general finite strain deformation. Naturally,they are applicable to small deformations, small deformation-large rotations, and stress stiffening as particularcases. The formulations are based on principle of virtual work. Minimal assumptions are used in arriving at theslope of nonlinear force-displacement relationship, i.e., element tangent stiffness. Hence, they are also calledconsistent formulations. These formulations have been implemented in PLANE182, PLANE183 , SOLID185, andSOLID186. SOLID187, SOLSH190, LINK180, SHELL181, BEAM188, BEAM189, SHELL208, and SHELL209 are furtherspecializations of the general theory.

Chapter 3: Structures with Geometric Nonlinearities

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In this section, the convention of index notation will be used. For example, repeated subscripts imply summationon the possible range of the subscript, usually the space dimension, so that σii = σ11 + σ22 + σ33, where 1, 2, and

3 refer to the three coordinate axes x1, x2, and x3, otherwise called x, y, and z.

3.5.1. Fundamental Equations

General finite strain deformation has the following characteristics:

• Geometry changes during deformation. The deformed domain at a particular time is generally differentfrom the undeformed domain and the domain at any other time.

• Strain is no longer infinitesimal so that a large strain definition has to be employed.

• Cauchy stress can not be updated simply by adding its increment. It has to be updated by a particular al-gorithm in order to take into account the finite deformation.

• Incremental analysis is necessary to simulate the nonlinear behaviors.

The updated Lagrangian method is applied to simulate geometric nonlinearities (accessed with NLGEOM,ON).Assuming all variables, such as coordinates xi, displacements ui, strains εij, stresses σij, velocities vi, volume V and

other material variables have been solved for and are known at time t; one solves for a set of linearized simultan-eous equations having displacements (and hydrostatic pressures in the mixed u-P formulation) as primary un-knowns to obtain the solution at time t + ∆t. These simultaneous equations are derived from the element for-mulations which are based on the principle of virtual work:

(3–84)σ δ δ δij ij

viB

is

is

is

e dV f u dV f u ds∫ ∫ ∫= +

where:

σij = Cauchy stress component

eux

u

xiji

j

j

i= ∂

∂+

∂∂

=1

2deformation tensor (Bathe(2))

ui = displacement

xi = current coordinate

fiB = component of body force

fiS = component of surface traction

V = volume of deformed bodyS = surface of deformed body on which tractions are prescribed

The internal virtual work can be indicated by:

(3–85)δ σ δW e dVij ij

v

= ∫

where:

W = internal virtual work

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Section 3.5: General Element Formulations

Element formulations are obtained by differentiating the virtual work (Bonet and Wood(236) and Gadala andWang(292)). In derivation, only linear differential terms are kept and all higher order terms are ignored so thatfinally a linear set of equations can be obtained.

In element formulation, material constitutive law has to be used to create the relation between stress incrementand strain increment. The constitutive law only reflects the stress increment due to straining. However, theCauchy stress is affected by the rigid body rotation and is not objective (not frame invariant). An objective stressis needed, therefore, to be able to be applied in constitutive law. One of these is Jaumann rate of Cauchy stressexpressed by McMeeking and Rice(293)

(3–86)& & & &σ σ σ ω σ ωij

Jij ik jk jk ik= − −

where:

&σijJ

= Jaumann rate of Cauchy stress

&ω υ υij

i

j

j

ix x= ∂

∂−

∂∂

=1

2spin tensor

&σij = time rate of Cauchy stress

Therefore, the Cauchy stress rate is:

(3–87)& & & &σ σ σ ω σ ωij ij

Jik jk jk ik= + +

Using the constitutive law, the stress change due to straining can be expressed as:

(3–88)&σij

Jijkl klc d=

where:

cijkl = material constitutive tensor

dvx

v

xiji

j

j

i= ∂

∂+

∂∂

=1

2rate of deformation tensor

vi = velocity

The Cauchy stress rate can be shown as:

(3–89)& & &σ σ ω σ ωij ijkl kl ik jk jk ikc d= + +

3.5.2. Classical Pure Displacement Formulation

Pure displacement formulation only takes displacements or velocities as primary unknown variables. All otherquantities such as strains, stresses and state variables in history-dependent material models are derived fromdisplacements. It is the most widely used formulation and is able to handle most nonlinear deformation problems.

The differentiation of δW:

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(3–90)D W D e dV D e dV e D dVij ij ij ij ij ij

vδ σ δ σ δ σ δ= + +∫ ( ( ))

From Equation 3–89, the stress differentiation can be derived as:

(3–91)D C De D Dij ijkl kl ik jk jk ikσ σ ω σ ω= + +

where:

D Du

x

u

xij

i

j

j

iω =

∂∂

−∂

∂

1

2

The differentiation of ωV is:

(3–92)D dV

Dux

dV De dVk

kv( ) = ∂

∂=

where:

ev = eii

Substitution of Equation 3–91 and Equation 3–92 into Equation 3–90 yields:

(3–93)

D W e C De dV

ux

Dux

e De d

ij ijkl klv

ijk

i

k

jik kj

δ δ

σ δ δ

=

+ ∂∂

∂∂

−

∫

2 VV

eDux

dV

v

ij ijk

kv

∫

∫+ ∂∂

δ σ

The third term is unsymmetric and is usually insignificant in most of deformation cases. Hence, it is ignored. Thefinal pure displacement formulation is:

(3–94)

D W e C De dV

ux

Dux

e De dV

ij ijkl klv

ijk

i

k

jik kj

δ δ

σ δ δ

=

+ ∂∂

∂∂

−

∫

vv∫

The above equation is a set of linear equations of Dui or displacement change. They can be solved out by linear

solvers. This formulation is exactly the same as the one published by McMeeking and Rice(293). The stiffness hastwo terms: the first one is material stiffness due to straining; the second one is stiffness due to geometric nonlin-earity (stress stiffness).

Since no other assumption is made on deformation, the formulation can be applied to any deformation problems(small deformation, finite deformation, small deformation-large rotation, stress stiffening, etc.) so it is called ageneral element formulation.

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Section 3.5: General Element Formulations

To achieve higher efficiency, the second term or stress stiffness is included only if requested for analyses withgeometric nonlinearities (NLGEOM,ON, PSTRES,ON, or SSTIF,ON) or buckling analysis (ANTYPE,BUCKLE).

3.5.3. Mixed u-P Formulations

The above pure displacement formulation is computationally efficient. However, the accuracy of any displacementformulation is dependent on Poisson's ratio or the bulk modulus. In such formulations, volumetric strain is de-termined from derivatives of displacements, which are not as accurately predicted as the displacements them-selves. Under nearly incompressible conditions (Poisson's ratio is close to 0.5 or bulk modulus approaches infinity),any small error in the predicted volumetric strain will appear as a large error in the hydrostatic pressure andsubsequently in the stresses. This error will, in turn, also affect the displacement prediction since external loadsare balanced by the stresses. This may result in displacements very much smaller than they should be for a givenmesh (this is called “locking”) or, in some cases, it will result in no convergence at all.

Another disadvantage of pure displacement formulation is that it is not to be able to handle fully incompressibledeformation, such as fully incompressible hyperelastic materials.

To overcome these difficulties, mixed u-P formulations were developed as an option in 18x solid elements usingKEYOPT(6) > 0. (HYPER56, HYPER58, HYPER74, HYPER84, HYPER86, and HYPER158 were also developed as mixedu-P elements for nearly incompressible hyperelastic material. These elements are discussed in Chapter 14 of thismanual. The formulation and description here are not applicable to these HYPER elements.) In these u-P formu-

lations, the hydrostatic pressure P is interpolated on the element level and solved on the global level independ-ently in the same way as displacements. The final stiffness matrix has the format of:

(3–95)

K K

K K

u

P

Fuu uP

Pu PP

=

∆∆

∆0

where:

∆u = displacement increment

∆P = hydrostatic pressure increment

Since hydrostatic pressure is obtained on a global level instead of being calculated from volumetric strain, thesolution accuracy is independent of Poisson's ratio and bulk modulus. Hence, it is more robust for nearly incom-pressible material. For fully incompressible material, mixed u-P formulation has to be employed in order to getsolutions.

The pressure DOFs are brought to global level by using internal nodes. The internal nodes are different from theregular (external) nodes in the following aspects:

• Each internal node is associated with only one element.

• The location of internal nodes is not important. They are used only to bring the pressure DOFs into theglobal equations.

• Internal nodes are created automatically and are not accessible by users.

The interpolation function of pressure is determined according to the order of elements. To remedy the lockingproblem, they are one order less than the interpolation function of strains or stresses. For 18x solid elements,the number of pressure DOFs, number of internal nodes, and interpolation functions are shown inTable 3.1: “Interpolation Functions of Hydrostatic Pressure”.

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Table 3.1 Interpolation Functions of Hydrostatic Pressure

FunctionsPInternal nodesKEYOPT(6)Element

P P= 1111182

P P P Ps t= + +1 2 3321183

P P= 1111185

P P P P Ps t r= + + +1 2 3 4421186

P P= 1111187

P P P P Ps t r= + + +1 2 3 4422187

In Table 3.1: “Interpolation Functions of Hydrostatic Pressure”, Pi , P1, P2 , P3 , and P4 are the pressure DOF atinternal node i. s, t, and r are the natural coordinates.

3.5.4. u-P Formulation I

This formulation is for nearly incompressible materials other than hyperelastic materials. For these materials, thevolumetric constraint equations or volumetric compatibility can be defined as (see Bathe(2) for details):

(3–96)P P

K− = 0

where:

P m ii= − = − =σ σ13

hydrostatic pressure from material constituti vve law

K = bulk modulus

P can also be defined as:

(3–97)DP KDev= −

In mixed formulation, stress is updated and reported by:

(3–98)σ σ δ σ δ δij ij ij ij ij ijP P P= − = + −′

where:

δij = Kronecker delta

σij = Cauchy stress from constitutive law

so that the internal virtual work Equation 3–85 can be expressed as:

(3–99)δ σ δW e dVa ij ij

v

= ∫

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Section 3.5: General Element Formulations

Introduce the constraint Equation 3–96 by Lagrangian multiplier P , the augmented internal virtual work is:

(3–100)δ σ δ δW e dV

P PK

PdVa ij ijv v

= + −

∫ ∫

Substitute Equation 3–98 into above; it is obtained:

(3–101)δ σ δ δ δW e dV P P e dV

P Pk

PdVa ij ijv

vv v

= + − + −

∫ ∫ ∫( )

where:

ev = δij eij = eii

Take differentiation of Equation 3–100, ignore all higher terms of Dui and DP than linear term, the final formu-

lation can be expressed as:

(3–102)

D W e C De dV KDe e dV

ux

Dux

e

a ij ijkl klv

v vv

ijk

i

k

jik

δ δ δ

σ δ δ

= −

+ ∂∂

∂∂

−

∫ ∫

2 DDe dV

DP e De P dVK

DP PdV

kjv

v vv

− + −

∫

∫ ∫( )δ δ δ1

This is a linear set of equations of Dui and DP (displacement and hydrostatic pressure changes). In the final

mixed u-P formulation, the third term is the stress stiffness and is included only if requested (NLGEOM,ON,PSTRES,ON, or SSTIF,ON). The rest of the terms are based on the material stiffness. The first term is from mater-ial constitutive law directly or from straining; the second term is because of the stress modification (Equation 3–98);the fourth and fifth terms are the extra rows and columns in stiffness matrix due to the introduction of the extraDOF: pressure, i.e., KuP, KPu and KPP as in Equation 3–95.

The stress stiffness in the above formulation is the same as the one in pure displacement formulation. All otherterms exist even for small deformation and are the same as the one derived by Bathe(2) for small deformationproblems.

It is worthwhile to indicate that in the mixed formulation of the higher order elements (PLANE183 , SOLID186and SOLID187 with KEYOPT(6) = 1), elastic strain only relates to the stress in the element on an averaged basis,

rather than pointwise. The reason is that the stress is updated by Equation 3–98 and pressure P is interpolatedindependently in an element with a function which is one order lower than the function for volumetric strain.For lower order elements (PLANE182, SOLID185), this problem is eliminated since either B bar technology oruniform reduced integration is used; volumetric strain is constant within an element, which is consistent with

the constant pressure P interpolation functions (see Table 3.1: “Interpolation Functions of Hydrostatic Pressure”).

In addition, this problem will not arise in element SOLID187 with linear interpolation function of P (KEYOPT(6)

= 2). This is because the order of interpolation function of P is the same as the one for volumetric strain. In

other words, the number of DOF P in one element is large enough to make P consistent with the volumetric

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strain at each integration point. Therefore, when mixed formulation of element SOLID187 is used with nearly

incompressible material, the linear interpolation function of P or KEYOPT(6) = 2 is recommended.

3.5.5. u-P Formulation II

A special formulation is necessary for fully incompressible hyperelastic material since the volume constraintequation is different and hydrostatic pressure can not be obtained from material constitutive law. Instead, it hasto be calculated separately. For these kinds of materials, the stress has to be updated by:

(3–103)σ σ δij ij ijP= −′

where:

σij′

= deviatoric component of Cauchy stress tensor

The deviatoric component of deformation tensor defined by the eij term of Equation 3–84 can be expressed as:

(3–104)e e eij ij ij v

′ = − 13

δ

The internal virtual work (Equation 3–85) can be shown using σij

′ and

eij′

:

(3–105)δ σ δ δW e P e dVij ij v

v

= −′ ′∫ ( )

The volume constraint is the incompressible condition. For a fully incompressible hyperelastic material, it canbe as defined by Sussman and Bathe(124), Bonet and Wood(236), Crisfield(294):

(3–106)1 0− =J

where:

J FxX

dVdVij

i

j o= = ∂

∂=

Fij = determinant of deformation gradient tensorXi = original coordinate

Vo = original volume

As in the mixed u-P formulation I (Section 3.5.4: u-P Formulation I), the constraint Equation 3–106 was introduced

to the internal virtual work by the Lagrangian multiplier P . Then, differentiating the augmented internal virtualwork, the final formulation is obtained.

This formulation is similar to the formulation for nearly incompressible materials, i.e. Equation 3–102. The onlymajor difference is that [KPP] = [0] in this formulation. This is because material in this formulation is fully incom-

pressible.

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Section 3.5: General Element Formulations

3.5.6. u-P Formulation III

When material behavior is almost incompressible, the pure displacement formulation may be applicable. Thebulk modulus of material, however, is usually very large and thus often results in a poorly ill-conditional matrix.To avoid this problem, a special mixed u-P formulation is therefore introduced. The almost incompressible ma-

terial usually has small volume changes at all material integration points. A new variable J is introduced toquantify this small volume change, and the constraint equation

(3–107)J J− = 0

is enforced by introduction of the modified potential:

(3–108)W Q W

WJ

J J+ = − ∂∂

−( )

where:

W = hyperelastic strain energy potentialQ = energy augmentation due to volume constraint condition

3.5.7. Volumetric Constraint Equations in u-P Formulations

The final set of linear equations of mixed formulations (see Equation 3–95) can be grouped into two:

(3–109)[ ] [ ] K u K P Fuu uP∆ ∆ ∆+ =

(3–110)[ ] [ ] K u K PPu PP∆ ∆+ = 0

Equation 3–109 are the equilibrium equations and Equation 3–110 are the volumetric constraint equations. Thetotal number of active equilibrium equations on a global level (indicated by Nd) is the total number of displacement

DOFs without any prescribed displacement boundary condition. The total number of volumetric constraintequations (indicated by Np) is the total number of pressure DOFs in all mixed u-P elements. The optimal ratio of

Nd/Np is 2 for 2-D elements and 3 for 3-D elements. When Nd/Np is too small, the system may have too many

constraint equations which may result in a severe locking problem. On the other hand, when Nd/Np is too large,

the system may have too few constraint equations which may result in too much deformation and loss of accuracy.

When Nd/Np < 1, the system has more volumetric constraint equations than equilibrium equations, thus the

system is over-constrained. In this case, if the u-P formulation I is used, the system equations will be very ill-conditioned so that it is hard to keep accuracy of solution and may cause divergence. If the u-P formulation II isused, the system equation will be singular because [KPP] = [0] in this formulation so that the system is not solvable.

Therefore, over-constrained models should be avoided as described in the ANSYS Elements Reference.

Volumetric constraint is incorporated into the final equations as extra conditions. A check is made at the elementlevel to see if the constraint equations are satisfied. The number of elements in which constraint equations havenot been satisfied is reported.

For u-P formulation I, the volumetric constraint is met if:

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(3–111)

P PK

dV

VtolV

V

−

≤∫

and for u-P formulation II, the volumetric constraint is met if:

(3–112)

JJ

dV

VtolV

V

−

≤∫

1

where:

tolV = tolerance for volumetric compatibility (input as Vtol on SOLCONTROL command)

and for u-P formulation III, the volumetric constraint is met if:

(3–113)

J JJ

dV

VtolV

V

−

≤∫

3.6. Constraints and Lagrange Multiplier Method

Constraints are generally implemented using the Lagrange Multiplier Method (See Belytschko(348)). This formu-lation has been implemented in MPC184 as described in the ANSYS Elements Reference. In this method, the internalenergy term given by Equation 3–85 is augmented by a set of constraints, imposed by the use of Lagrangemultipliers and integrated over the volume leading to an augmented form of the virtual work equation:

(3–114)δ δ δλ λ δ′ = + + ∫∫W W u dv u dvT TΦ Φ( ) ( )

where:

W' = augmented potential

and

(3–115)Φ( )u = 0

is the set of constraints to be imposed.

The variation of the augmented potential is zero provided Φ( )u = 0 (and, hence δΦ = 0 ) and, simultaneously:

(3–116)δW = 0

The equation for augmented potential (Equation 3–114) is a system of ntot equations, where:

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Section 3.6: Constraints and Lagrange Multiplier Method

(3–117)n n ntot dof c= +

where:

ndof = number of degrees of freedom in the model

nc = number of Lagrange multipliers

The solution vector consists of the displacement degrees of freedom u and the Lagrange multipliers.

The stiffness matrix is of the form:

(3–118)K H B

B

u r B

u

T T+

= − −

λ

λλ

0

∆∆ Φ( )

where:

r f f

e f u dv f u ds

ext

ij ij iB

iv

is

is

= −

= − −∫∫ ∫

int

σ δ δ δ

K r= δ

Bu

u= ∂

∂Φ( )

HBu

= ∂∂

∆ ∆u, λ = increments in displacements and Lagrange multiplier, respectively.

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Chapter 4: Structures with MaterialNonlinearitiesThis chapter discusses the structural material nonlinearities of plasticity, creep, nonlinear elasticity, hyperelasticity,viscoelasticity, concrete and swelling. Not included in this section are the slider, frictional, or other nonlinearelements (such as COMBIN7, COMBIN40, CONTAC12, etc. discussed in Chapter 14, “Element Library”) that canrepresent other nonlinear material behavior.

Material nonlinearities are due to the nonlinear relationship between stress and strain, that is, the stress is anonlinear function of the strain. The relationship is also path dependent (except for the case of nonlinear elasticityand hyperelasticity), so that the stress depends on the strain history as well as the strain itself.

The program can account for many material nonlinearities:

1. Rate-independent plasticity is characterized by the irreversible instantaneous straining that occurs in amaterial.

2. Rate-dependent plasticity allows the plastic-strains to develop over a time interval. This is also termedviscoplasticity.

3. Creep is also an irreversible straining that occurs in a material and is rate-dependent so that the strainsdevelop over time. The time frame for creep is usually much larger than that for rate-dependent plasticity.

4. Gasket material may be modelled using special relationships.

5. Nonlinear elasticity allows a nonlinear stress-strain relationship to be specified. All straining is reversible.

6. Hyperelasticity is defined by a strain energy density potential that characterizes elastomeric and foam-type materials. All straining is reversible.

7. Viscoelasticity is a rate-dependent material characterization that includes a viscous contribution to theelastic straining.

8. Concrete materials include cracking and crushing capability.

9. Swelling allows materials to enlarge in the presence of neutron flux.

Only the concrete element (SOLID65) supports the concrete model and only the viscoelastic elements (VISCO88,VISCO89) support the viscoelastic material model. Note that also listed in this table are how many stress andstrain components are involved. One component uses x (e.g., SX, EPELX, etc.), four components use X, Y, Z, XYand six components use X, Y, Z, XY, YZ, XZ.

The plastic pipe elements (PIPE20 and PIPE60) have four components, so that the nonlinear torsional and pressureeffects may be considered. If only one component is available, only the nonlinear stretching and bending effectscould be considered. This is relevant, for instance, to the 3-D thin-walled beam (BEAM24) which has only onecomponent. Thus linear torsional effects are included, but nonlinear torsional effects are not.

Strain Definitions

For the case of nonlinear materials, the definition of elastic strain given with Equation 2–1 has the form of:

(4–1) ε ε ε ε ε εel th pl cr sw= − − − −

where:

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εel = elastic strain vector (output as EPEL)ε = total strain vector

εth = thermal strain vector (output as EPTH)

εpl = plastic strain vector (output as EPPL)

εcr = creep strain vector (output as EPCR)

εsw = swelling strain vector (output as EPSW)

In this case, ε is the strain measured by a strain gauge. Equation 4–1 is only intended to show the relationshipsbetween the terms. See subsequent sections for more detail).

In POST1, total strain is reported as:

(4–2) ε ε ε εtot el pl cr= + +

where:

εtot = component total strain (output as EPTO)

Comparing the last two equations,

(4–3) ε ε ε εtot th sw= − −

The difference between these two “total” strains stems from the different usages: ε can be used to compare

strain gauge results and εtot can be used to plot nonlinear stress-strain curves.

4.1. Rate-Independent Plasticity

Rate-independent plasticity is characterized by the irreversible straining that occurs in a material once a certainlevel of stress is reached. The plastic strains are assumed to develop instantaneously, that is, independent oftime. The ANSYS program provides seven options to characterize different types of material behaviors. Theseoptions are:

• Material Behavior Option

• Bilinear Isotropic Hardening

• Multilinear Isotropic Hardening

• Nonlinear Isotropic Hardening

• Classical Bilinear Kinematic Hardening

• Multilinear Kinematic Hardening

• Nonlinear Kinematic Hardening

• Anisotropic

• Drucker-Prager

• Cast Iron

• User Specified Behavior (see User Routines and Non-Standard Uses of the ANSYS Advanced Analysis Tech-niques Guide and the Guide to ANSYS User Programmable Features)

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Except for User Specified Behavior (TB,USER), each of these is explained in greater detail later in this chapter.Figure 4.1: “Stress-Strain Behavior of Each of the Plasticity Options” represents the stress-strain behavior of eachof the options.

4.1.1. Theory

Plasticity theory provides a mathematical relationship that characterizes the elastoplastic response of materials.There are three ingredients in the rate-independent plasticity theory: the yield criterion, flow rule and thehardening rule. These will be discussed in detail subsequently. Table 4.1: “Notation” summarizes the notationused in the remainder of this chapter.

4.1.2. Yield Criterion

The yield criterion determines the stress level at which yielding is initiated. For multi-component stresses, thisis represented as a function of the individual components, f(σ), which can be interpreted as an equivalent stressσe:

(4–4)σ σe f= ( )

where:

σ = stress vector

Table 4.1 Notation

ANSYS Output LabelDefinitionVariable

EPELelastic strainsεel

EPPLplastic strainsεpl

trial strainεtr

EPEQ[1]equivalent plastic strain

εpl

Sstressesσ

equivalent stressσe

material yield parameterσy

HPRESmean or hydrostatic stressσm

SEPLequivalent stress parameter

σepl

plastic multiplierλ

yield surface translationα

plastic workκ

translation multiplierC

stress-strain matrix[D]

tangent modulusET

yield criterionF

SRATstress ratioN

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Section 4.1: Rate-Independent Plasticity

ANSYS Output LabelDefinitionVariable

plastic potentialQ

deviatoric stressS

1. In the large strain solids VISCO106, VISCO107, and VISCO108, EPEQ is labeled as PSV.

When the equivalent stress is equal to a material yield parameter σy,

(4–5)f y( )σ σ=

the material will develop plastic strains. If σe is less than σy, the material is elastic and the stresses will develop

according to the elastic stress-strain relations. Note that the equivalent stress can never exceed the material yieldsince in this case plastic strains would develop instantaneously, thereby reducing the stress to the material yield.Equation 4–5 can be plotted in stress space as shown in Figure 4.2: “Various Yield Surfaces” for some of theplasticity options. The surfaces in Figure 4.2: “Various Yield Surfaces” are known as the yield surfaces and anystress state inside the surface is elastic, that is, they do not cause plastic strains.

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Figure 4.1 Stress-Strain Behavior of Each of the Plasticity Options

σ

σ2σ

σσ

σ2σ

2σε ε

y

y

max

1

2

1

2

(a) Bilinear Kinematic (b) Multilinear Kinematic

σ σ

σmax σmaxσy σ

σ1

2

2σmax 2σmax

(c) Bilinear Isotropic (d) Multilinear Isotropicσ

σytσxt

σxc

σyc

τcr

σxy

εxy

σx σy σz( + + )=13

σy σm=σm

(e) Anisotropic (f) Drucker-Prager

ε

= mean stress (= constant)

εε

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Section 4.1: Rate-Independent Plasticity

Figure 4.2 Various Yield Surfaces

!"# $&%'() +*

-,. /10"#(203 $

$1 (-4$+5( 6 (*7(

4.1.3. Flow Rule

The flow rule determines the direction of plastic straining and is given as:

(4–6) d

Qplε λσ

= ∂∂

where:

λ = plastic multiplier (which determines the amount of plastic straining)Q = function of stress termed the plastic potential (which determines the direction of plastic straining)

If Q is the yield function (as is normally assumed), the flow rule is termed associative and the plastic strains occurin a direction normal to the yield surface.

4.1.4. Hardening Rule

The hardening rule describes the changing of the yield surface with progressive yielding, so that the conditions(i.e. stress states) for subsequent yielding can be established. Two hardening rules are available: work (or isotropic)hardening and kinematic hardening. In work hardening, the yield surface remains centered about its initialcenterline and expands in size as the plastic strains develop. For materials with isotropic plastic behavior this istermed isotropic hardening and is shown in Figure 4.3: “Types of Hardening Rules” (a). Kinematic hardening as-

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sumes that the yield surface remains constant in size and the surface translates in stress space with progressiveyielding, as shown in Figure 4.3: “Types of Hardening Rules” (b).

The yield criterion, flow rule and hardening rule for each option are summarized in Table 4.2: “Summary of Plas-ticity Options” and are discussed in detail later in this chapter.

Figure 4.3 Types of Hardening Rules

!

" #$ %&%!' )(*%!,+.-/ 0 " #12 34 -/ 0

Table 4.2 Summary of Plasticity Options

Material ResponseHardening RuleFlow RuleYield CriterionTB LabName

bilinearwork hardeningassociativevon Mises/HillBISOBilinear IsotropicHardening

multilinearwork hardeningassociativevon Mises/HillMISOMultilinear IsotropicHardening

nonlinearwork hardeningassociativevon Mises/HillNLISONonlinear IsotropicHardening

bilinearkinematichardening

associative(Prandtl- Reussequations)

von Mises/HillBKINClassical Bilinear Kin-ematic Hardening

multilinearkinematichardening

associativevon Mises/HillMKIN/KINHMultilinear KinematicHardening

nonlinearkinematichardening

associativevon Mises/HillCHABNonlinear KinematicHardening

bilinear, each direc-tion and tension andcompression differ-ent

work hardeningassociativemodified vonMises

ANISOAnisotropic

elastic- perfectlyplastic

noneassociative ornon- associative

von Mises withdependence onhydrostaticstress

DPDrucker- Prager

multilinearwork hardeningnon- associativevon Mises withdependence onhydrostaticstress

CASTCast Iron

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Section 4.1: Rate-Independent Plasticity

4.1.5. Plastic Strain Increment

If the equivalent stress computed using elastic properties exceeds the material yield, then plastic straining mustoccur. Plastic strains reduce the stress state so that it satisfies the yield criterion, Equation 4–5. Based on thetheory presented in the previous section, the plastic strain increment is readily calculated.

The hardening rule states that the yield criterion changes with work hardening and/or with kinematic hardening.Incorporating these dependencies into Equation 4–5, and recasting it into the following form:

(4–7)F( , , )σ κ α = 0

where:

κ = plastic workα = translation of yield surface

κ and α are termed internal or state variables. Specifically, the plastic work is the sum of the plastic work doneover the history of loading:

(4–8)κ σ ε= ∫ [ ] T plM d

where:

[ ]M =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

and translation (or shift) of the yield surface is also history dependent and is given as:

(4–9) α ε= ∫ C d pl

where:

C = material parameterα = back stress (location of the center of the yield surface)

Equation 4–7 can be differentiated so that the consistency condition is:

(4–10)dFF

M dF

dF

M dT T

= ∂∂

+ ∂∂

+ ∂∂

=σ

σκ

κα

α[ ] [ ] 0

Noting from Equation 4–8 that

(4–11)d M dT plκ σ ε= [ ]

and from Equation 4–9 that

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(4–12) d C d plα ε=

Equation 4–10 becomes

(4–13)∂∂

+ ∂∂

+ ∂∂

FM d

FM d C

FM d

TT pl

Tpl

σσ

κσ ε

αε[ ] [ ] [ ] == 0

The stress increment can be computed via the elastic stress-strain relations

(4–14) [ ] d D d elσ ε=

where:

[D] = stress-strain matrix

with

(4–15) d d del plε ε ε= −

since the total strain increment can be divided into an elastic and plastic part. Substituting Equation 4–6 intoEquation 4–13 and Equation 4–15 and combining Equation 4–13, Equation 4–14, and Equation 4–15 yields

(4–16)λ σ

ε

κσ

σ α

=

∂∂

− ∂∂

∂∂

− ∂∂

FM D d

FM

QC

F

T

T

[ ][ ]

[ ]

∂∂

+ ∂∂

∂∂

T TM

Q FM D

Q[ ] [ ][ ]

σ σ σ

The size of the plastic strain increment is therefore related to the total increment in strain, the current stressstate, and the specific forms of the yield and potential surfaces. The plastic strain increment is then computedusing Equation 4–6:

(4–17) d

Qplε λσ

= ∂∂

4.1.6. Implementation

An Euler backward scheme is used to enforce the consistency condition Equation 4–10. This ensures that theupdated stress, strains and internal variables are on the yield surface. The algorithm proceeds as follows:

1. The material parameter σy Equation 4–5 is determined for this time step (e.g., the yield stress at the current

temperature).

2. The stresses are computed based on the trial strain εtr, which is the total strain minus the plastic strainfrom the previous time point (thermal and other effects are ignored):

(4–18) ε ε εn

trn n

pl= − −1

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Section 4.1: Rate-Independent Plasticity

where the superscripts are described with Section 1.2: Notation and subscripts refer to the time point.Where all terms refer to the current time point, the subscript is dropped. The trial stress is then

(4–19) [ ] σ εtr trD−

3. The equivalent stress σe is evaluated at this stress level by Equation 4–4. If σe is less than σy the material

is elastic and no plastic strain increment is computed.

4. If the stress exceeds the material yield, the plastic multiplier λ is determined by a local Newton-Raphsoniteration procedure (Simo and Taylor(155)).

5. ∆εpl is computed via Equation 4–17.

6. The current plastic strain is updated

ε ε εnpl

npl pl= +−1 ∆

(4–20)

where:

εnpl

= current plastic strains (output as EPPL)

and the elastic strain computed

(4–21) ε ε εel tr pl= − ∆

where:

εel = elastic strains (output as EPEL)

The stress vector is:

(4–22) [ ] σ ε= D el

where:

σ = stresses (output as S)

7. The increments in the plastic work ∆κ and the center of the yield surface ∆α are computed via Equa-tion 4–11 and Equation 4–12 and the current values updated

(4–23)κ κ κn n= +−1 ∆

and

(4–24) α α αn n= +−1 ∆

where the subscript n-1 refers to the values at the previous time point.

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8.

For output purposes, an equivalent plastic strain εpl

(output as EPEQ), equivalent plastic strain increment

∆ εpl (output with the label “MAX PLASTIC STRAIN STEP”), equivalent stress parameter

σepl

(output asSEPL) and stress ratio N (output as SRAT) are computed. The stress ratio is given as

(4–25)N e

y= σ

σ

where σe is evaluated using the trial stress . N is therefore greater than or equal to one when yielding is

occurring and less than one when the stress state is elastic. The equivalent plastic strain increment isgiven as:

(4–26)∆ ∆ ∆ε ε ε^ [ ] pl pl T plM=

23

12

The equivalent plastic strain and equivalent stress parameters are developed for each option in the nextsections.

Note that the Euler backward integration scheme in step 4 is the radial return algorithm (Krieg(46)) for the vonMises yield criterion.

4.1.7. Elastoplastic Stress-Strain Matrix

The tangent or elastoplastic stress-strain matrix is derived from the local Newton-Raphson iteration scheme usedin step 4 above (Simo and Taylor(155)). It is therefore the consistent (or algorithmic) tangent. If the flow rule isnonassociative (F ≠ Q), then the tangent is unsymmetric. To preserve the symmetry of the matrix, for analyseswith a nonassociative flow rule (Drucker-Prager only), the matrix is evaluated using F only and again with Q onlyand the two matrices averaged.

4.1.8. Specialization for Hardening

Multilinear Isotropic Hardening and Bilinear Isotropic Hardening

These options use the von Mises yield criterion with the associated flow rule and isotropic (work) hardening(accessed with TB,MISO and TB,BISO).

The equivalent stress Equation 4–4 is:

(4–27)σeTs M s=

32

12 [ ]

where s is the deviatoric stress Equation 4–35. When σe is equal to the current yield stress σk the material is as-

sumed to yield. The yield criterion is:

(4–28)F s M sTk=

− =32

0

12 [ ] σ

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Section 4.1: Rate-Independent Plasticity

For work hardening, σk is a function of the amount of plastic work done. For the case of isotropic plasticity assumed

here, σk can be determined directly from the equivalent plastic strain εpl

of Equation 4–40 (output as EPEQ) and

the uniaxial stress-strain curve as depicted in Figure 4.4: “Uniaxial Behavior”. σk is output as the equivalent stress

parameter (output as SEPL). For temperature-dependent curves with the MISO option, σk is determined by

temperature interpolation of the input curves after they have been converted to stress-plastic strain format.

Figure 4.4 Uniaxial Behavior

For Multilinear Isotropic Hardening and σk Determination

4.1.9. Specification for Nonlinear Isotropic Hardening

In addition to the bilinear and multilinear isotropic hardening options, ANSYS also provides another nonlinearisotropic hardening option, which is also called the (Voce(253)) hardening law (accessed with TB,NLISO). Theisotropic hardening behavior of materials is specified by an equation:

(4–29)R k R R eopl b pl= + + −∞

−ε ε^ ( )^

1

where:

k = elastic limit

Ro, R∞ , b = material parameters characterizing the isotropic hardening behavior of materials

εpl = equivalent plastic strain

The constitutive equations are based on linear isotropic elasticity, the von Mises yield function and the associatedflow rule. The yield function is:

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(4–30)F s M s RT=

− =32

0

12

[ ]

The plastic strain increment is:

(4–31)

∆ε λσ

λσ

λσ

pl

e

Q F s= ∂∂

= ∂∂

= 32

where:

λ = plastic multiplier

The equivalent plastic strain increment is then:

(4–32)∆ ∆ ∆ε ε ε λ^ [ ] pl pl T plM= =2

3

The accumulated equivalent plastic strain is:

(4–33)ε εpl pl= ∑ ∆^

4.1.10. Specialization for Bilinear Kinematic Hardening

This option uses the von Mises yield criterion with the associated flow rule and kinematic hardening (accessedwith TB,BKIN).

The equivalent stress Equation 4–4 is therefore

(4–34)σ α αeTs M s= − −

32

12( ) [ ]( )

where: s = deviatoric stress vector

(4–35) s mT= − σ σ 1 1 1 0 0 0

where:

σ σ σ σm x y z= = + +mean or hydrostatic stress13

α = yield surface translation vector Equation 4–9

Note that since Equation 4–34 is dependent on the deviatoric stress, yielding is independent of the hydrostaticstress state. When σe is equal to the uniaxial yield stress, σy, the material is assumed to yield. The yield criterion

Equation 4–7 is therefore,

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Section 4.1: Rate-Independent Plasticity

(4–36)F s M sTy= − −

− =32

0

12

( ) [ ]( )α α σ

The associated flow rule yields

(4–37)∂∂

= ∂∂

= −Q Fs a

eσ σ σ3

2( )

so that the increment in plastic strain is normal to the yield surface. The associated flow rule with the von Misesyield criterion is known as the Prandtl-Reuss flow equation.

The yield surface translation is defined as:

(4–38) α ε= 2G sh

where:

G = shear modulus = E/(2 (1+n))E = Young's modulus (input as EX on MP command)ν = Poisson's ratio (input as PRXY or NUXY on MP command)

The shift strain is computed analogously to Equation 4–24:

(4–39) ε ε εn

shnsh sh= +−1 ∆

where:

∆ ∆ε εsh plCG

=2

(4–40)C

EEE E

T

T=

−23

where:

E = Young's modulus (input as EX on MP command)ET = tangent modulus from the bilinear uniaxial stress-strain curve

The yield surface translation εsh is initially zero and changes with subsequent plastic straining.

The equivalent plastic strain is dependent on the loading history and is defined to be:

(4–41)ε ε ε^ ^ ^npl

npl pl= +−1 ∆

where:

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εnpl

= equivalent plastic strain for this time point (output as EPEQ)

εnpl

−1 = equivalent plastic strain from the previous time point

The equivalent stress parameter is defined to be:

(4–42)σ σ ε^ ^epl

yT

TnplEE

E E= +

−

where:

σepl

= equivalent stress parameter (output as SEPL)

Note that if there is no plastic straining ( εpl = 0), then

σepl

is equal to the yield stress. σe

pl

only has meaning duringthe initial, monotonically increasing portion of the load history. If the load were to be reversed after plastic

loading, the stresses and therefore σe would fall below yield but σe

pl

would register above yield (since εpl

is

nonzero).

4.1.11. Specialization for Multilinear Kinematic Hardening

This option (accessed with TB,MKIN and TB,KINH) uses the Besseling(53) model also called the sublayer oroverlay model (Zienkiewicz(54)) to characterize the material behavior. The material behavior is assumed to becomposed of various portions (or subvolumes), all subjected to the same total strain, but each subvolume havinga different yield strength. (For a plane stress analysis, the material can be thought to be made up of a numberof different layers, each with a different thickness and yield stress.) Each subvolume has a simple stress-strainresponse but when combined the model can represent complex behavior. This allows a multilinear stress-straincurve that exhibits the Bauschinger (kinematic hardening) effect (Figure 4.1: “Stress-Strain Behavior of Each ofthe Plasticity Options” (b)).

The following steps are performed in the plasticity calculations:

1. The portion of total volume for each subvolume and its corresponding yield strength are determined.

2. The increment in plastic strain is determined for each subvolume assuming each subvolume is subjectedto the same total strain.

3. The individual increments in plastic strain are summed using the weighting factors determined in step1 to compute the overall or apparent increment in plastic strain.

4. The plastic strain is updated and the elastic strain is computed.

The portion of total volume (the weighting factor) and yield stress for each subvolume is determined bymatching the material response to the uniaxial stress-strain curve. A perfectly plastic von Mises material is assumedand this yields for the weighting factor for subvolume k

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Section 4.1: Rate-Independent Plasticity

(4–43)w

E E

E Ewk

Tk

Tk

ii

k= −

− −−

=

−∑1 2

31

1

ν

where:

wk = the weighting factor (portion of total volume) for subvolume k and is evaluated sequentially from 1 to

the number of subvolumesETk = the slope of the kth segment of the stress-strain curve (see Figure 4.5: “Uniaxial Behavior for Multilinear

Kinematic Hardening”)Σwi = the sum of the weighting factors for the previously evaluated subvolumes

Figure 4.5 Uniaxial Behavior for Multilinear Kinematic Hardening

The yield stress for each subvolume is given by

(4–44)σ

νε ν σyk k kE=

+− −1

2 13 1 2

( )( ( ) )

where (εk, σk) is the breakpoint in the stress-strain curve. The number of subvolumes corresponds to the number

of breakpoints specified.

The increment in plastic strain ∆εk

pl for each subvolume is computed using a von Mises yield criterion with

the associated flow rule. The section on specialization for bilinear kinematic hardening is followed but since eachsubvolume is elastic-perfectly plastic, C and therefore α is zero.

The plastic strain increment for the entire volume is the sum of the subvolume increments:

(4–45) ∆ ∆ε εpl

i ipl

i

Nw

sv=

=∑

1

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where:

Nsv = number of subvolumes

The current plastic strain and elastic strain can then be computed for the entire volume via Equation 4–20 andEquation 4–21.

The equivalent plastic strain εpl

(output as EPEQ) is defined by Equation 4–41 and equivalent stress parameter

σepl

(output as SEPL) is computed by evaluating the input stress-strain curve at εpl

(after adjusting the curve forthe elastic strain component).

4.1.12. Specialization for Nonlinear Kinematic Hardening

The material model considered is a rate-independent version of the nonlinear kinematic hardening model pro-posed by Chaboche(244, 245) (accessed with TB,CHAB). The constitutive equations are based on linear isotropicelasticity, a von Mises yield function and the associated flow rule. Like the bilinear and multilinear kinematichardening options, the model can be used to simulate the monotonic hardening and the Bauschinger effect.The model is also applicable to simulate the ratcheting effect of materials. In addition, the model allows the su-perposition of several kinematic models as well as isotropic hardening models. It is thus able to model thecomplicated cyclic plastic behavior of materials, such as cyclic hardening or softening and ratcheting or shakedown.

The model uses the von Mises yield criterion with the associated flow rule, the yield function is:

(4–46)F s a M s RT= − −

− =32

0

12

( ) [ ]( )α

where:

R = isotropic hardening variable

According to the normality rule, the flow rule is written:

(4–47) ∆ε λ

σpl Q= ∂

∂

where:

λ = plastic multiplier

The back stress α is superposition of several kinematic models as:

(4–48) α α=

=∑ ii

n

1

where:

n = number of kinematic models to be superposed.

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Section 4.1: Rate-Independent Plasticity

The evolution of the back stress (the kinematic hardening rule) for each component is defined as:

(4–49) ∆ ∆α ε γ α λi i

pli iC= −2

3

where:

Ci, γi, i = 1, 2, ... n = material constants for kinematic hardening

The associated flow rule yields:

(4–50)∂∂

= ∂∂

= −Q F s

eσ σα

σ32

The plastic strain increment, Equation 4–47 is rewritten as:

(4–51)

∆ε λ ασ

pl

e

s= −32

The equivalent plastic strain increment is then:

(4–52)∆ ∆ ∆ε ε ε λ^ [ ] pl pl T plM= =2

3

The accumulated equivalent plastic strain is:

(4–53)ε ε^ ^pl pl= ∑ ∆

The isotropic hardening variable, R, can be defined by:

(4–54)R k R R eopl b pl= + + −∞

−ε ε^ ( )^

1

where:

k = elastic limit

Ro, R∞ , b = material constants characterizing the material isotropic hardening behavior.

The material hardening behavior, R, in Equation 4–46 can also be defined through bilinear or multilinear isotropichardening options, which have been discussed early in Section 4.1.8: Specialization for Hardening.

The return mapping approach with consistent elastoplastic tangent moduli that was proposed by Simo andHughes(252) is used for numerical integration of the constitutive equation described above.

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4.1.13. Specialization for Anisotropic Plasticity

There are two anisotropic plasticity options in ANSYS. The first option uses Hill's(50) potential theory (accessedby TB,HILL command). The second option uses a generalized Hill potential theory (Shih and Lee(51)) (accessedby TB, ANISO command).

4.1.14. Hill Potential Theory

The anisotropic Hill potential theory (accessed by TB,HILL) uses Hill's(50) criterion. Hill's criterion is an extensionto the von Mises yield criterion to account for the anisotropic yield of the material. When this criterion is usedwith the isotropic hardening option, the yield function is given by:

(4–55)f MT p [ ] ( )σ σ σ σ ε= − 0

where:

σ0 = reference yield stress

εp = equivalent plastic strain

and when it is used with the kinematic hardening option, the yield function takes the form:

(4–56)f MT ( ) ( )σ σ α σ α σ= − [ ] − − 0

The material is assumed to have three orthogonal planes of symmetry. Assuming the material coordinate systemis perpendicular to these planes of symmetry, the plastic compliance matrix [M] can be written as:

(4–57)[ ]M

G H H G

H F H F

G F F G

N

L

M

=

+ − −− + −− − +

0 0 0

0 0 0

0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

F, G, H, L, M and N are material constants that can be determined experimentally. They are defined as:

(4–58)F

R R Ryy zz xx

= + −

12

1 1 12 2 2

(4–59)G

R R Rzz xx yy

= + −

12

1 1 12 2 2

(4–60)H

R R Rxx yy zz

= + −

12

1 1 12 2 2

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Section 4.1: Rate-Independent Plasticity

(4–61)L

Ryz

=

32

12

(4–62)M

Rxz

=

32

12

(4–63)N

Rxy

=

32

12

The yield stress ratios Rxx, Ryy, Rzz, Rxy, Ryz and Rxz are specified by the user and can be calculated as:

(4–64)Rxx

xxy

= σσ0

(4–65)Ryyyyy

=σσ0

(4–66)Rzz

zzy

= σσ0

(4–67)Rxyxyy

= 30

σσ

(4–68)Ryzyzy

= 30

σσ

(4–69)Rxz

xzy

= 30

σσ

where:

σijy

= yield stress values

Two notes:

• The inelastic compliance matrix should be positive definite in order for the yield function to exist.

• The plastic slope (see also Equation 4–40) is calculated as:

(4–70)E

E EE E

pl x t

x t=

−

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where:

Ex = elastic modulus in x-direction

Et = tangent modulus defined by the hardening input

4.1.15. Generalized Hill Potential Theory

The generalized anisotropic Hill potential theory (accessed by TB,ANISO) uses Hill's(50) yield criterion, whichaccounts for differences in yield strengths in orthogonal directions, as modified by Shih and Lee(51) accountingfor differences in yield strength in tension and compression. An associated flow rule is assumed and workhardening as presented by Valliappan et al.(52) is used to update the yield criterion. The yield surface is thereforea distorted circular cylinder that is initially shifted in stress space which expands in size with plastic straining asshown in Figure 4.2: “Various Yield Surfaces” (b).

The equivalent stress for this option is redefined to be:

(4–71)σ σ σ σeT TM L= −

13

13

12

[ ]

where [M] is a matrix which describes the variation of the yield stress with orientation and L accounts for thedifference between tension and compression yield strengths. L can be related to the yield surface translationα of Equation 4–34 (Shih and Lee(51)) and hence the equivalent stress function can be interpreted as havingan initial translation or shift. When σe is equal to a material parameter K, the material is assumed to yield. The

yield criterion Equation 4–7 is then

(4–72)3 0F M L KT T= − − = [ ] σ σ σ

The material is assumed to have three orthogonal planes of symmetry. The plastic behavior can then be charac-terized by the stress-strain behavior in the three element coordinate directions and the corresponding shearstress-shear strain behavior. Therefore [M] has the form:

(4–73)M

M M M

M M M

M M M

M

M

=

11 12 13

12 22 23

13 23 33

44

55

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 00 66M

By evaluating the yield criterion Equation 4–72 for all the possible uniaxial stress conditions the individual termsof [M] can be identified:

(4–74)M

Kjjj

j j= =

+ −σ σ, 1 to 6

where:

σ+j and σ-j = tensile and compressive yield strengths in direction j (j = x, y, z, xy, yz, xz)

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Section 4.1: Rate-Independent Plasticity

The compressive yield stress is handled as a positive number here. For the shear yields, σ+j = σ-j. Letting M11 = 1

defines K to be

(4–75)K x x= + −σ σ

The strength differential vector L has the form

(4–76) L L L LT= 1 2 3 0 0 0

and from the uniaxial conditions L is defined as

(4–77)L M jj jj j j= − =+ −( ),σ σ 1 to 3

Assuming plastic incompressibility (i.e. no increase in material volume due to plastic straining) yields the followingrelationships

(4–78)

M M M

M M M

M M M

11 12 13

12 22 23

13 23 33

0

0

0

+ + =+ + =+ + =

and

(4–79)L L L1 2 3 0+ + =

The off-diagonals of [M] are therefore

(4–80)

M M M M

M M M M

M M M M

12 11 22 33

13 11 22 33

23 11 22

121212

= − + −

= − − +

= − − + +

( )

( )

( 333 )

Note that Equation 4–79 (by means of Equation 4–74 and Equation 4–77) yields the consistency equation

(4–81)σ σ

σ σσ σ

σ σσ σ

σ σ+ −

+ −

+ −

+ −

+ −

+ −

− +−

+ − =x x

x x

y y

y y

z z

z z0

that must be satisfied due to the requirement of plastic incompressibility. Therefore the uniaxial yield strengthsare not completely independent.

The yield strengths must also define a closed yield surface, that is, elliptical in cross section. An elliptical yieldsurface is defined if the following criterion is met:

(4–82)M M M M M M M M M112

222

332

11 22 22 33 11 332 0+ + − + + <( )

Otherwise, the following message is output: “THE DATA TABLE DOES NOT REPRESENT A CLOSED YIELD SURFACE.THE YIELD STRESSES OR SLOPES MUST BE MADE MORE EQUAL”. This further restricts the independence of the

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uniaxial yield strengths. Since the yield strengths change with plastic straining (a consequence of work hardening),this condition must be satisfied throughout the history of loading. The program checks this condition throughan equivalent plastic strain level of 20% (.20).

For an isotropic material,

(4–83)

M M M

M M M

M M M

11 22 33

12 13 23

44 55 66

1

1 2

3

= = == = = −= = =

/

and

(4–84)L L L1 2 3 0= = =

and the yield criterion (Equation 4–72 reduces down to the von Mises yield criterion

Equation 4–36 with α = 0).

Work hardening is used for the hardening rule so that the subsequent yield strengths increase with increasingtotal plastic work done on the material. The total plastic work is defined by Equation 4–23 where the incrementin plastic work is

(4–85)∆ ∆κ σ ε= * pl

where:

*σ = average stress over the increment

Figure 4.6 Plastic Work for a Uniaxial Case

For the uniaxial case the total plastic work is simply

(4–86)κ ε σ σ= +12

plo( )

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Section 4.1: Rate-Independent Plasticity

where the terms are defined as shown in Figure 4.6: “Plastic Work for a Uniaxial Case”.

For bilinear stress-strain behavior,

(4–87)σ σ ε= +opl plE

where:

EEE

E Epl T

T=

− = plastic slope (see also Equation 4–40)E = elastic modulusET = tangent moulus

(4–88)E

EEE E

pl T

T=

−

Combining Equation 4–87 with Equation 4–86 and solving for the updated yield stress σ:

(4–89)σ κ σ= + 2 212Epl

o

Extending this result to the anisotropic case gives,

(4–90)σ κ σj jpl

ojE= + 2 212

where j refers to each of the input stress-strain curves. Equation 4–90 determines the updated yield stresses byequating the amount of plastic work done on the material to an equivalent amount of plastic work in each ofthe directions.

The parameters [M] and L can then be updated from their definitions Equation 4–74 and Equation 4–77 andthe new values of the yield stresses. For isotropic materials, this hardening rule reduces to the case of isotropichardening.

The equivalent plastic strain εpl

(output as EPEQ) is computed using the tensile x direction as the reference axisby substituting Equation 4–87 into Equation 4–86:

(4–91)ε σ σ κ^ ( )pl x x xpl

xpl

E

E= − + ++ + +

+

2122

where the yield stress in the tensile x direction σ+x refers to the initial (not updated) yield stress. The equivalent

stress parameter σe

pl

(output as SEPL) is defined as

(4–92)σ σ ε^ ^epl

xpl

xplE= ++ +

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where again σ+x is the initial yield stress.

4.1.16. Specialization for Drucker-Prager

This option uses the Drucker-Prager yield criterion with either an associated or nonassociated flow rule (accessedwith TB,DP). The yield surface does not change with progressive yielding, hence there is no hardening rule andthe material is elastic- perfectly plastic (Figure 4.1: “Stress-Strain Behavior of Each of the Plasticity Options”-f).The equivalent stress for Drucker-Prager is

(4–93)σ βσe mTs M s= +

312

12 [ ]

where:

σ σ σ σm x y z= = + +mean or hydrostatic stress13

s = deviatoric stress Equation 4–35β = material constant[M] = as defined with Equation 4–34

This is a modification of the von Mises yield criterion (Equation 4–34 with α = 0) that accounts for the influenceof the hydrostatic stress component: the higher the hydrostatic stress (confinement pressure) the higher theyield strength. β is a material constant which is given as

(4–94)β φ

φ=

−2

3 3

sin

sin( )

where:

φ = input angle of internal friction

The material yield parameter is defined as

(4–95)σ φ

φyc cos

sin=

−6

3 3( )

where:

c = input cohesion value

The yield criterion Equation 4–7 is then

(4–96)F s M smT

y= +

− =312

0

12βσ σ [ ]

This yield surface is a circular cone (Figure 4.2: “Various Yield Surfaces”-c) with the material parameters Equa-tion 4–94 and Equation 4–95 chosen such that it corresponds to the outer aspices of the hexagonal Mohr-Coulombyield surface, Figure 4.7: “Drucker-Prager and Mohr-Coulomb Yield Surfaces”.

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Section 4.1: Rate-Independent Plasticity

Figure 4.7 Drucker-Prager and Mohr-Coulomb Yield Surfaces

∂∂F

σ is readily computed as

(4–97)

∂∂

= +

F

s M s

sT

Tσ

β 1 1 1 0 0 01

12

12 [ ]

∂∂ Q

σ is similar, however β is evaluated using φf (the input “dilatancy” constant). When φf = φ, the flow rule is

associated and plastic straining occurs normal to the yield surface and there will be a volumetric expansion ofthe material with plastic strains. If φf is less than φ there will be less volumetric expansion and if φf is zero, there

will be no volumetric expansion.

The equivalent plastic strain εpl

(output as EPEQ) is defined by Equation 4–41 and the equivalent stress parameter

σepl

(output as SEPL) is defined as

(4–98)σ σ βσe

ply m= −3 3( )

The equivalent stress parameter is interpreted as the von Mises equivalent stress at yield at the current hydro-

static stress level. Therefore for any integration point undergoing yielding (stress ratio SRAT >1), σe

pl

should beclose to the actual von Mises equivalent stress (output as SIGE) at the converged solution.

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4.1.17. Cast Iron Material Model

The cast iron plasticity model is designed to model gray cast iron. The microstructure of gray cast iron can belooked at as a two-phase material, graphite flakes inserted into a steel matrix (Hjelm(334)). This microstructureleads to a substantial difference in behavior in tension and compression. In tension, the material is more brittlewith low strength and cracks form due to the graphite flakes. In compression, no cracks form, the graphite flakesbehave as incompressible media that transmit stress and the steel matrix only governs the overall behavior.

The model assumes isotropic elastic behavior, and the elastic behavior is assumed to be the same in tension andcompression. The plastic yielding and hardening in tension may be different from that in compression (see Fig-ure 4.8: “Idealized Response of Gray Cast Iron in Tension and Compression”). The plastic behavior is assumed toharden isotropically and that restricts the model to monotonic loading only.

Figure 4.8 Idealized Response of Gray Cast Iron in Tension and Compression

Compression

Tension

σ

σ

σ

ε

c0

t0

Yield Criteria

A composite yield surface is used to describe the different behavior in tension and compression. The tensionbehavior is pressure dependent and the Rankine maximum stress criterion is used. The compression behavioris pressure independent and the von Mises yield criterion is used. The yield surface is a cylinder with a tensioncutoff (cap). Figure 4.9: “Cross-Section of Yield Surface” shows a cross section of the yield surface on principaldeviatoric-stress space and Figure 4.10: “Meridian Section of Yield Surface” shows a meridional sections of theyield surface for two different stress states, compression (θ = 60) and tension (θ = 0).

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Section 4.1: Rate-Independent Plasticity

Figure 4.9 Cross-Section of Yield Surface

Rankine Triangle

Von Mises Circle

σ

σσ

'1

23

' '

(Viewed along the hydrostatic pressure axis)

Figure 4.10 Meridian Section of Yield Surface

= 0 (tension)

= 60 (compression)

σ

32

σt

3J 2

σσ

θ

θ

3

I1

t3

c

c

(von Mises cylinder with tension cutoff)

The yield surface for tension and compression “regimes” are described by Equation 4–99 and Equation 4–100(Chen and Han(332)).

The yield function for the tension cap is:

(4–99)f pt e t= + + =2

30cos( )θ σ σ

and the yield function for the compression regime is:

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(4–100)fc e c= − =σ σ 0

where:

p = I1 / 3 = σii / 3 = hydrostatic pressure

σe ij ijS S= =3 2/ von Mises equivalent stress

S = deviatoric stress tensor

θ = =13

3 3

23

23 2

arccos/

J

JLode angle

J S Sij ij212

= = second invariant of deviatoric stress tensor

J Sij3 = = third invariant of deviatoric stress tensor

σt = tension yield stress

σc = compression yield stress

Flow Rule

The plastic strain increments are defined as:

(4–101)& &ε λ

σpl Q= ∂

∂

where Q is the so-called plastic flow potential, which consists of the von Mises cylinder in compression andmodified to account for the plastic Poisson's ratio in tension, and takes the form:

(4–102)Q pe c c= − < −σ σ σfor / 3

(4–103)( )

/p Q

cQ pe c

− + = ≥ −2

22 29 3σ σ for

and

where:

cpl

pl= −

+9 1 2

5 2

( νν

νpl = plastic Poisson's ratio (input using TB,CAST)

Equation 4–103 is for less than 0.5. When νpl = 0.5, the equation reduces to the von Mises cylinder. This is shownbelow:

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Figure 4.11 Flow Potential for Cast Iron

σ

σ

σ

σ

As the flow potential is different from the yield function, nonassociated flow rule, the resulting material Jacobianis unsymmetric.

Hardening

The yield stress in uniaxial tension, σt, depends on the equivalent uniaxial plastic strain in tension, εt

pl

, and the

temperature T. Also the yield stress in uniaxial compression, σc, depends on the equivalent uniaxial plastic strain

in compression, εc

pl, and the temperature T.

To calculate the change in the equivalent plastic strain in tension, the plastic work expression in the uniaxialtension case is equated to general plastic work expression as:

(4–104)σ ε σ εt t

pl T pl∆ ∆=

where:

∆εpl = plastic strain vector increment

Equation 4–101 leads to:

(4–105)∆ ∆ε σ ε

σtpl

t

T pl= 1

In contrast, the change in the equivalent plastic strain in compression is defined as:

(4–106)∆ ∆ε εc

p pl= ^

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where:

∆ ε^pl = equivalent plastic strain increment

The yield and hardening in tension and compression are provided using the TB,UNIAXIAL command which hastwo options, tension and compression.

4.2. Rate-Dependent Plasticity

(Including Creep and Viscoplasticity)

Rate-dependent plasticity describes the flow rule of materials, which depends on time. The deformation of ma-terials is now assumed to develop as a function of the strain rate (or time). An important class of applications ofthis theory is high temperature “creep”. Several options are provided in ANSYS to characterize the different typesof rate-dependent material behaviors. The creep option is used for describing material “creep” over a relativelong period or at low strain. The rate-dependent plasticity option adopts a unified creep approach to describematerial behavior that is strain rate dependent. Anand's viscoplasticity option is another rate-dependent plasticitymodel for simulations such as metal forming. Other than other these built-in options, a rate-dependent plasticitymodel may be incorporated as user material option through the user programmable feature.

4.2.1. Creep Option

4.2.1.1. Definition and Limitations

Creep is defined as material deforming under load over time in such a way as to tend to relieve the stress. Creepmay also be a function of temperature and neutron flux level. The term “relaxation” has also been used inter-changeably with creep. The von Mises or Hill stress potentials can be used for creep analysis. For the von Misespotential, the material is assumed to be isotropic and the basic solution technique used is the initial-stiffnessNewton-Raphson method.

The options available for creep are described in Rate-Dependent Viscoplastic Materials of the ANSYS ElementsReference. Four different types of creep are available and the effects of the first three may be added togetherexcept as noted:

Primary creep is accessed with C6 (Ci values refer to the ith value given in the TBDATA command with TB,CREEP).

The creep calculations are bypassed if C1 = 0.

Secondary creep is accessed with C12. These creep calculations are bypassed if C7 = 0. They are also bypassed if a

primary creep strain was computed using the option C6 = 9, 10, 11, 13, 14, or 15, since they include secondary

creep in their formulations.

Irradiation induced creep is accessed with C66.

User-specified creep may be accessed with C6 = 100. See User Routines and Non-Standard Uses of the ANSYS Ad-

vanced Analysis Techniques Guide for more details.

The creep calculations are also bypassed if:

1. (change of time) ≤ 10-6

2. (input temperature + Toff) ≤ 0 where Toff = offset temperature (input on TOFFST command).

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3. For C6 = 0 case: A special effective strain based on εe and εcr is computed. A bypass occurs if it is equal

to zero.

4.2.1.2. Calculation of Creep

The creep equations are integrated with an explicit Euler forward algorithm, which is efficient for problemshaving small amounts of contained creep strains. A modified total strain is computed:

(4–107) ε ε ε ε εn n n

plnth

ncr′

−= − − − 1

This equation is analogous to Equation 4–18 for plasticity. The superscripts are described with Section 1.2:Notation and subscripts refer to the time point n. An equivalent modified total strain is defined as:

(4–108)

εν

ε ε ε ε ε ε

γ

et x y y z z x

xy

=+

− + − + −

+ +

′ ′ ′ ′ ′ ′

′

1

2 1

32

32

2 2 2

2

( )( ) ( ) ( )

( ) (( ) ( )γ γyz zx′ ′+

2 2

123

2

Also an equivalent stress is defined by:

(4–109)σ εe etE=

where:

E = Young's modulus (input as EX on MP command)ν = Poisson's ratio (input as PRXY or NUXY on MP command)

The equivalent creep strain increment (∆εcr) is computed as a scalar quantity from the relations given in Rate-Dependent Viscoplastic Materials of the ANSYS Elements Reference and is normally positive. If C11 = 1, a decaying

creep rate is used rather than a rate that is constant over the time interval. This option is normally not recommen-ded, as it can seriously underestimate the total creep strain where primary stresses dominate. The modified

equivalent creep strain increment ( )∆εmcr

, which would be used in place of the equivalent creep strain increment

(∆εcr) if C11 = 1, is computed as:

(4–110)∆ε εm

cret Ae

= −

1

1

where:

e = 2.718281828 (base of natural logarithms)

A = ∆εcr/εet

Next, the creep ratio (a measure of the increment of creep strain) for this integration point (Cs) is computed as:

(4–111)Cs

cr

et= ∆ε

ε

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The largest value of Cs for all elements at all integration points for this iteration is called Cmax and is output with

the label “CREEP RATIO”.

The creep strain increment is then converted to a full strain tensor. Nc is the number of strain components for a

particular type of element. If Nc = 1,

(4–112)∆ ∆ε ε ε

εxcr cr x

et=

′

Note that the term in brackets is either +1 or -1. If Nc = 4,

(4–113)∆ ∆ε εε

ε ε ενx

crcr

et

x y z=− −

+

′ ′ ′( )

( )

2

2 1

(4–114)∆ ∆ε εε

ε ε ενy

crcr

et

y z x=− −

+

′ ′ ′( )

( )

2

2 1

(4–115)∆ ∆ ∆ε ε εz

crxcr

ycr= − −

(4–116)∆ ∆ε ε

ε νγxy

crcr

etxy=

+′3

2 1( )

The first three components are the three normal strain components, and the fourth component is the shearcomponent. If Nc = 6, components 1 through 4 are the same as above, and the two additional shear components

are:

(4–117)∆ ∆ε ε

ε νγyz

crcr

etyz=

+′3

2 1( )

(4–118)∆ ∆ε ε

ε νγxz

crcr

etxz=

+′3

2 1( )

Next, the elastic strains and the total creep strains are calculated as follows, using the example of the x-component:

(4–119)( ) ( )ε ε εxel

n x n xcr= −′ ∆

(4–120)( ) ( )ε ε εxcr

n xcr

n xcr= +−1 ∆

Stresses are based on ( )εx n′

. This gives the correct stresses for imposed force problems and the maximum stressesduring the time step for imposed displacement problems.

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4.2.1.3. Time Step Size

A stability limit is placed on the time step size (Zienkiewicz and Cormeau(154)). This is because an explicit integ-ration procedure is used in which the stresses and strains are referred to time tn-1 (however, the temperature is

at time tn). The creep strain rate is calculated using time tn. It is recommended to use a time step such that the

creep ratio Cmax is less than 0.10. If the creep ratio exceeds 0.25, the run terminates with the message: “CREEP

RATIO OF . . . EXCEEDS STABILITY LIMIT OF .25.” Section 15.6: Automatic Time Stepping discusses the automatictime stepping algorithm which may be used with creep in order to increase or decrease the time step as neededfor an accurate yet efficient solution.

4.2.2. Rate-Dependent Plasticity

Currently this material option includes two models, Perzyna(296) and Peirce et al.(297). They are defined by fieldTBOPT (=PERZYNA, or PEIRCE) on the TB, RATE command. The material hardening behavior is assumed to beisotropic. The options (TB,RATE) are available with following elements: LINK180 , SHELL181, PLANE182, PLANE183, SOLID185, SOLID186 , SOLID187, SOLSH190, BEAM188 BEAM189, SHELL208, and SHELL209. The integration ofthe material constitutive equations are based a return mapping procedure (Simo and Hughes(252)) to enforceboth stress and material tangential stiffness matrix are consistent at the end of time step. A typical applicationof this material model is the simulation of material deformation at high strain rate, such as impact.

4.2.2.1. Perzyna Option

The Perzyna model has the form of

(4–121)σ ε

γσ= +

1&pl m

o

where:

σ = material yield stress

&εpl = equivalent plastic strain rate

m = strain rate hardening parameter (input as C1 in TBDATA command)γ = material viscosity parameter (input as C2 in TBDATA command)σo = static yield stress of material (defined using TB,BISO; TB,MISO; or TB,NLISO commands)

Note that σo is a function of some hardening parameters in general.

As γ tends to ∞ , or m tends to zero or &εpl

tends to zero, the solution converges to the static (rate-independent)solution. However, for this material option when m is very small (< 0.1), the solution shows difficulties in conver-gence (Peric and Owen(298)).

4.2.2.2. Peirce Option

The option of Peirce model takes form

(4–122)σ εγ

σ= +

1&pl m

o

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Similar to Perzyna model, the solution converges to the static (rate-independent) solution, as γ tends to ∞ , or

m tends to zero, or &εpl

tends to zero. For small value of m, this option shows much better convergency thanPERZYNA option (Peric and Owen(298)).

4.2.3. Anand Viscoplasticity

Viscoplasticity is characterized by the irreversible straining that occurs in a material over time. The plastic strainsare assumed to develop as a function of the strain rate. Two options are provided to characterize different typesof rate-dependent material behaviors, and they are only available with the large strain solids VISCO106, VISCO107,and VISCO108. These options are:

Material Behavior Option

Anand's Model

User Specified Behavior (see User Routines and Non-Standard Uses of the ANSYS Advanced Analysis Techniques Guide andthe Guide to ANSYS User Programmable Features)

(Note that a rate-dependent model may be incorporated in the USER option of the rate-independent plasticity).

4.2.3.1. Overview

For metals, it has long been recognized that the notion of rate-independence of plastic response is only a con-venient approximation at low temperature. Although in reality the plastic flow due to dislocation even at lowtemperature is not truly rate-independent, the use of rate-independent plastic models is quite common. Herewe present a rate-dependent plasticity model as proposed by Anand(159) and Brown et al.(147). This rate-de-pendent model differs from the rate-independent model in that there is no explicit yield condition, and noloading/unloading criterion is used. Instead, plastic flow is assumed to take place at all nonzero stress values,although at low stresses the rate of plastic flow may be immeasurably small. Further, the equivalent plastic strainrate, which is determined by the consistency condition in the rate-independent model, needs to be prescribedby an appropriate constitutive function in the rate-dependent model.

4.2.3.2. Theory

There are two basic features in Anand's model applicable to isotropic rate-dependent constitutive model formetals. First, there is no explicit yield surface, rather the instantaneous response of the material is dependenton its current state. Secondly, a single scalar internal variable “s”, called the deformation resistance, is used torepresent the isotropic resistance to inelastic flow of the material. The specifics of this constitutive equation arethe flow equation:

(4–123)d Ae sinhs

p mQR=

−θ ξ σ

1

and the evolution equation:

(4–124)&s h B

BB

doa p=

( )

Equation 4–124 allows modelling not only strain hardening, but also strain softening.

where:

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Bs

s= −

∗1

with

(4–125)s sdA

ep Q

R

n

∗ =

^ θ

where:

dp = effective inelastic deformation rateσ = effective Cauchy stresss = deformation resistance (output as PSV)s* = saturation value of deformation resistance&s = time derivative of deformation resistanceθ = absolute temperature

The remaining terms are defined in Table 4.3: “Material Parameter Units for Anand Model”. All terms must bepositive, except constant a, which must be 1.0 or greater. The inelastic strain rate in Anand's definition of mater-ial is temperature and stress dependent as well as dependent on the rate of loading. Determination of the ma-terial parameters is performed by curve-fitting a series of the stress-strain data at various temperatures and strainrates as in Anand(159) or Brown et al.(147).

4.2.3.3. Implementation

A consistent stress update procedure which is equivalent to the Euler backward scheme used to enforce theconsistency condition and the evolution Equation 4–124 at the end of the time step. The consistency conditionin rate-dependent plasticity requires that the stress and strain values are consistent via Equation 4–14, Equa-tion 4–15 and the rate-dependent counterpart to Equation 4–17.

The accumulated plastic work (see Equation 4–23) is available (output as PLWK).

Table 4.3 Material Parameter Units for Anand Model

UnitsMeaningParameterTBDATA Constant

stress, e.g. psi, MPaInitial value of deformation resist-ance

so1

energy / volume, e.g.kJ / mole

Q = activation energyQ/R2

energy / (volumetemperature), e.g. kJ/ (mole - °K

R = universal gas content

1 / time e.g. 1 /second

pre-exponential factorA3

dimensionlessmultiplier of stressξ4

dimensionlessstrain rate sensitivity of stressm5

stress e.g. psi, MPahardening/softening constantho6

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UnitsMeaningParameterTBDATA Constant

stress e.g. psi, MPacoefficient for deformation resist-ance saturation valueS

^7

dimensionlessstrain rate sensitivity of saturation(deformation resistance) value

n8

dimensionlessstrain rate sensitivity of hardeningor softening

a9

where:

kJ = kilojoules°K = degrees Kelvin

4.3. Gasket Material

Gasket joints are essential components in most of structural assemblies. Gaskets as sealing components betweenstructural components are usually very thin and made of many materials, such as steel, rubber and composites.From a mechanics point of view, gaskets act to transfer the force between mating components. The gasket ma-terial is usually under compression. The material under compression exhibits high nonlinearity. The gasket ma-terial also shows quite complicated unloading behavior. The primary deformation of a gasket is usually confinedto 1 direction, that is through-thickness. The stiffness contribution from membrane (in-plane) and transverseshear are much smaller, and are neglected.

The table option GASKET allows gasket joints to be simulated with the interface elements, in which the through-thickness deformation is decoupled from the in-plane deformation, see Section 14.192: INTER192 - 2-D 4-NodeGasket, Section 14.193: INTER193 - 2-D 6-Node Gasket, Section 14.194: INTER194 - 3-D 16-Node Gasket, andSection 14.195: INTER195 - 3-D 8-Node Gasket for detailed description of interface elements. The user can directlyinput the experimentally measured complex pressure-closure curve (compression curve) and several unloadingpressure-closure curves for characterizing the through thickness deformation of gasket material.

Figure 4.12: “Pressure vs. Deflection Behavior of a Gasket Material” shows the experimental pressure vs. closure(relative displacement of top and bottom gasket surfaces) data for a graphite composite gasket material. Thesample was unloaded and reloaded 5 times along the loading path and then unloaded at the end of the test todetermine the unloading stiffness of the material.

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Section 4.3: Gasket Material

Figure 4.12 Pressure vs. Deflection Behavior of a Gasket Material

4.3.1. Stress and Deformation

The gasket pressure and deformation are based on the local element coordinate systems. The gasket pressureis actually the stress normal to the gasket element midsurface in the gasket layer. Gasket deformation is charac-terized by the closure of top and bottom surfaces of gasket elements, and is defined as:

(4–126)d u u= −TOP BOTTOM

Where, uTOP and uBOTTOM are the displacement of top and bottom surfaces of interface elements in the local elementcoordinate system based on the mid-plane of element.

4.3.2. Material Definition

The input of material data of a gasket material is specified by the command (TB,GASKET). The input of materialdata considers of 2 main parts: general parameters and pressure closure behaviors. The general parametersdefines initial gasket gap, the stable stiffness for numerical stabilization, and the stress cap for gasket in tension.The pressure closure behavior includes gasket compression (loading) and tension data (unloading).

The GASKET option has followings sub-options:

DescriptionSub-option

Define gasket material general parametersPARA

Define gasket compression dataCOMP

Define gasket linear unloading dataLUNL

Define gasket nonlinear unloading dataNUNL

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A gasket material can have several options at the same time. When no unloading curves are defined, the mater-ial behavior follows the compression curve while it is unloaded.

4.3.3. Thermal Deformation

The thermal deformation is taken into account by using an additive decomposition in the total deformation, d,as:

(4–127)d = + +d d di th o

where:

d = relative total deformation between top and bottom surfaces of the interface elementdi = relative deformation between top and bottom surfaces causing by the applying stress, this can be also

defined as mechanical deformationdth = relative thermal deformation between top and bottom surfaces due to free thermal expansion

do = initial gap of the element and is defined by sub-option PARA

The thermal deformation causing by free thermal expansion is defined as:

(4–128)d T hth = ∆α * *

where:

α = coefficient of thermal expansion (input as ALPX on MP command)∆T = temperature change in the current load steph = thickness of layer at the integration point where thermal deformation is of interest

4.4. Nonlinear Elasticity

4.4.1. Overview and Guidelines for Use

The ANSYS program provides a capability to model nonlinear (multilinear) elastic materials (input using TB,MELAS).Unlike plasticity, no energy is lost (the process is conservative).

Figure 4.13: “Stress-Strain Behavior for Nonlinear Elasticity” represents the stress-strain behavior of this option.Note that the material unloads along the same curve, so that no permanent inelastic strains are induced.

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Section 4.4: Nonlinear Elasticity

Figure 4.13 Stress-Strain Behavior for Nonlinear Elasticity

The total strain components εn are used to compute an equivalent total strain measure:

(4–129)

εν

ε ε ε ε ε ε

ε ε

et

x y y z z x

xy yz

=+

− + − + −

+ + +

1

2 1

32

32

2 2 2

2 2

( )( ) ( ) ( )

( ) ( )332

2

12( )εxz

εet

is used with the input stress-strain curve to get an equivalent value of stress σe .

The elastic (linear) component of strain can then be computed:

(4–130) ε σ

εεn

el e

et n

E=

and the “plastic” or nonlinear portion is therefore:

(4–131) ε ε εnpl

n nel= −

In order to avoid an unsymmetric matrix, only the symmetric portion of the tangent stress-strain matrix is used:

(4–132)[ ] [ ]D

EDep

e

e= σ

ε

which is the secant stress-strain matrix.

4.5. Shape Memory Alloy Material Model

4.5.1. Background

The shape memory alloy (SMA) material model implemented (accessed with TB,SMA) is intended for modelingthe superelastic behavior of Nitinol alloys, in which the material undergoes large deformation without showing

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permanent deformation under isothermal conditions, as shown in Figure 4.14: “Typical Superelasticity Behavior”.In this figure the material is first loaded (ABC), showing a nonlinear behavior. When unloaded (CDA), the reversetransformation occurs. This behavior is hysteretic with no permanent strain (Auricchio et al.(347)).

Figure 4.14 Typical Superelasticity Behavior

σ

ε

Nitinol is a nickel titanium alloy that was discovered in 1960s, at the Naval Ordnance Laboratory. Hence, the ac-ronym NiTi-NOL (or nitinol) has been commonly used when referring to Ni-Ti based shape memory alloys.

The mechanism of superelasticity behavior of the shape memory alloy is due to the reversible phase transform-ation of austenite and martensite. Austenite is the crystallographically more-ordered phase and martensite isthe crystallographically less-ordered phase. Typically, the austenite is stable at high temperatures and low valuesof the stress, while the martensite is stable at low temperatures and high values of the stress. When the materialis at or above a threshold temperature and has a zero stress state, the stable phase is austenite. Increasing thestress of this material above the threshold temperature activates the phase transformation from austenite tomartensite. The formation of martensite within the austenite body induces internal stresses. These internalstresses are partially relieved by the formation of a number of different variants of martensite. If there is no pre-ferred direction for martensite orientation, the martensite tends to form a compact twinned structure and theproduct phase is called multiple-variant martensite. If there is a preferred direction for the occurrence of thephase transformation, the martensite tends to form a de-twinned structure and is called single-variantmartensite. This process usually associated with a nonzero state of stress. The conversion of a single-variantmartensite to another single-variant martensite is possible and is called re-orientation process (Auricchio etal.(347)).

4.5.2. The Continuum Mechanics Model

The phase transformation mechanisms involved in the superelastic behavior are:

a. Austenite to Martensite (A->S)b. Martensite to Austenite (S->A)c. Martensite reorientation (S->S)

We consider here two of the above phase transformations: that is A->S and S->A. The material is composed oftwo phases, the austenite (A) and the martensite (S). Two internal variables, the martensite fraction, ξS, and the

austenite fraction, ξA, are introduced. One of them is dependent variable, and they are assumed to satisfy the

following relation,

(4–133)ξ ξS A+ = 1

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Section 4.5: Shape Memory Alloy Material Model

The independent internal variable chosen here is ξS.

The material behavior is assumed to be isotropic. The pressure dependency of the phase transformation ismodeled by introducing the Drucker-Prager loading function:

(4–134)F q p= + 3α

where:

α = material parameter

(4–135)q M= σ σ: :

(4–136)p Tr= ( )/σ 3

where:

M = matrix defined with Equation 4–8σ = stress vectorTr = trace operator

The evolution of the martensite fraction, ξS, is then defined:

(4–137)&

&

&ξ

ξ

ξS

AS

fAS

SAS

fSA

H SF

F R

HF

F R

=

− −−

−

→

→

( )1 A S transformation

S A trransformation

where:

RfAS

fAS= +σ α( )1

RfSA

fSA= +σ α( )1

where:

σ σfAS

fSA and

= material parameters shown in Figure 4.15: “Idealized Stress-Strain Diagram of SuperelasticBehavior”

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Figure 4.15 Idealized Stress-Strain Diagram of Superelastic Behavior

σ

εε

σ

∫

σ

∫

σ

σ

(4–138)H

F R

FAS

fAS

=< <

>

10

0

if

otherwise

Rs

AS

&

(4–139)H

F R

FSA

sSA

=< <

<

10

0

if

otherwise

Rf

SA

&

(4–140)RsAS

sAS= +σ α( )1

(4–141)RsSA

sSA= +σ α( )1

where:

σ σsAS

sSA and = material parameters shown in Figure 4.15: “Idealized Stress-Strain Diagram of Superelastic

Behavior”

The incremental stress-strain relation is:

(4–142) ( )∆ = ∆ − ∆σ ε εD tr

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Section 4.5: Shape Memory Alloy Material Model

(4–143)

∆ = ∆ ∂

∂ε ξ ε

σtr

s LF

where:

[D] = stress-stain matrix

∆εtr = incremental transformation strain

εL = material parameter shown in Figure 4.15: “Idealized Stress-Strain Diagram of Superelastic Behavior”.

4.6. Hyperelasticity

4.6.1. Introduction

Hyperelasticity refers to materials which can experience large elastic strain that is recoverable. Rubber-like andmany other polymer materials fall in this category. The constitutive behavior of hyperelastic materials are usuallyderived from the strain energy potentials. Also, hyperelastic materials generally have very small compressibility.This is often referred to incompressibility.

The hyperelastic material models assume that materials response is isotropic and isothermal. This assumptionallows that the strain energy potentials are expressed in terms of strain invariants or principal stretch ratios. Exceptas otherwise indicated, the materials are also assumed to be nearly or purely incompressible. Material thermalexpansion is also assumed to be isotropic.

The hyperelastic material models include:

1. Several forms of strain energy potential, such as Neo-Hookean, Mooney-Rivlin, Polynomial Form, OgdenPotential, Arruda-Boyce, Gent, and Yeoh are defined through data tables (accessed with TB,HYPER). Thisoption works with following elements SHELL181, PLANE182, PLANE183, SOLID185, SOLID186 , SOLID187,SOLSH190, SHELL208, and SHELL209.

2. Blatz-Ko and Ogden Compressible Foam options are applicable to compressible foam or foam-typematerials.

3. Mooney-Rivlin option (accessed with *MOONEY or TB,MOONEY command) to define Mooney-Rivlinmodels. This option works with hyperelastic elements HYPER56, HYPER58, HYPER74, and HYPER158.

4.6.2. Finite Strain Elasticity

A material is said to be hyperelastic if there exists an elastic potential function W (or strain energy density function)which is a scalar function of one of the strain or deformation tensors, whose derivative with respect to a straincomponent determines the corresponding stress component. This can be expressed by:

(4–144)S

WE

WCij

ij ij= ∂

∂≡ ∂

∂2

where:

Sij = components of the second Piola-Kirchhoff stress tensor

W = strain energy function per unit undeformed volumeEij = components of the Lagrangian strain tensor

Cij = components of the right Cauchy-Green deformation tensor

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The Lagrangian strain may be expressed as follows:

(4–145)E Cij ij ij= −1

2( )δ

where:

δij = Kronecker delta (δij = 1, i = j; δij = 0, i ≠ j)

The deformation tensor Cij is comprised of the products of the deformation gradients fij

(4–146)C F Fij ik kj= = component of the Cauchy-Green deformation tensorr

where:

Fij = components of the second Piola-Kirchhoff stress tensor

Xi = undeformed position of a point in direction i

xi = Xi + ui = deformed position of a point in direction i

ui = displacement of a point in direction i

The eigenvalues (principal stretch ratios) of Cij are λ12

, λ22

, and λ32

, and exist only if:

(4–147)det Cij p ij−

=λ δ2 0

which can be re-expressed as:

(4–148)λ λ λp p pI I I6

14

22

3 0− + − =

where:

I1, I2, and I3 = invariants of Cij,

(4–149)

I

I

I J

1 12

22

32

2 12

22

22

32

32

12

3 12

22

32 2

= + +

= + +

= =

λ λ λ

λ λ λ λ λ λ

λ λ λ

and

(4–150)J det Fij=

J is also the ratio of the deformed elastic volume over the reference (undeformed) volume of materials (Ogden(295)and Crisfield(294)).

4.6.3. Deviatoric-Volumetric Multiplicative Split

Under the assumption that material response is isotropic, it is convenient to express the strain energy functionin terms of strain invariants or principal stretches (Simo and Hughes(252)).

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Section 4.6: Hyperelasticity

(4–151)W W I I I W I I J= =( , , ) ( , , )1 2 3 1 2

or

(4–152)W W= ( , , )λ λ λ1 2 3

Define the volume-preserving part of the deformation gradient, Fij , as:

(4–153)F J Fij ij= −1 3/

and thus

(4–154)J det Fij= = 1

The modified principal stretch ratios and invariants are then:

(4–155)λ λp pJ p= =−1 3 1 2 3/ ( , , )

(4–156)I J Ip p= −2 3/

The strain energy potential can then be defined as:

(4–157)W W I I J W J= =( , , ) ( , , , )1 2 1 2 3λ λ λ

4.6.4. Strain Energy Potentials

Following are several forms of strain energy potential (W) provided (as options TBOPT in TB,HYPER) for the sim-ulation of incompressible or nearly incompressible hyperelastic materials.

4.6.4.1. Neo-Hookean

The form Neo-Hookean strain energy potential is:

(4–158)W I

dJ= − + −µ

23

111

2( ) ( )

where:

µ = initial shear modulus of materials (input on TBDATA commands with TB,HYPER)d = material incompressibility parameter (input on TBDATA commands with TB,HYPER)

The initial bulk modulus is related to the material incompressibility parameter by:

(4–159)K

d= 2

where:

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K = initial bulk modulus

4.6.4.2. Mooney-Rivlin

This option includes 2, 3, 5, and 9 terms Mooney-Rivlin models. The form of the strain energy potential for 2parameter Mooney-Rivlin model is:

(4–160)W c I c I

dJ= − + − + −10 1 01 2

23 31

1( ) ( ) ( )

where:

c10, c01, d = material constants (input on TBDATA commands with TB,HYPER)

The form of the strain energy potential for 3 parameter Mooney-Rivlin model is

(4–161)W c I c I c I I

dJ= − + − + − − + −10 1 01 2 11 1 2

23 3 3 31

1( ) ( ) ( )( ) ( )

where:

c10, c01, c11, d = material constants (input on TBDATA commands with TB,HYPER)

The form of the strain energy potential for 5 parameter Mooney-Rivlin model is:

(4–162)

W c I c I c I

c I I c I

= − + − + −

+ − − + −

10 1 01 2 20 12

11 1 2 02 1

3 3 3

3 3 3

( ) ( ) ( )

( )( ) ( )22 211+ −

dJ( )

where:

c10, c01, c20, c11, c02, d = material constants (input on TBDATA commands with TB,HYPER)

The form of the strain energy potential for 9 parameter Mooney-Rivlin model is:

(4–163)

W c I c I c I

c I I c I

= − + − + −

+ − − + −

10 1 01 2 20 12

11 1 2 02 2

3 3 3

3 3 3

( ) ( ) ( )

( )( ) ( )22 30 23

21 12

2 12 1 22

03 23

3

3 3 3 3 3

+ −

+ − − + − − + −

c I

c I I c I I c I

( )

( ) ( ) ( )( ) ( ) ++ −11 2

dJ( )

where:

c10, c01, c20, c11, c02, c30, c21, c12, c03, d = material constants (input on TBDATA commands with TB,HYPER)

The initial shear modulus is given by:

(4–164)µ = +2 10 01( )c c

The initial bulk modulus is:

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Section 4.6: Hyperelasticity

(4–165)K

d= 2

4.6.4.3. Polynomial Form

The polynomial form of strain energy potential is

(4–166)W c I I

dJij

i j

i j

N

k

k

k

N= − − + −

+ = =∑ ∑( ) ( ) ( )1 2

1

2

13 3

11

where:

N = material constant (input as NPTS on TB,HYPER)cij, dk = material constants (input on TBDATA commands with TB,HYPER)

In general, there is no limitation on N in ANSYS program (see TB command). A higher N may provide better fitthe exact solution, however, it may, on the other hand, cause numerical difficulty in fitting the material constantsand requires enough data to cover the entire range of interest of deformation. Therefore a very higher N valueis not usually recommended.

The Neo-Hookean model can be obtained by setting N = 1 and c01 = 0. Also for N = 1, the two parameters Mooney-

Rivlin model is obtained, for N = 2, the five parameters Mooney-Rivlin model is obtained and for N = 3, the nineparameters Mooney-Rivlin model is obtained.

The initial shear modulus is defined:

(4–167)µ = +2 10 01( )c c

The initial bulk modulus is:

(4–168)K

d= 2

1

4.6.4.4. Ogden Potential

The Ogden form of strain energy potential is based on the principal stretches of left-Cauchy strain tensor, whichhas the form:

(4–169)W

dJi

ii

N

k

k

k

Ni i i= + + − + −

= =∑ ∑µ

αλ λ λα α α( ) ( )1 2 3

1

2

13

11

where:

N = material constant (input as NPTS on TB,HYPER)µi, αi, dk = material constants (input on TBDATA commands with TB,HYPER)

Similar to the Polynomial form, there is no limitation on N. A higher N can provide better fit the exact solution,however, it may, on the other hand, cause numerical difficulty in fitting the material constants and also it requeststo have enough data to cover the entire range of interest of the deformation. Therefore a value of N > 3 is notusually recommended.

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The initial shear modulus, µ, is given as:

(4–170)µ α µ=

=∑1

2 1i i

i

N

The initial bulk modulus is:

(4–171)K

d= 2

1

For N = 1 and α1 = 2, the Ogden potential is equivalent to the Neo-Hookean potential. For N = 2, α1 = 2 and α2

= -2, the Ogden potential can be converted to the 2 parameter Mooney-Rivlin model.

4.6.4.5. Arruda-Boyce Model

The form of the strain energy potential for Arruda-Boyce model is:

(4–172)

W I I IL L

L

= − + − + −

+

µλ λ

λ

12

31

209

11

105027

19

7000

1 2 12

4 13

6

( ) ( ) ( )

( II Id

JInJ

L14

8 15

281

519

673750243

1 12

− + −

+ − −

) ( )

λ

where:

µ = initial shear modulus of material (input on TBDATA commands with TB,HYPER)λL = limiting network stretch (input on TBDATA commands with TB,HYPER)

d = material incompressibility parameter (input on TBDATA commands with TB,HYPER)

The initial bulk modulus is:

(4–173)K

d= 2

As the parameter λL goes to infinity, the model is converted to Neo-Hookean form.

4.6.4.6. Gent Model

The form of the strain energy potential for the Gent model is:

(4–174)WJ I

J dJ

J= −−

+ − −

−µ m

mln ln

21

3 1 12

11 2

where:

µ = initial shear modulus of material (input on TBDATA commands with TB,HYPER)

Jm = limiting value of I1 3−

(input on TBDATA commands with TB,HYPER)

d = material incompressibility parameter (input on TBDATA commands with TB,HYPER)

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Section 4.6: Hyperelasticity

The initial bulk modulus is:

(4–175)K

d= 2

As the parameter Jm goes to infinity, the model is converted to Neo-Hookean form.

4.6.4.7. Yeoh Model

The Yeoh model is also called the reduced polynomial form. The strain energy potential is:

(4–176)W c I

dJi

i

Ni

kk

Nk= − + −

= =∑ ∑0

11

1

231

1( ) ( )

where:

N = material constant (input as NPTS on TB,HYPER)Ci0 = material constants (input on TBDATA commands with TB,HYPER)

dk = material constants (input on TBDATA commands with TB,HYPER)

The Neo-Hookean model can be obtained by setting N = 1.

The initial shear modulus is defined:

(4–177)µ = 2 10c

The initial bulk modulus is:

(4–178)K

d= 2

1

4.6.4.8. Ogden Compressible Foam Model

The strain energy potential of the Ogden compressible foam model is based on the principal stretches of left-Cauchy strain tensor, which has the form:

(4–179)W J Ji

ii

Ni

i ii

Ni i i i i i= + + − + −

= =

−∑ ∑µα

λ λ λ µα β

α α α α α β

1

31 2 3

13 1( ( ) ) ( )/

where:

N = material constant (input as NPTS on TB,HYPER)µi, αi, βi = material constants (input on TBDATA commands with TB,HYPER)

The initial shear modulus, µ, is given as:

(4–180)µµ α

= =∑ i ii

N

12

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The initial bulk modulus K is defined by:

(4–181)K i i

i

N

i= +

=

∑ µ α β1

13

For N = 1, α1 = -2, µ1= –µ, and β = 0.5, the Ogden option is equivalent to the Blatz-Ko option.

4.6.4.9. Blatz-Ko Model

The form of strain energy potential for the Blatz-Ko model is:

(4–182)W

II

I= + −

µ2

2 52

33

where:

µ = initial shear modulus of material (input on TBDATA commands with TB,HYPER)

The initial bulk modulus is defined as:

(4–183)k = 5

3µ

This model is also used in HYPER84 and HYPER86.

4.6.5. USER Subroutine

The option of user subroutine allows users to define their own strain energy potential. A user subroutine userhy-per.F is need to provide the derivatives of the strain energy potential with respect to the strain invariants. Referto the Guide to ANSYS User Programmable Features for more information on writing a user hyperelasticity subroutine.

4.6.6. Mooney-Rivlin (Using TB,MOONEY Command)

HYPER56, HYPER58, HYPER74, HYPER84, HYPER86, and HYPER158 elements use the Mooney-Rivlin material lawdefined by the TB,MOONEY command.

The Mooney-Rivlin constitutive law is a reasonable model for representing the stress-strain behavior of somenearly incompressible natural rubbers (Rivlin(89), Mooney(91)). The Mooney-Rivlin strain energy density function,for HYPER84 and HYPER86, has the following form:

(4–184)W a I a I I I= − + − + − −

10 1 01 2 32

32 23 3( ) ( ) ( )β

where:

I I I Ii = = −reduced in ith direction which are given by: 1 1 31 3/ ,, ,/ /I I I I I I2 1 3

2 33 1 3

1 2= =− and

a10, a01 = Mooney-Rivlin material constants (input on TBDATA commands with TB,HYPER)

β νν

= +−

+11 2 24

10 01( )

a a

ν = Poisson's ratio (input as PRXY or NUXY on MP command, must be less than 0.50)

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Section 4.6: Hyperelasticity

Ii = invariants of the right Cauchy-Green deformation tensor Cij given as:

(4–185)I Cii1 =

(4–186)I I C Cij ij2 1

212

= −( )

(4–187)I Cij3= =det volume change ratio

Note that for small strains, 2(a10+a01) represents the shear modulus and 6(a10+a01) represents the Young's mod-

ulus.

For the elements HYPER56, HYPER58, HYPER74, and HYPER158 the two, five, and nine parameter Mooney-Rivlinmodels are available. The strain energy density function for these elements is given in polynomial form by:

(4–188)W a I I Ik

k

k

N= − − + −∑

+ =l

l

l( ) ( ) / ( )1 2 3

2

13 3 1 2 1κ

where:

akl = constants of the nine-parameter cubic Mooney-Rivlin relationship (input with TB,MOONEY using theTBDATA command or the *MOONEY command with experimental data input on STRS and STRN fields)

κν

= = +−

bulk modulus2

1 210 01( )

( )a a

Setting N = 1 in Equation 4–188, we obtain the strain energy density function for the two parameter Mooney-Rivlin model:

(4–189)W a I a I I= − + − + −10 1 01 2 3

23 3 1 2 1( ) ( ) / ( )κ

Likewise, setting N = 2 and N = 3 in Equation 4–188, we obtain analogous strain energy density functions for thefive Equation 4–190 and nine Equation 4–191 parameter Mooney-Rivlin models:

(4–190)

W a I a I a I a I I

a I

= − + − + − + − −

+ −

10 1 01 2 20 12

11 1 2

02 2

3 3 3 3 3

3

( ) ( ) ( ) ( )( )

( )22 321 2 1+ −/ ( )κ I

(4–191)

W a I a I a I a I I

a I

= − + − + − + − −

+ −

10 1 01 2 20 12

11 1 2

02 2

3 3 3 3 3

3

( ) ( ) ( ) ( )( )

( )22 30 13

21 12

2

12 1 22

03 23

3 3 3

3 3 3

+ − + − −

+ − − + −

a I a I I

a I I a I

( ) ( ) ( )

( )( ) ( ) ++ −1 2 132/ ( )κ I

Note that the last term in Equation 4–151 through Equation 4–154 always represents the hydrostatic (volumetric)work.

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4.6.7. Output Quantities

Stresses (output quantities S) are true (Cauchy) stresses in the element coordinate system. They are computedfrom the second Piola-Kirchhoff stresses using:

(4–192)σ ρ

ρijo

ik kl jl ik kl jlf S fI

f S f= = 1

3

where:

ρ, ρo = mass densities in the current and initial configurations

Strains (output as EPEL) are the Hencky (logarithmic) strains (see Equation 3–6). They are in the element coordinatesystem.

4.6.8. Determining Mooney-Rivlin Material Constants

The hyperelastic constants in the strain energy density function of a material determine its mechanical response.Therefore, in order to obtain successful results during a hyperelastic analysis, it is necessary to accurately assessthe Mooney-Rivlin constants of the materials being examined. Mooney-Rivlin constants are generally derivedfor a material using experimental stress-strain data. It is recommended that this test data be taken from severalmodes of deformation over a wide range of strain values. In fact, it has been observed that to achieve stability,the Mooney-Rivlin constants should be fit using test data in at least as many deformation states as will be exper-ienced in the analysis.

For hyperelastic materials, simple deformation tests (consisting of six deformation modes) can be used to accuratelycharacterize the Mooney-Rivlin constants (using the *MOONEY command). All the available laboratory test datawill be used to determine the Mooney-Rivlin hyperelastic material constants. The six different deformation modesare graphically illustrated in Figure 4.16: “Illustration of Deformation Modes”. Combinations of data from multipletests will enhance the characterization of the hyperelastic behavior of a material.

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Section 4.6: Hyperelasticity

Figure 4.16 Illustration of Deformation Modes

! !

"# $ "# %

Although the algorithm accepts up to six different deformation states, it can be shown that apparently differentloading conditions have identical deformations, and are thus equivalent. Superposition of tensile or compressivehydrostatic stresses on a loaded incompressible body results in different stresses, but does not alter deformationof a material. As depicted in Figure 4.17: “Equivalent Deformation Modes”, we find that upon the addition ofhydrostatic stresses, the following modes of deformation are identical:

1. Uniaxial Tension and Equibiaxial Compression.

2. Uniaxial Compression and Equibiaxial Tension.

3. Planar Tension and Planar Compression.

With several equivalent modes of testing, we are left with only three independent deformation states for whichone can obtain experimental data.

Chapter 4: Structures with Material Nonlinearities

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Figure 4.17 Equivalent Deformation Modes

"!#$% &'( ) "!#$%

"!#$% * &'( ) $

+ "!#$% , + .-/# 102%( "!3 4

+ 5 "!#$%

The following sections outline the development of hyperelastic stress relationships for each independent testingmode. In the analyses, the coordinate system is chosen to coincide with the principal directions of deformation.Thus, the right Cauchy-Green strain tensor can be written in matrix form by:

(4–193)[ ]C =

λ

λ

λ

12

22

32

0 0

0 0

0 0

where:

λi = 1 + εi ≡ principal stretch ratio in the ith direction

εi = principal value of the engineering strain tensor in the ith direction

The principal invariants of Cij, Equation 4–185 through Equation 4–187, then become:

(4–194)I1 12

22

32= + +λ λ λ

(4–195)I2 12

22

12

32

22

32= + +λ λ λ λ λ λ

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Section 4.6: Hyperelasticity

(4–196)I3 12

22

32= λ λ λ

For each mode of deformation, fully incompressible material behavior is also assumed so that third principal in-variant, I3, is identically one:

(4–197)λ λ λ12

22

32 1=

Finally, the hyperelastic Piola-Kirchhoff stress tensor, Equation 4–144 can be algebraically manipulated to de-termine components of the Cauchy (true) stress tensor. In terms of the principal invariants of the right Cauchy-Green strain tensor, the Cauchy stress components, as determined from Equation 4–144, can be shown to be:

(4–198)σ δij ij ij ijp W I C W I C= − + ∂ ∂ − ∂ ∂ −2 21 2

1

where:

p = pressure

4.6.9. Uniaxial Tension (Equivalently, Equibiaxial Compression)

As shown in Figure 4.16: “Illustration of Deformation Modes”, a hyperelastic specimen is loaded along one of itsaxis during a uniaxial tension test. For this deformation state, the principal stretch ratios in the directions ortho-gonal to the 'pulling' axis will be identical. Therefore, during uniaxial tension, the principal stretches, λi, are given

by:

(4–199)λ1 = stretch in direction being loaded

(4–200)λ λ2 3= = stretch in directions not being loaded

Due to incompressibility Equation 4–197:

(4–201)λ λ λ2 3 11= −

and with Equation 4–200,

(4–202)λ λ λ2 3 1

1 2= = −

For uniaxial tension, the first and second strain invariants then become:

(4–203)I1 12

112= + −λ λ

and

(4–204)I2 1 122= + −λ λ

Substituting the uniaxial tension principal stretch ratio values into the Equation 4–198, we obtain the followingstresses in the 1 and 2 directions:

(4–205)σ λ λ11 1 12

2 122 2= − + ∂ ∂ − ∂ ∂ −p W I W I

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and

(4–206)σ λ λ22 1 11

2 12 2 0= − + ∂ ∂ − ∂ ∂ =−p W I W I

Subtracting Equation 4–206 from Equation 4–205, we obtain the principal true stress for uniaxial tension:

(4–207)σ λ λ λ11 12

11

1 11

22= − ∂ ∂ + ∂ ∂− −( )[ ]W I W I

4.6.10. Equibiaxial Tension (Equivalently, Uniaxial Compression)

During an equibiaxial tension test, a hyperelastic specimen is equally loaded along two of its axes, as shown inFigure 4.16: “Illustration of Deformation Modes”. For this case, the principal stretch ratios in the directions beingloaded are identical. Hence, for equibiaxial tension, the principal stretches, λi, are given by:

(4–208)λ λ1 2= = stretch ratio in direction being loaded

(4–209)λ3 = stretch in direction not being loaded

Utilizing incompressibility Equation 4–197, we find:

(4–210)λ λ3 12= −

For equibiaxial tension, the first and second strain invariants then become:

(4–211)I1 12

142= + −λ λ

and

(4–212)I2 14

122= + −λ λ

Substituting the principal stretch ratio values for equibiaxial tension into the Cauchy stress Equation 4–198, weobtain the stresses in the 1 and 3 directions:

(4–213)σ λ λ11 1 12

2 122 2= − + ∂ ∂ − ∂ ∂ −p W I W I

and

(4–214)σ λ λ33 1 14

2 142 2 0= − + ∂ ∂ − ∂ ∂ =−p W I W I

Subtracting Equation 4–214 from Equation 4–213, we obtain the principal true stress for equibiaxial tension:

(4–215)σ λ λ λ11 12

14

1 12

22= − ∂ ∂ + ∂ ∂−( )[ ]W I W I

4.6.11. Pure Shear

(Uniaxial Tension and Uniaxial Compression in Orthogonal Directions)

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Section 4.6: Hyperelasticity

Pure shear deformation experiments on hyperelastic materials are generally performed by loading thin, shortand wide rectangular specimens, as shown in Figure 4.18: “Pure Shear from Direct Components”. For pure shear,plane strain is generally assumed so that there is no deformation in the 'wide' direction of the specimen: λ2 = 1.

Figure 4.18 Pure Shear from Direct Components

Due to incompressibility Equation 4–197, it is found that:

(4–216)λ λ3 11= −

For pure shear, the first and second strain invariants are:

(4–217)I1 12

12 1= + +−λ λ

and

(4–218)I2 12

12 1= + +−λ λ

Substituting the principal stretch ratio values for pure shear into the Cauchy stress Equation 4–198, we obtainthe following stresses in the 1 and 3 directions:

(4–219)σ λ λ11 1 12

2 122 2= − + ∂ ∂ − ∂ ∂ −p W I W I

and

(4–220)σ λ λ33 1 12

2 122 2 0= − + ∂ ∂ − ∂ ∂ =−p W I W I

Subtracting Equation 4–220 from Equation 4–219, we obtain the principal pure shear true stress equation:

(4–221)σ λ λ11 12

12

1 22= − ∂ ∂ + ∂ ∂−( )[ ]W I W I

4.6.12. Least Squares Fit Analysis

By performing a least squares fit analysis the Mooney-Rivlin constants can be determined from experimentalstress-strain data and Equation 4–206, Equation 4–215, and Equation 4–221. Briefly, the least squares fit minimizesthe sum of squared error between experimental and Cauchy predicted stress values. The sum of the squarederror is defined by:

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(4–222)E ai i ij

i

n2 2

1= −

=∑( ( ))σ σ%

where:

E = least squares residual errorσi = experimental stress values

%σi ija( ) = Cauchy stress values (function of Mooney-Rivlin coonstants)

n = number of experimental data points

Equation 4–222 is minimized by setting the variation of the squared error to zero: δ E2 = 0. This yields a set ofsimultaneous equations which can be used to solve for the Mooney-Rivlin constants:

(4–223)

∂ ∂ =

∂ ∂ =

E a

E a

etc

2

210

10

0

0

iii

.

It should be noted that for the pure shear case, the Mooney-Rivlin constants cannot be uniquely determinedfrom Equation 4–221. In this case, the shear data must by supplemented by either or both of the other two typesof test data to determine the constants.

4.6.13. Material Stability Check

Stability checks are provided for the Mooney-Rivlin hyperelastic materials. A nonlinear material is stable if thesecondary work required for an arbitrary change in the deformation is always positive. Mathematically, this isequivalent to:

(4–224)d dij ijσ ε > 0

where:

dσ = change in the Cauchy stress tensor corresponding to a change in the logarithmic strain

Since the change in stress is related to the change in strain through the material stiffness tensor, checking forstability of a material can be more conveniently accomplished by checking for the positive definiteness of thematerial stiffness.

The material stability checks are done at two levels. The first stability check occurs at the end of preprocessingbut before an analysis actually begins. At that time, the program checks for the loss of stability for six typicalstress paths (uniaxial tension and compression, equibiaxial tension and compression, and planar tension andcompression). The range of the stretch ratio over which the stability is checked is chosen from 0.1 to 10. If thematerial is stable over the range then no message will appear. Otherwise, a warning message appears that liststhe Mooney-Rivlin constants and the critical values of the nominal strains where the material first becomes un-stable. The second stability check is optional and occurs during the analysis. This check will be made for eachelement for each iteration. This optional stability check is available for all hyperelastic elements with mixed u-Pformulation (HYPER56, HYPER58, HYPER74, HYPER158) by setting KEYOPT(8) = 1.

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Section 4.6: Hyperelasticity

4.7. Viscoelasticity

A material is said to be viscoelastic if the material has an elastic (recoverable) part as well as a viscous (nonrecov-erable) part. Upon application of a load, the elastic deformation is instantaneous while the viscous part occursover time.

The viscoelastic model usually depicts the deformation behavior of glass or glass-like materials and may simulatecooling and heating sequences of such material. These materials at high temperatures turn into viscous fluidsand at low temperatures behave as solids. Further, the material is restricted to be thermorheologically simple(TRS), which assumes the material response to a load at a high temperature over a short duration is identical tothat at a lower temperature but over a longer duration. The material model is available with the viscoelasticelements VISCO88, VISCO89 for small deformation viscoelasticity and elements LINK180, SHELL181, PLANE182,PLANE183, SOLID185, SOLID186, SOLID187, SOLSH190, BEAM188, BEAM189, SHELL208, and SHELL209 for smallas well as large deformation viscoelasticity.

4.7.1. Small Strain Viscoelasticity

In this section, the constitutive equations and the numerical integration scheme for small strain viscoelasticityare discussed. Large strain viscoelasticity will be presented in Section 4.7.5: Large Deformation Viscoelasticity.

4.7.2. Constitutive Equations

A material is viscoelastic if its stress response consists of an elastic part and viscous part. Upon application of aload, the elastic response is instantaneous while the viscous part occurs over time. Generally, the stress functionof a viscoelastic material is given in an integral form. Within the context of small strain theory, the constitutiveequation for an isotropic viscoelastic material can be written as:

(4–225)σ τ

ττ τ

ττ= − + −∫ ∫2

0 0G t

dd

d K tdd

dt t

( ) ( )e

I ∆

where:

σ = Cauchy stresse = deviatoric part of the strain∆ = volumetric part of the strainG(t) = shear relaxation kernel functionK(t) = bulk relaxation kernel functiont = current timeτ = past timeI = unit tensor

For the viscoelastic elements VISCO88 and VISCO89 the material properties are expressed in integral form usingthe kernel function of the generalized Maxwell elements as:

(4–226)G G G ei

i

nGiG

( )( / )ξ ξ λ= +∞

=

−∑

1

(4–227)K K K ei

i

nKiK

( )( / )ξ ξ λ= +∞

=

−∑

1

(4–228)G C G Gi i= − ∞( )0

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(4–229)K D K Ki i= − ∞( )0

where:

ξ = reduced or pseudo timeG(ξ) = shear relaxation kernel functionK(ξ) = bulk relaxation kernel functionnG = number of Maxwell elements used to approximate the shear relaxation kernel (input constant 50)

nK = number of Maxwell elements used to approximate the bulk relaxation kernel (input constant 71)

Ci = constants associated with the instantaneous response for shear behavior (input constants 51–60)

Di = constants associated with the instantaneous response for bulk behavior (input constants 76–85)

G0 = initial shear modulus (input constant 46)

G∞ = final shear modulus (input constant 47)

K0 = initial bulk modulus (input constant 48)

K∞ = final bulk modulus (input constant 49)

λ iG = constants associated with a discrete relaxation spectruum in shear

(input constants 61-70)

λ iK = constants associated with a discrete relaxation spectruum in bulk

(input constants 86-95)

For the elements LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, SOLSH190, BEAM188,SHELL208, and SHELL209, the kernel functions are represented in terms of Prony series, which assumes that:

(4–230)G G G

ti

i

n

iG

G= + ∑ −

∞

=

1exp

τ

(4–231)K K K

ti

i

n

iK

K= + ∑ −

∞

=

1exp

τ

In the above, G∞ and Gi are shear elastic moduli, K∞ and Ki are bulk elastic moduli and τ i

G and τ i

K are the re-

laxation times for each Prony component. Introducing the relative moduli

(4–232)αi

GiG G= / 0

(4–233)αiK

iK K= / 0

where:

G G G ii

nG0

1= + ∑∞

=

K K K ii

nK0

1= + ∑∞

=

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The kernel functions can be equivalently expressed as:

(4–234)G G

tK KG

iG

i

n

iG

KiK

i

nG K= + ∑ −

= +∞=

∞=

01

01

α ατ

α αexp , ∑∑ −

expt

iKτ

The integral function Equation 4–225 can recover the elastic behavior at the limits of very slow and very fastload. Here, G0 and K0 are, respectively, the shear and bulk moduli at the fast load limit (i.e. the instantaneous

moduli), and G∞ and K∞ are the moduli at the slow limit. The elasticity parameters input correspond to thoseof the fast load limit. Moreover by admitting Equation 4–230, the deviatoric and volumetric parts of the stressare assumed to follow different relaxation behavior. The number of Prony terms for shear nG and for volumetric

behavior nK need not be the same nor do the relaxation times τ i

G and τ i

K.

The Prony representation has a prevailing physical meaning, that it corresponds to the solution of the classicaldifferential model, parallel Maxwell model, of viscoelasticity. This physical rooting is the key to understand theextension of the above constitutive equations to large deformation cases as well as the appearance of the time-scaling law (e.g. pseudo time) at the presence of time-dependent viscous parameters.

4.7.3. Numerical Integration

To perform finite element analysis, the integral Equation 4–225 need to be integrated. The integration schemeproposed by Taylor(112) and subsequently modified by Simo(327) is adapted. We will delineate the integrationprocedure for the deviatoric stress. The pressure response can be handled in an analogous way. To integrate thedeviatoric part of Equation 4–225, first, break the stress response into components and write:

(4–235)s s s= + ∑∞ i

i

nG

where:

s = deviatoric stress

S e∞ ∞= 2G

In addition,

(4–236)s

ei i

iG

tG

t dd

d= − −

∫ 2

0exp

ττ τ

τ

One should note that

(4–237)

( ) exp

exp

se

i n in

iG

t

in

Gt d

dd

Gt t

n

++= − −

∫

= − + −

+1

1

02

2

1 ττ τ

τ

∆ τττ τ

τ

ττ

iG

t

in

iGt

t

n

n

n

dd

d

Gt d

d

∫

+ − −

∫

+

0

21

e

e

expττ

τd

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where:

∆t = tn+1 - tn.

The first term of Equation 4–237 is readily recognized as:

exp( )( )− ∆ti n

iGτ

s.

Using the middle point rule for time integration for the second term, a recursive formula can be obtained as:

(4–238)( ) exp ( ) exps s ei n

iG i n

iG i

t tG+ = −

+ −

1

2

∆ ∆ ∆τ τ

where:

∆e = en+1 - en.

4.7.4. Thermorheological Simplicity

Materials viscous property depends strongly on temperature. For example, glass-like materials turn into viscousfluids at high temperatures while behave like solids at low temperatures. In reality, the temperature effects canbe complicated. The so called thermorheological simplicity is an assumption based on the observations for manyglass-like materials, of which the relaxation curve at high temperature is identical to that at a low temperatureif the time is properly scaled (Scherer(326)). In essence, it stipulates that the relaxation times (of all Prony com-ponents) obey the scaling law

(4–239)ττ

ττ

iG i

Gr

riG i

Gr

rT

T

A T TT

T

A T T( )

( )

( , ), ( )

( )

( , )= =

Here, A(T, Tr) is called the shift function. Under this assumption (and in conjunction with the differential model),

the deviatoric stress function can be shown to take the form

(4–240)s

e= + − −

∞

=∑∫ 2

10G G

dd

dit s

iGi

nt Gexp

ξ ξτ τ

τ

likewise for the pressure part. Here, notably, the Prony representation still holds with the time t, τ in the integrandbeing replaced by:

ξ τ ξ τt

t

s

tAt d At d= ∫ = ∫exp( ) exp( )

0 0and

here ξ is called pseudo (or reduced) time. In Equation 4–240, τ i

G is the decay time at a given temperature.

The assumption of thermorheological simplicity allows for not only the prediction of the relaxation time overtemperature, but also the simulation of mechanical response under prescribed temperature histories. In thelatter situation, A is an implicit function of time t through T = T(t). In either case, the stress equation can be integ-rated in a manner similar to Equation 4–235. Indeed,

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(4–241)

( ) exp

exp

se

i n in n

iG

t

in

Gdd

d

G

n

++= − −

∫

= −+

+1

1

02

2

1 ξ ξτ τ

τ

ξ ξ∆ −−

∫

+ − −

+

ξτ τ

τ

ξ ξτ τ

s

iG

t

in s

iGt

dd

d

Gdd

n

n

e

e

0

12 expttn

d+∫

1τ

Using the middle point rule for time integration on Equation 4–241 yields

(4–242)( ) exp ( ) exps s ei n

iG i n

iG iG+ = −

+ −

1 212∆

∆∆ξ

τ

ξ

τ

where:

∆ξ τ τ= ∫+

A T dt

t

n

n( ( ))

1

∆ξ τ τ12 1

2

1= ∫

+

+A T d

t

t

n

n( ( ))

Two widely used shift functions, namely the William-Landel-Ferry shift function and the Tool-Narayanaswamyshift function, are available. The form of the functions are given in Section 4.7.8: Shift Functions.

4.7.5. Large Deformation Viscoelasticity

Two types of large deformation viscoelasticity models are implemented: large deformation, small strain andlarge deformation, large strain viscoelasticity. The first is associated with hypo-type constitutive equations andthe latter is based on hyperelasticity.

4.7.6. Visco-Hypoelasticity

For visco-hypoelasticity model, the constitutive equations are formulated in terms of the rotated stress RTσR,

here R is the rotation arising from the polar decomposition of the deformation gradient F. Let RTσR = Σ + pI

where Σ is the deviatoric part and p is the pressure. It is evident that Σ = RTSR. The stress response function isgiven by:

(4–243)Σ = + − −

∞

=∑∫ 2

10G G

tdi

ii

ntT

G

Gexp ( )

ττ

τR dR

(4–244)p K K

ttr di

iKi

nt K= + − −

∞=∑∫ exp ( )

ττ

τ10

D

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where:

d = deviatoric part of the rate of deformation tensor D.

This stress function is consistent with the generalized differential model in which the stress rate is replaced byGreen-Naghdi rate.

To integrate the stress function, one perform the same integration scheme in Equation 4–235 to the rotatedstress Equation 4–243 to yield:

(4–245)( ) exp ( ) expΣ Σ∆ ∆

i niG i n

iG i n

Tt tG+ +

= −

+ −

1 2

212τ τ

R (( )d Rn n+ +12

12

where:

Rn+ 12 = rotation tensor arising from the polar decomposition of the middle point deformation gradient

F F Fn n n+ += +12

12 1( )

In the actual implementations, the rate of deformation tensor is replaced by the strain increment and we have

(4–246)D µn n nt u+ + +≈ = ∇1

212

12

∆ ∆ ∆symm( )

where:

symm[.] = symmetric part of the tensor.

From Σ = RTsR and using Equation 4–245 and Equation 4–246, it follows that the deviatoric Cauchy stress is givenby

(4–247)( ) exp ( ) expS R S Ri n

iG i n

T

iG i

t tG+ = −

+ −

1 2

2

∆ ∆ ∆ ∆ ∆τ τ

RR e R12

12

12

( )∆ ∆n nT

+ +

where:

∆R R R = n nT

+1

∆R R R12

12

1 = + +n nT

∆ ∆en n+ +=12

12

deviatoric part of ε

The pressure response can be integrated in a similar manner and the details are omitted.

4.7.7. Large Strain Viscoelasticity

The large strain viscoelasticity implemented is based on the formulation proposed by (Simo(327)), amendedhere to take into account the viscous volumetric response and the thermorheological simplicity. Simo's formu-

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lation is and extension of the small strain theory. Again, the viscoelastic behavior is specified separately by theunderlying elasticity and relaxation behavior.

(4–248)Φ( ) ( ) ( )C C= +φ U J

where:

J = det (F)

C C= =J23 isochoric part of the right Cauchy-Green deformationn tensor C

This decomposition of the energy function is consistent with hyperelasticity described in Section 4.6: Hyperelasti-city.

As is well known, the constitutive equations for hyperelastic material with strain energy function Φ is given by:

(4–249)S

C2 2d = ∂

∂Φ

where:

S2d = second Piola-Kirchhoff stress tensor

The true stress can be obtained as:

(4–250)σ = = ∂

∂1 12

J Jd T TFS F F

CF

Φ

Using Equation 4–248 in Equation 4–250 results

(4–251)σ ϕ= ∂∂

+ ∂∂

2J

U JJ

TFC

CF I

( ) ( )

It has been shown elsewhere that F

CC

F∂

∂ϕ( ) T

is deviatoric, therefore Equation 4–251 already assumes the formof deviatoric/pressure decomposition.

Following Simo(327) and Holzapfel(328), the viscoelastic constitutive equations, in terms of the second Piola-Kirchhoff stress, is given by

(4–252)

SC

2

102d G

iG

iGi

n t dd

dd

G= + − −

∞=∑α α τ

τ τexp

Φtt

KiK

iGi

n

d

t dd

dUdJ

K

∫

∑+ + − −

∞

=

τ

α α ττ τ

exp1

2

∫ −

0

1t

dτC

Denote

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(4–253)S

Cid G

iG

iGi

n t dd

dd

G2

12= + − −

∞=∑α α τ

τ τexp

Φ

00

td∫ τ

(4–254)p

t dd

dUdJi

KiK

iGi

nt K= + − −

∞=∑α α τ

τ τexp

102∫∫ −dτC 1

and applying the recursive formula to Equation 4–253 and Equation 4–254 yields,

(4–255)( ) exp ( ) expS Si

dn

iG i

dn i

G

iG

t t d21

2

2+ = −

+ −

∆ ∆ Φτ

ατ dd

ddn nC C+

−

1

Φ

(4–256)( ) exp ( ) expp

tp

t dUdJi n

iK i n i

G

iK

n+

+= −

+ −

1

2

∆ ∆τ

ατ 11

−

dUdJn

The above are the updating formulas used in the implementation. Cauchy stress can be obtained using Equa-tion 4–250.

4.7.8. Shift Functions

For VISCO88, VISCO89 viscoelastic elements, the Tool-Narayanaswamy shift function (defined in Naray-anaswamy(110)) is accessed with constant 5 = 0, and is calculated as:

(4–257)A

HR T

x

T t

x

T tref f

= − − −

′ ′exp

( )

( )

( )

1 1

where:

H

R = activation energy divided by the ideal gas constant (input constant 1)Tref = reference temperature (input on TREF command)

x = input constant 2T(t') = temperature at time t'Tf(t') = fictive temperature at time t' (see below for more information)

The Williams-Landau-Ferry shift function (defined in Williams et al.(277)) is accessed with constant 5 = 1, and isbased on logarithm to the base 10. For consistency, this is converted to base e internally and is calculated as:

(4–258)log A

C T t T

C T t Tbase

base10

1

2

( )( ( ) )

( ( ) )=

−+ −

′

′

where:

c1 = first material constant (input constant 1)

c2 = second material constant (input constant 2)

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T(t') = temperature at time t'Tbase = base temperature for TRS material properties (input constant 4)

Initial fictive temperature is defined using input constants 6 thru 15 and 36 thru 45. Subsequent fictive temper-atures use input constants 16 thru 25. These usages are explained in (Markovsky(108)), and (Narayanaswamy(110)).(The fictive temperature is output as FICT TEMP.)

The incremental change in growth (volumetric) strain follows the relationship:

(4–259)∆ ∆ ∆ε α α αgr

g f g fT T T T T= + − ( ) ( ) ( )l

where:

∆εgr = incremental change in growth strainα(T)g = coefficient of thermal expansion for the glass state, which is a function of the actual temperature T

(input constants 31 thru 35)∆T = change of actual temperature

α( )Tf l = coefficient of thermal expansion for the liquid state, which is a function of the fictive temperatureTf (input constants 26 thru 30)

∆Tf = change of fictive temperature

The incremental changes in growth strain are summed to give:

(4–260)ε εgr gr

j

Nt=

=∑ ( )∆

1

where:

εgr = summed total strains (output as GR STRAIN)Nt = total number of time steps up to and including the current time point

For elements LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, SOLSH190, BEAM188,BEAM189, SHELL208, and SHELL209, the shift functions the William-Landel-Ferry shift function takes the form:

(4–261)log ( ( ( )))

( )10

1

2A T

c T Tc T T

r

rτ = −

+ −

where:

c1 = first material constant

c2 = second material constant

T = temperature at time tTr = base temperature

and the Tool-Narayanaswamy shift function takes the form:

(4–262)A T d

T Tr( ( )) expτ = −

1

1 1

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where:

d1 = first material constant

Other shift functions may be accommodated for elements LINK180, SHELL181, PLANE182, PLANE183, SOLID185,SOLID186, SOLID187, SOLSH190, BEAM188, BEAM189, SHELL208, and SHELL209, through the user providedsubroutine USRSHIFT. The inputs for this subroutine are the user-defined parameters, the current value of timeand temperature, their increments and the current value of user state variables (if any). The outputs from thesubroutine are ∆ξ, ∆ξ1/2 as well as the current value of user state variables.

4.8. Concrete

The concrete material model predicts the failure of brittle materials. Both cracking and crushing failure modesare accounted for. TB,CONCR accesses this material model, which is available with the reinforced concrete elementSOLID65.

The criterion for failure of concrete due to a multiaxial stress state can be expressed in the form (Willam andWarnke(37)):

(4–263)

Ff

Sc

− ≥ 0

where:

F = a function (to be discussed) of the principal stress state (σxp, σyp, σzp)

S = failure surface (to be discussed) expressed in terms of principal stresses and five input parameters ft, fc,

fcb, f1 and f2 defined in Table 4.4: “Concrete Material Table”

fc = uniaxial crushing strength

σxp, σyp, σzp = principal stresses in principal directions

If Equation 4–263 is satisfied, the material will crack or crush.

A total of five input strength parameters (each of which can be temperature dependent) are needed to definethe failure surface as well as an ambient hydrostatic stress state. These are presented in Table 4.4: “ConcreteMaterial Table”.

Table 4.4 Concrete Material Table

(Input on TBDATA Commands with TB,CONCR)

ConstantDescriptionLabel

3Ultimate uniaxial tensile strengthft

4Ultimate uniaxial compressive strengthfc

5Ultimate biaxial compressive strengthfcb

6Ambient hydrostatic stress stateσh

a

7Ultimate compressive strength for a state of biaxial compression

superimposed on hydrostatic stress state σh

af1

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(Input on TBDATA Commands with TB,CONCR)

ConstantDescriptionLabel

8Ultimate compressive strength for a state of uniaxial compression

superimposed on hydrostatic stress state σh

af2

However, the failure surface can be specified with a minimum of two constants, ft and fc. The other three constants

default to Willam and Warnke(37):

(4–264)f fcb c= 1 2.

(4–265)f fc1 1 45= .

(4–266)f fc2 1 725= .

However, these default values are valid only for stress states where the condition

(4–267)σh cf≤ 3

(4–268)σ σ σ σh xp yp zp= = + +

hydrostatic stress state13

( )

is satisfied. Thus condition Equation 4–267 applies to stress situations with a low hydrostatic stress component.All five failure parameters should be specified when a large hydrostatic stress component is expected. If conditionEquation 4–267 is not satisfied and the default values shown in Equation 4–264 thru Equation 4–266 are assumed,the strength of the concrete material may be incorrectly evaluated.

When the crushing capability is suppressed with fc = -1.0, the material cracks whenever a principal stress com-

ponent exceeds ft.

Both the function F and the failure surface S are expressed in terms of principal stresses denoted as σ1, σ2, and

σ3 where:

(4–269)σ σ σ σ1 = max xp yp zp( , , )

(4–270)σ σ σ σ3 = min( , , )xp yp zp

and σ1 ≥ σ2 ≥ σ3. The failure of concrete is categorized into four domains:

1. 0 ≥ σ1 ≥ σ2 ≥ σ3 (compression - compression - compression)

2. σ1 ≥ 0 ≥ σ2 ≥ σ3 (tensile - compression - compression)

3. σ1 ≥ σ2 ≥ 0 ≥ σ3 (tensile - tensile - compression)

4. σ1 ≥ σ2 ≥ σ3 ≥ 0 (tensile - tensile - tensile)

In each domain, independent functions describe F and the failure surface S. The four functions describing thegeneral function F are denoted as F1, F2, F3, and F4 while the functions describing S are denoted as S1, S2, S3, and

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S4. The functions Si (i = 1,4) have the properties that the surface they describe is continuous while the surface

gradients are not continuous when any one of the principal stresses changes sign. The surface will be shown inFigure 4.19: “3-D Failure Surface in Principal Stress Space” and Figure 4.21: “Failure Surface in Principal StressSpace with Nearly Biaxial Stress”. These functions are discussed in detail below for each domain.

4.8.1. The Domain (Compression - Compression - Compression)

0 ≥ σ1 ≥ σ2 ≥ σ3

In the compression - compression - compression regime, the failure criterion of Willam and Warnke(37) is imple-mented. In this case, F takes the form

(4–271)F F= = − + − + −

1 1 2

22 3

23 1

2121

15( ) ( ) ( )σ σ σ σ σ σ

and S is defined as

(4–272)S Sr r r r r r r r r r r

= =− + − − + −

12 2

212

2 1 2 22

12 2

12

1 22 2 4 5 4( )cos ( ) ( )cosη η

− + −

12

4 222

12 2

2 12( )cos ( )r r r rη

Terms used to define S are:

(4–273)

cos

( ) ( ) ( )

ησ σ σ

σ σ σ σ σ σ

=− −

− + − + −

2

2

1 2 3

1 22

2 32

3 12

12

(4–274)r a a a1 0 1 22= + +ξ ξ

(4–275)r b b b2 0 1 22= + +ξ ξ

(4–276)ξ σ= h

cf

σh is defined by Equation 4–268 and the undetermined coefficients a0, a1, a2, b0, b1, and b2 are discussed below.

This failure surface is shown as Figure 4.19: “3-D Failure Surface in Principal Stress Space”. The angle of similarityη describes the relative magnitudes of the principal stresses. From Equation 4–273, η = 0° refers to any stressstate such that σ3 = σ2 > σ1 (e.g. uniaxial compression, biaxial tension) while ξ = 60° for any stress state where σ3

>σ2 = σ1 (e.g. uniaxial tension, biaxial compression). All other multiaxial stress states have angles of similarity

such that 0° ≤ η ≤ 60°. When η = 0°, S1 Equation 4–272 equals r1 while if η = 60°, S1 equals r2. Therefore, the

function r1 represents the failure surface of all stress states with η = 0°. The functions r1, r2 and the angle η are

depicted on Figure 4.19: “3-D Failure Surface in Principal Stress Space”.

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Figure 4.19 3-D Failure Surface in Principal Stress Space

!#"$

&%

(' %

'

&%

('

)!*"$

!+"$

It may be seen that the cross-section of the failure plane has cyclic symmetry about each 120° sector of the octa-hedral plane due to the range 0° < η < 60° of the angle of similitude. The function r1 is determined by adjusting

a0, a1, and a2 such that ft, fcb, and f1 all lie on the failure surface. The proper values for these coefficients are de-

termined through solution of the simultaneous equations:

(4–277)

Ff

f

Ff

f

Ff

ct

ccb

cha

11 2 3

11 2 3

11 2 3

0

0

( , )

( , )

( ,

σ σ σ

σ σ σ

σ σ σ σ

= = =

= = = −

= − = == − −

=

σ

ξ ξ

ξ ξ

ξ ξha

t t

cb cb

f1

2

2

1 12

1

1

1)

a

a

a

0

1

2

with

(4–278)ξ ξ ξσ

tt

ccb

cb

c

ha

c c

ff

ff f

ff

= = − = − −3

23

231

1, ,

The function r2 is calculated by adjusting b0, b1, and b2 to satisfy the conditions:

(4–279)

Ff

f

Ff

f

Ff

cc

cha

ha

c

11 2 3

11 2 3 2

1

0

0

( , )

( , )

σ σ σ

σ σ σ σ σ

= = = −

= = − = − −

=

−

113

19

1

1

2 22

0 02

0

1

2

ξ ξ

ξ ξ

b

b

b

ξ2 is defined by:

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(4–280)ξσ

22

3= − −h

a

c cfff

and ξ0 is the positive root of the equation

(4–281)r a a a2 0 0 1 0 2 02 0( )ξ ξ ξ= + + =

where a0, a1, and a2 are evaluated by Equation 4–277.

Since the failure surface must remain convex, the ratio r1 / r2 is restricted to the range

(4–282). .5 1 251 2< <r r

although the upper bound is not considered to be restrictive since r1 / r2 < 1 for most materials (Willam(36)). Also,

the coefficients a0, a1, a2, b0, b1, and b2 must satisfy the conditions (Willam and Warnke(37)):

(4–283)a a a0 1 20 0 0> ≤ ≤, ,

(4–284)b b b0 1 20 0 0> ≤ ≤, ,

Therefore, the failure surface is closed and predicts failure under high hydrostatic pressure (ξ > ξ2). This closure

of the failure surface has not been verified experimentally and it has been suggested that a von Mises type cyl-inder is a more valid failure surface for large compressive σh values (Willam(36)). Consequently, it is recommended

that values of f1 and f2 are selected at a hydrostatic stress level ( )σh

a in the vicinity of or above the expected

maximum hydrostatic stress encountered in the structure.

Equation 4–281 expresses the condition that the failure surface has an apex at ξ = ξ0. A profile of r1 and r2 as a

function of ξ is shown in Figure 4.20: “A Profile of the Failure Surface”.

Figure 4.20 A Profile of the Failure Surface

As a Function of ξα

The lower curve represents all stress states such that η = 0° while the upper curve represents stress states suchthat η = 60°. If the failure criterion is satisfied, the material is assumed to crush.

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Section 4.8: Concrete

4.8.2. The Domain (Tension - Compression - Compression)

σ1 ≥ 0 ≥ σ2 ≥ σ3

In the regime, F takes the form

(4–285)F F= = − + +

2 2 3

222

321

15

12( )σ σ σ σ

and S is defined as

(4–286)S = S = 1-ft

2p p -p cos +p 2p -p 4 p -p2

1 2 22

12

2 1 2 22

12

σ η

( ) ( ) ( ))

( ) ( )

cos +5p -4p p

4 p -p cos + p -2p

212

1 2

22

12 2

2 12

η

η

12

where cos η is defined by Equation 4–273 and

(4–287)p a a a1 0 1 22= + +χ χ

(4–288)p b b b2 0 1 22= + +χ χ

The coefficients a0, a1, a2, b0, b1, b2 are defined by Equation 4–277 and Equation 4–279 while

(4–289)χ σ σ= +1

3 2 3( )

If the failure criterion is satisfied, cracking occurs in the plane perpendicular to principal stress σ1.

This domain can also crush. See (Willam and Warnke(37)) for details.

4.8.3. The Domain (Tension - Tension - Compression)

σ1 ≥ σ2 ≥ 0 ≥ σ3

In the tension - tension - compression regime, F takes the form

(4–290)F F ii= = =3 1 2σ ; ,

and S is defined as

(4–291)S S

ff f

it

c c= = +

=3

31 1 2σ

; ,

If the failure criterion for both i = 1, 2 is satisfied, cracking occurs in the planes perpendicular to principal stressesσ1 and σ2. If the failure criterion is satisfied only for i = 1, cracking occurs only in the plane perpendicular to

principal stress σ1.

This domain can also crush. See (Willam and Warnke(37)) for details.

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4.8.4. The Domain (Tension - Tension - Tension)

σ1 ≥ σ2 ≥ σ3 ≥ 0

In the tension - tension - tension regimes, F takes the form

(4–292)F F ii= = =4 1 2 3σ ; , ,

and S is defined as

(4–293)S S

fft

c= =4

If the failure criterion is satisfied in directions 1, 2, and 3, cracking occurs in the planes perpendicular to principalstresses σ1, σ2, and σ3.

If the failure criterion is satisfied in directions 1 and 2, cracking occurs in the plane perpendicular to principalstresses σ1 and σ2.

If the failure criterion is satisfied only in direction 1, cracking occurs in the plane perpendicular to principal stressσ1.

Figure 4.21 Failure Surface in Principal Stress Space with Nearly Biaxial Stress

σ

σ

σ ! #"%$ & #'(()+*&,.- (/σ !0#"%$ 1*&,- (/σ !2#"%$ 1*&,- (/

Figure 4.21: “Failure Surface in Principal Stress Space with Nearly Biaxial Stress” represents the 3-D failure surfacefor states of stress that are biaxial or nearly biaxial. If the most significant nonzero principal stresses are in theσxp and σyp directions, the three surfaces presented are for σzp slightly greater than zero, σzp equal to zero, and

σzp slightly less than zero. Although the three surfaces, shown as projections on the σxp - σyp plane, are nearly

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Section 4.8: Concrete

equivalent and the 3-D failure surface is continuous, the mode of material failure is a function of the sign of σzp.

For example, if σxp and σyp are both negative and σzp is slightly positive, cracking would be predicted in a direction

perpendicular to the σzp direction. However, if σzp is zero or slightly negative, the material is assumed to crush.

4.9. Swelling

The ANSYS program provides a capability of irradiation induced swelling (accessed with TB,SWELL). Swelling isdefined as a material enlarging volumetrically in the presence of neutron flux. The amount of swelling may alsobe a function of temperature. The material is assumed to be isotropic and the basic solution technique used isthe initial stress method. Swelling calculations are available only through the user swelling subroutine. See UserRoutines and Non-Standard Uses of the ANSYS Advanced Analysis Techniques Guide and the Guide to ANSYS UserProgrammable Features for more details. Input must have C72 set to 10. Constants C67 through C71 are used to-

gether with fluence and temperature, as well as possibly strain, stress and time, to develop an expression forswelling rate.

Any of the following three conditions cause the swelling calculations to be bypassed:

1. If C67 ≤ 0. and C68 ≤ 0.

2. If (input temperature + Toff) U ≤ 0, where Toff = offset temperature (input on TOFFST command).

3. If Fluencen ≤ Fluencen-1 (n refers to current time step).

The total swelling strain is computed in subroutine USERSW as:

(4–294)ε ε εn

swnsw sw= +−1 ∆

where:

εnsw

= swelling strain at end of substep n

∆εsw = r∆f = swelling strain incrementr = swelling rate∆f = fn - fn-1 = change of fluence

fn = fluence at end of substep n (input as VAL1, etc. on the BFE,,FLUE command)

For a solid element, the swelling strain vector is simply:

(4–295) ε ε ε εswnsw

nsw

nsw T

=

0 0 0

It is seen that the swelling strains are handled in a manner totally analogous to temperature strains in an isotropicmedium and that shearing strains are not used.

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Chapter 5: Electromagnetics

5.1. Electromagnetic Field Fundamentals

Electromagnetic fields are governed by the following Maxwell's equations (Smythe(150)):

(5–1)∇ = + ∂

∂

= + + + ∂∂

x H JDt

J J JDts e v

(5–2)∇ = − ∂

∂

x EBt

(5–3)∇ ⋅ = B 0

(5–4)∇ ⋅ = D ρ

where:

∇ x = curl operator∇ ⋅ = divergence operatorH = magnetic field intensity vectorJ = total current density vectorJs = applied source current density vector

Je = induced eddy current density vector

Jvs = velocity current density vector

D = electric flux density vector (Maxwell referred to this as the displacement vector, but to avoid misunder-standing with mechanical displacement, the name electric flux density is used here.)t = timeE = electric field intensity vectorB = magnetic flux density vectorρ = electric charge density

The continuity equation follows from taking the divergence of both sides of Equation 5–1.

(5–5)∇ ⋅ + ∂

∂

= J

Dt

0

The continuity equation must be satisfied for the proper setting of Maxwell's equations. Users should prescribeJs taking this into account.

The above field equations are supplemented by the constitutive relation that describes the behavior of electro-magnetic materials. For problems considering saturable material without permanent magnets, the constitutiverelation for the magnetic fields is:

(5–6) [ ] B H= µ

where:

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µ = magnetic permeability matrix, in general a function of H

The magnetic permeability matrix [µ] may be input either as a function of temperature or field. Specifically, if [µ]is only a function of temperature,

(5–7)[ ]µ µ

µµ

µ

=

o

rx

ry

rz

0 0

0 0

0 0

where:

µo = permeability of free space (input on EMUNIT command)

µrx = relative permeability in the x-direction (input as MURX on MP command)

If [µ] is only a function of field,

(5–8)[ ]µ µ=

h

1 0 0

0 1 0

0 0 1

where:

µh = permeability derived from the input B versus H curve (input with TB,BH).

Mixed usage is also permitted, e.g.:

(5–9)[ ]µ

µµ µ

µ

=

h

o ry

h

0 0

0 0

0 0

When permanent magnets are considered, the constitutive relation becomes:

(5–10) [ ] B H Mo o= +µ µ

where:

Mo = remanent intrinsic magnetization vector

Rewriting the general constitutive equation in terms of reluctivity it becomes:

(5–11) [ ] [ ] H B M

oo= −ν

νν1

where:

[ν] = reluctivity matrix = [µ]-1

νµo

o= =reluctivity of free space

1

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The constitutive relations for the related electric fields are:

(5–12) [ ][ ]J E v B= + ×σ

(5–13) [ ] D E= ε

where:

[ ]σσ

σ

σ

=

=xx

yy

zz

0 0

0 0

0 0

electrical conductivity matriix

[ ]εε

ε

ε

=

=xx

yy

zz

0 0

0 0

0 0

permittivity matrix

v

v

v

v

x

y

z

=

= velocity vector

σxx = conductivity in the x-direction (input as inverse of RSVX on MP command)

εxx = permittivity in the x-direction (input as PERX on MP command)

The solution of magnetic field problems is commonly obtained using potential functions. Two kinds of potentialfunctions, the magnetic vector potential and the magnetic scalar potential are used depending on the problemto be solved. Factors affecting the choice of potential include: field dynamics, field dimensionality, source currentconfiguration, domain size and discretization.

The applicable regions are shown below. These will be referred to with each solution procedure discussed below.

Figure 5.1 Electromagnetic Field Regions

! #"

"

$&%$&'

$)(" (

*+

* +

, %

- (

where:

Ω0 = free space region

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Section 5.1: Electromagnetic Field Fundamentals

Ω1 = nonconducting permeable region

Ω2 = conducting region

µ = permeability of ironµo = permeability of air

Mo = permanent magnets

S1 = boundary of W1

σ = conductivityΩ = Ω1 + Ω2 + Ω0

5.1.1. Magnetic Scalar Potential

The scalar potential method as implemented in SOLID5, SOLID96, and SOLID98 for 3-D magnetostatic fields isdiscussed in this section. Magnetostatics means that time varying effects are ignored. This reduces Maxwell'sequations for magnetic fields to:

(5–14)∇ =x H Js

(5–15)∇ ⋅ = B 0

5.1.2. Solution Strategies

In the domain Ω0 and Ω1 of a magnetostatic field problem (Ω2 is not considered for magnetostatics) a solution

is sought which satisfies the relevant Maxwell's Equation 5–14 and Equation 5–15 and the constitutive relationEquation 5–10 in the following form (Gyimesi(141) and Gyimesi(149)):

(5–16) H Hg g= − ∇φ

(5–17)∇ ⋅ ∇ − ∇ ⋅ − ∇ ⋅ =[ ] [ ] µ φ µ µg g o oH M 0

where:

Hg = preliminary or “guess” magnetic field

φg = generalized potential

The development of Hg varies depending on the problem and the formulation. Basically, Hg must satisfy

Ampere's law (Equation 5–14) so that the remaining part of the field can be derived as the gradient of the gen-eralized scalar potential φg. This ensures that φg is singly valued. Additionally, the absolute value of Hg must be

greater than that of ∆φg. In other words, Hg should be a good approximation of the total field. This avoids

difficulties with cancellation errors (Gyimesi(149)).

This framework allows for a variety of scalar potential formulation to be used. The appropriate formulation dependson the characteristics of the problem to be solved. The process of obtaining a final solution may involve severalsteps (controlled by the MAGOPT solution option).

As mentioned above, the selection of Hg is essential to the development of any of the following scalar potential

strategies. The development of Hg always involves the Biot-Savart field Hs which satisfies Ampere's law and

is a function of source current Js. Hs is obtained by evaluating the integral:

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(5–18)

( )H

J r

rd volcs

svolc

= ×∫

14 3π

where:

Js = current source density vector at d(volc)

r = position vector from current source to node pointvolc = volume of current source

The above volume integral can be reduced to the following surface integral (Gyimesi et al.(173))

(5–19)

( )HJr

d surfcss

surfc= ×∫

14π

where:

surfc = surface of the current source

Evaluation of this integral is automatically performed upon initial solution execution or explicitly (controlled bythe BIOT command). The values of Js are obtained either directly as input by:

SOURC36 - Current Source

or indirectly calculated by electric field calculation using:

SOLID5 - 3-D Coupled-Field SolidLINK68 - Coupled Thermal-Electric LineSOLID69 - 3-D Coupled Thermal-Electric SolidSOLID98 - Tetrahedral Coupled-Field Solid

Depending upon the current configuration, the integral given in Equation 5–19 is evaluated in a closed formand/or a numerical fashion (Smythe(150)).

Three different solution strategies emerge from the general framework discussed above:

Reduced Scalar Potential (RSP) StrategyDifference Scalar Potential (DSP) StrategyGeneral Scalar Potential (GSP) Strategy

5.1.2.1. RSP Strategy

Applicability

If there are no current sources (Js = 0) the RSP strategy is applicable. Also, in general, if there are current sources

and there is no iron ([µ] = [µo]) within the problem domain, the RSP strategy is also applicable. This formulation

is developed by Zienkiewicz(75).

Procedure

The RSP strategy uses a one-step procedure (MAGOPT,0). Equation 5–16 and Equation 5–17 are solved makingthe following substitution:

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Section 5.1: Electromagnetic Field Fundamentals

(5–20) H Hg s= in ando 1Ω Ω

Saturation is considered if the magnetic material is nonlinear. Permanent magnets are also considered.

5.1.2.2. DSP Strategy

Applicability

The DSP strategy is applicable when current sources and singly connected iron regions exist within the problemdomain (Js ≠ 0) and ([µ] ≠ [µo]). A singly connected iron region does not enclose a current. In other words a

contour integral of H through the iron must approach zero as u → ∞ .

(5–21)o H d u∫ ⋅ → → ∞ in as1 l 0 Ω

This formulation is developed by Mayergoyz(119).

Procedure

The DSP strategy uses a two-step solution procedure. The first step (MAGOPT,2) makes the following substitutioninto Equation 5–16 and Equation 5–17:

(5–22) H Hg s= in ando 1Ω Ω

subject to:

(5–23) n H Sg× = 0 1 on

This boundary condition is satisfied by using a very large value of permeability in the iron (internally set by theprogram). Saturation and permanent magnets are not considered. This step produces a near zero field in theiron region which is subsequently taken to be zero according to:

(5–24) H1 10= in Ω

and in the air region:

(5–25) H Ho s g o= − ∇φ in Ω

The second step (MAGOPT,3) uses the fields calculated on the first step as the preliminary field for Equation 5–16and Equation 5–17:

(5–26) Hg = 0 1 in Ω

(5–27) H Hg o o= in Ω

Here saturation and permanent magnets are considered. This step produces the following fields:

(5–28) H g1 1= −∇φ in Ω

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and

(5–29) H Ho g g o= − ∇φ in Ω

which are the final results to the applicable problems.

5.1.2.3. GSP Strategy

Applicability

The GSP strategy is applicable when current sources (Js ≠ 0) in conjunction with a multiply connected iron ([µ]

≠ [µo]) region exist within the problem domain. A multiply connected iron region encloses some current source.

This means that a contour integral of H through the iron region is not zero:

(5–30)o H d∫ ⋅ → l 0 in 1Ω

where:

⋅ = refers to the dot product

This formulation is developed by Gyimesi(141, 149, 201).

Procedure

The GSP strategy uses a three-step solution procedure. The first step (MAGOPT,1) performs a solution only inthe iron with the following substitution into Equation 5–16 and Equation 5–17:

(5–31) H Hg s o= in Ω

subject to:

(5–32) [ ]( )n H Sg g⋅ − ∇ =µ φ 0 1 on

Here S1 is the surface of the iron air interface. Saturation can optimally be considered for an improved approxim-

ation of the generalized field but permanent magnets are not. The resulting field is:

(5–33) H Hs g1 = − ∇φ

The second step (MAGOPT,2) performs a solution only in the air with the following substitution into Equation 5–16and Equation 5–17:

(5–34) H Hg s o= in Ω

subject to:

(5–35) n H n H Sg× = × 1 1 in

This boundary condition is satisfied by automatically constraining the potential solution φg at the surface of the

iron to be what it was on the first step (MAGOPT,1). This step produces the following field:

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Section 5.1: Electromagnetic Field Fundamentals

(5–36) H Ho s g o= − ∇φ in Ω

Saturation or permanent magnets are of no consequence since this step obtains a solution only in air.

The third step (MAGOPT,3) uses the fields calculated on the first two steps as the preliminary field for Equa-tion 5–16 and Equation 5–17:

(5–37) H Hg = 1 1 in Ω

(5–38) H Hg o o= in Ω

Here saturation and permanent magnets are considered. The final step allows for the total field to be computedthroughout the domain as:

(5–39) H Hg g= − ∇φ in Ω

5.1.3. Magnetic Vector Potential

The vector potential method as implemented in PLANE13, PLANE53 and SOLID97 for both 2-D and 3-D electro-magnetic fields is discussed in this section. Considering static and dynamic fields and neglecting displacementcurrents (quasi-stationary limit), the following subset of Maxwell's equations apply:

(5–40)∇ × = H J

(5–41)∇ × = − ∂

∂ E

Bt

(5–42)∇ ⋅ = B 0

The usual constitutive equation for magnetic and electric field apply as described by Equation 5–11 and Equa-tion 5–12. Although some restriction on anisotropy and nonlinearity do occur in the formulations mentionedbelow.

In the entire domain, Ω, of an electromagnetic field problem a solution is sought which satisfies the relevantMaxwell's Equation 5–40 thru Equation 5–41. See Figure 5.1: “Electromagnetic Field Regions” for a representationof the problem domain Ω.

A solution can be obtained by introducing potentials which allow the magnetic field B and the electric field Eto be expressed as (Biro(120)):

(5–43) B A= ∇ ×

(5–44) E

At

V= − ∂∂

− ∇

where:

A = magnetic vector potentialV = electric scalar potential

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These specifications ensure the satisfaction of two of Maxwell's equations, Equation 5–41 and Equation 5–42.What remains to be solved is Ampere's law, Equation 5–40 in conjunction with the constitutive relations, Equa-tion 5–11, and the divergence free property of current density. Additionally, to ensure uniqueness of the vectorpotential, the Coulomb gauge condition is employed. The resulting differential equations are:

(5–45)

∇ × ∇ × − ∇ ∇ ⋅ + ∂∂

+ ∇

− × ∇ × =

[ ] [ ] [ ]

[ ]

ν ν σ σ

σ

A AAt

V

v A

e

0 inn Ω2

(5–46)∇ ⋅ ∂

∂

− ∇ + × ∇ ×

=[ ] [ ] [ ] σ σ σA

tV v A 0 2 in Ω

(5–47)∇ × ∇ × − ∇ ∇ ⋅ = + ∇ × +[ ] [ ] inν ν

νν A A J Me s

oo o

11Ω Ω

where:

ν ν ν ν νe tr= = + +13

13

11 2 2 3 3[ ] ( ( , ) ( , ) ( , ))

Of course these equations are subject to the appropriate boundary conditions.

This system of simplified Maxwell's equations with the introduction of potential functions has been used for thesolutions of 2-D and 3-D, static and dynamic fields. Silvester(72) presents a 2-D static formulation and Demer-dash(151) develops the 3-D static formulation. Chari(69), Brauer(70) and Tandon(71) discuss the 2-D eddy currentproblem and Weiss(94) and Garg(95) discuss 2-D eddy current problems which allow for skin effects (eddy currentspresent in the source conductor). The development of 3-D eddy current problems is found in Biro(120). In manyof these references the important issues of appropriate boundary conditions, gauging and uniqueness are dis-cussed. The edge-flux formulation with tree gauging (Gyimesi and Ostergaard(202), (221), Ostergaard and Gy-imesi(222), (223)) is discussed in Section 14.117: SOLID117 - 3-D 20-Node Magnetic Edge.

For models containing materials with different permeabilities, the 3-D vector potential formulation is not recom-mended. The solution has been found to be incorrect when the normal component of the vector potential issignificant at the interface between elements of different permeability. A further discussion on this limitation isfound in Biro et al.(200).

5.1.4. Edge Flux Degrees of Freedom

Biro et al.(200) and Preis et al.(203) observed inaccuracies in the finite element analysis of 3-D magnetic fieldproblems with the nodal based continuous vector potential, A, in the presence of inhomogeneous medium. Thistheoretical shortcomings of the nodal vector potential, A, has been demonstrated by Gyimesi and Ostergaard(201,221), Ostergaard and Gyimesi(222, 223).

The shortcomings of the nodal based continuous vector potential formulation is demonstrated below. Theseshortcomings can be eliminated by the edge element method. The edge element formulation constitutes thetheoretical foundation of low-frequency electromagnetic element, SOLID117. Section 12.9: ElectromagneticEdge Elements describes the pertinent edge shape functions. Section 12.9: Electromagnetic Edge Elements dis-cusses topics related to the matrix formulation and gauging. Section 5.6: Inductance, Flux and Energy Computationby LMATRIX and SENERGY Macros presents the details of the high frequency edge formulation.

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Section 5.1: Electromagnetic Field Fundamentals

To eliminate these inaccuracies, edge elements with a discontinuous normal component have been proposed.Section 14.117: SOLID117 - 3-D 20-Node Magnetic Edge and Section 12.9: Electromagnetic Edge Elements aredevoted to discuss this topic.

5.1.5. Limitation of the Nodal Vector Potential

Consider a volume bounded by planes, x = ± -1, y = ± 1, and z = ± 1. See Figure 5.2: “Patch Test Geometry”.Subdivide the volume into four elements by planes, x = 0 and y = 0. The element numbers are set according tothe space quadrant they occupy. The permeability, µ, of the elements is µ1, µ2, µ3, and µ4, respectively. Denote

unit vectors by 1x, 1y, and 1z. Consider a patch test with a known field, Hk = 1z, Bk = µHk changes in the

volume according to µ.

Figure 5.2 Patch Test Geometry

Since Bk is constant within the elements, one would expect that even a first order element could pass the patch

test. This is really the case with edge element but not with nodal elements. For example, A = µ x 1y provides

a perfect edge solution but not a nodal one because the normal component of A in not continuous.

The underlying reason is that the partials of a continuous A do not exist; not even in a piece-wise manner. Toprove this statement, assume that they exist. Denote the partials at the origin by:

(5–48)

Ay

A Ay

A

Ax

A A

x x x x

y y y

+

+

= ∂∂

> = ∂∂

<

= ∂∂

> = ∂∂

y y

x

for for

for

0 0

0

; ;

;xx

Ay xfor < 0;

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Note that there are only four independent partials because of A continuity. The following equations follow fromBk = curl A.

(5–49)

A A A A

A A A A

y x y x

y x y x

+ + +

+

− = − =

− = − =

µ µ

µ µ

1 2

3 4

;

;

Since the equation system, (Equation 5–49) is singular, a solution does not exist for arbitrary µ. This contradictionconcludes the proof.

5.1.6. Harmonic Analysis Using Complex Formalism

In a general dynamic problem, any field quantity, q(r,t) depends on the space, r, and time, t, variables. In a har-monic analysis, the time dependence can be described by periodic functions:

(5–50)q r t a r cos t r( , ) ( ) ( ( ))= +ω φ

or

(5–51)q r t c r cos t s r sin t( , ) ( ) ( ) ( ) ( )= −ω ω

where:

r = location vector in spacet = timew = angular frequency of time change.a(r) = amplitude (peak)φ(r) = phase anglec(r) = measurable field at ωt = 0 degreess(r) = measurable field at ωt = -90 degrees

In an electromagnetic analysis, q(r,t) can be the flux density, B, the magnetic field, H, the electric field, E, thecurrent density, J, the vector potential, A, or the scalar potential, V. Note, however, that q(r,t) can not be the

Joule heat, Qj, the magnetic energy, W, or the force, Fjb, because they include a time-constant term.

The quantities in Equation 5–50 and Equation 5–51 are related by

(5–52)c r a r cos r( ) ( ) ( ( ))= φ

(5–53)s r a r sin r( ) ( ) ( ( ))= φ

(5–54)a r c r s r2 2 2( ) ( ) ( )= +

(5–55)tan r s r c r( ( )) ( ) ( )φ =

In Equation 5–50) a(r), φ(r), c(r) and s(r) depend on space coordinates but not on time. This separation of spaceand time is taken advantage of to minimize the computational cost. The originally 4 (3 space + 1 time) dimen-sional real problem can be reduced to a 3 (space) dimensional complex problem. This can be achieved by thecomplex formalism.

The measurable quantity, q(r,t), is described as the real part of a complex function:

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Section 5.1: Electromagnetic Field Fundamentals

(5–56)q r t Re Q r exp j t( , ) ( ) ( )= ω

Q(r) is defined as:

(5–57)Q r Q r jQ rr i( ) ( ) ( )= +

where:

j = imaginary unitRe = denotes real part of a complex quantityQr(r) and Qi(r) = real and imaginary parts of Q(r). Note that Q depends only on the space coordinates.

The complex exponential in Equation 5–56 can be expressed by sine and cosine as

(5–58)exp j t cos t jsin t( ) ( ) ( )ω ω ω= +

Substituting Equation 5–58 into Equation 5–56 provides Equation 5–57

(5–59)q r t Q r cos t Q r sin tr i( , ) ( ) ( ) ( ) ( )= −ω ω

Comparing Equation 5–50 with Equation 5–59 reveals:

(5–60)c r Q rr( ) ( )=

(5–61)s r Q ri( ) ( )=

In words, the complex real, Qr(r), and imaginary, Qi(r), parts are the same as the measurable cosine, c(r), and sine,

s(r), amplitudes.

A harmonic analysis provides two sets of solution: the real and imaginary components of a complex solution.According to Equation 5–50, and Equation 5–60 the magnitude of the real and imaginary sets describe themeasurable field at t = 0 and at ωt = -90 degrees, respectively. Comparing Equation 5–51 and Equation 5–60provides:

(5–62)a r Q r Q rr i( ) ( ) ( )2 2 2= +

(5–63)tan r Q r Q ri r( ( )) ( ) ( )φ =

Equation 5–62 expresses the amplitude (peak) and phase angle of the measurable harmonic field quantities bythe complex real and imaginary parts.

The time average of harmonic fields such as A, E, B, H, J, or V is zero at point r. This is not the case for P, W, or Fbecause they are quadratic functions of B, H, or J. To derive the time dependence of a quadratic function - forthe sake of simplicity - we deal only with a Lorentz force, F, which is product of J and B. (This is a cross product;but components are not shown to simplify writing. The space dependence is also omitted.)

(5–64)

F t J t B t J cos t J sin t B cos t B sin tjbr i r i( ) ( ) ( ) ( ( ) ( ))( ( ) ( ))= = − −ω ω ω ω

== + − +J B cos t JB sin t JB J B sin t cos tr r i i i r r i( ) ( ) ( ) ( ) ( )ω ω ω ω2 2

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where:

Fjb = Lorentz Force density (output as FMAG on PRESOL command)

The time average of cos2 and sin2 terms is 1/2 whereas that of the sin cos term is zero. Therefore, the time averageforce is:

(5–65)F J B JBjbr r i i= +1 2/ ( )

Thus, the force can be obtained as the sum of “real” and “imaginary” forces. In a similar manner the time averaged

Joule power density, Qj, and magnetic energy density, W, can be obtained as:

(5–66)Q J E JEjr r i i= +1 2/ ( )

(5–67)W B H B Hr r i i= +1 4/ ( )

where:

W = magnetic energy density (output as SENE on PRESOL command)

Qj = Joule Power density heating per unit volume (output as JHEAT on PRESOL command)

The time average values of these quadratic quantities can be obtained as the sum of real and imaginary setsolutions.

The element returns the integrated value of Fjb is output as FJB and W is output as SENE. Qj is the average element

Joule heating and is output as JHEAT. For F and Qj the 1/2 time averaging factor is taken into account at printout.For W the 1/2 time factor is ignored to preserve the printout of the real and imaginary energy values as the in-stantaneous stored magnetic energy at t = 0 and at ωt = -90 degrees, respectively. The element force, F, is distrib-uted among nodes to prepare a magneto-structural coupling. The average Joule heat can be directly applied tothermoelectric coupling.

5.1.7. Nonlinear Time-Harmonic Magnetic Analysis

Many electromagnetic devices operate with a time-harmonic source at a typical power frequency. Although thepower source is time-harmonic, numerical modeling of such devices can not be assumed as a linear harmonicmagnetic field problem in general, since the magnetic materials used in these devices have nonlinear B-H curves.A time-stepping procedure should be used instead. This nonlinear transient procedure provides correct solutionsfor electromagnetic field distribution and waveforms, as well as global quantities such as force and torque. Theonly problem is that the procedure is often computationally intensive. In a typical case, it takes about 4-5 timecycles to reach a sinusoidal steady state. Since in each cycle, at least 10 time steps should be used, the analysiswould require 40-50 nonlinear solution steps.

In many cases, an analyst is often more interested in obtaining global electromagnetic torque and power lossesin a magnetic device at sinusoidal steady state, but less concerned with the actual flux density waveform. Undersuch circumstances, an approximate time-harmonic analysis procedure may be pursued. If posed properly, thisprocedure can predict the time-averaged torque and power losses with good accuracy, and yet at much reducedcomputational cost.

The basic principle of the present nonlinear time-harmonic analysis is briefly explained next. First of all, the actualnonlinear ferromagnetic material is represented by another fictitious material based on energy equivalence. This

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amounts to replacing the DC B-H curve with a fictitious or effective B-H curve based on the following equationfor a time period cycle T (Demerdash and Gillott(231)):

(5–68)12

4

00

4H dB

TH sin t dB dtm eff

o

B

m

BT

eff

∫ ∫∫=

( )ω

where:

Hm = peak value of magnetic field

B = magnetic flux densityBeff = effective magnetic flux density

T = time periodω = angular velocityt = time

With the effective B-H curve, the time transient is suppressed, and the nonlinear transient problem is reducedto a nonlinear time-harmonic one. In this nonlinear analysis, all field quantities are all sinusoidal at a given fre-quency, similar to the linear harmonic analysis, except that a nonlinear solution has to be pursued.

It should be emphasized that in a nonlinear transient analysis, given a sinusoidal power source, the magneticflux density B has a non-sinusoidal waveform. While in the nonlinear harmonic analysis, B is assumed sinusoidal.Therefore, it is not the true waveform, but rather represents an approximation of the fundamental time harmonicof the true flux density waveform. The time-averaged global force, torque and loss, which are determined bythe approximate fundamental harmonics of fields, are then subsequently approximation to the true values, Nu-merical benchmarks show that the approximation is of satisfactory engineering accuracy.

5.1.8. Electric Scalar Potential

Neglecting the time-derivative of magnetic flux density

∂∂

Bt (the quasistatic approximation), the system of

Maxwell's equations (Equation 5–1 through Equation 5–4) reduces to:

(5–69)∇ × = + ∂

∂

H JDt

(5–70)∇ × = E 0

(5–71)∇ =i B 0

(5–72)∇ =i D ρ

As follows from Equation 5–70, the electric field E is irrotational, and can be derived from:

(5–73) E V= −∇

where:

V = electric scalar potential

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In the time-varying electromagnetic field governed by Equation 5–69 through Equation 5–72, the electric andmagnetic fields are uncoupled. If only electric solution is of interest, replacing Equation 5–69 by the continuityEquation 5–5 and eliminating Equation 5–71 produces the system of differential equations governing thequasistatic electric field.

Repeating Equation 5–12 and Equation 5–13 without velocity effects, the constitutive equations for the electricfields become:

(5–74) [ ] J E= σ

(5–75) [ ] D E= ε

where:

[ ]σ

ρ

ρ

ρ

=

=

10 0

01

0

0 01

xx

yy

zz

electrical conduuctivity matrix

[ ]εε

ε

ε

=

=xx

yy

zz

0

permittivity matrix

0

0 0

0 0

ρxx = resistivity in the x-direction (input as RSVX on MP command)

εxx = permittivity in the x-direction (input as PERX on MP command)

The conditions for E, J, and D on an electric material interface are:

(5–76)E Et t1 2 0− =

(5–77)J

Dt

JD

tnn

nn

11

22+ ∂

∂= + ∂

∂

(5–78)D Dn n s1 2− = ρ

where:

Et1, Et2 = tangential components of E on both sides of the interface

Jn1, Jn2 = normal components of J on both sides of the interface

Dn1, Dn2 = normal components of D on both sides of the interface

ρs = surface charge density

Two cases of the electric scalar potential approximation are considered below.

5.1.8.1. Quasistatic Electric Analysis

In this analysis, the relevant governing equations are Equation 5–73 and the continuity equation (below):

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(5–79)∇ + ∂

∂

=i

J

Dt

0

Substituting the constitutive Equation 5–74 and Equation 5–75 into Equation 5–79, and taking into accountEquation 5–73, one obtain the differential equation for electric scalar potential:

(5–80)−∇ ∇ − ∇ ∇ ∂

∂

=i i([ ] ) [ ]σ εVVt

0

Equation 5–80 is used to approximate a time-varying electric field in elements PLANE230, SOLID231, and SOLID232.It takes into account both the conductive and dielectric effects in electric materials. Neglecting time-variationof electric potential Equation 5–80 reduces to the governing equation for steady-state electric conduction:

(5–81)−∇ ∇ =i([ ] )σ V 0

In the case of a time-harmonic electric field analysis, the complex formalism allows Equation 5–80 to be re-writtenas:

(5–82)−∇ ∇ + ∇ ∇ =i i([ ] ) ([ ] )ε

ωσV

jV 0

where:

j = imaginary unitω = angular frequency

Equation 5–82 is the governing equation for a time-harmonic electric analysis using elements PLANE121, SOLID122,and SOLID123.

In a time-harmonic analysis, the loss tangent tanδ can be used instead of or in addition to the electrical conduct-ivity [σ] to characterize losses in dielectric materials. In this case, the conductivity matrix [σ] is replaced by theeffective conductivity [σeff] defined as:

(5–83)[ ] [ ] [ ] tanσ σ ω ε δeff = +

where:

tanδ = loss tangent (input as LSST on MP command)

5.1.8.2. Electrostatic Analysis

Electric scalar potential equation for electrostatic analysis is derived from governing Equation 5–72 and Equa-tion 5–73, and constitutive Equation 5–75:

(5–84)−∇ ∇ =i([ ] )ε ρV

Equation 5–84, subject to appropriate boundary conditions, is solved in an electrostatic field analysis of dielectricsusing elements PLANE121, SOLID122, and SOLID123.

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5.2. Derivation of Electromagnetic Matrices

The finite element matrix equations can be derived by variational principles. These equations exist for linear andnonlinear material behavior as well as static and transient response. Based on the presence of linear or nonlinearmaterials (as well as other factors), the program chooses the appropriate Newton-Raphson method. The usermay select another method with the (NROPT command (see Section 15.11: Newton-Raphson Procedure)). Whentransient affects are to be considered a first order time integration scheme must be involved (TIMINT command(see Section 17.2: Transient Analysis)).

5.2.1. Magnetic Scalar Potential

The scalar potential formulations are restricted to static field analysis with partial orthotropic nonlinear permeab-ility. The degrees of freedom (DOFs), element matrices, and load vectors are presented here in the followingform (Zienkiewicz(75), Chari(73), and Gyimesi(141)):

5.2.1.1. Degrees of freedom

φe = magnetic scalar potentials at the nodes of the element (input/output as MAG)

5.2.1.2. Coefficient Matrix

(5–85)[ ] [ ] [ ]K K Km L N= +

(5–86)[ ] ( ) [ ]( ) ( )K N N d volL T T T

vol= ∇ ∇∫ µ

(5–87)[ ] ( ) ( )

( )K

HH N H N

d volH

N h T T Tvol

T T= ∂∂

∇ ∇∫µ

5.2.1.3. Applied Loads

(5–88)[ ] ( ) [ ]( ) ( )J N H H d voli

T Tg cvol

= ∇ +∫ µ

where:

N = element shape functions (φ = NTφe)

∇ = =

∂∂

∂∂

∂∂

Tx y z

gradient operator

vol = volume of the elementHg = preliminary or “guess” magnetic field (see Section 5.1: Electromagnetic Field Fundamentals)

Hc = coercive force vector (input as MGXX, MGYY, MGZZ on MP command))

[µ] = permeability matrix (derived from input material property MURX, MURY, and MURZ (MP command)and/or material curve B versus H (accessed with TB,BH))(see Equation 5–7, Equation 5–8 and Equation 5–9)

dd H

hµ

= derivative of permeability with respect to magnitude of the magnetic field intensity (derived fromthe input material property curve B versus H (accessed with TB,BH))

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The material property curve is input in the form of B values versus H values and is then converted to a spline fit

curve of µ versus H from which the permeability terms µh and

dd H

hµ

are evaluated.

The coercive force vector is related to the remanent intrinsic magnetization vector as:

(5–89)[ ] µ µH Mc o o=

where:

µo = permeability of free space (input as MUZRO on EMUNIT command)

The Newton-Raphson solution technique (Option on the NROPT command) is necessary for nonlinear analyses.Adaptive descent is also recommended (Adaptky on the NROPT command). When adaptive descent is usedEquation 5–85 becomes:

(5–90)[ ] [ ] ( )[ ]K K Km L N= + −1 ξ

where:

ξ = descent parameter (see Section 15.11: Newton-Raphson Procedure)

5.2.2. Magnetic Vector Potential

The vector potential formulation is applicable to both static and dynamic fields with partial orthotropic nonlinearpermeability. The basic equation to be solved is of the form:

(5–91)[ ] [ ] C u K u Ji& &+ =

The terms of this equation are defined below (Biro(120)); the edge-flux formulation matrices are obtained fromthese terms in Section 14.117: SOLID117 - 3-D 20-Node Magnetic Edge following Gyimesi and Ostergaard(201).

5.2.2.1. Degrees of Freedom

(5–92)

u

Ae

e=

ν

where:

Ae = magnetic vector potentials (input/output as AX, AY, AZ)

νe = time integrated electric scalar potential (ν = Vdt) (input/output as VOLT)

The VOLT DOF is a time integrated electric potential to allow for symmetric matrices.

5.2.2.2. Coefficient Matrices

(5–93)[ ]

[ ] [ ]

[ ] [ ]K

K

K

AA

vA=

0

0

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(5–94)[ ] [ ] [ ] [ ]K K K KAA L N G= + +

(5–95)[ ] ( [ ] ) [ ]( [ ] [ ][ ]( [ ] )) ( )K N N N v N d volL

AT T

AT

A AT

vol

= ∇ × ∇ × − × ∇ ×∫ ν σ

(5–96)[ ] ( [ ] ) [ ]( [ ] ) ( )K N N d volG

AT

AT

vol

T= ∇ ⋅ ∇ ⋅∫ ν

(5–97)[ ]

( )( ( [ ] )) ( ( [ ] )) ( )K

d

d BB N B N d volN h T

AT T T

AT

vol= ∇ × ∇ ×∫2

2ν

(5–98)[ ] ( [ ] ) [ ] [ ] ( )K N v N d volVA T T

AT= − ∇ × ∇ ×∫ σ

(5–99)[ ]

[ ] [ ]

[ ] [ ]C

C C

C C

AA Av

Av T vv=

(5–100)[ ] [ ][ ][ ] ( )C N N d volAA

A AT

vol

= ∫ σ

(5–101)[ ] [ ][ ] ( )C N N d volAv

AT

vol= ∇∫ σ

(5–102)[ ] ( ) [ ] ( )C N N d volv v T T T

vol

= ∇ ∇∫ σ

5.2.2.3. Applied Loads

(5–103)

J

J

Ii

A

t=

(5–104) J J JA S pm= +

(5–105) [ ] ( )J J N d volS

s AT

vol

= ∫

(5–106) ( [ ] ) ( )J x N H d volpm

AT T

cvol

= ∇∫

(5–107) [ ] ( )I J N d volt

t AT

vol

= ∫

where:

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[NA] = matrix of element shape functions for A ( [ ] ; )A N A A A A AA

Te e

Txe

Tye

Tze

T= =

[N] = vector of element shape functions for V (V = NTVe)

Js = source current density vector (input as JS on BFE command)

Jt = total current density vector (input as JS on BFE command) (valid for 2-D analysis only)

vol = volume of the elementHc = coercive force vector (input as MGXX, MGYY, MGZZ on MP command)

νo = reluctivity of free space (derived from value using MUZRO on EMUNIT command)

[ν] = partially orthotropic reluctivity matrix (inverse of [µ], derived from input material property curve B versusH (input using TB,BH command))

d

d Bhν

( )2= derivative of reluctivity with respect to the magnitude of magnetic flux squared (derived from input

material property curve B versus H (input using TB,BH command))[σ] = orthotropic conductivity (input as RSVX, RSVY, RSVZ on MP command (inverse)) (see Equation 5–12).v = velocity vector

The coercive force vector is related to the remanent intrinsic magnetization vector as:

(5–108) [ ] H Mc

oo= 1

νν

The material property curve is input in the form of B values versus H values and is then converted to a spline fit

curve of ν versus |B|2 from which the isotropic reluctivity terms νh and

d

d Bhν

( )2 are evaluated.

The above element matrices and load vectors are presented for the most general case of a vector potentialanalysis. Many simplifications can be made depending on the conditions of the specific problem. In 2-D there isonly one component of the vector potential as opposed to three for 3-D problems (AX, AY, AZ).

Combining some of the above equations, the variational equilibrium equations may be written as:

(5–109) ( [ ] [ ] [ ] [ ] )A K A K C d dt A C d dt JeT AA

eAV

eAA

eAV

eA+ + + − =ν ν 00

(5–110) ( [ ] [ ] [ ] [ ] )ν ν νeT VA

eVV

eVA

eVV

etK A K C d dt A C d dt l+ + + − = 00

Here T denotes transposition.

Static analyses require only the magnetic vector potential DOFs (KEYOPT controlled) and the K coefficient matrices.If the material behavior is nonlinear then the Newton-Raphson solution procedure is required (Option on theNROPT command (see Section 15.11: Newton-Raphson Procedure)).

For 2-D dynamic analyses a current density load of either source (Js) or total Jt current density is valid. Jt input

represents the impressed current expressed in terms of a uniformly applied current density. This loading is onlyvalid in a skin-effect analysis with proper coupling of the VOLT DOFs. In 3-D only source current density is allowed.The electric scalar potential must be constrained properly in order to satisfy the fundamentals of electromagneticfield theory. This can be achieved by direct specification of the potential value (using the D command) as wellas with coupling and constraining (using the CP and CE commands).

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The general transient analysis (ANTYPE,TRANS (see Section 15.4: Element Reordering)) accepts nonlinear mater-ial behavior (field dependent [ν] and permanent magnets (MGXX, MGYY, MGZZ). Harmonic transient analyses(ANTYPE,HARMIC (see Section 17.4: Harmonic Response Analyses)) is a linear analyses with sinusoidal loads;therefore, it is restricted to linear material behavior without permanent magnets.

5.2.3. Electric Scalar Potential

The electric scalar potential V is approximated over the element as follows:

(5–111)V N VTe=

where:

N = element shape functionsVe = nodal electric scalar potential (input/output as VOLT)

5.2.3.1. Quasistatic Electric Analysis

The application of the variational principle and finite element discretization to the differential Equation 5–80produces the matrix equation of the form:

(5–112)[ ] [ ] C V K V Ive

ve e

& + =

where:

[ ] ( ) [ ]( ) ( )K N N d volv T

vol

T T= ∇ ∇ =∫ σ element electrical conducctivity coefficient matrix

[ ] ( ) [ ]( ) ( )C N N d volv T

vol

T T= ∇ ∇ =∫ ε element dielectric permitttivity coefficient matrix

vol = element volumeIe = nodal current vector (input/output as AMPS)

Equation 5–112 is used in the finite element formulation of PLANE230, SOLID231, and SOLID232. These elementsmodel both static (steady-state electric conduction) and dynamic (time-transient and time-harmonic) electric

fields. In the former case, matrix [Cv] is ignored.

A time-harmonic electric analysis can also be performed using elements PLANE121, SOLID122, and SOLID123.In this case, the variational principle and finite element discretization are applied to the differential Equation 5–82to produce:

(5–113)( [ ] [ ]) j C K V Qvh vhe e

nω + =

where:

[ ] [ ]K Cvh v=

[ ] [ ]C Kvh v= − 12ω

Qen = nodal charge vector (input/output as CHRG)

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Section 5.2: Derivation of Electromagnetic Matrices

5.2.3.2. Electrostatic Analysis

The matrix equation for an electrostatic analysis using elements PLANE121, SOLID122, and SOLID123 is derivedfrom Equation 5–84:

(5–114)[ ] K V Qvse e=

[ ] ( ) [ ]( ) ( )K N N d volvs T

vol

T T= ∇ ∇ =∫ ε dielectric permittivity coefficient matrix

Q Q Q Qe en

ec

esc= + +

( )Q N d volec T

vol= ∫ ρ

( )Q N d volesc

sT

s= ∫ ρ

ρ = charge density vector (input as CHRGD on BF command)ρs = surface charge density vector (input as CHRGS on SF command)

5.3. Electromagnetic Field Evaluations

The basic magnetic analysis results include magnetic field intensity, magnetic flux density, magnetic forces andcurrent densities. These types of evaluations are somewhat different for magnetic scalar and vector formulations.The basic electric analysis results include electric field intensity, electric current densities, electric flux density,Joule heat and stored electric energy.

5.3.1. Magnetic Scalar Potential Results

The first derived result is the magnetic field intensity which is divided into two parts (see Section 5.1: Electromag-

netic Field Fundamentals); a generalized field Hg and the gradient of the generalized potential - ∇ φg. This

gradient (referred to here as Hφ) is evaluated at the integration points using the element shape function as:

(5–115) H N T

gφ φ= −∇

where:

∇ = =

∂∂

∂∂

∂∂

Tx y z

gradient operator

N = shape functionsωg = nodal generalized potential vector

The magnetic field intensity is then:

(5–116) H H Hg= + φ

where:

H = magnetic field intensity (output as H)

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Then the magnetic flux density is computed from the field intensity:

(5–117) [ ] B H= µ

where:

B = magnetic flux density (output as B)[µ] = permeability matrix (defined in Equation 5–7, Equation 5–8, and Equation 5–9)

Nodal values of field intensity and flux density are computed from the integration points values as described inSection 13.6: Nodal and Centroidal Data Evaluation.

Magnetic forces are also available and are discussed below.

5.3.2. Magnetic Vector Potential Results

The magnetic flux density is the first derived result. It is defined as the curl of the magnetic vector potential. Thisevaluation is performed at the integration points using the element shape functions:

(5–118) [ ] B N AAT

e= ∇ ×

where:

B = magnetic flux density (output as B)∇ x = curl operator[NA] = shape functions

Ae = nodal magnetic vector potential

Then the magnetic field intensity is computed from the flux density:

(5–119) [ ] H B= ν

where:

H = magnetic field intensity (output as H)[ν] = reluctivity matrix

Nodal values of field intensity and flux density are computed from the integration point value as described inSection 13.6: Nodal and Centroidal Data Evaluation.

Magnetic forces are also available and are discussed below.

For a vector potential transient analysis current densities are also calculated.

(5–120) J J J Jt e s v= + +

where:

Jt = total current density

(5–121) [ ] [ ] [ ] J

At n

N Ae AT

ei

n= − ∂

∂

= −=∑σ σ 1

1

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Section 5.3: Electromagnetic Field Evaluations

where:

Je = current density component due to A

[σ] = conductivity matrixn = number of integration points[NA] = element shape functions for A evaluated at the integration points

Ae = time derivative of magnetic vector potential

and

(5–122) [ ] [ ] J V

nN Vs

Te

i

n= − ∇ = ∇

=∑σ σ 1

1

where:

Js = current density component due to V

∇ = divergence operatorVe = electric scalar potential

N = element shape functions for V evaluated at the integration points

and

(5–123) J v Bv = ×

where:

Jv = velocity current density vector

v = applied velocity vectorB = magnetic flux density (see Equation 5–118)

5.3.3. Magnetic Forces

Magnetic forces are computed by elements using the vector potential method (PLANE13, PLANE53, and SOLID97)and the scalar potential method (SOLID5, SOLID96, and SOLID98). Three different techniques are used to calculatemagnetic forces at the element level.

5.3.3.1. Lorentz forces

Magnetic forces in current carrying conductors (element output quantity FJB) are numerically integrated using:

(5–124) ( ) ( )F N J B d voljb T

vol= ×∫

where:

N = vector of shape functions

For a 2-D analysis, the corresponding electromagnetic torque about +Z is given by:

(5–125)T Z r J B d voljb

vol= ⋅ × ×∫ ( ) ( )

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where:

Z = unit vector along +Z axisr = position vector in the global Cartesian coordinate system

In a time-harmonic analysis, the time-averaged Lorentz force and torque are computed by:

(5–126) ( ) ( )F N J B d volav

jb Tvol

= ×∗∫12

and

(5–127)T Z r J B d volav

jbvol

= ⋅ × ×∫ ( ) ( )

respectively.

where:

J* = complex conjugate of J

5.3.3.2. Maxwell Forces

The Maxwell stress tensor is used to determine forces on ferromagnetic regions (element output quantity FMX).This force calculation is performed on surfaces of air material elements which have a nonzero face loading specified(MXWF on SF commands) (Moon(77)). For the 2-D application, this method uses extrapolated field values andresults in the following numerically integrated surface integral:

(5–128) F

T T

T T

n

ndsmx

o s=

∫1 11 12

21 22

1

2µ

where:

µo = permeability of free space (input on EMUNIT command)

T B Bx112 21

2= −

T12 = Bx By

T21 = Bx By

T B By222 21

2= −

3-D applications are an extension of the 2-D case.

For a 2-D analysis, the corresponding electromagnetic torque about +Z axis is given by:

(5–129)T Z r n B B B B n dsmx

o s= ⋅ × ⋅ − ⋅

∫ ( ) ( )^ ^1 1

2µ

where:

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Section 5.3: Electromagnetic Field Evaluations

n^ = unit surface normal in the global Cartesian coordinate system

In a time-harmonic analysis, the time-averaged Maxwell stress tensor force and torque are computed by:

(5–130) Re ( ) ( )^ ^F n B B B B n dsav

mx

o s= ⋅ − ⋅

∗ ∗∫1

212µ

and

(5–131)T Z r n B B B B n dsav

mx

o s= ⋅ × ⋅ − ⋅

∗ ∗ Re ( ) ( )^ ^12

12µ ∫∫

respectively.

where:

B* = complex conjugate of BRe = denotes real part of a complex quantity

5.3.3.3. Virtual Work Forces

Electromagnetic nodal forces (including electrostatic forces) are calculated using the virtual work principle. Thetwo formulations currently used for force calculations are the element shape method (magnetic forces) andnodal perturbations method (electromagnetic forces).

5.3.3.3.1. Element Shape Method

Magnetic forces calculated using the virtual work method (element output quantity FVW) are obtained as thederivative of the energy versus the displacement (MVDI on BF commands) of the movable part. This calculationis valid for a layer of air elements surrounding a movable part (Coulomb(76)). To determine the total force actingon the body, the forces in the air layer surrounding it can be summed. The basic equation for force of an air ma-terial element in the s direction is:

(5–132)F B

Hs

d vol B dHs

d volsT

volT

vol= ∂

∂

+ ∂∂∫ ∫∫ ( ) ( ) ( )

where:

Fs = force in element in the s direction

∂∂

=H

sderivative of field intensity with respect to diisplacements

s = virtual displacement of the nodal coordinates taken alternately to be in the X, Y, Z global directionsvol = volume of the element

For a 2-D analysis, the corresponding electromagnetic torque about +Z axis is given by:

(5–133)T Z r B B s B s B d volvw

o vo= ⋅ × ⋅ ∇ − ⋅ ∇

( ) ( ) ( )1 1

2µ ll∫

Chapter 5: Electromagnetics

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In a time-harmonic analysis, the time-averaged virtual work force and torque are computed by:

(5–134) ( ) Re ( ) ( )F B B s B s B d volav

vw

o v= ⋅ ∇ − ⋅ ∇

∗ ∗12

12µ ool∫

and

(5–135)T Z R B B s B s B dav

vw

o= ⋅ × ⋅ ∇ − ⋅ ∇

∗ ∗ ( ) Re ( ) 1

212µ

(( )volvol∫

respectively.

5.3.3.3.2. Nodal Perturbation Method

Electromagnetic forces are calculated as the derivatives of the total element coenergy (sum of electrostatic andmagnetic coenergies) with respect to the element nodal coordinates(Gyimesi et al.(346)):

(5–136)F

xd E B H d volxi

i

T T

vol= ∂

∂+

∫

12

( ) ( )

where:

Fxi = x-component (y- or z-) of electromagnetic force calculated in node i

xi = nodal coordinate (x-, y-, or z-coordinate of node i)

vol = volume of the element

Nodal electromagnetic forces are calculated for each node in each element. In an assembled model the nodalforces are added up from all adjacent to the node elements.

5.3.4. Joule Heat in a Magnetic Analysis

Joule heat is computed by elements using the vector potential method (PLANE13, PLANE53, and SOLID97) if theelement has a nonzero resistivity (material property RSVX) and a nonzero current density (either applied Js or

resultant Jt). It is available as the output power loss (output as JHEAT) or as the coupled field heat generation

load (LDREAD,HGEN).

Joule heat per element is computed as:

1. Static or Transient Magnetic Analysis

(5–137)Q

nJ Jj

ti tii

n= ⋅

=∑1

1[ ] ρ

where:

Qj = Joule heat per unit volumen = number of integration points[ρ] = resistivity matrix (input as RSVX, RSVY, RSVZ on MP command)Jti = total current density in the element at integration point i

2. Harmonic Magnetic Analysis

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Section 5.3: Electromagnetic Field Evaluations

Q Ren

J Jjti ti

i

n= ⋅

∗

=∑1

2 1[ ] ρ

(5–138)

where:

Re = real componentJti = complex total current density in the element at integration point i

Jti* = complex conjugate of Jti

5.3.5. Electric Scalar Potential Results

The first derived result in this analysis is the electric field. By definition (Equation 5–73), it is calculated as thenegative gradient of the electric scalar potential. This evaluation is performed at the integration points usingthe element shape functions:

(5–139) E N VTe= −∇

Nodal values of electric field (output as EF) are computed from the integration points values as described inSection 13.6: Nodal and Centroidal Data Evaluation. The derivation of other output quantities depends on theanalysis types described below.

5.3.5.1. Quasistatic Electric Analysis

The conduction current and electric flux densities are computed from the electric field (see Equation 5–74 andEquation 5–75):

(5–140) [ ] J E= σ

(5–141) [ ] D E= ε

Both the conduction current J and electric flux D densities are evaluated at the integration point locations;however, whether these values are then moved to nodal or centroidal locations depends on the element typeused to do a quasistatic electric analysis:

• In a current-based electric analysis using elements PLANE230, SOLID231, and SOLID232, the conductioncurrent density is stored at both the nodal (output as JC) and centoidal (output as JT) locations. The electricflux density vector components are stored at the element centroidal location and output as nonsummablemiscellaneous items;

• In a charge-based analysis using elements PLANE121, SOLID122, and SOLID123 (harmonic analysis), theconduction current density is stored at the element centroidal location (output as JT), while the electricflux density is moved to the nodal locations (output as D).

The total electric current Jtot density is calculated as a sum of conduction J and displacement current

∂∂

Dt

densities:

(5–142) J J

Dttot = + ∂

∂

Chapter 5: Electromagnetics

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The total electric current density is stored at the element centroidal location (output as JS). It can be used as asource current density in a subsequent magnetic analysis (LDREAD,JS).

The Joule heat is computed from the centroidal values of electric field and conduction current density. In asteady-state or transient electric analysis, the Joule heat is calculated as:

(5–143)Q J ET=

where:

Q = Joule heat generation rate per unit volume (output as JHEAT)

In a harmonic electric analysis, the Joule heat is time-averaged over a one period and calculated as:

(5–144)Q J ET= 1

2Re( *)

where:

Re = real componentE* = complex conjugate of E

The value of Joule heat can be used as heat generation load in a subsequent thermal analysis (LDREAD,HGEN).

In a transient electric analysis, the element stored electric energy is calculated as:

(5–145)W D E d volT

vol= ∫

12

( )

where:

W = stored electric energy (output as SENE)

In a harmonic electric analysis, the time-averaged electric energy is calculated as:

(5–146)W D E d volT

vol= ∫

14

( *) ( )

5.3.5.2. Electrostatic Analysis

The derived results in an electrostatic analysis are:

Electric field (see Equation 5–139) at nodal locations (output as EF);Electric flux density (see Equation 5–141) at nodal locations (output as D);Element stored electric energy (see Equation 5–145) output as SENE

Electrostatic forces are also available and are discussed below.

5.3.6. Electrostatic Forces

Electrostatic forces are determined using the Maxwell stress tensor. This force calculation is performed on surfacesof elements which have a nonzero face loading specified (MXWF on SF commands). For the 2-D application, thismethod uses extrapolated field values and results in the following numerically integrated surface integral:

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Section 5.3: Electromagnetic Field Evaluations

(5–147) F

T T

T T

n

ndsmx

os

=

∫ε 11 12

21 22

1

2

where:

εo = free space permittivity (input as PERX on MP command)

T E Ex112 21

2=

T12 = Ex Ey

T21 = Ey Ex

T E Ey222 21

2=

n1 = component of unit normal in x-direction

n2 = component of unit normal in y-direction

s = surface area of the element face

E E Ex y2 2 2=

3-D applications are an extension of the 2-D case.

5.3.7. Electric Constitutive Error

The dual constitutive error estimation procedure as implemented for the electrostatic p-elements SOLID127 andSOLID128 is activated (with the PEMOPTS command) and is briefly discussed in this section. Suppose a field pair

^ ^E D which verifies the Maxwell's Equation 5–70 and Equation 5–72, can be found for a given problem. This

couple is the true solution if the pair also verifies the constitutive relation (Equation 5–75). Or, the couple is justan approximate solution to the problem, and the quantity

(5–148) [ ] e D E= ⋅ε

is called error in constitutive relation, as originally suggested by Ladeveze(274) for linear elasticity. To measure

the error ^e , the energy norm over the whole domain Ω is used:

(5–149) [ ] ^ ^ ^e D E

Ω Ω= − ⋅ε

with

(5–150) [ ] σ σ ε σΩΩ

Ω=

−∫ T d1

12

By virtue of Synge's hypercircle theorem(275), it is possible to define a relative error for the problem:

Chapter 5: Electromagnetics

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(5–151)ε

ε

εΩ

Ω

Ω

=

− ⋅

+ ⋅

[ ]

[ ]

^ ^

^ ^

D E

D E

The global relative error (Equation 5–151) is seen as sum of element contributions:

(5–152)ε εΩ

2 2= ∑ EE

where the relative error for an element E is given by

(5–153)ε

ε

εE

E

D E

D E

=

− ⋅

+ ⋅

[ ]

[ ]

^ ^

^ ^

Ω

The global error εΩ allows to quantify the quality of the approximate solution pair ^ ^E D and local error εE allows

to localize the error distribution in the solution domain as required in a p-adaptive analysis.

5.4. Voltage Forced and Circuit-Coupled Magnetic Field

The magnetic vector potential formulation discussed in Chapter 5, “Electromagnetics” requires electric currentdensity as input. In many industrial applications, a magnetic device is often energized by an applied voltage orby a controlling electric circuit. In this section, a brief outline of the theoretical foundation for modeling suchvoltage forced and circuit-coupled magnetic field problems is provided. The formulations apply to static, transientand harmonic analysis types.

To make the discussion simpler, a few definitions are introduced first. A stranded coil refers to a coil consistingof many turns of conducting wires. A massive conductor refers to an electric conductor where eddy currentsmust be accounted for. When a stranded coil is connected directly to an applied voltage source, we have avoltage forced problem. If a stranded coil or a massive conductor is connected to an electric circuit, we have acircuit-coupled problem. A common feature in both voltage forced and circuit-coupled problems is that theelectric current in the coil or conductor must be treated as an additional unknown.

For general circuit and reduced order modeling capabilities refer to Section 5.12: Circuit Analysis, Reduced OrderModeling. To obtain parameters of circuit elements one may either compute them using a handbook formula,use LMATRIX (Section 5.6: Inductance, Flux and Energy Computation by LMATRIX and SENERGY Macros) and/orCMATRIX (Section 5.10: Capacitance Computation), or another numerical package and/or GMATRIX (Section 5.13:Conductance Computation)

5.4.1. Voltage Forced Magnetic Field

Assume that a stranded coil has an isotropic and constant magnetic permeability and electric conductivity. Then,by using the magnetic vector potential approach from Chapter 5, “Electromagnetics”, the following elementmatrix equation is derived.

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Section 5.4: Voltage Forced and Circuit-Coupled Magnetic Field

(5–154)

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

0 0

0 0 0C

A K K

KiA

AA Ai

ii

+

&

=

A

i Vo

0

where:

A = nodal magnetic vector potential vector (AX, AY, AZ)⋅ = time derivativei = nodal electric current vector (input/output as CURR)

[KAA] = potential stiffness matrix

[Kii] = resistive stiffness matrix

[KAi] = potential-current coupling stiffness matrix

[CiA] = inductive damping matrixVo = applied voltage drop vector

The magnetic flux density B, the magnetic field intensity H, magnetic forces, and Joule heat can be calculatedfrom the nodal magnetic vector potential A using Equation 5–117 and Equation 5–118.

The nodal electric current represents the current in a wire of the stranded coil. Therefore, there is only one inde-pendent electric current unknown in each stranded coil. In addition, there is no gradient or flux calculation asso-ciated with the nodal electric current vector.

5.4.2. Circuit-Coupled Magnetic Field

When a stranded coil or a massive conductor is connected to an electric circuit, both the electric current andvoltage (not the time-integrated voltage) should be treated as unknowns. To achieve a solution for this problem,the finite element equation and electric circuit equations must be solved simultaneously.

The modified nodal analysis method (McCalla(188)) is used to build circuit equations for the following linearelectric circuit element options:

1. resistor

2. inductor

3. capacitor

4. voltage source

5. current source

6. stranded coil current source

7. 2-D massive conductor voltage source

8. 3-D massive conductor voltage source

9. mutual inductor

10. voltage-controlled current source

11. voltage-controlled voltage source

12. current-controlled voltage source

13. current-controlled current source

These circuit elements are implemented in element CIRCU124.

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Assuming an isotropic and constant magnetic permeability and electric conductivity, the following elementmatrix equation is derived for a circuit-coupled stranded coil:

(5–155)

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

0 0 0

0 0

0 0 0

0

0

CiA

&A

+

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

K K

K K

A

i

AA Ai

ii ie

0

0

0 0 0

e

=

0

0

0

where:

e = nodal electromotive force drop (EMF)

[Kie] = current-emf coupling stiffness

For a circuit-coupled massive conductor, the matrix equation is:

(5–156)

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

C

C

AA

VA

0 0

0 0 0

0 0

0

0

&A

+

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

K K

K K

AAA AV

iV VV

0

0 0 0

0

ii

V

=

0

0

0

where:

V = nodal electric voltage vector (input/output as VOLT)

[KVV] = voltage stiffness matrix

[KiV] = current-voltage coupling stiffness matrix

[CAA] = potential damping matrix

[CVA] = voltage-potential damping matrix

The magnetic flux density B, the magnetic field intensity H, magnetic forces and Joule heat can be calculatedfrom the nodal magnetic vector potential A using Equation 5–117 and Equation 5–118.

5.5. High-Frequency Electromagnetic Field Simulation

In previous sections, it has been assumed that the electromagnetic field problem under consideration is eitherstatic or quasi-static. For quasi-static or low-frequency problem, the displacement current in Maxwell's equationsis ignored, and Maxwell's Equation 5–1 through Equation 5–4 are simplified as Equation 5–40 through Equa-tion 5–42. This approach is valid when the working wavelength is much larger than the geometric dimensionsof structure or the electromagnetic interactions are not obvious in the system. Otherwise, the full set of Maxwell'sequations must be solved. The underlying problems are defined as high-frequency/full-wave electromagneticfield problem (Volakis et al.(299) and Itoh et al.(300)), in contrast to the quasi-static/low-frequency problems inprevious sections. The purpose of this section is to introduce full-wave FEA formulations, and define useful outputquantities.

5.5.1. High-Frequency Electromagnetic Field FEA Principle

A typical electromagnetic FEA configuration is shown in Figure 5.3: “A Typical FEA Configuration for Electromag-netic Field Simulation”. A closed surface Γ0 truncates the infinite open domain into a finite numerical domain Ωwhere FEA is applied to simulate high frequency electromagnetic fields. An electromagnetic plane wave fromthe infinite may project into the finite FEA domain, and the FEA domain may contain radiation sources, inhomo-geneous materials and conductors, etc.

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Section 5.5: High-Frequency Electromagnetic Field Simulation

Figure 5.3 A Typical FEA Configuration for Electromagnetic Field Simulation

!!" #$%&')(!&*+,.-

/ (&0&*!1 (!2+4365

78 9) 1&: $%"!;.(!&=<>!;.+ ,.?

@A !;9)&) ;1 (B ; 1.!"DCE F,HG

I / 1&:IJ /

K (&'<*!;. ; 1! # LM"!1

,ON

3

Based on Maxwell's Equation 5–1 and Equation 5–2 with the time-harmonic assumption ejωt, the electric fieldvector Helmholtz equation is cast:

(5–157)∇ × ⋅ ∇ ×

− ⋅ = −−µ ε ωµ= =r r sE k E j J1

02

0( )ur ur r

where:

Eur

= electric field vector

ε=r = complex tensor associated with the relative permittivity and conductivity of material (input as PERX,PERY, PERZ, and RSVX, TSVY, RSVZ on MP command)µ0 = free space permeability

µ=r = complex relative permeability tensor of material (input as MURX, MURY, MURZ on MP command)k0 = vacuum wave number

ω = operating angular frequency

Jsr

= excitation current density (input as JS on BF command)

Test the residual Rur

of the electric field vector Helmholtz equation with vector function Tur

and integrate overthe FEA domain to obtain the “weak” form formulation:

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(5–158)

R T T E k T Er rur ur ur ur ur

, ( ) ( )= ∇ × ⋅ ⋅ ∇ ×

− ⋅ ⋅

−µ ε= =1

02

+ ⋅

− ⋅ × +

∫∫∫ ∫∫∫

∫∫+

d j T J d

j T n H d j

s ss

o

Ω Ω

Γ

Ω Ω

Γ Γ

ωµ

ωµ ω

0

01

ur r

ur ur( )^ µµ0 Y n T n E d r

r

( ) ( )^ ^× ⋅ ×∫∫ur ur

ΓΓ

where:

n^ = outward directed normal unit of surface

Hur

= magnetic fieldY = surface admittance

Assume that the electric field Eur

is approximated by:

(5–159)E W Ei i

i

Nur u ru=

=∑

1

where:

Ei = degree of freedom that is the projection of vector electric field at edge, on face or in volume of element.

Wu ru

= vector basis function

Representing the testing vector Tur

as vector basis function Wu ru

(Galerkin's approach) and rewriting Equation 5–158in FEA matrix notation yields:

(5–160)( [ ] [ ] [ ]) − + + =k M jk C K E F02

0

where:

M W W dij i r j= ⋅ ⋅∫∫∫u ru u ru

ε= ,Re ΩΩ

Ck

W W d k W Wij i r j i r j

w

= ∇ × ⋅ ⋅ ∇ × − ⋅ ⋅−∫∫∫1

0

10

u ru u ru u ruµ ε= =

,Im ,Im( ) ΩΩ

dd

Z Y n W n W dRe i j rr

Ω

Γ

Ω

Γ

∫∫∫

∫∫+ × ⋅ ×0 ( ) ( )^ ^u ru u ru

K W W d k Z Y n Wij i r j Im i= ∇ × ⋅ ⋅ ∇ × − × ⋅−∫∫∫ ( ) ( ) ( ) (,Im^

u ru u ru u ruµ= 1

0 0ΩΩ

nn W dj rr

^ )×∫∫u ru

ΓΓ

F jk Z W J d jk Z W n H di i s s i

s

= − ⋅ + ⋅ ×∫∫∫ ∫∫+

0 0 0 00 1

u ru r u ru urΩ Γ

Ω Γ Γ( )^

Re = real part of a complex numberIm = imaginary part of a complex number

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Section 5.5: High-Frequency Electromagnetic Field Simulation

For electromagnetic scattering simulation, a pure scattered field formulation should be used to ensure the nu-merical accuracy of solution, since the difference between total field and incident field leads to serious round-off numerical errors when the scattering fields are required. Since the total electric field is the sum of incident

field Eur

inc and scattered field Eur

sc , i.e.Eur

tot = Eur

inc + Eur

sc, the “weak” form formulation for scattered field is:

(5–161)

R T T E k T Ersc

rscur ur ur ur ur ur

, ( ) ( )= ∇ × ⋅ ⋅ ∇ ×

− ⋅ ⋅ −µ ε= =1

02

+ × ⋅ × + ⋅

∫∫∫

∫∫

d

j Y n T n E d j T J dscr i

r

Ω

Γ

Ω

Γωµ ωµ0 0( ) ( )^ ^

r r ur rΩΩ

Ωs

rinc

rin

s

T E k T E

∫∫∫

+ ∇ × ⋅ ⋅ ∇ ×

− ⋅ ⋅−( ) ( )ur ur ur ur

µ ε= =102 cc

d

dinc

d

T n E d j Y n

d

d o

− ⋅ × ∇ × + ×

∫∫∫

∫∫+

Ω

Γ

Ω

Γ Γ

ur ur( ) (^ ^ωµ0 TT n E d

j T n H d

incr

r

r

r

ur ur

ur ur

) ( )

( )

^

^

⋅ ×

− ⋅ ×

∫∫

∫∫

Γ

Γ

Γ

Γωµ0

where:

n^d = outward directed normal unit of surface of dielectric volume

Rewriting the scattering field formulation (Equation 5–161) in FEA matrix notation again yields:

(5–162)− + + =k M jk C K E Fsc02

0[ ] [ ] [ ]

where matrix [M], [C], [K] are the same as matrix notations for total field formulation (Equation 5–160) and:

(5–163)

F jk Z W J d jk Z W n H di i i s i

s

= − ⋅ + ⋅ ×

+ ∇

∫∫∫ ∫∫+

0 0 0 00 1

u ru r u ru urΩ Γ

Ω Γ Γ( )

(

^

×× ⋅ ⋅ ∇ ×

− ⋅ ⋅

−W E k W Ei r

inci r

incu ru ur u ru ur) ( )µ ε= =1

02

− ⋅ × ∇ × + ×

∫∫∫

∫∫+

d

W n E d jk Z Y n W

d

i dinc

s

d

Ω

Γ

Ω

Γ Γ

u ru ur u r( ) (^ ^

0

0 0uu ur

iinc

rn E dr

) ( )^⋅ ×∫∫ ΓΓ

It should be noticed that the total tangential electric field is zero on the perfect electric conductor (PEC) boundary,

and the boundary condition for Eur

sc of Equation 5–6 will be imposed automatically.

For a resonant structure, a generalized eigenvalue system is involved. The matrix notation for the cavity analysisis written as:

(5–164)[ ] [ ] K E k M E= 02

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where:

M W W dij i r j= ⋅ ⋅∫∫∫u ru u ru

ε= ,Re ΩΩ

K W W dij i r j= ∇ × ⋅ ⋅ ∇ ×−∫∫∫ ( ) ( ),Re

u ru u ruµ= 1 Ω

Ω

Here the real generalized eigen-equation will be solved, and the damping matrix [C] is not included in the eigen-equation. The lossy property of non-PEC cavity wall and material filled in cavity will be post-processed if thequality factor of cavity is calculated.

If the electromagnetic wave propagates in a guided-wave structure, the electromagnetic fields will vary with thepropagating factor exp(-jγz) in longitude direction, γ = β - jγ. Here γ is the propagating constant, and α is theattenuation coefficient of guided-wave structure if exists. When a guided-wave structure is under consideration,

the electric field is split into the transverse component Eur

t and longitudinal component Ez, i.e., E E zEt zur ur ur

= + ^.

The variable transformation is implemented to construct the eigen-equation using e j Et tr ur

= γ and ez = Ez. The

“weak” form formulation for the guided-wave structure is:

(5–165)

R W W W z e e z k Wt z t r t z t z r zur u ru u ru u ru r

, ( ) ( )^ ^,= ∇ + × ⋅ ⋅ ∇ + × −

−γ µ ε2 1

02

zz z

t t r t t t rt t

e d

W e k W e

]

+ ∇ × ⋅ ⋅ ∇ × − ⋅ ⋅

∫∫

−

ΩΩ

( ) ( )u ru r u ru r

µ ε= =102

∫∫ dΩ

Ω

where:

∇t = transverse components of ∇ operator

The FEA matrix notation of Equation 5–165 is:

(5–166)

k S k G

k G k Q S

E

Ez z

t t t

z

t

max max

max max

[ ] [ ]

[ ] [ ] [ ]

2 2

2 2 +

= −

( )[ ] [ ]

[ ] [ ]

maxkS G

G Q

E

Ez z

t t

z

t

2 2γ

where:

kmax = maximum wave number in the material

[ ] [ ] [ ]S S k Tt t t= − 02

[ ] [ ] [ ]S S k Tz z z= − 02

and the matrix elements are:

(5–167)S W W dt ij t t i r t t j, , ,( ) ( )= ∇ × ⋅ ⋅ ∇ ×−∫∫

u ru u ruµ= 1 Ω

Ω

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Section 5.5: High-Frequency Electromagnetic Field Simulation

(5–168)Q W z W z dt ij t i r t j, , ,( ) ( )^ ^= × ⋅ ⋅ ×−∫∫

u ruµ= 1 Ω

Ω

(5–169)G W z W z dz ij z i r t j, , ,( ) ( )^ ^= ∇ × ⋅ ⋅ ×−∫∫ µ= 1

u ruΩ

Ω

(5–170)S W z W z dz ij z i r z j, , ,( ) ( )^ ^= ∇ × ⋅ ⋅ ∇ ×−∫∫ µ= 1 Ω

Ω

(5–171)T W W dt ij t i r t t j, , , ,= ⋅ ⋅∫∫

u ru u ruε= Ω

Ω

(5–172)G W z W z dt ij t i r z j, , ,( ) ( )^ ^= × ⋅ ⋅ ×−∫∫

u ru u ruµ= 1 Ω

Ω

(5–173)T W W dz ij z i r z z j, , , ,= ∫∫ ε Ω

Ω

Refer to Section 12.9: Electromagnetic Edge Elements for high-frequency electromagnetic vector shapes.

5.5.2. Boundary Conditions and Perfectly Matched Layers (PML)

5.5.2.1. PEC Boundary Condition

On a Perfect Electric Conductor (PEC) boundary, the tangential components of the electric field Eur

will vanish,i.e.:

(5–174)n E^ × =ur

0

A PEC condition exists typically in two cases. One is the surface of electrical conductor with high conductance if

the skin depth effect can be ignored. Another is on an antisymmetric plane for electric field Eur

. It should be statedthat the degree of freedom must be constrained to zero on PEC.

5.5.2.2. PMC Boundary Condition

On the Perfect Magnetic Conductor (PMC) boundary, the tangential components of electric field Hur

will vanish,i.e.:

(5–175)n H^ × =ur

0

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A PMC condition exists typically either on the surface of high permeable material or on the symmetric plane of

magnetic field Hur

. No special constraint conditions are required on PMC when electric field “weak” form formu-lation is used.

5.5.2.3. Impedance Boundary Condition

A Standard Impedance Boundary Condition (SIBC) exists on the surface (Figure 5.4: “Impedance Boundary Con-dition”) where the electric field is related to the magnetic field by

(5–176)n n E Zn Hout out^ ^ ^′ ′ ′× × = − ×

ur ur

(5–177)n n E Zn Hinc inc

^ ^ ^× × = − ×ur ur

where:

n^ = outward directed normal unit

n^ ′ = inward directed normal unit

Eur

inc, Hur

inc = fields of the normal incoming wave

Eur

out, Hur

out = fields of the outgoing waveZ = complex wave impedance (input as IMPD on SF or SFE command)

Figure 5.4 Impedance Boundary Condition

The SIBC can be used to approximate the far-field radiation boundary, a thin dielectric layer, skin effect of non-perfect conductor and resistive surface, where a very fine mesh is required. Also, SIBC can be used to match thesingle mode in the waveguide.

On the far-field radiation boundary, the relation between the electric field and the magnetic field of incidentplane wave, Equation 5–176, is modified to:

(5–178)n k E Z n Hinc inc^ ^ ^× × = − ×

ur ur0

where:

k^

= unit wave vector

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and the impedance on the boundary is the free-space plane wave impedance, i.e.:

(5–179)Z0 0 0= µ ε

where:

ε0 = free-space permittivity

For air-dielectric interface, the surface impedance on the boundary is:

(5–180)Z Z r r= 0 µ ε

For a dielectric layer with thickness τ coating on PEC, the surface impedance on the boundary is approximatedas:

(5–181)Z jZ tan kr

rr r= 0 0

µε

µ ε τ( )

For a non-perfect electric conductor, after considering the skin effect, the complex surface impedance is definedas:

(5–182)Z j= +ωµ

σ21( )

where:

σ = conductivity of conductor

For a traditional waveguide structure, such as a rectangular, cylindrical coaxial or circular waveguide, where theanalytic solution of electromagnetic wave is known, the wave impedance (not the characteristics impedance)of the mode can be used to terminate the waveguide port with matching the associated single mode. The surfaceintegration of Equation 5–158 is cast into

(5–183)

W n Hd n W n E d

n W

IBC IBC

u ru ur u ru ur⋅ × = − × ⋅ ×

+ ×

∫∫ ∫∫^ ^ ^

^

( ) ( )

(

Γ ΓΓ Γ

1

21

η

η

uu ru ur) ( )^⋅ ×∫∫ n E d

inc

IBCΓ

Γ

where:

Einc = incident wave defined by a waveguide fieldη = wave impedance corresponding to the guided wave

5.5.2.4. Perfectly Matched Layers

Perfectly Matched Layers (PML) is an artificial anisotropic material that is transparent and heavily lossy to incomingelectromagnetic waves so that the PML is considered as a super absorbing boundary condition for the meshtruncation of an open FEA domain, and superior to conventional radiation absorbing boundary conditions. Thecomputational domain can be reduced significantly using PML. It is easy to implement PML in FEA for complicatedmaterials, and the sparseness of the FEA matrices will not be destroyed, which leads to an efficient solution.

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Figure 5.5 PML Configuration

(5–184)∇ × = ⋅H j E

ur urωε[ ]Λ

(5–185)∇ × = − ⋅E j H

ur urωµ[ ]Λ

where:

[Λ] = anisotropic diagonal complex material defined in different PML regions

For the face PML region PMLx to which the x-axis is normal (PMLy, PMLz), the matrix [Λ]x is specified as:

(5–186)[ ] , ,Λ x

xx xdiag

WW W=

1

where:

Wx = frequency-dependent complex number representing the property of the artificial material

The indices and the elements of diagonal matrix are permuted for other regions.

For the edge PML region PMLyz sharing the region PMLy and PMLz (PMLzx, PMLxy), the matrix [Λ]yz is defined as

(5–187)[ ] , , ,Λ yz y z

z

y

y

zdiag W W

WW

W

W=

where:

Wy, Wz = frequency-dependent complex number representing the property of the artificial material.

The indices and the elements of diagonal matrix are permuted for other regions.

For corner PML region Pxyz, the matrix [Λ]xyz is:

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(5–188)[ ] , ,Λ xyz

y z

x

z x

y

x y

zdiag

W W

WW W

W

W W

W=

See Zhao and Cangellaris(301) for details about PML.

5.5.2.5. Periodic Boundary Condition

The periodic boundary condition is necessary for the numerical modeling of the time-harmonic electromagneticscattering, radiation, and absorption characteristics of general doubly-periodic array structures. The periodicarray is assumed to extend infinitely as shown in Figure 5.6: “Arbitrary Infinite Periodic Structure”. Without lossof the generality, the direction normal to the periodic plane is selected as the z-direction of a global Cartesiancoordinate system.

Figure 5.6 Arbitrary Infinite Periodic Structure

From the theorem of Floquet, the electromagnetic fields on the cellular sidewalls exhibit the following dependency:

(5–189)f s D s D z e f s s zs sj( , , ) ( , , )( )

1 1 2 2 1 21 2+ + = − +φ φ

where:

φ1 = phase shift of electromagnetic wave in the s1 direction

φ2 = phase shift of electromagnetic wave in the s2 direction

5.5.3. Excitation Sources

In terms of applications, several excitation sources, waveguide modal sources, current sources, a plane wavesource, electric field source and surface magnetic field source, can be defined in high frequency simulator.

5.5.3.1. Waveguide Modal Sources

The waveguide modal sources exist in the waveguide structures where the analytic electromagnetic field solutionsare available. In high frequency simulator, TEM modal source in cylindrical coaxial waveguide, TEmn/TMmn modal

source in either rectangular waveguide or circular waveguide and TEM/TE0n/TM0n modal source in parallel-plate

waveguide are available. See ANSYS High-Frequency Electromagnetic Analysis Guide for details about commandsand usage.

5.5.3.2. Current Excitation Source

The current source can be used to excite electromagnetic fields in high-frequency structures by contribution toEquation 5–158:

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(5–190)W J ds s

s

u ru r⋅∫∫∫ Ω

Ω

where:

Jsr

= electric current density

5.5.3.3. Plane Wave Source

A plane incident wave in Cartesian coordinate is written by:

(5–191)E E exp jk x cos sin y sin sin zcosur ur

= + +[ ]0 0( )φ θ φ θ θ

where:

Eur

0 = polarization of incident wave

(x, y, z) = coordinate valuesφ = angle between x-axis and wave vectorθ = angle between z-axis and wave vector

5.5.3.4. Surface Magnetic Field Source

A surface magnetic field source on the exterior surface of computational domain is a “hard” magnetic field sourcethat has a fixed magnetic field distribution no matter what kind of electromagnetic wave projects on the sourcesurface. Under this circumstance the surface integration in Equation 5–158 becomes on exterior magnetic fieldsource surface

(5–192)W n Hd W n H d

feed feed

feedu ru ur u ru ur

⋅ × = ⋅ ×∫∫ ∫∫^ ^Γ ΓΓ Γ

When a surface magnetic field source locates on the interior surface of the computational domain, the surfaceexcitation magnetic field becomes a “soft” source that radiates electromagnetic wave into the space and allowsvarious waves to go through source surface without any reflection. Such a “soft” source can be realized bytransforming surface excitation magnetic field into an equivalent current density source (Figure 5.7: “Soft Excit-ation Source”), i.e.:

(5–193)J n Hsincr ur

= ×2 ^

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Figure 5.7 “Soft” Excitation Source

5.5.3.5. Electric Field Source

Electric field source is a “hard” source. The DOF that is the projection of electric field at the element edge for 1st-order element will be imposed to the fixed value so that a voltage source can be defined.

5.5.4. High-Frequency Parameters Evaluations

A time-harmonic complex solution of the full-wave formulations in Section 5.5.1: High-Frequency Electromag-netic Field FEA Principle yields the solution for all degrees of freedom in FEA computational domain. However,those DOF solutions are not immediately transparent to the needs of analyst. It is necessary to compute theconcerned electromagnetic parameters, in terms of the DOF solution.

5.5.4.1. Electric Field

The electric field Hur

is calculated at the element level using the vector shape functions Wu ru

:

(5–194)E W Ei i

i

Nur u ru=

=∑

1

5.5.4.2. Magnetic Field

The magnetic field Hur

is calculated at the element level using the curl of the vector shape functions Wu ru

:

(5–195)Hj

W Er i ii

Nur u ru= ⋅ ∇ ×−

=∑ωµ

µ0

1

1

=

5.5.4.3. Poynting Vector

The time-average Poynting vector (i.e., average power density) over one period is defined by:

(5–196)P Re E Havur ur ur

= ×∗1

2

where:

* = complex conjugate

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5.5.4.4. Power Flow

The complex power flow through an area is defined by

(5–197)P E H ndsf

s= × ⋅∫∫

12

ur ur* ^

5.5.4.5. Stored Energy

The time-average stored electric and magnetic energy are given by:

(5–198)W E E dve r

v= ⋅ ⋅

∗∫∫∫

εε0

4

ur ur=

(5–199)W H H dvm

vr= ⋅ ⋅

∗∫∫∫

µ µ04

ur ur=

5.5.4.6. Dielectric Loss

For a lossy dielectric, the incurred time-average volumetric power loss is:

(5–200)P E E dvd

v= ⋅ ⋅

∗∫∫∫

12

ur urσ=

where:

σ = conductivity tensor of the dielectric material

5.5.4.7. Surface Loss

On the resistive surface, the incurred time-average surface loss is calculated:

(5–201)P R H H dsL s

s

= ⋅∗

∫∫12

ur ur

where:

Rs = surface resistivity

5.5.4.8. Quality Factor

Taking into account dielectric and surface loss, the quality factor (Q-factor) of a resonant structure at certainresonant frequency is calculated (using the QFACT command macro) by:

(5–202)1 1 1Q Q QL d

= +

where:

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QW

PLr e

L= 2ω

QW

Pdr e

d= 2ω

ωr = resonant frequency of structure

5.5.4.9. Voltage

The voltage Vba (computed by the EMF command macro) is defined as the line integration of the electric field

Eur

projection along a path from point a to b by:

(5–203)V E dIba

a

b

= − ⋅∫ur r

where:

dIr

= differential vector line element of the path

5.5.4.10. Current

The electrical current (computed by the MMF command macro) is defined as the line integration of the magnetic

field Hur

projection along an enclosed path containing the conductor by:

(5–204)I o H dI

c= ⋅∫

ur r

5.5.4.11. Characteristic Impedance

The characteristic impedance (computed by the IMPD command macro) of a circuit is defined by:

(5–205)Z

VIba=

5.5.4.12. Scattering Matrix (S-Parameter)

Scattering matrix of a network with multiple ports is defined as (Figure 5.8: “Two Ports Network”):

(5–206) [ ] b S a=

A typical term of [S] is:

(5–207)S

b

ajij

i=

where:

ai = normalized incoming wave at port i

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bj = normalized outgoing wave at port j

Figure 5.8 Two Ports Network

Assume port i as the driven port and port j as matched port in a guided-wave structure, if the transverse eigen

electric field ren is known at port i, the coefficients are written as:

(5–208)a

E e ds

e e dsi

t inc ns

n ns

i

i

=

⋅

⋅

∫∫

∫∫

ur r

r r

,

(5–209)b

E E e ds

e e dsi

t tot t inc ns

n ns

i

i

=

⋅ ⋅

⋅

∫∫

∫∫

( ), ,ur ur r

r r

where:

Eur

t,tot = transverse total electric field

Eur

t,inc = transverse incident electric field

For port j, we have aj = 0, and theEur

t,inc = 0 in above formulations. The coefficients must be normalized by the

power relation

(5–210)P aa bb= −∗ ∗1

2( )

S-parameters of rectangular, circular, cylindrical coaxial and parallel-plate waveguide can be calculated (bySPARM command macro).

If the transverse eigen electric field is not available in a guided-wave structure, an alternative for S-parametercan be defined as:

(5–211)S

V

VZZji

j

i

i

j=

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where:

Vi = voltage at port i

Vj = voltage at port j

Zi = characteristic impedance at port i

Zj = characteristic impedance at port j

The conducting current density on Perfect Electric Conductor (PEC) surface is:

(5–212)

r r vJ n H= ×

where:

rJ = current density

Hur

= magnetic field

The conducting current density in lossy material is:

(5–213)

r rJ E= σ

where:

σ = conductivity of material

Eur

= electric field

5.5.4.13. Surface Equivalence Principle

The surface equivalence principle states that the electromagnetic fields exterior to a given (possibly fictitious)surface is exactly represented by equivalent currents (electric and magnetic) placed on that surface and allowedradiating into the region external to that surface (see figure below). The radiated fields due to these equivalentcurrents are given by the integral expressions

(5–214)

r r r r r rE(r) = - G E( ) G H( r

S Sc c

∇ × ′ × ′ ′ + ′ × ′∫∫ ∫∫⋅ ⋅( ) ( )^ ^R n r ds jk Z R n0 0 )) ds′

(5–215)

r r r r r rH(r) = - G H r ) - G( R) E r

S Sc c

∇ × ′ × ′ ′ ′ × ′∫∫ ∫∫⋅ ⋅( ) ( (^ ^R n ds jk Y n0 0 )) ds′

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Figure 5.9 Surface Equivalent Currents

!

"#$&% !

' )(%)*

+-,

where:

R r r= −r r

¢rr = observation pointrr ¢ = integration point

n^ = outward directed unit normal at point rr ¢

When Jsr

, Msur

are radiating in free space, the dyadic Green's function is given in closed form by:

(5–216)G R I

kG R( ) ( )= − + ∇∇

02 0

where:

I x x y y z z=

= + +^ ^ ^ ^ ^ ^

The scalar Green's function is given by:

(5–217)G R G r re

R

jk R

0 00

4( ) ( , )= =′

−

π

The surface equivalence principle is necessary for the calculation of either near or far electromagnetic field beyondFEA computational domain.

5.5.4.14. Radar Cross Section (RCS)

Radar Cross Section (RCS) is used to measure the scattering characteristics of target projected by incident planewave, and depends on the object dimension, material, wavelength and incident angles of plane wave etc. In dBunits, RCS is defined by:

(5–218)R logCS = =10 10σ Radar Cross Section

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σ is given by:

(5–219)σ π=

→∞lim r

E

Er

sc

inc4 2

2

2

ur

ur

where:

Eur

inc = incident electric field

Eur

sc = scattered electric field

If RCS is normalized by wavelength square, the definition is written by

(5–220)RCSN Normalized Radar Cross Section= =10 102log dB( )( )σ λ

For RCS due to the pth component of the scattered field for a q-polarized incident plane wave, the scatteringcross section is defined as:

(5–221)σ πpqD

r

sc

qinc

rE p

E

3 2

2

24=

→∞

⋅lim

^r

where p and q represent either φ or θ spherical components with φ measured in the xy plane from the x-axis andθ measured from the z-axis.

For 2-D case, RCS is defined as:

(5–222)σ π

ρ φ2

2

22D r

sc

inc

rE

E=

→∞lim

( , )r

r

or

(5–223)RCS dBmD= 10 10 2log ( )σ

If RCS is normalized by the wavelength, it is given by:

(5–224)RCSN dBD= 10 10 2log ( / ) ( )σ λ

5.5.4.15. Antenna Pattern

The far-field radiation pattern of the antenna measures the radiation direction of antenna. The normalized antennapattern is defined by:

(5–225)S

E

Emax=

ur

ur( , )

( , )

φ θ

φ θ

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where:

φ = angle between position vector and x-axisθ = angle between position vector and z-axis

5.5.4.16. Antenna Radiation Power

The total time-average power radiated by an antenna is:

(5–226)P E H ds E H r r d d Udr = × = × =⋅ ⋅∫∫ ∫∫ ∫∫12

12

2Re( ) Re( ) sin* * ^r r r r r

θ θ φ Ω

where:

dΩ = differential solid angledΩ = sinθdθdφ

and the radiation intensity is defined by:

(5–227)U E H r r= × ⋅12

2Re( )* ^r r

5.5.4.17. Antenna Directive Gain

The directive gain, GD (φ, θ), of an antenna is the ration of the radiation intensity in the direction (φ, θ) to the av-

erage radiation intensity:

(5–228)G

UP

U

UdD

r( , )

( , )/

( , )φ θ φ θ φ θ= =∫∫Ω

ΩΩ

where:

Ω Ω= ∫∫ =d solid angle of radiation surface

The maximum directive gain of an antenna is called the directivity of the antenna. It is the ratio of the maximumradiation intensity to the average radiation intensity and is usually denoted by D:

(5–229)D

UU

U

av r= =max max

PΩ

5.5.4.18. Antenna Power Gain

The power gain, Gp, is used to measure the efficiency of an antenna. It is defined as:

(5–230)G

UPp

i= Ω max

where:

Pi = input power

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5.5.4.19. Antenna Radiation Efficiency

The ratio of the power gain to the directivity of an antenna is the radiation efficiency, ηr:

(5–231)ηr

p r

i

G

DPP

= =

5.5.4.20. Electromagnetic Field of Phased Array Antenna

The total electromagnetic field of a phased array antenna is equal to the product of an array factor and the unitcell field:

(5–232)

r rE E e etotal unit

j m

m

M j n

n

N= × ∑ ∑

− +

=

− +

=

( )( ) ( )( )1

1

1

11 1 2 2φ β φ β

where:

M = number of array units in the s1 direction

φ1 = phase shift of electromagnetic wave in the unit in s1 direction

β1 = initial phase in the s1 direction

N = number of array units in the s2 direction

φ2 = phase shift of electromagnetic wave in the unit in s2 direction

β2 = initial phase in the s2 direction

5.5.4.21. Specific Absorption Rate (SAR)

The time-average specific absorption rate of electromagnetic field in lossy material is defined by :

(5–233)SE

W kgAR =σ

ρ

r 2

( / )

where:

SAR = specific absorption rate (output using PRESOL and PLESOL commands)rE = r.m.s. electric field strength inside material (V/m)

σ = conductivity of material (S/m) (input as KXX on MP command)

ρ = mass density of material (kg/m3) (input as DENS on MP command)

5.5.4.22. Power Reflection and Transmission Coefficient

The Power reflection coefficient (Reflectance) of a system is defined by:

(5–234)Γp

iP= Pr

where:

Γp = power reflection coefficient (output using HFPOWER command)

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Pi = input power (W) (Figure 5.10: “Input, Reflection, and Transmission Power in the System”)

Pr = reflection power (W) (Figure 5.10: “Input, Reflection, and Transmission Power in the System”)

The Power transmission coefficient (Transmittance) of a system is defined by:

(5–235)T

Ppt

i=

P

where:

Tp = power transmission coefficient (output using HFPOWER command)

Pt = transmission power (W) (Figure 5.10: “Input, Reflection, and Transmission Power in the System”)

The Return Loss of a system is defined by:

(5–236)L

PdBR

i= −10log

P( )r

where:

LR = return loss (output using HFPOWER command)

The Insertion Loss of a system is defined by:

(5–237)I

PdBL i

t= −10log

P( )

where:

IL = insertion loss (output using HFPOWER command)

Figure 5.10 Input, Reflection, and Transmission Power in the System

5.5.4.23. Reflection and Transmission Coefficient in Periodic Structure

The reflection coefficient in a periodic structure under plane wave excitation is defined by:

(5–238)Γ =

r

rE

Etr

ti

where:

Γ = reflection coefficient (output with FSSPARM command)rEt

i = tangential electric field of incident wave (Figure 5.11: “Periodic Structure Under Plane Wave Excitation”)

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rEt

r = tangential electric field of reflection wave (Figure 5.11: “Periodic Structure Under Plane Wave Excitation”)

In general the electric fields are referred to the plane of periodic structure.

The transmission coefficient in a periodic structure under plane wave excitation is defined by:

(5–239)T

E

Ett

ti

=r

r

where:

T = transmission coefficient (output with FSSPARM command)rEt

t = tangential electric field of transmission wave (Figure 5.11: “Periodic Structure Under Plane Wave Excit-

ation”)

Figure 5.11 Periodic Structure Under Plane Wave Excitation

5.5.4.24. The Smith Chart

In the complex wave w = u + jv, the Smith Chart is constructed by two equations:

(5–240)

ur

r r

ux x

−+

+ =+

− + −

=

11

1

11 1

22

2

22 2

ν

ν( )

where:

r and x = determined by Z/Zo = r + jx and Y/Yo = r + jx

Z = complex impedanceY = complex admittanceZo = reference characteristic impedance

Yo = 1/Zo

The Smith Chart is generated by PLSCH command.

5.5.4.25. Conversion Among Scattering Matrix (S-parameter), Admittance Matrix(Y-parameter), and Impedance Matrix (Z-parameter)

For a N-port network the conversion between matrices can be written by:

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(5–241)

[ ] [ ] ([ ] [ ])([ ] [ ]) [ ]

[ ] [ ] ([ ] [ ]

Y Z I S I S Z

S Z I Z

o o

o o

= − +

= −

− − −

−

12 1

12

12 [[ ])([ ] [ ][ ]) [ ]

[ ] [ ]

Y I Z Y Z

Z Y

o o+

=

−

−

112

1

where:

[S] = scattering matrix of the N-port network[Y] = admittance matrix of the N-port network[Z] = impedance matrix of the N-port network[Zo] = diagonal matrix with reference characteristic impedances at ports

[I] = identity matrix

Use PLSYZ and PRSYZ commands to convert, display, and plot network parameters.

5.6. Inductance, Flux and Energy Computation by LMATRIX and SENERGYMacros

For general circuit and reduced order modeling capabilities refer to Section 5.12: Circuit Analysis, Reduced OrderModeling. The capacitance may be obtained using the CMATRIX command macro (Section 5.10: CapacitanceComputation).

Inductance plays an important role in the characterization of magnetic devices, electrical machines, sensors andactuators. The concept of a non-variant (time-independent), linear inductance of wire-like coils is discussed inevery electrical engineering book. However, its extension to variant, nonlinear, distributed coil cases is far fromobvious. The LMATRIX command macro accomplishes this goal for a multi-coil, potentially distributed systemby the most robust and accurate energy based method.

Time-variance is essential when the geometry of the device is changing: for example actuators, electrical machines.In this case, the inductance depends on a stroke (in a 1-D motion case) which, in turn, depends on time.

Many magnetic devices apply iron for the conductance of magnetic flux. Most iron has a nonlinear B-H curve.Because of this nonlinear feature, two kinds of inductance must be differentiated: differential and secant. Thesecant inductance is the ratio of the total flux over current. The differential inductance is the ratio of flux changeover a current excitation change.

The flux of a single wire coil can be defined as the surface integral of the flux density. However, when the size ofthe wire is not negligible, it is not clear which contour spans the surface. The field within the coil must be takeninto account. Even larger difficulties occur when the current is not constant: for example solid rotor or squirrel-caged induction machines.

The energy-based methodology implemented in the LMATRIX macro takes care of all of these difficulties.Moreover, energy is one of the most accurate qualities of finite element analysis - after all it is energy-based -thus the energy perturbation methodology is not only general but also accurate and robust.

The voltage induced in a variant coil can be decomposed into two major components: transformer voltage andmotion induced voltage.

The transformer voltage is induced in coils by the rate change of exciting currents. It is present even if the geometryof the system is constant, the coils don't move or expand. To obtain the transformer voltage, the knowledge of

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flux change (i.e., that of differential flux) is necessary when the exciting currents are perturbed. This is characterizedby the differential inductance provided by the LMATRIX command macro.

The motion induced voltage (sometimes called back-EMF) is related to the geometry change of the system. It ispresent even if the currents are kept constant. To obtain the motion induced voltage, the knowledge of absoluteflux in the coils is necessary as a function of stroke. The LMATRIX command macro provides the absolute fluxtogether with the incremental inductance.

Obtaining the proper differential and absolute flux values needs consistent computations of magnetic absoluteand incremental energies and co-energies. This is provided by the SENERGY command macro. The macro usesan “energy perturbation” consistent energy and co-energy definition.

5.6.1. Differential Inductance Definition

Consider a magnetic excitation system consisting of n coils each fed by a current, Ii. The flux linkage ψi of the

coils is defined as the surface integral of the flux density over the area multiplied by the number of turns, Ni, of

the of the pertinent coil. The relationship between the flux linkage and currents can be described by the secantinductance matrix, [Ls]:

(5–242) ( , ) ψ ψ= [ ] +L t I Is o

where:

ψ = vector of coil flux linkagest = timeI = vector of coil currents.ψo = vector of flux linkages for zero coil currents (effect of permanent magnets)

Main diagonal element terms of [Ls] are called self inductance, whereas off diagonal terms are the mutual induct-

ance coefficients. [Ls] is symmetric which can be proved by the principle of energy conservation.

In general, the inductance coefficients depend on time, t, and on the currents. The time dependent case is calledtime variant which is characteristic when the coils move. The inductance computation used by the program isrestricted to time invariant cases. Note that time variant problems may be reduced to a series of invariant analyseswith fixed coil positions. The inductance coefficient depends on the currents when nonlinear magnetic materialis present in the domain.

The voltage vector, U, of the coils can be expressed as:

(5–243) U

t= ∂

∂ψ

In the time invariant nonlinear case

(5–244)

U

d L

d II L

tI L I

tIs

s d=[ ]

+ [ ]

∂∂

= [ ]∂∂

The expression in the bracket is called the differential inductance matrix, [Ld]. The circuit behavior of a coil system

is governed by [Ld]: the induced voltage is directly proportional to the differential inductance matrix and the

time derivative of the coil currents. In general, [Ld] depends on the currents, therefore it should be evaluated for

each operating point.

Chapter 5: Electromagnetics

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5.6.2. Review of Inductance Computation Methods

After a magnetic field analysis, the secant inductance matrix coefficients, Lsij, of a coupled coil system could be

calculated at postprocessing by computing flux linkage as the surface integral of the flux density, B. The differ-ential inductance coefficients could be obtained by perturbing the operating currents with some current incre-ments and calculating numerical derivatives. However, this method is cumbersome, neither accurate nor efficient.A much more convenient and efficient method is offered by the energy perturbation method developed byDemerdash and Arkadan(225), Demerdash and Nehl(226) and Nehl et al.(227). The energy perturbation methodis based on the following formula:

(5–245)L

d WdIdIdij

i j=

2

where W is the magnetic energy, Ii and Ij are the currents of coils i and j. The first step of this procedure is to obtain

an operating point solution for nominal current loads by a nonlinear analysis. In the second step linear analysesare carried out with properly perturbed current loads and a tangent reluctivity tensor, νt, evaluated at the oper-

ating point. For a self coefficient, two, for a mutual coefficient, four, incremental analyses are required. In thethird step the magnetic energies are obtained from the incremental solutions and the coefficients are calculatedaccording to Equation 5–245.

5.6.3. Inductance Computation Method Used

The inductance computation method used by the program is based on Gyimesi and Ostergaard(229) who revivedSmythe's procedure(150).

The incremental energy Wij is defined by

(5–246)W H B dVij = ∫

12

∆ ∆

where ∆H and ∆B denote the increase of magnetic field and flux density due to current increments, ∆Ii and

∆Ij. The coefficients can be obtained from

(5–247)W L I Iij dij i j= 1

2∆ ∆

This allows an efficient method that has the following advantages:

1. For any coefficient, self or mutual, only one incremental analysis is required.

2. There is no need to evaluate the absolute magnetic energy. Instead, an “incremental energy” is calculatedaccording to a simple expression.

3. The calculation of incremental analysis is more efficient: The factorized stiffness matrix can be applied.(No inversion is needed.) Only incremental load vectors should be evaluated.

5.6.4. Transformer and Motion Induced Voltages

The absolute flux linkages of a time-variant multi-coil system can be written in general:

(5–248) ( ( ), ( ))ψ ψ= X t I t

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where:

X = vector of strokes

The induced voltages in the coils are the time derivative of the flux linkages, according to Equation 5–243. Afterdifferentiation:

(5–249)

U

dd I

d Idt

dd X

d Xdt

= +ψ ψ

(5–250) ( , )

U Ld I Xd I

dtdd X

V= [ ]

+ ψ

where:

V = vector of stroke velocities

The first term is called transformer voltage (it is related to the change of the exciting current). The proportionalterm between the transformer voltage and current rate is the differential inductance matrix according to Equa-tion 5–244.

The second term is the motion included voltage or back EMF (it is related to the change of strokes). The timederivative of the stroke is the velocity, hence the motion induced voltage is proportional to the velocity.

5.6.5. Absolute Flux Computation

Whereas the differential inductance can be obtained from the differential flux due to current perturbation asdescribed in Section 5.6.1: Differential Inductance Definition, Section 5.6.2: Review of Inductance ComputationMethods, and Section 5.6.3: Inductance Computation Method Used. The computation of the motion inducedvoltage requires the knowledge of absolute flux. In order to apply Equation 5–250, the absolute flux should be

mapped out as a function of strokes for a given current excitation ad the derivative

dd X ψ

provides the matrixlink between back EMF and velocity.

The absolute flux is related to the system co-energy by:

(5–251)

ψ =′d W

d I

According to Equation 5–251, the absolute flux can be obtained with an energy perturbation method by changingthe excitation current for a given stroke position and taking the derivative of the system co-energy.

The increment of co-energy can be obtained by:

(5–252)∆ ∆W B H dVi i

′ = ∫

where:

Wi′ = change of co-energy due to change of current Ii

∆Hi = change of magnetic field due to change of current Ii

Chapter 5: Electromagnetics

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5.6.6. Inductance Computations

The differential inductance matrix and the absolute flux linkages of coils can be computed (with the LMATRIXcommand macro).

The differential inductance computation is based on the energy perturbation procedure using Equation 5–246and Equation 5–247.

The absolute flux computation is based on the co-energy perturbation procedure using Equation 5–251 andEquation 5–252.

The output can be applied to compute the voltages induced in the coils using Equation 5–250.

5.6.7. Absolute Energy Computation

The absolute magnetic energy is defined by:

(5–253)W H d Bs

B= ∫

0

and the absolute magnetic co-energy is defined by:

(5–254)W B d Hc

H

H

c

=−∫

See Figure 5.12: “Energy and Co-energy for Non-Permanent Magnets” and Figure 5.13: “Energy and Co-energyfor Permanent Magnets” for the graphical representation of these energy definitions. Equations and provide theincremental magnetic energy and incremental magnetic co-energy definitions used for inductance and absoluteflux computations.

The absolute magnetic energy and co-energy can be computed (with the LMATRIX command macro).

Figure 5.12 Energy and Co-energy for Non-Permanent Magnets

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Figure 5.13 Energy and Co-energy for Permanent Magnets

! !

!

"

!"

Equation 5–246 and Equation 5–252 provide the incremental magnetic energy and incremental magnetic co-energy definitions used for inductance and absolute flux computations.

5.7. Electromagnetic Particle Tracing

Once the electromagnetic field is computed, particle trajectories can be evaluated by solving the equations ofmotion:

(5–255)m a F q E v B ( )= = + ×

where:

m = mass of particleq = charge of particleE = electric field vectorB = magnetic field vectorF = Lorentz force vectora = acceleration vectorv = velocity vector

The tracing follows from element to element: the exit point of an old element becomes the entry point of a newelement. Given the entry location and velocity for an element, the exit location and velocity can be obtained byintegrating the equations of motion.

Chapter 5: Electromagnetics

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ANSYS particle tracing algorithm is based on Gyimesi et al.(228) exploiting the following assumptions:

1. No relativistic effects (Velocity is much smaller than speed of light).

2. Pure electric tracing (B = 0), pure magnetic tracing (E = 0), or combined E-B tracing.

3. Electrostatic and/or magnetostatic analysis

4. Constant E and/or B within an element.

5. Quadrangle, triangle, hexahedron, tetrahedron, wedge or pyramid element shapes bounded by planarsurfaces.

These simplifications significantly reduce the computation time of the tracing algorithm because the trajectorycan be given in an analytic form:

1. parabola in the case of electric tracing

2. helix in the case of magnetic tracing.

3. generalized helix in the case of coupled E-B tracing.

The exit point from an element is the point where the particle trajectory meets the plane of bounding surfaceof the element. It can be easily computed when the trajectory is a parabola. However, to compute the exit pointwhen the trajectory is a helix, a transcendental equation must be solved. A Newton Raphson algorithm is imple-mented to obtain the solution. The starting point is carefully selected to ensure convergence to the correctsolution. This is far from obvious: about 70 sub-cases are differentiated by the algorithm. This tool allows particletracing within an element accurate up to machine precision. This does not mean that the tracing is exact sincethe element field solution may be inexact. However, with mesh refinement, this error can be controlled.

Once a trajectory is computed, any available physical items can be printed or plotted along the path (using thePLTRAC command). For example, elapsed time, traveled distance, particle velocity components, temperature,field components, potential values, fluid velocity, acoustic pressure, mechanical strain, etc. Animation is alsoavailable.

The plotted particle traces consist of two branches: the first is a trajectory for a given starting point at a givenvelocity (forward ballistic); the second is a trajectory for a particle to hit a given target location at a given velocity(backward ballistics).

5.8. Maxwell Stress Tensor

The Maxwell stress tensor provides a convenient way of computing forces acting on bodies by evaluating asurface integral. The Maxwell stress tensor is output in various ways (e.g., ETABLE and ESOL commands, andFMAGBC and FMAGSUM command macros.)

Following Vago and Gyimesi(239), this section derives the Maxwell stress tensor from Maxwell's equations(Equation 5–1 thru Equation 5–4). The derivation requires involved mathematical operations. Section 5.8.1:Notation summarizes the vector and tensor algebraic notations. The fundamental identities of vector and tensoranalysis are given in Section 5.8.2: Fundamental Relations. Using these identities, Equation 5–270 is derived inSection 5.8.3: Derived Relations. Section 5.8.4: Maxwell Tensor From Maxwell's Equations derives the Maxwellstress tensor from Maxwell's equations using Equation 5–270. The fundamental vector and tensor algebraicequations can be found in Flugge(240) and Legally(241).

5.8.1. Notation

This section summarizes the notations of vector and tensor algebraic notations used to derive the Maxwell stresstensor.

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Section 5.8: Maxwell Stress Tensor

where:

a = scalarA = vector[A] = 2nd order tensor1x, 1y, 1z = unit vectors

[1] = 2nd order unit tensor

A ⋅ B = dot product vectors resulting in a scalarA * B = cross product of vectors resulting in a vectorA @ B = dyadic product of vectors resulting in a 2nd order tensorgrad = gradient of a scalar resulting in a vectorcurl = rotation of a vector resulting in a scalardiv = divergence of a vector resulting in a scalargrad = gradient of a vector resulting in a tensor

grad u v w

ux

uy

uz

vx

vy

vz

wx

wy

w

x y z1 1 1+ + =

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂∂

z

div = divergence of a tensor resulting in a vector

div

a b c

d e f

g h i

=

div(a1 +b1 +c1 )

div(d1 +

x y z

x

ee1 +f1 )

div(g1 +h1 +i1 )

y z

x y z

5.8.2. Fundamental Relations

This section provides the fundamental identities of vector and tensor analysis. See Vago & Gyimesi(239),Flugge(240), and Lagally(241).

(5–256) * * A B C B A C A B= ⋅ − ⋅

(5–257)div a B adiv B B grada( ) = +

(5–258)div A B B curl A A curl B * = −

(5–259)curl a B acurl B grada B * = +

(5–260)

grad A B Grad A B Grad B A

A curl B B curl A

( ) [ ] [ ]

* *

⋅ = + ++

(5–261)

curl A B Grad A B Grad B A

A div B B div A

[ ] [ ]

⋅ = + ++

(5–262)Grad a B aGrad B B grada @= +

Chapter 5: Electromagnetics

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(5–263)Div A B A div B GradA B[ @ ] [ ] = +

from Equation 5–263:

(5–264)Div B A B div A GradB A[ @ ] [ ] = +

(5–265)Div a B aDiv B B grada[ [ ]] [ ] [ ]= +

5.8.3. Derived Relations

This section proves Equation 5–270 using the fundamental identities of vector and tensor analysis.

From Equation 5–265:

(5–266)Div A B grad A B grad A B[( )[ ]] [ ] ( ) ( )⋅ = ⋅ = ⋅1 1

From Equation 5–260, Equation 5–263, and Equation 5–264:

(5–267)

grad A B Div A B A div B

Div B A B div A

( ) [ @ ]

[ @ ]

⋅ = − +−

* * A curl B B curl A+

From Equation 5–266 and Equation 5–267:

(5–268)

Div A B B A A B

A div B B div A A cu

[ @ @ [ ]]

*

+ − ⋅ =+ −

1

rrl B curl A *

Substitute B = a A in Equation 5–268 and apply Equation 5–256, Equation 5–257, and Equation 5–259

(5–269)

Div A B A B

A div B B diva

B A c

[ @ [ ]]

*

2 1

1

− ⋅ =

+

− uurl a A B curl A

A div B Ba

div Ba

B grad a

*

⋅ =

+ − ⋅ 1 1

2

−

+ − =−

* * *

A a curl A grad a A B curl A

A div B A2 (( )

* * *

(

A grad a

B curl A A grad a A

A div B A

⋅ −− =

−

2

2 AA grad a B curl A

A A grad a A A grad a

A

) *

( )

⋅ − −⋅( ) + ⋅ =

2

2 ddiv B B curl A A A grad a * ( )− − ⋅2

After arrangement,

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(5–270)

a A curl A A div a AA A

grada

Div A a AA

* ( )

@ (

= − ⋅

− −

2⋅⋅

a A )[ ]

21

5.8.4. Maxwell Tensor From Maxwell's Equations

This section derives the Maxwell stress tensor from Maxwell's equations using Equation 5–16. Equation 5–20constitutes the fundamental relation of Maxwell stress tensor, and Equation 5–22 shows its application to eval-uate forces by a surface integral.

Maxwell's equations are described in Section 5.1: Electromagnetic Field Fundamentals. The definitions usedthere are the same as used here. Take the vector product of Equation 5–1 by B and Equation 5–2 by D andadd up these equations, providing:

(5–271)

* *

* *

D curl E B curl H

DdBdt

B JdDdt

+ =

−

+ +

=

− − * * J B d dt E Hµυ

In a homogeneous linear constitutive case, Equation 5–5 and Equation 5–13 simplify:

(5–272) B H= µ

(5–273) D E= υ

Combining Equation 5–270 and Equation 5–271 provides:

(5–274)

( )

[ ]

( )

E div DE E

grad Div T

H div BH H

grad

e− ⋅ −

+ − ⋅2

2

υ

µµ

µυ

− =

− −

Div T

J Bddt

E H

m[ ]

* *

After arrangement

(5–275)

Div T T f

E J BE E

grad

H Hgra

e m[[ ] [ ]]

* ( )

+ = =

+ − ⋅

⋅

ρ υ2

2dd

dSdt

µ µυ+

where:

[ ] @ ( )

[ ]T E DE D

e = − ⋅ =2

1 electric Maxwell stress tensor

Chapter 5: Electromagnetics

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[ ] @ ( )

[ ]T H BH B

m = − ⋅ =2

1 magnetic Maxwell stress tensor

f = force densityρ = charge densityS = Poynting vector

Equation 5–275 is the same as equations (1.255) and (1.257) in Vago and Gyimesi(239). The terms in Equation 5–275can be interpreted as:

ρ E + J * B = Lorentz force density

µυ dSdt

= light radiation pressure

grad terms = force due to material inhomogeneity

The force over a volume V:

(5–276) [[ ] [ ]]F f dV Div T T dVe m= = +∫ ∫

Applying Gauss theorem:

(5–277) [[ ] [ ]] F T T dSe m= + ⋅∫

i.e., the force is the surface integral of Maxwell stress tensor.

5.9. Electromechanical Transducers

For general circuit and reduced order modeling capabilities refer to Section 5.12: Circuit Analysis, Reduced OrderModeling. To obtain the capacitance of the transducer element one may either compute the capacitance usinga handbook formula, use CMATRIX (Section 5.10: Capacitance Computation), or another numerical package.

A review of electromechanical coupling methods and available transducers are given in:

Section 11.5: Review of Coupled Electromechanical MethodsSection 14.109: TRANS109 - 2-D Electromechanical TransducerSection 14.126: TRANS126 - Electromechanical Transducer

5.10. Capacitance Computation

Capacitance computation is one of the primary goals of an electrostatic analysis. For the definition of ground(partial) and lumped capacitance matrices see Vago and Gyimesi(239). The knowledge of capacitance is essentialin the design of electrostatic devices, Micro Electro Mechanical Systems (MEMS), transmission lines, printed circuitboards (PCB), electromagnetic interference and compatibility (EMI/EMC) etc. The computed capacitance can bethe input of a subsequent MEMS analysis by an electrostructural transducer element TRANS126; for theory seeSection 14.126: TRANS126 - Electromechanical Transducer.

For general circuit and reduced order modeling capabilities refer to Section 5.12: Circuit Analysis, Reduced OrderModeling. To obtain inductance and flux using the LMATRIX command macro see Section 5.6: Inductance, Fluxand Energy Computation by LMATRIX and SENERGY Macros.

The capacitance matrix of an electrostatic system can be computed (by the CMATRIX command macro). Thecapacitance calculation is based on the energy principle. For details see Gyimesi and Ostergaard(249) and its

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successful application Hieke(251). The energy principle constitutes the basis for inductance matrix computation,as shown in Section 5.6: Inductance, Flux and Energy Computation by LMATRIX and SENERGY Macros.

The electrostatic energy of a linear three electrode (the third is ground) system is:

(5–278)W C V C V C V Vg g g= + +1

21211 1

222 2

212 1 2

where:

W = electrostatic energyV1 = potential of first electrode with respect to ground

V2 = potential of second electrode with respect to ground

Cg11 = self ground capacitance of first electrode

Cg22 = self ground capacitance of second electrode

Cg12 = mutual ground capacitance between electrodes

By applying appropriate voltages on electrodes, the coefficients of the ground capacitance matrix can be calculatedfrom the stored static energy.

The charges on the conductors are:

(5–279)Q C V C Vg g

1 11 1 12 2= +

(5–280)Q C V C Vg g

2 12 1 22 2= +

where:

Q1 = charge of first electrode

Q2 = charge of second electrode

The charge can be expressed by potential differences, too:

(5–281)Q C V C V V1 11 1 12 1 2= + −l l ( )

(5–282)Q C V C V V2 22 2 12 2 1= + −l l ( )

where:

C11l = self lumped capacitance of first electrode

C22l = self lumped capacitance of second electrode

C12l = mutual lumped capacitance between electrode

The lumped capacitances can be realized by lumped capacitors as shown in Figure 5.14: “Lumped CapacitorModel of Two Conductors and Ground”. Lumped capacitances are suitable for use in circuit simulators.

Chapter 5: Electromagnetics

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Figure 5.14 Lumped Capacitor Model of Two Conductors and Ground

l

l!l

In some cases, one of the electrodes may be located very far from the other electrodes. This can be modeled asan open electrode problem with one electrode at infinity. The open boundary region can be modeled by infiniteelements, Trefftz method (see Section 5.11: Open Boundary Analysis with a Trefftz Domain) or simply closingthe FEM region far enough by an artificial Dirichlet boundary condition. In this case the ground key parameter(GRNDKEY on the CMATRIX command macro) should be activated. This key assumes that there is a groundelectrode at infinity.

The previous case should be distinguished from an open boundary problem without an electrode at infinity. Inthis case the ground electrode is one of the modeled electrodes. The FEM model size can be minimized in thiscase, too, by infinite elements or the Trefftz method. When performing the capacitance calculation, however,the ground key (GRNDKEY on the CMATRIX command macro) should not be activated since there is no electrodeat infinity.

Figure 5.15 Trefftz and Multiple Finite Element Domains

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3# 4'5-5'%-,76 8:9*6 8-6 (%2;=<')9>4'&:%-;

?$@A"B33-?C

?0)%=99(D28+','%-;3E"BFG. 3E"BFH1

The FEM region can be multiply connected. See for example Figure 5.15: “Trefftz and Multiple Finite ElementDomains”. The electrodes are far from each other: Meshing of the space between the electrodes would becomputationally expensive and highly ineffective. Instead, a small region is meshed around each electrode and

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the rest of the region is modeled by the Trefftz method (see Section 5.11: Open Boundary Analysis with a TrefftzDomain).

5.11. Open Boundary Analysis with a Trefftz Domain

The Trefftz method was introduced in 1926 by the founder of boundary element techniques, E. Trefftz(259, 260).The generation of Trefftz complete function systems was analyzed by Herrera(261). Zienkiewicz et al.(262), Ziel-inski and Zienkiewicz(263), Zienkiewicz et al.(264, 265, 266) exploited the energy property of the Trefftz methodby introducing the Generalized Finite Element Method with the marriage a la mode: best of both worlds (finiteand boundary elements) and successfully applied it to mechanical problems. Mayergoyz et al.(267), Chari(268),and Chari and Bedrosian(269) successfully applied the Trefftz method with analytic Trefftz functions to electro-magnetic problems. Gyimesi et al.(255), Gyimesi and Lavers(256), and Gyimesi and Lavers(257) introduced theTrefftz method with multiple multipole Trefftz functions to electromagnetic and acoustic problems. This lastapproach successfully preserves the FEM-like positive definite matrix structure of the Trefftz stiffness matrix whilemaking no restriction to the geometry (as opposed to analytic functions) and inheriting the excellent accuracyof multipole expansion.

Figure 5.16 Typical Hybrid FEM-Trefftz Domain

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Figure 5.16: “Typical Hybrid FEM-Trefftz Domain” shows a typical hybrid FEM-Trefftz domain. The FEM domainlies between the electrode and exterior surface. The Trefftz region lies outside the exterior surface. Within thefinite element domain, Trefftz multiple multipole sources are placed to describe the electrostatic field in theTrefftz region according to Green's representation theorem. The FEM domain can be multiply connected asshown in Figure 5.17: “Multiple FE Domains Connected by One Trefftz Domain”. There is minimal restriction re-garding the geometry of the exterior surface. The FEM domain should be convex (ignoring void region interiorto the model from conductors) and it should be far enough away so that a sufficiently thick cushion distributesthe singularities at the electrodes and the Trefftz sources.

The energy of the total system is

(5–283)W u K u w L wT T= +1

212

[ ] [ ]

where:

W = energy

Chapter 5: Electromagnetics

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u = vector of FEM DOFsw = vector of Trefftz DOFs[K] = FEM stiffness matrix[L] = Trefftz stiffness matrix

At the exterior surface, the potential continuity can be described by the following constraint equations:

(5–284)[ ] [ ] Q u P w+ = 0

where:

[Q] = FEM side of constraint equations[P] = Trefftz side of constraint equations

The continuity conditions are obtained by a Galerkin procedure. The conditional energy minimum can be foundby the Lagrangian multiplier's method. This minimization process provides the (weak) satisfaction of the governingdifferential equations and continuity of the normal derivative (natural Neumann boundary condition.)

To treat the Trefftz region, creates a superelement and using the constraint equations are created (using theTZEGEN command macro). The user needs to define only the Trefftz nodes (using the TZAMESH commandmacro).

Figure 5.17 Multiple FE Domains Connected by One Trefftz Domain

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&$'

5.12. Circuit Analysis, Reduced Order Modeling

In many analysis problems, a distributed finite element analysis is not practical; a reduced order with lumpedcircuit elements is more efficient or practical. ANSYS provides several circuit elements to exploit this simplification.An ANSYS model may be built using circuit elements only or, distributed finite elements and lumped circuitelements can be used together.

ANSYS has pure mechanical and electrical circuit elements as well as coupled field circuit elements. Coupledfield elements permit strong coupling between different physics domains.

5.12.1. Mechanical Circuit Elements

• COMBIN14: spring, dashpot

• MASS21: lumped mass

• COMBIN39: nonlinear spring

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Section 5.12: Circuit Analysis, Reduced Order Modeling

5.12.2. Electrical Circuit Elements

• CIRCU124: resistor, inductor, capacitor, sources, etc.

• CIRCU125: common diode, Zener diode

5.12.3. Coupled Field Circuit Elements

• TRANS126 : electromechanical transducer

• CIRCU124: voltage fed and circuit coupled magnetic field

The parameters of the lumped circuit elements may be obtained from an ANSYS field analysis using the commandmacros (CMATRIX and LMATRIX) described in Section 5.6: Inductance, Flux and Energy Computation byLMATRIX and SENERGY Macros and Section 5.10: Capacitance Computation of the ANSYS, Inc. Theory Reference.

For details on these elements, see the following sections of the ANSYS, Inc. Theory Reference: Section 5.4: VoltageForced and Circuit-Coupled Magnetic Field (Voltage Forced and Circuit Coupled Magnetic Field); Section 5.9:Electromechanical Transducers; Section 14.14: COMBIN14 - Spring-Damper; Section 14.21: MASS21 - StructuralMass; Section 14.39: COMBIN39 - Nonlinear Spring; Section 14.124: CIRCU124 - Electric Circuit; Section 14.125:CIRCU125 - Diode, and Section 14.126: TRANS126 - Electromechanical Transducer.

For a more detailed explanation and demonstration, refer to Demerdash and Arkadan(225), Demerdash andNehl(226), Nehl, Faud, et al.(227), Gyimesi and Ostergaard(229, 248, 286, 289, 290), Vago and Gyimesi (239), Heike,et al.(251), Ostergaard, Gyimesi, et al.(287), Gyimesi, Wang, et al.(288), and Hieke(291).

5.13. Conductance Computation

Conductance computation is one of the primary goals of an electrostatic analysis. For the definition of ground(partial) and lumped conductance matrices see Vago and Gyimesi(239). The knowledge of conductance is essentialin the design of electrostatic devices, Micro Electro Mechanical Systems (MEMS), transmission lines, printed circuitboards (PCB), electromagnetic interference and compatibility (EMI/EMC) etc. The computed conductance canbe the input of a subsequent MEMS analysis by an electrostructural transducer element TRANS126; for theorysee Section 14.126: TRANS126 - Electromechanical Transducer.

For general circuit and reduced order modeling capabilities refer to Section 5.12: Circuit Analysis, Reduced OrderModeling. To obtain inductance and flux using the LMATRIX command macro see Section 5.6: Inductance, Fluxand Energy Computation by LMATRIX and SENERGY Macros.

The conductance matrix of an electrostatic system can be computed (by the GMATRIX command macro). Theconductance calculation is based on the energy principle. For details see Gyimesi and Ostergaard(249) and itssuccessful application Hieke(251). The energy principle constitutes the basis for inductance matrix computation,as shown in Section 5.6: Inductance, Flux and Energy Computation by LMATRIX and SENERGY Macros.

The electrostatic energy of a linear three conductor (the third is ground) system is:

(5–285)W G V G V G V Vg g g= + +1

21211 1

222 2

212 1 2

where:

W = electrostatic energyV1 = potential of first conductor with respect to ground

V2 = potential of second conductor with respect to ground

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Gg11 = self ground conductance of first conductor

Gg22 = self ground conductance of second conductor

Gg12 = mutual ground conductance between conductors

By applying appropriate voltages on conductors, the coefficients of the ground conductance matrix can be cal-culated from the stored static energy.

The currents in the conductors are:

(5–286)I G V G Vg g1 11 1 12 2= +

(5–287)I G V G Vg g2 12 1 22 2= +

where:

I1 = current in first conductor

I2 = current in second conductor

The currents can be expressed by potential differences, too:

(5–288)I G V G V V1 11 1 12 1 2= + −l l ( )

(5–289)I G V G V V2 22 2 12 2 1= + −l l ( )

where:

G11 = self lumped capacitance of first conductor

G22 = self lumped capacitance of second conductor

G12 = mutual lumped capacitance between conductor

G11l = self lumped conductance of first conductor

G22l = self lumped conductance of second conductor

G12l = mutual lumped conductance between conductors

The lumped conductances can be realized by lumped conductors as shown in Figure 5.18: “Lumped CapacitorModel of Two Conductors and Ground”. Lumped conductances are suitable for use in circuit simulators.

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Section 5.13: Conductance Computation

Figure 5.18 Lumped Capacitor Model of Two Conductors and Ground

l

!l"l

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Chapter 6: Heat Flow

6.1. Heat Flow Fundamentals

6.1.1. Conduction and Convection

The first law of thermodynamics states that thermal energy is conserved. Specializing this to a differential controlvolume:

(6–1)ρc

Tt

v L T L q qT T∂∂

+

+ = &&&

where:

ρ = density (input as DENS on MP command)c = specific heat (input as C on MP command)T = temperature (=T(x,y,z,t))t = time

L

x

y

z

=

=

∂∂∂

∂∂

∂

vector operator

v

v

v

v

x

y

z

=

=velocity vector for mass transport of heat(input as VX, VY, VZ on command,PLANE55 and SOLID7

R00 only).

q = heat flux vector (output as TFX, TFY, and TFZ)&&&q = heat generation rate per unit volume (input on BF or BFE commands)

It should be realized that the terms LT and LTq may also be interpreted as ∇ T and ∇ ⋅ q, respectively,

where ∇ represents the grad operator and ∇ ⋅ represents the divergence operator.

Next, Fourier's law is used to relate the heat flux vector to the thermal gradients:

(6–2) [ ] q D L T= −

where:

[ ]D

K

K

K

xx

yy

zz

=

=0 0

0 0

0 0

conductivity matrix

Kxx, Kyy, Kzz = conductivity in the element x, y, and z directions, respectively (input as KXX, KYY, KZZ on MPcommand)

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Combining Equation 6–1 and Equation 6–2,

(6–3)ρc

Tt

v L T L D L T qT T∂∂

+

= + ([ ] ) &&&

Expanding Equation 6–3 to its more familiar form:

(6–4)

ρcTt

vTx

vTy

vTz

qx

KTx y

x y z

x

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

=

+ ∂∂

∂∂

+ ∂∂

&&& KKTy z

KTzy z

∂∂

+ ∂

∂∂∂

It will be assumed that all effects are in the global Cartesian system.

Three types of boundary conditions are considered. It is presumed that these cover the entire element.

1. Specified temperatures acting over surface S1:

(6–5)T T= *

where T* is the specified temperature (input on D command).

2. Specified heat flows acting over surface S2:

(6–6) q qT η = − ∗

where:

η = unit outward normal vectorq* = specified heat flow (input on SF or SFE commands)

3. Specified convection surfaces acting over surface S3 (Newton's law of cooling):

(6–7) ( )q h T TTf S Bη = −

where:

hf = film coefficient (input on SF or SFE commands) Evaluated at (TB + TS)/2 unless otherwise specified

for the elementTB = bulk temperature of the adjacent fluid (input on SF or SFE commands)

TS = temperature at the surface of the model

Note that positive specified heat flow is into the boundary (i.e., in the direction opposite of η), which accountsfor the negative signs in Equation 6–6 and Equation 6–7.

Combining Equation 6–2 with Equation 6–6 and Equation 6–7

(6–8) [ ] η T D L T q= ∗

(6–9) [ ] ( )η Tf BD L T h T T= −

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Premultiplying Equation 6–3 by a virtual change in temperature, integrating over the volume of the element,and combining with Equation 6–8 and Equation 6–9 with some manipulation yields:

(6–10)

ρ δ δc TTt

v L T L T D L T d volT Tvol

∂∂

+

+

=∫ ( )([ ] ) ( )

δδ δ δT q d S Th T T d S T qd volS f BS vol

∗∫ ∫ ∫+ − +( ) ( ) ( ) ( )2 32 3

&&&

where:

vol = volume of the elementδT = an allowable virtual temperature (=δT(x,y,z,t))

6.1.2. Radiation

Radiant energy exchange between neighboring surfaces of a region or between a region and its surroundingscan produce large effects in the overall heat transfer problem. Though the radiation effects generally enter theheat transfer problem only through the boundary conditions, the coupling is especially strong due to nonlineardependence of radiation on surface temperature.

Extending the Stefan-Boltzmann Law for a system of N enclosures, the energy balance for each surface in theenclosure for a gray diffuse body is given by Siegal and Howell(88(Equation 8-19)) , which relates the energylosses to the surface temperatures:

(6–11)

δε

εε

δ σji

iji

i

i ii

i

N

ji ji ii

NF

AQ F T− −

= −

= =∑ ∑1 1

1

4

1( )

where:

N = number of radiating surfacesδji = Kronecker delta

εi = effective emissivity (input on EMIS or MP command) of surface i

Fji = radiation view factors (see below)

Ai = area of surface i

Qi = energy loss of surface i

σ = Stefan-Boltzmann constant (input on STEF or R command)Ti = absolute temperature of surface i

For a system of two surfaces radiating to each other, Equation 6–11 can be simplified to give the heat transferrate between surfaces i and j as (see Chapman(356)):

(6–12)

Q

A A F A

T Tii

i i i ij

j

j j

i j=− + +

−

−1

1 1 14 4

εε

εε

σ( )

where:

Ti, Tj = absolute temperature at surface i and j, respectively

If Aj is much greater than Ai, Equation 6–12 reduces to:

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Section 6.1: Heat Flow Fundamentals

(6–13)Q A F T Ti i i ij i j= −ε σ’ ( )4 4

where:

FFij

Fijij i i

’

( )=

− +1 ε ε

6.1.2.1. View Factors

The view factor, Fij, is defined as the fraction of total radiant energy that leaves surface i which arrives directly

on surface j, as shown in Figure 6.1: “View Factor Calculation Terms”. It can be expressed by the following equation:

Figure 6.1 View Factor Calculation Terms

(6–14)F

A rd A d Aij

i

i jjA iA ji

= ∫∫1

2

cos cos( ) ( )

θ θ

π

where:

Ai,Aj = area of surface i and surface j

r = distance between differential surfaces i and jθi = angle between Ni and the radius line to surface d(Aj)

θj = angle between Nj and the radius line to surface d(Ai)

Ni,Nj = surface normal of d(Ai) and d(Aj)

6.1.2.2. Radiation Usage

Four methods for analysis of radiation problems are included:

1. Radiation link element LINK31(Section 14.31: LINK31 - Radiation Link). For simple problems involvingradiation between two points or several pairs of points. The effective radiating surface area, the formfactor and emissivity can be specified as real constants for each radiating point.

2. Surface effect elements - SURF151 in 2-D and SURF152 in 3-D for radiating between a surface and a point(Section 14.151: SURF151 - 2-D Thermal Surface Effect and Section 14.152: SURF152 - 3-D Thermal Surface

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Effect ). The form factor between a surface and the point can be specified as a real constant or can becalculated from the basic element orientation and the extra node location.

3. Radiation matrix method (Section 6.4: Radiation Matrix Method). For more generalized radiation problemsinvolving two or more surfaces. The method involves generating a matrix of view factors between radi-ating surfaces and using the matrix as a superelement in the thermal analysis.

4. Radiosity solver method (Section 6.5: Radiosity Solution Method). For generalized problems in 3-D in-volving two or more surfaces. The method involves calculating the view factor for the flagged radiatingsurfaces using the hemicube method and then solving the radiosity matrix coupled with the conductionproblem.

6.2. Derivation of Heat Flow Matrices

As stated before, the variable T was allowed to vary in both space and time. This dependency is separated as:

(6–15)T N TTe=

where:

T = T(x,y,z,t) = temperatureN = N(x,y,z) = element shape functionsTe = Te(t) = nodal temperature vector of element

Thus, the time derivatives of Equation 6–15 may be written as:

(6–16)&T

Tt

N TTe= ∂

∂=

δT has the same form as T:

(6–17)δ δT T NeT=

The combination LT is written as

(6–18) [ ] L T B Te=

where:

[B] = LNT

Now, the variational statement of Equation 6–10 can be combined with Equation 6–15 thru Equation 6–18 toyield:

(6–19)

ρ δ ρ δc T N N T d vol c T N v B T d voleT T

evol eT T

e ( ) [ ] (&∫ + ))

[ ] [ ][ ] ( ) ( )

vol

eT T

evol eT

ST B D B T d vol T N q d S

∫∫ ∫+ =

+

∗δ δ 22

( ) ( ) ( )δ δT N h T N T d S T N qd voleT

f BT

eS eT

vol− +∫ ∫3

3&&&

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Section 6.2: Derivation of Heat Flow Matrices

Terms are defined in Section 6.1: Heat Flow Fundamentals. ρ is assumed to remain constant over the volume of

the element. On the other hand, c and &&&q may vary over the element. Finally, Te, &Te , and δTe are nodal

quantities and do not vary over the element, so that they also may be removed from the integral. Now, since allquantities are seen to be premultiplied by the arbitrary vector δTe, this term may be dropped from the resulting

equation. Thus, Equation 6–19 may be reduced to:

(6–20)

ρ ρ

ρ

c N N d vol T c N v B d vol T

B

Tevol

Tevol

T

( ) [ ] ( )

[ ] [

&∫ ∫+

+ DD B d vol T N q d S

T h N d S h N N

vol e S

B fS f

][ ] ( ) ( )

( )

∫ ∫∫

= +

−

∗2

3

2

3 ( ) ( )T

eS volT d S q N d vol3

3∫ ∫+ &&&

Equation 6–20 may be rewritten as:

(6–21)[ ] ([ ] [ ] [ ]) C T K K K T Q Q Qet

e etm

etb

etc

e e ec

eg& + + + = + +

where:

[ ] ( )C c N N d volet

volT= =∫ρ element specific heat (thermal daamping) matrix

[ ] [ ] ( )K c N v B d voletm

volT= =∫ρ element mass transport conducctivity matrix

[ ] [ ] [ ][ ] ( )K B D B d voletb

vol

T= =∫ element diffusion conductivity matrix

[ ] ( )K h N N d Setc

fST= =∫

33 element convection surface conducttivity matrix

* ( )Q N q d Sef

S= =∫ 2

2element mass flux vector

( )Q T h N d Sec

B fS= =∫ 3

3element convection surface heat flow vector

( )Q q N d voleg

vol= =∫ &&& element heat generation load

Comments on and modifications of the above definitions:

1. [ ]Ketm

is not symmetric.

2. [ ]Ketc

is calculated as defined above, for SOLID90 only. All other elements use a diagonal matrix, with

the diagonal terms defined by the vector h N d SfS3 3∫ ( )

.

3. [ ]Cet

is frequently diagonalized, as described in Section 13.2: Lumped Matrices.

4.If [ ]Ce

t exists and has been diagonalized and also the analysis is a transient (Key = ON on the TIMINT

command), Qeg

has its terms adjusted so that they are proportioned to the main diagonal terms of

[ ]Cet

. Qej

, the heat generation rate vector for Joule heating is treated similarly, if present. This adjust-

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ment ensures that elements subjected to uniform heating will have a uniform temperature rise. However,this adjustment also changes nonuniform input of heat generation to an average value over the element.

5.For phase change problems, [ ]Ce

t is evaluated from the enthalpy curve (Tamma and Namnuru(42)) if

enthalpy is input (input as ENTH on MP command). This option should be used for phase change problems.

6.3. Heat Flow Evaluations

6.3.1. Integration Point Output

The element thermal gradients at the integration points are:

(6–22) a L TTx

Ty

Tz

T

= = ∂∂

∂∂

∂∂

where:

a = thermal gradient vector (output as TG)L = vector operatorT = temperature

Using shape functions, Equation 6–22 may be written as:

(6–23) [ ] a B Te=

where:

[B] = shape function derivative matrix evaluated at the integration pointsTe = nodal temperature vector of element

Then, the heat flux vector at the integration points may be computed from the thermal gradients:

(6–24) [ ] [ ][ ] q D a D B Te= − = −

where:

q = heat flux vector (output as TF)[D] = conductivity matrix (see Equation 6–2)

Nodal gradient and flux vectors may be computed from the integration point values as described in Section 13.6:Nodal and Centroidal Data Evaluation.

6.3.2. Surface Output

The convection surface output is:

(6–25)q h T Tcf S B= −( )

where:

qc = heat flow per unit area due to convection

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Section 6.3: Heat Flow Evaluations

hf = film coefficient (input on SF or SFE commands)

TS = temperature at surface of model

TB = bulk temperature of the adjacent fluid (input on SF or SFE commands)

6.4. Radiation Matrix Method

In the radiation matrix method, for a system of two radiating surfaces, Equation 6–13 can be expanded as:

(6–26)Q F A T T T T T Ti i ij i i j i j i j= + + −σ ε ( )( )( )2 2

or

(6–27)Q K T Ti i j= −′( )

where:

′ = + +K F A T T T Ti ij i i j i jσ ε ( )( )2 2

K' cannot be calculated directly since it is a function of the unknowns Ti and Tj. The temperatures from previous

iterations are used to calculate K' and the solution is computed iteratively.

For a more general case, Equation 6–11 can be used to construct a single row in the following matrix equation:

(6–28)[ ] [ ] C Q D T= 4

such that:

(6–29)each row j in C F

Ai Nji

iji

i

i i[ ]= − −

= …

δε

εε

1 11 2, ,

(6–30)each row j in [D] = − = …( ) , ,δ σji jiF i N1 2

Solving for Q:

(6–31) [ ] Q K Tts= 4

and therefore:

(6–32)[ ] [ ] [ ]K C Dts = −1

Equation 6–31 is analogous to Equation 6–11 and can be set up for standard matrix equation solution by theprocess similar to the steps shown in Equation 6–26 and Equation 6–27.

(6–33) [ ] Q K T= ′

[K'] now includes T3 terms and is calculated in the same manner as in Equation 6–27). To be able to include radiationeffects in elements other than LINK31, MATRIX50 (the substructure element) is used to bring in the radiation

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matrix. MATRIX50 has an option that instructs the solution phase to calculate [K']. The AUX12 utility is used to

create the substructure radiation matrix. AUX12 calculates the effective conductivity matrix, [Kts], in Equation 6–31,

as well as the view factors required for finding [Kts]. The user defines flat surfaces to be used in AUX12 by over-laying nodes and elements on the radiating edge of a 2-D model or the radiating face of a 3-D model.

Two methods are available in the radiation matrix method to calculate the view factors (VTYPE command), thenon-hidden method and the hidden method.

6.4.1. Non-Hidden Method

The non-hidden procedure calculates a view factor for every surface to every other surface whether the view isblocked by an element or not. In this procedure, the following equation is used and the integration is performedadaptively.

For a finite element discretized model, Equation 6–14 for the view factor Fij between two surfaces i and j can be

written as:

(6–34)F

A

cos cos

rA Aij

i

ip jqip jq

q

n

p

m=

==∑∑1

211

θ θ

π

where:

m = number of integration points on surface in = number of integration points on surface j

When the dimensionless distance between two viewing surfaces D, defined in Equation 6–35, is less than 0.1,the accuracy of computed view factors is known to be poor (Siegal and Howell(88)).

(6–35)D

d

Amin

max=

where:

dmin = minimum distance between the viewing surfaces A1 and A2

Amax = max (A1, A2)

So, the order of surface integration is adaptively increased from order one to higher orders as the value of D fallsbelow 8. The area integration is changed to contour integration when D becomes less than 0.5 to maintain theaccuracy. The contour integration order is adaptively increased as D approaches zero.

6.4.2. Hidden Method

The hidden procedure is a simplified method which uses Equation 6–14 and assumes that all the variables areconstant, so that the equation becomes:

(6–36)F

A

rcos cosij

ji j=

πθ θ

2

The hidden procedure numerically calculates the view factor in the following conceptual manner. The hidden-line algorithm is first used to determine which surfaces are visible to every other surface. Then, each radiating,or “viewing”, surface (i) is enclosed with a hemisphere of unit radius. This hemisphere is oriented in a local co-

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Section 6.4: Radiation Matrix Method

ordinate system (x' y' z'), whose center is at the centroid of the surface with the z axis normal to the surface, thex axis is from node I to node J, and the y axis orthogonal to the other axes. The receiving, or “viewed”, surface (j)is projected onto the hemisphere exactly as it would appear to an observer on surface i.

As shown in Figure 6.2: “Receiving Surface Projection”, the projected area is defined by first extending a line fromthe center of the hemisphere to each node defining the surface or element. That node is then projected to thepoint where the line intersects the hemisphere and transformed into the local system x' y' z', as described inKreyszig(23)

Figure 6.2 Receiving Surface Projection

The view factor, Fij, is determined by counting the number of rays striking the projected surface j and dividing

by the total number of rays (Nr) emitted by surface i. This method may violate the radiation reciprocity rule, that

is, AiFi-j ≠ Aj Fj-i.

6.4.3. View Factors of Axisymmetric Bodies

When the radiation view factors between the surfaces of axisymmetric bodies are computed (GEOM,1,n command),special logic is used. In this logic, the axisymmetric nature of the body is exploited to reduce the amount ofcomputations. The user, therefore, needs only to build a model in plane 2-D representing the axisymmetricbodies as line “elements”.

Consider two axisymmetric bodies A and B as shown in Figure 6.3: “Axisymmetric Geometry”.

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Figure 6.3 Axisymmetric Geometry

The view factor of body A to body B is computed by expanding the line “element” model into a full 3-D modelof n circumferential segments (GEOM,1,n command) as shown in Figure 6.4: “End View of Showing n = 8 Seg-ments”.

Figure 6.4 End View of Showing n = 8 Segments

View factor of body A to B is given by

(6–37)F Fk

n

k

n= −

==∑∑ ll 11

where:

Fk - l = view factor of segment k on body A to segment l on body B

The form factors between the segments of the axisymmetric bodies are computed using the method describedin the previous section. Since the coefficients are symmetric, the summation Equation 6–37 may be simplifiedas:

(6–38)F n F

n= −

=∑ 1

1l

l

Both hidden and non-hidden methods are applicable in the computation of axisymmetric view factors. However,the non-hidden method should be used if and only if there are no blocking surfaces. For example, if radiationbetween concentric cylinders are considered, the outer cylinder can not see part of itself without obstruction

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Section 6.4: Radiation Matrix Method

from the inner cylinder. For this case, the hidden method must be used, as the non-hidden method would def-initely give rise to inaccurate view factor calculations.

6.4.4. Space Node

A space node may be defined (SPACE command) to absorb all energy not radiated to other elements. Any radiantenergy not incident on any other part of the model will be directed to the space node. If the model is not a closedsystem, then the user must define a space node with its appropriate boundary conditions.

6.5. Radiosity Solution Method

In the radiosity solution method for the analysis of gray diffuse radiation between N surfaces, Equation 6–11 issolved in conjunction with the basic conduction problem.

For the purpose of computation it is convenient to rearrange Equation 6–11 into the following series of equations

(6–39)δ ε ε σij i ij j

oi i

j

NF q T− − =

=∑ ( )1 4

1

and

(6–40)q q F qi i

oij j

o

j

N= −

=∑

1

Equation 6–39 and Equation 6–40 are expressed in terms of the outgoing radiative fluxes (radiosity) for each

surface, q j

o

, and the net flux from each surface qi. For known surface temperatures, Ti, in the enclosure, Equa-

tion 6–40 forms a set of linear algebraic equations for the unknown, outgoing radiative flux (radiosity) at eachsurface. Equation 6–40 can be written as

(6–41)[ ] A q Do =

where:

A Fij ij i ij= − −δ ε( )1

qjo = radiosity flux for surface i

D Ti i i= ε σ 4

[A] is a full matrix due to the surface to surface coupling represented by the view factors and is a function oftemperature due to the possible dependence of surface emissivities on temperature. Equation 6–41 is solved

using a Newton-Raphson procedure for the radiosity flux qo.

When the qo values are available, Equation 6–40 then allows the net flux at each surface to be evaluated. Thenet flux calculated during each iteration cycle is under-relaxed, before being updated using

(6–42)q q qi

netik

ik= + −+φ φ1 1( )

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where:

φ = radiosity flux relaxation factork = iteration number

The net surface fluxes provide boundary conditions to the finite element model for the conduction process. Theradiosity Equation 6–41 is solved coupled with the conduction Equation 6–11 using a segregated solution pro-cedure until convergence of the radiosity flux and temperature for each time step or load step.

The surface temperatures used in the above computation must be uniform over each surface in order to satisfyconditions of the radiation model. In the finite element model, each surface in the radiation problem correspondsto a face or edge of a finite element. The uniform surface temperatures needed for use in Equation 6–41 are ob-tained by averaging the nodal point temperatures on the appropriate element face.

For open enclosure problems using the radiosity method, an ambient temperature needs to be specified usinga space temperature (SPCTEMP command) or a space node (SPCNOD command), to account for energy balancebetween the radiating surfaces and the ambient.

6.5.1. View Factor Calculation - Hemicube Method

For solution of radiation problems in 3-D, the radiosity method calculates the view factors using the hemicubemethod as compared to the traditional double area integration method for 3-D geometry. Details using theHemicube method for view factor calculation are given in Glass(272) and Cohen and Greenberg(276).

The hemicube method is based upon Nusselt's hemisphere analogy. Nusselt's analogy shows that any surface,which covers the same area on the hemisphere, has the same view factor. From this it is evident that any inter-mediate surface geometry can be used without changing the value of the view factors. In the hemicube method,instead of projecting onto a sphere, an imaginary cube is constructed around the center of the receiving patch.A patch in a finite element model corresponds to an element face of a radiating surface in an enclosure. Theenvironment is transformed to set the center of the patch at the origin with the normal to the patch coincidingwith the positive Z axis. In this orientation, the imaginary cube is the upper half of the surface of a cube, the lowerhalf being below the 'horizon' of the patch. One full face is facing in the Z direction and four half faces are facingin the +X, -X, +Y, and -Y directions. These faces are divided into square 'pixels' at a given resolution, and the en-vironment is then projected onto the five planar surfaces. Figure 6.5: “The Hemicube” shows the hemicube dis-cretized over a receiving patch from the environment.

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Section 6.5: Radiosity Solution Method

Figure 6.5 The Hemicube

Figure 6.6 Derivation of Delta-View Factors for Hemicube Method

The contribution of each pixel on the cube's surface to the form-factor value varies and is dependent on thepixel location and orientation as shown in Figure 6.6: “Derivation of Delta-View Factors for Hemicube Method ”.A specific delta form-factor value for each pixel on the cube is found from modified form of Equation 6–14 forthe differential area to differential area form-factor. If two patches project on the same pixel on the cube, a depthdetermination is made as to which patch is seen in that particular direction by comparing distances to eachpatch and selecting the nearer one. After determining which patch (j) is visible at each pixel on the hemicube,a summation of the delta form-factors for each pixel occupied by patch (j) determines the form-factor from patch(i) at the center of the cube to patch (j). This summation is performed for each patch (j) and a complete row ofN form-factors is found.

Chapter 6: Heat Flow

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At this point the hemicube is positioned around the center of another patch and the process is repeated for eachpatch in the environment. The result is a complete set of form-factors for complex environments containingoccluded surfaces. The overall view factor for each surface on the hemicube is given by:

(6–43)F F

cos cos

rAij n

n

Ni j

j= ==

∑ ∆ ∆1

2

φ φ

π

where:

N = number of pixels∆F = delta-view factor for each pixel

The hemicube resolution (input on the HEMIOPT command) determines the accuracy of the view factor calcu-lation and the speed at which they are calculated using the hemicube method. Default is set to 10. Higher valuesincrease accuracy of the view factor calculation.

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Section 6.5: Radiosity Solution Method

6–16

Chapter 7: Fluid Flow

7.1. Fluid Flow Fundamentals

This chapter discusses the FLOTRAN solution method used with elements FLUID141 and FLUID142. These elementsare used for the calculation of 2-D and 3-D velocity and pressure distributions in a single phase, Newtonian fluid.Thermal effects, if present, can be modeled as well.

The fluid flow problem is defined by the laws of conservation of mass, momentum, and energy. These laws areexpressed in terms of partial differential equations which are discretized with a finite element based technique.

Assumptions about the fluid and the analysis are as follows:

1. There is only one phase.

2. The user must determine: (a) if the problem is laminar (default) or turbulent; (b) if the incompressible(default) or the compressible algorithm must be invoked.

7.1.1. Continuity Equation

From the law of conservation of mass law comes the continuity equation:

(7–1)∂∂

+ ∂∂

+∂

∂+ ∂

∂=ρ ρ ρ ρ

tvx

v

yvz

x y z( ) ( ) ( )0

where:

vx, vy and vz = components of the velocity vector in the x, y and z directions, respectively

ρ = density (see Section 7.6.1: Density)x, y, z = global Cartesian coordinatest = time

The rate of change of density can be replaced by the rate of change of pressure and the rate at which densitychanges with pressure:

(7–2)∂∂

= ∂∂

∂∂

ρ ρt P

Pt

where:

P = pressure

The evaluation of the derivative of the density with respect to pressure comes from the equation of state. If thecompressible algorithm is used, an ideal gas is assumed:

(7–3)ρ ρ= ⇒ ∂

∂=P

RT P RT1

where:

R = gas constantT = temperature

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If the incompressible solution algorithm is used (the default), the user can control the specification of the valuewith:

(7–4)ddP

ρβ

= 1

where:

β = bulk modulus (input on the FLDATA16 command)

The default value of 1015 for β implies that for a perfectly incompressible fluid, pressure waves will travel infinitelyfast throughout the entire problem domain, e.g. a change in mass flow will be seen downstream immediately .

7.1.2. Momentum Equation

In a Newtonian fluid, the relationship between the stress and rate of deformation of the fluid (in index notation)is:

(7–5)τ δ µ δ λij ij

i

j

j

iij

i

iP

ux

u

xux

= − + ∂∂

+∂∂

+ ∂

∂

where:

tij = stress tensor

ui = orthogonal velocities (u1 = vx, u2 = vy, u3 = vz)

µ = dynamic viscosityλ = second coefficient of viscosity

The final term, the product of the second coefficient of viscosity and the divergence of the velocity, is zero for aconstant density fluid and is considered small enough to neglect in a compressible fluid.

Equation 7–5 transforms the momentum equations to the Navier-Stokes equations; however, these will still bereferred to as the momentum equations elsewhere in this chapter.

The momentum equations, without further assumptions regarding the properties, are as follows:

(7–6)

∂∂

+ ∂∂

+∂

∂+ ∂

∂= − ∂

∂

+ + ∂∂

∂

ρ ρ ρ ρ ρ

µ

vt

v vx

v v

yv vz

gPx

Rx

x x x y x z xx

x e

( ) ( ) ( )

vvx y

vy z

vz

Txe

xe

xx∂

+ ∂∂

∂∂

+ ∂

∂∂∂

+µ µ

(7–7)

∂∂

+∂

∂+

∂∂

+∂

∂= − ∂

∂

+ + ∂∂

∂

ρ ρ ρ ρρ

µ

v

t

v v

x

v v

y

v v

zg

Py

Rx

y x y y y z yy

y e

( ) ( ) ( )

vv

x y

v

y z

v

zTy

ey

ey

y∂

+ ∂

∂∂∂

+ ∂

∂∂∂

+µ µ

Chapter 7: Fluid Flow

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(7–8)

∂∂

+ ∂∂

+∂

∂+ ∂

∂= − ∂

∂

+ + ∂∂

∂

ρ ρ ρ ρ ρ

µ

vt

v vx

v v

yv vz

gPz

Rx

z x z y z z zz

z e

( ) ( ) ( )

vvx y

vy z

vz

Tze

ze

zz∂

+ ∂∂

∂∂

+ ∂

∂∂∂

+µ µ

where:

gx, gy, gz = components of acceleration due to gravity (input on ACEL command)

ρ = density (input as described in Section 7.6: Fluid Properties)µe = effective viscosity (discussed below)

Rx, Ry, Rz = distributed resistances (discussed below)

Tx, Ty, Tz = viscous loss terms (discussed below)

For a laminar case, the effective viscosity is merely the dynamic viscosity, a fluid property (input as described inSection 7.6: Fluid Properties). The effective viscosity for the turbulence model is described later in this section.

The terms Rx, Ry Rz represent any source terms the user may wish to add. An example is distributed resistance,

used to model the effect of some geometric feature without modeling its geometry. Examples of this includeflow through screens and porous media.

The terms Tx, Ty Tz are viscous loss terms which are eliminated in the incompressible, constant property case.

The order of the differentiation is reversed in each term, reducing the term to a derivative of the continuityequation, which is zero.

(7–9)T

xvx y

v

x zvxx

x y z= ∂∂

∂∂

+ ∂∂

∂∂

+ ∂

∂∂∂

µ µ µ

(7–10)T

xvy y

v

y zvyy

x y z= ∂∂

∂∂

+ ∂

∂∂∂

+ ∂

∂∂∂

µ µ µ

(7–11)T

xvz y

v

z zvzz

x y z= ∂∂

∂∂

+ ∂∂

∂∂

+ ∂

∂∂∂

µ µ µ

The conservation of energy can be expressed in terms of the stagnation (total) temperature, often useful in highlycompressible flows, or the static temperature, appropriate for low speed incompressible analyses.

7.1.3. Compressible Energy Equation

The complete energy equation is solved in the compressible case with heat transfer (using the FLDATA1 com-mand).

In terms of the total (or stagnation) temperature, the energy equation is:

(7–12)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

=

∂∂

∂∂

tC T

xv C T

yv C T

zv C T

xK

T

p o x p o y p o z p o

o

( ) ( ) ( ) ( )ρ ρ ρ ρ

xx yK

Ty z

KTz

W E QPt

o o v kv

+ ∂∂

∂∂

+ ∂

∂∂∂

+ + + + + ∂∂

Φ

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Section 7.1: Fluid Flow Fundamentals

where:

Cp = specific heat (input with FLDATA8 command for fluid, MP command for non-fluid element)

To = total (or stagnation) temperature (input and output as TTOT)

K = thermal conductivity (input with FLDATA8 command for fluid, MP command for non-fluid element)

Wv = viscous work termQv = volumetric heat source (input with BFE or BF command)

Φ = viscous heat generation term

Ek = kinetic energy (defined later)

The static temperature is calculated from the total temperature from the kinetic energy:

(7–13)T T

vCo

p= −

2

2

where:

T = static temperature (output as TEMP)v = magnitude of the fluid velocity vector

The static and total temperatures for the non-fluid nodes will be the same.

The Wv, Ek and Φ terms are described next.

The viscous work term using tensor notation is:

(7–14)W u

x

u

x xux

vj

i

j

i k

k

j= ∂

∂∂∂

+ ∂∂

∂∂

µ

where the repetition of a subscript implies a summation over the three orthogonal directions.

The kinetic energy term is

(7–15)E

xKC x

vy

KC y

vk

p p= − ∂

∂∂

∂

− ∂∂

∂∂

12

12

2 2

− ∂

∂∂∂

z

KC z

vp

12

2

Finally, the viscous dissipation term in tensor notation is

(7–16)Φ = ∂

∂+ ∂

∂

∂∂

µ ux

ux

ux

i

k

k

i

i

k

In the absence of heat transfer (i.e., the adiabatic compressible case), Equation 7–13 is used to calculate thestatic temperature from the total temperature specified (with the FLDATA14 command).

7.1.4. Incompressible Energy Equation

The energy equation for the incompressible case may be derived from the one for the compressible case by

neglecting the viscous work (Wv), the pressure work, viscous dissipation (f), and the kinetic energy (Ek). As the

Chapter 7: Fluid Flow

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kinetic energy is neglected, the static temperature (T) and the total temperature (To) are the same. The energy

equation now takes the form of a thermal transport equation for the static temperature:

(7–17)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

= ∂∂

∂∂

tC T

xv C T

yv C T

zv C T

xK

Tx

p x p y p z p( ) ( ) ( ) ( )ρ ρ ρ ρ

+ ∂

∂∂∂

+ ∂

∂∂∂

+y

KTy z

KTz

Qv

7.1.5. Turbulence

If inertial effects are great enough with respect to viscous effects, the flow may be turbulent. The user is responsiblefor deciding whether or not the flow is turbulent (using the FLDATA1 command). Turbulence means that theinstantaneous velocity is fluctuating at every point in the flow field. The velocity is thus expressed in terms of amean value and a fluctuating component:

(7–18)v v vx x x= + ′

where:

vx = mean component of velocity in x-direction

vx′

= fluctuating component of velocity in x-direction

If an expression such as this is used for the instantaneous velocity in the Navier-Stokes equations, the equationsmay then be time averaged, noting that the time average of the fluctuating component is zero, and the timeaverage of the instantaneous value is the average value. The time interval for the integration is arbitrarily chosenas long enough for this to be true and short enough so that “real time” transient effects do not affect this integ-ration.

(7–19)1

01

0 0δ δ

δ δ

tx

tx xv dt v dt v

t t′∫ ∫= =;

After the substitution of Equation 7–18 into the momentum equations, the time averaging leads to additionalterms. The velocities in the momentum equations are the averaged ones, and we drop the bar in the subsequentexpression of the momentum equations, so that the absence of a bar now means the mean value. The extraterms are:

(7–20)σ ρ ρ ρx

Rx x x y x zx

v vy

v vz

v v= − ∂∂

− ∂∂

− ∂∂

′ ′ ′ ′ ′ ′( ) ( ) ( )

(7–21)σ ρ ρ ρy

Ry x y y y zx

v vy

v vz

v v= − ∂∂

− ∂∂

− ∂∂

′ ′ ′ ′ ′ ′( ) ( ) ( )

(7–22)σ ρ ρ ρz

Rz x z y z zx

v vy

v vz

v v= − ∂∂

− ∂∂

− ∂∂

′ ′ ′ ′ ′ ′( ) ( ) ( )

where:

σR = Reynolds stress terms

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Section 7.1: Fluid Flow Fundamentals

In the eddy viscosity approach to turbulence modeling one puts these terms into the form of a viscous stressterm with an unknown coefficient, the turbulent viscosity. For example:

(7–23)− = ∂

∂ρ µv v

vyx y tx

The main advantage of this strategy comes from the observation that the representation of σR is of exactly the

same form as that of the diffusion terms in the original equations. The two terms can be combined if an effectiveviscosity is defined as the sum of the laminar viscosity and the turbulent viscosity:

(7–24)µ µ µe t= +

The solution to the turbulence problem then revolves around the solution of the turbulent viscosity.

Note that neither the Reynolds stress nor turbulent heat flux terms contain a fluctuating density because of theapplication of Favre averaging to Equation 7–20 to Equation 7–22. Bilger(187) gives an excellent description ofFavre averaging. Basically this technique weights each term by the mean density to create a Favre averagedvalue for variable φ which does not contain a fluctuating density:

(7–25)%φ ρφ

ρ≡

The tilde indicates the Favre averaged variable. For brevity, reference is made to Bilger(187) for further details.

There are eight turbulence models available in FLOTRAN (selected with the FLDATA24 command). The modelacronyms and names are as follows:

• Standard k-ε Model

• Zero Equation Model

• RNG - (Re-normalized Group Model)

• NKE - (New k-ε Model due to Shih)

• GIR - (Model due to Girimaji)

• SZL - (Shi, Zhu, Lumley Model)

• Standard k-ω Model

• SST - (Shear Stress Transport Model)

The simplest model is the Zero Equation Model, and the other five models are the two equation standard k-εmodel and four extensions of it. The final two models are the Standard k-ω Model and SST model. In the ZeroEquation Model, the turbulent viscosity is calculated as:

(7–26)µ ρt sL= 2 Φ

where:

µt = turbulent viscosity

Φ = viscous dissipation (Equation 7–16)

Chapter 7: Fluid Flow

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L

L L

L

Ls

x x

n

c

=>

≤

if

if Lminimum x

0 0

4

090 0

.

.

..

Lx = length scale (input on FLDATA24 command)

Ln = shortest distance from the node to the closest wall

Lc = characteristic length scale (largest value of Ln encountered)

In the k-ε model and its extensions, the turbulent viscosity is calculated as a function of the turbulence parameterskinetic energy k and its dissipation rate ε using Equation 7–27. In the RNG and standard models, Cµ is constant,

while it varies in the other models.

(7–27)µ ρεµt C

k=2

where:

Cµ = turbulence constant (input on FLDATA24 command)

k = turbulent kinetic energy (input/output as ENKE)ε = turbulent kinetic energy dissipation rate (input/output as ENDS)

In the k-ω model and SST model, the turbulent viscosity is calculated as:

(7–28)µ ρ

ωtk=

Here ω is defined as:

(7–29)ω ε

µ=

C k

where:

ω = specific dissipation rate

The k-ε model and its extensions entail solving partial differential equations for turbulent kinetic energy and itsdissipation rate whereas the k-ω and SST models entail solving partial differential equations for the turbulentkinetic energy and the specific dissipation rate. The equations below are for the standard k-ε model. The differentcalculations for the other k-ε models will be discussed in turn. The basic equations are as follows:

7.1.5.1. Standard k-ε Model

The reader is referred to Spalding and Launder(178) for details.

The Turbulent Kinetic Energy equation is:

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Section 7.1: Fluid Flow Fundamentals

(7–30)

∂∂

+ ∂∂

+∂

∂+ ∂

∂

= ∂∂

∂∂

+ ∂

∂

ρ ρ ρ ρ

µσ

µ

kt

v kx

v k

yv kz

xkx y

x y z

t

k

( ) ( ) ( )

tt

k

t

k

tt

tx y

ky z

kz

Cg

Tx

gT

σµσ

µ ρεβµ

σ

∂∂

+ ∂

∂∂∂

+ − + ∂∂

+ ∂Φ 4∂∂

+ ∂∂

y

gTzz

The Dissipation Rate equation is:

(7–31)

∂∂

+ ∂∂

+∂

∂+ ∂

∂

= ∂∂

∂∂

+ ∂

∂

ρε ρ ε ρ ε ρ ε

µσ

ε

ε

tvx

v

yvz

x x

x y z

t

( ) ( ) ( )

yy y z z

Ck

Ck

C C

t t

t

µσ

ε µσ

ε

µ ε ρ ε

ε ε

εµ

∂∂

+ ∂

∂∂∂

+ − +−

1 2

2 31Φ

( ))βρ

σ

kg

Tx

gTy

gTzt

x y z∂∂

+ ∂∂

+ ∂∂

The final term in each equation are terms used to model the effect of buoyancy and are described by Viollet(177).Default values for the various constants in the standard model are provided by Lauder and Spalding(178) andare given in Table 7.1: “Standard Model Coefficients”.

Table 7.1 Standard Model Coefficients

CommandDefaultValue

(FLDATA24,TURB,C1,Value)1.44C1, C1ε

(FLDATA24,TURB,C2,Value)1.92C2

(FLDATA24,TURB,CMU,Value)0.09Cµ

(FLDATA24,TURB,SCTK,Value)1.0σk

(FLDATA24,TURB,SCTD,Value)1.3σε

(FLDATA24,TURB,SCTT,Value)0.85σt

(FLDATA24,TURB,BUC3,Value)1.0C3

(FLDATA24,TURB,BUC4,Value)0.0C4

(FLDATA24,TURB,BETA,Value)0.0β

The solution to the turbulence equations is used to calculate the effective viscosity and the effective thermalconductivity:

(7–32)µ µ ρεe C

k= + ∝2

(7–33)K K

Ce

t p

t= +

µσ

where:

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µe = effective viscosity

Ke = effective conductivity

σt = Turbulent Prandtl (Schmidt) Number

The four extensions to the standard k-ε model have changes in either the Cµ term or in the source term of the

dissipation equation. The new functions utilize two invariants constructed from the symmetric deformationtensor Sij, and the antisymmetric rotation tensor Wij. These are based on the velocity components vk in the flow

field.

(7–34)S v vij i j j i= +1

2( ), ,

(7–35)W v v Cij i j j i r m mij= − +1

2( ), , Ω ε

where:

Cr = constant depending on turbulence model used

Ωm = angular velocity of the coordinate system

εmij = alternating tensor operator

The invariants are:

(7–36)η

ε= k

S Sij ij2

and

(7–37)ζ

ε= k

W Wij ij2

7.1.5.2. RNG Turbulence Model

In the RNG model, the constant C1ε in the dissipation Equation 7–31, is replaced by a function of one of the in-

variants.

(7–38)C1 31 42

1

1ε

η ηη

βη= −

−

+∞.

Table 7.2 RNG Model Coefficients

CommandDefaultValue

(FLDATA24A,RNGT,BETA,Value)0.12β ∞

(FLDATA24A,RNGT,C2,Value)1.68C2

(FLDATA24A,RNGT,CMU,Value)0.085Cµ

(FLDATA24A,RNGT,SCTK,Value)0.72σk

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Section 7.1: Fluid Flow Fundamentals

CommandDefaultValue

(FLDATA24A,RNGT,SCTD,Value)0.72σε

(FLDATA24A,RNGT,ETAI,Value)4.38η ∞

In the RNG model a constant Cµ is used. The value is specified with a separate command than the one used to

specify the Cµ in the standard model. The same is true of the constant C2. As shown in the above table, the diffusion

multipliers have different values than the default model, and these parameters also have their own commandsfor the RNG model. The value of the rotational constant Cr in the RNG model is 0.0. Quantities in Equation 7–31

not specified in Table 7.2: “RNG Model Coefficients” are covered by Table 7.1: “Standard Model Coefficients”.

7.1.5.3. NKE Turbulence Model

The NKE Turbulence model uses both a variable Cµ term and a new dissipation source term.

The Cµ function used by the NKE model is a function of the invariants.

(7–39)Cµ

η ζ=

+ +

1

4 1 5 2 2.

The production term for dissipation takes on a different form. From Equation 7–31, the production term for thestandard model is:

(7–40)C

kt1εµ ε Φ

The NKE model replaces this with:

(7–41)ρ εεC S Sij ij1 2

The constant in the dissipation rate Equation 7–31 is modified in the NKE model to be:

(7–42)C max C M1 1 5ε

ηη

=+

The constant C2 in the dissipation Equation 7–31 of the NKE model has a different value than that for the corres-

ponding term in the standard model. Also, the values for the diffusion multipliers are different. Commands areprovided for these variables to distinguish them from the standard model parameters. So for the NKE model,the input parameters are as follows:

Table 7.3 NKE Turbulence Model Coefficients

CommandDefaultValue

(FLDATA24B,NKET,C1MX,Value)0.43C1M

(FLDATA24B,NKET,C2,Value)1.90C2

(FLDATA24B,NKET,SCTK,Value)1.0σk

(FLDATA24B,NKET,SCTD,Value)1.2σε

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The value of the rotational constant Cr in the NKE model is 3.0. All parameters in Equation 7–30 and Equation 7–31

not covered by this table are covered in Table 7.1: “Standard Model Coefficients”

7.1.5.4. GIR Turbulence Model

The Girimaji model relies on a complex function for the calculation of the Cµ coefficient. The coefficients in

Table 7.4: “GIR Turbulence Model Coefficients” are used.

Table 7.4 GIR Turbulence Model Coefficients

CommandDefaultValue

(FLDATA24C,GIRT,G0,Value)3.6C10

(FLDATA24C,GIRT,G1,Value)0.0C11

(FLDATA24C,GIRT,G2,Value)0.8C2

(FLDATA24C,GIRT,G3,Value)1.94C3

(FLDATA24C,GIRT,G4,Value)1.16C4

These input values are used in a series of calculations as follows

First of all, the coefficients L10

to L4 have to be determined from the input coefficients. Note, these coefficients

are also needed for the coefficients of the nonlinear terms of this model, which will be discussed later.

(7–43)LC

L C LC

LC

LC

10 1

0

11

11

22

33

44

21 1

223 2

12

1= − = + = − = − = −; ; ; ;

Secondly, the following coefficients have to be calculated:

(7–44)

pL

Lr

L L

L

arccosb

a

q

= − =

= −

−

=

212

12

2

27

1

12

10

211

10

2

211

2 3η η

; ; Θ

ηη

η η ζ2

11

2 10 2 2

11

22

32 2

421

213

L

L L L L L

a qp

+ − +

= −

( ) ( ) ( )

223

2 3

31

272 9 27

4 27; ( );b p pq r D

b a= − + = +

With these coefficients we can now determine the coefficient Cµ from the following set of equations:

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Section 7.1: Fluid Flow Fundamentals

(7–45)C

L L L L L L

p

µ

η ζ η

= −

− +

= =

− +

10

2 10 2

32 2

42

111

30

3

( ) ( ) ( ) if 0 or

−− +

+ − +

− + −

bD

bD

p acos

2 2

32

3 3

1 3 1 3/ /

if D>0

if Θ

DD b

p acos D b

< <

− + − +

< >

0 0

32

3 323

0 0

,

if , Θ π

and for the GIR model, the rotational term constant Cr is

(7–46)C

CCr = −

−4

4

42

7.1.5.5. SZL Turbulence Model

The Shi-Zhu-Lemley turbulence model uses a simple expression for the Cµ coefficient and uses the standard

dissipation source terms.

The user controls three constants in the calculation of the coefficients:

(7–47)C

AA A

s

s sµ η ζ

=+ +

1

2 3

The constants and their defaults are as follows:

Table 7.5 SZL Turbulence Model Coefficients

CommandDefaultValue

(FLDATA24D,SZLT,SZL1,Value)0.66666As1

(FLDATA24D,SZLT,SZL2,Value)1.25As2

(FLDATA24D,SZLT,SZL3,Value)0.90As3

The value of the rotational constant Cr for the SZL model is 4.0.

7.1.5.6. Standard k-ω Model

The k-ω model solves for the turbulent kinetic energy k and the specific dissipation rate ω (Wilcox(349)). As inthe k-ε based turbulence models, the quantity k represents the exact kinetic energy of turbulence. The otherquantity ω represents the ratio of the turbulent dissipation rate ε to the turbulent kinetic energy k, i.e., is the rateof dissipation of turbulence per unit energy (see Equation 7–29).

The turbulent kinetic energy equation is:

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(7–48)

∂∂

+ ∂∂

+∂

∂+ ∂

∂

= ∂∂

+ ∂∂

+ ∂

∂+

ρ ρ ρ ρ

µ µσ

µ

kt

V kx

V k

yV kz

xkx y

x y z

t

k( ) (

µµσ

µ µσ

µ ρ ωβµ

σµ

t

k

t

k

tt

kx

ky z

kz

C kC

g

) ( )∂∂

+ ∂

∂+ ∂

∂

+ − +Φ 4 ∂∂∂

+ ∂∂

+ ∂∂

Tx

gTy

gTzy z

The specific dissipation rate equation is:

(7–49)

∂∂

+ ∂∂

+∂

∂+ ∂

∂

= ∂∂

+ ∂∂

+ ∂

∂+

ρω ρ ω ρ ω ρ ω

µµσ

ω µω

tVx

V

yVz

x x y

x y z

t( ) (µµσ

ω µµσ

ω

γρ β ρωβρ

σ

ω ω

t ty z z

C

) ( )

( )

∂∂

+ ∂

∂+ ∂

∂

+ − ′ +−

Φ 2 31

ttx y zg

Tx

gTy

gTz

∂∂

+ ∂∂

+ ∂∂

The final term in Equation 7–48 and Equation 7–49 is derived from the standard k-ε model to model the effectof buoyancy. Default values for the model constants in the k-ω model are provided by Wilcox(349). Some valuesare the same with the standard k-ε model and are thus given in Table 7.1: “Standard Model Coefficients”,whereas the other values are given in Table 7.6: “The k-ω Model Coefficients”.

Table 7.6 The k-ω Model Coefficients

CommandDefaultValue

(FLDATA24E,SKWT,SCTK,Value)2.0σk

(FLDATA24E,SKWT,SCTW,Value)2.0σω

(FLDATA24E,SKWT,BUC3,Value)0.5555γ

(FLDATA24E,SKWT,BETA,Value)0.075′β

The k-ω model has the advantage near the walls to predict the turbulence length scale accurately in the presenceof adverse pressure gradient, but it suffers from strong sensitivity to the free-stream turbulence levels. Its deficiencyaway from the walls can be overcome by switching to the k-ε model away from the walls with the use of the SSTmodel.

7.1.5.7. SST Turbulence Model

The SST turbulence model combines advantages of both the standard k-ε model and the k-ω model. As comparedto the turbulence equations in the k-ω model, the SST model first modifies the turbulence production term inthe turbulent kinetic energy equation. From Equation 7–48, the production term from the k-ω model is:

(7–50)Pt t= µ Φ

The SST model replaces it with:

(7–51)P Ct t lmt= min( , )µ εΦ

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By default, the limiting value of Clmt is set to 1015, so Equation 7–51 is essentially the same with Equation 7–50.

However, Equation 7–51 allows the SST model to eliminate the excessive build-up of turbulence in stagnationregions for some flow problems with the use of a moderate value of Clmt.

Further, the SST model adds a new dissipation source term in the specific dissipation rate equation:

(7–52)( )1 21 2− ∂

∂∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

F kx x

ky y

kz z

ρσω

ω ω ωω

Here, F1 is a blending function that is one near the wall surface and zero far away from the wall. The expression

of the bending function F1 is given by Menter(350), and with the help of F1, the SST model automatically switches

to the k-ω model in the near region and the k-ε model away from the walls. The model coefficients are all calculatedas functions of F1:

(7–53)ϕ ϕ ϕ= + −F F1 1 1 21( )

Here, φ stands for the model coefficient (σk, σω, ′β , γ) of the SST model, and φ1 and φ2 stand for the model coef-

ficient of the k-ω model and the k-ε model respectively. Default values for the various constants in the SSTmodel are provided by Menter(350), and are given in Table 7.7: “The SST Model Coefficients”.

Table 7.7 The SST Model Coefficients

CommandDefaultValue

(FLDATA24F,SST1,CLMT,Value)1015Clmt

(FLDATA24G,SST1,SCTK,Value)1.176σk1

(FLDATA24G,SST1,SCTW,Value)2.0σω1

(FLDATA24G,SST1,GAMA,Value)0.5532γ1

(FLDATA24G,SST1,BETA,Value)0.075′β1

(FLDATA24H,SST2,SCTK,Value)1.0σk2

(FLDATA24H,SST2,SCTW,Value)1.168σω2

(FLDATA24H,SST2,GAMA,Value)0.4403γ2

(FLDATA24EH,SST2,BETA,Value)0.0828′β2

7.1.5.8. Near-Wall Treatment

All of the above turbulence models except the Zero Equation Model use the near-wall treatment discussed here.The near-wall treatment for the k-ω model and SST model are slightly different from the following discussions.Refer to Wilcox (349) and Menter (350) for differences for those two models.

The k-ε models are not valid immediately adjacent to the walls. A wall turbulence model is used for the wallelements. Given the current value of the velocity parallel to the wall at a certain distance from the wall, an ap-proximate iterative solution is obtained for the wall shear stress. The equation is known as the “Log-Law of theWall” and is discussed in White(181) and Launder and Spalding(178).

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(7–54)

vln

Etan

τρ

κδ

ντρ

=

1

where:

vtan = velocity parallel to the wall

τ = shear stressν = kinematic viscosity (m/r)κ = slope parameter of law of the wall (FLDATA24,TURB,KAPP,Value)E = law of the wall constant (FLDATA24,TURB,EWLL,Value)δ = distance from the wall

The default values of κ and E are 0.4 and 9.0 respectively, the latter corresponding to a smooth wall condition.

From the shear stress comes the calculation of a viscosity:

(7–55)µ δ τ

wtanv

=

The wall element viscosity value is the larger of the laminar viscosity and that calculated from Equation 7–55.

Near wall values of the turbulent kinetic energy are obtained from the k-ε model. The near wall value of the dis-sipation rate is dominated by the length scale and is given by Equation 7–56.

(7–56)εκδ

µnw

nwC k=

(. ) ( . )75 1 5

where:

εnw = near wall dissipation rate

knw = near wall kinetic energy

The user may elect to use an alternative wall formulation (accessed with the FLDATA24,TURB,WALL,EQLB com-mand) directly based on the equality of turbulence production and dissipation. This condition leads to the fol-lowing expression for the wall parameter y+ (see White(181) for more background):

(7–57)yC knw+ = µ ρ δ

µ

1 4 1 2

The wall element effective viscosity and thermal conductivity are then based directly on the value of y+. The

laminar sublayer extends to yt+

(input on the FLDATA24,TURB,TRAN command) with the default being 11.5.

For y+ < yt+

:

(7–58)

µ µeff

effK K

==

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For y+ ≥ yt+

:

(7–59)µ µ

κ

effy

n Ey=

+

+1 l ( )

(7–60)K

C y

nEy Peff

p

tfn

=+

+

+σµ

κ1 l

where:

l n = natural logarithm

Psin

A Pr Prfn

t t=

−

( )( )

// /

ππ κ σ σ4

41

1 2 1 4

Pr = Prandtl number

Although the wall treatment should not affect the laminar solution, the shear stress calculation is part of the wallalgorithm. Thus, shear stresses from the equilibrium model will differ slightly from those obtained from the defaulttreatment, as described in Equation 7–54 thru Equation 7–56.

7.1.6. Pressure

For numerical accuracy reasons, the algorithm solves for a relative pressure rather than an absolute pressure.

Considering the possibility that the equations are solved in a rotating coordinate system, the defining expressionfor the relative pressure is:

(7–61)P P P g r r rabs ref rel o o= + − ⋅ + × × ⋅ρ ρ ω ω ( )

12

where:

ρo = reference density (calculated from the equation of state defined by the property type using the nominal

temperature (input using FLDATA14 command))Pref = reference pressure (input using FLDATA15 command)

g = acceleration vector due to gravity (input using ACEL command)Pabs = absolute pressure

Prel = relative pressure

r = position vector of the fluid particle with respect to the rotating coordinate systemω = constant angular velocity vector of the coordinate system (input using CGOMGA command)

Combining the momentum equations (Equation 7–6 through Equation 7–8) into vector form and again consid-ering a rotating coordinate system, the result is:

(7–62)

ρ ρ ω ρ ω ω

ρ µ

D vDt

v r

g P vabs

+ × + × ×

= − ∇ + ∇

2

2

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where:

v = vector velocity in the rotating coordinate systemµ = fluid viscosity (assumed constant for simplicity)ρ = fluid density

In the absence of rotation, v is simply the velocity vector in the global coordinate system.

The negative of the gradient of the absolute pressure is:

(7–63)−∇ = −∇ − + × ×P P g rabs rel o oρ ρ ω ω

Inserting this expression into the vector form of the momentum equation puts it in terms of the relative pressureand the density differences.

(7–64)

ρ ρ ω ρ ρ ω ω

ρ ρ µ

D vDt

v r

g P v

o

o rel

( )

( )

+ × + − × ×

= − − ∇ + ∇

2

2

This form has the desirable feature (from a numerical precision standpoint) of expressing the forcing functiondue to gravity and the centrifugal acceleration in terms of density differences.

For convenience, the relative pressure output is that measured in the stationary global coordinate system. Thatis, the rotational terms are subtracted from the pressure calculated by the algorithm.

Conversely, the total pressure is output in terms of the rotating coordinate system frame. This is done for theconvenience of those working in turbomachinery applications.

7.1.7. Multiple Species Transport

Several different fluids, each with different properties, are tracked if the multiple species option is invoked (withthe FLDATA1 command).

A single momentum equation is solved for the flow field. The properties for this equation are calculated fromthose of the species fluids and their respective mass fractions if the user specifies the composite gas option(FLDATA7,PROT,DENS,CGAS) for density or the composite mixture option (FLDATA7,PROT,DENS,CMIX). CGASonly applies for density, but CMIX applies to density, viscosity or conductivity. If these options are not invoked,the species fluids are carried by a bulk fluid, with the momentum equation solved with properties of a singlefluid.

The governing equations for species transport are the mass balance equations for each of the species.

For i = 1, . . . , n-1 (where n is the number of species)

(7–65)∂

∂+ ∇ ⋅ − ∇ ⋅ ∇ =( )

( ) ( )ρ ρ ρYt

Yv D Yii mi i 0

where:

Yi = mass fraction for the ith species

ρ = bulk density (mass/length3)v = velocity vector (length/time)

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Section 7.1: Fluid Flow Fundamentals

Dmi = mass diffusion coefficient (length2/time) (input on MSPROP command)

The equation for the nth species, selected by the user as the “algebraic species”, is not solved directly. The massfraction for the nth species is calculated at each node from the identity:

(7–66)Y YN i

i

n= −

=

−∑1

1

1

The diffusion information available for the species fluid is sometimes cast in terms of a Schmidt number for aspecies (not to be confused with the turbulent Schmidt number). The relationship between the Schmidt numberand the mass diffusion coefficient is as follows:

(7–67)Sc

Dimi

= µρ

In the above expression, the density and the viscosity are those of the bulk carrier fluid, or the “average” propertiesof the flow.

As with the general “bulk” momentum equation, the effect of turbulence is to increase the diffusion and ismodeled with an eddy viscosity approach. First note that the laminar diffusion term can be cast in terms of the“laminar” Schmidt number associated with the species diffusion:

(7–68)∇ ⋅ ∇ = ∇ ⋅ ∇

( )ρ µ

D YSc

Ymi ii

i

In the presence of turbulence, an additional term is added:

(7–69)∇ ⋅ ∇

→ ∇ ⋅ +

∇

µ µ µSc

YSc Sc

Yi

ii

t

Tii

where:

µt = turbulent viscosity (from the turbulence model)

ScTi = turbulent Schmidt number (input on MSSPEC command)

7.1.8. Arbitrary Lagrangian-Eulerian (ALE) Formulation

The equations of motion described in the previous sections were based on an Eulerian (fixed) frame of reference.The governing equations may also be formulated in a Lagrangian frame of reference, i.e. the reference framemoves with the fluid particles. Both formulations have their advantages and disadvantages. With the Eulerianframework it is not straightforward to solve problems involving moving boundaries or deforming domains. Whilesuch problems are more suitable for a Lagrangian framework, in practice the mesh distortions can be quite severeleading to mesh entanglement and other inaccuracies. A pragmatic way around this problem is to move themesh independent of the fluid particles in such a way as to minimize the distortions. This is the ALE formulationwhich involves moving the mesh nodal points in some heuristic fashion so as to track the boundary motion/domaindeformation and at the same time minimizing the mesh degradation.

The Eulerian equations of motion described in the previous sections need to be modified to reflect the movingframe of reference. Essentially the time derivative terms need to be rewritten in terms of the moving frame ofreference.

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(7–70)

∂∂

= ∂∂

− ⋅ ∇φ φ φt t

wfixed frame moving frame

uru

where:

φ = any degree of freedom

wur

= velocity of the moving frame of reference

For example, Equation 7–65 is rewritten as:

(7–71)

∂∂

− ⋅ ∇ + ∇ ⋅ + ∇ ⋅ ∇ =( )( ) ( ) ( )

ρ ρ ρ ρYt

w Y Y v D Yii i mi i

moving frame

uru r0

A complete and detailed description of the ALE formulation may be found in Huerta and Liu(278).

Note that a steady state solution in an Eulerian sense requires,

(7–72)

∂∂

=φt fixed frame

0

In order to have the same interpretation of a steady solution in an ALE formulation we require that,

(7–73)

∂∂

= − ⋅ ∇ =φ φt

wmoving frame

uru0

In practice, this can be achieved for the following two cases:

(7–74)

∂∂

= =φt

wmoving frame

0 0,uru r

(7–75)φ = constant

7.2. Derivation of Fluid Flow Matrices

A segregated, sequential solution algorithm is used. This means that element matrices are formed, assembledand the resulting system solved for each degree of freedom separately. Development of the matrices proceedsin two parts. In the first, the form of the equations is achieved and an approach taken towards evaluating all theterms. Next, the segregated solution algorithm is outlined and the element matrices are developed from theequations.

7.2.1. Discretization of Equations

The momentum, energy, species transport, and turbulence equations all have the form of a scalar transportequation. There are four types of terms: transient, advection, diffusion, and source. For the purposes of describingthe discretization methods, let us refer to the variable considered as φ. The form of the scalar transport equationis:

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Section 7.2: Derivation of Fluid Flow Matrices

(7–76)

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

=

∂∂

∂∂

tC

xv C

yv C

zv C

x x

x y z( ) ( ) ( ) ( )ρ φ ρ φ ρ φ ρ φ

φ

φ φ φ φ

φΓ

+ ∂∂

∂∂

+ ∂

∂∂∂

+y y z z

SΓ Γφ φ φφ φ

where:

Cφ = transient and advection coefficient

Γφ = diffusion coefficient

Sφ = source terms

Table 7.8: “Transport Equation Representation” below shows what the variables, coefficients, and source termsare for the transport equations. The pressure equation is derived using the continuity equation. Its form will unfoldduring the discussion of the segregated solver. The terms are defined in the previous section.

Since the approach is the same for each equation, only the generic transport equation need be treated. Each ofthe four types of terms will be outlined in turn. Since the complete derivation of the discretization method wouldrequire too much space, the methods will be outlined and the reader referred to more detailed expositions onthe subjects.

Table 7.8 Transport Equation Representation

SφΓφCφDOFMeaningφ

ρg p x Rx x− ∂ ∂ +/µe1VXx-velocityvx

ρg p y Ry y− ∂ ∂ +/µe1VYy-velocityvy

ρg p z Rz z− ∂ ∂ +/µe1VZz-velocityvz

µ µ ρε βµ σt t i i tC g T xΦ / ( / )− + ∂ ∂4KCpTEMPtemperatureT

Q E W p tvk v+ + + + ∂ ∂µΦ /µt/σk1ENKEkinematic energyk

C k C k

C C C kg T xt

i i t

1 22

1 3

µ ε ρε

β σµ

Φ / /

( / ) /

− +

∂ ∂

µt/σε1ENDSdissipation rateε

0ρ Dmi1SP01-06species mass fractionYi

The discretization process, therefore, consists of deriving the element matrices to put together the matrixequation:

(7–77)([ ] [ ] [ ]) A A A Setransient

eadvection

ediffusion

e e+ + =φ φ

Galerkin's method of weighted residuals is used to form the element integrals. Denote by We the weightingfunction for the element, which is also the shape function.

7.2.2. Transient Term

The first of the element matrix contributions is from the transient term. The general form is simply:

Chapter 7: Fluid Flow

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(7–78)[ ]( )

( )A WC

td vole

transient ee

=∂

∂∫ρ φφ

For node i:

(7–79)WC

td vol W C W d vol

tW

C

tWi

e eje j

ee

je

i i

∂∂

=∂

∂+

∂∂∫ ∫ ∫

( )( ) ( )

( )ρ φρ

φ ρφφ

φ dd vol je( )φ

Subscripts i and j indicate the node number. If the second part in Equation 7–79 is neglected, the consistentmass matrix can be expressed as:

(7–80)M W C W d volij i

eje= ∫ ρ φ ( )

If a lumped mass approximation is used (accessed with the FLDATA38 command for fluid, and the MSMASScommand for multiple species).

(7–81)M W C d volij ij i

e= ∫δ ρ φ ( )

where:

δij = Kronecker delta (0 if i ≠ j, 1 if i = j)

There are two time integration methods available (selected on the FLDATA4 command): Newmark and backwarddifference. If the Newmark time integration method is selected, the following nodal basis implicit formulationis used. The current time step is the nth time step and the expression involves the previous one time step results.

(7–82)( ) ( )

( )( )

( )ρφ ρφ δ ρφ δ ρφn n

tt tn n

= + ∆ ∂∂

+ − ∂∂

−

−1

11

where:

δ = time integration coefficient for the Newmark method (input on the FLDATA4 command).

Equation 7–82 can be rewritten as:

(7–83)∂

∂

=∆

−∆

+ − ∂∂

−

−

( )( ) ( ) ( )

( )ρφδ

ρφδ

ρφδ

ρφt t t tn n

n n

1 11

11

11

If the backward difference method is selected, the following nodal basis implicit formulation is used. The currenttime step is the nth time step and the expression involves the previous two time step results.

(7–84)∂

∂= − +− −( ) ( ) ( ) ( )ρφ ρφ ρφ ρφ

t t t tn n n2 1

24

23

2∆ ∆ ∆

For a Volume of Fluid (VOF) analysis, the above equation is modified as only the results at one previous time stepare needed:

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(7–85)∂

∂= − −( ) ( ) ( )ρφ ρφ ρφ

t t tn n

∆ ∆1

The above first-order time difference scheme is chosen to be consistent with the current VOF advection algorithm.

The nth time step produces a contribution to the diagonal of the element matrix, while the derivatives from theprevious time step form contributions to the source term.

7.2.3. Advection Term

Currently FLOTRAN has three approaches to discretize the advection term. The monotone streamline upwind(MSU) approach is first order accurate and tends to produce smooth and monotone solutions. The streamlineupwind/Petro-Galerkin (SUPG) and the collocated Galerkin (COLG) approaches are second order accurate andtend to produce oscillatory solutions.

7.2.4. Monotone Streamline Upwind Approach (MSU)

The advection term is handled through a monotone streamline approach based on the idea that pure advectiontransport is along characteristic lines. It is useful to think of the advection transport formulation in terms of aquantity being transported in a known velocity field. See Figure 7.1: “Streamline Upwind Approach”.

Figure 7.1 Streamline Upwind Approach

!

"#

$&% '

$&(

The velocity field itself can be envisioned as a set of streamlines everywhere tangent to the velocity vectors. Theadvection terms can therefore be expressed in terms of the streamline velocities.

In pure advection transport, one assumes that no transfer occurs across characteristic lines, i.e. all transfer occursalong streamlines. Therefore one may assume that the advection term,

(7–86)

∂∂

+∂

∂+

∂∂

=∂

∂( ) ( ) ( ) ( )ρ φ ρ φ ρ φ ρ φφ φ φ φC v

x

C v

y

C v

z

C v

sx y z s

when expressed along a streamline, is constant throughout an element:

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(7–87)[ ]( )

( )Ad C v

dsW d vole

advection s e= ∫ρ φφ

This formulation is made for every element, each of which will have only one node which gets contributionsfrom inside the element. The derivative is calculated using a simple difference:

(7–88)d C v

ds

C v C v

Dss s U s D( ) ( ) ( )ρ ρ φ ρ φφ φ φ=

−

where:

D = subscript for value at the downstream nodeU = subscript for value taken at the location at which the streamline through the downwind node enters theelement∆s = distance from the upstream point to the downstream node

The value at the upstream location is unknown but can be expressed in terms of the unknown nodal values it isbetween. See Figure 7.1: “Streamline Upwind Approach” again.

The process consists of cycling through all the elements and identifying the downwind nodes. A calculation ismade based on the velocities to see where the streamline through the downwind node came from. Weightingfactors are calculated based on the proximity of the upwind location to the neighboring nodes.

Consult Rice and Schnipke(179) for more details .

7.2.5. Streamline Upwind/Petro-Galerkin Approach (SUPG)

The SUPG approach consists of a Galerkin discretization of the advection term and an additional diffusion-likeperturbation term which acts only in the advection direction.

(7–89)

[ ]( ) ( ) ( )

A Wv C

x

v C

y

v C

zeadvection e x y z=

∂∂

+∂

∂+

∂∂

ρ φ ρ φ ρ φφ φ φ

+

∂∂

+∂∂

+ ∂∂

∫

∫

d vol

Czh

Uv W

x

v W

yv W

z

v

mag

xe

ye

ze

x

( )

2 2τ

∂∂∂

+∂

∂+

∂∂

( ) ( ) ( )( )

ρ φ ρ φ ρ φφ φ φC

x

v C

y

v C

zd voly z

where:

C2τ = global coefficient set to 1.0

h = element length along advection direction

U v v vmag x y z= + +2 2 2

zPe

Pe Pe=

≤ <≥

1 0 3

3 3

if

if

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PeC U hmag= =

ρ φ

φ2ΓPeclet number

It is clear from the SUPG approach that as the mesh is refined, the perturbation terms goes to zero and theGalerkin formulation approaches second order accuracy. The perturbation term provides the necessary stabilitywhich is missing in the pure Galerkin discretization. Consult Brooks and Hughes(224) for more details.

7.2.6. Collocated Galerkin Approach (COLG)

The COLG approach uses the same discretization scheme with the SUPG approach with a collocated concept.In this scheme, a second set of velocities, namely, the element-based nodal velocities are introduced. The element-based nodal velocities are made to satisfy the continuity equation, whereas the traditional velocities are madeto satisfy the momentum equations.

(7–90)

[ ]( ) ( ) ( )

A Wv C

x

v C

y

v C

zeadvection e x

eye

ze

=∂

∂+

∂∂

+∂

∂

ρ φ ρ φ ρ φφ φ φ

+

∂∂

+∂∂

+ ∂∂

∫ d vol

Czh

U

v Wx

v W

yv W

zmage

xe e

ye e

ze e

( )

22

τ

∂∂

+∂

∂+

∂∂

∫

v C

x

v C

y

v C

zd vox

eye

ze( ) ( ) ( )

(ρ φ ρ φ ρ φφ φ φ ll)

Where all the parameters are defined similar to those in the SUPG approach.

In this approach, the pressure equation is derived from the element-based nodal velocities, and it is generallyasymmetric even for incompressible flow problems. The collocated Galerkin approach is formulated in such away that, for steady-state incompressible flows, exact conservation is preserved even on coarse meshes uponthe convergence of the overall system.

7.2.7. Diffusion Terms

The expression for the diffusion terms comes from an integration over the problem domain after the multiplicationby the weighting function.

(7–91)

Diffusion contribution = ∂∂

∂∂

+ ∂∂

∂∂∫ W

x xd vol W

ye eΓ Γφ φ

φ φ( )

yyd vol

Wz z

d vole

∂∂

∂∂

∫

∫

( )

( )Γφφ

The x, y and z terms are all treated in similar fashion. Therefore, the illustration is with the term in the x direction.An integration by parts is applied:

(7–92)W

x xd vol

Wx x

d volee∂

∂∂∂

= ∂∂

∂∂∫ ∫Γ Γφ φ

φ φ( ) ( )

Once the derivative of φ is replaced by the nodal values and the derivatives of the weighting function, the nodalvalues will be removed from the integrals

Chapter 7: Fluid Flow

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.7–24

(7–93)∂∂

=φ φx

Wxe

(7–94)WWxx

ee

= ∂∂

The diffusion matrix may now be expressed as:

(7–95)[ ] ( )A W W W W W W d vole

diffusionxe

xe

ye

ye

ze

ze= + +∫ Γ Γ Γφ φ φ

7.2.8. Source Terms

The evaluation of the source terms consists of merely multiplying the source terms as depicted in Fig-ure 7.1: “Streamline Upwind Approach” by the weighting function and integrating over the volume.

(7–96)S W S d vole e

φ φ= ∫ ( )

7.2.9. Segregated Solution Algorithm

Each degree of freedom is solved in sequential fashion. The equations are coupled, so that each equation issolved with intermediate values of the other degrees of freedom. The process of solving all the equations in turnand then updating the properties is called a global iteration. Before showing the entire global iteration structure,it is necessary to see how each equation is formed.

The preceding section outlined the approach for every equation except the pressure equation, which comesfrom the segregated velocity-pressure solution algorithm. In this approach, the momentum equation is used togenerate an expression for the velocity in terms of the pressure gradient. This is used in the continuity equationafter it has been integrated by parts. This nonlinear solution procedure used in FLOTRAN belongs to a generalclass of Semi-Implicit Method for Pressure Linked Equations (SIMPLE). There are currently two segregated solutionalgorithms available. One is the original SIMPLEF algorithm, and the other is the enhanced SIMPLEN algorithm.

The incompressible algorithm is a special case of the compressible algorithm. The change in the product ofdensity and velocity from iteration to the next is approximating by considering the changes separately througha linearization process. Denoting by the superscript * values from the previous iteration, in the x direction, forexample, results:

(7–97)ρ ρ ρ ρv v v vx x x x= + −∗ ∗ ∗ ∗

The continuity equation becomes:

(7–98)

∂∂

+ ∂∂

+ ∂∂

+∂

∂+

∂∂

+

∂∂

+ ∂

∗ ∗

∗

∗ ∗ρ ρ ρ ρ ρ

ρ ρ

tvx

vx

v

y

v

y

vz

x x y y

z

( ) ( ) ( ) ( )

( ) ( vvz

vx

v

yvz

z x y z∗ ∗ ∗ ∗

∂− ∂

∂−

∂∂

− ∂∂

=∗ ∗ ∗) ( ) ( ) ( )ρ ρ ρ

0

The transient term in the continuity equation can be expressed in terms of pressure immediately by employingthe ideal gas relationship:

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Section 7.2: Derivation of Fluid Flow Matrices

(7–99)W

td vol

tW

PRT

d vole e∂∂

= ∂∂∫ ∫

ρ( ) ( )

The backward differencing process is then applied directly to this term.

Application of Galerkin's method to the remaining terms yields:

(7–100)

Wvx

v

yvz

d vol

Wvx

x y z

x

∂∂

+∂

∂+ ∂

∂

+ ∂∂

+∂

∗ ∗ ∗

∗

∫( ) ( ) ( )

( )

( )

ρ ρ ρ

ρ (( ) ( )( )

( ) ( )

ρ ρ

ρ ρ

v

yvz

d vol

Wvx

v

y

y z

x y

∗ ∗

∗ ∗ ∗ ∗

∂+ ∂

∂

− ∂∂

+∂

∂+

∫

∂∂∂

∗ ∗∫

( )( )

ρ vz

d volz

There are thus three groups of terms. In the first group, terms with the derivatives of the unknown new velocitiesmust be integrated by parts to remove the derivative. The integration by parts of just these terms becomes:

(7–101)

Wvx

v

yvz

d vol

W v v

x y z

x y

∂∂

+∂

∂+ ∂

∂

= + +

∗ ∗ ∗

∗ ∗

∫( ) ( ) ( )

( )ρ ρ ρ

ρ ρ ρ∗∗

∗ ∗ ∗

− ∂∂

+ ∂∂

+ ∂∂

∫ v d area

vWx

vWy

vWz

d v

z

x y z

( )

( ) ( ) ( ) (ρ ρ ρ ool)∫

Illustrating with the x direction, the unknown densities in the second group expressed in terms of the pressuresare:

(7–102)W

xv d vol

WR x

vPT

d volx x∂

∂= ∂

∂

∗ ∗∫ ∫( ) ( ) ( )ρ

In the third group, the values from the previous iteration are used to evaluate the integrals.

The next step is the derivation of an expression for the velocities in terms of the pressure gradient. When themomentum equations are solved, it is with a previous value of pressure. Write the algebraic expressions of themomentum equations assuming that the coefficient matrices consist of the transient, advection and diffusioncontributions as before, and all the source terms are evaluated except the pressure gradient term.

(7–103)Av s WPx

d volx

e

e

E= − ∂

∂

=

∑φ ( )1

(7–104)Av s WPy

d voly

e

e

E= − ∂

∂

=∑φ ( )

1

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ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.7–26

(7–105)Av s WPz

d volz

e

e

E= − ∂

∂

=

∑φ ( )1

Each of these sets represents a system of N algebraic equations for N unknown velocities. It is possible, after thesummation of all the element quantities, to show an expression for each velocity component at each node interms of the velocities of its neighbors, the source terms which have been evaluated, and the pressure drop.Using the subscript “i” to denote the nodal equation, for i =1 to N, where N is the number of fluid nodes andsubscript “j” to denote its neighboring node:

For SIMPLEF algorithm:

(7–106)v v

aW

px

d volx xiixi i= − ∂

∂

∫^ ( )1

(7–107)v v

aW

py

d voly y

iiyi i= − ∂

∂

∫^ ( )

1

(7–108)v v

aW

pz

d volz ziizi i= − ∂

∂

∫^ ( )1

For SIMPLEN algorithm:

(7–109)

v va

ra

Wpx

d volx xiix

x ijx

j

j ii i= −+ ∑

∂∂

≠ ∫^ ( )

1Ω

(7–110)

v va

ra

Wpy

d voly y

iiy

y ijy

j

j ii i= −

+ ∑

∂∂

≠∫^ ( )

1Ω

(7–111)

v va

ra

Wpz

d volz ziiz

z ijz

j

j ii i= −+ ∑

∂∂

≠ ∫^ ( )

1Ω

where for SIMPLEF algorithm:

v

a v S

ax

ijx

x xj

j i

iixi

j^ =

− +≠∑

v

a v S

ay

ijy

y yj

j i

iiyi

j^ =

− +≠∑

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Section 7.2: Derivation of Fluid Flow Matrices

v

a v S

az

ijz

z zj

j i

iizi

j^ =

− +≠∑

and or SIMPLEN algorithm:

v

a v v b

a

ra

x

ijx

x x ix

j

j i

iix

x ijx

j

j ii

j i^

( )

=− − +

+ ∑

≠

≠

∑

v

a v v b

a

ra

y

ijy

y y iy

j

j i

iiy

y ijy

j

j ii

j i^

( )

=− − +

+ ∑

≠

≠

∑

v

a v v b

a

ra

z

ijz

z z iz

j

j i

iiz

z ijz

j

j ii

j i^

( )

=− − +

+ ∑

≠

≠

∑

Here the aij represent the values in the x, y, and z coefficient matrices for the three momentum equations, r is

the relaxation factor, and bi is the modified source term taking into effect the relaxation factors.

For the purposes of this expression, the neighboring velocities for each node are considered as being knownfrom the momentum equation solution. At this point, the assumption is made that the pressure gradient isconstant over the element, allowing it to be removed from the integral. This means that only the weightingfunction is left in the integral, allowing a pressure coefficient to be defined in terms of the main diagonal of themomentum equations and the integral of the weighting function:

For SIMPLEF algorithm:

(7–112)M

aW d volx

iix e

N=

=∑1

1( )

(7–113)M

aW d voly

iiy e

N=

=∑1

1( )

(7–114)M

aW d volz

iiz e

N=

=∑1

1( )

For SIMPLEN algorithm:

Chapter 7: Fluid Flow

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.7–28

(7–115)

Ma

ra

W d volxiix

x ijx

j

j i e

N=

+≠ =∑

∑1

1( )

(7–116)

Ma

ra

W d volyiiy

y ijy

j

j i e

N=

+≠ =∑

∑1

1( )

(7–117)

Ma

ra

W d volziiz

z ijz

j

j i e

N=

+≠ =∑

∑1

1( )

Therefore, expressions for unknown nodal velocities have been obtained in terms of the pressure drop and apressure coefficient.

(7–118)v v MPxx x x= − ∂

∂^

(7–119)v v M

Pyy y y= − ∂

∂^

(7–120)v v MPzz z z= − ∂

∂^

These expressions are used to replace the unknown velocities in the continuity equation to convert it into apressure equation. The terms coming from the unknown velocities (replaced with the pressure gradient term)and with the unknown density (expressed in terms of the pressure) contribute to the coefficient matrix of thepressure equation while all the remaining terms will contribute to the forcing function.

The entire pressure equation can be written on an element basis, replacing the pressure gradient by the nodalpressures and the derivatives of the weighting function, putting all the pressure terms on the left hand side andthe remaining terms on the right hand side (Equation 7–121).

(7–121)

[ ] ( )PWx

MWx

Wy

MWy

Wz

MWz

d vol

W

ex y z

e∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

+

∗ ∗ ∗∫ ρ ρ ρ

RR xv

PT y

vPT z

vPT

d volx y z∂

∂

+ ∂∂

+ ∂∂

∗ ∗ ∗ ( )ee

x y zeW

xv

Wy

vWz

v d vol

Wx

v

∫

∫= ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

∗ ∗ ∗

∗

ρ ρ ρ

ρ

^ ^ ^ ( )

( xx y ze

xs s

yv

zv d vol

W v d area

∗ ∗ ∗ ∗ ∗

∗

+ ∂∂

+ ∂∂

−

∫

∫

) ( ) ( ) ( )

[ ] ( )

ρ ρ

ρ −− −∗ ∗∫ ∫W v d area W v d areays s

zs s[ ] ( ) [ ] ( )ρ ρ

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Section 7.2: Derivation of Fluid Flow Matrices

It is in the development of the forcing function that the solution to the momentum equation comes into play:the “hat” velocities contribute to the source term of the pressure equation.

In the incompressible case, the second and fourth lines of the above equation disappear because the linearizationdefined in Equation 7–97 is unnecessary. The second line is treated with the same advection routines that areused for the momentum equation.

The final step is the velocity update. After the solution for pressure equation, the known pressures are used toevaluate the pressure gradients. In order to ensure that a velocity field exists which conserves mass, the pressureterm is added back into the “hat” velocities:

For SIMPLEF algorithm:

(7–122)v v

aW

Wx

d vol Px xiix

e e= − ∂∂

∫^ ( ) [ ]1

(7–123)v v

aW

Wy

d vol Py y

iiy

e e= − ∂∂

∫^ ( ) [ ]

1

(7–124)v v

aW

Wz

d vol Pz ziiz

e e= − ∂∂

∫^ ( ) [ ]1

For SIMPLEN algorithm:

(7–125)

v va

ra

WWx

d vol Px xiix

x ijx

j

j ie e= −

+ ∑

∂∂

≠ ∫^ ( ) [ ]

1

(7–126)

v va

ra

WWy

d vol Py y

iiy

y ijy

j

j i

e e= −

+ ∑

∂∂

≠∫^ ( ) [ ]

1

(7–127)

v va

ra

WWz

d vol Pz ziiz

z ijz

j

j ie e= −

+ ∑

∂∂

≠ ∫^ ( ) [ ]

1

The global iterative procedure is summarized below.

•Formulate and solve vx

^

equation approximately

•

Formulate and solve vy^

equation approximately

•Formulate and solve vz

^

equation approximately

Chapter 7: Fluid Flow

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•

Formulate pressure equation using vx^

, vy^

, and vz^

• Solve pressure equation for P

•

Update velocities based on vx^

, vy^

, vz^

, and P

• Formulate and solve energy equation for T

• Solve species transport equations

• Update temperature dependent properties

• Solve turbulence equations for k and ε

• Update effective properties based on turbulence solution

• Check rate of change of the solution (convergence monitors)

• End of global iteration

7.3. Volume of Fluid Method for Free Surface Flows

7.3.1. Overview

A free surface refers to an interface between a gas and a liquid where the difference in the densities betweenthe two is quite large. Due to a low density, the inertia of the gas is usually negligible, so the only influence ofthe gas is the pressure acted on the interface. Hence, the region of gas need not be modeled, and the free surfaceis simply modeled as a boundary with constant pressure.

The volume of fluid (VOF) method (activated with the FLDATA1 command) determines the shape and locationof free surface based on the concept of a fractional volume of fluid. A unity value of the volume fraction (VFRC)corresponds to a full element occupied by the fluid (or liquid), and a zero value indicates an empty elementcontaining no fluid (or gas). The VFRC value between zero and one indicates that the corresponding element isthe partial (or surface) element. In general, the evolution of the free surface is computed either through a VOFadvection algorithm or through the following equation:

(7–128)∂∂

+ ⋅ ∇ =Ft

u Fr

0

where:

F = volume fraction (or VFRC)

In order to study complex flow problems, an original VOF algorithm has been developed that is applicable tothe unstructured mesh.

7.3.2. CLEAR-VOF Advection

Here, CLEAR stands for Computational Lagrangian-Eulerian Advection Remap. This algorithm takes a new approachto compute the fluxes of fluid originating from a home element towards each of its immediate neighboringelements. Here, these fluxes are referred to as the VFRC fluxes. The idea behind the computation of the VFRCfluxes is to move the fluid portion of an element in a Lagrangian sense, and compute how much of the fluid remainsin the home element, and how much of it passes into each of its neighboring elements. This process is illustratedin Figure 7.2: “Typical Advection Step in CLEAR-VOF Algorithm”(a-d).

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Section 7.3: Volume of Fluid Method for Free Surface Flows

Figure 7.2 Typical Advection Step in CLEAR-VOF Algorithm

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First, the fluid portion inside each non-empty element is used to define a polygon in that element as shown inFigure 7.2: “Typical Advection Step in CLEAR-VOF Algorithm”(a). If the element is full, the polygon of fluid coincideswith the element. The vertices of this polygon are material points in the fluid flow. Each material point undergoesa Lagrangian displacement (ξ, η) which define the velocity components (vx, vy):

(7–129)v

ddtx = ζ

(7–130)v

ddty = η

After the velocity field is obtained through the normal FLOTRAN solution procedure, the Equation 7–129 andEquation 7–130 can be used to compute the Lagrangian displacements:

(7–131)ζ

δ=

+∫ v dtxt

t t

(7–132)η

δ=

+∫ v dtyt

t t

Chapter 7: Fluid Flow

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.7–32

After the computation of the displacements for each vertex of the polygon, the new locations of these verticescan be obtained, as shown in Figure 7.2: “Typical Advection Step in CLEAR-VOF Algorithm”(b). A portion of thenew polygon of fluid will remains inside of the home element (Sii), and several other parts will cross into the

neighboring elements (Sij, Sil and Sim) as illustrated in Figure 7.2: “Typical Advection Step in CLEAR-VOF Al-

gorithm”(c). The exact amount of fluid volume portions belonging to each element is determined by an algorithmfor intersection of the advected polygon and the home element (or its immediate neighboring elements) withtheoretical basis in computational geometry. For efficiency, algorithms are developed to compute the intersectionof two convex polygons. The assumption of convexity holds by the grid generation characteristics for quadrilat-eral 2-D elements, and the advected polygons of fluid are maintained to convex shape through an automaticprocedure for selecting the time step. In summary, this algorithm uses the following geometric calculations:

• Computation of the polygon area

• Relative location of a point with respect to a line segment

• Intersection of two line segments

• Relative location of a point with respect to a polygon

• Intersection of the two polygons

With the above geometric tools available, we can proceed to compute exactly how much of the advected fluidis still in the home element, and how much of it is located in the immediate neighboring elements. At this moment,a local conservation of the volume (or area) is checked, by comparing the volume of fluid in the initial polygonand the sum of all VFRC fluxes originating from the home element. A systematic error will occur if the time stepis too large, where either the immediate neighbors of the home element fail to cover all the elements touchedby the advected polygon, or the advected polygon lose the convexity. In either case, the time increment for VOFadvection will be automatically reduced by half. This automatic reduction will continue until the local balanceof volume is preserved.

After the advected polygons of fluid from all non-empty elements have been redistributed locally in the Eulerianfixed mesh, a sweep through all elements is necessary to update the volume fraction field. The new volume offluid in each home element can be obtained by the sum of all VFRC fluxes originating from itself (Sii) and its im-

mediate neighboring elements (Spi, Sqi and Ski), and the new volume fraction can simply obtained by dividing

this sum by the volume of this home element as illustrated in Figure 7.2: “Typical Advection Step in CLEAR-VOFAlgorithm”(d).

7.3.3. CLEAR-VOF Reconstruction

In order to continue the VOF advection in the next time step, the new volume fraction is needed to reconstructthe new polygon of fluid in each non-empty element. In the present implementation, a piecewise linear recon-struction method is used where the interface is reconstructed as a line segment inside each partial element.Since the polygon of fluid coincides with the home element for every full element, there is no need for interfacereconstruction for full elements. This process is illustrated in Figure 7.3: “Types of VFRC Boundary Conditions”.

In order to combine the unstructured mesh capability of the CLEAR-VOF with a piecewise linear method, thefollowing procedure has been adopted for the interface reconstruction:

• Store the local distribution of updated volume fraction field and mesh geometry. Here, local means thehome element and its immediate neighbors.

•Compute the unit normal vector n

^ to the interface line inside the home element as the unit gradient

vector of the volume fraction field in its neighborhood

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Section 7.3: Volume of Fluid Method for Free Surface Flows

•The equation of line in the home element is g( x

r) = n

^ ⋅ x

r + c = 0. Once the unit vector n

^ is found, the

constant c is computed by requiring the volume fraction of the polygon of fluid delimited by the corres-ponding line interface to be equal to the given volume fraction for the home element.

• When a given value for c is computed, the volume fraction inside the home element is determined byconstructing the polygon of fluid delimited by the line of equation inside the home element. It is thus

necessary to retain the vertices of the home element inside the fluid, i.e., the vertices that verify g( xr

) > 0,and the intersection points lie between the interface line and the edges of the home element.

In the present algorithm, the least squares gradient method has been chosen to compute the unit normal vector

n^ = ∇ f / | ∇ f |. This method is essentially independent of any mesh topology or dimensionality, and is thus ableto handle any unstructured meshes. Further, the line constant c is obtained by solving an additional equationthat imposes the conservation of fluid volume in the home element. The idea is that volume of the polygon offluid, delimited inside the home element by the interface line, must correspond to the known VFRC value. Thesolution of this equation can be obtained iteratively by halving iteration of the interval [cmin, cmax]. The limits are

found by allowing the interface line to pass through each of the home element vertices, computing the volumefraction and isolating the extreme cases F = 0 and F = 1.

7.3.4. Treatment of Finite Element Equations

In a VOF (Volume of Fluid) analysis, each element can be identified as full, partially full, or empty. Full elementsrepresent the fluid, and empty elements represent the void. Partial elements are regions of transition betweenthe fluid and the void. In the present solution algorithm, the finite element equations are assembled only forpartial and full elements, because empty elements have no effect on the motion of the fluid. The contributionsof the full elements are treated in the usual manner as in other flow analyses, whereas those of the partial elementsare modified to reflect the absence of fluid in parts of the elements.

In the solution algorithm, partial elements are reconstructed differently from the CLEAR-VOF reconstructionscheme. The nodes are moved towards the center of the element so that the reduced element preserves thesame shape as the original element, and the ratio between the two is kept to be equal to the volume fraction ofthis partial element. The modified nodal coordinates are then used to evaluate the integration of the finite elementequations over a reduced integration limit. It shall be noted that this modification is only intended for the eval-uation of the finite element equations, and the actual spatial coordinates of the nodes are not changed.

For a VOF analysis, boundary conditions are required for boundary nodes that belong to at least one non-empty(partial or full) element. For boundary nodes belonging to only empty elements, on the other hand, the prescribedboundary conditions will remain inactive until those nodes are touched by fluid. Finally, boundary conditionsare also applied to nodes that belong to at least one empty element and at least one non-empty element. Thesenodes represent the transition region between the fluid and the void. This free surface is treated as naturalboundary conditions for all degrees of freedom except pressure. For the pressure, a constant value (using theFLDATA36 command) is imposed on the free surface.

In order to impose proper boundary conditions on the element-based volume fraction (VFRC), imaginary elementsare created along the exterior boundary to act as neighbors to the elements forming the boundary. Two typesof boundary conditions are applied on these imaginary elements. The imaginary elements can be specified aseither full or empty depending on the imposed volume fraction value as shown in Figure 7.3: “Types of VFRCBoundary Conditions”(a and b).

Chapter 7: Fluid Flow

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Figure 7.3 Types of VFRC Boundary Conditions

!#"$ &% % !% '()*#+$ ',- !% /. 0 !#" &% % !% 1(

2354+6879$ (: * !; #1% &%& !,%# ";* <,=>%; ? (@ *' $ ("$ A% / /B<

3C4+687D (: * !; #1% &%& %0 ' E1% '"F*' ' <,=>#%G H$ (: *' $ (" A'% 0 0B<

Partial imaginary elements are not allowed on boundaries. These boundary volume fraction will serve as aneighbor value when determine the interface normal vector. For the full imaginary elements, a second boundarycondition is specified to determine whether the fluid is advected into the computational domain. The boundaryis then further identified as either wetting or non-wetting as shown in Figure 7.3: “Types of VFRC BoundaryConditions”(c and d).

For the wetting boundary, the imaginary elements have to be full, and the fluid is advected into the domain. Forthe non-wetting boundary, the fluid or void can not be advected into the domain.

7.3.5. Treatment of Volume Fraction Field

In summary, the advection of the reconstructed polygon of fluid consists of the following steps:

1. Compute the new locations of the polygon vertices in the Lagrangian displacement step.

2. Determine the distribution of the advected fluid volume into the neighborhood using an algorithm forintersection of polygons.

3. Update the volume fraction at the new time step.

In the last step, the VFRC fluxes are regrouped to evaluate the total volume flowing into each home element.Since the volume fraction is just this volume divided by the volume of the home element, this evaluation ofvolume fraction is exact, and there exists no error in this step.

In the second step, the polygon of fluid at the new time level is only redistributed into its neighborhood, and nofluid shall be created or destroyed in this process. Therefore, the volume of fluid in the advected polygon shallbe equal to the sum of all VFRC fluxes originating from this polygon. This conservation of the fluid volume willbe violated only in two cases. The first one involves the failure of the polygon intersection algorithm. This willoccur when the deformation of the advected polygon is too large during the Lagrangian step such that theconvexity of the polygon is lost. The second one involves an incomplete coverage of the advected polygon bythe immediate neighbors of the home element. In this case, some VFRC fluxes will flow into its far neighbors andwill not be taken into account by the present algorithm. In either case, the time increment in the Lagrangian

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Section 7.3: Volume of Fluid Method for Free Surface Flows

step will be reduced by half in order to reduce the Lagrangian deformation and the traveling distance of theadvected polygon. This automatic reduction in time increment will continue until the local balance of fluidvolume is preserved. You can also specify the number of VOF advection steps per solution step (using theFLDATA4 command).

In the Lagrangian step, the polygon of fluid undergoes a Lagrangian movement. The Lagrangian velocity is takento be the same with the Eulerian velocity at a particular instance in time. The Lagrangian velocity is then usedto calculate the displacements and the new locations of the polygon vertices. This new polygon is then used tointersect with the immediate neighbors of the home element in the next step. There do exist some potentialproblems in the numerical approximation of this algorithm. Consider a bulk of fluid flows along a no-slip wallemptying the elements behind it as time advances. In reality, however, there exist certain cases where thepolygon may have two vertices lie on the no-slip wall during the reconstruction stage. In such cases, there willalways a certain amount of volume left in the home element, which make it practically impossible to empty thesewall elements. As time advances, the bulk of fluid may leave behind a row of partial elements rather empty ele-ments. This phenomenon is usually referred to as the artificial formation and accumulation of droplets. In otherwords, a droplet is never reattached to the main fluid once it is formed. To eliminate those isolated droplets, thestatus of partial element's neighbors are always checked, and if necessary, a local adjustment will be performed.A partial element is reset to be empty if it is not adjacent to at least one full element. Similarly, a partial elementis reset to be full if its immediate neighbors are all full elements to avoid an isolated partial element inside a bulkof fluid.

Another type of error introduced in the Lagrangian advection step is due to the imperfection of Eulerian velocityfield. In the solution algorithm, the continuity equation is expressed in a Galerkin weak form. As a result, diver-gence-free condition is not satisfied exactly, and the error is usually in the same order with the discretizationerror. This error will further result in artificial compressibility of the polygon of fluid during the Lagrangian advectionstep, and thus introduce local and global imbalance in the fluid volume. Fortunately, both this type of error andthat in the local adjustment of volume fraction field are very small compared to the total fluid volume. Unfortu-nately, the error due to the velocity divergence can accumulate exponentially as time advances. Hence a globaladjustment is necessary to retain the global balance of the fluid volume. Currently, the volume fraction of partialelements are increased or decreased proportionally according to the global imbalance.

(7–133)

F FV

F V

Fpnew

pold imb

qold

qq

N pold

q= +

=∑

1

where:

Fp, Fq = volume fraction of a given partial element

old = superscript for the value before the adjustmentnew = superscript for the value after the adjustmentNq = total number of partial elements

Vimb = amount of the total volume imbalance = difference between the volume flowing across the external

boundary (in - out) and the change of total volume inside the domain.Vq = volume of a given partial element

In the above practice, the volume fraction of a nearly full element may be artificially adjusted to an unphysicalvalue greater than one, and will thus be reset to one. Although this global adjustment for partial elements intro-duces a numerical diffusion effect, it is believed that the benefit of global conservation of the fluid volume willcertainly outweigh this effect. Hence, the global balance of the fluid volume is always checked, and if an imbalanceoccurs, it will adjust the volume fraction to enforce the global balance.

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7.3.6. Treatment of Surface Tension Field

In a VOF analysis, the surface tension is modeled through a continuum-surface force (CSF) method (accessedwith the FLDATA1 command). There are two components in this surface force. The first one is normal to the in-terface due to the local curvature, and the second one is tangential to the interface due to local variations of thesurface tension coefficient (accessed with FLDATA13 command). In this approach, the surface force localizedat the fluid interface is replaced by a continuous volume force to fluid elements everywhere within a thin transitionregion near the interface. The CSF method removes the topological restrictions without losing accuracy (Brack-bill(281)), and it has thus been used widely and successfully in a variety of studies (Koth and Mjolsness(282);Richards(283); Sasmal and Hochstein(284); Wang(285)).

The surface tension is a force per unit area given by:

(7–134)f ns t

r= + ∇σκ σ^

^

where:

fsr

= surface forceσ = surface tension coefficientκ = surface curvature

n^ = unit normal vector

∇ t^

= surface gradient

Refer to Section 7.6.7: Multiple Species Property Options on details on surface tension coefficient. Here, the surfacecurvature and unit normal vector are respectively given by:

(7–135)κ = −∇ ⋅ = ⋅ ∇

− ∇ ⋅

nn

n

nn n^ ( )

1r

r

rr r

(7–136)n

n

n

FF

^ = = ∇∇

r

r

The surface gradient is given by:

(7–137)∇ = ⋅ ∇

tt t^^ ^( )

where:

t^

= unit tangent vector at the surface

In Equation 7–134, the first term is acting normal to the interface, and is directed toward the center of the localcurvature of the interface. The second term is acting tangential to the interface, and is directed toward the regionof higher surface tension coefficient σ.

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Section 7.3: Volume of Fluid Method for Free Surface Flows

In the CSF method, the surface force is reformulated into a volumetric force Fsr

as follows:

(7–138)F f

FF

s s sr r

=< >

δ

where:

< F > = averaged volume fraction across the interfaceδs = surface delta function

(7–139)δs n F= = ∇

r

The δs function is only nonzero within a finite thickness transition region near the interface, and the corresponding

volumetric force Fsr

will only act within this transition region.

In this model, the surface curvature depends on the second derivatives of the volume fraction. On the otherhand, the volume fraction from the CLEAR-VOF algorithm will usually jump from zero to one within a single layerof partial elements. As a result, there may exist large variations in the κ values near the interface, which in turnmay introduce artificial numerical noises in the surface pressure. One remedy is to introduce spatial smoothingoperations for the volume fraction and the surface curvature. In order to minimize any unphysical smearing of

the interface shape, only one pass of least square smoothing is performed for F, n^

and κ values, and under-relax-ation is used with its value set to one half.

7.4. Fluid Solvers

The algorithm requires repeated solutions to the matrix equations during every global iteration. In some cases,exact solutions to the equations must be obtained, while in others approximate solutions are adequate. In certainsituations, the equation need not be solved at all. It has been found that for the momentum equations, the timesaved by calculating fast approximate solutions offsets the slightly slower convergence rates one obtains withan exact solution. In the case of the pressure equation, exact solutions are required to ensure conservation ofmass. In a thermal problem with constant properties, there is no need to solve the energy equation at all untilthe flow problem has been converged.

To accommodate the varying accuracy requirements, three types of solvers are provided. Two types of solversare iterative and the other one is direct. The direct solver used here is the Boeing sparse direct method. The firstiterative solver is a sweeping method known as the Tri-Diagonal Matrix Algorithm (TDMA), and the rest are semi-direct including the conjugate direction methods, the preconditioned generalized minimal residual method,and the preconditioned bi-conjugate gradient stabilized method. TDMA is used to obtain the approximatesolution and the other methods are used when exact solutions are needed. The user has control over whichmethod is applied to which degree of freedom (using the FLDATA18 command).

The TDMA method is described in detail in Patankar(182). The method consists of breaking the problem into aseries of tri-diagonal problems where any entries outside the tri-diagonal portion are treated as source termsusing the previous values. For a completely unstructured mesh, or an arbitrarily numbered system, the methodreduces to the Gauss-Seidel iterative method.

Since it is considered an approximate method, TDMA is not executed to convergence. Rather, the number ofTDMA sweeps that should be executed is input (using the FLDATA19 command).

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The conjugate direction methods are the conjugate gradient (for symmetric systems) method and the conjugateresidual method (for non-symmetric systems). These are iterative methods used to attempt an exact solution tothe equation of interest. The conjugate gradient method is preconditioned with an incomplete Choleski decom-position and is used only for the pressure equation in incompressible flows. The sequential solution algorithmmust allow space for a non-symmetric coefficient matrix for the momentum and energy equations. Only halfthis storage is required for the symmetric matrix and the other half is used to store the decomposition. Theconjugate residual method can be used with or without preconditioning, the latter approach requiring significantlyless computer memory. A convergence criterion and a maximum number of iterations are specified by the user(using the FLDATA21 and FLDATA22 commands).

The conjugate direction method develop a solution as a linear combination of orthogonal vectors. These vectorsare generated one at a time during an iteration. In the case of the conjugate gradient method, the symmetry ofthe coefficient matrix and the process generating the vectors ensures that each one is automatically orthogonalto all of the previous vectors. In the non-symmetric case, the new vector at each iteration is made orthogonal tosome user specified number of previous vectors (search directions). The user has control of the number (usingthe FLDATA20 command).

More information on the conjugate directions is available from Hestenes and Stiefel(183) , Reid(184), and El-man(185).

7.5. Overall Convergence and Stability

7.5.1. Convergence

The fluid problem is nonlinear in nature and convergence is not guaranteed. Some problems are transient innature, and a steady state algorithm may not yield satisfactory results. Instabilities can result from a number offactors: the matrices may have poor condition numbers because of the finite element mesh or very large gradientsin the actual solution. The fluid phenomena being observed could be unstable in nature.

Overall convergence of the segregated solver is measured through the convergence monitoring parameters. Aconvergence monitor is calculated for each degree of freedom at each global iteration. It is loosely normalizedrate of change of the solution from one global iteration to the next and is calculated for each DOF as follows:

(7–140)M

ik

ik

i

N

ik

i

Nφ

φ φ

φ=

− −

=

=

∑

∑

1

1

1

where:

Mφ = convergence monitor for degree of freedom f

N = total number of finite element nodesφ = degree of freedomk = current global iteration number

It is thus the sum of the absolute value of the changes over the sum of the absolute values of the degree offreedom.

The user may elect to terminate the calculations when the convergence monitors for pressure and temperaturereach very small values. The convergence monitors are adjusted (with FLDATA3 command). Reduction of therate of change to these values is not guaranteed. In some cases the problem is too unstable and in others thefinite element mesh chosen leads to solution oscillation.

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Section 7.5: Overall Convergence and Stability

7.5.2. Stability

Three techniques are available to slow down and stabilize a solution. These are relaxation, inertial relaxation,and artificial viscosity.

7.5.2.1. Relaxation

Relaxation is simply taking as the answer some fraction of the difference between the previous global iterationresult and the newly calculated values. In addition to the degrees of freedom, relaxation can be applied to thelaminar properties (which may be a function of temperature and, in the case of the density of a gas, pressure)and the effective viscosity and effective conductivity calculated through the turbulence equations. Denoting byφi the nodal value of interest, the expression for relaxation is as follows:

(7–141)φ φ φφ φ

inew

iold

icalcr r= − +( )1

where:

rφ = relaxation factor for the variable.

7.5.2.2. Inertial Relaxation

Inertial relaxation is used to make a system of equations more diagonally dominant. It is similar to a transientsolution. It is most commonly used in the solution of the compressible pressure equation and in the turbulenceequations. It is only applied to the DOF.

The algebraic system of equations to be solved may be represented as, for i = 1 to the number of nodes:

(7–142)a a fii i ij j

j iiφ φ+ =

≠∑

With inertial relaxation, the system of equations becomes:

(7–143)( )a A a f Aii ii

di ij j

j ii ii

diold+ + = +

≠∑φ φ φ

where:

AWd vol

Biid

rf= ∫ ρ ( )

Brf = inertial relaxation factor (input on the FLDATA26 command)

At convergence, φiold

(i.e. the value of the φi from the previous global iteration) and φi will be identical, so the

same value will have been added to both sides of the equation. This form of relaxation is always applied to the

equations, but the default value of Brf = 1.0 x 1015 effectively defeats it.

7.5.2.3. Artificial Viscosity

Artificial viscosity is a stabilization technique that has been found useful in compressible problems and incom-pressible problems involving distributed resistance. The technique serves to increase the diagonal dominance

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of the equations where the gradients in the momentum solution are the highest. Artificial viscosity enters theequations in the same fashion as the fluid viscosity. The additional terms are:

(7–144)R

xvx

v

yvzx a

x y z= ∂∂

∂∂

+∂∂

+ ∂∂

µ

(7–145)R

yvx

v

yvzy a

x y z= ∂∂

∂∂

+∂∂

+ ∂∂

µ

(7–146)R

zvx

v

yvzz a

x y z= ∂∂

∂∂

+∂∂

+ ∂∂

µ

where:

µa = artificial viscosity

This formulation is slightly different from that of Harlow and Amsden(180) in that here µa is adjustable (using

the FLDATA26 command).

In each of the momentum equations, the terms resulting from the discretization of the derivative of the velocityin the direction of interest are additions to the main diagonal, while the terms resulting from the other gradientsare added as source terms.

Note that since the artificial viscosity is multiplied by the divergence of the velocity, (zero for an incompressiblefluid), it should not impact the final solution. For compressible flows, the divergence of the velocity is not zeroand artificial viscosity must be regarded as a temporary convergence tool, to be removed for the final solution.

7.5.3. Residual File

One measure of how well the solution is converged is the magnitude of the nodal residuals throughout thesolution domain. The residuals are calculated based on the “old” solution and the “new” coefficient matrices andforcing functions. Residuals are calculated for each degree of freedom (VX, VY, VZ, PRES, TEMP, ENKE, ENDS).

Denoting the DOF by φ, the matrix equation for the residual vector r may be written as follows:

(7–147)[ ] A b rn n n

φ φ φφ 1 =

where the superscript refers to the global iteration number and the subscript associates the matrix and the forcingfunction with the degree of freedom φ.

The residuals provide information about where a solution may be oscillating.

The values at each node are normalized by the main diagonal value for that node in the coefficient matrix. Thisenables direct comparison between the value of the residual and value of the degree of freedom at the node.

7.5.4. Modified Inertial Relaxation

Similar to inertial relaxation, modified inertial relaxation (MIR) is used to make the system of equations more di-agonally dominant. It is most commonly used to make the solution procedure by SUPG scheme more stable.

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Section 7.5: Overall Convergence and Stability

The algebraic system of equations with modified inertial relaxation has the same form with Equation 7–143, butthe definition of the added diagonal term is different:

(7–148)A B

uh h

d voliid MIR= +

∫

ρ Γ2

( )

where:

ρ = densityΓ = generalized diffusion coefficientu = local velocity scaleh = local length scale

BMIR = modified inertial relaxation factor (input on the FLDATA34 or MSMIR command).

7.6. Fluid Properties

Specific relationships are implemented for the temperature variation of the viscosity and thermal conductivityfor both gases and liquids. These relationships were proposed by Sutherland and are discussed in White(181).The equation of state for a gas is assumed to be the ideal gas law. Density in a liquid may vary as a function oftemperature through a polynomial. Fluid properties are isotropic. In addition to gas and liquid-type variations,non-Newtonian variations of viscosity are also included (Gartling(197) and Crochet et al.(198)).

The relationships are:

7.6.1. Density

Constant: For the constant type, the density is:

(7–149)ρ ρ= N

where:

ρ = densityρN = nominal density (input on FLDATA8 command)

Liquid: For the liquid type, the density is:

(7–150)ρ ρ ρ ρ ρ ρ= + − + −N C T C C T C2 1 3 1

2( ) ( )

where:

P = absolute pressureT = absolute temperature

C1ρ

= first density coefficient (input on FLDATA9 command)

= absolute temperature at which ρ ρ ρ= =N C P( )if 2

C2ρ

= second density coefficient (input on FLDATA10 command)

C3ρ

= third density coefficient (input on FLDATA11 command)

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Gas: For the gas type, the density is:

(7–151)ρ ρ ρ

ρ= N

P

C

C

T2

1

Table: For the table type, you enter density data as a function of temperature (using the MPTEMP and MPDATAcommands).

User-Defined Density: In recognition of the fact that the density models described above can not satisfy therequests of all users, a user-programmable subroutine (UserDens) is also provided with access to the followingvariables: position, time, pressure, temperature, etc. See the Guide to ANSYS User Programmable Features andUser Routines and Non-Standard Uses in the ANSYS Advanced Analysis Techniques Guide for information aboutuser written subroutines.

7.6.2. Viscosity

Constant: For the constant type, the viscosity is:

(7–152)µ µ= N

where:

µ = viscosityµN = nominal viscosity (input on FLDATA8 command)

Liquid: For the liquid type, the viscosity is:

(7–153)µ µ= NAe

where:

A CT C

CT C

= −

+ −

2

13

1

21 1 1 1µ

µµ

µ

C1µ

= first viscosity coefficient (input on FLDATA9 command)= absolute temperature at which µ = µN

C2µ

= second viscosity coefficient (input on FLDATA10 command)

C3µ

= third viscosity coefficient (input on FLDATA11 command)

Gas: For the gas type, the viscosity is:

(7–154)µ µµ

µ µ

µ=

+

+

N

T

C

C C

T C1

1 5

1 2

2

.

In addition for non-Newtonian flows, additional viscosity types are available (selected with FLDATA7 command).A viscosity type is considered non-Newtonian if it displays dependence on the velocity gradient.

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Section 7.6: Fluid Properties

Power Law: For the power law model, the viscosity is:

(7–155)µ

µ

µ=

>

≤

−

−o

no

o on

o

KD D D

KD D D

1

1

for

for

where:

µo = nominal viscosity (input on FLDATA8 command)

K = consistency index (input on FLDATA10 command)

D = I2

Do = cutoff value for D (input on FLDATA9 command)

n = power (input as value on FLDATA11 command)I2 = second invariant of strain rate tensor

= ∑∑1

2L Lij ij

ji

L v vij i j j i= +1

2( ), ,

vi,j = ith velocity component gradient in jth direction

This relationship is used for modeling polymers, blood, rubber solution, etc. The units of K depend on the valueof n.

Carreau Model: For the Carreau Model, the viscosity is:

(7–156)µ µ µ µ λ= − +∞+ ∞

−

( )( ( ) )o

n

D1 21

2

µ∞ = viscosity at infinite shear rate (input on FLDATA9 command)µo = viscosity at zero shear rate (input on FLDATA8 command)

λ = time constant (input on FLDATA10 command)n = power (input on FLDATA11 command)

Typically the fluid viscosity behaves like a Power Law model for intermediate values of shear rate while remainingbounded for zero/infinite shear rates. This model removes some of the deficiencies associated with the PowerLaw model. The fluid is assumed to have lower and upper bounds on the viscosity.

Bingham Model: For the “ideal” Bingham model, the viscosity is:

(7–157)µ

µ ττ

=+ ≥

∞ <

o G D G

G

/ if

if

where:

µo = plastic viscosity (input on FLDATA8 command)

G = yield stress (input on FLDATA9 command)

τ τ τ= = ∑∑stress level1

2ij ij

ji

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τij = extra stress on ith face in the jth direction

Figure 7.4 Stress vs. Strain Rate Relationship for “Ideal” Bingham Model

Figure 7.4: “Stress vs. Strain Rate Relationship for “Ideal” Bingham Model” shows the stress-strain rate relationship.

So long as the stress is below the plastic level, the fluid behaves as a rigid body. When the stress exceeds theplastic level the additional stress is proportional to the strain rate, i.e., the behavior is Newtonian. Numerically,it is difficult to model. In practice it is modelled as a “biviscosity” model:

(7–158)µ

µµ µ

µµ µ

=+ >

−

≤−

or o

rr o

G D DG

DG

if

if

where:

µr = Newtonian viscosity (input on FLDATA10 command)

Figure 7.5: “Stress vs. Strain Rate Relationship for “Biviscosity” Bingham Model” shows the stress-strain rate rela-tionship for the “biviscosity” Bingham model.

µr is chosen to at least an order of magnitude larger than µo. Typically µr is approximately 100 µo in order to

replicate true Bingham fluid behavior.

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Section 7.6: Fluid Properties

Figure 7.5 Stress vs. Strain Rate Relationship for “Biviscosity” Bingham Model

Table: For the table type, you enter viscosity data as a function of temperature (using the MPTEMP and MPDATAcommands).

User-Defined Viscosity: In recognition of the fact that the viscosity models described above can not satisfy therequests of all users, a user-programmable subroutine (UserVisLaw) is also provided with access to the followingvariables: position, time, pressure, temperature, velocity component, velocity gradient component. See the Guideto ANSYS User Programmable Features and User Routines and Non-Standard Uses in the ANSYS Advanced AnalysisTechniques Guide for information about user written subroutines.

7.6.3. Thermal Conductivity

Constant: For the constant type, the conductivity is:

(7–159)K KN=

where:

K = conductivityKN = nominal conductivity (input on FLDATA8 command)

Liquid: For a liquid type, the conductivity is:

(7–160)K K eNB=

where:

B CT C

CT C

KK

KK

= −

+ −

2

13

1

21 1 1 1

CK1 = first conductivity coefficient (input on FLDATA9 command)

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= absolute temperature at which K = KN

CK2 = second conductivity coefficient (input on FLDATA10 command)

CK3 = third conductivity coefficient (input on FLDATA11 command)

Gas: For a gas type, the conductivity is:

(7–161)K KT

C

C C

T CN K

K K

K=

++

1

1 51 2

2

.

Table: For the table type, you enter conductivity data as a function of temperature (using the MPTEMP andMPDATA commands).

User-Defined Conductivity: In recognition of the fact that the conductivity models described above can notsatisfy the requests of all users, a user-programmable subroutine (UserCond) is also provided with access to thefollowing variables: position, time, pressure, temperature, etc. See the Guide to ANSYS User Programmable Featuresand User Routines and Non-Standard Uses in the ANSYS Advanced Analysis Techniques Guide for informationabout user written subroutines.

7.6.4. Specific Heat

Constant: For the constant type, the specific heat is:

(7–162)C Cp pN=

where:

CpN = nominal specific heat (input on FLDATA8 command)

Table: For the table type, you specify specific heat data as a function of temperature (using the MPTEMP andMPDATA commands).

User-Defined Specific Heat: In recognition of the fact that the specific heat models described above can notsatisfy the requests of all users, a user-programmable subroutine (UserSpht) is also provided with access to thefollowing variables: position, time, pressure, temperature, etc. See the Guide to ANSYS User Programmable Featuresand User Routines and Non-Standard Uses in the ANSYS Advanced Analysis Techniques Guide for informationabout user written subroutines.

7.6.5. Surface Tension Coefficient

Constant: For the constant type, the surface tension coefficient is:

(7–163)σ σ= N

where:

σ = surface tension coefficientσN = nominal surface tension coefficient (input on FLDATA8 command)

Liquid: For the liquid type, the surface tension is:

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Section 7.6: Fluid Properties

(7–164)σ σ σ σ σ σ= + − + −N C T C C T C2 1 3 2

2( ) ( )

where:

T = absolute temperature

C1σ

= first coefficient for surface tension coefficient (input as value on FLDATA9 command)

C2σ

= second coefficient for surface tension coefficient (input on FLDATA10 command)

C3σ

= third coefficient for surface tension coefficient (input on FLDATA11 command)

Table: For the table type, you enter density data as a function of temperature (using the MPTEMP and MPDATAcommands).

User-Defined Surface Tension Coefficient: In recognition of the fact that the surface tension models describedabove can not satisfy the requests of all users, a user-programmable subroutine (UserSfTs) is also provided withaccess to the following variables: position, time, pressure, temperature, etc. See the Guide to ANSYS User Program-mable Features and User Routines and Non-Standard Uses in the ANSYS Advanced Analysis Techniques Guide forinformation about user written subroutines.

7.6.6. Wall Static Contact Angle

The wall static contact angle θw describes the effect of wall adhesion at the solid boundary. It is defined as the

angle between the tangent to the fluid interface and the tangent to the wall. The angle is not only a materialproperty of the fluid but also depends on the local conditions of both the fluid and the wall. For simplicity, it isinput as a constant value between 0° and 180° (on the FLDATA8 command). The wall adhesion force is thencalculated in the same manner with the surface tension volume force using Equation 7–138 except that the unitnormal vector at the wall is modified as follows (Brackbill(281)):

(7–165)n n cos n sinw w t w^ ^ ^= +θ θ

where:

nw^

= unit wall normal vector directed into the wall

nt^

= unit vector normal to the interface near the wall

7.6.7. Multiple Species Property Options

For multiple species problems, the bulk properties can be calculated as a combination of the species propertiesby appropriate specification of the bulk property type. Choices are composite mixture, available for the density,viscosity, thermal conductivity, specific heat and composite gas, available only for the density.

Composite Mixture: For the composite mixture (input with FLDATA7,PROT,property,CMIX) each of the propertiesis a combination of the species properties:

(7–166)α αbulk i i

i

NY=

=∑

1

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where:

αbulk = bulk density, viscosity, conductivity or specific heat

αi = values of density, viscosity, conductivity or specific heat for each of the species

Composite Gas: For a composite gas (input with FLDATA7,PROT,DENS,CGAS), the bulk density is calculated asa function of the ideal gas law and the molecular weights and mass fractions.

(7–167)

ρ =

=∑

P

RTYM

i

ii

N

1

where:

R = universal gas constant (input on MSDATA command)Mi = molecular weights of each species (input on MSSPEC command)

The most important properties in simulating species transport are the mass diffusion coefficient and the bulkproperties. Typically, in problems with dilute species transport, the global properties will not be affected by thedilute species and can be assumed to be dependent only on the temperature (and pressure for gas density).

7.7. Derived Quantities

The derived quantities are total pressure, pressure coefficient, mach number, stream function, the wall parametery-plus, and the wall shear stress. These quantities are calculated from the nodal unknowns and stored on anodal basis.

7.7.1. Mach Number

The Mach number is ratio of the speed of the fluid to the speed of sound in that fluid. Since the speed of soundis a function of the equation of state of the fluid, it can be calculated for a gas regardless of whether or not thecompressible algorithm is used.

(7–168)M

v

RT=

( ) /γ 1 2

where:

M = Mach number (output as MACH)γ = ratio of specific heats| v | = magnitude of velocityR = ideal gas constantT = absolute temperature

7.7.2. Total Pressure

The calculation differs, depending on whether the compressible option has been activated (on the FLDATA1command).

Compressible:

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Section 7.7: Derived Quantities

(7–169)P P P M Ptot ref ref= + + −

−−( ) 11

22 1γ

γγ

Incompressible:

(7–170)P P vtot = + 1

22ρ

where:

Ptot = total pressure (output as PTOT)

P = relative pressurePref = reference pressure

ρ = density

The calculation is the same for compressible and incompressible cases.

(7–171)P

P P

vcoef

f

f f

= −22

( )

ρ

where:

Pcoef = pressure coefficient (output as PCOEF)

subscript f = free stream conditions

7.7.3. Y-Plus and Wall Shear Stress

These quantities are part of the turbulence modeling of the wall conditions. First, solving iteratively for τw:

(7–172)

vln

Etan

w

w

τρ

κδρµ

τρ

=

1

where:

µ = viscosityδ = distance of the near wall node from the wallvtan = velocity at the near wall node parallel to the wall

E = constant in the turbulence model (defaults to 9.0)κ = constant in the turbulence model (defaults to 0.4)τw = wall shear stress (output as TAUW)

Then, using τw:

(7–173)y w+ = δ ρ

µτρ

where:

Chapter 7: Fluid Flow

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y+ = nondimensional distance from the wall (output as YPLU)

7.7.4. Stream Function

The stream function is computed for 2-D structures and is defined by way of its derivatives:

7.7.4.1. Cartesian Geometry

(7–174)∂∂

= −ψ ρx

vy

(7–175)∂∂

=ψ ρy

vx

7.7.4.2. Axisymmetric Geometry (about x)

(7–176)∂∂

=ψ ρx

y vy

(7–177)∂∂

= −ψ ρy

y vx

7.7.4.3. Axisymmetric Geometry (about y)

(7–178)∂∂

= −ψ ρx

x vy

(7–179)∂∂

=ψ ρy

x vx

7.7.4.4. Polar Coordinates

(7–180)∂∂

= −ψ ρ θrv

(7–181)∂∂

=ψθ

ρr vr

where:

y = stream function (output as STRM)x, y = global Cartesian coordinates

r = radial coordinate (= x2 + y2)θ = circumferential coordinatevx, vy = global Cartesian velocity components

vr, vθ = polar velocity components

The stream function is zero at points where both vx and vy are zero. Thus, a zero value of the stream function

would bound a recirculation region.

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Section 7.7: Derived Quantities

7.7.5. Heat Transfer Film Coefficient

7.7.5.1. Matrix Procedure

To calculate the heat flux and film coefficient, the matrix procedure (accessed using FLDATA37,ALGR,HFLM,MATX)first calculates the sum of heat transfer rate from the boundary face using the sum of the residual of the right-hand side:

(7–182) [ ] Q K Tnt= −

where:

Qn = nodal heat rate

[Kt] = conductivity matrix for entire modelT = nodal temperature vector

See Section 6.1: Heat Flow Fundamentals for more information.

The nodal heat flux at each node on the wall is defined as:

(7–183)q

QAn

n

n=

where:

qn = nodal heat flux

Qn = a value of the vector Qn

An = surface area associated with the node (depends on all of its neighboring surface elements)

7.7.5.2. Thermal Gradient Procedure

The thermal gradient procedure (accessed with FLDATA37,ALGR,HFLM,TEMP) does not use a saved thermalconductivity matrix. Instead, it uses the temperature solution at each node and uses a numerical interpolationmethod to calculate the temperature gradient normal to the wall.

(7–184)T N Ta a

a

L=

=∑ ( )ξ

1

where:

n = direction normal to the surfaceD = material conductivity matrix at a point

7.7.5.3. Film Coefficient Evaluation

For both procedures the film coefficient is evaluated at each node on the wall by:

(7–185)h

qT Tn

n

n B=

−

where:

Chapter 7: Fluid Flow

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hn = nodal film coefficient

Tn = nodal temperature

TB = free stream or bulk fluid temperature (input on SF or SFE commands)

7.8. Squeeze Film Theory

Reynolds equation known from lubrication technology and theory of rarified gas physics are the theoreticalbackground to analyze fluid structural interactions of microstructures (Blech(337), Griffin(338), Langlois(339)).FLUID136 and FLUID138 can by applied to structures where a small gap between two plates opens and closeswith respect to time. This happens in case of accelerometers where the seismic mass moves perpendicular to afixed wall, in micromirror displays where the mirror plate tilts around an horizontal axis, and for clamped beamssuch as RF filters where a flexible structure moves with respect to a fixed wall. Other examples are published inliterature (Mehner(340)).

FLUID136 and FLUID138 can be used to determine the fluidic response for given wall velocities. Both elementsallow for static, harmonic and transient types of analyses. Static analyses can be used to compute dampingparameter for low driving frequencies (compression effects are neglected). Harmonic response analysis can beused to compute damping and squeeze effects at the higher frequencies. Transient analysis can be used for non-harmonic load functions.

7.8.1. Flow Between Flat Surfaces

FLUID136 is used to model the thin-film fluid behavior between flat surfaces and is based in the linearizedReynolds squeeze film equation known from lubrication theory (Blech(337), Yang(341)):

(7–186)d P

x

P

y

dP

Pto

z

3 2

2

2

212ην∂

∂+ ∂

∂

= ∂

∂+

where:

P = pressure changex, y = coordinatesη = dynamic viscosityd = local gap separationPo = ambient pressure

t = timeνz = wall velocity in normal direction

Reynolds squeeze film equations are restricted to structures with lateral dimensions much larger than the gapseparation. Furthermore the pressure change must be small compared to Po, and viscous friction may not cause

a significant temperature change. Continuum theory (KEYOPT(1) = 0) is valid for Knudsen numbers smaller than0.01.

The Knudsen number Kn of the squeeze film problem can be estimated by:

(7–187)Kn

L PP do ref

o=

Lo = mean free path length of the fluid

Pref = reference pressure for the mean free path Lo

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Section 7.8: Squeeze Film Theory

For systems which operate at high Knudsen numbers, an effective viscosity ηeff is introduced in order to consider

slip flow boundary conditions and models derived from Boltzmann equation. This assumption holds for Knudsennumbers up to 880 (Veijola(342)):

(7–188)

η ηeff

o o

o

L Pp d

=

+

1 9 6381 159

..

The surface accommodation coefficient, α, distinguishes between diffuse reflection (α = 1), specular reflection(α = 0), and molecular reflection (0 < α < 1) of the molecules at the walls of the squeeze film. Typical accommod-ation factors for silicon are reported between 0.8 and 0.9, those of metal surfaces are almost 1. Specular reflectiondecreases the effective viscosity at high Knudsen numbers compared to diffuse reflection. Different accommod-ation factors might be specified for each wall by using α1 and α2 (input as A1 and A2 on R command). The fit

functions for the effective viscosity are found in Veijola(342).

7.8.2. Flow in Channels

FLUID138 can be used to model the fluid flow though short circular and rectangular channels of micrometersize. The element assumes isothermal viscous flow at low Reynolds numbers, the channel length to be smallcompared to the acoustic wave length, and a small pressure drop with respect to ambient pressure.

In contrast to FLUID116, FLUID138 considers gas rarefaction, is more accurate for channels of rectangular crosssections, allows channel dimensions to be small compared to the mean free path, allows evacuated systems,and considers fringe effects at the inlet and outlet which considerably increase the damping force in case ofshort channel length. FLUID138 can be used to model the stiffening and damping effects of fluid flow in channelsof micro-electromechanical systems (MEMS).

Using continuum theory (KEYOPT(1) = 0) the flow rate Q of channels with circular cross-section (KEYOPT(3) = 0)is given by the Hagen-Poiseuille equation:

(7–189)Q

r Al

Pc

=2

8η∆

Q = flow rate in units of volume/timer = radiuslc = channel length

A = cross-sectional area∆P = pressure difference along channel length

This assumption holds for low Reynolds numbers (Re < 2300), for l >> r and r >> Lm where Lm is the mean freepath at the current pressure.

(7–190)L P

L PPm oo o

o( ) =

In case of rectangular cross sections (KEYOPT(3) = 1) the channel resistance depends on the aspect ratio ofchannel. The flow rate is defined by:

(7–191)Qr A

lPh

c=

8 2

ηχ∆

Chapter 7: Fluid Flow

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where:

rh = hydraulic radius (defined below)

A = true cross-sectional area (not that corresponding to the hydraulic radius)χ = so-called friction factor (defined below)

The hydraulic radius is defined by:

(7–192)r

AU

HWH Wh= =

+2 2

2( )

and the friction factor χ is approximated by:

(7–193)χ = − + +

−1 0 63 0 052

31

32

5 2 1. . ( )n n n

where:

H = height of channelW = width of channel (must be greater than H)n = H/W

A special treatment is necessary to consider high Knudsen numbers and short channel length (KEYOPT(1) = 1)(Sharipov(343)).

7.9. Slide Film Theory

Slide film damping occurs when surfaces move tangentially with respect to each other. Typical applications ofslide film models are damping between fingers of a comb drive and damping between large horizontally movingplates (seismic mass) and the silicon substrate. Slide film damping can be described by a nodal force displacementrelationship. FLUID139 is used to model slide film fluid behavior.

Slide film problems are defined by:

(7–194)ρ ν η ν∂

∂= ∂

∂t z

2

2

where:

P = pressureν = plate fluid velocityη = dynamic viscosityz = normal direction of the laterally moving platest = time

Slide film problems can be represented by a series connection of mass-damper elements with internal nodeswhere each damper represents the viscous shear stress between two fluid layers and each mass represents itsinertial force. The damper elements are defined by:

(7–195)C

Adi

= η

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Section 7.9: Slide Film Theory

where:

C = damping coefficientA = actual overlapping plate areadi = separation between two internal nodes (not the gap separation)

The mass of each internal node is given by:

(7–196)M Adi= ρ

where:

ρ = fluid density

In case of slip flow boundary conditions (KEYOPT(3) = 1) the fluid velocity at the moving plate is somewhatsmaller than the plate velocity itself. Slip flow BC can be considered by additional damper elements which areplaced outside the slide film whereby the damping coefficient must be:

(7–197)C

ALm

= η

where:

Lm = mean free path length of the fluid at the current pressure

In case of second order slip flow (KEYOPT(3) = 2) the damping coefficient is:

(7–198)CL

AdA

Kn emKn

= +

−−

η η0 1 0 788 10

1

. .

where Kn is defined with Equation 7–187

Note that all internal nodes are placed automatically by FLUID139.

Two node models are sufficient for systems where the operating frequency is below the cut-off frequency whichis defined by:

(7–199)f

dc = η

πρ2 2

where:

fc = cut-off frequency

d = gap separation

In this special case, damping coefficients are almost constant, regardless of the frequency, and inertial effectsare negligible. At higher frequencies, the damping ratio increases significantly up to a so-called maximum fre-quency, which is defined by:

(7–200)fLm

max = ηπρ2 2

Chapter 7: Fluid Flow

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where:

fmax = maximum frequency

The meaning of both numbers is illustrated below:

Figure 7.6 Flow Theory, Cut-off, and Maximum Frequency Interrelation

In case of large signal damping, the current overlapping plate are as defined by:

(7–201)A A

dAdu

u unew init n i= + −( )

where:

Anew = actual area

Ainit = initial area

ui = nodal displacement in operating direction of the first interface node

un = nodal displacement of the second interface node

For rectangular plates which move parallel to its edge, the area change with respect to the plate displacement(dA/du) is equal to the plate width. These applications are typical for micro-electromechanical systems as combdrives where the overlapping area changes with respect to deflection.

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Section 7.9: Slide Film Theory

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Chapter 8: Acoustics

8.1. Acoustic Fluid Fundamentals

8.1.1. Governing Equations

In acoustical fluid-structure interaction problems, the structural dynamics equation needs to be considered alongwith the Navier-Stokes equations of fluid momentum and the flow continuity equation. The discretized structuraldynamics equation can be formulated using the structural elements as shown in Equation 17–5. The fluid mo-mentum (Navier-Stokes) and continuity equations (Equation 7–1 and Equation 7–6 through Equation 7–8) aresimplified to get the acoustic wave equation using the following assumptions (Kinsler(84)):

1. The fluid is compressible (density changes due to pressure variations).

2. The fluid is inviscid (no viscous dissipation).

3. There is no mean flow of the fluid.

4. The mean density and pressure are uniform throughout the fluid.

The acoustic wave equation is given by:

(8–1)1

02

2

22

c

P

tP

∂ − ∇ =δ

where:

c = speed of sound ( )k oρ

in fluid medium (input as SONC on MP command)ρo = mean fluid density (input as DENS on MP command)

k = bulk modulus of fluidP = acoustic pressure (=P(x, y, z, t))t = time

Since the viscous dissipation has been neglected, Equation 8–1 is referred to as the lossless wave equation forpropagation of sound in fluids. The discretized structural Equation 17–5 and the lossless wave Equation 8–1 haveto be considered simultaneously in fluid-structure interaction problems. The lossless wave equation will be dis-cretized in the next subsection followed by the derivation of the damping matrix to account for the dissipationat the fluid-structure interface. The fluid pressure acting on the structure at the fluid-structure interface will beconsidered in the final subsection to form the coupling stiffness matrix.

For harmonically varying pressure, i.e.

(8–2)P Pe j t= ω

where:

P = amplitude of the pressure

j = −1

ω = 2πff = frequency of oscillations of the pressure

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Equation 8–1 reduces to the Helmholtz equation:

(8–3)ω2

22 0

cP P+ ∇ =

8.1.2. Discretization of the Lossless Wave Equation

The following matrix operators (gradient and divergence) are introduced for use in Equation 8–1:

(8–4)∇ ⋅ = = ∂

∂∂

∂∂∂

() L

x y zT

(8–5)∇ =() L

Equation 8–1 is rewritten as follows:

(8–6)1

02

2

2c

P

tP

∂∂

− ∇ ⋅ ∇ =

Using the notations given in Equation 8–4 and Equation 8–5, Equation 8–6 becomes in matrix notation:

(8–7)1

02

2

2c

P

tL L PT∂

∂− = ( )

The element matrices are obtained by discretizing the wave Equation 8–7 using the Galerkin procedure (Bathe(2)).Multiplying Equation 8–7 by a virtual change in pressure and integrating over the volume of the domain (Zien-kiewicz(86)) with some manipulation yields:

(8–8)12

2

2cP

P

td vol L P L P d vol n P L P

volT

volTδ δ δ∂

∂+ =∫ ∫( ) ( )( ) ( ) ( )) ( )d S

S∫

where:

vol = volume of domainδP = a virtual change in pressure (=δP(x, y, z, t))S = surface where the derivative of pressure normal to the surface is applied (a natural boundary condition)n = unit normal to the interface S

In the fluid-structure interaction problem, the surface S is treated as the interface. For the simplifying assumptionsmade, the fluid momentum equations yield the following relationships between the normal pressure gradientof the fluid and the normal acceleration of the structure at the fluid-structure interface S (Zienkiewicz(86)):

(8–9)

n P n

u

to⋅ ∇ = − ⋅ ∂

∂ρ

2

2

where:

u = displacement vector of the structure at the interface

Chapter 8: Acoustics

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In matrix notation, Equation 8–9 is given by:

(8–10) ( ) n L P n

tuT

oT= − ∂

∂

ρ

2

2

After substituting Equation 8–10 into Equation 8–8, the integral is given by:

(8–11)12

2

2

2

cP

P

td vol L P L P d vol P n

volT

vol oTδ δ ρ δ∂

∂+ = − ∂

∫ ∫( ) ( )( ) ( ) ∂∂

∫

tu d S

S2

( )

8.2. Derivation of Acoustics Fluid Matrices

Equation 8–11 contains the fluid pressure P and the structural displacement components ux, uy, and uz as the

dependent variables to solve. The finite element approximating shape functions for the spatial variation of thepressure and displacement components are given by:

(8–12)P N PTe=

(8–13)u N uTe= ′

where:

N = element shape function for pressureN' = element shape function for displacementsPe = nodal pressure vector

ue = uxe,uye,uze = nodal displacement component vectors

From Equation 8–12 and Equation 8–13, the second time derivative of the variables and the virtual change inthe pressure can be written as follows:

(8–14)∂∂

=2

2P

tN PT

e &&

(8–15)∂∂

= ′2

2tu N uT

e &&

(8–16)δ δP N PTe=

Let the matrix operator L applied to the element shape functions N be denoted by:

(8–17)[ ] B L N T=

Substituting Equation 8–12 through Equation 8–17 into Equation 8–11, the finite element statement of the waveEquation 8–1 is given by:

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Section 8.2: Derivation of Acoustics Fluid Matrices

(8–18)

12c

P N N d vol P P B B d vol PeT T

evol eT T

evo ( ) [ ] [ ] ( ) δ δ&&∫ +

ll

o eT T T

eS

P N n N d S u

∫

∫+ =′ρ δ ( ) && 0

where:

n = normal at the fluid boundary

Other terms are defined in Section 8.1: Acoustic Fluid Fundamentals. Terms which do not vary over the elementare taken out of the integration sign. δPe is an arbitrarily introduced virtual change in nodal pressure, and it can

be factored out in Equation 8–18. Since δPe is not equal to zero, Equation 8–18 becomes:

(8–19)

12c

N N d vol P B B d vol P

N n

Tvol e

Tevol

oT

( ) [ ] [ ] ( )

∫ ∫+

+

&&

ρ ( ) N d S uTe

S

′∫ =&& 0

Equation 8–19 can be written in matrix notation to get the discretized wave equation:

(8–20)[ ] [ ] [ ] M P K P R ueP

e eP

e o eT

e&& &&+ + =ρ 0

where:

[ ] ( )Mc

N N d voleP T

vol= =∫

12

fluid mass matrix (fluid)

[ ] [ ] [ ] ( )K B B d voleP T

vol= =∫ fluid stiffness matrix (fluid)

ρ ρo e oT T

S

R N n N d S[ ] ( )= =′∫ coupling mass matrix (fluid-strructure interface)

8.3. Absorption of Acoustical Pressure Wave

8.3.1. Addition of Dissipation due to Damping at the Boundary

In order to account for the dissipation of energy due to damping, if any, present at the fluid boundary, a dissipationterm is added to the lossless Equation 8–1 to get (Craggs(85)):

(8–21)δ δ δ

ρP

c

P

td vol P L L P d vol P

rcvol

Tvol o

1 12

2

2∂∂

− +

∫ ∫( ) ( ) ( )

ccPt

d SS∫

∂∂

=( ) 0

where:

r = characteristic impedance of the material at the boundary

Other terms are defined in Section 8.1: Acoustic Fluid Fundamentals.

Chapter 8: Acoustics

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Since it is assumed that the dissipation occurs only at the boundary surface S, the dissipation term in Equation 8–21is integrated over the surface S:

(8–22)D P

rc c

Pt

d SoS

=

∂∂∫ δ

ρ1

( )

where:

D = dissipation term

Using the finite element approximation for P given by Equation 8–15:

(8–23)D P N

rc c

N d SPte

T

o

TS

e=

∂∂

∫ ( )δ

ρ1

Using the following notations:

βρ

= =rco

boundary absorption coefficient (input as MU on MP ccommand)

&PPtee= ∂

∂

βc and δPe are constant over the surface of the element and can be taken out of the integration. Equation 8–23

is rewritten as:

(8–24)D P

cN N d S Pe

T TS e= ∫ ( ) δ β &

The dissipation term given by Equation 8–24 is added to Equation 8–18 to account for the energy loss at theabsorbing boundary surface.

(8–25)[ ] ( ) C P

cN N d S Pe

Pe

TeS

& &= ∫β

where:

[ ] ( )Cc

N N d SeP T

S= =∫

β(fluid damping matrix)

Finally, combining Equation 8–20 and Equation 8–25, the discretized wave equation accounting for losses at theinterface is given by:

(8–26)[ ] [ ] [ ] [ ] M P C P K P R ueP

e eP

e eP

e o eT

e&& & &&+ + + =ρ 0

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Section 8.3: Absorption of Acoustical Pressure Wave

8.4. Acoustics Fluid-Structure Coupling

In order to completely describe the fluid-structure interaction problem, the fluid pressure load acting at the in-terface is now added to Equation 17–5. This effect is included in FLUID29 and FLUID30 only if KEYOPT(2) = 0. So,the structural equation is rewritten here:

(8–27)[ ] [ ] [ ] M u C u K u F Fe e e e e e e epr&& &+ + = +

The fluid pressure load vector Fepr

at the interface S is obtained by integrating the pressure over the area ofthe surface:

(8–28) ( )F N P n d Se

pr

S

= ′∫

where:

N' = shape functions employed to discretize the displacement components u, v, and w (obtained from thestructural element)n = normal at the fluid boundary

Substituting the finite element approximating function for pressure given by Equation 8–12 into Equation 8–19:

(8–29) ( ) F N N n d S Pe

pr T

Se= ′∫

By comparing the integral in Equation 8–29 with the matrix definition of ρo [Re]T in Equation 8–20, it becomes

clear that:

(8–30) [ ] F R Pepr

e e=

where:

[ ] ( )R N N n d SeT T

S= ′∫

The substitution of Equation 8–30 into Equation 8–27 results in the dynamic elemental equation of the structure:

(8–31)[ ] [ ] [ ] [ ] M u C u K u R P Fe e e e e e e e e&& &+ + − =

Equation 8–26 and Equation 8–31 describe the complete finite element discretized equations for the fluid-structure interaction problem and are written in assembled form as:

(8–32)

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [

M

M M

u

P

C

C

efs

ep

e

e

e

ep

0 0

0

+&&&& ]]

[ ] [ ]

[ ] [ ]

+

&&u

P

K K

K

u

P

e

e

efs

ep

e

0 ee

eF

=

0

Chapter 8: Acoustics

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where:

[Mfs] = ρo [Re]T

[Kfs] = -[Re]

For a problem involving fluid-structure interaction, therefore, the acoustic fluid element will generate all the

submatrices with superscript p in addition to the coupling submatrices ρo [Re]T and [Re]. Submatrices without a

superscript will be generated by the compatible structural element used in the model.

8.5. Acoustics Output Quantities

The pressure gradient is evaluated at the element centroid using the computed nodal pressure values.

(8–33)∂∂

= ∂∂

Px

Nx

PT

e

(8–34)∂∂

= ∂∂

Py

Ny

PT

e

(8–35)∂∂

= ∂∂

Pz

Nz

PT

e

where:

∂∂

∂∂

∂∂

=Px

Py

Pz

, , andgradients in x, y and z directions, reespectively,

(output quantities PGX, PGY and PGZ)

Other terms are defined in Section 8.1: Acoustic Fluid Fundamentals and Section 8.2: Derivation of AcousticsFluid Matrices.

The element fluid velocity is computed at the element centroid for the full harmonic analysis (ANTYPE,HARMwith HROPT,FULL) by:

(8–36)V

j Pxx

o= ∂

∂ρ ω

(8–37)V

j Pyy

o= ∂

∂ρ ω

(8–38)V

j Pzz

o= ∂

∂ρ ω

where:

Vx, Vy, and Vz = components of the fluid velocity in the x, y, and z directions, respectively (output quantities

VLX, VLY and VLZ)ω = 2πf

8–7ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 8.5: Acoustics Output Quantities

f = frequency of oscillations of the pressure wave (input on HARFRQ command)

j = −1

The sound pressure level is computed by:

(8–39)L

PPsprms

ref=

20 log

where:

Lsp = sound pressure level (output as SOUND PR. LEVEL)log = logarithm to the base 10

Pref = reference pressure (input as PREF on R command, defaults to 20 x 10-6)

Prms = root mean square pressure (Prms = P /2

)

Chapter 8: Acoustics

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Chapter 11: Coupling

11.1. Coupled Effects

11.1.1. Introduction

Coupled-field analyses are useful for solving problems where the coupled interaction of phenomena from variousdisciplines of physical science is significant. Several examples of this include: an electric field interacting with amagnetic field, a magnetic field producing structural forces, a temperature field influencing fluid flow, a temper-ature field giving rise to thermal strains and the usual influence of temperature dependent material properties.The latter two examples can be modeled with most non-coupled-field elements, as well as with coupled-fieldelements. The following elements have coupled-field capability:

Table 11.1 Elements Used for Coupled Effects

3-D Coupled-Field Solid (Section 5.2: Derivation of Electromagnetic Matrices, Section 11.1: CoupledEffects, Section 14.5: SOLID5 - 3-D Coupled-Field Solid)

SOLID5

2-D Coupled-Field Solid (Section 5.2: Derivation of Electromagnetic Matrices, Section 11.1: CoupledEffects, Section 14.5: SOLID5 - 3-D Coupled-Field Solid)

PLANE13

2-D Acoustic Fluid (Section 8.2: Derivation of Acoustics Fluid Matrices, Section 14.29: FLUID29 - 2-DAcoustic Fluid)

FLUID29

3-D Acoustic Fluid (Section 8.2: Derivation of Acoustics Fluid Matrices, Section 14.30: FLUID30 - 3-DAcoustic Fluid)

FLUID30

2-D 8-Node Magnetic Solid (Section 5.2: Derivation of Electromagnetic Matrices, Section 5.3: Electro-magnetic Field Evaluations, Section 14.53: PLANE53 - 2-D 8-Node Magnetic Solid)

PLANE53

3-D Magneto-Structural Solid (Section 14.62: SOLID62 - 3-D Magneto-Structural Solid)SOLID62

2-D Coupled Thermal-Electric Solid (Section 14.67: PLANE67 - 2-D Coupled Thermal-Electric Solid)PLANE67

Coupled Thermal-Electric Line (Section 14.68: LINK68 - Coupled Thermal-Electric Line)LINK68

3-D Coupled Thermal-Electric Solid (Section 14.69: SOLID69 - 3-D Coupled Thermal-Electric Solid)SOLID69

3-D Magnetic Solid (Section 14.97: SOLID97 - 3-D Magnetic Solid)SOLID97

Tetrahedral Coupled-Field Solid (Section 5.2: Derivation of Electromagnetic Matrices, Section 11.1:Coupled Effects, Section 14.98: SOLID98 - Tetrahedral Coupled-Field Solid)

SOLID98

2-D Electromechanical Transducer (Section 5.9: Electromechanical Transducers, Section 11.5: Reviewof Coupled Electromechanical Methods, Section 14.109: TRANS109 - 2-D Electromechanical Trans-ducer)

TRANS109

Coupled Thermal-Fluid Pipe (Section 14.116: FLUID116 - Coupled Thermal-Fluid Pipe)FLUID116

Electric Circuit Element (Section 5.4: Voltage Forced and Circuit-Coupled Magnetic Field, Sec-tion 14.124: CIRCU124 - Electric Circuit)

CIRCU124

Electromechanical Transducer (Section 5.9: Electromechanical Transducers, Section 5.10: CapacitanceComputation, Section 5.11: Open Boundary Analysis with a Trefftz Domain, Section 11.5: Review ofCoupled Electromechanical Methods, Section 14.126: TRANS126 - Electromechanical Transducer)

TRANS126

2-D Fluid (Section 7.2: Derivation of Fluid Flow Matrices, Section 14.141: FLUID141 - 2-D Fluid-Thermal)FLUID141

3-D Fluid (Section 7.2: Derivation of Fluid Flow Matrices, Section 14.142: FLUID142 - 3-D Fluid-Thermal)FLUID142

Coupled Thermal-Electric Shell (Section 14.157: SHELL157 - Thermal-Electric Shell)SHELL157

2-D 8-Node Coupled-Field Solid (Section 14.223: PLANE223 - 2-D 8-Node Coupled-Field Solid)PLANE223

3-D 20-Node Coupled-Field Solid (Section 14.226: SOLID226 - 3-D 20-Node Coupled-Field Solid)SOLID226

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3-D 10-Node Coupled-Field Solid (Section 14.227: SOLID227 - 3-D 10-Node Coupled-Field Solid)SOLID227

There are certain advantages and disadvantages inherent with coupled-field formulations:

11.1.1.1. Advantages

1. Allows for solutions to problems otherwise not possible with usual finite elements.

2. Simplifies modeling of coupled-field problems by permitting one element type to be used in a singleanalysis pass.

11.1.1.2. Disadvantages

1. Increases wavefront (unless a segregated solver is used).

2. Inefficient matrix reformulation (if a section of a matrix associated with one phenomena is reformed, theentire matrix will be reformed).

3. Larger storage requirements.

11.1.2. Coupling

There are basically two methods of coupling distinguished by the finite element formulation techniques usedto develop the matrix equations. These are illustrated here with two types of degrees of freedom (X1, X2):

1. Strong (simultaneous, full) coupling - where the matrix equation is of the form:

[ ] [ ]

[ ] [ ]

K K

K K

X

X

F

F11 12

21 22

1

2

1

2

=

(11–1)

and the coupled effect is accounted for by the presence of the off-diagonal submatrices [K12] and [K21].

This method provides for a coupled response in the solution after one iteration.

2. Weak (sequential) coupling - where the coupling in the matrix equation is shown in the most generalform:

(11–2)

[ ( , )] [ ]

[ ] [ ( , )]

K X X

K X X

X

X11 1 2

22 1 2

1

2

0

0

=F X X

F X X1 1 2

2 1 2

( , )

( , )

and the coupled effect is accounted for in the dependency of [K11] and F1 on X2 as well as [K22] and

F2 on X1. At least two iterations are required to achieve a coupled response.

The following is a list of the types of coupled-field analyses including methods of coupling present in each:

Table 11.2 Coupling Methods

Example ApplicationCouplingMethod

UsedAnalysis Category

High temperature turbineWSection 11.1.2.1: Thermal-Structural Analysis

Solenoid, high energy magnets (MRI)WSection 11.1.2.2: Magneto-Structural Analysis (VectorPotential)

Chapter 11: Coupling

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Section 11.1.2.3: Magneto-Structural Analysis (ScalarPotential)

Example ApplicationCouplingMethod

UsedAnalysis Category

Current fed massive conductorsSSection 11.1.2.4: Electromagnetic Analysis

Direct current electromechanical devices ingeneral

WSection 11.1.2.5: Electro-Magneto-Thermo-StructuralAnalysis

Section 11.1.2.6: Electro-Magneto-Thermal Analysis

Transducers, resonatorsSSection 11.1.2.7: Piezoelectric Analysis

MEMSWElectromechanical - ESSOLV

Pressure and force sensorsWSection 11.1.2.8: Piezoresistive Analysis

MEMSSSection 5.9: Electromechanical Transducers

Piping networksS, WSection 11.1.2.9: Thermo-Pressure Analysis

Fluid structure interactionWSection 11.1.2.10: Velocity-Thermo-Pressure Analysis

AcousticsSSection 11.1.2.11: Pressure-Structural (Acoustic) Analysis

High temperature electronics, Peltier cool-ers, thermoelectric generators

S, WSection 11.1.2.12: Thermo-Electric Analysis

Direct current transients: power interrupts,surge protection

WSection 11.1.2.13: Magnetic-Thermal Analysis

Circuit-fed solenoids, transformers, andmotors

SSection 11.1.2.14: Circuit-Magnetic Analysis

where:

S = strong couplingW = weak coupling

The solution sequence follows the standard finite element methodology. Convergence is achieved when changesin all unknowns (i.e. DOF) and knowns, regardless of units, are less than the values specified (on the CNVTOLcommand) (except for FLUID141 and FLUID142). Some of the coupling described above is always or usually one-way. For example, in Category A, the temperatures affect the displacements of the structure by way of the thermalstrains, but the displacements usually do not affect the temperatures.

The following descriptions of coupled phenomena will include:

1. Applicable element types

2. Basic matrix equation indicating coupling terms in bold print. In addition to the terms indicated in boldprint, any equation with temperature as a degree of freedom can have temperature-dependency in allterms. FLUID141 and FLUID142 have coupling indicated with a different method.

3. Applicable analysis types, including the matrix and/or vector terms possible in each analysis type.

The nomenclature used on the following pages is given in Table 11.3: “Nomenclature of Coefficient Matrices” atthe end of the section. In some cases, element KEYOPTS are used to select the DOF of the element. DOF will notbe fully active unless the appropriate material properties are specified. Some of the elements listed may not beapplicable for a particular use as it may be only 1-D, whereas a 3-D element is needed (e.g. FLUID116).

11–3ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 11.1: Coupled Effects

11.1.2.1. Thermal-Structural Analysis

(see Section 2.2: Derivation of Structural Matrices and Section 6.2: Derivation of Heat Flow Matrices)

1. Element type: SOLID5, PLANE13, SOLID98

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

M u0

0 0

0

0

+

&&&&

&

T

C

Ct

uu K

[ ]

[ ] [ ]

&T K

u

T

F

Qt

+[ ]

=

0

0 (11–3)

where:

[Kt] = [Ktb] + [Ktc]

F = Fnd + Fth + Fpr + Fac

Q = Qnd + Qg + Qc

3. Analysis types: Static or Transient

11.1.2.2. Magneto-Structural Analysis (Vector Potential)

(see Section 5.2: Derivation of Electromagnetic Matrices and Section 11.2: Piezoelectrics)

1. Element type: PLANE13, SOLID62

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

M

A

C

Cm

0

0 0

0

0

+

&&&&

&u uu K u

[ ] [ ]

[ ] [ ]

&A K A

Fm

i

+

=0

0 ψ

(11–4)

where:

F = Fnd + Fpr + Fac + Fth + Fjb + Fmx

Ψi ind s pm= + +ψ ψ ψ

3. Analysis types: Static or Transient

11.1.2.3. Magneto-Structural Analysis (Scalar Potential)

1. Element type: SOLID5, SOLID98

2. Matrix equation:

[ ] [ ]

[ ] [ ]

K

K

u Fm

f

0

0

=

φ ψ (11–5)

where:

F = Fnd + Fpr + Fac + Fth + Fmx

Ψf fnd b pm= + +ψ ψ ψ

Chapter 11: Coupling

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3. Analysis types: Static

11.1.2.4. Electromagnetic Analysis

(see Section 5.2: Derivation of Electromagnetic Matrices and Section 5.4: Voltage Forced and Circuit-CoupledMagnetic Field)

1. Element type: PLANE13, PLANE53, SOLID97

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[

C AAA C

C C

KAv

Av T vv

AA

+

&

&ν0

0]] [ ]

0

=

A

Ii

νψ

(11–6)

where:

Ψi ind s pm= + +ψ ψ ψ

I = Ind

3. Analysis types: Harmonic or Transient

11.1.2.5. Electro-Magneto-Thermo-Structural Analysis

(see Section 5.2: Derivation of Electromagnetic Matrices and Section 11.2: Piezoelectrics)

1. Element types: SOLID5, SOLID98

2. Matrix equation:

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

M 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

+

[ ] [ ] [ ] [ ]

[ ] [ ]

&&&&&&&

u

T

V

C

Ct

φ

0 0 0

0 00 0

0 0 0 0

0 0 0 0

[ ][ ][ ]

[ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

&&&

u

T

V&&φ

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[

+

K

K

K

t

v

0 0 0

0 0 0

0 0 0

00 0 0] [ ] [ ] [ ]

K

u

T

V

F

m

=

φ

Q

I

fψ

(11–7)

where:

[Kt] = [Ktb] + [Ktc]

F = Fnd + Fth + Fac + Fjb + Fpr + Fmx

Q = Qnd + Qg + Qj + Qc

I = Ind

Ψf fnd g pm= + +ψ ψ ψ

3. Analysis types: Static or Transient

11–5ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 11.1: Coupled Effects

11.1.2.6. Electro-Magneto-Thermal Analysis

(see Section 5.2: Derivation of Electromagnetic Matrices)

1. Element types: SOLID5, SOLID98

2. Matrix equation:

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

C T

V

t 0 0

0 0 0

0 0 0

&&&φ

+

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

K

K

K

Q

I

t

v

m f

0 0

0 0

0 0 ψ

(11–8)

where:

[Kt] = [Ktb] + [Ktc]

Q = Qnd + Qg + Qj + Qc

I = Ind

Ψf fnd g pm= + +ψ ψ ψ

3. Analysis types: Static or Transient

11.1.2.7. Piezoelectric Analysis

(see Section 11.2: Piezoelectrics)

1. Element types: SOLID5, PLANE13, SOLID98, PLANE223, SOLID226, and SOLID227.

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

M u

V

C u0

0 0

0

0 0

+

&&&&

&&&V

K K

K K

u

V

F

L

z

z T d

[ ] [ ]

[ ] [ ]

+

=

(11–9)

where:

F = Fnd + Fth + Fac + Fpr

L = Lnd + Lc + Lsc

Note — Lc and Lsc are applicable to only PLANE223, SOLID226, and SOLID227.

3. Analysis types: Static, modal, harmonic, or transient

11.1.2.8. Piezoresistive Analysis

(see Section 2.2: Derivation of Structural Matrices, Section 5.2: Derivation of Electromagnetic Matrices, and Sec-tion 6.5: Piezoresistive Analysis in the ANSYS Coupled-Field Analysis Guide)

1. Element type: PLANE223, SOLID226, SOLID227

2. Matrix equation:

Chapter 11: Coupling

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[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

M u

V

C u0

0 0

0

0 0

+

&&&&

&&&V

K

K

u

V

F

Iv

[ ] [ ]

[ ] [ ]

+

=

0

0 (11–10)

where:

[Kv] = conductivity matrix (see Equation 11–55) updated for piezoresistive effects

F = Fnd + Fth + Fpr + Fac)

I = Ind

3. Analysis types: Static or transient

11.1.2.9. Thermo-Pressure Analysis

(see Section 14.116: FLUID116 - Coupled Thermal-Fluid Pipe)

1. Element type: FLUID116

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

C T

P

K

K

t t

p

0

0 0

0

0

+

&&

=

T

P

Q

W (11–11)

where:

[Kt] = [Ktb] + [Ktc] + [Ktm]

Q = Qnd + Qc + Qg

W = Wnd + Wh

3. Analysis types: Static or Transient

11.1.2.10. Velocity-Thermo-Pressure Analysis

(See Section 7.2: Derivation of Fluid Flow Matrices)

1. Element type: FLUID141 and FLUID142

2. Matrix equation ([A] matrices combine effects of [C] and [K] matrices):

[ ] A V FVXx

NX− (11–12)

(11–13)[ ] A V FVY

yNY=

(11–14)[ ] A V FVZz

NZ=

(11–15)[ ] A P FP P=

(11–16)[ ] A T FT T=

(11–17)[ ] A k FK K=

11–7ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 11.1: Coupled Effects

(11–18)[ ] A Fε εε =

where:

[AVX] = advection-diffusion matrix for Vx velocities = function of previous Vx, Vy, Vz, T, k, and

ε

[AVY] = advection-diffusion matrix for Vy velocities = function of previous Vx, Vy, Vz, T, k, and

ε

[AVZ] = advection-diffusion matrix for Vz velocities = function of previous Vx, Vy, Vz, T, k, and ε

[AP] = pressure coefficient matrix = function of previous Vx, Vy, Vz, T, k, and ε

[AT] = advection-diffusion matrix for temperature = function of previous Vx, Vy, Vz, and T

[Ak] = advection-diffusion matrix for turbulent kinetic energy = function of previous Vx, Vy, Vz,

k, and ε

[Aε] = advection-diffusion matrix for dissipation energy = function of previous Vx, Vy, Vz, k, and

ε

FVX = load vector for Vx velocities = function of previous P and T

FVY = load vector for Vy velocities = function of previous P and T

FVZ = load vector for Vz velocities = function of previous P and T

FP = pressure load vector = function of previous Vx, Vy and Vz

FT = heat flow vector = function of previous T

Fk = turbulent kinetic energy load vector = function of previous Vx, Vy, Vz, T, k, and ε

Fε = dissipation rate load vector = function of previous Vx, Vy, Vz, k, and ε

3. Analysis types: Static or Transient

11.1.2.11. Pressure-Structural (Acoustic) Analysis

(see Section 8.2: Derivation of Acoustics Fluid Matrices)

1. Element type: FLUID29 and FLUID30 (with other structural elements)

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

M

M

u

P

C

Cp p

0 0

0Mfs

+

&&&&

+

=

[ ] [ ]

[ ] [ ]

&&u

P

K

K

u

P

Fp

Kfs

0 W

(11–19)

where:

F = Fnd

W = Wnd

Note that [M], [C], and [K] are provided by other elements.

3. Analysis types: Transient, harmonic and modal analyses can be performed. Applicable matrices are shownin the following table:

Chapter 11: Coupling

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HarmonicTransient

Modal

Sym.Unsym.Sym.Unsym.Damped

****[M]

****[Mfs]

******[Mp]

***[C]

****[Cp]

****[K]

****[Kfs]

******[Kp]

**Fnd

11.1.2.12. Thermo-Electric Analysis

1. Element types: SOLID5, PLANE67, LINK68, SOLID69, SOLID98, SHELL157, PLANE223, SOLID226, andSOLID227

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

C

C

T

V

K

K K

t

v

t

vt v

0

0

0

+

&&

=

T

V

Q

I (11–20)

where:

[Kt] = [Ktb] + [Ktc]

Q = Qnd + Qc + Qg + Qj + Qp

I = Ind

Note — Qp, [Kvt], and [Cv] are used only for PLANE223, SOLID226, and SOLID227.

3. Analysis types: Static or Transient

11.1.2.13. Magnetic-Thermal Analysis

(see Section 5.2: Derivation of Electromagnetic Matrices)

1. Element type: PLANE13

2. Matrix equation:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

C

C

A

T

K

K

AA

t

AA

t

0

0

0

0

+

&&

=

A

T Qiψ

(11–21)

where:

[Kt] = [Ktb] + [Ktc]

11–9ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 11.1: Coupled Effects

Ψi ind s pm= + +ψ ψ ψ

Q = Qnd + Qg + Qj + Qc

3. Analysis types: Static or Transient

11.1.2.14. Circuit-Magnetic Analysis

(see Section 5.4: Voltage Forced and Circuit-Coupled Magnetic Field)

1. Element type: PLANE53, SOLID97, CIRCU124

2. Matrix equation:

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

0 0 0

0 0

0 0 0

0

0

C

AiA

&

+

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

K K

K K

A

i

AA Ai

ii ie

0

0

0 0 0

e

=

0

0

0(11–22)

3. Analysis types: Static, Transient, or Harmonic

Table 11.3 Nomenclature of Coefficient Matrices

UsageMeaningSymbol

[1]structural mass matrix (discussed in Section 2.2: Derivation of Structural Matrices)[M]

[1]fluid-structure coupling mass matrix (discussed in Section 8.2: Derivation of AcousticsFluid Matrices)

[Mfs]

[1]acoustic mass matrix (discussed in Section 8.2: Derivation of Acoustics Fluid Matrices)[Mp]

[2]structural damping matrix (discussed in Section 2.2: Derivation of Structural Matrices)[C]

[2]thermal specific heat matrix (discussed in Section 6.2: Derivation of Heat Flow Matrices)[Ct]

[2]magnetic damping matrix (discussed in Section 5.3: Electromagnetic Field Evaluations)[CAA]

[2]acoustic damping matrix (discussed in Section 8.2: Derivation of Acoustics Fluid Matrices)[Cp]

[2]magnetic-electric damping matrix (discussed in Section 5.2: Derivation of Electromag-netic Matrices)

[CAv]

[2]electric damping matrix (discussed in Section 5.2: Derivation of Electromagnetic Matrices)[Cvv]

[2]inductive damping matrix (discussed in Section 5.4: Voltage Forced and Circuit-CoupledMagnetic Field)

[CiA]

[2]dielectric permittivity coefficient matrix (discussed in Section 5.2.3.1: Quasistatic ElectricAnalysis)

[Cv]

[3]structural stiffness matrix (discussed in Section 2.2: Derivation of Structural Matrices)[K]

[3]thermal conductivity matrix (may consist of 1, 2, or 3 of the following 3 matrices) (dis-cussed in Section 6.2: Derivation of Heat Flow Matrices)

[Kt]

[3]thermal conductivity matrix of material (discussed in Section 6.2: Derivation of HeatFlow Matrices)

[Ktb]

[3]thermal conductivity matrix of convection surface (discussed in Section 6.2: Derivationof Heat Flow Matrices)

[Ktc]

[3]thermal conductivity matrix associated with mass transport (discussed in Section 6.2:Derivation of Heat Flow Matrices)

[Ktm]

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UsageMeaningSymbol

[3]scalar magnetic potential coefficient matrix (discussed in Section 5.2: Derivation ofElectromagnetic Matrices)

[Km]

[3]vector magnetic potential coefficient matrix (discussed in Section 5.2: Derivation ofElectromagnetic Matrices)

[KAA]

[3]potential-current coupling stiffness matrix (discussed in Section 5.4: Voltage Forcedand Circuit-Coupled Magnetic Field)

[KAi]

[3]resistive stiffness matrix (discussed in Section 5.4: Voltage Forced and Circuit-CoupledMagnetic Field)

[Kii]

[3]current-emf coupling stiffness (discussed in Section 5.4: Voltage Forced and Circuit-Coupled Magnetic Field)

[Kie]

[3]electrical conductivity coefficient matrix (discussed in Section 5.2: Derivation of Electro-magnetic Matrices)

[Kv]

[3]piezoelectric stiffness matrix (discussed in Section 11.2: Piezoelectrics)[Kz]

[3]dielectric coefficient matrix (discussed in Section 11.2: Piezoelectrics)[Kd]

[3]momentum matrix due to diffusion (discussed in Section 7.2: Derivation of Fluid FlowMatrices)

[Kf]

[3]buoyancy matrix (discussed in Section 7.2: Derivation of Fluid Flow Matrices)[Kg]

[3]pressure gradient matrix (discussed in Section 7.2: Derivation of Fluid Flow Matrices)[Kc]

[3]pressure coefficient or fluid stiffness matrix (discussed in Section 7.2: Derivation of FluidFlow Matrices)

[Kp]

[3]fluid-structure coupling stiffness matrix (discussed in Section 7.2: Derivation of FluidFlow Matrices)

[Kfs]

[3]Seebeck coefficient coupling matrix[Kvt]

1. Coefficient matrices of second time derivatives of unknowns.

2. Coefficient matrices of first time derivative of unknowns

3. Coefficient matrices of unknowns

Vectors of Knowns

Associated Input / Out-put Label

MeaningSymbol

FX ... MZapplied nodal force vector (discussed in Section 2.2: Derivation of Struc-tural Matrices)

Fnd

FX ... MZNewton-Raphson restoring load vector (discussed in Section 15.11:Newton-Raphson Procedure

Fnr

FX ... MZthermal strain force vector (discussed in Section 2.2: Derivation of Struc-tural Matrices)

Fth

FX ... MZpressure load vector (discussed in Section 2.2: Derivation of StructuralMatrices)

Fpr

FX ... MZforce vector due to acceleration effects (i.e., gravity) (discussed in Sec-tion 2.2: Derivation of Structural Matrices)

Fac

FX ... FZLorentz force vector (discussed in Section 5.2: Derivation of Electromag-netic Matrices)

Fjb

FX ... FZMaxwell force vector (discussed in Section 5.2: Derivation of Electromag-netic Matrices)

Fmx

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Section 11.1: Coupled Effects

Associated Input / Out-put Label

MeaningSymbol

FX ... MZbody force load vector due to non-gravity effects (discussed in Section 6.2:Derivation of Heat Flow Matrices)

Fb

HEAT, HBOT, HE2, ... HTOPapplied nodal heat flow rate vector (discussed in Section 6.2: Derivationof Heat Flow Matrices)

Qnd

HEAT, HBOT, HE2, ... HTOPheat flux vector (discussed in Section 6.2: Derivation of Heat Flow Matrices)Qf

HEAT, HBOT, HE2, ... HTOPconvection surface vector (discussed in Section 6.2: Derivation of HeatFlow Matrices)

Qc

HEAT, HBOT, HE2, ... HTOPheat generation rate vector for causes other than Joule heating (discussedin Section 6.2: Derivation of Heat Flow Matrices)

Qg

HEATheat generation rate vector for Joule heating (discussed in Section 5.3:Electromagnetic Field Evaluations)

Qj

HEATPeltier heat flux vectorQp

CSGX, CSGY, CSGZapplied nodal source current vector (associated with A) (discussed inSection 5.2: Derivation of Electromagnetic Matrices)

ψind

FLUXapplied nodal flux vector (associated with φ) (discussed in Section 5.2:Derivation of Electromagnetic Matrices)

ψ fnd

FLUXSource (Biot-Savart) vector (discussed in Section 5.2: Derivation of Electro-magnetic Matrices)

Ψg

FLUXcoercive force (permanent magnet) vector (discussed in Section 5.2: De-rivation of Electromagnetic Matrices)

Ψpm

FLUXsource current vector (discussed in Section 5.2: Derivation of Electromag-netic Matrices)

Ψs

AMPSapplied nodal electric current vector (discussed in Section 5.2: Derivationof Electromagnetic Matrices)

Ind

AMPS (CHRG forPLANE223, SOLID226, andSOLID227)

applied nodal charge vector (discussed in Section 11.2: Piezoelectrics)Lnd

CHRGDcharge density load vector (discussed in Section 5.2: Derivation of Electro-magnetic Matrices)

Lc

CHRGSsurface charge density load vector (discussed in Section 5.2: Derivationof Electromagnetic Matrices)

Lsc

FLOWapplied nodal fluid flow vector (discussed in Section 14.116: FLUID116 -Coupled Thermal-Fluid Pipe)

Wnd

FLOWstatic head vector (discussed in Section 14.116: FLUID116 - CoupledThermal-Fluid Pipe)

Wh

Vectors of Unknowns

UX ... ROTZdisplacement vector (discussed in Section 2.2: Derivation of StructuralMatrices)

u

TEMP, TBOT, TE2, ...TTOP

thermal potential (temperature) vector (discussed in (discussed in Sec-tion 6.2: Derivation of Heat Flow Matrices and Section 7.2: Derivation ofFluid Flow Matrices)

T

VOLTelectric potential vector (discussed in Section 5.2: Derivation of Electromag-netic Matrices)

V

VOLTtime integrated electric potential vector (discussed in Section 5.2: Derivationof Electromagnetic Matrices)

ν

Chapter 11: Coupling

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MAGmagnetic scalar potential vector (discussed in Section 5.2: Derivation ofElectromagnetic Matrices)

φ

AX, AY, AZmagnetic vector potential vector (discussed in Section 5.2: Derivation ofElectromagnetic Matrices)

A

CURRelectric current vector (discussed in Section 5.4: Voltage Forced and Circuit-Coupled Magnetic Field)

i

EMFelectromagnetic force drop vector (discussed in Section 5.4: Voltage Forcedand Circuit-Coupled Magnetic Field)

e

PRESpressure vector (discussed in Section 7.2: Derivation of Fluid Flow Matricesand Section 8.2: Derivation of Acoustics Fluid Matrices)

P

VX, VY, VZvelocity (discussed in Section 7.2: Derivation of Fluid Flow Matrices)v

ENKEturbulent kinetic energy (discussed in Section 7.2: Derivation of Fluid FlowMatrices)

k

ENDSturbulent dissipation energy (discussed in Section 7.2: Derivation of FluidFlow Matrices)

ε

time derivative.

second time derivative. .

11.2. Piezoelectrics

The capability of modeling piezoelectric response exists in the following elements:

SOLID5 - Coupled-Field Solid ElementPLANE13 - 2-D Coupled-Field Solid ElementSOLID98 - Tetrahedral Coupled-Field Solid ElementPLANE223 - 2-D 8-Node Coupled-Field Solid ElementSOLID226 - 3-D 20-Node Coupled-Field Solid ElementSOLID227 - 3-D 10-Node Coupled-Field Solid Element

Variational principles are used to develop the finite element equations which incorporate the piezoelectric effect(Allik(81)).

The electromechanical constitutive equations for linear material behavior are:

(11–23) [ ] [ ] T c S e E= −

(11–24) [ ] [ ] D e S ET= + ε

or equivalently

(11–25)

[ ] [ ]

[ ] [ ]

T

D

c e

e

S

ET

=−

−

ε

where:

T = stress vector (referred to as σ elsewhere in this manual)D = electric flux density vectorS = strain vector (referred to as ε elsewhere in this manual)E = electric field vector

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Section 11.2: Piezoelectrics

[c] = elasticity matrix (evaluated at constant electric field (referred to as [D] elsewhere in this manual))[e] = piezoelectric stress matrix[ε] = dielectric matrix (evaluated at constant mechanical strain)

Equation 11–23 and Equation 11–24 are the usual constitutive equations for structural and electrical fields, re-spectively, except for the coupling terms involving the piezoelectric matrix [e].

The elasticity matrix [c] is the usual [D] matrix described in Section 2.1: Structural Fundamentals (input using the

MP commands). It can also be input directly in uninverted form [c] or in inverted form [c]-1 as a general anisotropicsymmetric matrix (input using TB,ANEL):

(11–26)[ ]c

c c

c c

c

c c c

c c c

c c c=

c

Symmetric

11 12 13

22 23

33

14 15 16

24 25 26

34 35 36

cc44 c c

c c

c

45 46

55 56

66

The piezoelectric stress matrix [e] (input using TB,PIEZ with TBOPT = 0) relates the electric field vector E in theorder X, Y, Z to the stress vector T in the order X, Y, Z, XY, YZ, XZ and is of the form:

(11–27)[ ]e

e e e

e e e

e e e

e e e

e e e

e e e

=

11 12 13

21 22 23

31 32 33

41 42 43

51 52 53

61 62 63

The piezoelectric matrix can also be input as a piezoelectric strain matrix [d] (input using TB,PIEZ with TBOPT =1). ANSYS will automatically convert the piezoelectric strain matrix [d] to a piezoelectric stress matrix [e] usingthe elasticity matrix [c] at the first defined temperature:

(11–28)[ ] [ ][ ]e c d=

The orthotropic dielectric matrix [ε] uses the electrical permittivities (input as PERX, PERY and PERZ on the MPcommands) and is of the form:

(11–29)[ ]ε

εε

ε=

11

22

33

0 0

0 0

0 0

The anisotropic dielectric matrix at constant strain [εS] (input used by (used by TB,DPER,,,,0 command) PLANE223,SOLID226, and SOLID227) and is of the form:

(11–30)[ ]ε

ε ε εε ε

ε=

11 12 13

22 23

33Symm

Chapter 11: Coupling

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.11–14

The dielectric matrix can also be input as a dielectric permittivity matrix at constant stress [εT] (input usingTB,DPER,,,,1). The program will automatically convert the dielectric matrix at constant stress to a dielectric matrixat constant strain:

(11–31)[ ] [ ] [ ] [ ]ε εS T Te d= −

where:

[εS] = dielectric permittivity matrix at constant strain

[εT] = dielectric permittivity matrix at constant stress[e] = piezoelectric stress matrix[d] = piezoelectric strain matrix

The finite element discretization is performed by establishing nodal solution variables and element shape functionsover an element domain which approximate the solution.

(11–32) [ ] u N ucu T=

(11–33)V N VcV T=

where:

uc = displacements within element domain in the x, y, z directions

Vc = electrical potential within element domain

[Nu] = matrix of displacement shape functions

NV = vector of electrical potential shape functionu = vector of nodal displacementsV = vector of nodal electrical potential

Expanding these definitions:

(11–34)[ ]N

N N

N N

N N

u Tn

n

n

=

1

1

1

0 0 0 0

0 0 0 0

0 0 0 0

……

(11–35) ( )N N N NV Tn= …1 2

where:

Ni = shape function for node i

(11–36) u UX UY UZ UX UY UZn n nT= … 1 2 3

(11–37) V

V

V

Vn

=

1

2

M

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Section 11.2: Piezoelectrics

where:

n = number of nodes of the element

Then the strain S and electric field E are related to the displacements and potentials, respectively, as:

(11–38) [ ] S B uu=

(11–39) [ ] E B VV= −

(11–40)[ ]B

x

y

z

y x

z y

z x

u =

∂∂

∂∂

∂∂

∂∂

∂∂∂∂

∂∂

∂∂

∂∂

0 0

0 0

0 0

0

0

0

(11–41)[ ] B

x

y

z

NVV T=

∂∂∂

∂∂∂

After the application of the variational principle and finite element discretization (Allik(81)), the coupled finiteelement matrix equation derived for a one element model is:

(11–42)

[ ] [ ]

[ ] [ ]

[ ]

[ ] [ ]

M u

V

C u0

0 0

0

0 0

+ [ ]

&&&&

&&&V

K K

K K

u

V

F

L

z

z T d

[ ] [ ]

[ ] [ ]

+

=

where a dot above a variable denotes a time derivative. The following equations provide an explanation of thesubmatrices in Equation 11–42:

11.2.1. Structural Mass

(11–43)[ ] [ ][ ] ( )M N N d volu u T

vol= ∫ ρ

where:

ρ = mass density (input as DENS on MP command)

Chapter 11: Coupling

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11.2.2. Structural Damping

Explanation of [C] found in Section 15.3: Damping Matrices (valid for displacement DOF only).

11.2.3. Structural Stiffness

(11–44)[ ] [ ] [ ][ ] ( )K B c B d volu T

uvol= ∫

11.2.4. Dielectric Conductivity

(11–45)[ ] [ ] [ ][ ] ( )K B B d vold

VT

Vvol= −∫ ε

11.2.5. Piezoelectric Coupling Matrix

(11–46)[ ] [ ] [ ][ ] ( )K B e B d volz

uT

Vvol= ∫

11.2.6. Structural Load Vector

F = vector of nodal forces, surface forces, and body forces (see Chapter 17, “Analysis Procedures”).

11.2.7. Electrical Load Vector

L = vector of nodal, surface, and body charges.

In the reduced mode-frequency analysis (ANTYPE,MODAL), the potential DOF is not usable as a master DOF inthe reduction process since it has no mass and is, therefore, condensed into the master DOF.

In a harmonic response analysis (ANTYPE,HARMIC), the potential DOF is allowed as a master DOF.

Energy coefficients are calculated for each piezoelectric element as follows:

11.2.8. Elastic Energy

(11–47)U S c SE

T= 12

[ ]

11.2.9. Dielectric Energy

(11–48)U E ED

T= 12

[ ] ε

11.2.10. Electromechanical Coupling Energy

(11–49)U S e EM

T= − 12

[ ]

The potential energy (output as OUTPR,VENG command) is calculated as:

11–17ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 11.2: Piezoelectrics

(11–50)E U UpoE D= +

11.3. Piezoresistivity

The capability to model piezoresistive effect exists in the following elements:

PLANE223 - 2-D 8-Node Coupled-Field SolidSOLID226 - 3-D 20-Node Coupled-Field SolidSOLID227 - 3-D 10-Node Coupled-Field Solid

In piezoresistive materials, stress or strain cause a change of electric resistivity:

(11–51)[ ] [ ]([ ] [ ])ρ ρ= +o I r

where:

[ ]ρ

ρ ρ

= =electric resistivity matrix of a loaded material

xx xyy xz

yy yz

symm zz

ρ

ρ ρ

ρ

[ ]ρ

ρo = =electric resistivity matrix of an unloaded material

xxxo

yyo

zzo

0 0

0 0

0 0

ρ

ρ

ρ ρ ρxxo

yyo

zzo, , = electrical resistivities (input as RSVX, RSVVY, RSVZ on command)MP

[ ]I = =

identity matrix

1 0 0

0 1 0

0 0 1

[ ]r

r r r

r r

r

x xy xz

y yz

symm z

= =

relative change in resistivity

calculated as:

(11–52) [ ] r = π σ

where:

r = vector of matrix [r] components = [rx ry rz rxy ryz rxz]T

Chapter 11: Coupling

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[ ]π

π π π π π ππ π π

= =piezoresistive stress matrix

11 12 13 14 15 16

21 22 223 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53

π π ππ π π π π ππ π π π π ππ π π π554 55 56

61 62 63 64 65 66

π ππ π π π π π

(input on TBB,PZRS commandwith = 0)TBOPT

[ ]σ σ σ σ σ σ σ= =stress vector x y z xy yz xzT

Similarly, for strains:

(11–53) [ ] r m el= ε

where:

[m] = piezoresistive strain matrix (input on TB,PZRS command with TBOPT = 1)

εel = elastic strain vector

The coupled-field finite element matrix equation for the piezoresistive analysis is given by:

(11–54)[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

M u

V

C u0

0 0

0

0 0

+

&&&&

&&&V

K

K

u

V

F

Iv

[ ] [ ]

[ ] [ ]

+

=

0

0

The terms used in the above equation are explained in Section 11.1.2.8: Piezoresistive Analysis where the con-

ductivity matrix [Kv] is derived as:

(11–55)[ ] ( ) [ ] ( ) ( )K N N d volv T T T

vol

= ∇ ∇−∫ ρ 1

11.4. Thermoelectrics

The capability to model thermoelectric effects exists in the following elements:

PLANE223 - 2-D 8-Node Coupled-Field SolidSOLID226 - 3-D 20-Node Coupled-Field SolidSOLID227 - 3-D 10-Node Coupled-Field Solid

These elements support the Joule heating effect (irreversible), and the Seebeck, Peltier, and Thomson effects(reversible).

In addition to the above, the following elements suport a basic thermoelectric analysis that takes into consider-ation Joule heating effect only:

SOLID5 - 3-D 8-Node Coupled-Field SolidPLANE67 - 2-D 4-Node Coupled Thermal-Electric SolidLINK68 - 3-D 2-Node Coupled Thermal-Electric LineSOLID69 - 3-D 8-Node Coupled Thermal-Electric SolidSOLID98 - 3-D 10-Node Coupled-Field Solid

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Section 11.4: Thermoelectrics

SHELL157 - 3-D 4-Node Thermal-Electric Shell

The coupled thermoelectric constitutive equations (Landau and Lifshitz(358)) are:

(11–56) [ ] [ ] q J K T= − ∇Π

(11–57) [ ]( [ ] )J E T= − ∇σ α

Substituting [Π] with T[α] to further demonstrate the coupling between the above two equations,

(11–58) [ ] [ ] q T J K T= − ∇α

(11–59) [ ]( [ ] )J E T= − ∇σ α

where:

[Π] = Peltier coefficient matrix = T[α]T = absolute temperature

[ ]αα

α

α

=

xx

yy

zz

0 0

0 0

0 0

q = heat flux vector (output as TF)J = electric current density (output as JC for elements that support conduction current calculation)

[ ]K

k

k

k

xx

yy

zz

=

=0 0

0 0

0 0

thermal conductivity matrix eevaluated at zero electric current ( )J = 0

∇ =T thermal gradient (output as TG)

[ ]σ

ρ

ρ

ρ

=

=

10 0

01

0

0 01

xx

yy

zz

electrical conduuctivity matrix evaluated at zero temperature gradient (∇∇ =T )0

E = electric field (output as EF)αxx, αyy, αzz = Seebeck coefficients (input as SBKX, SBKY, SBKZ on MP command)

Kxx, Kyy, Kzz = thermal conductivities (input as KXX, KYY, KZZ on MP command)

ρxx, ρyy, ρzz = resistivity coefficients (input as RSVX, RSVY, RSVZ on MP command)

The heat generation rate per unit volume is:

(11–60)Q q J ET= −∇ +i

Substituting q and E from Equation 11–58 and Equation 11–59 into Equation 11–60 we get:

(11–61)Q K T J J T JT= ∇ ∇ + − ∇−i([ ] ) [ ] ([ ] )σ α1

Chapter 11: Coupling

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The first term in Equation 11–61 is related to thermal conductivity; the second term is the Joule heat (output asJHEAT). The third term respresents the thermoelectric effects, and can be expanded further into Peltier, Thomson,

and Bridgman heat terms (Nye(359)). If the gradient ∇ ([α]J) in Equation 11–61 is associated with the temper-ature dependence of the Seebeck coefficient matrix (input as SBKX, SBKY, SBKZ on MPDATA command), thethird term in Equation 11–61 represents the Thomson heat:

(11–62)Q J TTh T= ∇ [ ] τ

where:

QTh = Thomson heat

[ ][ ]τ α= = −Thomson heat tensor T

ddT

11.5. Review of Coupled Electromechanical Methods

The sequential coupling between electrical and mechanical finite element physics domains for coupled Elec-tromechanical analysis can be performed by the ESSOLV command macro. ESSOLV allows the most generaltreatment of individual physics domains. However, it can not be applied to small signal modal and harmonicanalyses because a total system eigen frequency analysis requires matrix coupling. Moreover, sequential couplinggenerally converges slower.

Strong Electromechanical coupling can be performed by transducer elements:

TRANS126, Gyimesi and Ostergaard(248), Gyimesi and Ostergaard(330), Section 14.126: TRANS126 - Elec-tromechanical Transducer (also see Section 5.9: Electromechanical TransducersTRANS109, Section 14.109: TRANS109 - 2-D Electromechanical Transducer

Both TRANS126 and TRANS109 completely model the fully coupled system, converting electrostatic energy intomechanical energy and vise versa as well as storing electrostatic energy. Coupling between electrostatic forcesand mechanical forces is obtained from virtual work principles (Gyimesi and Ostergaard(248), Gyimesi et al.(329)).

TRANS126 takes on the form of a 2-node line element with electrical voltage and mechanical displacement DOFsas across variables and electric current and mechanical force as through variables. Input for the element consistsof a capacitance-stroke relationship that can be derived from electrostatic field solutions and using the CMATRIXcommand macro (Gyimesi et al.(288), Gyimesi and Ostergaard(289), (Section 5.10: Capacitance Computation)).

The element can characterize up to three independent translation degrees of freedom at any point to simulate3-D coupling. Thus, the electrostatic mesh is removed from the problem domain and replaced by a set of TRANS126elements hooked to the mechanical and electrical model providing a reduced order modeling of a coupledelectromechanical system (Gyimesi and Ostergaard (286), Gyimesi et al.(287), (Section 5.11: Open BoundaryAnalysis with a Trefftz Domain)).

TRANS126 allows treatment of all kinds of analysis types, including prestressed modal and harmonic analyses.However, TRANS126 is limited geometrically to problems when the capacitance can be accurately described asa function of a single degree of freedom, usually the stroke of a comb drive. In a bending electrode problem, likean optical switch, obviously, a single TRANS126 element can not be applied. When the gap is small and fringingis not significant, the capacitance between deforming electrodes can be practically modeled reasonably well byseveral capacitors connected parallel. The EMTGEN (electromechanical transducer generator) command macrocan be applied to this case.

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Section 11.5: Review of Coupled Electromechanical Methods

For more general 2-D geometries the 3-node transducer element TRANS109 (Gyimesi et al.(329)) is recommended(Section 14.109: TRANS109 - 2-D Electromechanical Transducer). TRANS109 has electrical voltage and mechanicaldisplacements as degrees of freedom. TRANS109 has electrical charge and mechanical force as reaction solution.TRANS109 can model geometries where it would be difficult to obtain a capacitance-stroke relationship, however,TRANS109 can be applied only in static and transient analyses - prestressed modal and harmonic analyses arenot supported.

The Newton-Raphson nonlinear iteration converges more quickly and robustly with TRANS126 than withTRANS109. Convergence issues may be experienced even with TRANS126 when applied to the difficult hystericpull-in and release analysis (Gyimesi et al.(329), Avdeev et al.(331)) because of the negative total system stiffnessmatrix. The issue is resolved when the augmented stiffness method is applied in TRANS126. TRANS109 Laplacianmesh morphing algorithm may result in convergence problems. See the Magnetic User Guides for their treatment.

Chapter 11: Coupling

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Chapter 12: Shape FunctionsThe shape functions for the elements are given in this chapter. They are referred to by the individual elementdescriptions in Chapter 14, “Element Library”. All subheadings for this chapter are included in the table of contentsto aid in finding a certain type of shape function.

The given functions are related to the nodal quantities by:

Table 12.1 Shape Function Labels

MeaningInput/Out-put Label

Variable

Translation in the x (or s) directionUXu

Translation in the y (or t) directionUYv

Translation in the x (or r) directionUZw

Rotation about the x directionROTXθx

Rotation about the y directionROTYθy

Rotation about the z directionROTZθz

X-component of vector magnetic potentialAXAx

Y-component of vector magnetic potentialAYAy

Z-component of vector magnetic potentialAZAz

Velocity in the x directionVXVx

Velocity in the y directionVYVy

Velocity in the z directionVZVz

Unused

PressurePRESP

TemperatureTEMP, TBOT,TE2, ... TTOP

T

Electric potential or source currentVOLTV

Scalar magnetic potentialMAGφ

Turbulent kinetic energyENKEEk

Energy dissipationENDSED

The vector correspondences are not exact, since, for example, u, v, and w are in the element coordinate system,whereas UX, UY, UZ represent motions in the nodal coordinate system. Generally, the element coordinate systemis the same as the global Cartesian system, except for:

1. Line elements (Section 12.1: 2-D Lines to Section 12.4: Axisymmetric Harmonic Shells), where u motionsare axial motions, and v and w are transverse motions.

2. Shell elements (Section 12.5: 3-D Shells), where u and v are in-plane motions and w is the out-of-planemotion.

Subscripted variables such as uJ refer to the u motion at node J. When these same variables have numbers for

subscripts (e.g. u1), nodeless variables for extra shape functions are being referred to. Coordinates s, t, and r are

normalized, going from -1.0 on one side of the element to +1.0 on the other, and are not necessarily orthogonal

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

to one another. L1, L2, L3, and L4 are also normalized coordinates, going from 0.0 at a vertex to 1.0 at the opposite

side or face.

Elements with midside nodes allow those midside nodes to be dropped in most cases. A dropped midside nodeimplies that the edge is and remains straight, and that any other effects vary linearly along that edge.

Gaps are left in the equation numbering to allow for additions. Labels given in subsection titles within parenthesesare used to relate the given shape functions to their popular names, where applicable.

Some elements in Chapter 14, “Element Library” (notably the 8 node solids) imply that reduced element geometries(e.g., wedge) are not available. However, the tables in Chapter 14, “Element Library” refer only to the availableshape functions. In other words, the shape functions used for the 8-node brick is the same as the 6-node wedge.

12.1. 2-D Lines

This section contains shape functions for line elements without and with rotational degrees of freedom (RDOF).

Figure 12.1 2–D Line Element

12.1.1. 2-D Lines without RDOF

These shape functions are for 2-D line elements without RDOF, such as LINK1 or LINK32.

(12–1)u u s u sI J= − + +1

21 1( ( ) ( ))

(12–2)v v s v sI J= − + +1

21 1( ( ) ( ))

(12–3)T T s T sI J= − + +1

21 1( ( ) ( ))

12.1.2. 2-D Lines with RDOF

These shape functions are for 2-D line elements with RDOF, such as BEAM3.

(12–4)u u s u sI J= − + +1

21 1( ( ) ( ))

Chapter 12: Shape Functions

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(12–5)

v vs

s vs

s

L

I J

z I

= − −

+ + −

+ −

12

12

3 12

3

81

2 2( ) ( )

( (,θ ss s s sz J2 21 1 1)( ) ( )( )),− + − +θ

12.2. 3-D Lines

This section contains shape functions for line elements without and with rotational degrees of freedom (RDOF).

Figure 12.2 3–D Line Element

12.2.1. 3-D 2 Node Lines without RDOF

These shape functions are for 3-D 2 node line elements without RDOF, such as LINK8, LINK33, LINK68 or BEAM188.

(12–6)u u s u sI J= − + +1

21 1( ( ) ( ))

(12–7)v v s v sI J= − + +1

21 1( ( ) ( ))

(12–8)w w s w sI J= − + +1

21 1( ( ) ( ))

(12–9)θ θ θx xI xJs s= − + +1

21 1( ( ) ( ))

(12–10)θ θ θy yI yJs s= − + +1

21 1( ( ) ( ))

(12–11)θ θ θz zI zJs s= − + +1

21 1( ( ) ( ))

(12–12)P P s P sI J= − + +12

1 1( ( ) ( ))

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Section 12.2: 3-D Lines

(12–13)T T s T sI J= − + +1

21 1( ( ) ( ))

(12–14)V V s V sI J= − + +1

21 1( ( ) ( ))

12.2.2. 3-D 2 Node Lines with RDOF

These shape functions are for 3-D 2-node line elements with RDOF, such as BEAM4.

(12–15)u u s u sI J= − + +1

21 1( ( ) ( ))

(12–16)

v vs

s vs

s

L

I J

z I

= − −

+ + −

+ −

12

12

3 12

3

81

2 2( ) ( )

( (,θ ss s s sz J2 21 1 1)( ) ( )( )),− − − +θ

(12–17)

w ws

s ws

s

L

I J

y I

= − −

+ + −

− −

12

12

3 12

3

81

2 2( ) ( )

( (,θ ss s s sy J2 21 1 1)( ) ( )( )),− − − +θ

(12–18)θ θ θx x I x Js s= − + +1

21 1( ( ) ( )), ,

12.2.3. 3-D 3 Node Lines

These shape functions are for 3-D 3 node line elements such as BEAM189.

(12–19)u u s s u s s u sH I J= − + + + + −1

212 2 2( ( ) ( )) ( )

(12–20)v v s s v s s v sH I J= − + + + + −1

212 2 2( ( ) ( )) ( )

(12–21)w w s s w s s w sH I J= − + + + + −1

212 2 2( ( ) ( )) ( )

(12–22)θ θ θ θx xH xI xJs s s s s= − + + + + −1

212 2 2( ( ) ( )) ( )

(12–23)θ θ θ θy yH yI yJs s s s s= − + + + + −1

212 2 2( ( ) ( )) ( )

(12–24)θ θ θ θz zH zI zJs s s s s= − + + + + −12

12 2 2( ( ) ( )) ( )

Chapter 12: Shape Functions

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(12–25)T T s s T s s T sH I J= − + + + + −1

212 2 2( ( ) ( )) ( )

12.3. Axisymmetric Shells

This section contains shape functions for 2-node axisymmetric shell elements under axisymmetric load. Theseelements may have extra shape functions (ESF).

12.3.1. Axisymmetric Shell without ESF

These shape functions are for 2-node axisymmetric shell elements without extra shape functions, such as SHELL51with KEYOPT(3) = 1.

(12–26)u u s u sI J= − + +1

21 1( ( ) ( ))

(12–27)v v s v sI J= − + +1

21 1( ( ) ( ))

(12–28)

w ws

s ws

s

Ls

I J

I

= − −

+ + −

+ −

12

12

3 12

3

81

2 2

2

( ) ( )

( (θ ))( ) ( )( ))1 1 12− − − +s s sJθ

12.3.2. Axisymmetric Shell with ESF

These shape functions are for 2-node axisymmetric shell elements with extra displacement shape functions,such as SHELL51 with KEYOPT(3) = 0.

(12–29)

u us

s us

s

Lu s

I J= − −

+ + −

+ −

12

12

3 12

3

81

2 2

12

( ) ( )

( ( ))( ) ( )( ))1 1 122− − − +s u s s

(12–30)

v vs

s vs

s

Lv s

I J= − −

+ + −

+ −

12

12

3 12

3

81

2 2

12

( ) ( )

( ( ))( ) ( )( ))1 1 122− − − +s v s s

(12–31)

w ws

s ws

s

Ls

I J

I

= − −

+ + −

+ −

12

12

3 12

3

81

2 2

2

( ) ( )

( (θ ))( ) ( )( ))1 1 12− − − +s s sJθ

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Section 12.3: Axisymmetric Shells

12.4. Axisymmetric Harmonic Shells

This section contains shape functions for 2-node axisymmetric shell elements under nonaxisymmetric (harmonic)load. These elements may have extra shape functions (ESF).

Figure 12.3 Axisymmetric Harmonic Shell Element

The shape functions of this section use the quantities sin l β and cos l β, where l = input quantity MODE on the

MODE command. The sin l β and cos l β are interchanged if Is = -1, where Is = input quantity ISYM on the MODE

command. If l = 0, both sin l β and cos l β are set equal to 1.0.

12.4.1. Axisymmetric Harmonic Shells without ESF

These shape functions are for 2-node axisymmetric harmonic shell elements without extra shape functions, suchas SHELL61 with KEYOPT(3) = 1.

(12–32)u u s u s cosI J= − + +1

21 1( ( ) ( )) lβ

(12–33)v v s v sI J= − + +1

21 1( ( ) ( ))sinlβ

(12–34)

w ws

s ws

s

L

I J

I

= − −

+ + −

+

12

12

3 12

3

8

2 2( ) ( )

( (θ 11 1 1 12 2− − − − +

s s s sJ)( ) ( )( )) cosθ βl

12.4.2. Axisymmetric Harmonic Shells with ESF

These shape functions are for 2-node axisymmetric harmonic shell elements with extra shape functions, such asSHELL61 with KEYOPT(3) = 0.

Chapter 12: Shape Functions

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(12–35)

u us

s us

s

Lu

I J= − −

+ + −

+

12

12

3 12

3

8

2 2

1

( ) ( )

( (11 1 1 122

2− − − − +

s s u s s)( ) ( )( )) cos lβ

(12–36)

v vs

s vs

s

Lv

I J= − −

+ + −

+

12

12

3 12

3

8

2 2

1

( ) ( )

( (11 1 1 122

2− − − − +

s s v s s)( ) ( )( )) sinlβ

(12–37)

w ws

s ws

s

L

I J

I

= − −

+ + −

+

12

12

3 12

3

8

2 2( ) ( )

( (θ 11 1 1 12 2− − − − +

s s s sJ)( ) ( )( )) cosθ βl

12.5. 3-D Shells

This section contains shape functions for 3-D shell elements. These elements are available in a number of config-urations, including certain combinations of the following features:

• triangular or quadrilateral.

- if quadrilateral, with or without extra shape functions (ESF).

• with or without rotational degrees of freedom (RDOF).

- if with RDOF, with or without shear deflections (SD).

• with or without midside nodes.

Figure 12.4 3-D Shell Elements

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Section 12.5: 3-D Shells

12.5.1. 3-D 3-Node Triangular Shells without RDOF (CST)

These shape functions are for 3-D 3-node triangular shell elements without RDOF, such as SHELL41, SHELL131,or SHELL132:

(12–38)u uL u L u LI J K= + +1 2 3

(12–39)v v L v L v LI J K= + +1 2 3

(12–40)w w L w L w LI J K= + +1 2 3

(12–41)A A L A L A Lx xI xJ xK= + +1 2 3

(12–42)A A L A L A Ly yI yJ yK= + +1 2 3

(12–43)A A L A L A Lz zI zJ zK= + +1 2 3

(12–44)T TL T L T LI J K= + +1 2 3

(12–45)φ φ φ φ= + +I J KL L L1 2 3

12.5.2. 3-D 6-Node Triangular Shells without RDOF (LST)

These shape functions are for 3-D 6-node triangular shell elements without RDOF, such as the mass or stressstiffening matrix of SHELL93:

(12–46)

u u L L u L L u L L

u L L u L LI J K

L M

= − + − + −+ + +( ) ( ) ( )

( ) ( )

2 1 2 1 2 1

4 41 1 2 2 3 3

1 2 2 3 uu L LN( )4 3 1

(12–47)v v LI= −( ) . . . (analogous to u)2 11

(12–48)w w LI= −( ) . . . (analogous to u)2 11

(12–49)T T LI= −( ) . . . (analogous to u)2 11

(12–50)V V LI= −( ) . . . (analogous to u)2 11

12.5.3. 3-D 3-Node Triangular Shells with RDOF but without SD

These shape functions are for the 3-D 3-node triangular shell elements with RDOF, but without shear deflection,such as SHELL63 when used as a triangle.

(12–51)u uL u L u LI J K= + +1 2 3

(12–52)v v L v L v LI J K= + +1 2 3

(12–53)w = not explicitly defined. A DKT element is used

Chapter 12: Shape Functions

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12.5.4. 3-D 3-Node Triangular Shells with RDOF and with SD

These shape functions are for 3-D 3-node triangular shell elements with RDOF and with shear deflection, suchas SHELL43 when used as a triangle.

(12–54)

u

v

w

N

u

v

w

Nrt

a b

ai

i

i

ii

ii

i

i i

=

+= =∑ ∑

1

3

1

3 1 1

2

, ,

22 2

3 3

, ,

, ,

,

,

i i

i i

x i

y i

b

a b

θ

θ

where:

Ni = shape functions given with u for PLANE42 (Equation 12–84)

ui = motion of node i

r = thickness coordinateti = thickness at node i

a = unit vector in s directionb = unit vector in plane of element and normal to aθx,i = rotation of node i about vector a

θy,i = rotation of node i about vector b

Note that the nodal translations are in global Cartesian space, and the nodal rotations are based on element (s-t) space. Transverse shear strains in natural space (see Figure 12.5: “Interpolation Functions for Transverse Strainsfor Triangles”) are assumed as:

(12–55)

ε ε ε

ε ε ε

% % %

% % %

13 13 13

23 23 2

12

112

1

12

112

1

= + + −

= + + −

( ) ( )

( ) ( )

t t

t t

A C

D33

B

where:

ε ε% %13 13A DI

= at A

ε ε% %13 13C DI

= at C

ε ε% %23 23D DI

= at D

ε ε% %23 23B DI

= at B

DI stands for values computed from assumed displacement fields directly. These assumptions can be seen tocause geometric anisotropy. See Section 14.43: SHELL43 - 4-Node Plastic Large Strain Shell for more details. Thein-plane RDOF (KEYOPT(3) = 2) logic is based on Allman(113).

12–9ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 12.5: 3-D Shells

Figure 12.5 Interpolation Functions for Transverse Strains for Triangles

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"#

12.5.5. 3-D 6-Node Triangular Shells with RDOF and with SD

These shape functions are for 3-D 6-node triangular shell elements with RDOF and with shear deflection, suchas for the stiffness matrix of SHELL93:

(12–56)

u

v

w

N

u

v

w

Nrt

a b

ai

i

i

ii

ii

i

i i

=

+= =∑ ∑

1

6

1

6 1 1

2

, ,

22 2

3 3

, ,

, ,

,

,

i i

i i

x i

y i

b

a b

θ

θ

where:

Ni = shape functions given with u for PLANE2 (Equation 12–96)

ui, vi, wi = motion of node i

r = thickness coordinateti = thickness at node i

a = unit vector in s directionb = unit vector in plane of element and normal to aθx,i = rotation of node i about vector a

θy,i = rotation of node i about vector b

Chapter 12: Shape Functions

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Note that the nodal translations are in global Cartesian space, and the nodal rotations are based on element (s-t) space.

12.5.6. 3-D 4-Node Quadrilateral Shells without RDOF and without ESF (Q4)

These shape functions are for 3-D 4-node triangular shell elements without RDOF and without extra displacementshapes, such as SHELL41 with KEYOPT(2) = 1 and the magnetic interface element INTER115.

(12–57)u u s t u s t

u s t u s t

I J

K L

= − − + + −

+ + + + − +

14

1 1 1 1

1 1 1 1

( ( )( ) ( )( )

( )( ) ( )( ))

(12–58)v v sI= −1

41( ( ) . . . (analogous to u)

(12–59)w w sI= −1

41( ( ) . . . (analogous to u)

(12–60)A A sx xI= −1

41( ( ) . . . (analogous to u)

(12–61)A A sy yI= −1

41( ( ) . . . (analogous to u)

(12–62)A A sz zI= −1

41( ( ) . . . (analogous to u)

(12–63)P P sI= −1

41( ( ) . . . (analogous to u)

(12–64)T T sI= −1

41( ( ) . . . (analogous to u)

(12–65)V V sI= −1

41( ( ) . . . (analogous to u)

(12–66)φ φ= −1

41( ( )I s . . . (analogous to u)

12.5.7. 3-D 4-Node Quadrilateral Shells without RDOF but with ESF (QM6)

These shape functions are for 3-D 4-node quadrilateral shell elements without RDOF but with extra shape functions,such as SHELL41 with KEYOPT(2) = 0:

(12–67)

u u s t u s t

u s t u s t

I J

K L

= − − + + −

+ + + + − +

+

14

1 1 1 1

1 1 1 1

( ( )( ) ( )( )

( )( ) ( )( ))

uu s u t12

221 1( ) ( )− + −

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Section 12.5: 3-D Shells

(12–68)v v sI= −1

41( ( ) . . . (analogous to u)

12.5.8. 3-D 8-Node Quadrilateral Shells without RDOF

These shape functions are for 3-D 8-node quadrilateral shell elements without RDOF such as the mass or stressstiffening matrix of SHELL93:

(12–69)

u u s t s t u s t s t

u s t

I J

K

= − − − − − + + − − −

+ + +

14

1 1 1 1 1 1

1 1

( ( )( )( ) ( )( )( )

( )( )(( ) ( )( )( ))

( ( )( ) ( )(

s t u s t s t

u s t u s

L

M N

+ − + − + − + −

+ − − + + −

1 1 1 1

12

1 1 1 12 tt

u s t u s tO P

2

2 21 1 1 1

)

( )( ) ( )( ))+ − + + − −

(12–70)v v sI= −1

41( ( ) . . . (analogous to u)

(12–71)w w sI= −1

41( ( ) . . . (analogous to u)

(12–72)P P sI= −1

41( ( ) . . . (analogous to u)

(12–73)T T sI= −1

41( ( ) . . . (analogous to u)

(12–74)V V sI= −1

41( ( ) . . . (analogous to u)

12.5.9. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and withoutESF

These shape functions are for 3-D 4-node quadrilateral shell elements with RDOF but without shear deflectionand without extra shape functions, such as SHELL63 with KEYOPT(3) = 1 when used as a quadrilateral:

(12–75)u u s t u s t

u s t u s t

I J

K L

= − − + + −

+ + + + − +

14

1 1 1 1

1 1 1 1

( ( )( ) ( )( )

( )( ) ( )( ))

(12–76)v v sI= −1

41( ( ) . . . (analogous to u)

(12–77)w = not explicitly defined. Four overlaid triangles

Chapter 12: Shape Functions

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12.5.10. 3-D 4-Node Quadrilateral Shells with RDOF but without SD and withESF

These shape functions are for 3-D 4-node quadrilateral shell elements with RDOF but without shear deflectionand with extra shape functions, such as SHELL63 with KEYOPT(3) = 0 when used as a quadrilateral:

(12–78)

u u s t u s t

u s t u s t

I J

K L

= − − + + −

+ + + + − +

+

14

1 1 1 1

1 1 1 1

( ( )( ) ( )( )

( )( ) ( )( ))

uu s u t12

221 1( ) ( )− + −

(12–79)v v sI= −1

41( ( ) . . . (analogous to u)

(12–80)w = not explicitly defined. Four overlaid triangles

12.5.11. 3-D 4-Node Quadrilateral Shells with RDOF and with SD

These shape functions are for 3-D 4-node quadrilateral shell with RDOF and with shear deflection such as SHELL43.Both use and nonuse of extra shape functions are considered.

(12–81)

u

v

w

N

u

v

w

Nrt

a b

ai

i

i

ii

ii

i

i i

=

+= =∑ ∑

1

4

1

4 1 1

2

, ,

22 2

3 3

, ,

, ,

,

,

i i

i i

x i

y i

b

a b

θ

θ

where:

Ni = shape functions given with u for in Equation 12–103. Extra shapes, if requested, use the shape functions

of Equation 12–115ui, vi, wi = motion of node i

r = thickness coordinateti = thickness at node i

a = unit vector in s directionb = unit vector in plane of element and normal to aθx,i = rotation of node i about vector a

θy,i = rotation of node i about vector b

Note that the nodal translations are in global Cartesian space, and the nodal rotations are based on element (s-t) space. Transverse shear strains in natural space (Figure 12.6: “Interpolation Functions for the Transverse Strainsfor Quadrilaterals”) are assumed as:

(12–82)

ε ε ε

ε ε ε

% % %

% % %

13 13 13

23 23 2

12

112

1

12

112

1

= + + −

= + + −

( ) ( )

( ) ( )

t t

t t

A C

D33

B

where:

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Section 12.5: 3-D Shells

ε ε% %13 13A DI

= at A

ε ε% %13 13C DI

= at C

ε ε% %23 23D DI

= at D

ε ε% %23 23B DI

= at B

DI stands for values computed from assumed displacement fields directly (see Figure 12.6: “Interpolation Functionsfor the Transverse Strains for Quadrilaterals”). See Section 14.43: SHELL43 - 4-Node Plastic Large Strain Shell formore details. The in-plane RDOF logic is based on Yunus(117).

Figure 12.6 Interpolation Functions for the Transverse Strains for Quadrilaterals

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Chapter 12: Shape Functions

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12.5.12. 3-D 8-Node Quadrilateral Shells with RDOF and with SD

These shape functions are for 3-D 8-node quadrilateral shell elements with RDOF and with shear deflection, suchas for the stiffness matrix of SHELL93:

(12–83)

u

v

w

N

u

v

w

Nrt

a b

ai

i

i

ii

ii

i

i i

=

+= =∑ ∑

1

8

1

8 1 1

2

, ,

22 2

3 3

, ,

, ,

,

,

i i

i i

x i

y i

b

a b

θ

θ

where:

Ni = shape functions given with u for PLANE82, Equation 12–117

ui, vi, wi = motion of node i

r = thickness coordinateti = thickness at node i

a = unit vector in s directionb = unit vector in plane of element and normal to aθx,i = rotation of node i about vector a

θy,i = rotation of node i about vector b

Note that the nodal translations are in global Cartesian space, and the nodal rotations are based on element (s-t) space.

12.6. 2-D and Axisymmetric Solids

This section contains shape functions for 2-D and axisymmetric solid elements. These elements are available ina number of configurations, including certain combinations of the following features:

• triangular or quadrilateral.

- if quadrilateral, with or without extra shape functions (ESF).

• with or without midside nodes.

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Section 12.6: 2-D and Axisymmetric Solids

Figure 12.7 2-D and Axisymmetric Solid Element

12.6.1. 2-D and Axisymmetric 3 Node Triangular Solids (CST)

These shape functions are for 2-D 3 node and axisymmetric triangular solid elements, such as PLANE13, PLANE42,PLANE67, or FLUID141 with only 3 nodes input:

(12–84)u uL u L u LI J K= + +1 2 3

(12–85)v v L v L v LI J K= + +1 2 3

(12–86)w w L w L w LI J K= + +1 2 3

(12–87)A A L A L A Lz zI zJ zK= + +1 2 3

(12–88)V V L A L A Lx xI zJ zK= + +1 2 3

(12–89)V V L A L A Ly yI zJ zK= + +1 2 3

(12–90)V V L A L A Lz zI zJ zK= + +1 2 3

(12–91)P PL A L A LI zJ zK= + +1 2 3

(12–92)T TL T L T LI J K= + +1 2 3

(12–93)V VL V L V LI J K= + +1 2 3

(12–94)E E L V L V LKIK

J K= + +1 2 3

(12–95)E E L V L V LDID

J K= + +1 2 3

Chapter 12: Shape Functions

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12.6.2. 2-D and Axisymmetric 6 Node Triangular Solids (LST)

These shape functions are for 2-D 6 node and axisymmetric triangular solids, such as PLANE2 or PLANE35 (orPLANE77 or PLANE82 reduced to a triangle):

(12–96)

u u L L u L L u L

u L L u L L uI J K

L M N

= − + − + −+ + +( ) ( ) ( )

( ) ( )

2 1 2 1 2 1

4 41 1 2 2 3

1 2 2 3 (( )4 3 1L L

(12–97)v v L LI= − +( )2 11 1 . . . (analogous to u)

(12–98)w w L LI= − +( )2 11 1 . . . (analogous to u)

(12–99)A A L Lz zI= −( )2 11 1 . . . (analogous to u)

(12–100)P P L LI= − +( )2 11 1 . . . (analogous to u)

(12–101)T T L LI= − +( )2 11 1 . . . (analogous to u)

(12–102)V V L LI= − +( )2 11 1 . . . (analogous to u)

12.6.3. 2-D and Axisymmetric 4 Node Quadrilateral Solid without ESF (Q4)

These shape functions are for the 2-D 4 node and axisymmetric quadrilateral solid elements without extra shapefunctions, such as PLANE13 with KEYOPT(2) = 1, PLANE42 with KEYOPT(2) = 1, LINK68, or FLUID141.

(12–103)u u s t u s t

u s t u s t

I J

K L

= − − + + −

+ + + + − +

14

1 1 1 1

1 1 1 1

( ( )( ) ( )( )

( )( ) ( )( ))

(12–104)v v sI= −1

41( ( ) . . . (analogous to u)

(12–105)w w sI= −1

41( ( ) . . . (analogous to u)

(12–106)A A sz zI= −1

41( ( ) . . . (analogous to u)

(12–107)V V sx xI= −1

41( ) . . . (analogous to u)

(12–108)V V sy yI= −1

41( ) . . . (analogous to u)

(12–109)V V sz zI= −1

41( ) . . . (analogous to u)

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Section 12.6: 2-D and Axisymmetric Solids

(12–110)P P sI= −1

41( ) . . . (analogous to u)

(12–111)T T sI= −1

41( ( ) . . . (analogous to u)

(12–112)V V sI= −1

41( ( ) . . . (analogous to u)

(12–113)E E sK

IK= −1

41( ( ) ( ) . . . analogous to u

(12–114)E E sD

ID= −1

41( ( ) ( ) . . . analogous to u

12.6.4. 2-D and Axisymmetric 4 Node Quadrilateral Solids with ESF (QM6)

These shape functions are for the 2-D 4 node and axisymmetric solid elements with extra shape functions, suchas PLANE13 with KEYOPT(2) = 0 or PLANE42 with KEYOPT(2) = 0. (Taylor et al.(49))

(12–115)

u u s t u s t

u s t u s t

I J

K L

= − − + + −

+ + + + − +

+

14

1 1 1 1

1 1 1 1

( ( )( ) ( )( )

( )( ) ( )( ))

uu s u t12

221 1( ) ( )− + −

(12–116)v v sI= −1

41( ( ) . . . (analogous to u)

Equation 12–115 is adjusted for axisymmetric situations by removing the u1 or u2 term for elements near the

centerline, in order to avoid holes or “doubled” material at the centerline.

12.6.5. 2-D and Axisymmetric 8 Node Quadrilateral Solids (Q8)

These shape functions are for the 2-D 8 node and axisymmetric quadrilateral elements such as PLANE77 andPLANE82:

(12–117)

u u s t s t u s t s t

u s t

I J

K

= − − − − − + + − − −

+ + +

14

1 1 1 1 1 1

1 1

( ( )( )( ) ( )( )( )

( )( )(( ) ( )( )( ))

( ( )( ) ( )(

s t u s t s t

u s t u s

L

M N

+ − + − + − + −

+ − − + + −

1 1 1 1

12

1 1 1 12 tt

u s t u s tO P

2

2 21 1 1 1

)

( )( ) ( )( ))+ − + + − −

(12–118)v v sI= −1

41( ( ) . . . (analogous to u)

Chapter 12: Shape Functions

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(12–119)w w sI= −1

41( ( ) . . . (analogous to u)

(12–120)A A sz zI= −1

41( ( ) . . . (analogous to u)

(12–121)T T sI= −1

41( ( ) . . . (analogous to u)

(12–122)V V sI= −1

41( ( ) . . . (analogous to u)

12.6.6. 2-D and Axisymmetric 4 Node Quadrilateral Infinite Solids

Figure 12.8 4 Node Quadrilateral Infinite Solid Element

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%& "('

)*+

,-) ,/..1032! 546 78:980

;

2

<

.

7

=

>

These Lagrangian isoparametric shape functions and “mapping” functions are for the 2-D and axisymmetric 4node quadrilateral solid infinite elements such as INFIN110:

12.6.6.1. Lagrangian Isoparametric Shape Functions

(12–123)

A A s t t A s t t

A s t A

z zI zJ

zK zL

= − − + + −

+ + − +

14

1 1

12

1 1

2 2

2

( ( )( ) ( )( ))

( ( )( ) (( )( ))1 1 2− −s t

(12–124)T T sI= −1

41( ( ) . . . (analogous to A )z

(12–125)V V sI= −1

41( ( ) . . . (analogous to A )z

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Section 12.6: 2-D and Axisymmetric Solids

12.6.6.2. Mapping Functions

(12–126)

x x s t t x s t t

X s t t

I J

K

= − − − + + − −

+ + + −

( )( ) / ( ) ( )( ) / ( )

( )( ) / ( )

1 1 1 1

12

1 1 1 ++ − + −12

1 1 1X s t tL( )( ) / ( )

(12–127)y y sI= −( ) . . . (analogous to x)1

12.6.7. 2-D and Axisymmetric 8 Node Quadrilateral Infinite Solids

Figure 12.9 8 Node Quadrilateral Infinite Solid Element

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$% '&&)( %* +,.- &

These Lagrangian isoparametric shape functions and “mapping” functions are for the 2-D and axisymmetric 8node quadrilateral infinite solid elements such as INFIN110:

12.6.7.1. Lagrangian Isoparametric Shape Functions

(12–128)

A A s t s t A s t

A s

z zI zJ

zK

= − − − − − + − −

+ +

14

1 1 112

1 1

14

1

2( ( )( )( )) ( ( )( ))

( ( ))( )( )) ( ( )( ))

( ( )( ))

1 112

1 1

12

1 1

2

2

− − + − + + −

+ − −

t s t A s t

A s t

zL

zM

(12–129)T T sI= −( ( )1 . . . (analogous to A )z

(12–130)V V sI= −( ( )1 . . . (analogous to A )z

Chapter 12: Shape Functions

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12.6.7.2. Mapping Functions

(12–131)

x x s s t t x s t

x s s t t

I J

K

= − − − − − + − −

+ + − + − −

( )( ) ( ) ( ) ( )

( )( ) ( )

1 1 1 2 1 1

1 1 1

2

++ + + −

+ + + −

12

1 1 1

12

1 1 1

x s t t

x s t t

L

M

( )( ) ( )

( )( ) ( )

(12–132)y y sI= −( ) . . . (analogous to x)1

The shape and mapping functions for the nodes N, O and P are deliberately set to zero.

12.7. Axisymmetric Harmonic Solids

This section contains shape functions for axisymmetric harmonic solid elements. These elements are availablein a number of configurations, including certain combinations of the following features:

• triangular or quadrilateral.

- if quadrilateral, with or without extra shape functions (ESF).

• with or without midside nodes.

The shape functions of this section use the quantities sin l β and cos l β (where l = input as MODE on the MODE

command). sin l β and cos l β are interchanged if Is = -1 (where Is = input as ISYM on the MODE command). If l

= 0, sin l β = cos l β = 1.0.

Figure 12.10 Axisymmetric Harmonic Solid Elements

12.7.1. Axisymmetric Harmonic 3 Node Triangular Solids

These shape functions are for the 3 node axisymmetric triangular solid elements, such as PLANE25 with only 3nodes input:

12–21ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 12.7: Axisymmetric Harmonic Solids

(12–133)u uL u L u L cosI J K= + +( )1 2 3 lβ

(12–134)v v L v L v L cosI J K= + +( )1 2 3 lβ

(12–135)w w L w L w L sinI J K= + +( )1 2 3 lβ

(12–136)T TL T L T L cosI J K= + +( )1 2 3 lβ

12.7.2. Axisymmetric Harmonic 6 Node Triangular Solids

These shape functions are for the 6 node axisymmetric triangular solids elements, such as PLANE83 input as atriangle:

(12–137)

u u L L u L L u L

u L L u L L uI J K

L M

= − + − + −+ + +

( ( ) ( ) ( )

( ) ( )

2 1 2 1 2 1

4 41 1 2 2 3

1 2 2 3 NN L L( ))cos4 3 1 lβ

(12–138)v v LI= −( ( ) cos2 11 . . . . . .)(analogous to u) lβ

(12–139)w w LI= −( ( ) cos2 11 . . . . . .)(analogous to u) lβ

(12–140)T T LI= −( ( ) cos2 11 . . . . . .)(analogous to u) lβ

12.7.3. Axisymmetric Harmonic 4 Node Quadrilateral Solids without ESF

These shape functions are for the 4 node axisymmetric harmonic quadrilateral solid elements without extrashape functions, such as PLANE25 with KEYOPT(2) = 1, or PLANE75:

(12–141)u u s t u s t

u s t u s t

I J

K L

= − − + + −

+ + + + − +

14

1 1 1 1

1 1 1 1

( ( )( ) ( )( )

( )( ) ( )( ))cooslβ

(12–142)v v sI= −1

41( ( ) cos. . . . . .)(analogous to u) lβ

(12–143)w w sI= −1

41( ( ) )sin. . . . . .(analogous to u) lβ

(12–144)T T sI= −1

41( ( ) cos. . . . . .)(analogous to u) lβ

12.7.4. Axisymmetric Harmonic 4 Node Quadrilateral Solids with ESF

These shape functions are for the 4 node axisymmetric harmonic quadrilateral elements with extra shape functions,such as PLANE25 with KEYOPT(2) = 0.

Chapter 12: Shape Functions

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(12–145)

u u s t u s t u s t

u s t

I J K

L

= − − + + − + + +

+ − + )

( ( ( )( ) ( )( ) ( )( )

( )( )

14

1 1 1 1 1 1

1 1 ++ − + −u s u t12

221 1( ) ( )))coslβ

(12–146)v v sI= −( ( ( ) cos

14

1 . . . . . .)(analogous to u) lβ

(12–147)w w sI= −( ( ( ) sin

14

1 . . . . . .)(analogous to u) lβ

Unless l (MODE) = 1, u1 or u2 and w1 or w2 motions are suppressed for elements near the centerline.

12.7.5. Axisymmetric Harmonic 8 Node Quadrilateral Solids

These shape functions are for the 8 node axisymmetric harmonic quadrilateral solid elements such as PLANE78or PLANE83.

(12–148)

u u s t s t u s t s t

u s t

I J

K

= − − − − − + + − − −

+ + +

( ( ( )( )( ) ( )( )( )

( )(

14

1 1 1 1 1 1

1 1 ))( ) ( )( )( ))

( ( )( ) ( )(

s t u s t s t

u s t u s

L

M N

+ − + − + − + −

+ − − + +

1 1 1 1

12

1 1 1 12 −−

+ − + + − −

t

u s t u s tO P

2

2 21 1 1 1

)

( )( ) ( )( )))coslβ

(12–149)v v sI= −( ( ( ) cos

14

1 . . . . . .)(analogous to u) lβ

(12–150)w w sI= −( ( ( ) sin

14

1 . . . . . .)(analogous to u) lβ

(12–151)T T sI= −1

41( ( ) cos. . . . . .)(analogous to u) lβ

12.8. 3-D Solids

This section contains shape functions for 3-D solid elements. These elements are available in a number of config-urations, including certain combinations of the following features:

• element shapes may be tetrahedra, pyramids, wedges, or bricks (hexahedra).

- if wedges or bricks, with or without extra shape functions (ESF)

• with or without rotational degrees of freedom (RDOF)

• with or without midside nodes

The wedge elements with midside nodes (15 node wedges) are either a condensation of the 20 node brick elementor are based on wedge shape functions.

12–23ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 12.8: 3-D Solids

12.8.1. 4 Node Tetrahedra

This element is a condensation of an 8 node brick element such as SOLID5, FLUID30, SOLID45, or FLUID142

Figure 12.11 3-D Solid Elements

The resulting effective shape functions are:

(12–152)u uL u L u L u LI J K M= + + +1 2 3 4

(12–153)v v L v L v L v LI J K M= + + +1 2 3 4

(12–154)w w L w L w L w LI J K M= + + +1 2 3 4

(12–155)V V L w L w L w Lx xI J K M= + + +1 2 3 4

(12–156)V V L w L w L w Ly yI J K M= + + +1 2 3 4

(12–157)V V L w L w L w Lz zI J K M= + + +1 2 3 4

(12–158)P PL P L P L P LI J K M= + + +1 2 3 4

(12–159)T TL T L T L T LI J K M= + + +1 2 3 4

(12–160)V VL V L V L V LI J K M= + + +1 2 3 4

(12–161)φ φ φ φ φ= + + +I J K ML L L L1 2 3 4

(12–162)E E L E L E L E LKIK

JK

KK

MK= + + +1 2 3 4

(12–163)E E L E L E L E LDID

JD

KD

MD= + + +1 2 3 4

12.8.2. 10 Node Tetrahedra

These shape functions are for 10 node tetrahedron elements such as SOLID98 and SOLID92:

Chapter 12: Shape Functions

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Figure 12.12 10 Node Tetrahedra Element

(12–164)

u u L L u L L u L L

u L L u L LI J K

L M

= − + − + −+ − + +

( ) ( ) ( )

( )

2 1 2 1 2 1

2 1 41 1 2 2 3 3

4 4 1 2 uu L L u L L

u L L u L L u L LN O

P Q R

2 3 1 3

1 4 2 4 3 4

++ + +

(12–165)v v L LI= − +( )2 11 1 . . . (analogous to u)

(12–166)w w L LI= − +( )2 11 1 . . . (analogous to u)

(12–167)T T L LI= − +( )2 11 1 . . . (analogous to u)

(12–168)V V L LI= − +( )2 11 1 . . . (analogous to u)

(12–169)φ φ= − +I L L( )2 11 1 . . . (analogous to u)

12.8.3. 5 Node Pyramids

This element is a condensation of an 8 node brick element.

Figure 12.13 8 Node Brick Element

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Section 12.8: 3-D Solids

The resulting effective shape functions are:

(12–170)

T T s t r T s t r

T s t r

I J

K

= − − − + + − −

+ + + − +

18

1 1 1 1 1 1

1 1 1

( )( )( ) ( )( )( )

( )( )( ) TT s t r

T r

L

M

( )( )( )

( )

1 1 1

12

1

− + −

+ +

12.8.4. 13 Node Pyramids

These shape functions are for 13 node pyramid elements which are based on a condensation of a 20 node brickelement such as SOLID95:

Figure 12.14 13 Node Pyramid Element

!

"$# &%(' )*

(12–171)

uq

u s t qs qt u s t qs qt

u s

I J

K

= − − − − − + + − − + −

+ +4

1 1 1 1 1 1

1

( ( )( )( ) ( )( )( )

( )(( )( ) ( )( )( ))

( )( )

1 1 1 1 1

1 1 2

2

2

+ − + + + − + − − ++ − −

+

t sq qt u s t qs qt

u q q

q

L

M

(( ( )( ) ( )( ) ( )( )

( )( ))

u t s u s t u t s

u s t

Q R S

T

1 1 1 1 1 1

1 1

2 2 2

2

− − + + − + + −

+ − −++ − − − + + + − − + + + ++ − + −

q q u s t st u s t st u s t st

u s tY Z A

B

( )( ( ) ( ) )( )

(

1 1 1 1

1 sst))

(12–172)v

qv sI= −

41( ( ). . . (analogous to u)

Chapter 12: Shape Functions

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(12–173)w

qw sI= −

41( ( ). . . (analogous to u)

(12–174)T

qT sI= −

41( ( ). . . (analogous to u)

(12–175)V

qV sI= −

41( ( ). . . (analogous to u)

12.8.5. 6 Node Wedges without ESF

Figure 12.15 6 Node Wedge Element

The 6 node wedge elements are a condensation of an 8 node brick such as SOLID5, FLUID30, or SOLID45. Theseshape functions are for 6 node wedge elements without extra shape functions:

(12–176)u uL r u L r u L r

u L r u L r u L

I J K

M N O

= − + − + −

+ + + + +

12

1 1 1

1 1

1 2 3

1 2 3

( ) ( ) ( )

( ) ( ) (( )1+ r

(12–177)v v L rI= −1

211( ( ). . . (analogous to u)

(12–178)w w L rI= −1

211( ( ). . . (analogous to u)

(12–179)P PL rI= −1

211( ( ). . . (analogous to u)

(12–180)T TL rI= −1

211( ( ). . . (analogous to u)

(12–181)V VL rI= −1

211( ( ). . . (analogous to u)

12–27ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 12.8: 3-D Solids

(12–182)φ φ= −1

211( ( )IL r . . . (analogous to u)

12.8.6. 6 Node Wedges with ESF

The 6 node wedge elements are a condensation of an 8 node brick such as SOLID5, FLUID30, or SOLID45. (Pleasesee Figure 12.15: “6 Node Wedge Element”.) These shape functions are for 6 node wedge elements with extrashape functions:

(12–183)

u uL r u L r u L r

u L r u L r u L

I J K

M N O

= − + − + −

+ + + + +

12

1 1 1

1 1

1 2 3

1 2

( ( ) ( ) ( )

( ) ( ) 33 121 1( ) ( ))+ + −r u r

(12–184)v v L rI= −1

211( ( ). . . (analogous to u)

(12–185)w w L rI= −1

211( ( ). . . (analogous to u)

12.8.7. 15 Node Wedges as a Condensation of 20 Node Brick

Figure 12.16 15 Node Wedge Element (SOLID90)

!

These shape functions are for 15 node wedge elements such as SOLID90 that are based on a condensation of a20 node brick element Equation 12–209.

Chapter 12: Shape Functions

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12.8.8. 15 Node Wedges Based on Wedge Shape Functions

Figure 12.17 15 Node Wedge Element (SOLID95)

!

"

#$

% &

"

(' *),+

Elements such as SOLID95 in a wedge configuration use shape functions based on triangular coordinates andthe r coordinate going from -1.0 to +1.0.

(12–186)

u u L L r L r u L r L r

u

I J= − − − − + − − − −

+

12

2 1 1 1 1 1 11 1 12

2 22( ( ( )( ) ( )) ( )( ) ( ))

KK M

N

L L r L r u L L r

L r u

( ( )( ) ( )) ( ( )( )

( ))

3 3 32

1 1

12

2 1 1 1 2 1 1

1

− − − − + − +

− − + (( ( ))( ) ( ))

( ( )( ) ( )) (

L L r L r

u L L r L rO

2 2 22

3 3 32

2 1 1 1

2 1 1 1 2

− + − −

+ − + − − + uu L L r

u L L r u L L r u L L r

u L L

Q

R T U

V

1 2

2 3 3 1 1 2

2 3

1

1 1 1

1

( ))

( ) ( ) ( )

(

−+ − + − + +

+ ++ + + )+ −

+ − + −

r u L L r u L r

u L r u L r

X Y

Z A

) ( ) ( )

( ) ( )

3 1 12

22

32

1 1

1 1

(12–187)v v L LI= −1

22 11 1( ( ). . . (analogous to u)

(12–188)w w L LI= −1

22 11 1( ( ). . . (analogous to u)

(12–189)T TL LI= −1

22 11 1( ( ). . . (analogous to u)

(12–190)V VL LI= −1

22 11 1( ( ). . . (analogous to u)

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Section 12.8: 3-D Solids

12.8.9. 8 Node Bricks without ESF

Figure 12.18 8 Node Brick Element

These shape functions are for 8 node brick elements without extra shape functions such as SOLID5 with KEYOPT(3)= 1, FLUID30, SOLID45 with KEYOPT(1) = 1, or FLUID142:

(12–191)

u u s t r u s t r

u s t r

I J

K

= − − − + + − −

+ + + −

18

1 1 1 1 1 1

1 1 1

( ( )( )( ) ( )( )( )

( )( )( ) ++ − + −+ − − + + + − ++

u s t r

u s t r u s t r

u

L

M N

O

( )( )( )

( )( )( ) ( )( )( )

1 1 1

1 1 1 1 1 1

(( )( )( ) ( )( )( ))1 1 1 1 1 1+ + + + − + +s t r u s t rP

(12–192)v v sI= −1

81( ( ). . . (analogous to u)

(12–193)w w sI= −1

81( ( ). . . (analogous to u)

(12–194)A A sx xI= −1

81( ( ). . . (analogous to u)

(12–195)A A sy yI= −1

81( ( ). . . (analogous to u)

(12–196)A A sz zI= −1

81( ( ). . . (analogous to u)

(12–197)V V sx xI= −1

81( ( ). . . (analogous to u)

Chapter 12: Shape Functions

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(12–198)V V sy yI= −1

81( ( ). . . (analogous to u)

(12–199)V V sz zI= −1

81( ( ). . . (analogous to u)

(12–200)P P sI= −1

81( ( ). . . (analogous to u)

(12–201)T T sI= −1

81( ( ). . . (analogous to u)

(12–202)V V sI= −1

81( ( ). . . (analogous to u)

(12–203)φ φ= −1

81( ( )I s . . . (analogous to u)

(12–204)E E sK

IK= −1

81( ( ). . . (analogous to u)

(12–205)E E sD

ID= −1

81( ( ). . . (analogous to u)

12.8.10. 8 Node Bricks with ESF

(Please see Figure 12.18: “8 Node Brick Element”) These shape functions are for 8 node brick elements with extrashape functions such as SOLID5 with KEYOPT(3) = 0 or SOLID45 with KEYOPT(1) = 0:

(12–206)

u u s t r u s t r

u s t r

I J

K

= − − − + + − −

+ + + −

18

1 1 1 1 1 1

1 1 1

( ( )( )( ) ( )( )( )

( )( )( ) ++ − + −+ − − + + + − ++

u s t r

u s t r u s t r

u

L

M N

O

( )( )( )

( )( )( ) ( )( )( )

1 1 1

1 1 1 1 1 1

(( )( )( ) ( )( )( ))

( ) ( ) (

1 1 1 1 1 1

1 1 112

22

3

+ + + + − + +

+ − + − +

s t r u s t r

u s u t u

P

−− r2 )

(12–207)v v sI= −1

81( ( ). . . (analogous to u)

(12–208)w w sI= −1

81( ( ). . . (analogous to u)

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Section 12.8: 3-D Solids

12.8.11. 20 Node Bricks

Figure 12.19 20 Node Brick Element

These shape functions are used for 20 node solid elements such as SOLID90 or SOLID95.

(12–209)

u u s t r s t r u s t r s t rI J= − − − − − − − + + − − − − −18

1 1 1 2 1 1 1 2( ( )( )( )( ) ( )( )( )( ))

( )( )( )( ) ( )( )( )( )+ + + − + − − + − + − − + − −+

u s t r s t r u s t r s t rK L1 1 1 2 1 1 1 2

uu s t r s t r u s t r s t r

uM N

O

( )( )( )( ) ( )( )( )( )1 1 1 2 1 1 1 2− − + − − + − + + − + − + −+ (( )( )( )( ) ( )( )( )( ))

(

1 1 1 2 1 1 1 2

14

+ + + + + − + − + + − + + −

+

s t r s t r u s t r s t rP

uu s t r u s t r

u s t r

Q R

S

( )( )( ) ( )( )( )

( )( )( )

1 1 1 1 1 1

1 1 1

2 2

2

− − − + + − −

+ − + − + uu s t r

u s t r u s t r

T

U V

( )( )( )

( )( )( ) ( )( )( )

1 1 1

1 1 1 1 1 1

2

2 2

− − −

+ − − + + + − +

+uu s t r u s t r

u s t r

W X

Y

( )( )( ) ( )( )( )

( )( )( )

1 1 1 1 1 1

1 1 1

2 2

2

− + + + − − +

+ − − − + uu s t r

u s t r u s t r

Z

A B

( )( )( )

( )( )( ) ( )( )( ))

1 1 1

1 1 1 1 1 1

2

2 2

+ − −

+ + + − + − + −

(12–210)v v sI= −1

81( ( ). . . (analogous to u)

(12–211)w w sI= −1

81( ( ). . . (analogous to u)

(12–212)T T sI= −1

81( ( ). . . (analogous to u)

(12–213)V V sI= −1

81( ( ). . . (analogous to u)

(12–214)φ φ= −18

1( ( )I s . . . (analogous to u)

Chapter 12: Shape Functions

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12.8.12. 8 Node Infinite Bricks

Figure 12.20 3-D 8 Node Brick Element

These Lagrangian isoparametric shape functions and “mapping” functions are for the 3-D 8 node solid brick in-finite elements such as INFIN111:

12.8.12.1. Lagrangian Isoparametric Shape Functions

(12–215)

A A s t r r

A s t r r

A s t

x xI

xJ

xK

= − − −

+ + − −

+ + +

18

1 1

1 1

1 1

2

2

( ( )( )( )

( )( )( )

( )( ))( )

( )( )( ))

( ( )( )( )

(

r r

A s t r r

A s t r

A

xL

xM

xN

2

2

2

1 1

14

1 1 1

1

−

+ − + −

+ − − −

+ ++ − −

+ + + −

+ − + −

s t r

A s t r

A s t r

xO

xP

)( )( )

( )( )( )

( )( )( ))

1 1

1 1 1

1 1 1

2

2

2

(12–216)Ay = 1

8( ( )A 1 - s . . . xl (analogous to A )x

(12–217)Az = 1

8( ( )A 1 - s . . . zl (analogous to A )x

(12–218)T = 1

8( ( )A 1 - s . . . Tl (analogous to A )x

(12–219)V = 18

( ( )A 1 - s . . . Vl (analogous to A )x

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Section 12.8: 3-D Solids

(12–220)φ φ= 1

8( ( )l 1 - s . . . (analogous to A )x

12.8.12.2. Mapping Functions

(12–221)

x x s t r r

x s t r r

x s

I

J

K

= − − − −

+ + − − −+ +

12

1 1 1

1 1 1

1

( ( )( )( ) /( )

( )( )( ) /( )

( )(( )( ) /( )

( )( )( ) /( ))

( ( )( )(

1 1

1 1 1

14

1 1 1

+ − −+ − + − −

+ − −

t r r

x s t r r

x s t

L

M ++ −

+ + − + −+ + + + −

r r

x s t r r

x s t r rN

O

) /( )

( )( )( ) /( )

( )( )( ) /( )

1

1 1 1 1

1 1 1 1

++ − + + −x s t r rP( )( )( ) /( ))1 1 1 1

(12–222)y y sI= −1

21( ( ). . . (analogous to x)

(12–223)z z sI= −1

21( ( ). . . (analogous to x)

12.8.13. 3-D 20 Node Infinite Bricks

Figure 12.21 20 Node Solid Brick Infinite Element

These Lagrangian isoparametric shape functions and “mapping” functions are for the 3-D 20 node solid brickinfinite elements such as INFIN111:

Chapter 12: Shape Functions

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12.8.13.1. Lagrangian Isoparametric Shape Functions

(12–224)

A A s t r s t r

A s t r

x xI

xJ

= − − − − − − −

+ − − −

18

1 1 1 2

14

1 1 12

( ( )( )( )( ))

( ( )( )( )))

( ( )( )( )( ))

( ( )( )( ))

+ + − − − − −

+ + − −

18

1 1 1 2

14

1 1 12

A s t r s t r

A s t r

xK

xL

++ + + − + − −

+ − + −

+

18

1 1 1 2

14

1 1 12

( ( )( )( )( ))

( ( )( )( ))

A s t r s t r

A s t r

xM

xN

118

1 1 1 2

14

1 1 12

( ( )( )( )( ))

( ( )( )( ))

A s t r s t r

A s t r

xO

xP

− + − − + − −

+ − − −

+ 114

1 1 1

14

1 1 1

14

1

2

2

( ( )( )( ))

( ( )( )( ))

( (

A s t r

A s t r

A

xQ

xR

xS

− − −

+ + − −

+ + ss t r

A s t rxT

)( )( ))

( ( )( )( ))

1 1

14

1 1 1

2

2

+ −

+ − + −

(12–225)Ay = 1

8( ( )A 1 - s . . . xl (analogous to A )x

(12–226)Az = 1

8( ( )A 1 - s . . . zl (analogous to A )x

(12–227)T = 1

8( ( )A 1 - s . . . Tl (analogous to A )x

(12–228)V = 1

8( ( )A 1 - s . . . Vl (analogous to A )x

(12–229)φ φ= 1

8( ( )l 1 - s . . . (analogous to A )x

12–35ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 12.8: 3-D Solids

12.8.13.2. Mapping Functions

(12–230)

x x s t s t r r

x s t r

x

I

J

K

= − − − − − − −

+ − − −+

( )( )( ) /( ( ))

( )( ) /( )

(

1 1 2 2 1

1 1 1

1

2

++ − − − − − −

+ + − −+ +

s t s t r r

x s t r

x sL

M

)( )( ) /( ( ))

( )( ) /( )

( )(

1 2 2 1

1 1 1

1 1

2

++ + − − −

+ − + −+ − + −

t s t r r

x s t r

x s t sN

O

)( ) /( ( ))

( )( ) /( )

( )( )(

2 2 1

1 1 1

1 1

2

++ − − −

+ − − −+ − − +

t r r

x s t r

x s t rP

Q

2 2 1

1 1 1

1 1 1 4

2

) /( ( ))

( )( ) /( )

( )( )( ) /( (( ))

( )( )( ) /( ( ))

( )( )( ) /( (

1

1 1 1 4 1

1 1 1 4 1

−+ + − + −+ + + + −

r

x s t r r

x s t rR

S rr

x s t r rT

))

( )( )( ) /( ( ))+ − + + −1 1 1 4 1

(12–231)y y sI= −( ) . . . (analogous to x)1

(12–232)z z sI= −( )1 . . . (analogous to x)

The shape and mapping functions for the nodes U, V, W, X, Y, Z, A, and B are deliberately set to zero.

12.9. Electromagnetic Edge Elements

The shortcomings of electromagnetic analysis by nodal based continuous vector potential is discussed in Sec-tion 5.1.4: Edge Flux Degrees of Freedom. These can be eliminated by edge shape functions described in thissection. The edge element formulation constitutes the theoretical foundation of low-frequency electromagneticelement, Section 14.117: SOLID117 - 3-D 20-Node Magnetic Edge.

Edge elements on tetrahedra and rectangular blocks have been introduced by Nedelec(204); first order andquadratic isoparametric hexahedra by van Welij(205) and Kameari(206), respectively. Difficulty with distortedhexahedral edge elements is reported by Jin(207) without appropriate resolution. Gyimesi and Ostergaard(201),(221), Ostergaard and Gyimesi(222, 223) explained the theoretical shortage of isoparametric hexahedra. Theirnonconforming edge shape functions are implemented, eliminating the negative effect of element distortion.The extension of brick shapes to tetrahedra, wedge and pyramid geometries is given in Gyimesi and Oster-gaard(221).

Section 12.9.1: 2-D 8 Node Quad Geometry and DOFs and Section 12.9.2: 3-D 20 Node Brick Geometry and DOFsdescribe the 2-D and 3-D electromagnetic edge elements, respectively.

Chapter 12: Shape Functions

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12.9.1. 2-D 8 Node Quad Geometry and DOFs

Figure 12.22 2-D 8 Node Quad Edge Element

Figure 12.22: “2-D 8 Node Quad Edge Element” shows the geometry of 2-D 8-node electromagnetic edge elements.The corner nodes, I, J, K, and L are used to:

• describe the geometry

• orient the edges

• support time integrated electric potential DOFs, labeled VOLT

The side nodes, M, N, O, and P are used to:

• support the edge-flux DOFs, labeled as AZ. The positive orientation of an edge is defined to point fromthe adjacent (to the edge) corner node with lower node number to the other adjacent node with highernode number. For example, edge, M, is oriented from node I to J if I has a smaller node number than J;otherwise it is oriented from J to I.

The edge-flux DOFs are used in both magnetostatic and dynamic analyses; the VOLT DOFs are used only for dy-namic analysis.

The vector potential, A, and time integrated electric scalar potential, V, can be described as

(12–233)A A E A E A E A EM M N N O O P P= + + +

(12–234)V VN V N V N V NI I J J K K L L= + + +

where:

AM, AN, AO, AP = edge-flux

AZ = nodal DOFs supported by the side nodesVI, VJ, VK, VL = time integrated electric scalar potential

VOLT = nodal DOFs supported by corner nodesEM, EN, EO, EP = vector edge shape functions

NI, NJ, NK, NL = scalar nodal shape functions

The following subsections describe these shape functions.

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Section 12.9: Electromagnetic Edge Elements

The global Cartesian coordinates, X and Y, can be expressed by the isoparametric coordinates, r and s.

(12–235)X N r s X N r s X N r s X N r s XI I J J K K L L= + + +( , ) ( , ) ( , ) ( , )

(12–236)Y N r s Y N r s Y N r s Y N r s YI I J J K K L L= + + +( , ) ( , ) ( , ) ( , )

where:

XI thru YL = global Cartesian coordinates of the corner nodes

NI, NJ, NK, and NL = first order scalar nodal shape functions

(12–237)N r sI = − −( )( )1 1

(12–238)N r sJ = −( )1

(12–239)N rsK =

(12–240)N r sL = −( )1

The isoparametric vector edge shape functions are defined as

(12–241)E sM = + −( )1 grad r

(12–242)EN = r grad s

(12–243)EO = −s grad r

(12–244)E rP = − −( )1 grads

Note that the tangential component (the dot product with a unit vector pointing in the edge direction) of thevector edge shape functions disappears on all edges but one. The one on which the tangential component ofan edge shape function is not zero is called a supporting edge which is associated with the pertinent side node.

Note also that the line integral of an edge shape function along the supporting edge is unity. The flux crossinga face is the closed line integral of the vector potential, A. Thus, the sum of the DOFs supported by side nodesaround a face is the flux crossing the face. Therefore, these DOFs are called edge-flux DOFs.

Chapter 12: Shape Functions

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12.9.2. 3-D 20 Node Brick Geometry and DOFs

Figure 12.23 3-D 20 Node Brick Edge Element

Figure 12.23: “3-D 20 Node Brick Edge Element” shows the geometry of 3-D 20-node electromagnetic edge ele-ments. The corner nodes, I ... P are used to:

• describe the geometry

• orient the edges

• support time integrated electric potential DOFs (labeled VOLT)

The side nodes, Q ... A are used to:

• support the edge-flux DOFs, labeled as AZ

• define the positive orientation of an edge to point from the adjacent (to the edge) corner node with lowernode number to the other adjacent node with higher node number. For example, edge, M, is orientedfrom node I to J if I has a smaller node number than J; otherwise it is oriented from J to I.

The edge-flux DOFs are used in both magnetostatic and dynamic analyses; the VOLT DOFs are used only for dy-namic analysis.

The vector potential, A, and time integrated electric scalar potential, V, can be described as

(12–245)A A E A EQ Q B B= + ⋅ ⋅ ⋅ +

(12–246)V VN V NI I P P= + ⋅ ⋅ ⋅ +

where:

AQ . . . AB = edge-flux

AZ = nodal DOFs supported by the side nodesVI . . . VP = time integrated electric scalar potential

VOLT = nodal DOFs supported by corner nodesEQ . . . EB = vector edge shape functions

NI . . . NP = scalar nodal shape functions

Do not confuse edge-flux DOF label, AZ, with the actual value of the DOF at node Z, AZ.

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Section 12.9: Electromagnetic Edge Elements

The following subsections describe these shape functions.

The global Cartesian coordinates, X, Y and Z, can be expressed by the master coordinates, r, s and t.

(12–247)X N r s t X N r s t XI I P P= + ⋅ ⋅ ⋅ +( , , ) ( , , )

(12–248)Y N r s t Y N r s t YI I P P= + ⋅ ⋅ ⋅ +( , , ) ( , , )

(12–249)Z N r s t Z N r s t ZI I P P= + ⋅ ⋅ ⋅ +( , , ) ( , , )

where:

XI, YI, ZI . . . = global Cartesian coordinates of the corner nodes

NI . . . NP = first order scalar nodal shape functions

(12–250)N r s tI = − − −( )( )( )1 1 1

(12–251)N r s tJ = − −( )( )1 1

(12–252)N rs tK = −( )1

(12–253)N r s tL = − −( ) ( )1 1

(12–254)N r s tM = − −( )( )1 1

(12–255)N r s tN = −( )1

(12–256)N rstO =

(12–257)N r stP = −( )1

The isoparametric vector edge shape functions are defined as

(12–258)E s tQ = + − −( )( )1 1 grad r

(12–259)E r tR = + −( )1 grad s

(12–260)E s tS = − −( )1 grad r

(12–261)E r tT = − − −( )( )1 1 grad s

(12–262)E sU = + −( )1 t grad r

(12–263)EV = + r t grads

(12–264)EW = −s t grad r

Chapter 12: Shape Functions

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(12–265)E rX = − −( )1 t grads

(12–266)E s rY = + − −( )( )1 1 grad t

(12–267)E s rZ = + −( )1 grad t

(12–268)EA = +s t grad t

(12–269)E sB = + −( )1 r grad t

Note that the tangential component (the dot product with a unit vector pointing in the edge direction) of thevector edge shape functions disappears on all edges but one. The one on which the tangential component ofan edge shape function is not zero is called a supporting edge which is associated with the pertinent side node.

Note also that the line integral of an edge shape function along the supporting edge is unity. The flux crossinga face is the closed line integral of the vector potential, A. Thus, the sum of the DOFs supported by side nodesaround a face is the flux crossing the face. Therefore, these DOFs are called edge-flux DOFs.

The 20 node brick geometry is allowed to degenerate to 10-node tetrahedron, 13-node pyramid or 15-nodewedge shapes as described in Gyimesi and Ostergaard(221). The numerical bench-working shows that tetrahedrashapes are advantageous in air (no current) domains, whereas hexahedra are recommended for current carryingregions. Pyramids are applied to maintain efficient meshing between hexahedra and tetrahedra regions. Wedgesare generally applied for 2-D like geometries, when longitudinal dimensions are longer than transverse sizes. Inthis case the cross-section can be meshed by area meshing and wedges are generated by extrusion.

12.10. High Frequency Electromagnetic Tangential Vector Elements

In electromagnetics, we encounter serious problems when node-based elements are used to represent vectorelectric or magnetic fields. First, the spurious modes can not be avoided when modeling cavity problems usingnode-based elements. This limitation can also jeopardize the near-field results of a scattering problem, the far-field simulation typically has no such a limitation, since the spurious modes do not radiate. Secondly, node-basedelements require special treatment for enforcing boundary conditions of electromagnetic field at material inter-faces, conducting surfaces and geometric corners. Tangential vector elements, whose degrees of freedom areassociated with the edges, faces and volumes of the finite element mesh, have been shown to be free of theabove shortcomings (Volakis, et al.(299), Itoh, et al.(300)).

12.10.1. Tetrahedral Elements (HF119)

The tetrahedral element is the simplest tessellated shape and is able to model arbitrary 3-D geometric structures.It is also well suited for automatic mesh generation. The tetrahedral element, by far, is the most popular elementshape for 3-D applications in FEA.

For the 1st-order tetrahedral element (KEYOPT(1) = 1), the degrees of freedom (DOF) are at the edges of elementi.e., (DOFs = 6) (Figure 12.24: “1st-Order Tetrahedral Element”). In terms of volume coordinates, the vector basisfunctions are defined as:

(12–270)

rW hIJ IJ I J J I= ∇ − ∇( )λ λ λ λ

(12–271)

rW hJK JK J K K J= ∇ − ∇( )λ λ λ λ

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Section 12.10: High Frequency Electromagnetic Tangential Vector Elements

(12–272)

rW hKI KI K I I K= ∇ − ∇( )λ λ λ λ

(12–273)

rW hIL IL I L L I= ∇ − ∇( )λ λ λ λ

(12–274)

rW hJL JL J L L J= ∇ − ∇( )λ λ λ λ

(12–275)

rW hKL KL K L L K= ∇ − ∇( )λ λ λ λ

where:

hIJ = edge length between node I and J

λI, λJ, λK, λL = volume coordinates (λK = 1 - λI - λJ - λL)

∇ λI, ∇ λJ, ∇ λK, ∇ λL = the gradient of volume coordinates

Figure 12.24 1st-Order Tetrahedral Element

The tangential component of electric field is constant along the edge. The normal component of field varieslinearly.

For the 2nd-order tetrahedral element (KEYOPT(1) = 2), the degrees of freedom (DOF) are at the edges and onthe faces of element. Each edge and face have two degrees of freedom (DOFs = 20) (Figure 12.25: “2nd-OrderTetrahedral Element”). The vector basis functions are defined by:

(12–276)

rW WIJ I J JI J I= ∇ = ∇λ λ λ λ (on edge IJ)

(12–277)

rW WJK J K KJ K J= ∇ = ∇λ λ λ λ (on edge JK)

(12–278)

rW WKI K I IK I K= ∇ = ∇λ λ λ λ (on edge KI)

(12–279)

rW WIL I L LI L I= ∇ = ∇λ λ λ λ (on edge IL)

(12–280)

rW WJL J L LJ L J= ∇ = ∇λ λ λ λ (on edge JL)

Chapter 12: Shape Functions

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(12–281)

rW WKL K L LK L K= ∇ = ∇λ λ λ λ (on edge KL)

(12–282)

r rF FIJK I J K K J IJK K J I I J

1 2= ∇ − ∇ = ∇ − ∇λ λ λ λ λ λ λ λ λ λ( ) ( ) (on face IJK)

(12–283)

r rF FIJL I J L L J IJL L J I I J

1 2= ∇ − ∇ = ∇ − ∇λ λ λ λ λ λ λ λ λ λ( ) ( ) (on face IJL)

(12–284)

r rF FJKL J K L L K JKL L K J J K

1 2= ∇ − ∇ = ∇ − ∇λ λ λ λ λ λ λ λ λ λ( ) ( ) (on face JKL)

(12–285)

r rF FKIL I K L L K KIL L K I I K

1 2= ∇ − ∇ = ∇ − ∇λ λ λ λ λ λ λ λ λ λ( ) ( ) (on face KIL)

Figure 12.25 2nd-Order Tetrahedral Element

12.10.2. Hexahedral Elements (HF120)

Tangential vector bases for hexahedral elements can be derived by carrying out the transformation mapping ahexahedral element in the global xyz coordinate to a brick element in local str coordinate.

For the 1st-order brick element (KEYOPT(1) = 1), the degrees of freedom (DOF) are at the edges of element (DOFs= 12) (Figure 12.26: “1st-Order Brick Element”). The vector basis functions are cast in the local coordinate

(12–286)

rW

ht r ss

e s= ± ± ∇8

1 1( )( ) parallel to s-axis

(12–287)

rW

hr s tt

e t= ± ± ∇8

1 1( )( ) parallel to t-axis

(12–288)

rW

hs t rr

e r= ± ± ∇8

1 1( )( ) parallel to r-axis

where:

hs, ht, hr = length of element edge

∇ s, ∇ t, ∇ r = gradient of local coordinates

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Section 12.10: High Frequency Electromagnetic Tangential Vector Elements

Figure 12.26 1st-Order Brick Element

For the 2nd-order brick element (KEYOPT(1) = 2), 24 DOFs are edge-based (2 DOFs/per edge), 24 DOFs are face-based (4 DOFs/per face) and 6 DOFs are volume-based (6 DOFs/per volume) (DOFs = 54) (Figure 12.27: “2nd-Order Brick Element”). The edge-based vector basis functions can be derived by:

(12–289)

rW s t t r r ss

e = ± ± ± ∇( ) ( ) ( )1 1 1 parallel to s-axis

(12–290)

rW t r r s s tt

e = ± ± ± ∇( ) ( ) ( )1 1 1 parallel to t-axis

(12–291)

rW t r r s s rr

e = ± ± ± ∇( ) ( ) ( )1 1 1 parallel to r-axis

The face-based vector basis functions are given by:

(12–292)W s t r r ss

f, ( )( ) ( )1

21 1 1= ± − ± ∇ parallel to s-axis

(12–293)W s t t r ss

f, ( ) ( )( )2

21 1 1= ± ± − ∇ parallel to s-axis

(12–294)W t r s s tt

f, ( )( ) ( )1

21 1 1= ± − ± ∇ parallel to t-axis

(12–295)W t r r s tt

f, ( ) ( )( )2

21 1 1= ± ± − ∇ parallel to t-axis

(12–296)W r s t t rr

f, ( )( ) ( )1

21 1 1= ± − ± ∇ parallel to r-axis

(12–297)W r s s t rr

f, ( ) ( )( )2

21 1 1= ± ± − ∇ parallel to r-axis

The volume-based vector basis functions are cast into:

(12–298)W s t r ssv = ± − − ∇( )( )( )1 1 12 2 parallel to s-axis

(12–299)W t r s ttv = ± − − ∇( )( )( )1 1 12 2 parallel to t-axis

Chapter 12: Shape Functions

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(12–300)W r s t rrv = ± − − ∇( )( )( )1 1 12 2 parallel to t-axis

Figure 12.27 2nd-Order Brick Element

12.10.3. Triangular Elements (HF118)

Triangular elements can be used to model electromagnetic problems in 2-D arbitrary geometric structures, es-pecially for guided-wave structure whose either cutoff frequencies or relations between the longitudepropagating constant and working frequency are required, while the mixed scalar-vector basis functions mustbe used.

For the 1st-order mixed scalar-vector triangular element (KEYOPT(1) = 1), there are three edge-based vector basisfunctions for transverse electric field, and three node-based scalar basis functions for longitude component ofelectric field (DOFs = 6) (see Figure 12.28: “Mixed 1st-Order Triangular Element”). The edge-based vector basisfunctions are defined as:

(12–301)

rW hIJ IJ I J J I= ∇ − ∇( )λ λ λ λ (at edge IJ)

(12–302)

rW hJK JK J K K J= ∇ − ∇( )λ λ λ λ (at edge JK)

(12–303)

rW hKI KI K I I K= ∇ − ∇( )λ λ λ λ (at edge KI)

The node-based scalar basis functions are given by

(12–304)NI I= λ (at node I)

(12–305)NJ J= λ (at node J)

(12–306)NK K= λ (at node K)

where:

hIJ = edge length between node I and J

λI, λJ, λK = area coordinates (λK = 1 - λI - λJ)

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Section 12.10: High Frequency Electromagnetic Tangential Vector Elements

∇ λI, ∇ λJ, ∇ λK = gradient of area coordinate

Figure 12.28 Mixed 1st-Order Triangular Element

For the 2nd-order mixed scalar-vector triangular element (KEYOPT(1) = 2), there are six edge-based, two face-based vector basis functions for transverse components of electric field, and six node-based scalar basis functionsfor longitude component of electric field (DOFs = 14) (see Figure 12.29: “Mixed 2nd-Order Triangular Element”).The edge-based vector basis functions can be written by:

(12–307)

rW WIJ I J JI J I= ∇ = ∇λ λ λ λ (on edge IJ)

(12–308)

rW WJK J K KJ K J= ∇ = ∇λ λ λ λ (on edge JK)

(12–309)

rW WKI K I IK I K= ∇ = ∇λ λ λ λ (on edge KI)

The face-based vector basis functions are similar to those in 3-D tetrahedron, i.e.:

(12–310)

rFIJK I J K K J

1 = ∇ − ∇λ λ λ λ λ( )

(12–311)

rFIJK K J I I J

2 = ∇ − ∇λ λ λ λ λ( )

The node-based scalar basis functions are given by:

(12–312)NI I I= −λ λ( )2 1 (at node I)

(12–313)NJ J J= −λ λ( )2 1 (at node J)

(12–314)NK K K= −λ λ( )2 1 (at node K)

(12–315)NL I J= 4λ λ (at node L)

(12–316)NM J K= 4λ λ (at node M)

(12–317)NN K I= 4λ λ (at node N)

Chapter 12: Shape Functions

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Figure 12.29 Mixed 2nd-Order Triangular Element

12.10.4. Quadrilateral Elements (HF118)

Tangential vector bases for quadrilateral elements can be derived by carrying out the transformation mappinga quadrilateral element in the global xy coordinate to a square element in local st coordinate.

For the 1st-order mixed scalar-vector quadrilateral element (KEYOPT(1) = 1), there are four edge-based vectorbasis functions and four node-based scalar basis functions (DOFs = 8) (Figure 12.30: “Mixed 1st-Order Quadrilat-eral Element”). Four edge-based vector basis functions are cast into:

(12–318)

rW

ht ss

e s= ± ∇4

1( ) parallel to s-axis

(12–319)

rW

hs tt

e t= ± ∇4

1( ) parallel to t-axis

Four node-based scalar basis functions are given by

(12–320)N s tI = − −1

41 1( )( ) (at node I)

(12–321)N s tJ = + −1

41 1( )( ) (at node J)

(12–322)N s tK = + +1

41 1( )( ) (at node K)

(12–323)N s tL = − +1

41 1( )( ) (at node L)

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Section 12.10: High Frequency Electromagnetic Tangential Vector Elements

Figure 12.30 Mixed 1st-Order Quadrilateral Element

For the 2nd-order mixed scalar-vector quadrilateral element (KEYOPT(1) = 2), there are 8 edge-based, 4 face-based vector basis functions and 8 node-based scalar basis functions (DOFs = 20) (Figure 12.31: “Mixed 2nd-OrderQuadrilateral Element”). The edge-based vector basis functions are derived by:

(12–324)

rW s t t ss

e = ± ± ∇( ) ( )1 1 parallel to s-axis

(12–325)

rW t s s tt

e = ± ± ∇( ) ( )1 1 parallel to t-axis

Four face-based vector basis functions can also be defined by:

(12–326)

rW s t ss

f = ± − ∇( )( )1 1 2 parallel to s-axis

(12–327)

rW t s tt

f = ± − ∇( )( )1 1 2 parallel to t-axis

The node-based scalar basis functions are given by:

(12–328)N s t s tI = − − − + +1

41 1 1( )( )( ) (at node I)

(12–329)N s t s tJ = − + − − +1

41 1 1( )( )( ) (at node J)

(12–330)N s t s tK = − + + − −1

41 1 1( )( )( ) (at node K)

(12–331)N s t s tL = − − + + −1

41 1 1( )( )( ) (at node L)

(12–332)N t sM = − − −1

21 12( )( ) (at node M)

(12–333)N s tN = − + −1

21 12( )( ) (at node N)

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(12–334)N t sO = − + −1

21 12( )( ) (at node O)

(12–335)N s tP = − − −1

21 12( )( ) (at node P)

Figure 12.31 Mixed 2nd-Order Quadrilateral Element

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Section 12.10: High Frequency Electromagnetic Tangential Vector Elements

12–50

Chapter 13: Element Tools

13.1. Integration Point Locations

The ANSYS program makes use of both standard and nonstandard numerical integration formulas. The particularintegration scheme used for each matrix or load vector is given with each element description in Chapter 14,“Element Library”. Both standard and nonstandard integration formulas are described in this section. The numbersafter the subsection titles are labels used to identify the integration point rule. For example, line (1, 2, or 3 points)represents the 1, 2, and 3 point integration schemes along line elements. Midside nodes, if applicable, are notshown in the figures in this section.

13.1.1. Lines (1, 2, or 3 Points)

The standard 1-D numerical integration formulas which are used in the element library are of the form:

(13–1)f x dx H f xi i

i( ) ( )

− =∫ = ∑1

1

1

l

where:

f(x) = function to be integratedHi = weighting factor (see Table 13.1: “Gauss Numerical Integration Constants”)

xi = locations to evaluate function (see Table 13.1: “Gauss Numerical Integration Constants”; these locations

are usually the s, t, or r coordinates)l = number of integration (Gauss) points

Table 13.1 Gauss Numerical Integration Constants

Weighting Factors (Hi)Integration Point Locations (xi)No. Integration Points

2.00000.00000.000000.00000.00000.000001

1.00000.00000.00000±0.57735 02691 896262

0.55555 55555 55556±0.77459 66692 414833

0.88888 88888 888890.00000.00000.00000

For some integrations of multi-dimensional regions, the method of Equation 13–1 is simply expanded, as shownbelow.

13.1.2. Quadrilaterals (2 x 2 or 3 x 3 Points)

The numerical integration of 2-D quadrilaterals gives:

(13–2)f x y dxdy H H f x yj i i j

ij

m( , ) ( , )

−− ==∫∫ ∑∑=1

1

1

1

11

l

and the integration point locations are shown in Figure 13.1: “Integration Point Locations for Quadrilaterals”.

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Figure 13.1 Integration Point Locations for Quadrilaterals

One element models with midside nodes (e.g., PLANE82) using a 2 x 2 mesh of integration points have beenseen to generate spurious zero energy (hourglassing) modes.

13.1.3. Bricks and Pyramids (2 x 2 x 2 Points)

The 3-D integration of bricks and pyramids gives:

(13–3)f x y z dxdydz H H H f x y zk j i i j k

ij

m

k

n( , , ) ( , , )

−−− ===∫∫∫ ∑∑=1

1

1

1

1

1

111

l∑∑

and the integration point locations are shown in Figure 13.2: “Integration Point Locations for Bricks and Pyramids”.

Figure 13.2 Integration Point Locations for Bricks and Pyramids

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One element models with midside nodes using a 2 x 2 x 2 mesh of integration points have been seen to generatespurious zero energy (hourglassing) modes.

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13.1.4. Triangles (1, 3, or 6 Points)

The integration points used for these triangles are given in Table 13.2: “Numerical Integration for Triangles” andappear as shown in Figure 13.3: “Integration Point Locations for Triangles”. L varies from 0.0 at an edge to 1.0 atthe opposite vertex.

Table 13.2 Numerical Integration for Triangles

Weighting FactorIntegration Point LocationType

1.000000L1=L2=L3=.33333331 Point Rule

0.33333 33333 33333L1=.66666 66666 66666

L2=L3=.16666 66666 66666

Permute L1, L2, and L3 for other locations)

3 Point Rule

0.10995 17436 55322L1=0.81684 75729 80459

L2=L3=0.09157 62135 09661

Permute L1, L2, and L3 for other locations)

CornerPoints

6 Point Rule0.22338 15896 78011L1=0.10810 30181 6807

L2=L3=0.44594 84909 15965

Permute L1, L2, and L3 for other locations)

Edge CenterPoints

Figure 13.3 Integration Point Locations for Triangles

13.1.5. Tetrahedra (1, 4, 5, or 11 Points)

The integration points used for tetrahedra are given in Table 13.3: “Numerical Integration for Tetrahedra”.

Table 13.3 Numerical Integration for Tetrahedra

Weighting FactorIntegration Point LocationType

1.00000 00000 00000L1=L2=L3=L4=.25000 00000 00000CenterPoint

1 Point Rule

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Section 13.1: Integration Point Locations

Weighting FactorIntegration Point LocationType

0.25000 00000 00000L1=.58541 01966 24968

L2=L3=L4=.13819 66011 25010

Permute L1, L2, L3, and L4 for other loca-

tions)

CornerPoints

4 Point Rule

-0.80000 00000 00000L1=L2=L3=L4=.25000 00000 00000CenterPoint

5 Point Rule

0.45000 00000 00000L1=.50000 00000 00000

L2=L3=L4=.16666 66666 66666

Permute L1, L2, L3, and L4 for other loca-

tions)

CornerPoints

0.01315 55555 55555L1=L2=L3=L4=.25000 00000 00000CenterPoint

11 Point Rule

0.00762 22222 22222L1=L2=L3=.0714285714285714

L4=.78571 42857 14286

(Permute L1, L2, L3 and L4 for other three

locations)

CornerPoint

0.02488 88888 88888L1=L2=0.39940 35761 66799

L3=L4=0.10059 64238 33201

Permute L1, L2, L3 and L4 such that two

of L1, L2, L3 and L4 equal 0.39940 35761

66799 and the other two equal 0.1005964238 33201 for other five locations

Edge CenterPoints

These appear as shown in Figure 13.4: “Integration Point Locations for Tetrahedra”. L varies from 0.0 at a face to1.0 at the opposite vertex.

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Figure 13.4 Integration Point Locations for Tetrahedra

13.1.6. Triangles and Tetrahedra (2 x 2 or 2 x 2 x 2 Points)

These elements use the same integration point scheme as for 4-node quadrilaterals and 8-node solids, as shownin Figure 13.5: “Integration Point Locations for Triangles and Tetrahedra”:

Figure 13.5 Integration Point Locations for Triangles and Tetrahedra

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3x3 and 3x3x3 cases are handled similarly.

13.1.7. Wedges (3 x 2 or 3 x 3 Points)

These wedge elements use an integration scheme that combines linear and triangular integrations, as shownin Figure 13.6: “6 and 9 Integration Point Locations for Wedges”

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Section 13.1: Integration Point Locations

Figure 13.6 6 and 9 Integration Point Locations for Wedges

13.1.8. Wedges (2 x 2 x 2 Points)

These wedge elements use the same integration point scheme as for 8-node solid elements as shown by twoorthogonal views in Figure 13.7: “8 Integration Point Locations for Wedges”:

Figure 13.7 8 Integration Point Locations for Wedges

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13.1.9. Bricks (14 Points)

The 20-node solid uses a different type of integration point scheme. This scheme places points close to each ofthe 8 corner nodes and close to the centers of the 6 faces for a total of 14 points. These locations are given inTable 13.4: “Numerical Integration for 20-Node Brick”:

Table 13.4 Numerical Integration for 20-Node Brick

Weighting FactorIntegration Point LocationType

.33518 00554 01662s = ±.75868 69106 39328

t = ±75878 69106 39329

r = ±75878 69106 39329

CornerPoints

14 Point Rule

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.88642 65927 97784s = ±.79582 24257 54222, t=r=0.0

t = ±79582 24257 54222, s=r=0.0

r = ±79582 24257 54222, s=t=0.0

CenterPoints

and are shown in Figure 13.8: “Integration Point Locations for 14 Point Rule”.

Figure 13.8 Integration Point Locations for 14 Point Rule

13.1.10. Nonlinear Bending (5 Points)

Both beam and shell elements that have nonlinear materials must have their effects accumulated thru thethickness. This uses nonstandard integration point locations, as both the top and bottom surfaces have an integ-ration point in order to immediately detect the onset of the nonlinear effects.

Table 13.5 Thru-Thickness Numerical Integration

Weighting FactorIntegration Point Location[1]Type

0.1250000±0.500

5 0.5787036±0.300

0.59259260.000

1. Thickness coordinate going from -0.5 to 0.5.

These locations are shown in Figure 13.9: “Nonlinear Bending Integration Point Locations”.

Figure 13.9 Nonlinear Bending Integration Point Locations

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Section 13.1: Integration Point Locations

13.2. Lumped Matrices

Some of the elements allow their consistent mass or specific heat matrices to be reduced to diagonal matrices(accessed with the LUMPM,ON command). This is referred to as “lumping”.

13.2.1. Diagonalization Procedure

One of two procedures is used for the diagonalization, depending on the order of the element shape functions.The mass matrix is used as an example.

For lower order elements (linear or bilinear) the diagonalized matrix is computed by summing rows (or columns).The steps are:

1.Compute the consistent mass matrix ([ ])Me

′ in the usual manner.

2. Compute:

S i M i jej

n( ) ( , )= ′

=∑

1 for i =1, n

(13–4)

where:

n = number of degrees of freedom (DOFs) in the element

3. Set

M i je( , ) .= ≠0 0 for i j(13–5)

(13–6)M i j S ie( , ) ( )= for i = 1, n

For higher order elements the procedure suggested by Hinton, et al.(45), is used. The steps are:

1.Compute the consistent mass matrix ([ ])Me

′ in the usual manner.

2. Compute:

S M i jej

n

i

n= ′

==∑∑ ( , )

11 (13–7)

(13–8)D M i ie

i

n= ′

=∑ ( , )

1

3. Set:

M i je( , ) .= ≠0 0 if i j(13–9)

(13–10)M i i

SD

M i ie e( , ) ( , )= ′

Note that this method ensures that:

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1. The element mass is preserved

2. The element mass matrix is positive definite

It may be observed that if the diagonalization is performed by simply summing rows or columns in higher orderelements, the resulting element mass matrix is not always positive definite.

13.2.2. Limitations of Lumped Mass Matrices

Lumped mass matrices have the following limitations:

1. Elements containing both translational and rotational degrees of freedom will have mass contributionsonly for the translational degrees of freedom. Rotational degrees of freedom are included for:

• SHELL181, SHELL208, and SHELL209 unless an unbalanced laminate construction is used.

• BEAM188 and BEAM189 if there are no offsets.

2. Lumping, by its very nature, eliminates the concept of mass coupling between degrees of freedom.Therefore, the following restrictions exist:

• Lumping is not allowed for FLUID29, FLUID30, or FLUID38 elements.

• Lumping is not allowed for BEAM44 elements when using member releases in the element UY or UZdirections.

• Lumping is not allowed for PIPE59 elements when using 'added mass' on the outside of the pipe. Inthis case, the implied coupling exists when the element x-axis is not parallel to one of the threenodal axes.

• A warning message will be output if BEAM23, BEAM24, BEAM44, or BEAM54 elements are used withexplicit or implied offsets.

• The effect of the implied offsets is ignored by the lumping logic when used with warped SHELL63elements.

• Lumping is not allowed for the mass matrix option of MATRIX27 elements if it is defined with nonzerooff-diagonal terms.

• The use of lumping with constraint equations may effectively cause the loss of some mass for dynamicanalyses, resulting in higher frequencies. This loss comes about because of the generation of off-di-agonal terms by the constraint equations, which then are ignored. (The exception to this loss isPowerDynamics, which uses lumped mass matrices (without using the LUMPM,ON command) andloses no mass.)

13.3. Reuse of Matrices

Matrices are reused automatically as often as possible in order to decrease running time. The information belowis made available for use in running time estimates.

13.3.1. Element Matrices

For static (ANTYPE,STATIC) or full transient dynamic (ANTYPE,TRANS with TRNOPT,FULL) analyses, elementstiffness/conductivity, mass, and damping/specific heat, matrices ([Ke], [Me], [Ce]) are always reused from iteration

to iteration, except when:

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Section 13.3: Reuse of Matrices

1. The full Newton-Raphson option (NROPT,FULL) is used, or for the first equilibrium iteration of a timestep when the modified Newton-Raphson option (NROPT,MODI) is used and the element has eithernonlinear materials or large deformation (NLGEOM,ON) is active.

2. The element is nonlinear (e.g. gap, radiation, or control element) and its status changes.

3. MODE or ISYM (MODE command) have changed from the previous load step for elements PLANE25,SHELL61, PLANE75, PLANE78, FLUID81, or PLANE83.

4. [ ]Ket

will be reformulated if a convective film coefficient (input on the SF or SFE commands) on an elementface changes. Such a change could occur as a ramp (KBC,0) within a load step.

5. The materials or real constants are changed by new input, or if the material properties have changeddue to temperature changes for temperature-dependent input.

Element stress stiffness matrices [Se] are never reused, as the stress normally varies from iteration to iteration.

13.3.2. Structure Matrices

The overall structure matrices are reused from iteration to iteration except when:

1. An included element matrix is reformed (see above).

2. The set of specified degrees of freedom (DOFs) is changed.

3. The integration time step size changes from that used in the previous substep for the transient (AN-TYPE,TRANS) analysis.

4. The stress stiffening option (SSTIF,ON) has been activated.

5. Spin softening (KSPIN on the OMEGA Command) is active.

and/or

6. The first iteration of a restart is being performed.

13.3.3. Override Option

The above tests are all performed automatically by the program. The user can select to override the program'sdecision with respect to whether the matrices should be reformed or not. For example, if the user has temperature-dependent input as the only cause which is forcing the reformulation of the matrices, and there is a load stepwhere the temperature dependency is not significant, the user can select that the matrices will not be reformedat that load step (KUSE,1). (Normally, the user would want to return control back to the program for the followingload step (KUSE,0)). On the other hand, the user can select that all element matrices are to be reformed each it-eration (KUSE,-1).

13.4. Temperature-Dependent Material Properties

Temperature-dependent material properties are evaluated at each integration point. Elements for which thisapplies include PLANE2, PLANE42, SOLID45, PLANE82, SOLID92, SOLID95, SHELL181, PLANE182, PLANE183 ,SOLID185, SOLID186 , SOLID187, SOLSH190, BEAM188, BEAM189, SHELL208, and SHELL209. Elements using aclosed form solution (without integration points) have their material properties evaluated at the average tem-perature of the element. Elements for which this applies include LINK1, BEAM3, BEAM4, LINK8, PIPE16, PIPE17,PIPE18, SHELL28, BEAM44, BEAM54, PIPE59, and LINK180 .

Other cases:

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1. Hyperelastic elements HYPER56, HYPER58, HYPER74, and HYPER158 have all material properties evaluatedat the average element temperature.

2. For the structural elements PLANE13, PIPE20, BEAM23, BEAM24, SHELL43, SHELL51, PIPE60, SOLID62,SOLID65, SHELL91, SHELL93, SHELL143, and SOLID191, the nonlinear material properties (TB commands)are evaluated at the integration points, but the linear material properties (MP commands) are evaluatedat the average element temperature.

3. Layered structural elements SOLID46, SOLID191, SHELL91, and SHELL99 have their linear materialproperties evaluated at the average temperature of the layer.

4. Numerically integrated structural elements PLANE25, SHELL41, SHELL61, SHELL63, SOLID64, and PLANE83have their linear material properties evaluated at the average element temperature.

5. Non-structural elements have their material properties evaluated only at the average element temper-ature, except for the specific heat (Cp) which is evaluated at each integration point.

Whether shape functions are used or not, materials are evaluated at the temperature given, i.e. no account ismade of the temperature offset (TOFFST command).

For a stress analysis, the temperatures used are based directly on the input. As temperature is the unknown ina heat transfer analysis, the material property evaluation cannot be handled in the same direct manner. For thefirst iteration of a heat transfer analysis, the material properties are evaluated at the uniform temperature (inputon BFUNIF command). The properties of the second iteration are based on the temperatures of the first iteration.The properties of the third iteration are based on the temperatures of the second iteration, etc.

See Section 2.1.3: Temperature-Dependent Coefficient of Thermal Expansion for a special discussion about thecoefficient of thermal expansion.

13.5. Positive Definite Matrices

By definition, a matrix [D] (as well as its inverse [D]-1) is positive definite if the determinants of all submatrices ofthe series:

(13–11)[ ], ,,

, ,

, ,

, , ,

, , ,

,

DD D

D D

D D D

D D D

D11

11 12

2 1 2 2

11 12 13

2 1 2 2 2 3

3 1

DD D3 2 3 3, ,

,

etc.

including the determinant of the full matrix [D], are positive. The series could have started out at any other diag-onal term and then had row and column sets added in any order. Thus, two necessary (but not sufficient) conditionsfor a symmetric matrix to be positive definite are given here for convenience:

(13–12)Di i, .> 0 0

(13–13)D D Di j i i j j, , ,<

If any of the above determinants are zero (and the rest positive), the matrix is said to be positive semidefinite. Ifall of the above determinants are negative, the matrix is said to be negative definite.

13.5.1. Matrices Representing the Complete Structure

In virtually all circumstances, matrices representing the complete structure with the appropriate boundaryconditions must be positive definite. If they are not, the message “NEGATIVE PIVOT . . .” appears. This usually

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Section 13.5: Positive Definite Matrices

means that insufficient boundary conditions were specified. An exception is a piezoelectric analysis, which workswith negative definite matrices, but does not generate any error messages.

13.5.2. Element Matrices

Element matrices are often positive semidefinite, but sometimes they are either negative or positive definite.For most cases where a negative definite matrix could inappropriately be created, the program will abort witha descriptive message.

13.6. Nodal and Centroidal Data Evaluation

Area and volume elements normally compute results most accurately at the integration points. The location ofthese data, which includes structural stresses, elastic and thermal strains, field gradients, and fluxes, can then bemoved to nodal or centroidal locations for further study. This is done with extrapolation or interpolation, basedon the element shape functions or simplified shape functions given in Table 13.6: “Assumed Data Variation ofStresses”.

Table 13.6 Assumed Data Variation of Stresses

Assumed Data VariationNo. Integration PointsGeometry

a + bs + ct3Triangles

a + bs + ct + dst4Quadrilaterals

a + bs + ct + dr4Tetrahedra

a + bs + ct + dr + est + ftr + gsr + hstr8Hexahedra

where:

a, b, c, d, e, f, g, h = coefficientss, t, r = element natural coordinates

The extrapolation is done or the integration point results are simply moved to the nodes, based on the user'srequest (input on the ERESX command). If material nonlinearities exist in an element, the least squares fit cancause inaccuracies in the extrapolated nodal data or interpolated centroidal data. These inaccuracies are normallyminor for plasticity, creep, or swelling, but are more pronounced in elements where an integration point maychange status, such as SHELL41, SOLID65, etc.

There are a number of adjustments and special cases:

1. SOLID90 and SOLID95 use only the eight corner integration points.

2. SHELL63 uses a least squares fitting procedure for the bending stresses. Data from all three integrationpoints of each of the four triangles is used.

3. SHELL43, SOLID46, SHELL91, SHELL93, SHELL99, and SOLID191 use the procedure for quadrilaterals re-peatedly at various levels through the element thickness.

4. Uniform stress cases, like a constant stress triangle, do not require the above processing.

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13.7. Element Shape Testing

13.7.1. Overview

All continuum elements (2-D and 3-D solids, 3-D shells) are tested for acceptable shape as they are defined bythe E, EGEN, AMESH, VMESH, or similar commands. This testing, described in the following sections, is performedby computing shape parameters (such as Jacobian ratio) which are functions of geometry, then comparing themto element shape limits whose default values are functions of element type and settings (but can be modifiedby the user on the SHPP command with Lab = MODIFY as described below). Nothing may be said about an ele-ment, one or more warnings may be issued, or it may be rejected with an error.

13.7.2. 3-D Solid Element Faces and Cross-Sections

Some shape testing of 3-D solid elements (bricks [hexahedra], wedges, pyramids, and tetrahedra) is performedindirectly. Aspect ratio, parallel deviation, and maximum corner angle are computed for 3-D solid elements usingthe following steps:

1. Each of these 3 quantities is computed, as applicable, for each face of the element as though it were aquadrilateral or triangle in 3-D space, by the methods described in sections Section 13.7.3: Aspect Ratio,Section 13.7.8: Parallel Deviation, and Section 13.7.10: Maximum Corner Angle.

2. Because some types of 3-D solid element distortion are not revealed by examination of the faces, cross-sections through the solid are constructed. Then, each of the 3 quantities is computed, as applicable, foreach cross-section as though it were a quadrilateral or triangle in 3-D space.

3. The metric for the element is assigned as the worst value computed for any face or cross-section.

A brick element has 6 quadrilateral faces and 3 quadrilateral cross-sections (Figure 13.10: “Brick Element”). Thecross-sections are connected to midside nodes, or to edge midpoints where midside nodes are not defined.

Figure 13.10 Brick Element

A pyramid element has 1 quadrilateral face and 4 triangle faces, and 8 triangle cross-sections (Figure 13.11: “Pyr-amid Element”).

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Section 13.7: Element Shape Testing

Figure 13.11 Pyramid Element

As shown in Figure 13.12: “Pyramid Element Cross-Section Construction”, each pyramid cross-section is constructedby passing a plane through one of the base edges and the closest point on the straight line containing one ofthe opposite edges. (Midside nodes, if any, are ignored.)

Figure 13.12 Pyramid Element Cross-Section Construction

A wedge element has 3 quadrilateral and 2 triangle faces, and has 3 quadrilateral and 1 triangle cross-sections.As shown in Figure 13.13: “Wedge Element”, the cross-sections are connected to midside nodes, or to edgemidpoints where midside nodes are not defined.

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Figure 13.13 Wedge Element

A tetrahedron element has 4 triangle faces and 6 triangle cross-sections (Figure 13.14: “Tetrahedron Element”).

Figure 13.14 Tetrahedron Element

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As shown in Figure 13.15: “Tetrahedron Element Cross-Section Construction”, each tetrahedron cross-section isconstructed by passing a plane through one of the edges and the closest point on the straight line containingthe opposite edge. (Midside nodes, if any, are ignored.)

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Section 13.7: Element Shape Testing

Figure 13.15 Tetrahedron Element Cross-Section Construction

13.7.3. Aspect Ratio

Aspect ratio is computed and tested for all except Emag or FLOTRAN elements (see Table 13.7: “Aspect RatioLimits”). This shape measure has been reported in finite element literature for decades (Robinson(121)), and isone of the easiest ones to understand. Some analysts want to be warned about high aspect ratio so they canverify that the creation of any stretched elements was intentional. Many other analysts routinely ignore it.

Unless elements are so stretched that numeric round off could become a factor (aspect ratio > 1000), aspect ratioalone has little correlation with analysis accuracy. Finite element meshes should be tailored to the physics of thegiven problem; i.e., fine in the direction of rapidly changing field gradients, relatively coarse in directions withless rapidly changing fields. Sometimes this calls for elements having aspect ratios of 10, 100, or in extreme cases1000. (Examples include shell or thin coating analyses using solid elements, thermal shock “skin” stress analyses,and fluid boundary layer analyses.) Attempts to artificially restrict aspect ratio could compromise analysis qualityin some cases.

13.7.4. Aspect Ratio Calculation for Triangles

Figure 13.16 Triangle Aspect Ratio Calculation

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The aspect ratio for a triangle is computed in the following manner, using only the corner nodes of the element(Figure 13.16: “Triangle Aspect Ratio Calculation”):

1. A line is constructed from one node of the element to the midpoint of the opposite edge, and anotherthrough the midpoints of the other 2 edges. In general, these lines are not perpendicular to each otheror to any of the element edges.

2. Rectangles are constructed centered about each of these 2 lines, with edges passing through the elementedge midpoints and the triangle apex.

3. These constructions are repeated using each of the other 2 corners as the apex.

4. The aspect ratio of the triangle is the ratio of the longer side to the shorter side of whichever of the 6rectangles is most stretched, divided by the square root of 3.

The best possible triangle aspect ratio, for an equilateral triangle, is 1. A triangle having an aspect ratio of 20 isshown in Figure 13.17: “Aspect Ratios for Triangles”.

Figure 13.17 Aspect Ratios for Triangles

13.7.5. Aspect Ratio Calculation for Quadrilaterals

Figure 13.18 Quadrilateral Aspect Ratio Calculation

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& ! '%

( !" )

The aspect ratio for a quadrilateral is computed by the following steps, using only the corner nodes of the element(Figure 13.18: “Quadrilateral Aspect Ratio Calculation”):

1. If the element is not flat, the nodes are projected onto a plane passing through the average of the cornerlocations and perpendicular to the average of the corner normals. The remaining steps are performedon these projected locations.

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Section 13.7: Element Shape Testing

2. Two lines are constructed that bisect the opposing pairs of element edges and which meet at the elementcenter. In general, these lines are not perpendicular to each other or to any of the element edges.

3. Rectangles are constructed centered about each of the 2 lines, with edges passing through the elementedge midpoints. The aspect ratio of the quadrilateral is the ratio of a longer side to a shorter side ofwhichever rectangle is most stretched.

4. The best possible quadrilateral aspect ratio, for a square, is one. A quadrilateral having an aspect ratioof 20 is shown in Figure 13.19: “Aspect Ratios for Quadrilaterals”.

Figure 13.19 Aspect Ratios for Quadrilaterals

Table 13.7 Aspect Ratio Limits

Why default is this looseWhy default is this tightDefaultType of LimitCommand to modify

Disturbance of analysisresults has not beenproven

It is difficult to avoidwarnings even with a lim-it of 20.

Elements this stretchedlook to many users likethey deserve warnings.

20warningSHPP,MODIFY,1

Threshold of round offproblems depends onwhat computer is beingused.

Valid analyses should notbe blocked.

Informal testing hasdemonstrated solutionerror attributable to com-puter round off at aspectratios of 1,000 to 100,000.

106errorSHPP,MODIFY,2

13.7.6. Angle Deviation

Angle deviation from 90° corner angle is computed and tested only for the SHELL28 shear/twist panel quadrilat-eral (see Table 13.8: “Angle Deviation Limits”). It is an important measure because the element derivation assumesa rectangle.

13.7.7. Angle Deviation Calculation

The angle deviation is based on the angle between each pair of adjacent edges, computed using corner nodepositions in 3-D space. It is simply the largest deviation from 90° of any of the 4 corner angles of the element.

The best possible deviation is 0° (Figure 13.20: “Angle Deviations for SHELL28”). Figure 13.20: “Angle Deviationsfor SHELL28” also shows angle deviations of 5° and 30°, respectively.

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Figure 13.20 Angle Deviations for SHELL28

Table 13.8 Angle Deviation Limits

Why default is this looseWhy default is this tightDefaultType of LimitCommand to Modify

It is difficult to avoidwarnings even with a lim-it of 5°

Results degrade as theelement deviates from arectangular shape.

5°warningSHPP,MODIFY,7

Valid analyses should notbe blocked.

Pushing the limit furtherdoes not seem prudent.

30°errorSHPP,MODIFY,8

13.7.8. Parallel Deviation

Parallel deviation is computed and tested for all quadrilaterals or 3-D solid elements having quadrilateral facesor cross-sections, except Emag or FLOTRAN elements (see Table 13.9: “Parallel Deviation Limits”). Formal testinghas demonstrated degradation of stress convergence in linear displacement quadrilaterals as opposite edgesbecome less parallel to each other.

13.7.9. Parallel Deviation Calculation

Parallel deviation is computed using the following steps:

1. Ignoring midside nodes, unit vectors are constructed in 3-D space along each element edge, adjustedfor consistent direction, as demonstrated in Figure 13.21: “Parallel Deviation Unit Vectors”.

Figure 13.21 Parallel Deviation Unit Vectors

2. For each pair of opposite edges, the dot product of the unit vectors is computed, then the angle (in de-grees) whose cosine is that dot product. The parallel deviation is the larger of these 2 angles. (In the illus-tration above, the dot product of the 2 horizontal unit vectors is 1, and acos (1) = 0°. The dot product ofthe 2 vertical vectors is 0.342, and acos (0.342) = 70°. Therefore, this element’s parallel deviation is 70°.)

3. The best possible deviation, for a flat rectangle, is 0°. Figure 13.22: “Parallel Deviations for Quadrilaterals”shows quadrilaterals having deviations of 0°, 70°, 100°, 150°, and 170°.

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Section 13.7: Element Shape Testing

Figure 13.22 Parallel Deviations for Quadrilaterals

Table 13.9 Parallel Deviation Limits

Why default is this looseWhy default is this tightDefaultType of LimitCommand to Modify

It is difficult to avoidwarnings even with a lim-it of 70°

Testing has shown resultsare degraded by thismuch distortion

70°warning for ele-ments withoutmidside nodes

SHPP,MODIFY,11

Valid analyses should notbe blocked.

Pushing the limit furtherdoes not seem prudent

150°error for ele-ments withoutmidside nodes

SHPP,MODIFY,12

Disturbance of analysisresults for quadratic ele-ments has not beenproven.

Elements having devi-ations > 100° look likethey deserve warnings.

100°warning for ele-ments with mid-side nodes

SHPP,MODIFY,13

Valid analyses should notbe blocked.

Pushing the limit furtherdoes not seem prudent

170°error for ele-ments with mid-side nodes

SHPP,MODIFY,14

13.7.10. Maximum Corner Angle

Maximum corner angle is computed and tested for all except Emag or FLOTRAN elements (seeTable 13.10: “Maximum Corner Angle Limits”). Some in the finite element community have reported that largeangles (approaching 180°) degrade element performance, while small angles don’t.

13.7.11. Maximum Corner Angle Calculation

The maximum angle between adjacent edges is computed using corner node positions in 3-D space. (Midsidenodes, if any, are ignored.) The best possible triangle maximum angle, for an equilateral triangle, is 60°. Fig-ure 13.23: “Maximum Corner Angles for Triangles” shows a triangle having a maximum corner angle of 165°. Thebest possible quadrilateral maximum angle, for a flat rectangle, is 90°. Figure 13.24: “Maximum Corner Anglesfor Quadrilaterals” shows quadrilaterals having maximum corner angles of 90°, 140° and 180°.

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Figure 13.23 Maximum Corner Angles for Triangles

Figure 13.24 Maximum Corner Angles for Quadrilaterals

Table 13.10 Maximum Corner Angle Limits

Why default is this looseWhy default is this tightDefaultType of LimitCommand to Modify

Disturbance of analysisresults has not beenproven.

It is difficult to avoidwarnings even with a lim-it of 165°.

Any element this distor-ted looks like it deservesa warning.

165°warnings for tri-angles

SHPP,MODIFY,15

Valid analyses should notbe blocked.

We can not allow 180°179.9°error for tri-angles

SHPP,MODIFY,16

Disturbance of analysisresults has not beenproven.

It is difficult to avoidwarnings even with a lim-it of 155°.

Any element this distor-ted looks like it deservesa warning.

155°warning forquadrilateralswithout midsidenodes

SHPP,MODIFY,17

Valid analyses should notbe blocked.

We can not allow 180°179.9°error for quadri-laterals withoutmidside nodes

SHPP,MODIFY,18

Disturbance of analysisresults has not beenproven.

It is difficult to avoidwarnings even with a lim-it of 165°.

Any element this distor-ted looks like it deservesa warning.

165°warning forquadrilateralswith midsidenodes

SHPP,MODIFY,19

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Why default is this looseWhy default is this tightDefaultType of LimitCommand to Modify

Valid analyses should notbe blocked.

We can not allow 180°179.9°error for quadri-laterals withmidside nodes

SHPP,MODIFY,20

13.7.12. Jacobian Ratio

Jacobian ratio is computed and tested for all elements except triangles and tetrahedra that (a) are linear (haveno midside nodes) or (b) have perfectly centered midside nodes (see Table 13.11: “Jacobian Ratio Limits”). A highratio indicates that the mapping between element space and real space is becoming computationally unreliable.

13.7.12.1. Jacobian Ratio Calculation

An element's Jacobian ratio is computed by the following steps, using the full set of nodes for the element:

1. At each sampling location listed in the table below, the determinant of the Jacobian matrix is computedand called RJ. RJ at a given point represents the magnitude of the mapping function between element

natural coordinates and real space. In an ideally-shaped element, RJ is relatively constant over the element,

and does not change sign.

RJ Sampling LocationsElement Shape

corner nodes10-node tetrahedra - SHPP,LSTET,OFF

integration points10-node tetrahedra - SHPP,LSTET,ON

base corner nodes and near apex node (apex RJ factored so

that a pyramid having all edges the same length will producea Jacobian ratio of 1)

5-node or 13-node pyramids

corner nodes and centroid8-node quadrilaterals

all nodes and centroid20-node bricks

corner nodesall other elements

2. The Jacobian ratio of the element is the ratio of the maximum to the minimum sampled value of RJ. If

the maximum and minimum have opposite signs, the Jacobian ratio is arbitrarily assigned to be -100(and the element is clearly unacceptable).

3. If the element is a midside-node tetrahedron, an additional RJ is computed for a fictitious straight-sided

tetrahedron connected to the 4 corner nodes. If that RJ differs in sign from any nodal RJ (an extremely

rare occurrence), the Jacobian ratio is arbitrarily assigned to be -100.

4. The sampling locations for midside-node tetrahedra depend upon the setting of the linear stress tetrahedrakey on the SHPP command. The default behavior (SHPP,LSTET,OFF) is to sample at the corner nodes,while the optional behavior (SHPP,LSTET.ON) is to sample at the integration points (similar to what wasdone for the DesignSpace product). Sampling at the integration points will result in a lower Jacobianratio than sampling at the nodes, but that ratio is compared to more restrictive default limits (seeTable 13.11: “Jacobian Ratio Limits” below). Nevertheless, some elements which pass the LSTET,ON testfail the LSTET,OFF test - especially those having zero RJ at a corner node. Testing has shown that such

elements have no negative effect on linear elastic stress accuracy. Their effect on other types of solutionshas not been studied, which is why the more conservative test is recommended for general ANSYS usage.Brick elements (i.e. SOLID95 and SOLID186) degenerated into tetrahedra are tested in the same manneras are 'native' tetrahedra (SOLID92 and SOLID187). In most cases, this produces conservative results.However, for SOLID185 and SOLID186 when using the non-recommended tetrahedron shape, it is possible

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that such a degenerate element may produce an error during solution, even though it produced nowarnings during shape testing.

5. If the element is a line element having a midside node, the Jacobian matrix is not square (because themapping is from one natural coordinate to 2-D or 3-D space) and has no determinant. For this case, avector calculation is used to compute a number which behaves like a Jacobian ratio. This calculation hasthe effect of limiting the arc spanned by a single element to about 106°

A triangle or tetrahedron has a Jacobian ratio of 1 if each midside node, if any, is positioned at the average ofthe corresponding corner node locations. This is true no matter how otherwise distorted the element may be.Hence, this calculation is skipped entirely for such elements. Moving a midside node away from the edge midpointposition will increase the Jacobian ratio. Eventually, even very slight further movement will break the element(Figure 13.25: “Jacobian Ratios for Triangles”). We describe this as “breaking” the element because it suddenlychanges from acceptable to unacceptable- “broken”.

Figure 13.25 Jacobian Ratios for Triangles

Any rectangle or rectangular parallelepiped having no midside nodes, or having midside nodes at the midpointsof its edges, has a Jacobian ratio of 1. Moving midside nodes toward or away from each other can increase theJacobian ratio. Eventually, even very slight further movement will break the element (Figure 13.26: “JacobianRatios for Quadrilaterals”).

Figure 13.26 Jacobian Ratios for Quadrilaterals

A quadrilateral or brick has a Jacobian ratio of 1 if (a) its opposing faces are all parallel to each other, and (b) eachmidside node, if any, is positioned at the average of the corresponding corner node locations. As a corner nodemoves near the center, the Jacobian ratio climbs. Eventually, any further movement will break the element(Figure 13.27: “Jacobian Ratios for Quadrilaterals”).

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Section 13.7: Element Shape Testing

Figure 13.27 Jacobian Ratios for Quadrilaterals

Table 13.11 Jacobian Ratio Limits

Why default is this looseWhy default is this tightDefaultType of limitCommand to modify

Disturbance of analysisresults has not beenproven. It is difficult toavoid warnings even witha limit of 30.

A ratio this high indicatesthat the mappingbetween element andreal space is becomingcomputationally unreli-able.

30 if SHPP,LSTET,OFF

warning for h-elements

SHPP,MODIFY,31

10 if SHPP,LSTET,ON

Valid analyses should notbe blocked.

Pushing the limit furtherdoes not seem prudent.

1,000 if SHPP,LSTET,OFF

SHPP,MODIFY,32

40 if SHPP,LSTET,ON

A ratio this high indicatesthat the mappingbetween element andreal space is becomingcomputationally unreli-able.

30warning for p-elements

SHPP,MODIFY,33

Valid analyses should notbe blocked.

The mapping is morecritical for p- than h- ele-ments

40warning for p-elements

SHPP,MODIFY,34

13.7.13. Warping Factor

Warping factor is computed and tested for some quadrilateral shell elements, and the quadrilateral faces ofbricks, wedges, and pyramids (see Table 13.12: “Applicability of Warping Tests” and Table 13.13: “Warping FactorLimits”). A high factor may indicate a condition the underlying element formulation cannot handle well, or maysimply hint at a mesh generation flaw.

13.7.13.1. Warping Factor Calculation for Quadrilateral Shell Elements

A quadrilateral element's warping factor is computed from its corner node positions and other available databy the following steps:

1. An average element normal is computed as the vector (cross) product of the 2 diagonals (Fig-ure 13.28: “Shell Average Normal Calculation”).

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Figure 13.28 Shell Average Normal Calculation

2. The projected area of the element is computed on a plane through the average normal (the dottedoutline on Figure 13.29: “Shell Element Projected onto a Plane”).

3. The difference in height of the ends of an element edge is computed, parallel to the average normal. InFigure 13.29: “Shell Element Projected onto a Plane”, this distance is 2h. Because of the way the averagenormal is constructed, h is the same at all four corners. For a flat quadrilateral, the distance is zero.

Figure 13.29 Shell Element Projected onto a Plane

4.The “area warping factor” ( Fa

w) for the element is computed as the edge height difference divided by

the square root of the projected area.

5. For all shells except those in the “membrane stiffness only” group, if the thickness is available, the“thickness warping factor” is computed as the edge height difference divided by the average elementthickness. This could be substantially higher than the area warping factor computed in 4 (above).

6. The warping factor tested against warning and error limits (and reported in warning and error messages)is the larger of the area factor and, if available, the thickness factor.

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Section 13.7: Element Shape Testing

7. The best possible quadrilateral warping factor, for a flat quadrilateral, is zero.

8. The warning and error limits for SHELL63 quadrilaterals in a large deflection analysis are much tighterthan if these same elements are used with small deflection theory, so existing SHELL63 elements areretested any time the nonlinear geometry key is changed. However, in a large deflection analysis it ispossible for warping to develop after deformation, causing impairment of nonlinear convergence and/ordegradation of results. Element shapes are not retested during an analysis.

Figure 13.30: “Quadrilateral Shell Having Warping Factor” shows a “warped” element plotted on top of a flat one.Only the right-hand node of the upper element is moved. The element is a unit square, with a real constantthickness of 0.1.

When the upper element is warped by a factor of 0.01, it cannot be visibly distinguished from the underlyingflat one.

When the upper element is warped by a factor of 0.04, it just begins to visibly separate from the flat one.

Figure 13.30 Quadrilateral Shell Having Warping Factor

Warping of 0.1 is visible given the flat reference, but seems trivial. However, it is well beyond the error limit fora membrane shell or a SHELL63 in a large deflection environment. Warping of 1.0 is visually unappealing. Thisis the error limit for most shells.

Warping beyond 1.0 would appear to be obviously unacceptable. However, SHELL43 and SHELL181 permit eventhis much distortion. Furthermore, the warping factor calculation seems to peak at about 7.0. Moving the nodefurther off the original plane, even by much larger distances than shown here, does not further increase thewarping factor for this geometry. Users are cautioned that manually increasing the error limit beyond its defaultof 5.0 for these elements could mean no real limit on element distortion.

13.7.13.2. Warping Factor Calculation for 3-D Solid Elements

The warping factor for a 3-D solid element face is computed as though the 4 nodes make up a quadrilateral shellelement with no real constant thickness available, using the square root of the projected area of the face as de-scribed in 4 (above).

The warping factor for the element is the largest of the warping factors computed for the 6 quadrilateral facesof a brick, 3 quadrilateral faces of a wedge, or 1 quadrilateral face of a pyramid.

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Any brick element having all flat faces has a warping factor of zero (Figure 13.31: “Warping Factor for Bricks”).

Figure 13.31 Warping Factor for Bricks

Twisting the top face of a unit cube by 22.5° and 45° relative to the base produces warping factors of about 0.2and 0.4, respectively.

Table 13.12 Applicability of Warping Tests

ANSYS internal key ielc(JSHELL)Limits Group from Warping Factor LimitsElement Name

7“shear / twist”SHELL28

4“membrane stiffness only”SHELL41

2“bending with high warping limit”SHELL43

11“non-stress”INFIN47

11“non-stress”SHELL57

3“bending stiffness included” if KEYOPT(1) = 0 or 2SHELL63

4“membrane stiffness only” if KEYOPT(1) = 1

1none ... element can curve out of planeSHELL91

1none ... element can curve out of planeSHELL93

1none ... element can curve out of planeSHELL99

11“non-stress”INTER115

11“non-stress”SHELL131

11“non-stress”SHELL132

2“bending with high warping limit”SHELL143

1none ... element can curve out of planeSHELL150

11“non-stress”SHELL157

2“bending with high warping limit”SHELL163

2“bending with high warping limit ” if KEYOPT(1) = 0SHELL181

4“membrane stiffness only” if KEYOPT(1) = 1

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Section 13.7: Element Shape Testing

Table 13.13 Warping Factor Limits

Why default is this looseWhy default is this tightDefaultType of limitCommand to modify

Element formulation de-rived from 8-node solidisn't disturbed by warp-ing.

Disturbance of analysisresults has not beenproven

Elements having warpingfactors > 1 look like theydeserve warnings

1warning for“bending withhigh warpinglimit” shells ielc(JSHELL)=2

SHPP,MODIFY,51

Valid analyses should notbe blocked.

Pushing this limit furtherdoes not seem prudent

5same as above,error limit

SHPP,MODIFY,52

It is difficult to avoid thesewarnings even with a lim-it of 0.1.

The element formulationis based on flat shell the-ory, with rigid beam off-sets for moment compat-ibility.

Informal testing hasshown that result errorbecame significant forwarping factor > 0.1.

0.1warning for“non-stress”shells or “bend-ing stiffness in-cluded” shellswithout geomet-ric nonlinearities3, 11

SHPP,MODIFY,53

Valid analyses should notbe blocked.

Pushing this limit furtherdoes not seem prudent.

1same as above,error limit

SHPP,MODIFY,54

Informal testing hasshown that the effect ofwarping < 0.02 is negli-gible.

The element formulationis based on flat shell the-ory, without any correc-tion for moment compat-ibility. The element can-not handle forces not inthe plane of the element.

0.02warning for“membrane stiff-ness only” shells4

SHPP,MODIFY,55

Valid analyses should notbe blocked.

Pushing this limit furtherdoes not seem prudent

0.2same as above,error limit

SHPP,MODIFY,56

It is difficult to avoid thesewarnings even with a lim-it of 0.1.

The element formulationis based on flat shell the-ory, with rigid beam off-sets for moment compat-ibility.

Informal testing hasshown that result errorbecame significant forwarping factor > 0.1.

0.1warning for“shear / twist”shells 7

SHPP,MODIFY,57

Valid analyses should notbe blocked.

Pushing this limit furtherdoes not seem prudent

1same as above,error limit

SHPP,MODIFY,58

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Why default is this looseWhy default is this tightDefaultType of limitCommand to modify

The element formulationis based on flat shell the-ory. The rigid beam off-sets added to warpedelements for momentcompatibility do not workwell with geometric non-linearities.

Informal testing hasshown that nonlinearconvergence was im-paired and/or result errorbecame significant forwarping factors >0.00001.

0.00001warning for“bending stiff-ness included”shells with geo-metric nonlinear-ities 3

SHPP,MODIFY,59

Valid analyses should notbe blocked.

Pushing this limit furtherdoes not seem prudent

0.01same as above,error limit

SHPP,MODIFY,60

Disturbance of analysisresults has not beenproven.

A warping factor of 0.2corresponds to about a22.5° rotation of the topface of a unit cube. Brickelements distorted thismuch look like they de-serve warnings.

0.2warning for 3-Dsolid elementquadrilateralface

SHPP,MODIFY,67

Valid analyses should notbe blocked.

A warping factor of 0.4corresponds to about a45° rotation of the topface of a unit cube. Push-ing this limit further doesnot seem prudent.

0.4same as above,error limit

SHPP,MODIFY,68

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13–30

Chapter 14: Element Library

Introduction

This chapter describes each element. The explanations are augmented by other sections referred to in thismanual as well as the external references.

The table below the introductory figure of each element is intended to be complete, except that the Newton-Raphson load vector is omitted. This load vector always uses the same shape functions and integration pointsas the applicable stiffness, conductivity and/or coefficient matrix. Exceptions associated mostly with some non-linear line elements are noted with the element description.

14.1. LINK1 - 2-D Spar (or Truss)

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–1Stiffness Matrix and Thermal Load Vector

NoneEquation 12–1 and Equation 12–2Mass Matrix

NoneEquation 12–2Stress Stiffness Matrix

DistributionLoad Type

Linear along lengthElement Temperature

Linear along lengthNodal Temperature

14.1.1. Assumptions and Restrictions

The element is not capable of carrying bending loads. The stress is assumed to be uniform over the entire element.

14.1.2. Other Applicable Sections

LINK8, the 3-D Spar, has analogous element matrices and load vectors described, as well as the stress printout.

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

14.2. PLANE2 - 2-D 6-Node Triangular Structural Solid

Integration PointsShape FunctionsMatrix or Vector

3Equation 12–96 and Equation 12–97Stiffness, Mass, and Stress StiffnessMatrices; and Thermal Load Vector

2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functionsElement Temperature

Same as shape functionsNodal Temperature

Linear along each facePressure

Reference: Zienkiewicz(39)

14.2.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.3. BEAM3 - 2-D Elastic Beam

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–2

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–4 and Equation 12–5Stiffness and Mass Matrices; andThermal and Pressure Load Vector

NoneEquation 12–5Stress Stiffness Matrix

DistributionLoad Type

Linear thru thickness and along lengthElement Temperature

Constant thru thickness, linear along lengthNodal Temperature

Linear along lengthPressure

14.3.1. Element Matrices and Load Vectors

The element stiffness matrix in element coordinates is (Przemieniecki(28)):

(14–1)[ ]

( ) ( ) ( ) (

K

AEL

AEL

EI

L

EI

L

EI

L

EI

L

l =

−

+ +−

+

0 0 0 0

012

1

6

10

12

1

6

13 2 3 2φ φ φ ++

++

+−

+−

+

−

φ

φφ

φ φφ

φ

)

( )

( )( ) ( )

( )( )

06

1

41

06

1

21

0

2 2EI

L

EIL

EI

L

EIL

AEL

00 0 0

012

1

6

10

12

1

6

1

06

3 2 3 2

AEL

EI

L

EI

L

EI

L

EI

L

EI

−+

−+ +

−+( ) ( ) ( ) ( )φ φ φ φ

LL

EIL

EI

L

EIL2 21

21

06

1

41( )

( )( ) ( )

( )( )+

−+

−+

++

φφ

φ φφ

φ

where:

A = cross-section area (input as AREA on R command)E = Young's modulus (input as EX on MP command)L = element lengthI = moment of inertia (input as IZZ on R command)

φ = 122

EI

GA Ls

G = shear modulus (input as GXY on MP command)

AA

Fs

s= = shear area

Fs = shear deflection constant (input as SHEARZ on R command)

The consistent element mass matrix (LUMPM,OFF) in element coordinates is (Yokoyama(167)):

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Section 14.3: BEAM3 - 2-D Elastic Beam

(14–2)[ ] ( ) ( )

( , ) ( , ) ( , ) ( , )

(M A m L

A r C r B r D r

C rinl = + −

−

ρ ε

φ φ φ φ

1

1 3 0 0 1 6 0 0

0 0

0 ,, ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( ,

φ φ φ φ

φ φ φ φ

E r D r F r

B r D r A r C r

0

1 6 0 0 1 3 0 0

0 0

−

− ))

( , ) ( , ) ( , ) ( , )0 0− − −

D r F r C r E rφ φ φ φ

where:

ρ = density (input as DENS on MP command)m = added mass per unit length (input as ADDMAS on R command)

εin = prestrain (input as ISTRN on R command)

A rr L

( , )( )

( )φ

φ φ

φ=

+ + +

+

1335

710

13

65

1

2 2

2

B rr L

( , )( )

( )φ

φ φ

φ=

+ +

+

970

310

16

65

1

2 2

2

C r

r L L

( , )

( )

( )φ

φ φ φ

φ=

+ + + −

+

11210

11120

124

110

12

1

2 2

2

D r

r L L

( , )

( )

( )φ

φ φ φ

φ=

+ + −

+

13420

340

124

110

12

1

2 2

2

E r

r L L

( , )

( )

(φ

φ φ φ φ=

+ + + + +

+

1105

160

1120

215

16

13

1

2 2 2 2

φφ)2

F r

r L L

( , )

( )

(φ

φ φ φ φ=

+ + + + −

+

1140

160

1120

130

16

16

1

2 2 2 2

φφ)2

rIA

= = radius of gyration

The lumped element mass matrix (LUMPM,ON) in element coordinates is:

(14–3)[ ]

( ) ( )M

A m L in

l = + −

ρ ε12

1 0 0 0 0 0

0 1 0 0 0 0

0 0 0 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–4

The element pressure load vector in element coordinates is:

(14–4) Fpr Tl = P P P P P P1 2 3 4 5 6

For uniform lateral pressure,

(14–5)P P1 4 0= =

(14–6)P P

PL2 5 2

= = −

(14–7)P PPL

3 6

2

12= − = −

where:

P = uniform applied pressure (units = force/length) (input on SFE command)

Other standard formulas (Roark(48)) for P1 through P6 are used for linearly varying loads, partially loaded elements,

and point loads.

14.3.2. Stress Calculation

The centroidal stress at end i is:

(14–8)σi

dir x iF

A= ,

where:

σidir = centroidal stress (output as SDIR)

Fx,i = axial force (output as FORCE)

The bending stress is

(14–9)σi

bnd iM tI

=2

where:

σibnd = bending stress at end i (output as SBEND)

Mi = moment at end i

t = thickness of beam in element y direction (input as HEIGHT on R command)

The presumption has been made that the cross-section is symmetric.

14–5ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.3: BEAM3 - 2-D Elastic Beam

14.4. BEAM4 - 3-D Elastic Beam

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–6, Equation 12–7, Equation 12–8, andEquation 12–9

Stiffness and Mass Matrices

NoneEquation 12–7 and Equation 12–8Stress Stiffness and DampingMatrices

NoneEquation 12–6, Equation 12–7, and Equation 12–8Pressure Load Vector and Temper-atures

DistributionLoad Type

Bilinear across cross-section, linear along lengthElement Temperature

Constant across cross-section, linear along lengthNodal Temperature

Linear along lengthPressure

14.4.1. Stiffness and Mass Matrices

The order of degrees of freedom (DOFs) is shown in Figure 14.1: “Order of Degrees of Freedom”.

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–6

Figure 14.1 Order of Degrees of Freedom

The stiffness matrix in element coordinates is (Przemieniecki(28)):

(14–10)[ ]K

AE L

a

a

GJ L

c e

c e

AE L

a

z

y

y y

z zl =

−

−−

0

0 0

0 0 0

0 0 0

0 0 0 0

0 0 0 0 0

0

Symmetric

zz z

y y

y y

z z

z

y

c

a c

GJ L

c f

c f

AE L

a

a

GJ

0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0

0

0 0

0 0 0

−−

−−

LL

c e

c e

y y

z z

0 0 0

0 0 0 0−

where:

A = cross-section area (input as AREA on R command)E = Young's modulus (input as EX on MP command)L = element lengthG = shear modulus (input as GXY on MP command)

JJ I

I Ix x

x x= =

=≠

torsional moment of inertiaif

if

0

0

Ix = input torsional moment of inertia (input as IXX on RMORE command)

Jx = polar moment of inertia = Iy + Iz

az = a(Iz,φy)

14–7ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.4: BEAM4 - 3-D Elastic Beam

ay = a(Iy,φz)

bz = b(Iz,φy)

Mfz = f(Iz,φy)

fy = f(Iy,φz)

a IEI

L( , )

( )φ

φ=

+12

13

c IEI

L( , )

( )φ

φ=

+6

12

e IEI

L( , )

( )( )

φ φφ

= ++

41

f IEI

L( , )

( )( )

φ φφ

= −+

21

φyz

zsEI

GA L= 12

2

φzy

ys

EI

GA L=

122

Ii = moment of inertia normal to direction i (input as Iii on R command)

A A Fis

is= =shear area normal to direction i /

F is = shear coefficient (input as SHEARi on command)RMORE

The consistent mass matrix (LUMPM,OFF) in element coordinates LUMPM,OFF is (Yokoyama(167)):

(14–11)[ ]M M

A

A

J A

C E

C E

B

t

z

y

x

y y

z z

z

l =

−

1 3

0

0 0

0 0 0 3

0 0 0

0 0 0 0

1 6 0 0 0 0 0

0

Symmetric

00 0 0

0 0 0 0

0 0 0 6 0 0

0 0 0 0

0 0 0 0

1 3

0

0 0

0 0 0 3

0

D

B D

J A

D F

D F

A

A

J A

z

y y

x

y y

z z

z

y

x

−

−

00 0

0 0 0 0

−

C E

C E

y y

z z

where:

Mt = (ρA+m)L(1-εin)

ρ = density (input as DENS on MP command)m = added mass per unit length (input as ADDMAS on RMORE command)

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–8

εin = prestrain (input as ISTRN on RMORE command)Az = A(rz,φy)

Ay = A(ry,φz)

Bz = B(rz,φy)

MFz = F(rz,φy)

Fy = F(ry,φz)

A rr L

( , )( )

( )φ

φ φ

φ=

+ + +

+

1335

710

13

65

1

2 2

2

B rr L

( , )( )

( )φ

φ φ

φ=

+ +

+

970

310

16

65

1

2 2

2

C r

r L L

( , )

( )

( )φ

φ φ φ

φ=

+ + + −

+

11210

11120

124

110

12

1

2 2

2

D r

r L L

( , )

( )

( )φ

φ φ φ

φ=

+ + −

+

13420

340

124

110

12

1

2 2

2

E r

r L L

( , )

( )

(φ

φ φ φ φ=

+ + + + +

+

1105

160

1120

215

16

13

1

2 2 2 2

φφ)2

F r

r L L

( , )

( )

(φ

φ φ φ φ=

+ + + + −

+

1140

160

1120

130

16

16

1

2 2 2 2

φφ)2

rI

Ayyy= = radius of gyration

rIAzzz= = radius of gyration

The mass matrix (LUMPM,ON) in element coordinates is:

14–9ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.4: BEAM4 - 3-D Elastic Beam

(14–12)[ ]M

Mtl =

2

1

0 1

0 0 1

0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0

Symmetric

00 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1

0 1

0 0 1

0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

14.4.2. Gyroscopic Damping Matrix

The element gyroscopic damping matrix is the same as for PIPE16.

14.4.3. Pressure and Temperature Load Vector

The pressure and temperature load vector are computed in a manner similar to that of BEAM3.

14.4.4. Local to Global Conversion

The element coordinates are related to the global coordinates by:

(14–13) [ ] u T uRl =

where:

ul = vector of displacements in element Cartesian coordinattes

u = vector of displacements in global Cartesian coordinates

[ ]T

T

T

T

T

R =

0 0 0

0 0 0

0 0 0

0 0 0

[T] is defined by:

(14–14)[ ] ( ) ( )

(

T

C C S C S

C S S S C S S S C C S C

C S C S S

= − − − +− −

1 2 1 2 2

1 2 3 1 3 1 2 3 1 3 3 2

1 2 3 1 33 1 2 3 1 3 3 2) ( )− −

S S C C S C C

where:

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–10

S

Y YL

L d

L d

xyxy

xy

1

2 1

=

− >

0.0 <

if

if

SZ Z

L22 1= −

S3 = sin (θ)

C

X XL

L d

L d

xyxy

xy

1

2 1

1 0

=

− >

<

if

if

.

CL

Lxy

2 =

C3 = cos (θ)

X1, etc. = x coordinate of node 1, etc.

Lxy = projection of length onto X-Y plane

d = .0001 Lθ = user-selected adjustment angle (input as THETA on R command)

If a third node is given, θ is not used. Rather C3 and S3 are defined using:

V1 = vector from origin to node 1

V2 = vector from origin to node 2

V3 = vector from origin to node 3

V4 = unit vector parallel to global Z axis, unless element is almost parallel to Z axis, in which case it is parallel

to the X axis.

Then,

(14–15) V V V5 3 1= − = vector between nodes I and K

(14–16) V V V6 2 1= − = vector along element X axis

(14–17) V V V7 6 4= ×

(14–18) V V V8 6 5= ×

and

(14–19)C

V VV V3

7 8

7 8=

⋅

(14–20)S

V V VV V V3

6 9 8

6 9 8= ⋅ × ( )

The x and • refer to vector cross and dot products, respectively. Thus, the element stiffness matrix in global co-ordinates becomes:

14–11ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.4: BEAM4 - 3-D Elastic Beam

(14–21)[ ] [ ] [ ][ ]K T K Te RT

R= l

(14–22)[ ] [ ] [ ][ ]M T M Te RT

R= l

(14–23)[ ] [ ] [ ][ ]S T S Te RT

R= l

(14–24) [ ] F T Fe RT= l

( [ ]Sl is defined in Section 3.1: Large Strain).

14.4.5. Stress Calculations

The centroidal stress at end i is:

(14–25)σi

dir x iF

A= ,

where:

σidir = centroidal stress (output as SDIR)

Fx,i = axial force (output as FX)

The bending stresses are

(14–26)σz i

bnd y i z

y

M t

I,,=

2

(14–27)σy i

bnd z i y

z

M t

I,,=

2

where:

σz ibnd, = bending stress in element x direction on the elemennt

+ z side of the beam at end i (output as SBZ)

σy ibnd

, = bending stess on the element in element x directionn- y side of the beam at end i (output as SBY)

My,i = moment about the element y axis at end i

Mz,i = moment about the element z axis at end i

tz = thickness of beam in element z direction (input as TKZ on R command)

ty = thickness of beam in element y direction (input as TKY on R command)

The maximum and minimum stresses are:

(14–28)σ σ σ σi i

dirz ibnd

y ibndmax

, ,= + +

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–12

(14–29)σ σ σ σi i

dirz ibnd

y ibndmin

, ,= − −

The presumption has been made that the cross-section is a rectangle, so that the maximum and minimumstresses of the cross-section occur at the corners. If the cross-section is of some other form, such as an ellipse,the user must replace Equation 14–28 and Equation 14–29 with other more appropriate expressions.

For long members, subjected to distributed loading (such as acceleration or pressure), it is possible that the peakstresses occur not at one end or the other, but somewhere in between. If this is of concern, the user should eitheruse more elements or compute the interior stresses outside of the program.

14.5. SOLID5 - 3-D Coupled-Field Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–205Magnetic Potential CoefficientMatrix

2 x 2 x 2Equation 12–204Electrical Conductivity Matrix

2 x 2 x 2Equation 12–203Thermal Conductivity Matrix

2 x 2 x 2Equation 12–191, Equation 12–192, and Equa-tion 12–193 or, if modified extra shapes are included(KEYOPT(3) = 0), Equation 12–206, Equation 12–207,and Equation 12–208

Stiffness Matrix and Thermal Expan-sion Load Vector

2 x 2 x 2Same as combination of stiffness matrix and conduct-ivity matrix.

Piezoelectric Coupling Matrix

2 x 2 x 2Same as conductivity matrix. Matrix is diagonalizedas described in Section 12.2: 3-D Lines

Specific Heat Matrix

2 x 2 x 2Equation 12–191, Equation 12–192, and Equa-tion 12–193

Mass and Stress Stiffening Matrices

2 x 2 x 2Same as coefficient or conductivity matrixLoad Vector due to ImposedThermal and Electric Gradients,Heat Generation, Joule Heating,Magnetic Forces, Magnetism dueto Source Currents and PermanentMagnets

2 x 2 x 2Same as stiffness or conductivity matrix specializedto the surface.

Load Vector due to ConvectionSurfaces and Pressures

14–13ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.5: SOLID5 - 3-D Coupled-Field Solid

References: Wilson(38), Taylor(49), Coulomb(76), Mayergoyz(119), Gyimesi(141,149)

14.5.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors aswell as heat flux evaluations. Section 5.2: Derivation of Electromagnetic Matrices discusses the scalar potentialmethod, which is used by this element. Section 11.2: Piezoelectrics discusses the piezoelectric capability usedby the element.

14.6. Not Documented

No detail or element available at this time.

14.7. COMBIN7 - Revolute Joint

!

"

$# %

&

'(

)*

+*

,*

Integration PointsShape FunctionsMatrix or Vector

NoneNoneStiffness and Damping Matrices; and Load Vector

NoneNone (lumped mass formulation)Mass Matrix

14.7.1. Element Description

COMBIN7 is a 5-node, 3-D structural element that is intended to represent a pin (or revolute) joint. The pin elementconnects two links of a kinematic assemblage. Nodes I and J are active and physically represent the pin joint.Node K defines the initial (first iteration) orientation of the moving joint coordinate system (x, y, z), while nodesL and M are control nodes that introduce a certain level of feedback to the behavior of the element.

In kinematic terms, a pin joint has only one primary DOF, which is a rotation (θz) about the pin axis (z). The joint

element has six DOFs per node (I and J) : three translations (u, v, w) and three rotations (θx, θy, θz) referenced to

element coordinates (x, y, z). Two of the DOFs (θz for nodes I and J) represent the pin rotation. The remaining 10

DOFs have a relatively high stiffness (see below). Among other options available are rotational limits, feedbackcontrol, friction, and viscous damping.

Chapter 14: Element Library

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Flexible behavior for the constrained DOF is defined by the following input quantities:

K1 = spring stiffness for translation in the element x-y plane (input as K1 on R command)

K2 = spring stiffness for translation in the element z direction (input as K2 on R command)

K3 = spring stiffness for rotation about the element x and y axes (input as K3 on R command)

Figure 14.2 Joint Element Dynamic Behavior About the Revolute Axis

The dynamics of the primary DOF (θz) of the pin is shown in Figure 14.2: “Joint Element Dynamic Behavior About

the Revolute Axis”. Input quantities are:

K4 = rotation spring stiffness about the pin axis when the element is “locked” (input as K4 on R command)

Tf = friction limit torque (input as TF on R command)

Ct = rotational viscous friction (input as CT on R command)

Ti = imposed element torque (input as TLOAD on RMORE command)

θ = reverse rotation limit (input as STOPL on RMORE command)

θ = forward rotation limit (input as STOPU on RMORE command)θi = imposed (or interference) rotation (input as ROT on RMORE command)

Im = joint mass (input as MASS on RMORE command)

A simple pin can be modeled by merely setting K4 = 0, along with Ki > 0 (i = 1 to 3). Alternately, when K4 > 0, a

simple pin is formed with zero friction (Tf = 0). The total differential rotation of the pin is given by:

(14–30)θ θ θt zJ zI= −

When friction is present (Tf = 0), this may be divided into two parts, namely:

(14–31)θ θ θt f K= +

where:

θf = the amount of rotation associated with friction

θK = the rotation associated with the spring (i.e., spring torque /K4)

One extreme condition occurs when Tf = 0, and it follows that θK = 0 and θt = θf. On the other hand, when a high

level of friction is specified to the extent that the spring torque never exceeds Tf, then it follows that θf = 0 and

14–15ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.7: COMBIN7 - Revolute Joint

θf = θK. When a negative friction torque is specified (Tf < 0), the pin axis is “locked” (or stuck) with revolute stiffness

K4. The pin also becomes locked when a stop is engaged, that is when:

(14–32)θ θf ≥ (forward stop engaged)

(14–33)θ θf ≤ − ( )reverse stop engaged

Stopping action is removed when θ = θ = 0.

Internal self-equilibrating element torques are imposed about the pin axis if either Ti or θi are specified. If Ti is

specified, the internal torques applied to the active nodes are:

(14–34)T T TJ I i= − =

If a local rotation θi is input, it is recommended that one should set Tf < 0, K4 > 0, and Ti = 0. Internal loads then

become

(14–35)T T KJ I i= − = 4θ

14.7.2. Element Matrices

For this element, nonlinear behavior arises when sliding friction is present, stops are specified, control featuresare active, or large rotations are represented.

As mentioned above, there are two active nodes and six DOFs per node. Thus, the size of the element mass,damping, and stiffness matrices in 12 x 12, with a 12 x 1 load vector.

The stiffness matrix is given by:

(14–36)K

K

K

K

K

K

K

K

K

K

p[ ]=

−−

−

1

1

2

3

3

1

1

2

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 −−−

−

K

K

K

K

K

K

K

K

K

p

p

Symmetry

3

3

1

1

2

3

3

0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–16

where:

KK

p =

≥ ≤ −

≠

4

0

,

;

if or and both

and (stop engaged)

or

T

f fθ θ θ θ

θ θ

ff

4

(locked)

or (not sliding)

if

K

0,

<<

< <

0

θ

θ θ θ

K f

f

T

- aand (sliding) K TK f4 0θ ≥ ≥

The mass matrix is lumped and given by:

(14–37)M

M

M

M

I

I

I

m

m

m[ ]= 12

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 00 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

Symmetry

M

M

M

I

I

I

m

m

m

where:

M = total mass (input as MASS on RMORE command)Im = total mass moment of inertia (input as IMASS on RMORE command)

The damping matrix, derived from rotational viscous damping about the pin axis is given as:

(14–38)C Ct[ ]=

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

1

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 00 0 0 0 1

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

1

−

Symmetry

The applied load vector for COMBIN7 is given by:

14–17ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.7: COMBIN7 - Revolute Joint

(14–39) ( ) ( )F T K T Ki i i iT= − + + 0 0 0 0 0 0 0 0 04 4θ θ

14.7.3. Modification of Real Constants

Four real constants (C1, C2, C3, C4) are used to modify other real constants for a dynamic analysis (ANTYPE,TRAN

with TRNOPT,FULL). The modification is performed only if either C1 ≠ 0 or C3 ≠ 0 and takes the form:

(14–40)R R M’ = +

where:

R' = modified real constant valueR = original real constant value

MC

f C C C C C

Cv

C

v

= +C C C1 v if KEYOPT(9) = 0

if KEYOPT

23

4

1 1 2 3 4( , , , , ) ((9) = 1

C1, C2, C3, C4 = user-selected constants (input as C1, C2, C3 and C4 on RMORE command)

Cv = control value (defined below)

f1 = function defined by subroutine USERRC

By means of KEYOPT(7), the quantity R is as follows:

(14–41)R

K

K

K=

1

2

3

MROT

if KEYOPT(7) = 0 to 1

if KEYOPT(7) = 2

if KEYOPT((7) = 3

if KEYOPT(7) = 13

Negative values for R' are set equal to zero for quantities Tf (KEYOPT(7) = 6), θ (KEYOPT(7) = 11), and θ (KEYOPT(7)

= 12).

The calculation for Cv depends of control nodes L and M, as well as KEYOPT(1), KEYOPT(3), and KEYOPT(4). The

general formulation is given by:

(14–42)C

u

udt

t

v

ot

=

∆∆

∆

∆∫

d( u)dt

d ( u)

dt

2

2

if KEYOPT(1) = 1 or 0

if KEYOP

,

,

TT(1) = 2

if KEYOPT(1) = 3

if KEYOPT(1) = 1 or 0

if KEYOPT(1) = 1 or 0

in which t is time and ∆u is determined from

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–18

(14–43)∆ =

−−−

u

u u

v v

w w

L M

L M

L M

,

,

,

if KEYOPT(3) = 0,1

if KEYOPT(3) = 2

if KEYOPT(33) = 3

if KEYOPT(3) = 4

if KEYOPT(3) = 4

if K

θ θθ θ

θ θ

xL xM

yL yM

zL zM

−−

−

,

,

, EEYOPT(3) = 4

If KEYOPT(4) = 0, then the DOFs above are in nodal coordinates. The DOFs are in the moving element coordinatesif KEYOPT(4) = 1.

14.8. LINK8 - 3-D Spar (or Truss)

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–6Stiffness Matrix and Thermal LoadVector

NoneEquation 12–6, Equation 12–7, and Equation 12–8Mass Matrix

NoneEquation 12–7 and Equation 12–8Stress Stiffening Matrix

DistributionLoad Type

Linear along lengthElement Temperature

Linear along lengthNodal Temperature

Reference: Cook et al.(117)

14.8.1. Assumptions and Restrictions

The element is not capable of carrying bending loads. The stress is assumed to be uniform over the entire element.

14.8.2. Element Matrices and Load Vector

All element matrices and load vectors described below are generated in the element coordinate system and arethen converted to the global coordinate system. The element stiffness matrix is:

14–19ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.8: LINK8 - 3-D Spar (or Truss)

(14–44)[ ]

^

KAELl =

−

−

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

where:

A = element cross-sectional area (input as AREA on R command)

E^ =

E, Young’s modulus (input as EX on

command) if linear

MP

..

E , tangent modulus (see Rate Independent Plasticity)

ifT

plasticity is present and the tangent matrix is

to be compputed (see Rate Independent Plasticity and

Nonlinear Elastticity).

L = element length

The consistent element mass matrix (LUMPM,OFF) is:

(14–45)[ ]

( )M

AL in

l = −

ρ ε16

2 0 0 1 0 0

0 2 0 0 1 0

0 0 2 0 0 1

1 0 0 2 0 0

0 1 0 0 2 0

0 0 1 0 0 2

where:

ρ = density (input as DENS on MP command)

εin = initial strain (input as ISTRN on R command)

The lumped element mass matrix (LUMPM,ON) is:

(14–46)[ ]

( )M

AL in

l = −

ρ ε12

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

The element stress stiffness matrix is:

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–20

(14–47)S

FLl[ ]=

−−

−−

0 0 0 0 0 0

0 1 0 0 1 0

0 0 1 0 0 1

0 0 0 0 0 0

0 1 0 0 1 0

0 0 1 0 0 1

where:

F

for the first iteration: A E

for all subsequent itera=

εin

ttions: the axial force

in the element as computed in the prrevious

stress pass of the element

The element load vector is:

(14–48) F F Fa nrl l l= −

where:

Fal = applied load vector

Fnrl = Newton-Raphson restoring force, if applicable

The applied load vector is:

(14–49) F AEanT T

l = ε -1 0 0 1 0 0

For a linear analysis or the first iteration of a nonlinear (Newton-Raphson) analysis εnT

is:

(14–50)ε ε εnT

nth in= −

with

(14–51)ε αnth

n n refT T= −( )

where:

αn = coefficient of thermal expansion (input as ALPX on MP command) evaluated at Tn

Tn = average temperature of the element in this iteration

Tref = reference temperature (input on TREF command)

For the subsequent iterations of a Newton-Raphson analysis:

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Section 14.8: LINK8 - 3-D Spar (or Truss)

(14–52)ε εnT

nth= ∆

with the thermal strain increment computed through:

(14–53)∆ε α αnth

n n ref n n refT T T T= − − −− −( ) ( )1 1

where:

αn, αn-1 = coefficients of thermal expansion evaluated at Tn and Tn-1, respectively

Tn, Tn-1 = average temperature of the element for this iteration and the previous iteration

The Newton-Raphson restoring force vector is:

(14–54) F AEnrnel T

l = − −ε 1 1 0 0 1 0 0

where:

εnel

− =1 elastic strain for the previous iteration

14.8.3. Force and Stress

For a linear analysis or the first iteration of a nonlinear (Newton-Raphson) analysis:

(14–55)ε ε ε εnel

n nth in= − +

where:

εnel = elastic strain (output as EPELAXL)

εnuL

= =total strain

u = difference of nodal displacements in axial direction

εnth = thermal strain (output as EPTHAXL)

For the subsequent iterations of a nonlinear (Newton-Raphson) analysis:

(14–56)ε ε ε ε ε ε εn

elnel th pl cr sw= + − − − −−1 ∆ ∆ ∆ ∆ ∆

where:

∆ = =ε strain increment∆uL

∆u = difference of nodal displacements increment in axial direction

∆εth = thermal strain increment

∆εpl = plastic strain increment

∆εcr = creep strain increment

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∆εsw = swelling strain increment

The stress is:

(14–57)σ ε= E a

where:

σ = stress (output as SAXL)

ε ε ε εanel cr sw= = + ∆ + ∆adjusted strain

Thus, the strain used to compute the stress has the creep and swelling effects as of the beginning of the substep,not the end. Finally,

(14–58)F A= σ

where:

F = force (output as MFORX)

14.9. INFIN9 - 2-D Infinite Boundary

Integration PointsShape FunctionsMatrix or Vector

NoneA = C1 + C2xMagnetic Potential Coefficient Matrix or ThermalConductivity Matrix

References: Kagawa, Yamabuchi and Kitagami(122)

14.9.1. Introduction

This boundary element (BE) models the exterior infinite domain of the far-field magnetic and thermal problems.This element is to be used in combination with elements having a magnetic potential (AZ) or temperature (TEMP)as the DOF.

14.9.2. Theory

The formulation of this element is based on a first order infinite boundary element (IBE) that is compatible withfirst order quadrilateral or triangular shaped finite elements, or higher order elements with dropped midside

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Section 14.9: INFIN9 - 2-D Infinite Boundary

nodes. For unbounded field problems, the model domain is set up to consist of an interior finite element domain,ΩF, and a series of exterior BE subdomains, ΩB, as shown in Figure 14.3: “Definition of BE Subdomain and the

Characteristics of the IBE”. Each subdomain, ΩB, is treated as an ordinary BE domain consisting of four segments:

the boundary element I-J, infinite elements J-K and I-L, and element K-L; element K-L is assumed to be locatedat infinity.

Figure 14.3 Definition of BE Subdomain and the Characteristics of the IBE

The approach used here is to write BE equations for ΩB, and then convert them into equivalent load vectors for

the nodes I and J. The procedure consists of four separate steps that are summarized below (see reference (122)for details).

First, a set of boundary integral equations is written for ΩB. To achieve this, linear shape functions are used for

the BE I-J:

(14–59)N s s1

12

1( ) ( )= −

(14–60)N s s2

12

1( ) ( )= +

Over the infinite elements J-K and I-L the potential (or temperature) φ and its derivative q (flux) are respectivelyassumed to be:

(14–61)φ φ( ) ,r

r

rii=

i = I,J

(14–62)q r qr

rii( ) ,=

2

i = I,J

The boundary integral equations are the same as presented in Equation 14–364 except that the Green's functionin this case would be:

(14–63)G xk

k

r( , ) lnξ

π=

12

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where:

x = field point in boundary elementξ = source point

k =

magnetic reluctivity (inverse of free space

permeability iinput on command) for

AZ DOF (KEYOPT(1) = 0)

or

therm

EMUNIT

aal conductivity (input as KXX on

command) for TEMPDOF (

MP

KKEYOPT(1) = 1)

Note that all the integrations in the present case are performed in closed form.

Second, in the absence of a source or sink in ΩB, the flux q(r) is integrated over the boundary ΓB of ΩB and set to

zero.

(14–64)qd

B

ΓΓ∫ = 0

Third, a geometric constraint condition that exists between the potential φ and its derivatives

∂∂

∂∂

=φ φτ τn

qand

at the nodes I and J is written as:

(14–65)q q

rn i ii

ii i

= +τ α φ αcos

sini = I,J

Fourth, the energy flow quantity from ΩB is written as:

(14–66)w q d

B

= ∫ φ ΓΓ

This energy flow is equated to that due to an equivalent nodal F defined below.

The four steps mentioned above are combined together to yield, after eliminating qn and qτ,

(14–67)[ ] K Fφ =

where:

[K] = 2 x 2 equivalent unsymmetric element coefficient matrixφ = 2 x 1 nodal DOFs, AZ or TEMPF = 2 x 1 equivalent nodal force vector

For linear problems, the INFIN9 element forms the coefficient matrix [K] only. The load vector F is not formed.The coefficient matrix multiplied by the nodal DOF's represents the nodal load vector which brings the effectsof the semi-infinite domain ΩB onto nodes I and J.

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Section 14.9: INFIN9 - 2-D Infinite Boundary

14.10. LINK10 - Tension-only or Compression-only Spar

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–6Stiffness Matrix and Thermal LoadVector

NoneEquation 12–6, Equation 12–7 , and Equation 12–8Mass Matrix

NoneEquation 12–7 and Equation 12–8Stress Stiffness Matrix

DistributionLoad Type

Linear along lengthElement Temperature

Linear along lengthNodal Temperature

14.10.1. Assumptions and Restrictions

The element is not capable of carrying bending loads. The stress is assumed to be uniform over the entire element.

14.10.2. Element Matrices and Load Vector

All element matrices and load vectors are generated in the element coordinate system and must subsequentlythen be converted to the global coordinate system. The element stiffness matrix is:

(14–68)[ ]K

AEL

C C

C Cl =

−

−

1 1

1 1

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

where:

A = element cross-sectional area (input as AREA on R command)E = Young's modulus (input as EX on MP command)L = element lengthC1 = value given in Table 14.1: “Value of Stiffness Coefficient (C1)”

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Table 14.1 Value of Stiffness Coefficient (C1)

Strain is Currently CompressiveStrain is Currently TensileUser Options

0.01.0KEYOPT(2) = 0

KEYOPT(3) = 0

1.0 x 10-61.0KEYOPT(2) > 0

KEYOPT(3) = 0

1.00.0KEYOPT(2) = 0

KEYOPT(3) = 1

1.01.0 x 10-6KEYOPT(2) > 0

KEYOPT(3) = 1

No extra stiffness for non-load carrying case

Has small stiffness for non-load carrying case

Tension-only spar

Compression-only spar

Meanings:

KEYOPT(2) = 0

KEYOPT(2) = 1,2

KEYOPT(3) = 0

KEYOPT(3) = 1

The element mass matrix is the same as for LINK8.

The element stress stiffness matrix is:

(14–69)[ ]S

FL

C C

C C

C C

C C

l =

−−

−−

0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0

2 2

2 2

2 2

2 2

where:

F

for the first iteration: A E

for all subsequent itera

in

=

ε

ttions: the axial force

in the element (output as FORC)

C2 = value given in Table 14.2: “Value of Stiffness Coefficient (C2)”.

Table 14.2 Value of Stiffness Coefficient (C2)

Strain is Currently CompressiveStrain is Currently TensileUser Options

0.01.0KEYOPT(2) < 2

KEYOPT(3) = 0

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Section 14.10: LINK10 - Tension-only or Compression-only Spar

Strain is Currently CompressiveStrain is Currently TensileUser Options

AE

F1061.0KEYOPT(2) = 2

KEYOPT(3) = 0

1.00.0KEYOPT(2) < 2

KEYOPT(3) = 1

1.0AE

F106

KEYOPT(2) = 2

KEYOPT(3) = 1

No extra stress stiffness value

Include extra stress stiffness value

Tension-only spar

Compression-only spar

Meanings:

KEYOPT(2) = 0,1

KEYOPT(2) = 2

KEYOPT(3) = 0

KEYOPT(3) = 1

The element applied load vector is:

(14–70) F AE C CT Tl = − ε 1 10 0 0 0

where:

εT = α∆T - εin

α = coefficient of thermal expansion (input as ALPX on MP command)∆T = Tave - TREF

Tave = average temperature of element

TREF = reference temperature (input on TREF command)

εin = prestrain (input as ISTRN on R command)

14.11. LINK11 - Linear Actuator

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–6Stiffness and Damping Matrices

NoneNone (lumped mass formulation)Mass Matrix

NoneEquation 12–7 and Equation 12–8Stress Stiffness Matrix

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14.11.1. Assumptions and Restrictions

The element is not capable of carrying bending or twist loads. The force is assumed to be constant over the entireelement.

14.11.2. Element Matrices and Load Vector

All element matrices and load vectors are described below. They are generated in the element coordinate systemand are then converted to the global coordinate system. The element stiffness matrix is:

(14–71)[ ]K Kl =

−

−

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

where:

K = element stiffness (input as K on R command)

The element mass matrix is:

(14–72)[ ]M

Ml =

2

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

where:

M = total element mass (input as M on R command)

The element damping matrix is:

(14–73)[ ]C Cl =

−

−

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

where:

C = element damping (input as C on R command)

The element stress stiffness matrix is:

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Section 14.11: LINK11 - Linear Actuator

(14–74)[ ]S

FLl =

−−

−−

0 0 0 0 0 0

0 1 0 0 1 0

0 0 1 0 0 1

0 0 0 0 0 0

0 1 0 0 1 0

0 0 1 0 0 1

where:

F = the axial force in the element (output as FORCE)L = current element length (output as CLENG)

The element load vector is:

(14–75) F F Fap nrl l l= −

where:

Fapl = applied force vector

Fnrl = Newton-Raphson restoring force, if applicable

The applied force vector is:

(14–76) F Fap Tl = ′ − 1 0 0 1 0 0

where:

F' = applied force thru surface load input using the PRES label

The Newton-Raphson restoring force vector is:

(14–77) F Fnr Tl = − 1 0 0 1 0 0

14.11.3. Force, Stroke, and Length

The element spring force is determined from

(14–78)F K S SM A= −( )

where:

F = element spring force (output as FORCE)SA = applied stroke (output as STROKE) thru surface load input using the PRES label

SM = computed or measured stroke (output as MSTROKE)

The lengths, shown in the figure at the beginning of this section, are:

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Lo = initial length (output as ILEN)

Lo + SM = current length (output as CLEN)

14.12. CONTAC12 - 2-D Point-to-Point Contact

!"

Integration PointsShape FunctionsMatrix or Vector

NoneNone (nodes may be coincident)Stiffness Matrix

DistributionLoad Type

None - average used for material property evaluationElement Temperature

None - average used for material property evaluationNodal Temperature

14.12.1. Element Matrices

CONTAC12 may have one of three conditions if the elastic Coulomb friction option (KEYOPT(1) = 0) is used: closedand stuck, closed and sliding, or open. The following matrices are derived assuming that θ is input as 0.0.

1. Closed and stuck. This occurs if:

µ F Fn s>(14–79)

where:

µ = coefficient of friction (input as MU on MP command)Fn = normal force across gap

Fs = sliding force parallel to gap

The normal force is:

(14–80)F k u un n n J n I= − −( ), , ∆

where:

kn = normal stiffness (input as KN on R command

un,I = displacement of node I in normal direction

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Section 14.12: CONTAC12 - 2-D Point-to-Point Contact

un,J = displacement of node J in normal direction

∆ = interferenceinput as INTF on command if KEYOPT(4) = 0

=

R

- d if KEYOPT(4) = 1

d = distance between nodes

The sliding force is:

(14–81)F k u u us s s J s I o= − −( ), ,

where:

ks = sticking stiffness (input as KS on R command)

us,I = displacement of node I in sliding direction

us,J = displacement of node J in sliding direction

uo = distance that nodes I and J have slid with respect to each other

The resulting element stiffness matrix (in element coordinates) is:

(14–82)[ ]K

k k

k k

k k

k k

s s

n n

s s

n n

l =

−−

−−

0 0

0 0

0 0

0 0

and the Newton-Raphson load vector (in element coordinates) is:

(14–83) F

F

F

F

F

nr

s

n

s

n

l =−−

2. Closed and sliding. This occurs if:

µ F Fn s=(14–84)

In this case, the element stiffness matrix (in element coordinates) is:

(14–85)[ ]K

k k

k k

n n

n n

l =−

−

0 0 0 0

0 0

0 0 0 0

0 0

and the Newton-Raphson load vector is the same as in Equation 14–83. If the unsymmetric option ischosen (NROPT,UNSYM), then the stiffness matrix includes the coupling between the normal and slidingdirections; which for STAT = 2 is:

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(14–86)[ ]K

k k

k k

k k

k k

n n

n n

n n

n n

l =

−−

−−

0 0

0 0

0 0

0 0

µ µ

µ µ

3. Open - When there is no contact between nodes I and J. There is no stiffness matrix or load vector.

Figure 14.4: “Force-Deflection Relations for Standard Case” shows the force-deflection relationships for this ele-ment. It may be seen in these figures that the element is nonlinear and therefore needs to be solved iteratively.Further, since energy lost in the slider cannot be recovered, the load needs to be applied gradually.

Figure 14.4 Force-Deflection Relations for Standard Case

!#"%$'&)(*&+ ,)-+, .,(*/ $('0 &)1

22

14.12.2. Orientation of the Element

The element is normally oriented based on θ (input as THETA on R command). If KEYOPT(2) = 1, however, θ is notused. Rather, the first iteration has θ equal to zero, and all subsequent iterations have the orientation of the elementbased on the displacements of the previous iteration. In no case does the element use its nodal coordinates.

14.12.3. Rigid Coulomb Friction

If the user knows that a gap element will be in sliding status for the life of the problem, and that the relativedisplacement of the two nodes will be monotonically increasing, the rigid Coulomb friction option (KEYOPT(1)= 1) can be used to avoid convergence problems. This option removes the stiffness in the sliding direction, asshown in Figure 14.5: “Force-Deflection Relations for Rigid Coulomb Option”. It should be noted that if the relativedisplacement does not increase monotonically, the convergence characteristics of KEYOPT(1) = 1 will be worsethan for KEYOPT(1) = 0.

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Section 14.12: CONTAC12 - 2-D Point-to-Point Contact

Figure 14.5 Force-Deflection Relations for Rigid Coulomb Option

! #"$&%'$&

(&)*(+ ,-(%'. "%/ $&0

11

14.13. PLANE13 - 2-D Coupled-Field Solid2

3

4

5

6

7

8:9<;=9?>

@:9 A

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–106QuadMagnetic Potential CoefficientMatrix; and Permanent Mag-net and Applied Current LoadVector

1 if planar3 if axisymmetricEquation 12–87Triangle

Same as coefficient matrixEquation 12–111Quad

Thermal Conductivity MatrixEquation 12–92Triangle

Same as coefficient matrix

Equation 12–103 and Equation 12–104 and,if modified extra shapes are included (KEY-OPT(2) = 0) and element has 4 uniquenodes) Equation 12–115 and Equa-tion 12–116.

QuadStiffness Matrix; and Thermaland Magnetic Force LoadVector

Equation 12–84 and Equation 12–85Triangle

Same as coefficient matrixEquation 12–103 and Equation 12–104QuadMass and Stress Stiffness

Matrices Equation 12–84 and Equation 12–85Triangle

Same as coefficient matrixSame as conductivity matrix. Matrix is diagonalized asdescribed in Section 13.2: Lumped Matrices

Specific Heat Matrix

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Integration PointsShape FunctionsGeometryMatrix or Vector

Same as coefficient matrixEquation 12–106 and Equation 12–112QuadDamping (Eddy Current) Mat-

rix Equation 12–87 and Equation 12–93Triangle

2Same as conductivity matrix, specialized to the surfaceConvection Surface Matrix andLoad Vector

2Same as mass matrix specialized to the facePressure Load Vector

DistributionLoad Type

Bilinear across elementCurrent Density

Bilinear across elementCurrent Phase Angle

Bilinear across elementHeat Generation

Linear along each facePressure

References: Wilson(38), Taylor, et al.(49), Silvester, et al.(72),Weiss, et al.(94), Garg, et al.(95)

14.13.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors aswell as heat flux evaluations. Section 5.2: Derivation of Electromagnetic Matrices and Section 5.3: ElectromagneticField Evaluations discuss the magnetic vector potential method, which is used by this element. The diagonalizationof the specific heat matrix is described in Section 13.2: Lumped Matrices. Section 14.42: PLANE42 - 2-D StructuralSolid provides additional information on the element coordinate system, extra displacement shapes, and stresscalculations.

14.14. COMBIN14 - Spring-Damper

Integration PointsShape Functions[1]OptionMatrix or Vector

NoneEquation 12–6LongitudinalStiffness and DampingMatrices NoneEquation 12–18Torsional

NoneEquation 12–7, and Equation 12–8LongitudinalStress Stiffening Matrix

1. There are no shape functions used if the element is input on a one DOF per node basis (KEYOPT(2) > 0)as the nodes may be coincident.

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Section 14.14: COMBIN14 - Spring-Damper

14.14.1. Types of Input

COMBIN14 essentially offers two types of elements, selected with KEYOPT(2).

1. Single DOF per node (KEYOPT(2) > 0). The orientation is defined by the value of KEYOPT(2) and the twonodes are usually coincident.

2. Multiple DOFs per node (KEYOPT(2) = 0). The orientation is defined by the location of the two nodes;therefore, the two nodes must not be coincident.

14.14.2. Stiffness Pass

Consider the case of a single DOF per node first. The orientation is selected with KEYOPT(2). If KEYOPT(2) = 7(pressure) or = 8 (temperature), the concept of orientation does not apply. The form of the element stiffness anddamping matrices are:

(14–87)[ ]K ke =

−−

1 1

1 1

(14–88)[ ]C Ce v=

−−

1 1

1 1

where:

k = stiffness (input as K on R command)Cv = Cv1 + Cv2 |v|

Cv1 = constant damping coefficient (input as CV1 on R command)

Cv2 = linear damping coefficient (input as CV2 on R command)

v = relative velocity between nodes computed from the nodal Newmark velocities

Next, consider the case of multiple DOFs per node. Only the case with three DOFs per node will be discussed, asthe case with two DOFs per node is simply a subset. The stiffness, damping, and stress stiffness matrices in elementcoordinates are developed as:

(14–89)[ ]K kl =

−

−

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

(14–90)[ ]C Cvl =

−

−

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

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(14–91)[ ]S

FLl =

−

−

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

where subscript l refers to element coordinates.

and where:

F = force in element from previous iterationL = distance between the two nodes

There are some special notes that apply to the torsion case (KEYOPT(3) = 1):

1. Rotations are simply treated as a vector quantity. No other effects (including displacements) are implied.

2. In a large rotation problem (NLGEOM,ON), the coordinates do not get updated, as the nodes only rotate.(They may translate on other elements, but this does not affect COMBIN14 with KEYOPT(3) = 1). Therefore,there are no large rotation effects.

3. Similarly, as there is no axial force computed, no stress stiffness matrix is computed.

14.14.3. Output Quantities

The stretch is computed as:

(14–92)εo

J I

J I

AL

u u

v v

=

−

−

′ ′

′ ′

if KEYOPT(2) = 0

if KEYOPT(2) = 1

if KEYOPPT(2) = 2

if KEYOPT(2) = 3

if KEYOPT(2) = 4

w wJ I

xJ xI

′ ′

′ ′

−

−θ θ

θθ θ

θ θ

yJ yI

zJ zI

J IP P

′ ′

′ ′

−

−−

if KEYOPT(2) = 5

if KEYOPT(2) = 6

if KEYOPT(2) = 7

if KEYOPT(2) = 8T TJ I−

= output as STRETCH

where:

A = (XJ - XI)(uJ - uI) + (YJ - YI)(vJ - vI) + (ZJ - ZI)(wJ - wI)

X, Y, Z = coordinates in global Cartesian coordinatesu, v, w = displacements in global Cartesian coordinatesu', v', w' = displacements in nodal Cartesian coordinates (UX, UY, UZ)

θ θ θx y z′ ′ ′ =, , rotations in nodal Cartesian coordinates (ROTX,, ROTY, ROTZ)

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Section 14.14: COMBIN14 - Spring-Damper

P = pressure (PRES)T = temperatures (TEMP)

If KEYOPT(3) = 1 (torsion), the expression for A has rotation instead of translations, and εo is output as TWIST.

Next, the static force (or torque) is computed:

(14–93)F ks o= ε

where:

Fs = static force (or torque) (output as FORC (TORQ if KEYOPT(3) = 1))

Finally, if a nonlinear transient dynamic (ANTYPE,TRANS, with TIMINT,ON) analysis is performed, a dampingforce is computed:

(14–94)F C vD v=

where:

FD = damping force (or torque) (output as DAMPING FORCE (DAMPING TORQUE if KEYOPT(3) = 1))

v = relative velocity

relative velocity is computed using Equation 14–92, where the nodal displacements u, v, w, etc. are replaced

with the nodal Newmark velocities & & &u v w, , , etc.

14.15. Not Documented

No detail or element available at this time.

14.16. PIPE16 - Elastic Straight Pipe

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–15, Equation 12–16, Equation 12–17, andEquation 12–18

Stiffness and Mass Matrices

NoneEquation 12–16 and Equation 12–17Stress Stiffness and DampingMatrices

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Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–15, Equation 12–16, and Equation 12–17Pressure and Thermal LoadVectors

DistributionLoad Type

Linear thru thickness or across diameter, and along lengthElement Temperature

Constant across cross-section, linear along lengthNodal Temperature

Internal and External: constant along length and around circumference. Lateral:constant along length

Pressure

14.16.1. Other Applicable Sections

The basic form of the element matrices is given with the 3-D beam element, BEAM4.

14.16.2. Assumptions and Restrictions

The element is assumed to be a thin-walled pipe except as noted. The corrosion allowance is used only in thestress evaluation, not in the matrix formulation.

14.16.3. Stiffness Matrix

The element stiffness matrix of PIPE16 is the same as for BEAM4, except that

(14–95)A A D Dw

o i= = − =π4

2 2( ) pipe wall cross-sectional area

(14–96)I I I D D

Cy z o If

= = = − =π64

14 4( ) bending moment of inertia

(14–97)J D Do i= − =π

324 4( ) torsional moment of inertia

and,

(14–98)A

Asi = =

2 0.shear area

where:

π = 3.141592653Do = outside diameter (input as OD on R command)

Di = inside diameter = Do - 2tw

tw = wall thickness (input as TKWALL on R command)

Cff =

1 0.

if f = 0.0

if f > 0.0

f = flexibility factor (input as FLEX on R command)

Further, the axial stiffness of the element is defined as

14–39ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.16: PIPE16 - Elastic Straight Pipe

(14–99)K

A EL

w

l( , )11 =

if k = 0.0

if k > 0.0k

where:

Kl( , )11 = axial stiffness of element

E = Young's modulus (input as EX on MP command)L = element lengthk = alternate axial pipe stiffness (input as STIFF on RMORE command)

14.16.4. Mass Matrix

The element mass matrix of PIPE16 is the same as for BEAM4, except total mass of the element is assumed to be:

(14–100)m m A A Le ew

flfl

inin= + +( )ρ ρ

where:

me = total mass of element

mA L

m ifew

w

w= =

>

=ρ if m

mw

wpipe wall mass

0 0

0 0

.

.

mw = alternate pipe wall mass (input as MWALL on RMORE command)

ρ = pipe wall density (input as DENS on MP command)ρfl = internal fluid density (input as DENSFL on R command)

A Dfli= π

42

ρin = insulation density (input as DENSIN on RMORE command)

AD D A

A tL

A

ino o s

in

in in

sin

=− =

>

=+

π4

0 0

0 0

2 2( ) if

if

ins

.

.

uulation cross-sectional area

Do+ = Do + 2tin

tin = insulation thickness (input as TKIN on RMORE command)

Asin

= alternate representation of the surface area of the outside of the pipe element (input as AREAIN onRMORE command)

Also, the bending moments of inertia (Equation 14–96) are used without the Cf term.

14.16.5. Gyroscopic Damping Matrix

The element gyroscopic damping matrix is:

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–40

(14–101)[ ]C AL

g

h

h i

g

e =

−

−− −

−

2

0

0 0

0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0

Ωρ

Antisymmetric

00 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0

0

0 0

0 0

0 0 0 0

0 0 0 0

0 0 0 0

−−

−− −

−

−

h

g h

h j

h j

g

h

h i

where:

Ω = rotation frequency about the positive x axis (input as SPIN on RMORE command)

gr

L=

+6 5

1

2

2 2/

( )φ

hr

L= − −

+( )

( )

1 10 1 2

1

2

2φ

φ

ir= + +

+( )

( )

2 15 1 6 1 3

1

2 2

2φ φφ

jr= − + −

+( )

( )

1 30 1 6 1 6

1

2 2

2φ φφ

r I A= /

φ = 122

EI

GA Ls

G = shear modulus (input as GXY on MP command)

As = shear area ( = Aw/2.0)

14.16.6. Stress Stiffness Matrix

The element stress stiffness matrix of PIPE16 is identical to that for BEAM4.

14.16.7. Load Vector

The element pressure load vector is

(14–102) F

F

F

F

l M=

1

2

12

14–41ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.16: PIPE16 - Elastic Straight Pipe

where:

F1 = FA + FP

F7 = -FA + FP

F A EAw

xpr= ε

εxpr = axial strain due to pressure load, defined below

F P LCp A=

0 0

21

. if KEYOPT(5) = 0

if KEYOPT(5) = 1

F FP LCA

2 82

2= =

F FP LCA

3 93

2= =

F4 = F10 = 0.0

F FP L CA

5 113

2

12= − =

F FP L CA

6 122

2

12= − =

P1 = parallel pressure component in element coordinate system (force/unit length)

P2, P3 = transverse pressure components in element coordinate system (force/unit length)

CA =

1.0

positive sine of the angle between

the axis of the eleement and the

direction of the pressures, as

defined by P ,1 P and P

if KEYOPT(5) = 0

if KEYOPT(5) = 1

2 3

The transverse pressures are assumed to act on the centerline, and not on the inner or outer surfaces. Thetransverse pressures in the element coordinate system are computed by

(14–103)

P

P

P

T

P

P

P

X

Y

Z

1

2

3

=

[ ]

where:

[T] = conversion matrix defined in Equation 14–14PX = transverse pressure acting in global Cartesian X direction) (input using face 2 on SFE command)

PY = transverse pressure acting in global Cartesian Y direction) (input using face 3 on SFE command)

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–42

PZ = transverse pressure acting in global Cartesian Z direction) (input using face 4 on SFE command)

εxpr

, the unrestrained axial strain caused by internal and external pressure effects, is needed to compute thepressure part of the element load vector (see Figure 14.6: “Thermal and Pressure Effects”).

Figure 14.6 Thermal and Pressure Effects

εxpr

is computed using thick wall (Lame') effects:

(14–104)ε νxpr i i o o

o iEPD P D

D D= −

−−

11 2

2 2

2 2( )

where:

ν = Poisson's ratio (input as PRXY or NUXY on MP command)Pi = internal pressure (input using face 1 on SFE command)

Po = external pressure (input using face 5 on SFE command)

An element thermal load vector is computed also, based on thick wall effects.

14.16.8. Stress Calculation

The output stresses, computed at the outside surface and illustrated in Figure 14.7: “Elastic Pipe Direct StressOutput” and Figure 14.8: “Elastic Pipe Shear Stress Output”, are calculated from the following definitions:

(14–105)σ

π

dirx i i o o

w

F PD P D

a=

+ −4

2 2( )

(14–106)σ σbendb o

rC

M rI

=

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Section 14.16: PIPE16 - Elastic Straight Pipe

(14–107)σtor

x oM rJ

=

(14–108)σh

i i o o i

o i

PD P D D

D D= − +

−2 2 2 2

2 2( )

(14–109)σlf

swF

A=

2

where:

σdir = direct stress (output as SDIR)

Fx = axial force

a d Dw o i= −π4

2 2( )

do = 2 ro

rD

too

c= −2

tc = corrosion allowance (input as TKCORR on RMORE command)

σbend = bending stress (output as SBEND)

Cσ = stress intensification factor, defined in Table 14.3: “Stress Intensification Factors”

M M Mb y z= = +bending moment 2 2

I d Dr o i= −π64

4 4( )

σtor = torsional shear stress (output as ST)

Mx = torsional moment

J = 2Ir

σh = hoop pressure stress at the outside surface of the pipe (output as SH)

RD

ii=

2te = tw - tc

σlf = lateral force shear stress (output as SSF)

F F Fs y z= = +shear force 2 2

Average values of Pi and Po are reported as first and fifth items of the output quantities ELEMENT PRESSURES.

Equation 14–108 is a specialization of Equation 14–408. The outside surface is chosen as the bending stressesusually dominate over pressure induced stresses.

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–44

Figure 14.7 Elastic Pipe Direct Stress Output

Figure 14.8 Elastic Pipe Shear Stress Output

Stress intensification factors are given in Table 14.3: “Stress Intensification Factors”.

Table 14.3 Stress Intensification Factors

CσKEYOPT(2)

at node Jat node I

C Jσ,C Iσ,0

1.0C Tσ,1

C Tσ,1.02

C Tσ,C Tσ,3

Any entry in Table 14.3: “Stress Intensification Factors” either input as or computed to be less than 1.0 is set to1.0. The entries are:

C Iσ, = stress intensification factor of end I of straight pipe (input as SIFI on R command)

C Jσ, = stress intensification factor of end J of straight pipe (input as SIFJ on R command)

14–45ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.16: PIPE16 - Elastic Straight Pipe

Ct

D d

T

w

i o

σ =

+

=0 9

42 3

.

( )

"T" stess intensification factor (ASME(40))

σth (output as STH), which is in the postprocessing file, represents the stress due to the thermal gradient thru

the thickness. If the temperatures are given as nodal temperatures, σth = 0.0. But, if the temperatures are input

as element temperatures,

(14–110)σ α

υtho aE T T= − −−

( )1

where:

To = temperature at outside surface

Ta = temperature midway thru wall

Equation 14–110 is derived as a special case of Equation 2–8, Equation 2–9 and Equation 2–11 with y as the hoopcoordinate (h) and z as the radial coordinate (r). Specifically, these equations

1. are specialized to an isotropic material

2. are premultiplied by [D] and -1

3. have all motions set to zero, hence εx = εh = εr = γxh = γhr = γxr = 0.0

4. have σr = τhr = τxr = 0.0 since r = Ro is a free surface.

This results in:

(14–111)

σ

σ

σ

νυ

νν

ν ν

xt

ht

xht

E E

E E

G

=

−−

−−

−−

−−

−

1 1

0

1 10

0 0

2 2

2 2

αα

∆∆

T

T

0

or

(14–112)σ σ α

νσx

tht

thE T= = −

−=∆

1

and

(14–113)σxh

t = 0

Finally, the axial and shear stresses are combined with:

(14–114)σ σ σ σx dir bend thA= + +

(14–115)σ σ σxh tor fB= + l

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–46

where:

A, B = sine and cosine functions at the appropriate angleσx = axial stress on outside surface (output as SAXL)

σxh = hoop stress on outside surface (output as SXH)

The maximum and minimum principal stresses, as well as the stress intensity and the equivalent stress, are basedon the stresses at two extreme points on opposite sides of the bending axis, as shown in Figure 14.9: “Stress

Point Locations”. If shear stresses due to lateral forces σlf are greater than the bending stresses, the two pointsof maximum shearing stresses due to those forces are reported instead. The stresses are calculated from thetypical Mohr's circle approach in Figure 14.10: “Mohr Circles”.

The equivalent stress for Point 1 is based on the three principal stresses which are designated by small circles inFigure 14.10: “Mohr Circles”. Note that one of the small circles is at the origin. This represents the radial stress onthe outside of the pipe, which is equal to zero (unless Po ≠ 0.0). Similarly, the points marked with an X represent

the principal stresses associated with Point 2, and a second equivalent stress is derived from them.

Next, the program selects the largest of the four maximum principal stresses (σ1, output as S1MX), the smallest

of the four minimum principal stresses (σ3, output as S3MN), the largest of the four stress intensities (σI, output

as SINTMX), and the largest of the four equivalent stresses (σe, output as SEQVMX). Finally, these are also compared

(and replaced as necessary) to the values at the right positions around the circumference at each end. These fourvalues are then printed out and put on the postprocessing file.

Figure 14.9 Stress Point Locations

14–47ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.16: PIPE16 - Elastic Straight Pipe

Figure 14.10 Mohr Circles

Three additional items are put on the postdata file for use with certain code checking. These are:

(14–116)σpr

c i o

w

P Dt

=4

(14–117)σMI

cXI YI ZI

oM M MD

I= + +2 2 2

2

(14–118)σMJ

cXJ YJ ZJ

oM M MD

I= + +2 2 2

2

where:

σprc = special hoop stress (output as SPR2)

σMIc = special bending stress at end I (output as SMI)

σMJc = special bending stress at end J (output as SMJ)

MXI = moment about the x axis at node I, etc.

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ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–48

14.17. PIPE17 - Elastic Pipe Tee

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–15, Equation 12–16, Equation 12–17, andEquation 12–18

Stiffness and Mass Matrices

NoneEquation 12–16 and Equation 12–17Stress Stiffness Matrix

NoneEquation 12–15, Equation 12–16, and Equation 12–17Pressure and Thermal LoadVectors

DistributionLoad Type

In each branch: linear thru thickness, constant along the lengthElement Temperature

In each branch: constant thru thickness, linear along the lengthNodal Temperature

Internal and External: constant on all branches along the length and aroundthe circumference Lateral: constant on each branch along the length

Pressure

14.17.1. Other Applicable Sections

PIPE17 is essentially the same as three PIPE16 (elastic straight pipe) elements.

14.18. PIPE18 - Elastic Curved Pipe (Elbow)

14–49ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.18: PIPE18 - Elastic Curved Pipe (Elbow)

Integration PointsShape FunctionsMatrix or Vector

NoneNo shape functions are explicitly used. Rather a flexibilitymatrix similar to that developed by Chen(4) is invertedand used.

Stiffness Matrix

NoneNo shape functions are used. Rather a lumped massmatrix using only translational degrees of freedom isused.

Mass Matrix

NoneEquation 12–15, Equation 12–16, and Equation 12–17Thermal and Pressure LoadVector

DistributionLoad Type

Linear thru thickness or across diameter, and along lengthElement Temperature

Constant across cross-section, linear along lengthNodal Temperature

Internal and External: constant along length and around the circumferenceLateral: varies trigonometrically along length (see below)

Pressure

14.18.1. Other Applicable Sections

Section 14.16: PIPE16 - Elastic Straight Pipe covers some of the applicable stress calculations.

14.18.2. Stiffness Matrix

The geometry in the plane of the element is given in Figure 14.11: “Plane Element”.

Figure 14.11 Plane Element

The stiffness matrix is developed based on an approach similar to that of Chen(4). The flexibility of one end withrespect to the other is:

(14–119)[ ]f

f f f

f f f

f f f

f f f

f f

=

11 13 15

22 24 26

31 33 35

42 44 46

51 5

0 0 0

0 0 0

0 0 0

0 0 0

0 33 55

62 64 66

0 0

0 0 0

f

f f f

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–50

where:

fR C

EIR

EAR

fiw11

3

232 2

2 1

= − +

+ +

+ +

θ θ θ θ θ θ θ

ν

cos sin ( cos sin )

( )

EEAw( cos sin )θ θ θ−

f fR C

EIsin

R

EAfi

w13 31

31

252

2= − = − +

+ +

cossinθ θ θ θ θ ν

f fR C

EIfi

15 51

2= = −(sin )θ θ

fR

EIsin

REI

C sin

R

fo

22

3

3

1

21

4 1

= + −

+ + + −

+ +

( )( )

( )( cos )

( ( )

ν θ θ

ν θ θ θ

θ ν ))

EAw

f fREI

Cfo24 42

2

21= = + + −( )( cos sin )ν θ θ θ

f fREI

cos Cfo26 62

21 1

21= − = + − + + +

( )( ( )) sin ( )ν θ θ θ ν

fR C

EIR

EA

cos sin

fiw33

3

212

212

= −

+

+ +

θ θ θ

θ θ θ

cos sin

+

4 1R

EAw( )ν

f fR C

EIfi

35 53

21= − = −(cos )θ

fREI

CREI

C sinfo fo44 21

21= + + + + −( ) cos ( )ν θ θ ν θ

f fREI

Cfo46 64 21= − = + +( ) sinν θ θ

fRCEI

fi55 = θ

fREI

C Cfo fo66 21 1= + + − + −(( ) cos ( )sin )ν θ θ ν θ

and where:

R = radius of curvature (input as RADCUR on R command) (see Figure 14.11: “Plane Element”)θ = included angle of element (see Figure 14.11: “Plane Element”)E = Young's modulus (input as EX on MP command)ν = Poisson's ratio (input as PRXY or NUXY on MP command)

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Section 14.18: PIPE18 - Elastic Curved Pipe (Elbow)

I D Do i= = −moment of inertia ofcross-sectionπ

644 4( )

A D Dwo i= = −area of cross-section

π4

2 2( )

Do = outside diameter (input as OD on R command)

Di = Do - 2t = inside diameter

t = wall thickness (input as TKWALL on R command)

C

C C

h C

fi

fi fi

fi

=

′ ′ >if 0.0

whichever is greater if

or 1.0,1 65.

′′ = 0.0 and KEYOPT(3) = 0(ASME flexibility factor, ASME Coode(40))

whichever is greater ifor 1 65

11 0

..

hPrX

tECK+

ffi′ = 0.0 and KEYOPT(3) = 1(ASME flexibility factor, ASME Code(40))

if = 0.0 and KEYOPT(3) = 2

(K 10 12

1 12

2

2+

+

′h

h

Cfiaarman flexibility factor)

Cfi′ = in-plane flexibility (input as FLXI on command)R

htR

r=

2

rD to=

−average radius

( )2

PP P P P

P Po i o

i o=

− − >− ≤

1 0 0

0 0 0 0

if

if

.

. .

Pi = internal pressure (input on SFE command)

Po = external pressure (input on SFE command)

X

rt

Rr

ifRr

ifRr

K =

≥

<

6 1 7

0 0 1 7

43

13 .

. .

′ =′ ′ >

′ =

Cif

if foC C

C Cfo fo

fi fo

0 0

0 0

.

.

′ =Cfo out-of-plane flexibility (output as FLXO on comRMORE mmand)

The user should not use the KEYOPT(3) = 1 option if:

(14–120)θcR r< 2

where:

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ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–52

θc = included angle of the complete elbow, not just the included angle for this element (θ)

Next, the 6 x 6 stiffness matrix is derived from the flexibility matrix by inversion:

(14–121)[ ] [ ]K fo = −1

The full 12 x 12 stiffness matrix (in element coordinates) is derived by expanding the 6 x 6 matrix derived aboveand transforming to the global coordinate system.

14.18.3. Mass Matrix

The element mass matrix is a diagonal (lumped) matrix with each translation term being defined as:

(14–122)m

mt

e=2

where:

mt = mass at each node in each translation direction

me= (ρAw + ρflAfl + ρinAin)Rθ = total mass of element

ρ = pipe wall density (input as DENS on MP command)ρfl = internal fluid density (input as DENSFL on RMORE command)

A Dfli= π

42

ρin = insulation density (input as DENSIN on RMORE command)

A D Din o o= − =+π4

2 2( ) insulation cross-section area

Do+ = Do + 2 tin

tin = insulation thickness (input as TKIN on RMORE command)

14.18.4. Load Vector

The load vector in element coordinates due to thermal and pressure effects is:

(14–123) [ ] , ,F F R K A Fth pr ix e

pr tl l l+ = +ε

where:

εx = strain caused by thermal as well as internal and external pressure effects (see Equation 14–104 )

[Ke] = element stiffness matrix in global coordinates

A T= 0 0 1 0 0 0 0 0 1 0 0 0M

,Fpr tl = element load vector due to transverse pressure

,Fpr tl is computed based on the transverse pressures acting in the global Cartesian directions (input using face

2, 3, and 4 on SFE command) and curved beam formulas from Roark(48). Table 18, reference no. (loading) 3, 4,and 5 and 5c was used for in-plane effects and Table 19, reference no. (end restraint) 4e was used for out-of-

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Section 14.18: PIPE18 - Elastic Curved Pipe (Elbow)

plane effects. As a radial load varying trigonometrically along the length of the element was not one of theavailable cases given in Roark(48), an integration of a point radial load was done, using Loading 5c.

14.18.5. Stress Calculations

In the stress pass, the stress evaluation is similar to that for Section 14.16: PIPE16 - Elastic Straight Pipe. It is notthe same as for PIPE60 . The wall thickness is diminished by tc, the corrosion allowance (input as TKCORR on Rcommand). The bending stress components are multiplied by stress intensification factors (Cσ). The “intensified”

stresses are used in the principal and combined stress calculations. The factors are:

(14–124)C

C

I

o

σ, =, if SIFI < 1.0

stress intensification factor at endII (input as SIFI on command) if SIFI > 1.0, R

(14–125)C

C

J

o

σ, =, if SIFJ < 1.0

stress intensification factor at endJJ (input as SIFJ on command) if SIFJ > 1.0, R

(14–126)C ho e=

0 9

1 0

2 3.

.

whichever is greater (ASME Code(40))

where:

ht R

D de

e

i o

=+

162( )

te = t - tc

do = Do - 2 tc

14.19. Not Documented

No detail or element available at this time.

14.20. PIPE20 - Plastic Straight Pipe

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ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–54

Integration PointsShape FunctionsMatrix or Vector

None for elastic matrix. Sameas Newton-Raphson load vec-tor for tangent matrix withplasticity

Equation 12–15, Equation 12–16, Equation 12–17,and Equation 12–18

Stiffness Matrix

NoneEquation 12–16 and Equation 12–17Stress Stiffness Matrix

NoneSame as stiffness matrixMass Matrix

NoneEquation 12–15, Equation 12–16, and Equa-tion 12–17

Pressure and Thermal LoadVector

2 along the length and 8points around circumference.The points are located midwaybetween the inside and out-side surfaces.

Same as stiffness matrixNewton-Raphson Load Vector

DistributionLoad Type

Linear across diameter and along lengthElement Temperature

Constant across cross-section, linear along lengthNodal Temperature

Internal and External: constant along length and around circumferenceLateral: constant along lengthPressure

14.20.1. Assumptions and Restrictions

The radius/thickness ratio is assumed to be large.

14.20.2. Other Applicable Sections

Section 14.4: BEAM4 - 3-D Elastic Beam has an elastic beam element stiffness and mass matrix explicitly writtenout. Section 14.16: PIPE16 - Elastic Straight Pipe discusses the effect of element pressure and the elastic stressprintout. Section 14.23: BEAM23 - 2-D Plastic Beam defines the tangent matrix with plasticity and the Newton-Raphson load vector.

14.20.3. Stress and Strain Calculation

PIPE20 uses four components of stress and strain in the stress calculation:

(14–127) σ

σσσ

σ

=

x

h

r

xh

where x, h, r are subscripts representing the axial, hoop and radial directions, respectively. Since only the axialand shear strains can be computed directly from the strain-displacement matrices, the strains are computedfrom the stresses as follows.

The stresses (before plasticity adjustment) are defined as:

(14–128)σ ε π

x w i i o oEA

D P D P= + −′

42 2( )

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Section 14.20: PIPE20 - Plastic Straight Pipe

(14–129)σh i i o ot

DP D P= −12

( )

(14–130)σr i oP P= − −1

2( )

(14–131)σ β βxh w y j z j

x m

AF F

M DJ

= − +22

( sin cos )

where:

ε' = modified axial strain (see Section 14.23: BEAM23 - 2-D Plastic Beam)E = Young's modulus (input as EX on MP command)Pi = internal pressure (input using face 1 of SFE command)

Po = external pressure (input using face 5 of SFE command)

Di = internal diameter = Do - 2t

Do = external diameter (input as OD on R command)

t = wall thickness (input as TKWALL on R command)

A D Dwo i= − =π

42 2( ) wall area

J D tm= π4

3

Dm = (Di + Do)/2 = average diameter

βj = angular position of integration point J (see Figure 14.12: “Integration Points for End J”) (output as ANGLE)

Fy, Fz, Mx = forces on element node by integration point

Figure 14.12 Integration Points for End J

The forces on the element (Fy, Fz, Mx) are computed from:

(14–132) [ ]([ ] )F T K u FR e e el = −∆

where:

Fl = member forces (output as FORCES ON MEMBER AT NODE)

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[TR] = global to local conversion matrix

[Ke] = element stiffness matrix

∆ue = element incremental displacement vector

Fe = element load vector from pressure, thermal and Newton-Raphson restoring force effects

The forces Fl are in element coordinates while the other terms are given in global Cartesian coordinates. Theforces used in Equation 14–131 correspond to either those at node I or node J, depending at which end thestresses are being evaluated.

The modified total strains for the axial and shear components are readily calculated by:

(14–133)′ = − +ε σ ν σ σx x h rE

1( ( ))

(14–134)′ =ε σxh

xhG

where:

ν = Poisson's ratio (input as PRXY or NUXY on MP command)G = shear modulus (input as GXY on MP command)

The hoop and radial modified total strains are computed through:

(14–135)′ = +−ε ε εh h n h, 1 ∆

(14–136)′ = +−ε ε εr r n r, 1 ∆

where:

εh,n-1 = hoop strain from the previous iteration

εr,n-1 = radial strain from the previous iteration

∆εh = increment in hoop strain

∆εr = increment in radial strain

The strains from the previous iterations are computed using:

(14–137)ε σ ν σ σh n h x n rE, ,( ( ))− −= − +1 1

1

(14–138)ε σ ν σ σr n r x n hE, ,( ( ))− −= − +1 1

1

where σx,n-1 is computed using Equation 14–128 with the modified total strain from the previous iteration. The

strain increments in Equation 14–135 and Equation 14–136 are computed from the strain increment in the axialdirection:

(14–139)∆ ∆ε εh nh

xD=

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Section 14.20: PIPE20 - Plastic Straight Pipe

(14–140)∆ ∆ε εr nr

xD=

where:

∆ = ′ − ′ =−ε ε εx n 1 axial strain increment

D Dnh

nr

, = factors relating axial strain increment to hoop andd radial strain increments, respectively

These factors are obtained from the static condensation of the 3-D elastoplastic stress-strain matrix to the 1-Dcomponent, which is done to form the tangent stiffness matrix for plasticity.

Equation 14–133 through Equation 14–136 define the four components of the modified total strain from whichthe plastic strain increment vector can be computed (see Section 4.1: Rate-Independent Plasticity). The elasticstrains are:

(14–141) ε ε εel pl= ′ − ∆

where:

εel = elastic strain components (output as EPELAXL, EPELRAD, EPELH, EPELXH)

∆εpl = plastic strain increment

The stresses are then:

(14–142) [ ] σ ε= D el

where:

σ = stress components (output as SAXL, SRAD, SH, SXH)[D] = elastic stress-strain matrix

The definition of σ given by Equation 14–142 is modified in that σh and σr are redefined by Equation 14–129and Equation 14–130 as the stress values and must be maintained, regardless of the amount of plastic strain.

As long as the element remains elastic, additional printout is given during the solution phase. The stress intens-ification factors (Cσ) of PIPE16 are used in this printout, but are not used in the printout associated with the plastic

stresses and strains. The maximum principal stresses, the stress intensity, and equivalent stresses are compared(and replaced if necessary) to the values of the plastic printout at the eight positions around the circumferenceat each end. Also, the elastic printout is based on stresses at the outer fiber, but the plastic printout is based onmidthickness stresses. Hence, some apparent inconsistency appears in the printout.

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14.21. MASS21 - Structural Mass

Integration PointsShape FunctionsMatrix or Vector

NoneNoneMass Matrix

The element mass matrix is:

(14–143)[ ]M

a

b

c

d

e

f

e =

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

where:

a

b

c

d

e

f

a

b

c

d

e

f

=

′′′′′′

′′′′′′

if KEYOPT(1) = 0

i

a

b

c

d

e

f

ρ ff KEYOPT(1) = 1

ρ = density (input as DENS on MP command)

where a', b', c', d', e', and f' are user input (input on the R command) in the locations shown in the following table:

KEYOPT(3) = 4KEYOPT(3) = 3KEYOPT(3) = 2KEYOPT(3) = 0

1111a'

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Section 14.21: MASS21 - Structural Mass

KEYOPT(3) = 4KEYOPT(3) = 3KEYOPT(3) = 2KEYOPT(3) = 0

1112b'

--13c'

---4d'

---5e'

-2-6f'

For the mass summary, only the first real constant is used, regardless of which option of KEYOPT(3) is used.Analyses with inertial relief use the complete matrix.

14.22. Not Documented

No detail or element available at this time.

14.23. BEAM23 - 2-D Plastic Beam

Integration PointsShape FunctionsMatrix or Vector

None for elastic case. Same asNewton-Raphson load vector fortangent matrix with plastic case

Equation 12–4 and Equation 12–5Stiffness Matrix

NoneEquation 12–5Mass and Stress StiffnessMatrices; and Thermal Loadand Pressure Load Vectors

3 along the length5 thru the thicknessSame as stiffness matrix

Newton-Raphson Load Vectorand Stress Evaluation

DistributionLoad Type

Linear thru thickness and along lengthElement Temperature

Constant thru thickness, linear along lengthNodal Temperature

Linear along lengthPressure

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14.23.1. Other Applicable Sections

The complete stiffness and mass matrices for an elastic 2-D beam element are given in Section 14.3: BEAM3 - 2-D Elastic Beam.

14.23.2. Integration Points

There are three sets of integration points along the length of the element, one at each end and one at the middle.

Figure 14.13 Integration Point Locations

h is defined as:

h = thickness or height of member (input as HEIGHT on R command)

The five integration points through the thickness are located at positions y = -0.5 h, -0.3 h, 0.0, 0.3 h, and 0.5 h.Each one of these points has a numerical integration factor associated with it, as well as an effective width, whichare different for each type of cross-section. These are derived here in order to explain the procedure used in theelement, as well as providing users with a good basis for selecting their own input values for the case of an arbitrarysection (KEYOPT(6) = 4).

The criteria used for the element are:

1. The element, when under simple tension or compression, should respond exactly for elastic or plasticsituations. That is, the area (A) of the element should be correct.

2. The first moment should be correct. This is nonzero only for unsymmetric cross-sections.

3. The element, when under pure bending, should respond correctly to elastic strains. That is, the (second)moment of inertia (I) of the element should be correct.

4. The third moment should be correct. This is nonzero only for unsymmetric cross-sections.

5. Finally, as is common for numerically integrated cross-sections, the fourth moment of the cross-section(I4) should be correct.

For symmetrical sections an additional criterion is that symmetry about the centerline of the beam must bemaintained. Thus, rather than five independent constants, there are only three. These three constants are sufficientto satisfy the previous three criteria exactly. Some other cases, such as plastic combinations of tension and

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Section 14.23: BEAM23 - 2-D Plastic Beam

bending, may not be satisfied exactly, but the discrepancy for actual problems is normally small. For the unsym-metric cross-section case, the user needs to solve five equations, not three. For this case, use of two additionalequations representing the first and third moments are recommended. This case is not discussed further here.

The five criteria may be set up in equation form:

(14–144)A dA

AREA= ∫

(14–145)I ydA

AREA1 = ∫

(14–146)I y dA

AREA22= ∫

(14–147)I y dA

AREA33= ∫

(14–148)I y dA

AREA44= ∫

where:

dA = differential areay = distance to centroid

These criteria can be rewritten in terms of the five integration points:

(14–149)A H i L i h

i= ∑

=( ) ( )

1

5

(14–150)I H i L i h hP i

i1

1

5= ∑

=( ) ( ) ( ( ))

(14–151)I H i L i h hP i

i2

1

5 2= ∑=

( ) ( ) ( ( ))

(14–152)I H i L i h hP i

i3

1

5 3= ∑=

( ) ( ) ( ( ))

(14–153)I H i L i h hP i

i4

1

5 4= ∑=

( ) ( ) ( ( ))

where:

H(i) = weighting factor at point iL(i) = effective width at point iP(i) = integration point locations in y direction (P(1) = -0.5, P(2) = -0.3, etc.)

The L(i) follows physical reasoning whenever possible as in Figure 14.14: “Beam Widths”.

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Figure 14.14 Beam Widths

Starting with the case of a rectangular beam, all values of L(i) are equal to the width of the beam, which is com-puted from

(14–154)L i

I

hzz( ) = 12

3

where:

Izz = moment of inertia (input as IZZ on R command)

Note that the area is not used in the computation of the width. As mentioned before, symmetry may be used toget H(1) = H(5) and H(2) = H(4). Thus, H(1), H(2), and H(3) may be derived by solving the simultaneous equationsdeveloped from the above three criteria. These weighting factors are used for all other cross-sections, with theappropriate adjustments made in L(i) based on the same criteria. The results are summarized in Table 14.4: “Cross-Sectional Computation Factors”.

One interesting case to study is that of a rectangular cross-section that has gone completely plastic in bending.The appropriate parameter is the first moment of the area or

(14–155)I y dAF = ∫

This results in

(14–156)I H i L i h hP iF

i= ∑

=( ) ( ) ( )

1

5

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Section 14.23: BEAM23 - 2-D Plastic Beam

Table 14.4 Cross-Sectional Computation Factors

Effective Width (L(i))Numerical Weight-ing Factor (H(i))

Location thruThickness (P(i))

Numerical Integra-tion Point (i) PipeRectangular

8.16445tp12Izz/h3.06250000-.51

2.64115tp12Izz/h3.28935185-.32

2.00000tp12Izz/h3.29629630.03

2.64115tp12Izz/h3.28935185.34

8.16445tp12Izz/h3.06250000.55

Effective Width (L(i))Numerical Weight-ing Factor (H(i))

Location thruThickness (P(i))

Numerical Integra-tion Point (i) Arbitrary SectionRound Bar

A(-0.5)/h0.25341Do.06250000-.51

A(-0.3)/h0.79043Do.28935185-.32

A(0.0)/h1.00000Do.29629630.03

A(0.3)/h0.79043Do.28935185.34

A(0.5)/h0.25341Do.06250000.55

where:

P(i) = location, defined as fraction of total thickness from centroidIzz = moment of inertia (input as IZZ on R command)

h = thickness (input as HEIGHT on R command)tp = pipe wall thickness (input as TKWALL on R command)

Do = outside diameter (input as OD on R command)

A(i) = effective area based on width at location i (input as A(i) on R command)

Substituting in the values from Table 14.4: “Cross-Sectional Computation Factors”, the ratio of the theoreticalvalue to the computed value is 18/17, so that an error of about 6% is present for this case.

Note that the input quantities for the arbitrary cross-section (KEYOPT(6) = 4) are h, hL(1)(=A(-50)), hL(2)(=A(-30)),hL(3)(=A(0)), hL(4)(=A(30)), and hL(5)(=A(50)). It is recommended that the user try to satisfy Equation 14–149through Equation 14–153 using this input option. These equations may be rewritten as:

(14–157)A A A A A= − + + − + +0 06250 50 50 0 2935185 30 30 0 29629630. ( ( ) ( )) . ( ( ) ( )) . AA( )0

(14–158)I A A A A h1 0 0312500 50 50 0 008680556 30 30= − − + + − − +( . ( ( ) ( )) . ( ( ) ( )))

(14–159)I A A A A h220 01562500 50 50 0 02604170 30 30= − + + − +( . ( ( ) ( )) . ( ( ) ( )))

(14–160)I A A A A h330 00781250 50 50 0 00781250 30 30= − − + + − − +( . ( ( ) ( )) . ( ( ) ( )))

(14–161)I A A A A h440 00390630 50 50 0 00234375 30 30= − + + − +( . ( ( ) ( )) . ( ( ) ( )))

Of course, I1 = I3 = 0.0 for symmetric sections.

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Alternative to one of the above five equations, Equation 14–156 can be used and rewritten as:

(14–162)I A A A A hF = − + + − +( . ( ( ) ( )) . ( ( ) ( )))0 031250 50 50 0 08680554 30 30

Remember that I2 is taken about the midpoint and that Izz is taken about the centroid. The relationship between

these two is:

(14–163)I I Adzz = −22

where:

dIA

h H i L i P I H i L ii i

= = ∑ ∑= =

1

1

5

1

5( ) ( ) ( ) ( ) ( ) = 0.0 for symmetric ccross-sections

14.23.3. Tangent Stiffness Matrix for Plasticity

The elastic stiffness, mass, and stress stiffness matrices are the same as those for a 2-D beam element (BEAM3 ).The tangent stiffness matrix for plasticity, however, is formed by numerical integration. This discussion of thetangent stiffness matrix as well as the Newton-Raphson restoring force of the next subsection has been generalizedto include the effects of 3-D plastic beams. The general form of the tangent stiffness matrix for plasticity is:

(14–164)[ ] [ ] [ ][ ] ( )K B D B d voln

Tnvol= ∫

where:

[B] = strain-displacement matrix[Dn] = elastoplastic stress-strain matrix

This stiffness matrix for a general beam can also be written symbolically as:

(14–165)[ ] [ ] [ ] [ ] [ ]K K K K KB S A T= + + +

[KB] = bending contribution

[KS] = transverse shear contribution

[KA] = axial contribution

[KT] = torsional contribution

where the subscript n has been left off for convenience. As each of these four matrices use only one component

of strain at a time, the integrand of Equation 14–165 can be simplified from [B]T[Dn][B] to B Dn B . Each of

these matrices will be subsequently described in detail.

1. Bending Contribution ([KB]). The strain-displacement matrix for the bending stiffness matrix for bendingabout the z axis can be written as:

(14–166)B y BB

xB

=

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Section 14.23: BEAM23 - 2-D Plastic Beam

where Bx

B

contains the terms of

BB

which are only a function of x (see Narayanaswami and Adel-

man(129)) :

(14–167) B

L

xL

x LL

xL

x LL

xB =

+

−

− −

− −

− +

1

12

126

6 412

126

6 212

2 Φ

Φ

Φ

where:

L = beam lengthΦ = shear deflection constant (see Section 14.14: COMBIN14 - Spring-Damper)

The elastoplastic stress-strain matrix has only one component relating the axial strain increment to theaxial stress increment:

(14–168)D En T=

where ET is the current tangent modulus from the stress-strain curve. Using these definitions Equa-

tion 14–164 reduces to:

(14–169)[ ] ( )K B E y B d volB

xB

T xB

vol=

∫ 2

The numerical integration of Equation 14–169 can be simplified by writing the integral as:

(14–170)[ ] ( ( ))K B E y d area B dxB

xB

Tarea xB

L=

∫∫ 2

The integration along the length uses a two or three point Gauss rule while the integration through thecross-sectional area of the beam is dependent on the definition of the cross-section. For BEAM23, theintegration through the thickness (area) is performed using the 5 point rule described in the previoussection. Note that if the tangent modulus is the elastic modulus, ET = E, the integration of Equation 14–170yields the exact linear bending stiffness matrix.

The Gaussian integration points along the length of the beam are interior, while the stress evaluationand, therefore, the tangent modulus evaluation is performed at the two ends and the middle of thebeam for BEAM23. The value of the tangent modulus used at the integration point in evaluating Equa-tion 14–170 therefore assumes ET is linearly distributed between the adjacent stress evaluation points.

2. Transverse Shear Contribution ([KS]). The strain-displacement vector for the shear deflection matrix is(see Narayanaswami and Adelman(129)):

(14–171) B

L L Ls

T

=+

− − −

6

12

21

21

2φ

φ

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A plasticity tangent matrix for shear deflection is not required because either the shear strain componentis ignored (BEAM23 and BEAM24) or where the shear strain component is computed (PIPE20), the plasticshear deflection is calculated with the initial-stiffness Newton-Raphson approach instead of the tangentstiffness approach. Therefore, since Dn = G (the elastic shear modulus) Equation 14–164 reduces to:

(14–172)[ ] ( )K B G B d volS s s

vol=

∫

Integrating over the shear area explicitly yields:

(14–173)[ ] K GA B B dxS

ss s

L=

∫

where As is the shear area (see Section 14.3: BEAM3 - 2-D Elastic Beam). As is not a function of x in Equa-

tion 14–171, the integral along the length of the beam in Equation 14–173 could also be easily performedexplicitly. However, it is numerically integrated with the two or three point Gauss rule along with the

bending matrix [KB].

3. Axial Contribution ([KA]). The strain-displacement vector for the axial contribution is:

(14–174) B

LA T= −

11 1

As with the bending matrix, Dn = ET and Equation 14–164 becomes:

(14–175)[ ] ( )K B E B d volA A

TA

vol=

∫

which simplifies to:

(14–176)[ ] ( ( ))K B E d area B dxA A

TareaA

L= ∫

∫

The numerical integration is performed using the same scheme BEAM3 as is used for the bending matrix.

4. Torsion Contribution ([KT]). Torsional plasticity (PIPE20 only) is computed using the initial-stiffnessNewton-Raphson approach. The elastic torsional matrix (needed only for the 3-D beams) is:

(14–177)[ ]K

GJLT =

−−

1 1

1 1

14.23.4. Newton-Raphson Load Vector

The Newton-Raphson restoring force is:

(14–178) [ ] [ ] ( )F B D d voln

nr Tnel

vol= ∫ ε

where:

[D] = elastic stress-strain matrix

εnel = elastic strain from previous iteration

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Section 14.23: BEAM23 - 2-D Plastic Beam

The load vector for a general beam can be written symbolically as:

(14–179) F F F F Fnr

Bnr

Snr

Anr

Tnr= + + +

where:

FBnr = bending restoring force

FSnr = shear deflection restoring force

FAnr = axial restoring force

FTnr = torsional restoring force

and where the subscript n has been left off for convenience. Again, as each of the four vectors use only one

component of strain at a time, the integrand of Equation 14–178 can be simplified from [B]T[D] εnel

to B D εnel

.The appropriate B vector for each contribution was given in the previous section. The following paragraphs

describe D and εnel

for each of the contributing load vectors.

1.Bending Restoring Force FB

nr. For this case, the elasticity matrix has only the axial component of stress

and strain, therefore D = E, the elastic modulus. Equation 14–178 for the bending load vector is:

(14–180)[ ] ( ( ))F E B y d area dxB

nrxB el

areaL= ∫∫ ε

The elastic axial strain is computed by:

(14–181)ε φ ε ε ε ε εel a th pl cr swy= + − − − −

where:

φ = total curvature (defined below)

εa = total strain from the axial deformation (defined below)

εth = axial thermal strain

εpl = axial plastic strain

εcr = axial creep strain

εsw = axial swelling strain

The total curvature is:

(14–182)φ =

B ux

B B

where uB is the bending components of the total nodal displacement vector u. The total strain fromthe axial deformation of the beam is:

(14–183)εa

A A XJ XIB uu u

L=

= −

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where:

uA = axial components for the total nodal displacement vector uuXI, uXJ = axial displacement of nodes I and J

Equation 14–180 is integrated numerically using the same scheme outlined in the previous section.Again, since the nonlinear strain evaluation points for the plastic, creep and swelling strains are not atthe same location as the integration points along the length of the beam, they are linearly interpolated.

2.Shear Deflection Restoring Force

FSnr

. The shear deflection contribution to the restoring force loadvector uses D = G, the elastic shear modulus and the strain vector is simply:

(14–184)ε γelS=

where γS is the average shear strain due to shear forces in the element:

(14–185)γS

S BB u=

The load vector is therefore:

(14–186) F GA B dxS

nrS S

SL

= ∫γ

3.Axial Restoring Force FA

nr. The axial load vector uses the axial elastic strain defined in Equation 14–181

for which the load vector integral reduces to:

(14–187) ( ( ))F E B d area dxA

nr A elareaL

= ∫∫ ε

4.Torsional Restoring Force FT

nr. The torsional restoring force load vector (needed only for 3-D beams)

uses D = G, the elastic shear modulus and the strain vector is:

(14–188)γ γ γ γT

el pl cr= − −

where:

γTel = elastic torsional strain

γ = total torsional strain (defined below)

γpl = plastic shear strain

γcr = creep shear strain

The total torsional shear strain is defined by:

(14–189)γ θ θ ρ= −( )XJ XI

L

where:

θXI, θXJ = total torsional rotations from u for nodes I, J, respectively.

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Section 14.23: BEAM23 - 2-D Plastic Beam

ρ = + =( )y z2 2 distance from shear center

The load vector is:

(14–190) ( ( ))F G B d area dxT

nr TTel

areaL= ∫∫ ρ γ2

where:

BT = strain-displacement vector for torsion (same as axial Equation 14–174)

14.23.5. Stress and Strain Calculation

The modified total axial strain at any point in the beam is given by:

(14–191)′ = + − − −− −ε φ ε ε ε εn

a anth

npl

nswy 1 1

where:

φa = adjusted total curvature

εa = adjusted total strain from the axial deformation

εnth = axial thermal strain

εnpl

− =1 axial plastic strain from previous substep

εncr

− =1 axial creep strain from previous substep

εnsw

− =1 axial swelling strain from previous substep

The total curvature and axial deformation strains are adjusted to account for the applied pressure and accelerationload vector terms. The adjusted curvature is:

(14–192)φ φ φa pa= −

where:

φ = [BB]uB = total curvature

φpa = pressure and acceleration contribution to the curvature

φpa is readily calculated through:

(14–193)φpapaM

EI=

Mpa is extracted from the moment terms of the applied load vector (in element coordinates):

(14–194) F F Fpa pr ac= +

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Fpr is given in Section 14.3: BEAM3 - 2-D Elastic Beam and Fac is given in Section 17.1: Static Analysis. The valueused depends on the location of the evaluation point:

(14–195)M

M

M Mpa

Ipa

Ipa

Jpa= −

,

14

,

if evaluation is at end I

if evalu( ) aation is at the middle

if evaluation is at end JM , Jpa

The adjusted axial deformation strain is:

(14–196)ε ε εa pa= −

where:

ε = [BA]uA = total axial deformation strain

εpa = pressure and acceleration contribution to the axial deformation strain

εpa is computed using:

(14–197)εpa xpaF

EA=

where Fxpa

is calculated in a similar manner to Mpa.

From the modified total strain (Equation 14–191) the plastic strain increment can be computed (see Section 4.1:Rate-Independent Plasticity), leaving the elastic strain as:

(14–198)ε ε εel pl= ′ − ∆

where ∆εpl is the plastic strain increment. The stress at this point in the beam is then:

(14–199)σ ε= E el

14.24. BEAM24 - 3-D Thin-walled Beam

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Section 14.24: BEAM24 - 3-D Thin-walled Beam

Integration PointsShape FunctionsMatrix or Vector

Locations on the cross-sectionare user defined. No integra-tion points are used along thelength for elastic matrix. Sameas Newton-Raphson load vec-tor for tangent matrix withplasticity.

Equation 12–15, Equation 12–16, Equation 12–17,and Equation 12–18

Stiffness Matrix

NoneSame as stress stiffness matrix.Mass and Stress StiffnessMatrices; and Pressure LoadVector

NoneEquation 12–15, Equation 12–16, and Equa-tion 12–17

Thermal Load Vector

2 along the length2 in each segment Same as thermal load vectorNewton-Raphson Load Vector

The user defined points on thecross-section are used at eachend of the element

Same as thermal load vectorStress Evaluation

DistributionLoad Type

Bilinear across cross-section and linear along length. See Section 14.24.3:Temperature Distribution Across Cross-Section for more details.

Element Temperature

Constant across cross-section, linear along lengthNodal Temperature

Linear along length. The pressure is assumed to act along the element x axis.Pressure

References: Oden(27), Galambos(13), Kollbrunner(21)

14.24.1. Assumptions and Restrictions

1. The wall thickness is small in comparison to the overall cross-section dimensions (thin-walled theory).

2. The cross-section does not change shape under deformation.

3. St. Venant's theory of torsion governs the torsional behavior. The cross-section is therefore assumed freeto warp.

4. Only axial stresses and strains are used in determining the nonlinear material effects. Shear and torsionalcomponents are neglected.

14.24.2. Other Applicable Sections

Section 14.4: BEAM4 - 3-D Elastic Beam has an elastic beam element stiffness and mass matrix explicitly writtenout. Section 14.23: BEAM23 - 2-D Plastic Beam defines the tangent matrix with plasticity, the Newton-Raphsonload vector and the stress and strain computation.

14.24.3. Temperature Distribution Across Cross-Section

As stated above, the temperature is assumed to vary bilinearly across the cross-section (as well as along thelength). Specifically,

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(14–200)

T x y z T yTy

zTz

xL

T

II I

J

( , , ) = + ∂∂

+ ∂

∂

−

+

1

++ ∂∂

+ ∂

∂

y

Ty

zTz

xLJ J

where:

T(x,y,z) = temperature at integration point located at x, y, zx, y, z = location of point in reference coordinate system (coordinate system defined by the nodesTi = temperature at node i (input as T1, T4 on BFE command)

∂∂

∂∂

=Ty

Tz

, temperature gradients defined below

L = length

The gradients are:

(14–201)

∂∂

= −T

yT T

iyi i

(14–202)∂∂

= −Tz

T Ti

zi i

where:

Tyi = temperature at one unit from the node i parallel to reference y axis (input as T2, T5 on BFE command)

Tzi = temperature at one unit from the node i parallel to reference z axis (input as T3, T6 on BFE command)

14.24.4. Calculation of Cross-Section Section Properties

The cross-section constants are determined by numerical integration, with the integration points (segmentpoints) input by the user. The area of the kth segment (Ak) is:

(14–203)A tk k k= l

where:

lk = length of segment k (input indirectly as Y and Z on R commands)

tk = thickness of segment k (input as TK on R commands)

The total cross-section area is therefore

(14–204)A Ak= ∑

where:

∑ = implies summation over all the segments

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Section 14.24: BEAM24 - 3-D Thin-walled Beam

The first moments of area with respect to the reference axes used to input the cross-section are

(14–205)q z z Ay i j k= +∑

12

( )

(14–206)q y y Az i j k= ∑ +1

2( )

where:

yi, zi = input coordinate locations at beginning of segment k

yj, zj = input coordinate locations at end of segment k

The centroidal location with respect to the origin of the reference axes is therefore

(14–207)y q Ac z= /

(14–208)z q Ac y= /

where:

yc, zc = coordinates of the centroid

The moments of inertia about axes parallel to the reference axes but whose origin is at the centroid (yc, zc) can

be computed by:

(14–209)I z z z z Ay i i j j k= ∑ + +1

32 2( )

(14–210)I y y y y Az i i j j k= ∑ + +1

32 2( )

where:

y y yc= −

z z zc= −

The product moment of inertia is

(14–211)I y z y z A y z y z Ayz i i j j k i j j i k= ∑ + + ∑ +1

316

( ) ( )

Note that these are simply Simpson's integration rule applied to the standard formulas. The principal momentsof inertia are at an angle θp with respect to the reference coordinate system Figure 14.15: “Cross-Section Input

and Principal Axes”, where θp (output as THETAP) is calculated from:

(14–212)θp

yz

z y

I

I I=

−

−12

21tan

Chapter 14: Element Library

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Figure 14.15 Cross-Section Input and Principal Axes

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8 91 " 91:'1;$<

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A 1

B

B

C

C C

C

C

B

DD

DE

F

F

G F

The principal moments of inertia with respect to the element coordinate system are therefore:

(14–213)I I I I I Iyp y z y z p yz p= + + + −1

212

2 2( ) ( )cos( ) sin( )θ θ

and

(14–214)

I I I Izp y z yp= + −

= principal moment of inertia about the z axip ss (output as IZP)

The torsional constant for a thin-walled beam of either open or closed (single cell only) cross-section is

(14–215)J

A

t

to

k

k

c k k

d=

∑+ ∑

4 13

23

ll

where:

J = torsional constant (output as J)

Ao = area enclosed by centerline of closed part of cross-secttion = ∑ + −12

( )( )c

i j j iz z y y

=∑ summation over the segments enclosing the area onlyc

=∑d

summation over the remaining segments (not included in c

)∑

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Section 14.24: BEAM24 - 3-D Thin-walled Beam

The shear center location with respect to the origin of the reference axes (Figure 14.15: “Cross-Section Input andPrincipal Axes”) is:

(14–216)

y yI I I I

I I Is c

yz y z z

yz y z

= +−

−

=

ω ω2

y-distance to shear center (outtput as SHEAR CENTER)

(14–217)

z zI I I I

I I Is c

yz y y

yz y z

z= +−

−

=

ω ω2

z-distance to shear center (outtput as SHEAR CENTER)

The sectorial products of inertia used to develop the above expressions are:

(14–218)I y y A y y Ay i i j j k i j j i kω ω ω ω ω= ∑ + + ∑ +1

316

( ) ( )

(14–219)I z z A z z Az i i j j k i j j i kω ω ω ω ω= ∑ + + ∑ +1

316

( ) ( )

The sectorial products of inertia are analogous to the moments of inertia, except that one of the coordinates inthe definition (such as Equation 14–211) is replaced with the sectorial coordinate ω. The sectorial coordinate ofa point p on the cross-section is defined as

(14–220)ωp

o

shds= ∫

where h is the distance from some reference point (here the centroid) to the cross-section centerline and s is thedistance along the centerline from an arbitrary starting point to the point p. h is considered positive when thecross-section is being transversed in the counterclockwise direction with respect to the centroid. Note that theabsolute value of the sectorial coordinate is twice the area enclosed by the sector indicated in Figure 14.16: “Defin-ition of Sectorial Coordinate”.

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Figure 14.16 Definition of Sectorial Coordinate

! #" $

%&" !

'

Equation 14–220 can be rewritten using Simpson's integration rule as

(14–221)ωp i j i i j iy z z z y y

s= ∑ − − −

1( ) ( )

where:

= summation from first segment input to first segment contaiining point ps∑

If the segment is part of a closed section or cell, the sectorial coordinate is defined as

(14–222)

ωp i j i i j io

k

k

ck

ky z z z y y

A

tt

s= ∑ − − − −

∑1

2( ) ( )

ll

The warping moment of inertia (output quantity IW) is computed as:

(14–223)I Ani ni nj nj kω ω ω ω ω= ∑ + +1

22 2( )

where the normalized sectorial coordinates ωni and ωnj are defined in general as ωnp below. As BEAM24 ignores

warping torsion, Iω is not used in the stiffness formulation but it is calculated and printed for the user's convenience.

A normalized sectorial coordinate is defined to be

(14–224)ω ω ω ωnp oi oj k opA

A= ∑ + −12

( )

where:

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Section 14.24: BEAM24 - 3-D Thin-walled Beam

ωop = sectorial coordinate with respect to the shear center for integration point p

ωop is defined as with the expressions for the sectorial coordinates Equation 14–221 and Equation 14–222 except

that y and z are replaced by %y and z . These are defined by:

(14–225)%y y ys= −

(14–226)%z z zs= −

Thus, these two equations have been written in terms of the shear center instead of the centroid.

The location of the reference coordinate system affects the line of application of nodal and pressure loadings aswell as the member force printout directions. By default, the reference coordinate system is coincident with they-z coordinate system used to input the cross-section geometry (Figure 14.17: “Reference Coordinate System”(a)).If KEYOPT(3) = 1, the reference coordinate system x axis is coincident with the centroidal line while the referencey and z axes remain parallel to the input y-z axes (Figure 14.17: “Reference Coordinate System”(b)). The shearcenter and centroidal locations with respect to this coordinate system are

(14–227)

y y y

z z zs s o c o

s s o c o

= −

= −, ,

, ,

and

(14–228)

y

zc

c

==

0

0

where the subscript o on the shear center and centroid on the right-hand side of Equation 14–227 refers todefinitions with respect to the input coordinate systems in Equation 14–207, Equation 14–208, Equation 14–216and Equation 14–217. Likewise, if KEYOPT(3) = 2, the reference x axis is coincident with the shear centerline andthe locations of the centroid and shear center are determined to be (Figure 14.17: “Reference Coordinate Sys-tem”(c)).

(14–229)

y y y

z z zc c o s o

c c o s o

= −

= −

, ,

, ,

and

(14–230)

y

zs

s

==

0

0

14.24.5. Offset Transformation

The stiffness matrix for a beam element (Section 14.4: BEAM4 - 3-D Elastic Beam) is formulated with respect tothe element coordinate (principal axis) system for the bending and axial behavior and the shear center for tor-

sional behavior. The stiffness matrix and load vector in this system are [ ]Kl and Fl . In general, the referencecoordinate system in BEAM24 is noncoincident with the element system, hence a transformation between thecoordinate systems is necessary. The transformation is composed of a rotational part that accounts for the angle

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between the reference y and z axes and the element y and z axes (principal axes) and a translational part thataccounts for the offsets of the centroid and shear center. The rotational part has the form

(14–231)[ ]R =

λλ

λλ

0 0 0

0 0 0

0 0 0

0 0 0

where:

(14–232)[ ] cos sin

sin cos

λ θ θ

θ θ

=

−

1 0 0

0

0

p p

p p

and θp is the angle defined in Equation 14–212. The translational part is

(14–233)[ ]T

I T

I

I T

I

=

1

2

0 0

0 0 0

0 0

0 0 0

where [I] is the 3 x 3 identity matrix and [Ti] is

(14–234)[ ]T

z y

z x

y xi

c c

s i

s i

= −−

0

0

0

in which yc, zc, ys, and zs are centroid and shear center locations with respect to the element coordinate system

and xi is the offset in the element x direction for end i. The material to element transformation matrix is then

(14–235)[ ] [ ][ ]O R Tf =

The transformation matrix [Of] is used to transform the element matrices and load vector from the element to

the reference coordinate system

(14–236)[ ] [ ] [ ][ ]K O K OfT

fl l¢ =

(14–237) [ ] [ ]F O FfT

l l¢ =

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Section 14.24: BEAM24 - 3-D Thin-walled Beam

Figure 14.17 Reference Coordinate System

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;

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;

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;

The standard local to global transformation (Section 14.4: BEAM4 - 3-D Elastic Beam) can then be used to calculatethe element matrices and load vector in the global system:

(14–238)[ ] [ ] [ ][ ]K T K Te RT

R= l¢

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and

(14–239) [ ] [ ]F T Fe RT= l

¢

The mass and stress stiffening matrices are similarly transformed. The material to element transformation(Equation 14–236) for the mass matrix, however, neglects the shear center terms ys and zs as the center of mass

coincides with the centroid, not the shear center.

14.25. PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid

!

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2

Equations Equation 12–141, Equa-tion 12–142 , and Equation 12–143 orif modified extra shape functions areincluded (KEYOPT(2) = 0) and elementhas 4 unique nodes: Equation 12–145,Equation 12–146 , and Equation 12–147

QuadStiffness Matrix and ThermalLoad Vector

3Equation 12–133, Equation 12–134 ,and Equation 12–135

Triangle

2 x 2Equation 12–103, Equation 12–104 ,and Equation 12–105

QuadMass and Stress StiffnessMatrices

3Equation 12–84, Equation 12–85 , andEquation 12–86

Triangle

2Same as stress stiffness matrix, specialized to the surfacePressure Load Vector

DistributionLoad Type

Bilinear across element, harmonic around circumferenceElement Temperature

Bilinear across element, harmonic around circumferenceNodal Temperature

Linear along each face, harmonic around circumferencePressure

Reference: Wilson(38), Zienkiewicz(39), Taylor(49)

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Section 14.25: PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid

14.25.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.25.2. Assumptions and Restrictions

The material properties are assumed to be constant around the entire circumference, regardless of temperature

dependent material properties or loading. For l (input as MODE on MODE command) > 0, the extreme values

for combined stresses are obtained by computing these stresses at every 10/ l degrees and selecting the extremevalues.

14.25.3. Use of Temperature

In general, temperatures have two effects on a stress analysis:

1. Temperature dependent material properties.

2. Thermal expansion

In the case of l = 0, there is no conflict between these two effects. However, if l > 0, questions arise. As statedin the assumptions, the material properties may not vary around the circumference, regardless of the temperature.

That is, one side cannot be soft and the other side hard. The input temperature for l > 0 varies sinusoidallyaround the circumference. As no other temperatures are available to the element, the material properties areevaluated at Tref (input on TREF command). The input temperature can therefore be used to model thermal

bending. An approximate application of this would be a chimney subjected to solar heating on one side only. Avariant on this basic procedure is provided by the temperature KEYOPT (KEYOPT(3) for PLANE25). This variantprovides that the input temperatures be used only for material property evaluation rather than for thermalbending. This second case requires that αx, αy, and αz (input on MP commands) all be input as zero. An application

of the latter case is a chimney, which is very hot at the bottom and relatively cool at the top, subjected to a windload.

14.26. CONTAC26 - 2-D Point-to-Ground Contact

This element is no longer supported.

14.27. MATRIX27 - Stiffness, Damping, or Mass Matrix

Integration PointsShape FunctionsMatrix or Vector

NoneNoneStiffness, Mass, and Damping Matrices

14.27.1. Assumptions and Restrictions

All MATRIX27 matrices should normally be positive definite or positive semidefinite (see Section 13.5: PositiveDefinite Matrices for definition) in order to be valid structural matrices. The only exception to this occurs whenother (positive definite) matrices dominate the involved DOFs and/or sufficient DOFs are removed by way ofimposed constraints, so that the total (structure) matrix is positive definite.

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14.28. SHELL28 - Shear/Twist Panel

Integration PointsShape FunctionsMatrix or Vector

NoneNone (see reference)Stiffness Matrix

NoneNone (one-sixth of the mass of each of the IJK, JKL, KLI,and LIJ subtriangles is put at the nodes)

Mass Matrix

NoneNo shape functions are used. Rather, the stress stiffnessmatrix is developed from the two diagonal forces usedas spars

Stress Stiffness Matrix

Reference: Garvey(116)

14.28.1. Assumptions and Restrictions

This element is based directly on the reference by Garvey(116). It uses the idea that shear effects can be repres-ented by a uniform shear flow and nodal forces in the direction of the diagonals. The element only resists shearstress; direct stresses will not be resisted.

The shear panel assumes that only shearing stresses are present along the element edges. Similarly, the twistpanel assumes only twisting moment, and no direct moment.

This element does not generate a consistent mass matrix; only the lumped mass matrix is available.

14.28.2. Commentary

The element loses validity when used in shapes other than rectangular. For non-rectangular cases, the resultingshear stress is nonuniform, so that the patch test cannot be satisfied. Consider a rectangular element underuniform shear:

Figure 14.18 Uniform Shear on Rectangular Element

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Section 14.28: SHELL28 - Shear/Twist Panel

Then, add a fictional cut at 45° to break the rectangular element into two trapezoidal regions (elements):

Figure 14.19 Uniform Shear on Separated Rectangular Element

As can be seen, shear forces as well as normal forces are required to hold each part of the rectangle in equilibriumfor the case of “uniform shear”. The above discussion for trapezoids can be extended to parallelograms. If thepresumption of uniform shear stress is dropped, it is possible to hold the parts in equilibrium using only shearstresses along all edges of the quadrilateral (the presumption used by Garvey) but a truly uniform shear statewill not exist.

14.28.3. Output Terms

The stresses are also computed using the approach of Garvey(116).

When all four nodes lie in a flat plane, the shear flows are related to the nodal forces by:

(14–240)S

F FIJfl JI IJ

IJ= −

l

where:

SIJkl = shear flow along edge IJ (output as SFLIJ)

FJI = force at node I from node J (output as FJI)

FIJ = force at node J from node I (output as FIJ)

lIJ = length of edge I-J

The forces in the element z direction (output quantities FZI, FZJ, FZK, FZL) are zero for the flat case. When theflat element is also rectangular, all shear flows are the same. The stresses are:

(14–241)σxyIJflSt

=

where:

σxy = shear stress (output as SXY)

t = thickness (input as THCK on R command)

The logic to compute the results for the cases where all four nodes do not lie in a flat plane or the element isnon-rectangular is more complicated and is not developed here.

The margin of safety calculation is:

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(14–242)Ms

xyu

xym xy

mxyu

xym

=− ≠

σ

σσ σ

σ

1 0 0

0 0

.

.

if both and

if either oor σxyu =

0

where:

Ms = margin of safety (output as SMARGN)

σxym = maximum nodal shear stress (output as SXY(MAX))

σxyu = maximum allowable shear stress (input as SULT on coR mmmand)

14.29. FLUID29 - 2-D Acoustic Fluid

Integration PointsShape FunctionsMatrix or Vector

2 x 2Equation 12–110Fluid Stiffness and MassMatrices

2Equation 12–103, Equation 12–104 , and Equation 12–110specialized to the interface

Coupling Stiffness, Mass, andDamping Matrices (fluid-structure interface)

14.29.1. Other Applicable Sections

Chapter 8, “Acoustics” describes the derivation of acoustic element matrices and load vectors.

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Section 14.29: FLUID29 - 2-D Acoustic Fluid

14.30. FLUID30 - 3-D Acoustic Fluid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–200Fluid Stiffness and MassMatrices

2 x 2Equation 12–191, Equation 12–192, Equation 12–193,and Equation 12–200 specialized to the interface

Coupling Stiffness and MassMatrices (fluid-structure inter-face)

NoneNo shape functions are used. Instead, the area associatedwith each node at the interface is computed for thedamping to act upon.

Fluid Damping Matrix (fluid atfluid-structure interface)

14.30.1. Other Applicable Sections

Chapter 8, “Acoustics” describes the derivation of acoustic element matrices and load vectors.

14.31. LINK31 - Radiation Link

Integration PointsShape FunctionsMatrix or Vector

NoneNone (nodes may be coincident)Conductivity Matrix

14.31.1. Standard Radiation (KEYOPT(3) = 0)

The two-surface radiation equation (from Equation 6–13) that is solved (iteratively) is:

(14–243)Q FA T TI J= −σε ( )4 4

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where:

Q = heat flow rate between nodes I and J (output as HEAT RATE)σ = Stefan-Boltzmann constant (input as SBC on R command)ε = emissivity (input as EMISSIVITY on R or EMIS on MP command)F = geometric form factor (input as FORM FACTOR on R command)A = area of element (input as AREA on R command)TI, TJ = absolute temperatures at nodes I and J

The program uses a linear equation solver. Therefore, Equation 14–243 is expanded as:

(14–244)Q FA T T T T T TI J I J I J= + + −σε ( )( )( )2 2

and then rewritten as:

(14–245)Q FA T T T T T TI n J n I n J n I n J n= + + −− − − −σε ( )( )( ), , , , , ,1

21

21 1

where the subscripts n and n-1 refer to the current and previous iterations, respectively. It is then recast into finiteelement form:

(14–246)

Q

QC

T

TI

Jo

I n

J n

=−

−

1 1

1 1,

,

with

(14–247)C FA T T T To I n J n I n J n= + +− − − −σε ( )( ), , , ,1

21

21 1

14.31.2. Empirical Radiation (KEYOPT(3) = 1)

The basic equation is:

(14–248)Q FT ATI= −σε( )4

instead of Equation 14–243. This form leads to

(14–249)C F T A T F T A To I n J n I n J n= +

+

− − − −σε

12

12

12

12

14

1

14

1, , , ,

instead of Equation 14–247. And, hence the matrix Equation 14–246 becomes:

(14–250)

Q

QC

F A

F A

T

TI

Jo

I n

J n

=−

−

14

14

14

14

,

,

14.31.3. Solution

If the emissivity is input on a temperature dependent basis, Equation 14–247 is rewritten to be:

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Section 14.31: LINK31 - Radiation Link

(14–251)C FAo I n J n I n J n= + +− − − −σ β β β β( )( ), , , ,1

21

21 1

where:

β εi i iT= =( )13 (i 1 or J)

ε i ifT= =emissivity at node i evaluated at temperature

T T Tif

i off= −

Toff = offset temperature (input on TOFFST command)

Equation 14–249 is handled analogously.

14.32. LINK32 - 2-D Conduction Bar

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–3Conductivity and Specific HeatMatrices; and Heat Generation LoadVector

14.32.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors as well as heatflux evaluations.

14.32.2. Matrices and Load Vectors

The matrices and load vectors described in Section 14.33: LINK33 - 3-D Conduction Bar apply here.

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14.33. LINK33 - 3-D Conduction Bar

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–13Conductivity and Specific HeatMatrices; and Heat Generation LoadVector

14.33.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors as well as heatflux evaluations.

14.33.2. Matrices and Load Vectors

The conductivity matrix is:

(14–252)[ ]K

AKLe

t x=−

−

1 1

1 1

where:

A = area (input as AREA on R command)Kx = conductivity (input as KXX on MP command)

L = distance between nodes

The specific heat matrix is:

(14–253)[ ]C

C ALet p=

ρ2

1 0

0 1

where:

ρ = density (input as DENS on MP command)Cp = specific heat (input as C on MP command)

This specific heat matrix is a diagonal matrix with each diagonal being the sum of the corresponding row of aconsistent specific heat matrix. The heat generation load vector is:

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Section 14.33: LINK33 - 3-D Conduction Bar

(14–254) Q

qALe =

&&&

2

1

1

where:

&&&q = heat generation rate (input on or command)BF BFE

14.33.3. Output

The output is computed as:

(14–255)q K

T TLx

I J= −( )

and

(14–256)Q qA=

where:

q = thermal flux (output as THERMAL FLUX)TI = temperature at node I

TJ = temperature at node J

Q = heat rate (output as HEAT RATE)

14.34. LINK34 - Convection Link

Integration PointsShape FunctionsMatrix or Vector

NoneNone (nodes may be coincident)Conductivity Matrix

14.34.1. Conductivity Matrix

The element conductivity (convection) matrix is

(14–257)[ ]K Ahe

tfeff=

−−

1 1

1 1

where:

A = area over which element acts (input as AREA on R command)

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hfeff = effective film coefficient, defined by equation beloww

The effective film coefficient is:

(14–258)h

h C

h Cfeff f c

f c=

′′ +

maximum of if KEYOPT(3) = 3

if KEYOPT(3)

( , )

≠≠ 3

where:

′ =hf partial film coefficient term defined by equation beloww

Cc = user input constant (input as CC on R command)

The partial film coefficient term is:

(14–259)′ =

=

≠ ≠

≠

h

Fh

Fh Tf

f

f pn

T

T

p

if n

if n and

if n and

0 0

0 0 0

0 00 0

.

.

..

∆ ∆

∆ pp =

0

where:

FT if T

if TB B

B=

>≤

≠

0 and KEYOPT(3) = 2

or KEYOPT(3) 21 0 0.

TB = bulk temperature (input as TBULK on SFE command)

hH me

h

f

fi

=

≠

( )

if KEYOPT(3) 2

or

if KEYOPT(3) = 2 and = 0.0 h fin

nn if KEYOPT(3) = 2 and > 0.0 h fin

H(x) = alternate film coefficient (input on MP,HF command for material x)me = material number for this element (input on MAT command)

hfin = primary film coefficient (input on ,,,CONV,1 commaSFE nnd)

∆Tp = Tp,J - Tp,I

Tp,J = temperature from previous iteration at node J

n = exponent on temperature change (input as EN on R command)

∆Tp must be thought of as unitless, even though it is obviously derived from temperatures.

14.34.2. Output

The output is computed as:

(14–260)Q Ah T Tf

effI J= −( )

where:

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Section 14.34: LINK34 - Convection Link

Q = heat rate (output as HEAT RATE)TI = temperature at node I

TJ = temperature at node J

14.35. PLANE35 - 2-D 6-Node Triangular Thermal Solid

Integration PointsShape FunctionsMatrix or Vector

6Equation 12–101Conductivity Matrix and HeatGeneration Load Vector

6Equation 12–101. If KEYOPT(1) = 1, matrix is diagonalizedas described in Section 13.2: Lumped Matrices

Specific Heat Matrix

2Equation 12–101, specialized to the faceConvection Surface Matrix andLoad Vector

14.35.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors as well as heatflux evaluations.

Chapter 14: Element Library

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14.36. SOURC36 - Current Source

CUR

DZ

DYz

y

x

I

J

K

CUR

a) Type 1 - Coil b) Type 2 - Bar

z

y

x

I

DZ

DY

K

J

c) Type 3 - Arc

DZ

DY

y

z

x

I

J

K

CUR

14.36.1. Description

The functionality of SOURC36 is basically one of user convenience. It provides a means of specifying the necessarydata to evaluate the Biot-Savart integral (Equation 5–18) for the simple current source configurations, coil, barand arc. The magnetic field Hs that results from this evaluation in turn becomes a load for the magnetic scalar

potential elements (SOLID5, SOLID96 and SOLID98) as discussed in Chapter 5, “Electromagnetics”.

14.37. COMBIN37 - Control

Integration PointsShape FunctionsMatrix or Vector

NoneNone (nodes may be coincident)Stiffness Matrix

NoneNone (lumped mass formulation)Mass Matrix

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Section 14.37: COMBIN37 - Control

Integration PointsShape FunctionsMatrix or Vector

NoneNoneDamping Matrix

14.37.1. Element Characteristics

COMBIN37 is a nonlinear, 1-D element with two active nodes and one or two control nodes. The element hasspring-damper-sliding capability similar to COMBIN40. The degree of freedom (DOF) for the active nodes is se-lected using KEYOPT(3) and the DOF for the control nodes is selected using KEYOPT(2).

The action of the element in the structure is based upon the value of the control parameter (Pcn) (explained later),

On and Of (input as ONVAL and OFFVAL on R command), and the behavior switches KEYOPT(4) and (5). Fig-

ure 14.20: “Element Behavior” illustrates the behavior of one of the more common modes of operation of theelement. It is analogous to the normal home thermostat during the winter.

The behavior of all possible combinations of KEYOPT(4) and (5) values is summarized in the following table. Pcn

represents the control parameter (output as CONTROL PARAM). The element is active where the figure indicateson, and inactive where it indicates off. For some options, the element may be either on or off for Pcn between

On and Of, depending upon the last status change.

Figure 14.20 Element Behavior

"!

KEYOPT(4) = 0, KEYOPT(5) = 1, and Of > On

KEYOPT(4) = 0, KEYOPT(5) = 0, Of > On:

#$ #$#%&%#%&%

#(' #*) +-, )

KEYOPT(4) = 0, KEYOPT(5) = 0, Of ≤ On:

./ ./.0&0.0&0

.&1 .*2 3-4 2

KEYOPT(4) = 0, KEYOPT(5) = 1, Of > On:

[1]

56 57875657&7

5(9 5: ;-< :

KEYOPT(4) = 0, KEYOPT(5) = 1, Of ≤ On:

=> =?8?=>=?&?

=(@ =A B-C A

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KEYOPT(4) = 1, KEYOPT(5) = 1:

KEYOPT(4) = 1, KEYOPT(5) = 0:

1. Analogous to Figure 14.20: “Element Behavior”

14.37.2. Element Matrices

When the element status is ON, the element matrices are:

(14–261)[ ]K ke o=

−−

1 1

1 1

(14–262)[ ]M

M

MeI

J=

0

0

(14–263)[ ]C Ce o=

−−

1 1

1 1

where:

ko = stiffness (input as STIF on R command)

MI = mass at node I (input as MASI on R command)

MJ = mass at node J (input as MASJ on R command)

Co = damping constant (input as DAMP on R command)

When the element status is OFF, all element matrices are set to zero.

14.37.3. Adjustment of Real Constants

If KEYOPT(6) > 0, a real constant is to be adjusted as a function of the control parameter as well as other realconstants. Specifically,

(14–264)if KEYOPT(6) = ′ = +0 1or k k Do o,

(14–265)if KEYOPT(6) = ′ = +2, C C Do o

(14–266)if KEYOPT(6) = ′ = +3, M M DJ J

(14–267)if KEYOPT(6) O= ′ = +4, n nO D

(14–268)if KEYOPT(6) = ′ = +5, O O Df f

(14–269)if KEYOPT(6) = ′ = +6, F F DA A

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Section 14.37: COMBIN37 - Control

(14–270)if KEYOPT(6) = ′ = +7, M M DI I

(14–271)if KEYOPT(6) = ′ = +8, F F DS S

where:

D C P C P

f C C C C Pcn

Ccn

C

cn

= +1 3

1 1 2 3 4

2 4 if KEYOPT(9) = 0

if KEY( , , , , ) OOPT(9) = 1

FA = element load (input as AFORCE ON R command)

FS = slider force (input as FSLIDE on RMORE command)

C1, C2, C3, C4 = input constants (input as C1, C2, C3, and C4 on RMORE command)

Pcn = control parameter (defined below)

f1 = function defined by subroutine USERRC

If ′FS (or FS, if KEYOPT(6) ≠ 8) is less than zero, it is reset to zero.

14.37.4. Evaluation of Control Parameter

The control parameter is defined as:

(14–272)P

V

dVdt

d V

dtcn =

if KEYOPT(1) = 0 or 1

if KEYOPT(1) = 2

if KEY2

2OOPT(1) = 3

if KEYOPT(1) = 4

if KEYOPT(1) = 5

Vdt

to

t

∫

where:

Vu K u L

u K=

−( ) ( )

( )

if node L is defined

if node L is not definedd

t = time (input on TIME command)u = degree of freedom as selected by KEYOPT(2)

The assumed value of the control parameter for the first iteration (Pcn1

) is defined as:

(14–273)P

O O

orcn

n f

1

2

=

+if S = 1 or -1

if S = 0 and KEYOPT(2) =

t

tTUNIF 8

all other cases

or

0

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where:

St = constant defining starting status where: 1 means ON, -1 means OFF (input as START on R command)TUNIF = uniform temperature (input on BFUNIF command)

14.38. FLUID38 - Dynamic Fluid Coupling

Integration PointsShape FunctionsMatrix or Vector

None

uC

rC= −

1

2 2 cos θ

wC

rC= −

12 2 sinθ

Mass Matrix

NoneNot definedDamping Matrix

Reference: Fritz(12)

14.38.1. Description

This element is used to represent a dynamic coupling between two points of a structure. The coupling is basedon the dynamic response of two points connected by a constrained mass of fluid. The points represent thecenterlines of concentric cylinders. The fluid is contained in the annular space between the cylinders. The cylindersmay be circular or have an arbitrary cross-section. The element has two DOFs per node: translations in the nodal

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Section 14.38: FLUID38 - Dynamic Fluid Coupling

x and z directions. The axes of the cylinders are assumed to be in the nodal y directions. These orientations maybe changed with KEYOPT(6).

14.38.2. Assumptions and Restrictions

1. The motions are assumed to be small with respect to the fluid channel thickness.

2. The fluid is assumed to be incompressible.

3. Fluid velocities should be less than 10% of the speed of sound in the fluid.

4. The flow channel length should be small compared to the wave length for propagating vibratory disturb-ances (less than about 10%), in order to avoid the possibility of standing wave effects.

14.38.3. Mass Matrix Formulation

The mass matrix formulation used in the element is of the following form:

(14–274)[ ]M

m m

m m

m m

m m

e =

11 13

22 24

31 33

42 44

0 0

0 0

0 0

0 0

The m values are dependent upon the KEYOPT(3) value selected. For KEYOPT(3) = 0 (concentric cylinder case):

(14–275)m m M R R R11 22 14

12

22= = +( )

(14–276)m m m m M R R13 31 24 42 12

222= = = = − ( )

(14–277)m m M R R R33 44 12

22

24= = +( )

where:

ML

R R=

−π ρ

22

12

(Mass Length )4

ρ = fluid mass density (input as DENS on MP command)R1 = radius of inner cylinder (input as R1 on R command)

R2 = radius of outer cylinder (input as R2 on R command)

L = length of cylinders (input as L on R command)

Note that the shape functions are similar to that for PLANE25 or FLUID81 with MODE = 1. The element mass used

in the evaluation of the total structure mass is π ρL R R( )22

12− .

For KEYOPT(3) = 2, which is a generalization of the above cylindrical values but for different geometries, the mvalues are as follows:

(14–278)m Mhx11 =

(14–279)m m M Mhx13 31 1= = − +( )

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(14–280)m M M Mhx33 1 2= + +( )

(14–281)m Mhz22 =

(14–282)m m M Mhz24 42 1= = − +( )

(14–283)m M M Mhz44 1 2= + +

where:

M1 = mass of fluid displaced by the inner boundary (Boundary 1) (input as M1 on R command)

M2 = mass of fluid that could be contained within the outer boundary (Boundary 2) in absence of the inner

boundary (input as M2 on R command)Mhx, Mhz = hydrodynamic mass for motion in the x and z directions, respectively (input as MHX and MHZ on

R command)

The element mass used in the evaluation of the total structure mass is M2 - M1.

The lumped mass option (LUMPM,ON) is not available.

14.38.4. Damping Matrix Formulation

The damping matrix formulation used in the element is of the following form:

(14–284)[ ]C

c c

c c

c c

c c

e =

11 13

22 24

31 33

42 44

0 0

0 0

0 0

0 0

The c values are dependent upon the KEYOPT(3) value selected. For KEYOPT(3) = 0:

(14–285)c c C xWx11 33= = ∆

(14–286)c c C xWx13 31= = − ∆

(14–287)c c C zWz22 44= = ∆

(14–288)c c C zWz24 42= = − ∆

where:

Cf LR R R

R R= +

−ρ 1

212

22

2 133

( )

( )(Mass Length )

Wx, Wz = estimate of resonant frequencies in the x and z response directions, respectively (input as WX, WZ

on RMORE command)f = Darcy friction factor for turbulent flow (input as F on R command)

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Section 14.38: FLUID38 - Dynamic Fluid Coupling

∆x, ∆z = estimate of peak relative amplitudes between inner and outer boundaries for the x and z motions,respectively (input as DX, DZ on R command)

For KEYOPT(3) = 2, the c values are as follows:

(14–289)c c C xWx x11 33= = ∆

(14–290)c c C xWx x13 31= = − ∆

(14–291)c c C zWz z22 44= = ∆

(14–292)c c C zWz z24 42= = − ∆

where:

Cx, Cz = flow and geometry constants for the x and z motions, respectively (input as CX, CZ on RMORE com-

mand)

14.39. COMBIN39 - Nonlinear Spring

Integration PointsShape Functions[1]OptionMatrix or Vector

NoneEquation 12–15LongitudinalStiffness Matrix

NoneEquation 12–18Torsional

NoneEquation 12–7 and Equation 12–8LongitudinalStress Stiffening Matrix

1. There are no shape functions used if the element is input as a one DOF per node basis (KEYOPT(4) = 0)as the nodes are coincident.

14.39.1. Input

The user explicitly defines the force-deflection curve for COMBIN39 by the input of discrete points of force versusdeflection. Up to 20 points on the curve may be defined, and are entered as real constants. The input curve mustpass through the origin and must lie within the unshaded regions, if KEYOPT(1) = 1.

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Figure 14.21 Input Force-Deflection Curve

The input deflections must be given in ascending order, with the minimum change of deflection of:

(14–293)u u ui i min+ − >1 ∆ , i=1,19

where:

ui = input deflections (input as D1, D2, ... D20 on R or RMORE commands)

∆uu umax min

min =−

107

umax = most positive input deflection

umin = most negative input deflection

14.39.2. Element Stiffness Matrix and Load Vector

During the stiffness pass of a given iteration, COMBIN39 will use the results of the previous iteration to determinewhich segment of the input force-deflection curve is active. The stiffness matrix and load vector of the elementare then:

(14–294)[ ]K Ke

tg=−

−

1 1

1 1

(14–295) F Fe

nr =−

11

1

where:

Ktg = slope of active segment from previous iteration (output as SLOPE)F1 = force in element from previous iteration (output as FORCE)

If KEYOPT(4) > 0, Equation 14–294 and Equation 14–295 are expanded to 2 or 3 dimensions.

During the stress pass, the deflections of the current equilibrium iteration will be examined to see whether adifferent segment of the force-deflection curve should be used in the next equilibrium iteration.

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Section 14.39: COMBIN39 - Nonlinear Spring

Figure 14.22 Stiffness Computation

"!

#

14.39.3. Choices for Element Behavior

If KEYOPT(2) = 0 and if no force-deflection points are input for deflection less than zero, the points in the firstquadrant are reflected through the origin (Figure 14.23: “Input Force-Deflection Curve Reflected Through Origin”).

Figure 14.23 Input Force-Deflection Curve Reflected Through Origin

$%&'( )(*+ ,-(.

)(-*/ (0'1+ %,

2(*/ ('1(0.

If KEYOPT(2) = 1, there will be no stiffness for the deflection less than zero (Figure 14.24: “Force-Deflection Curvewith KEYOPT(2) = 1”).

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Figure 14.24 Force-Deflection Curve with KEYOPT(2) = 1

If KEYOPT(1) = 0, COMBIN39 is conservative. This means that regardless of the number of loading reversals, theelement will remain on the originally defined force-deflection curve, and no energy loss will occur in the element.This also means that the solution is not path-dependent. If, however, KEYOPT(1) = 1, the element is nonconser-vative. With this option, energy losses can occur in the element, so that the solution is path-dependent. Theresulting behavior is illustrated in Figure 14.25: “Nonconservative Unloading (KEYOPT(1) = 1)”.

Figure 14.25 Nonconservative Unloading (KEYOPT(1) = 1)

!" #%$ &' (

)

*

+

When a load reversal occurs, the element will follow a new force-deflection line passing through the point ofreversal and with slope equal to the slope of the original curve on that side of the origin (0+ or 0-). If the reversaldoes not continue past the force = 0 line, reloading will follow the straight line back to the original curve (Fig-ure 14.26: “No Origin Shift on Reversed Loading (KEYOPT(1) = 1)”).

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Section 14.39: COMBIN39 - Nonlinear Spring

Figure 14.26 No Origin Shift on Reversed Loading (KEYOPT(1) = 1)

!

If the reversal continues past the force = 0 line, a type of origin shift occurs, and reloading will follow a curve thathas been shifted a distance uorig (output as UORIG) (Figure 14.27: “Origin Shift on Reversed Loading (KEYOPT(1)= 1)”).

Figure 14.27 Origin Shift on Reversed Loading (KEYOPT(1) = 1)

"#$%&

'&() &%*+ #,

- $+ .+ ,/0+ (*

1

2

3

4

A special option (KEYOPT(2) = 2) is included to model crushing behavior. With this option, the element will followthe defined tensile curve if it has never been loaded in compression. Otherwise, it will follow a reflection throughthe origin of the defined compressive curve (Figure 14.28: “Crush Option (KEYOPT(2) = 2)”).

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Figure 14.28 Crush Option (KEYOPT(2) = 2)

14.40. COMBIN40 - Combination!

" #

$&%'($*)+ $&%'($*)+

,

-

.!/

01

2

3

4

Integration PointsShape FunctionsMatrix or Vector

NoneNone (nodes may be coincident)Stiffness, Mass, and Damping Matrices

14.40.1. Characteristics of the Element

The force-deflection relationship for the combination element under initial loading is as shown below (for nodamping).

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Section 14.40: COMBIN40 - Combination

Figure 14.29 Force-Deflection Relationship

where:

F1 = force in spring 1 (output as F1)

F2 = force in spring 2 (output as F2)

K1 = stiffness of spring 1 (input as K1 on R command)

K2 = stiffness of spring 2 (input as K2 on R command)

ugap = initial gap size (input as GAP on R command) (if zero, gap capability removed)

uI = displacement at node I

uJ = displacement at node J

FS = force required in spring 1 to cause sliding (input as FSLIDE on R command)

14.40.2. Element Matrices for Structural Applications

The element mass matrix is:

(14–296)[ ]M Me =

1 0

0 0 if KEYOPT(6) = 0

(14–297)[ ]M

Me =

2

1 0

0 1 if KEYOPT(6) = 1

(14–298)[ ]M Me =

0 0

0 1 if KEYOPT(6) = 2

where:

M = element mass (input as M on R command)

If the gap is open during the previous iteration, all other matrices and load vectors are null vectors. Otherwise,the element damping matrix is:

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(14–299)[ ]C ce =

−−

1 1

1 1

where:

c = damping constant (input as C on R command)

The element stiffness matrix is:

(14–300)[ ]K ke =

−−

1 1

1 1

where:

kK K

=+1 2 if slider was not sliding in previous iteration

ifK2 slider was sliding in previous iteration

and the element Newton-Raphson load vector is:

(14–301) ( )F F Fe

nr = +−

1 21

1

F1 and F2 are the current forces in the element.

14.40.3. Determination of F1 and F2 for Structural Applications

1. If the gap is open,

F F1 2 0 0+ = .(14–302)

If no sliding has taken place, F1 = F2 = 0.0. However, if sliding has taken place during unidirectional motion,

(14–303)F

u K KK K

s1

1 2

1 2=

+

and thus

(14–304)F F2 1= −

where:

us = amount of sliding (output as SLIDE)

2. If the gap is closed and the slider is sliding,

F FS1 = ±(14–305)

and

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Section 14.40: COMBIN40 - Combination

(14–306)F K u2 2 2=

where:

u2 = uJ - uI + ugap = output as STR2

3. If the gap is closed and the slider is not sliding, but had slid before,

F K u1 1 1=(14–307)

where:

u1 = u2 - us = output as STR1

and

(14–308)F K u2 2 2=

14.40.4. Thermal Analysis

The above description refers to structural analysis only. When this element is used in a thermal analysis, the

conductivity matrix is [Ke], the specific heat matrix is [Ce] and the Newton-Raphson load vector is fenr

, where

F1 and F2 represent heat flow. The mass matrix [M] is not used. The gap size ugap is the temperature difference.

Sliding, Fslide, is the element heat flow limit for conductor K1.

14.41. SHELL41 - Membrane Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2

Equation 12–57 and Equation 12–58and, if modified extra shape functionsare included (KEYOPT(2) = 0) and ele-ment has 4 unique nodes Equa-tion 12–67 and Equation 12–68

QuadStiffness Matrix; and Thermaland Normal Pressure LoadVector

1Equation 12–38 and Equation 12–39Triangle

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Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–59QuadFoundation Stiffness Matrix

1Equation 12–40Triangle

2 x 2Equation 12–57, Equation 12–58 andEquation 12–59

QuadMass and Stress StiffnessMatrices

1Equation 12–38, Equation 12–39, andEquation 12–40

Triangle

2Same as mass matrix, specialized to the edgeEdge Pressure Load Vector

DistributionLoad Type

Bilinear in plane of element, constant thru thicknessElement Temperature

Bilinear in plane of element, constant thru thicknessNodal Temperature

Bilinear in plane of element and linear along each edgePressure

References: Wilson(38), Taylor(49)

14.41.1. Assumptions and Restrictions

There is no out-of-plane bending stiffness.

When the 4-node option of this element is used, it is possible to input these four nodes so they do not lie in anexact flat plane. This is called a warped element, and such a nodal pattern should be avoided because equilibriumis lost. The element assumes that the resisting stiffness is at one location (in the plane defined by the crossproduct of the diagonals) and the structure assumes that the resisting stiffnesses are at other locations (thenodes). This causes an imbalance of the moments. The warping factor is computed as:

(14–309)φ = D

A

where:

D = component of the vector from the first node to the fourth node parallel to the element normalA = element area

A warning message will print out if the warping factor exceeds 0.00004 and a fatal message occurs if it exceeds0.04. Rigid offsets of the type used with SHELL63 are not used.

14.41.2. Wrinkle Option

When the wrinkle option is requested (KEYOPT(1) = 2), the stiffness is removed when the previous iteration is incompression, which is similar to the logic of the gap elements. This is referred to as the wrinkle option or clothoption. The following logic is used. First, the membrane stresses at each integration point are resolved into theirprincipal directions so that shear is not directly considered. Then, three possibilities exist:

1. Both principal stresses are in tension. In this case, the program proceeds with the full stiffness at this in-tegration point in the usual manner.

2. Both principal stresses are in compression. In this case, the contribution of this integration point to thestiffness is ignored.

3. One of the principal stresses is in tension and one is in compression. In this case, the integration pointis treated as an orthotropic material with no stiffness in the compression direction and full stiffness in

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Section 14.41: SHELL41 - Membrane Shell

the tension direction. Then a tensor transformation is done to convert these material properties to theelement coordinate system. The rest of the development of the element is done in the same manner isif the option were not used.

14.42. PLANE42 - 2-D Structural Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2

Equation 12–103 and Equation 12–104and, if modified extra shapes are in-cluded (KEYOPT(2) ≠ 1) and elementhas 4 unique nodes, Equation 12–115and Equation 12–116

Quad

Stiffness Matrix

3 if axisymmetric1 if planeEquation 12–84 and Equation 12–85Triangle

Same as stiffness matrixEquation 12–103 and Equation 12–104QuadMass and Stress Stiffness

Matrices Equation 12–84 and Equation 12–85Triangle

2Same as mass matrix, specialized to facePressure Load Vector

DistributionLoad Type

Bilinear across element, constant thru thickness or around circumferenceElement Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

References: Wilson(38), Taylor(49)

14.42.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

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14.43. SHELL43 - 4-Node Plastic Large Strain Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

In-plane: 2 x 2Thru-the-thickness:2 (linear material), 5 (nonlinear material)

Equation 12–81Quad

Stiffness Matrix and ThermalLoad Vector In-plane: 1

Thru-the-thickness:2 (linear material), 5 (nonlinear material)

Equation 12–54Triangle

Same as stiffnessmatrix

Equation 12–57, Equation 12–58 , andEquation 12–59

QuadMass and Stress StiffnessMatrices Same as stiffness

matrixEquation 12–38, Equation 12–39 , andEquation 12–40

Triangle

2 x 2Equation 12–59QuadTransverse Pressure LoadVector 1Equation 12–40Triangle

2Equation 12–57 and Equation 12–58 special-ized to the edge

Quad

Edge Pressure Load Vector

2Equation 12–38 and Equation 12–39 special-ized to the edge

Triangle

DistributionLoad Type

Bilinear in plane of element, linear thru thicknessElement Temperature

Bilinear in plane of element, constant thru thicknessNodal Temperature

Bilinear in plane of element and linear along each edgePressure

References: Ahmad(1), Cook(5), Dvorkin(96), Dvorkin(97), Bathe and Dvorkin(98), Allman(113), Cook(114), MacNealand Harder(115)

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Section 14.43: SHELL43 - 4-Node Plastic Large Strain Shell

14.43.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.43.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to thecenterplane.

Each pair of integration points (in the r direction) is assumed to have the same element (material) orientation.

This element does not generate a consistent mass matrix; only the lumped mass matrix is available.

14.43.3. Assumed Displacement Shape Functions

The assumed displacement and transverse shear strain shape functions are given in Chapter 12, “Shape Functions”.The basic shape functions are essentially a condensation of those used for SHELL93. The basic functions for thetransverse shear strain have been changed to avoid shear locking (Dvorkin(96), Dvorkin(97), Bathe and Dvorkin(98))and are pictured in Figure 14.30: “Shape Functions for the Transverse Strains”. One result of the use of thesedisplacement and strain shapes is that elastic rectangular elements give constant curvature results for flat elements,and also, in the absence of membrane loads, for curved elements. Thus, for these cases, nodal stresses are thesame as centroidal stresses. Both SHELL63 and SHELL93 can have linearly varying curvatures.

14.43.4. Stress-Strain Relationships

The material property matrix [D] for the element is:

(14–310)[ ]

.

D

AE A E

A E AE

G

G

x xy x

xy x y

xy

yz

=

ν

ν

0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 01 2

0

0 0 0 00 01 2Gxz

.

where:

AE

E E

y

y xy x

=− ( )ν 2

Ex = Young's modulus in element x-direction (input as EX on MP command)

νxy = Poisson's ratio in element x-y plane (input as NUXY on MP command)

Gxy = shear modulus in element x-y plane (input as GXY on MP command)

Gyz = shear modulus in element y-z plane (input as GYZ on MP command)

Gxz = shear modulus in element x-z plane (input as GXZ on MP command)

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Figure 14.30 Shape Functions for the Transverse Strains

!

!

!

!

!

!

14.43.5. In-Plane Rotational DOF

If KEYOPT(3) is 0 or 1, there is no significant stiffness associated with the in-plane rotation DOF (rotation aboutthe element r axis). A nominal value of stiffness is present (as described with SHELL63), however, to prevent freerotation at the node. KEYOPT(3) = 2 is used to include the Allman-type rotational DOFs (as described by Allman(113)and Cook(114)). Such rotations improve the in-plane and general 3-D shell performance of the element. However,one of the outcomes of using the Allman rotation is that the element stiffness matrix contains up to two spuriouszero energy modes (discussed below).

14.43.6. Spurious Mode Control with Allman Rotation

The first spurious mode is associated with constant rotations (Figure 14.31: “Constant In-Plane Rotation SpuriousMode”). The second spurious mode coincides with the well-known hourglass mode induced by reduced orderintegration (Figure 14.32: “Hourglass Mode”). It is interesting to note that the hourglass spurious mode is elast-ically restrained for nonrectangular and multi-element configurations.

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Section 14.43: SHELL43 - 4-Node Plastic Large Strain Shell

Figure 14.31 Constant In-Plane Rotation Spurious Mode

(θz1 = θz2 = θz3 = θz4)

Figure 14.32 Hourglass Mode

(θz1 = -θz2 = θz3 = -θz4)

The spurious modes are controlled on an elemental level using the concept suggested by MacNeal and Harder(115).For the constant rotation (Figure 14.31: “Constant In-Plane Rotation Spurious Mode”) spurious mode control, anenergy penalty is defined as:

(14–311)P V GI I xy I= δ θ θ1

where:

PI = energy penalty I

δ1 = penalty parameter (input as ZSTIF1 on R command)

V = element volumeθI = relative rotation, defined below

The relative rotation is computed at the element center as,

(14–312)θ θ θI o zi

i

n

n= −

=∑1

1

where:

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θoo o

vx

uy

= ∂∂

− ∂∂

12

u, v = in-plane motions assuming edges remain straightθzi = in-plane rotation at node i

n = number of nodes per element|o = evaluated at center of element

For the hourglass spurious modes which occur only for 4-node elements, the energy penalty is taken as the innerproduct of the constraint force vector and the alternating rotational mode shapes as,

(14–313)P V GII II xy II= δ θ θ2

where:

PII = energy penalty II

δ2 = penalty parameter (input as ZSTIF2 on RMORE command)

θ θ θ θ θII z z z z= − + −14 1 2 3 4( )

Once the energy penalties (PI and PII) are defined, the associated stiffness augmentations can be calculated as,

(14–314)[ ]K

Pu u

Pu uij

ea

I

i j

II

i j= ∂

∂ ∂+ ∂

∂ ∂

2 2

where:

ui = nodal displacement vector

This augmented stiffness matrix when added to the regular element stiffness matrix results in an effective stiffnessmatrix with no spurious modes.

14.43.7. Natural Space Extra Shape Functions with Allman Rotation

One of the outcomes of the Allman rotation is the dissimilar displacement variation along the normal and tan-gential directions of the element edges. The result of such variation is that the in-plane bending stiffness of the

elements is too large by a factor 1/(1-ν2) and sometimes termed as Poisson's ratio locking. To overcome thisdifficulty, two natural space (s and t) nodeless in-plane displacement shape functions are added in the elementstiffness matrix formulation and then condensed out at the element level. The element thus generated is freeof Poisson's ratio locking. For details of a similar implementation, refer to Yunus et al. (117).

14.43.8. Warping

A warping factor is computed as:

(14–315)φ = D

t

where:

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Section 14.43: SHELL43 - 4-Node Plastic Large Strain Shell

D = component of the vector from the first node to the fourth node parallel to the element normalt = average thickness of the element

If φ > 1.0, a warning message is printed.

14.43.9. Stress Output

The stresses at the center of the element are computed by taking the average of the four integration points onthat plane.

The output forces and moments are computed as described in Section 2.3: Structural Strain and Stress Evaluations.

14.44. BEAM44 - 3-D Elastic Tapered Unsymmetric Beam

!

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–15, Equation 12–16, Equation 12–17, andEquation 12–18

Stiffness Matrix

None

If consistent mass matrix option is used (KEYOPT(2) = 0),same as stiffness matrix. If reduced mass matrix optionis used (KEYOPT(2) = 1), Equation 12–6, Equation 12–7,and Equation 12–8

Mass Matrix

NoneEquation 12–16 and Equation 12–17Stress Stiffness and Founda-tion Stiffness Matrices

NoneEquation 12–15, Equation 12–16, and Equation 12–17Pressure and TemperatureLoad Vectors

DistributionLoad Type

Bilinear across cross-section, linear along lengthElement Temperature

Constant across cross-section, linear along lengthNodal Temperature

Linear along lengthPressure

14.44.1. Other Applicable Sections

This element is an extension of BEAM4, so that the basic element formulation as well as the local to global matrixconversion logic is described in Section 14.4: BEAM4 - 3-D Elastic Beam.

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14.44.2. Assumptions and Restrictions

1. Normals before deformation remain straight and normal after deformation.

2. Offsets, if any, are assumed to be completely rigid.

3. If both offsets and also angular velocities or angular accelerations (input on OMEGA, DOMEGA, CGOMGA,or DCGOMG commands) are used, the radius used in the inertial force calculations does not account forthe offsets.

4. Foundation stiffness effects are applied on the flexible length (i.e., before offsets are used).

5. Shear deflection effects are not included in the mass matrix, as they are for BEAM4.

6. Thermal bending assumes an (average) uniform thickness.

14.44.3. Tapered Geometry

When a tapered geometry is input, the program has no “correct” form to follow as the program does not knowthe shape of the cross-section. The supplied thicknesses are used only for thermal bending and stress evaluation.Consider the case of a beam with an area of 1.0 at one end and 4.0 at the other. Assuming all tapers are straight,the small end is a square, the large end is a 1.0 × 4.0 rectangular, and the midpoint of the beam would then havean area of 2.50. But if the large end is also square (2.0 × 2.0), the midpoint area would then be 2.25. Thus, thereis no unique solution. All effects of approximations are reduced by ensuring that the ratios of the section prop-erties are as close to 1.0 as possible. The discussion below indicates what is done for this element.

The stiffness matrix is the same as for BEAM4 (Equation 14–10), except that an averaged area is used:

(14–316)A A A A AAV = + +( ) /1 1 2 2 3

and all three moments of inertia use averages of the form:

(14–317)I I I I I I I I IAV = + + + +

1 1

32

412 12

342 5

The mass matrix is also the same as for BEAM4 (Equation 14–11), except the upper left quadrant uses sectionproperties only from end I, the lower right quadrant uses section properties only from end J, and the other twoquadrants use averaged values. For example, assuming no prestrain or added mass, the axial mass terms wouldbe ρA1 L/3 for end I, ρA2 L/3 for end J, and ρ(A1 + A2) L/12 for both off-diagonal terms. Thus, the total mass of the

element is: ρ(A1 + A2) L/2.

The stress stiffness matrix assumes a constant area as determined in Equation 14–316.

Finally, the thermal load vector uses average thicknesses.

14.44.4. Shear Center Effects

The shear center effects affect only the torsional terms (Mx, θx). The rotation matrix [Rs] (used below) is:

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Section 14.44: BEAM44 - 3-D Elastic Tapered Unsymmetric Beam

(14–318)[ ]R

C

C

C

s =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 1

1

2

3

where:

CLLSC

G1 =

CL

L Lys

SC

SB G2 = −

∆

CL

zs

SB3 = − ∆

L LSC G ys

zs= + +( ) ( ) ( )2 2 2∆ ∆

L LSB G ys= +( ) ( )2 2∆

∆ ∆ ∆ys

ys

ys= −2 1

∆ ∆ ∆zs

zs

zs= −2 1

∆ys

2 = shear center offset in y-direction at end z (input as DYSC2 on RMORE command)LG = actual flexible length, as shown in Figure 14.33: “Offset Geometry”

Note that only rotation about the shear centerline (θx) is affected. The shear center translations at node I are ac-

counted for by:

(14–319)[ ]TI

s

zs

ys

=

−

1 0 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1

1

∆

∆

A similar matrix [ ]TJ

s is defined at node J based on

∆ys

2 and ∆z

s2 . These matrices are then combined to generate

the [Sc] matrix:

(14–320)[ ][ ] [ ]

[ ] [ ]S

R T

R Tc

sIs

sJs

=

0

0

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This combination of [R] and [T] results because shear center offsets are measured in the element coordinate

system (xe ye ze in Figure 14.33: “Offset Geometry”). The element matrices are then transformed by

(14–321)[ ] [ ] [ ][ ]′ =K S K ScT

cl l

(14–322)[ ] [ ] [ ][ ]′ =S S S ScT

cl l

(14–323) [ ] ′ =F S FcT

l l

where:

[ ]Kl = element stiffness matrix in element (centroidal) coordinate system, similar to Equation 14–10

[ ]Sl = element stress stiffness matrix in element (centroidal) coordinate system

Fl = element load vector in element (centroidal) coordinate system, similar to Equation 14–13.

Figure 14.33 Offset Geometry

!"#%$'& ( *),+ & !-/. 0$ + #

12!"# 3% 3%$4&*56 ),+ &*-/.*78 $4& )'- . 0$ + #

9:;<. 0$ + #3%$#>=>48 8?$#-/. ) ) #@-A3 8 + B7 $4C $48 8 48D- !-/. 78 $4&

EF + #-/.*!),8 + G8 B8 '& H-/.

14.44.5. Offset at the Ends of the Member

It is convenient to define

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Section 14.44: BEAM44 - 3-D Elastic Tapered Unsymmetric Beam

(14–324)∆ ∆ ∆xo

x x= −2 1

(14–325)∆ ∆ ∆y

oy y= −2 1

(14–326)∆ ∆ ∆zo

z z= −2 1

where:

∆x2 = offset in x-direction at end z (input as DX2 on RMORE command)

These definitions of ∆i

o may be thought of as simply setting the offsets at node I to zero and setting the differ-

ential offset to the offset at node J, as shown in Figure 14.33: “Offset Geometry”. The rotation matrix [Ro] impliedby the offsets is defined by:

(14–327)u u u R u u uxe

ye

ze

xe

ye

ze T o

xr

yr

zr

xr

yr

zr T

θ θ θ θ θ θ

=

[ ]

where:

u uxe

ye, ,etc. are in element coordinate system=

u uxr

yr, ,etc. are in reference coordinate system defined by= the nodes

[ ][ ] [ ]

[ ] [ ]R

r

r

oo

o=

0

0

[ ]r

LL L

LL L

LLL L L

LLL

o

A

N

yo

B

A zo

N B

yo

N

A

B

yo

zo

N B

zo

N

B

N

= − −

−

∆ ∆

∆ ∆ ∆

∆0

To account for the translation of forces and moments due to offsets at node I, matrix [ ]T i

o

is defined using Fig-ure 14.34: “Translation of Axes”.

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Figure 14.34 Translation of Axes

The two systems are related by:

(14–328)u u u T u u uxe

ye

ze

xe

ye

ze T o

xr

yr

zr

xr

yr

zr T

θ θ θ θ θ θ

=

[ ]1

where:

[ ]TIo

z y

z x

=

−

−

1 0 0 0

0 1 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1 1

1 1

∆ ∆

∆ ∆

A similar matrix [ ]TJ

o is defined at node J, based on ∆x2, ∆y2, and ∆z2. These matrices are then combined to

generate the [OF] matrix:

(14–329)[ ]

[ ][ ] [ ]

[ ] [ ][ ]O

T R

T RF

Io o

Io o

=

0

0

The basis for the above transformations is taken from Hall and Woodhead(15). The element matrices are thentransformed again by:

(14–330)[ ] [ ] [ ][ ]′′ = ′K O K OFT

Fl l

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Section 14.44: BEAM44 - 3-D Elastic Tapered Unsymmetric Beam

(14–331)[ ] [ ] [ ][ ]′′ = ′S O S OFT

Fl l

(14–332)[ ] [ ] [ ][ ]′ =M O M OFT

Fl l

(14–333) [ ] ′′ = ′F O FFT

l l

where:

[ ]Ml = element mass matrix in element (centroidal) coordinate system, similar to Equation 14–11.

14.44.6. End Moment Release

End moment release (or end rotational stiffness release) logic is activated if either KEYOPT(7) or KEYOPT(8) > 0.The release logic is analogous to that discussed in Section 17.6: Substructuring Analysis, with the dropped rota-tional DOF represented by the slave DOF. The processing of the matrices may be symbolized by:

(14–334)[ ] [ ]′′ => ′′K Kl l using static condensation (equation (17.77))

(14–335)[ ] [ ]′′ => ′′S Sl l

using Guyan reduction (equation (17.89))

for tthe case of linear buckling (Type =

BUCKLE on the cANTYPE oommand)

using static condensation (equation (17.77))

after bbeing combined with [K ] for the cases other

than linearl′′

buckling (Type BUCKLE on the

command)

≠

ANTYPE

(14–336)[ ] [ ]′ => ′M Ml l using Guyan reduction (equation (17.89))

(14–337) ′′ => ′′F Fl l using static condensation (equation (17.78))

14.44.7. Local to Global Conversion

The generation of the local to global transformation matrix [TR] is discussed in Section 14.4: BEAM4 - 3-D Elastic

Beam. Thus, the final matrix conversions are:

(14–338)[ ] [ ] [ ][ ]K T K Te RT

R= ′′l

(14–339)[ ] [ ] [ ][ ]S T S Te RT

R= ′′l

(14–340)[ ] [ ] [ ][ ]M T M Te RT

R= ′l

(14–341) [ ] F T Fe RT= ′′l

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14.44.8. Stress Calculations

The axial stresses are computed analogously to BEAM4. The maximum stress at cross section i is computed by:

(14–342)σ

σ σ σ

σ σ σi

idir

zt ibnd

yt ibnd

idir

zt ibnd

ybmax

, ,

, ,=

+ +

+ +maximum of

iibnd

idir

zb ibnd

yb ibnd

idir

zb ibnd

yt ibnd

σ σ σ

σ σ σ

+ +

+ +

, ,

, ,

where:

σdir = direct stress at centerline (output as SDIR)

σytbnd = bending stress at top in y-direction (output as SBYTT)

σybbnd = bending stress at bottom in y-direction (output as SSBYB)

σztbnd = bending stress at top in z-direction (output as SBZTT)

σzbbnd = bending stress at bottom in z-direction (output as SSBZB)

The minimum stress is analogously defined.

The shear stresses are computed as:

(14–343)τL

yy

sy

F

A=

(14–344)τL

zz

sz

F

A=

where:

τ τLy

Lz, = transverse shear stress (output as SXY, SXZ)

Fy, Fz = transverse shear forces

A Asy

sz, = transverse shear areas (input as ARESZ1, etc. on RMMORE command)

and

(14–345)τT xM C=

where:

τT = torsional shear stress (output as SYZ)

Mx = torsion moment

C = user-supplied constant (input as TSF1 and TSF2 on RMORE command)

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Section 14.44: BEAM44 - 3-D Elastic Tapered Unsymmetric Beam

14.45. SOLID45 - 3-D Structural Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2 if KEYOPT(2) = 01 if KEYOPT(2) = 1

Equation 12–191, Equation 12–192, and Equa-tion 12–193 or, if modified extra shape functionsare included (KEYOPT(1) = 0) and element has8 unique nodes, Equation 12–206, Equa-tion 12–207, and Equation 12–208

Stiffness Matrix and ThermalLoad Vector

Same as stiffness matrixEquation 12–191, Equation 12–192, and Equa-tion 12–193

Mass and Stress StiffnessMatrices

2 x 2Equation 12–57 and Equa-tion 12–58

Quad

Pressure Load Vector

3Equation 12–38 and Equa-tion 12–39

Triangle

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

Reference: Wilson(38), Taylor et al.(49)

14.45.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Uniform reduced integration technique (Flanagan and Belytschko(232)) can be chosen by usingKEYOPT(2) = 1.

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14.46. SOLID46 - 3-D 8-Node Layered Structural Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2

Equation 12–191, Equation 12–192, and Equation 12–193and, if modified extra shape functions are included(KEYOPT(1) ≠ 1) and element has 8 unique nodes Equa-tion 12–206, Equation 12–207, and Equation 12–208

Stiffness Matrix and ThermalLoad Vector

2 x 2 x 2Equation 12–191, Equation 12–192, and Equation 12–193Mass and Stress StiffnessMatrices

2 x 2Equation 12–57 and Equation 12–58QuadPressure Load Vector

3Equation 12–38 and Equation 12–39Triangle

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

References: Wilson(38), Taylor et al.(49)

14.46.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. The theory of SOLID46 is analogous to that given with SHELL99 (Section 14.99: SHELL99 - LinearLayered Structural Shell), except as given in this section.

14.46.2. Assumptions and Restrictions

All material orientations are assumed to be parallel to the reference plane, even if the element has nodes inferringwarped layers.

The numerical integration scheme for the thru-thickness effects are identical to that used in SHELL99. This mayyield a slight numerical inaccuracy for elements having a significant change of size of layer area in the thru-thickness direction. The main reason for such discrepancy stems from the approximation of the variation of the

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Section 14.46: SOLID46 - 3-D 8-Node Layered Structural Solid

determinant of the Jacobian in the thru-thickness direction. The error is usually insignificant. However, usersmay want to try a patch-test problem to assess accuracy for their particular circumstances.

Unlike shell elements, SOLID46 cannot assume a zero transverse shear stiffness at the top and bottom surfacesof the element. Hence the interlaminar shear stress must be computed without using this assumption, whichleads to relatively constant values thru the element.

The use of effective (“eff”) material properties developed below is based on heuristic arguments and numericalexperiences rather than on a rigorous theoretical formulation. The fundamental difficulty is that multilinear dis-placement fields are attempted to be modeled by a linear (or perhaps quadratic) displacement shape functionsince the number of DOF per element must be kept to a minimum. A more rigorous solution can always be ob-tained by using more elements in the thru-the-layer direction. Numerical experimentation across a variety ofproblems indicates that the techniques used with SOLID46 give reasonable answers in most cases.

Figure 14.35 Offset Geometry

"!$#&%'(!) &*+&,) ,-./$) 0 & 1 2 354

6 %87/ 9 : ;!$#%'(!) &*,+) +/.() =<) (!0& ) .!+) & + # 3+4

14.46.3. Stress-Strain Relationships

For layer j, the stress-strain relationships in the layer coordinate system are (from Equation 2–8 thru Equation 2–13:

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(14–346)

εε

εε

ε

ε

α

α

α

x

y

z

xy

yz

xz

x j

y j

z j

T

T

T

=

,

,

,

∆

∆

∆

0

0

00

10 0 0

+

− −

−

E E Ex j

xy j

y j

xz j

z j

xy

,

,

,

,

,

,

ν ν

ν jj

y j y j

yz j

z j

xz j

z j

yz j

z j z j

x

E E E

E E E

G

, ,

,

,

,

,

,

, ,

10 0 0

10 0 0

0 0 01

−

− −

ν

ν ν

yy j

yz j

xz j

G

G

,

,

,

0 0

0 0 0 01

0

0 0 0 0 01

σσ

σσ

σ

σ

x

y

z

xy

yz

xz

where:

αx,j = coefficient of thermal expansion of layer j in the layer x-direction (input as ALPX on MP command)

Ex,j = Young's modulus of layer j in the layer x-direction (input as EX on MP command)

Gxy,j = shear modulus of layer j in layer x-y plane (input as GXY on MP command)

νxy,j = Poisson's ratio of layer j in x-y plane (input as NUXY on MP command)

∆T = T - Tref

T = temperature at point being studiedTref = reference temperature (input on TREF command)

To help ensure continuity of stresses between the layers, Equation 14–346 is modified to yield:

(14–347)

εε

εε

ε

ε

α

α

α

x

y

z

xy

yz

xz

x j

y j

z j

T

T

T

=

,

,

,

∆

∆

∆

0

0

00

10 0 0

+

− −

−

E E Ex j

xy j

y j

xz jeff

zeff

,

,

,

,ν ν

νν ν

ν ν

xy j

y j y j

yz jeff

zeff

xz jeff

zeff

yz jeff

ze

E E E

E E

,

, ,

,

, ,

10 0 0

−

− −fff

zeff

xy j

jG

jG

jG

jG

E

G

D D

D D

10 0 0

0 0 01

0 0

0 0 0 0

0 0 0 0

11 21

11 11

,

, ,

, ,

σσ

σσ

σ

σ

x

y

z

xy

yz

xz

where:

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Section 14.46: SOLID46 - 3-D 8-Node Layered Structural Solid

αα

zeff

j z jj

N

TOT

t

t=

∑=

,1

l

(Presumes temperatures are fairly uniiform within element)

νν

xz jeff

xz j

C

or,

,

.

.

=<

≥

if

if

C

C

45

45

CEExz j

zeff

z j= ν ,

,

Et

t

E

zeff TOT

j

z jj

N=

∑= ,1

l

[ ] ([ ] ) [ ][ ], ,

, ,

DD D

D DT d TG

jj

Gj

G

jG

jG j

T Gj=

= −11 21

12 22

1 −− =1 effective inverted shear moduli in layer system

[ ] [ ]dt

t AG

TOTj j

j

N= ∑ =−

=

1 1

1l

leffective inverted shear moduli iin element system

[ ] [ ] [ ] [ ]A T D Tj jT

z j jl =

[T]j = transformation matrix to convert from layer to element systems

[ ],

,D

G

Gz jyz j

xz j=

0

0

tj = average thickness of layer j

tTOT = average total thickness of element

Nl = numbers of layers

As the temperatures themselves are not used in the definition of αzeff

, large changes in α∆T may need to bemodelled with a relatively fine mesh thru the thickness.

14.46.4. General Strain and Stress Calculations

The following steps are used to compute strains and stresses at a typical point within layer j:

1.The strain vector

( , , , , , )ε ε ε ε ε εx y z xy yz xz is determined from the displacements and the strain-displace-ment relationships, evaluated at the point of interest.

2. The strains are converted from element to layer coordinates.

3. The strains are adjusted for thermal effects, with the effective coefficient of thermal expansion in the z-direction being:

(14–348)αα

zeff

j z jj

N

TOT

t

t=

∑=

,1

l

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4. The normal strain is then adjusted with

′ =ε εz zzeff

z j

EE , (14–349)

5. The transverse shear strains are computed by way of the stresses all in the layer coordinate system.

σ

σ

ε

εyz

xz

G yz

xzD

=

−[ ] 1

(14–350)

where:

εyz, εxz = shear strains based on strain-displacement relationships

Then,

(14–351)′ =ε σyz j yz yz jG, ,/

(14–352)′ =ε σxz j xz xz jG, ,/

where:

′ ′ =ε εyz xz, shear strains based on continuity of shearing strresses

6. Finally, the strains are converted to stresses by the usual relationship:

[ ] ( )σ ε εj j jth

jD= −(14–353)

where:

[D]j = inverse of stress-strain matrix used in Equation 14–346

7.If the element has more than one layer and any layer has

νxz jeff

, or νyz j

eff, exceeding 0.45, the normal

stresses are computed based on nodal forces.

14.46.5. Interlaminar Shear Stress Calculation

It may be seen that the interlaminar shear stress will, in general, not be zero at a free surface. This is because theelement has no knowledge as to whether or not the top or bottom face is a free surface or if there is anotherelement attached to that face.

There are two methods of calculating interlaminar shear stress: by nodal forces and by evaluating stresses layer-by-layer.

Nodal Forces

The shear stresses over the entire volume are computed to be:

(14–354)σxzMx

Ix

I MNx

Jx

J NOx

Kx

K OPx

Lx

L P

F F

A

F F

A

F F

A

F F

A=

−+

−+

−+

−

− − − −

14

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Section 14.46: SOLID46 - 3-D 8-Node Layered Structural Solid

(14–355)σyz

My

Iy

I MNy

Jy

J NOy

Ky

K OPy

Ly

L P

F F

A

F F

A

F F

A

F F

A=

−+

−+

−+

−

− − − −14

where:

σxz, σyz = average transverse shear stress components

F F etcIx

Iy, , . = forces at node I (etc.) parallel to reference plane,with x being parallel to element x direction

AI-M, etc. = tributary area for node (evaluated by using the determinant of the Jacobian at the nearest integ-ration point in base plane)

Evaluating Stresses Layer-by-Layer

This option is available only if KEYOPT(2) = 0 or 1 and simply uses the layer shear stresses for the interlaminarstresses. Thus, the interlaminar shear stresses in the element x direction are:

(14–356)σ σxz xz1 = at bottom of layer I (in plane I-J-K-L)

(14–357)σ σxzN

xzl + =1 at top of layer NL (in plane M-N-O-P)

(14–358)σ σ σxz

jxz xz= 1

2( at top of layer j-1 + at bottom of layer j) where i < j < Nl

The σxz terms are the shear stresses computed from Equation 14–353, except that the stresses have been convertedto element coordinates. The interlaminar shear stresses in the element y-direction are analogous. The componentsare combined as:

(14–359)σ σ σil xz yz= +( ) ( )2 2

and the largest value of σil is output as the maximum interlaminar shear stress.

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14.47. INFIN47 - 3-D Infinite Boundary

!

!

!

!

Integration PointsShape FunctionsMatrix or Vector

None on the boundary element IJKitself, however, 16-point 1-DGaussian quadrature is applied forsome of the integration on each ofthe edges IJ, JK, and KI of the infin-ite elements IJML, JKNM, and KILN(see Figure 14.36: “A Semi-infiniteBoundary Element Zone and theCorresponding Boundary ElementIJK”)

φ φ φ φ= + +

= −

− − + −

=

N N N

NA

x y x y

y y x x x y

N

I I J J K K

Io

J K K J

K J K J

J

,

[( )

( ) ( ) ]

12

112

12

Ax y x y

y y x x x y

NA

x y x y

y

oK I I K

I K I K

Ko

I J J I

[( )

( ) ( ) ]

[( )

(

−

− − + −

= −

− JJ I J I

o

y x x x y

A

− + −=

) ( ) ]

area of triangle IJK

Magnetic Potential CoefficientMatrix or Thermal Conductiv-ity Matrix

Reference: Kaljevic', et al.(130)

14.47.1. Introduction

This boundary element (BE) models the exterior infinite domain of the far-field magnetic and thermal problems.This element is to be used in combination with 3-D scalar potential solid elements, and can have magnetic scalarpotential (MAG), or temperature (TEMP) as the DOF.

14.47.2. Theory

The formulation of this element is based on a first order triangular infinite boundary element (IBE), but the elementcan be used as a 4-node quadrilateral as well. For unbounded field problems, the model domain is set up toconsist of an interior volumetric finite element domain, ΩF, and a series of exterior volumetric BE subdomains,

ΩB, as shown in Figure 14.36: “A Semi-infinite Boundary Element Zone and the Corresponding Boundary Element

IJK”. Each subdomain, ΩB, is treated as an ordinary BE domain consisting of five segments: the boundary element

IJK, infinite elements IJML, JKNM and KILN, and element LMN; element LMN is assumed to be located at infinity.

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Section 14.47: INFIN47 - 3-D Infinite Boundary

Figure 14.36 A Semi-infinite Boundary Element Zone and the Corresponding BoundaryElement IJK

The approach used here is to write BE equations for ΩB, and then convert them into equivalent load vectors for

the nodes I, J and K. The procedure consists of four steps that are summarized below (see (Kaljevic', et al.130) fordetails).

First, a set of boundary integral equations is written for ΩB. To achieve this, the potential (or temperature) and

its normal derivatives (fluxes) are interpolated on the triangle IJK (Figure 14.36: “A Semi-infinite Boundary ElementZone and the Corresponding Boundary Element IJK”) by linear shape functions:

(14–360)φ φ φ φ( , )x y N N NI I J J K K= +

(14–361)q x y N q N q N qn I nI J nJ K nK( , ) = + +

where:

φ = potential (or temperature)

qnn = ∂

∂=φ

normal derivative or flux

NI, NJ, NK = linear shape functions defined earlier

φI, φJ, φK = nodal potentials (or temperatures)

qnI, qnJ, qnK = nodal normal derivatives (or fluxes)

n = normal to the surface IJK

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Figure 14.37 Infinite Element IJML and the Local Coordinate System

Over an infinite element, such as IJML (Figure 14.37: “Infinite Element IJML and the Local Coordinate System”),the dependent variables, i.e., potentials (or temperatures) and their normal derivatives (fluxes) are respectivelyassumed to be (Figure 14.37: “Infinite Element IJML and the Local Coordinate System”):

(14–362)φ β φ φ ρ( , )r

sL

sL rIJ

IIJ

J= −

+

12

(14–363)q r

sL

qs

Lq

rIJJ

IJJτ τ τβ ρ

( , ) = −

+

13

where:

qτφτ

= ∂∂

= normal derivative (or flux) to infinite elements; ee.g., IJML (see figure above)

qτI, qτJ = nodal (nodes I and J) normal derivatives for infinite element IJML

s = a variable length from node I towards node JLIJ = length of edge IJ

ρ = radial distance from the origin of the local coordinate system O to the edge IJr = radial distance from the edge IJ towards infinityβ = variable angle from x-axis for local polar coordinate systemτ = normal to infinite elements IJML

The boundary integral equations for ΩB are now written as:

(14–364)c G x q x F x x d x

B

( ) ( ) ( , ) ( ) ( , ) ( ) ( )ξ φ ξ ξ ξ φ= −[ ]∫Γ

Γ

where:

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Section 14.47: INFIN47 - 3-D Infinite Boundary

c(ξ) = jump term in boundary element method

G xkr

( , )ξπ

= =14

Green’s function or fundamental solution for LLaplace’s equation

F xn

G x( , ) [ ( , )]ξ ξ= ∂∂

(x,ξ) = field and source points, respectivelyr = distance between field and source points

K

Magnetic reluctivity (inverse of free space permeability

=

))

(input on command) for AZ DOF (KEYOPT(1) = 0)

or

is

EMUNIT

ootropic thermal conductivity (input as KXX on command)MP

ffor TEMP DOF (KEYOPT(1) = 1)

The integrations in Equation 14–364 are performed in closed form on the boundary element IJK. The integrationson the infinite elements IJML, JKNM and KILN in the 'r' direction (Figure 14.37: “Infinite Element IJML and theLocal Coordinate System”) are also performed in closed form. However, a 16-point Gaussian quadrature rule isused for the integrations on each of the edges IJ, JK and KI on the infinite elements.

Second, in the absence of a source or sink in ΩB, the flux q(r) is integrated over the boundary ΓB of ΩB and set to

zero:

(14–365)qdr

BΓ∫ = 0

Third, geometric constraint conditions that exist between the potential φ (or temperature) and its derivatives

∂∂

=φn

qn and

∂∂

=φτ τq

at the nodes I, J and K are written. These conditions would express the fact that the normalderivative qn at the node I, say, can be decomposed into components along the normals to the two infinite ele-

ments IJML and KILN meeting at I and along OI.

Fourth, the energy flow quantity from ΩB is written as:

(14–366)w q dr

B

= ∫Γ

φ

This energy flow is equated to that due to an equivalent nodal force vector F defined below.

The four steps mentioned above are combined together to yield, after eliminating qn and qτ,

(14–367)[ ] K F eqvφ ≡

where:

[K] = 3 x 3 equivalent unsymmetric element coefficient matrixφ = 3 x 1 nodal degrees of freedom, MAG or TEMPFeqv = 3 x 1 equivalent nodal force vector

The coefficient matrix [K] multiplied by the nodal DOF's φ represents the equivalent nodal load vector whichbrings the effects of the semi-infinite domain ΩB onto nodes I, J and K.

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As mentioned in the beginning, the INFIN47 can be used with magnetic scalar potential elements to solve 3-Dmagnetic scalar potential problems (MAG degree of freedom). Magnetic scalar potential elements incorporatethree different scalar potential formulations (see Section 5.1: Electromagnetic Field Fundamentals) selected withthe MAGOPT command:

1. Reduced Scalar Potential (accessed with MAGOPT,0)

2. Difference Scalar Potential (accessed with MAGOPT,2 and MAGOPT,3)

3. Generalized Scalar Potential (accessed with MAGOPT,1, MAGOPT,2, and then MAGOPT,3)

14.47.3. Reduced Scalar Potential

If there is no “iron” in the problem domain, the reduced scalar potential formulation can be used both in the FEand the BE regimes. In this case, the potential is continuous across FE-BE interface. If there is “iron” in the FE domain,the reduced potential formulation is likely to produce “cancellation errors”.

14.47.4. Difference Scalar Potential

If there is “iron” and current in the FE region and the problem domain is singly-connected, we can use the differ-ence potential formulation in order to avoid cancellation error. The formulation consists of two-step solutionprocedures:

1. Preliminary solution in the air domain (MAGOPT, 2)

Here the first step consists of computing a magnetic field Ho under the assumption that the magnetic

permeability of iron is infinity, thereby neglecting any saturation. The reduced scalar potential φ is usedin FE region and the total scalar potential ψ is used in BE region. In this case, the potential will be discon-tinuous across the FE-BE interface. The continuity condition of the magnetic field at the interface can bewritten as:

(14–368)−∇ ⋅ = −∇ ⋅ +ψ τ φ τ τ HsT

where:

τ = tangent vector at the interface along element edgeHs = magnetic field due to current sources

Integrating the above equation along the interface, we obtain

(14–369)ψ φ τp p s

T

p

pH dt

o

= − ∫

If we take ψ = φ at a convenient point po on the interface, then the above equation defines the potential

jump at any point p on the interface. Now, the total potential ψ can be eliminated from the problemusing this equation, leading to the computation of the additional load vector,

(14–370) [ ] f K gg =

where:

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Section 14.47: INFIN47 - 3-D Infinite Boundary

g H dti sT

p

p

o

i= ∫ τ

[K] = coefficient matrix defined with Equation 14–367

2. Total solution (air and iron) (MAGOPT, 3)

In this step the total field, H = Ho - ∇ ψ, is computed where H is the actual field and Ho is the field

computed in step 1 above. Note that the same relation given in Equation 5–39 uses φg in place of ψ. The

total potential ψ is used in both FE and BE regimes. As a result, no potential discontinuity exists at theinterface, but an additional load vector due to the field Ho must be computed. Since the magnetic flux

continuity condition at the interface of air and iron is:

(14–371)µ ψ µ ψ µI

Io

Ao o

T

n nH n

∂∂

− ∂∂

= −

where:

µo = magnetic permeability of free space (air)

µI = magnetic permeability of iron

The additional load vector may be computed as

(14–372) f N H n dsf o o

T

s= −∫ µ

where:

N = weighting functions

14.47.5. Generalized Scalar Potential

If there is iron and current in the FE domain and the domain is multiply-connected, the generalized potentialformulation can be used. It consists of three different steps.

1. Preliminary solution in the iron domain (MAGOPT, 1). This step computes a preliminary solution in theiron only. The boundary elements are not used for this step.

2. Preliminary solution in the air domain (MAGOPT, 2). This step is exactly the same as the step 1 of thedifference potential formulation.

3. Total solution (air and iron) (MAGOPT, 3) . This step is exactly the same as the step 2 of the differencepotential formulation.

14.48. CONTAC48 - 2-D Point-to-Surface Contact

This element is no longer supported.

14.49. CONTAC49 - 3-D Point-to-Surface Contact

This element is no longer supported.

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14.50. MATRIX50 - Superelement (or Substructure)

Integration PointsShape FunctionsMatrix or Vector

Same as the constituentelements

Same as the constituent elements

Stiffness, Conductivity, Stress Stiffness(used only when added to the StiffnessMatrix), Convection Surface Matrices;and Gravity, Thermal and Pressure/HeatGeneration and Convection SurfaceLoad Vectors

Same as the constituentelements

Same as the constituent elements reduceddown to the master degrees of freedom

Mass/Specific Heat and DampingMatrices

DistributionLoad Type

As input during generation runElement Temperature and Heat Generation Rate

As input during generation runPressure/Convection Surface Distribution

14.50.1. Other Applicable Sections

Superelements are discussed in Section 17.6: Substructuring Analysis.

14.51. SHELL51 - Axisymmetric Structural Shell

!

"

##

$

%

&

'

(*),+-./ -0 1

23)-4/ -50 1

67

Integration PointsShape FunctionsMatrix or Vector

3 along length9 thru thickness

Equation 12–29, Equation 12–30, and Equation 12–31. Ifextra shape functions are not included (KEYOPT(3) = 1):equations Equation 12–26, Equation 12–27, and Equa-tion 12–28.

Stiffness Matrix; and Thermal,Pressure, and Newton-Raph-son Load Vectors

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Section 14.51: SHELL51 - Axisymmetric Structural Shell

Integration PointsShape FunctionsMatrix or Vector

Same as stiffness matrixEquation 12–26, Equation 12–27, and Equation 12–28.Mass and Stress StiffnessMatrices

DistributionLoad Type

Linear thru thickness and along length, constant around circumferenceElement Temperature

Constant thru thickness, linear along length, constant around circumferenceNodal Temperature

Linear along length, constant around circumferencePressure

14.51.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 14.61: SHELL61 - Axisymmetric-Harmonic Structural Shell contains information also applicableto SHELL51.

14.51.2. Integration Point Locations for Nonlinear Material Effects

The locations and weighting factors for the nine point integration rule through the element thickness for thenonlinear material effects is given in Table 13.5: “Thru-Thickness Numerical Integration”. Nonlinear material valuesare only computed at the midpoint between the nodes. When these values are needed for other integrationpoints along the length, they are simply transferred from the midpoint. This is to avoid “sawtoothing effects".

14.51.3. Large Deflections

Unlike other line elements, SHELL51 uses the rotational strain approach (Kohnke(20)).

14.52. CONTAC52 - 3-D Point-to-Point Contact

Integration PointsShape FunctionsGeometryMatrix or Vector

NoneNoneNormal DirectionStiffness Matrix

NoneNoneSliding Direction

DistributionLoad Type

None - average used for material property evaluationElement Temperature

None - average used for material property evaluationNodal Temperature

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14.52.1. Other Applicable Sections

Section 14.12: CONTAC12 - 2-D Point-to-Point Contact has many aspects also valid for CONTAC52, includingnormal and sliding force determinations, rigid Coulomb friction (KEYOPT(1) = 1), and the force-deflection rela-tionship shown in Figure 14.4: “Force-Deflection Relations for Standard Case”.

14.52.2. Element Matrices

CONTAC52 may have one of three conditions: closed and stuck, closed and sliding, or open.

If the element is closed and stuck, the element stiffness matrix (in element coordinates) is:

(14–373)[ ]K

k k

k k

k k

k k

k k

k k

n n

s s

s s

n n

s s

s

l =

−−

−−

−−

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 ss

where:

kn = normal stiffness (input as KN on R command)

ks = sticking stiffness (input as KS on R command)

The Newton-Raphson load vector is:

(14–374) F

F

F

F

F

F

F

nr

n

sy

sz

n

sy

sz

l =−−

−

where:

Fn = normal force across gap (from previous iteration)

Fs = sticking force across gap (from previous iteration)

If the element is closed and sliding in both directions, the element stiffness matrix (in element coordinates) is:

(14–375)[ ]K

k k

k k

n n

n nl =

−

−

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

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Section 14.52: CONTAC52 - 3-D Point-to-Point Contact

and the Newton-Raphson load vector is the same as in Equation 14–374. For details on the unsymmetric option(NROPT,UNSYM), see Section 14.12: CONTAC12 - 2-D Point-to-Point Contact

If the element is open, there is no stiffness matrix or load vector.

14.52.3. Orientation of Element

For both small and large deformation analysis, the orientation of the element is unchanged. The element is orientedso that the normal force is in line with the original position of the two nodes.

14.53. PLANE53 - 2-D 8-Node Magnetic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–120QuadMagnetic Potential CoefficientMatrix; and Permanent Magnetand Applied Current Load Vectors 3Equation 12–99Triangle

Same as coefficient matrixEquation 12–120 and Equa-tion 12–122

Quad

Damping (Eddy Current) Matrix

Same as coefficient matrixEquation 12–99 and Equa-tion 12–102

Triangle

DistributionLoad Type

Bilinear across elementCurrent Density, Voltage Load andPhase Angle Distribution

References: Silvester et al.(72), Weiss et al.(94), Garg et al.(95)

14.53.1. Other Applicable Sections

Section 5.2: Derivation of Electromagnetic Matrices has a complete derivation of the matrices and load vectorsof a general magnetic analysis element. Section 11.1: Coupled Effects contains a discussion of coupled fieldanalyses.

14.53.2. Assumptions and Restrictions

A dropped midside node implies that the edge is straight and that the solution varies linearly along that edge.

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14.53.3. VOLT DOF in 2-D and Axisymmetric Skin Effect Analysis

KEYOPT(1) = 1 can be used to model skin effect problems. The corresponding DOFs are AZ and VOLT. Here, AZrepresents the z- or θ-component of the magnetic vector potential for 2-D or axisymmetric geometry, respectively.VOLT has different meanings for 2-D and axisymmetric geometry. The difference is explained below for a transientcase.

A skin effect analysis is used to find the eddy current distribution in a massive conductor when a source currentis applied to it. In a general 3-D case, the (total) current density J is given by

(14–376)

J

At t

= − ∂∂

− ∂ ∇∂

σ σ ν

where:

ν = (time-integrated) electric scalar potential

Refer to Section 5.3.2: Magnetic Vector Potential Results for definitions of other variables. For a 2-D massiveconductor, the z-component of J may be rewritten as:

(14–377)JAt

Vtz

z= − ∂∂

+ ∂ ∇∂

σ σ %

where ∆ %V may be termed as the (time-integrated) source voltage drop per unit length and is defined by:

(14–378)∆ %V z= − ⋅∇^ ν

For an axisymmetric massive conductor, the θ-component of J may be rewritten as

(14–379)JAt r

Vtθ

θσ σπ

= − ∂∂

+ ∂ ∇∂2

%

where the (time-integrated) source voltage drop in a full 2π radius is defined by

(14–380)∆ %V r= − ⋅∇2π θ ν^

When KEYOPT(1) = 1, the VOLT DOF represents the definition given by Equation 14–378 and Equation 14–380for a 2-D and axisymmetric conductor, respectively. Also, all VOLT DOFs in a massive conductor region must be

coupled together so that ∆ %V has a single value.

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Section 14.53: PLANE53 - 2-D 8-Node Magnetic Solid

14.54. BEAM54 - 2-D Elastic Tapered Unsymmetric Beam

Y

X

v x,u

Rigid offsets

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–4 and Equation 12–5Stiffness and Mass Matrices; andThermal Load Vector

NoneEquation 12–5Stress Stiffness and Foundation Stiff-ness Matrices; and Pressure Load Vec-tor

DistributionLoad Type

Linear thru thickness, linear along lengthElement Temperature

Constant thru thickness, linear along lengthNodal Temperature

Linear along lengthPressure

14.54.1. Derivation of Matrices

All matrices and load vectors are derived in the same way as for Section 14.44: BEAM44 - 3-D Elastic TaperedUnsymmetric Beam, except that they are reduced to 2-D. Further, the same assumptions and restrictions apply.

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14.55. PLANE55 - 2-D Thermal Solid

Y,v

X,R

Ks

J

I

L

t

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–111QuadConductivity Matrix and Heat Genera-tion Load Vector 1 if planar

3 if axisymmetricEquation 12–92Triangle

Same as conductivity matrixSame as conductivity matrix. Matrix is diag-onalized as described in Section 13.2:Lumped Matrices.

Specific Heat Matrix

2Same as conductivity matrix evaluated atthe face

Convection Surface Matrix and LoadVector

14.55.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of the element matrices and load vectors as well as heat fluxevaluations. Section 14.70: SOLID70 - 3-D Thermal Solid describes fluid flow in a porous medium, accessed inPLANE55 with KEYOPT(9) = 1.

14.55.2. Mass Transport Option

If KEYOPT(8) > 0, the mass transport option is included as described in Section 6.1: Heat Flow Fundamentals with

Equation 6–1 and by Ketm

of Equation 6–21. The solution accuracy is dependent on the element size. The accuracyis measured in terms of the non-dimensional criteria called the element Peclet number (Gresho(58)):

(14–381)PVL C

Kep=

ρ2

where:

V = magnitude of the velocity vectorL = element length dimension along the velocity vector directionρ = density of the fluid (input as DENS on MP command)Cp = specific heat of the fluid (input as C on MP command)

K = equivalent thermal conductivity along the velocity vector direction

The terms V, L, and K are explained more thoroughly below:

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Section 14.55: PLANE55 - 2-D Thermal Solid

(14–382)V V Vx y= +( ) /2 2 1 2

where:

Vx = fluid velocity (mass transport) in x direction (input as VX on R command)

Vy = fluid velocity (mass transport) in y direction (input as VY on R command)

Length L is calculated by finding the intersection points of the velocity vector which passes through the elementorigin and intersects at the element boundaries.

For orthotropic materials, the equivalent thermal conductivity K is given by:

(14–383)K K Km

K m Kx y

y x

= ++

( )/

1 2

2 2 2

1 2

where:

Kx, Ky = thermal conductivities in the x and y directions (input as KXX and KYY on MP command)

mVy= =slope of velocity vector in element coordinate systemVVx

(if KEYOPT(4) = 0)

For the solution to be physically valid, the following condition has to be satisfied (Gresho(58)):

(14–384)Pe < 1

This check is carried out during the element formulation and an error message is printed out if equation (14.431)is not satisfied. When this error occurs, the problem should be rerun after reducing the element size in the directionof the velocity vector.

14.56. HYPER56 - 2-D 4-Node Mixed u-P Hyperelastic Solid

Y,v

X,R,u

Ks

J

I

L

t

Z,w

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Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–103, Equation 12–104, andEquation 12–105

QuadStiffness and Mass Matrices;and Thermal Load Vector 3 if axisymmetric

1 if planeEquation 12–84, Equation 12–85, andEquation 12–86

Triangle

2Same as stiffness matrix, specialized to facePressure Load Vector

DistributionLoad Type

Bilinear across element, constant thru thickness or around circumferenceElement Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

References: Oden(123), Sussman(124)

14.56.1. Other Applicable Sections

For the basic formulation refer to Section 14.58: HYPER58 - 3-D 8-Node Mixed u-P Hyperelastic Solid. The hyper-elastic material model (Mooney-Rivlin) is described in Section 4.6: Hyperelasticity.

14.57. SHELL57 - Thermal Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–64. No variation thruthickness

QuadConductivity Matrix, HeatGeneration Load Vector, andConvection Surface Matrix andLoad Vector 1

Equation 12–90 No variation thruthickness

Triangle

Same as conductivitymatrix

Same as conductivity matrix. Matrix is diagonalized asdescribed in Section 13.2: Lumped Matrices

Specific Heat Matrix

14.57.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of the thermal element matrices and load vectors as well as heatflux evaluations.

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Section 14.57: SHELL57 - Thermal Shell

14.58. HYPER58 - 3-D 8-Node Mixed u-P Hyperelastic Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–191, Equation 12–192 , and Equation 12–193Stiffness and Mass Matrices;and Thermal Load Vector

2 x 2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

References: Oden(123), Sussman(124)

14.58.1. Other Applicable Sections

The hyperelastic material model (Mooney-Rivlin) is described in Section 4.6: Hyperelasticity.

14.58.2. Mixed Hyperelastic Element Derivation

A mixed formulation is used that utilizes a modified strain energy density containing hydrostatic pressure as anexplicit solution variable. Since it uses separate interpolations for the displacements and the hydrostatic pressure,it is referred to as the u-P (displacement-pressure) formulation. The essentials of the u-P formulation are summar-ized below. For details see references Oden and Kikuchi(123), Sussman and Bathe(124), and Zienkiewicz et al.(125).

14.58.3. Modified Strain Energy Density

The u-P formulation starts with a modified potential that explicitly includes the pressure variables:

(14–385)W Q W

KP P+ = − −1

22( )

where:

Q = energy augmentation due to volume constraint condition

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K = bulk modulusP = pressure obtainable from W alone

P = separately interpolated pressure (output as stress item HPRES)

The original potential, W, for a Mooney-Rivlin material, which would be applicable for slightly incompressiblerubber-like materials, is given by Equation 4–188. Note that the last term of Equation 4–188 provides the pressureP.

The displacements are discretized using standard isoparametric interpolations, whereas the pressure P is dis-cretized by a polynomial expansion of the following form without any association with any nodes.

(14–386)P P P s P t P st= + + + + − − −1 2 3 4

where:

s, t = element coordinates in natural spaceP1, P2, P3 = pressure degrees of freedom (DOFS)

Unlike the displacement DOFs, the pressure DOFs are not associated with any node, but exist only within anelement. The pressure DOFs are automatically introduced on element level and are condensed out when elementstiffness is created. They are not accessible to the users.

The number of pressure DOFS used by the interpolation function Equation 14–386 is one order lower than theone for strain calculation in the elements. They are listed in Table 14.5: “Number of Pressure DOFs and Interpol-ation Functions”.

Table 14.5 Number of Pressure DOFs and Interpolation Functions

Interpolation FunctionNumber of Pressure DOFsElement

Constant1HYPER56

Constant1HYPER58

Linear3HYPER74

Constant1HYPER158

14.58.4. Finite Element Matrices

The finite element matrices in terms of the incremental displacements and pressures are given by:

(14–387)

K K

K K

u

P

F R

R

uu uP

Pu PP

u

P

=

−

&& 0

where:

F = external nodal forces

, & &U P = displacement and pressure increments, respectivelly

Ru and Rp are the Newton-Raphson restoring force vectors (elsewhere referred to as Fnr):

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Section 14.58: HYPER58 - 3-D 8-Node Mixed u-P Hyperelastic Solid

(14–388)R

uW

KP P dvol S

Eu

diu

i volkl

vol

kl

i= ∂

∂− −

∫

= ∫

∂∂& &

12

2( ) (vvol)

(14–389)R

PW

KP P dvol

KP P

P

PiP

i vol vol i= ∂

∂− −

∫

= ∫ − ∂

∂& &1

212( ) ( ) dd vol( )

(14–390)

KRu

CE

ij

uu iu

j

klrsuu

vol

kl

= ∂∂

=

= ∫∂∂

displacement-only stiffness

&& & & &uEu

d vol SE

u ud vol

i

rs

jkl

vol

kl

i j

∂∂

+ ∫∂∂ ∂

( ) ( )2

(14–391)

KR

P

R

uK

ij ij

uP iu

j

jP

i

uP= ∂∂

=∂∂

= =& displacement-pressure coupled stiffness

= ∫∂

∂∂∂

∂∂

1K

PE

Eu

P

Pd vol

vol kl

kl

i j& & ( )

(14–392)

KR

P

P

P K

ij

PP iu

j

ivol

= ∂∂

=

= ∂∂

∫ −

&

&

pressure-only stiffness

1 ∂∂∂

P

Pd vol

j& ( )

In the above,

(14–393)S S

KP P

PE

kl klkl

= − − ∂∂

1( )

(14–394)C

WE E K

PE

PE K

P PP

E Eklrsuu

kl rs kl rs kl rs= ∂

∂ ∂− ∂

∂∂

∂− − ∂

∂ ∂

2 21 1( )

where:

Cklrsuu = augmented incremental moduli

The new augmented stress tensor Skl has the property that the pressure corresponding to these new stresses

Skl when added with the pressure computed directly from the displacement configuration equals the separatelyinterpolated pressure.

14.58.5. Incompressibility

The analysis of rubber-like materials poses computational difficulties in that these materials are almost incom-pressible. The fact that the volume changes very little while the material undergoes large strains often leads to

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displacement locking. In the u-P hyperelastic elements this difficulty is circumvented by enforcing the incom-pressibility constraint through a constraint equation. This constraint equation relates the separately interpolated

pressure (P ) (output as HPRES) to the pressure (P) computed from the displacements and attempts to maintainthe volume constraint in an average integrated sense over an element.

To be effective, there should be enough pressure DOFs P j , but the number of

P j DOFs in a model must besmaller than the number of unconstrained kinematic DOFs ui (UX, UY, etc.) in order to allow deformation to occur

at all. As a guideline, the number of unconstrained kinematic DOFs should be at least twice the number ofpressure DOFs for 2-D problems, and at least three times the number of pressure DOFs for axisymmetric or 3-Dproblems.

14.58.6. Instabilities in the Material Constitutive Law

Instability may sometimes occur due to real buckling, or it may occur due to the mathematical procedure usedin the formulation. For example, the application of a load in a single step that leads to a very large strain, say100% or more, may cause instability. Furthermore, if there is a complex variation of the hydrostatic pressure, thenumber of pressure DOFs may not be adequate to describe the behavior. This may lead to a local volume change,associated with a decrease in total energy. In those cases, local mesh refinement or the use of higher order ele-ments is recommended.

14.58.7. Existence of Multiple Solutions

For nonlinear problems, more than one stable solution may exist for a given set of boundary conditions. Thecase of a hollow hemisphere with zero prescribed loads is an example of such multiple solutions. Here the twoequilibrium solutions are: the undeformed stress-free state and the inverted self-equilibrating state. Stableequilibrium solutions do not pose any difficulty; however, if the equilibrium becomes unstable at some point(e.g., incipient buckling) during the analysis, the solution procedure might collapse.

14.59. PIPE59 - Immersed Pipe or Cable

θ

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Section 14.59: PIPE59 - Immersed Pipe or Cable

Integration PointsShape FunctionsOptionsMatrix or Vector

NoneEquation 12–15, Equa-tion 12–16, Equation 12–17,and Equation 12–18

Pipe Option (KEYOPT(1) ≠ 1)Stiffness Matrix; and Thermal,Pressure, and HydrostaticLoad Vectors

NoneEquation 12–6, Equation 12–7,and Equation 12–8

Cable Option (KEYOPT(1) = 1)

NoneEquation 12–16 and Equa-tion 12–17

Pipe Option (KEYOPT(1) ≠ 1)Stress Stiffness Matrix

NoneEquation 12–7 and Equa-tion 12–8

Cable Option (KEYOPT(1) = 1)

NoneEquation 12–15, Equa-tion 12–17, and Equa-tion 12–16

Pipe Option (KEYOPT(1) ≠ 1)with consistent mass matrix(KEYOPT(2) = 0)

Mass Matrix

NoneEquation 12–6, Equation 12–7,and Equation 12–8

Cable Option (KEYOPT(1) = 1)or reduced mass matrix (KEY-OPT(2) = 1)

2Same as stiffness matrixHydrodynamic Load Vector

DistributionLoad Type

Linear thru thickness or across diameter, and along lengthElement Temperature*

Constant across cross-section, linear along lengthNodal Temperature*

Linearly varying (in Z direction) internal and external pressure caused by hydro-static effects. Exponentially varying external overpressure (in Z direction) causedby hydrodynamic effects

Pressure

Note — * Immersed elements with no internal diameter assume the temperatures of the water.

14.59.1. Overview of the Element

PIPE59 is similar to PIPE16 (or LINK8 if the cable option (KEYOPT(1) = 1) is selected). The principal differences arethat the mass matrix includes the:

1. Outside mass of the fluid (“added mass”) (acts only normal to the axis of the element),

2. Internal structural components (pipe option only), and the load vector includes:

a. Hydrostatic effects

b. Hydrodynamic effects

14.59.2. Location of the Element

The origin for any problem containing PIPE59 must be at the free surface (mean sea level). Further, the Z axis isalways the vertical axis, pointing away from the center of the earth.

The element may be located in the fluid, above the fluid, or in both regimes simultaneously. There is a tolerance

of only

De8 below the mud line, for which

(14–395)D D te o i= + 2

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where:

ti = thickness of external insulation (input as TKIN on RMORE command)

Do = outside diameter of pipe/cable (input as DO on R command)

The mud line is located at distance d below the origin (input as DEPTH with TB,WATER (water motion table)).This condition is checked with:

(14–396)Z N d

De( ) > − +

←8

no error message

(14–397)Z N d

De( ) ≤ − +

←8

fatal error message

where Z(N) is the vertical location of node N. If it is desired to generate a structure below the mud line, the usercan set up a second material property for those elements using a greater d and deleting hydrodynamic effects.Alternatively, the user can use a second element type such as PIPE16, the elastic straight pipe element.

If the problem is a large deflection problem, greater tolerances apply for second and subsequent iterations:

(14–398)Z N d De( ) ( )> − + ←10 no error message

(14–399)− + ≥ > ←( ) ( ) ( )d D Z N de10 2 warning message

(14–400)− ≥ ←( ) ( )2d Z N fatal error message

where Z(N) is the present vertical location of node N. In other words, the element is allowed to sink into the mudfor 10 diameters before generating a warning message. If a node sinks into the mud a distance equal to the waterdepth, the run is terminated. If the element is supposed to lie on the ocean floor, gap elements must be provided.

14.59.3. Stiffness Matrix

The element stiffness matrix for the pipe option (KEYOPT(1) ≠ 1) is the same as for BEAM4 (Equation 14–10),except that:

[ ]( , ) [ ]( , ) [ ]( , ) [ ]( , ) [ ]( , ) [K K K K T KTl l l l l4 1 1 4 10 7 7 10 7 4= = = = =and KK K K TTl l l]( , ) [ ]( , ) [ ]( , )4 7 10 1 110= = = −

where:

TT =

0 if KEYOPT(1) = 0, 1 (standard option for torquebalanced cable or pipe)

if KEYOPT(1) = 2 (twist tentionG D D

LT o i( )3 3−

option for non-torque

balanced cable or pipe)

GT = twist-tension stiffness constant, which is a function of the helical winding of the armoring (input as

TWISTEN on RMORE command, may be negative)Di = inside diameter of pipe = Do - 2 tw

tw = wall thickness (input as TWALL on R command)

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Section 14.59: PIPE59 - Immersed Pipe or Cable

L = element length

A D Do i= − =π

42 2( ) cross-sectional area

I D Do i= − =π64

moment of inertia( )4 4

J = 2I

The element stiffness matrix for the cable option (KEYOPT(1) = 1) is the same as for LINK8.

14.59.4. Mass Matrix

The element mass matrix for the pipe option (KEYOPT(1) ≠ 1) and KEYOPT(2) = 0) is the same as for BEAM4

(Equation 14–11), except that [ ]Ml (1,1), [ ]Ml (7,7), [ ]Ml (1,7), and [ ]Ml (7,1), as well as M(4,4), M(10,10), M(4,10),and M(10,4), are multiplied by the factor (Ma /Mt).

where:

Mt = (mw + mint + mins + madd) L = mass/unit length for motion normal to axis of element

Ma = (mw + mint + mins) L= mass/unit length for motion parallel to axis of element

m ( ) ( )win

o iD D= −14

2 2ε ρ π

ρ = density of the pipe wall (input as DENS on MP command)

εin = initial strain (input as ISTR on RMORE command)mint = mass/unit length of the internal fluid and additional hardware (input as CENMPL on RMORE command)

m ( ) ( )ins in i e oD D= − −14

2 2ε ρ π

ρi = density of external insulation (input as DENSIN on RMORE command)

m C Dadd in I w e= −( )14

2ε ρ π

CI = coefficient of added mass of the external fluid (input as CI on RMORE command)

ρw = fluid density (input as DENSW with TB,WATER)

The element mass matrix for the cable option (KEYOPT(1) = 1) or the reduced mass matrix option (KEYOPT(2) ≠

0) is the same form as for LINK8 except that [ ]Ml (1,1), [ ]Ml (4,4), [ ]Ml (1,4) and [ ]Ml (4,1) are also multiplied bythe factor (Ma/Mt).

14.59.5. Load Vector

The element load vector consists of two parts:

1. Distributed force per unit length to account for hydrostatic (buoyancy) effects (F/Lb) as well as axial

nodal forces due to internal pressure and temperature effects Fx.

2. Distributed force per unit length to account for hydrodynamic effects (current and waves) (F/Ld).

The hydrostatic and hydrodynamic effects work with the original diameter and length, i.e., initial strain and largedeflection effects are not considered.

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14.59.6. Hydrostatic Effects

Hydrostatic effects may affect the outside and the inside of the pipe. Pressure on the outside crushes the pipeand buoyant forces on the outside tend to raise the pipe to the water surface. Pressure on the inside tends tostabilize the pipe cross-section.

The buoyant force for a totally submerged element acting in the positive z direction is:

(14–401) / F L C D gb b w e= ρ π

42

where: F/Lb = vector of loads per unit length due to buoyancy

Cb = coefficient of buoyancy (input as CB on RMORE command)

g = acceleration vector

Also, an adjustment for the added mass term is made.

The crushing pressure at a node is:

(14–402)P gz Pos

w oa= − +ρ

where:

Pos

= crushing pressure due to hydrostatic effectsg = acceleration due to gravityz = vertical coordinate of the node

Poa

= input external pressure (input on SFE command)

The internal (bursting) pressure is:

(14–403)P g z S Pi o fo i

a= − − +ρ ( )

where:

Pi = internal pressure

ρo = internal fluid density (input as DENSO on R command)

Sfo = z coordinate of free surface of fluid (input as FSO on R command)

Pia

= input internal pressure (input as SFE command)

To ensure that the problem is physically possible as input, a check is made at the element midpoint to see if thecross-section collapses under the hydrostatic effects. The cross-section is assumed to be unstable if:

(14–404)P PE t

Dos

iw

o− >

−

4 1

22

3

( )ν

where:

E = Young's modulus (input as EX on MP command)

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ν = Poisson's ratio (input as PRXY or NUXY on MP command)

The axial force correction term (Fx) is computed as

(14–405)F AEx x= ε

where εx, the axial strain (see Equation 2–12) is:

(14–406)ε α σ ν σ σx x h rT

E= + − +∆ 1

( ( ))

where:

α = coefficient of thermal expansion (input as ALPX on MP command)∆T = Ta - TREF

Ta = average element temperature

TREF = reference temperature (input on TREF command)

σx = axial stress, computed below

σh = hoop stress, computed below

σr = radial stress, computed below

The axial stress, assuming the ends are closed, is:

(14–407)σx

i i o o

o i

P D P D

D D= −

−

2 2

2 2

and using the Lamé stress distribution,

(14–408)σh

i i o oi o

i o

o i

P D P DD D

DP P

D D=

− + −

−

2 22 2

2

2 2

( )

(14–409)σr

i i o oi o

i o

o i

P D P DD D

DP P

D D=

− − −

−

2 22 2

2

2 2

( )

where:

P P Po os

od= +

Pod

= hydrodynamic pressure, described belowD = diameter being studied

Pi and Po are taken as average values along each element. Combining Equation 14–406 thru Equation 14–409.

(14–410)ε α ν

xi i o o

o i

TE

P D P D

D D= + − −

−∆ 1 2 2 2

2 2

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Note that if the cross-section is solid (Di = 0.), Equation 14–408 reduces to:

(14–411)ε α ν

x oTE

P= − −∆ 1 2

14.59.7. Hydrodynamic Effects

All input quantities referred to in this section not otherwise identified comes from the TBDATA commands usedwith TB,WATER. Hydrodynamic effects may occur because the structure moves in a motionless fluid, the structureis fixed but there is fluid motion, or both the structure and fluid are moving. The fluid motion consists of twoparts: current and wave motions. The current is input by giving the current velocity and direction (input as W(i)and θ(i)) at up to eight different vertical stations (input as Z(i)). The velocity and direction are interpolated linearlybetween stations. The current is assumed to flow horizontally only. The wave may be input using one of fourwave theories in Table 14.6: “Wave Theory Table” (input as KWAVE with TB,WATER).

Table 14.6 Wave Theory Table

Description of Wave TheoryKWAV

Small amplitude wave theory, modified with empirical depth decay function, (Wheeler(35))0

Small amplitude wave theory, unmodified (Airy wave theory), (Wheeler(35))1

Strokes fifth order wave theory, (Skjelbreia et al.(31))2

Steam function wave theory, (Dean(59))3

The free surface of the wave is defined by

(14–412)η η βs i

i

Ni

i

N

iw w H

cos= ∑ = ∑= =1 1 2

where:

ηs = total wave height

Nw = =≠

number of wave componentsnumber of waves K 2

5

if w

K 2if w =

Kw = wave theory key (input as KWAVE with TB,WATER)

ηi = wave height of component i

Hi = ==

surface coefficientnumber of waves if K 0 or 1

derived w

ffrom other input if K 2w =

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Section 14.59: PIPE59 - Immersed Pipe or Cable

β

πλ τ

φ

π

i

i i

iR t

=

+ +

2

360

2

if KEYOPT(5) = 0 and K = 0 or 1w

RR ti

i i

iλ τ

φ+ +

360( ) if KEYOPT(5) = 0 and K = 2 or 3w

0.0 iif KEYOPT(5) = 1

if KEYOPT(5) = 2

if KEYOPT(5) = 3

if

π

π

π

2

2−

KKEYOPT(5) = 4

R = radial distance to point on element from origin in the X-Y plane in the direction of the wave

λi = = wave lengthinput as WL(i) if WL(i) > 0.0 and if Kw = 0 or 1

otherwise derived from equation (14.460)

t = time elapsed (input as TIME on TIME command) (Note that the default value of TIME is usually not desired.If zero is desired, 10-12 can be used).

ττ

i = =≠

wave periodinput as (i) if K 3

derived from other inpw

uut if K 3 w =

φi = phase shift = input as φ(i)

If λi is not input (set to zero) and Kw < 2, λi is computed iteratively from:

(14–413)λ λ π

λi id

i

d=

tanh

2

where:

λi = output quantity small amplitude wave length

λ τπi

d ig= =( )2

2output quantity deep water wave length

g = acceleration due to gravity (Z direction) (input on ACEL command)d = water depth (input as DEPTH with TB,WATER)

Each component of wave height is checked that it satisfies the “Miche criterion” if Kw ≠3. This is to ensure thatthe wave is not a breaking wave, which the included wave theories do not cover. A breaking wave is one thatspills over its crest, normally in shallow water. A warning message is issued if:

(14–414)H Hi b>

where:

Hd

b ii

=

=0 142

2. tanhλ π

λheight of breaking wave

When using wave loading, there is an error check to ensure that the input acceleration does not change afterthe first load step, as this would imply a change in the wave behavior between load steps.

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For Kw = 0 or 1, the particle velocities at integration points are computed as a function of depth from:

(14–415)v

cosh k Zfsinh k d

vRi

ii

N

ii D

wr r= ∑ +

=

( )( )1

2πτ

η

(14–416)v

sinh k Zfsinh k d

Zi

ii

N

iwr

&= ∑=

( )( )1

η

where:

vRr

= radial particle velocity

vZr

= vertical particle velocityki = 2π/λi

Z = height of integration point above the ocean floor = d+Z&ηi = time derivative of ηi

vDr

= drift velocity (input as W with TB,WATER)

f

dd s= +

=

=

η

1.0

if K 0 (Wheeler(35))

if K 1 (small amplitude

w

w wwave theory)

The particle accelerations are computed by differentiating vRr

and vZr

with respect to time. Thus:

(14–417)&r

&vcosh k Zfsinh k d

CRi

ii

N

ii i

w= ∑

=

( )( )

( )1

2πτ

η η

(14–418)&r

&vsinh k Zfsinh k d

CZi

ii

N

i ii i

w= ∑

−

=

( )( )1

2 22

πτ

πτ

η η τπ

where:

C

Zd

ds

i s= +=

=

&ηλ η2

0 0

2Π

( )

.

if K 0(Wheeler(35))

if K 1(small ampli

w

w ttude wave theory)

Expanding equation 2.29 of the Shore Protection Manual(43) for a multiple component wave, the wave hydro-dynamic pressure is:

(14–419)P g

coshZ

coshd

od

w ii

N i

i

w= ∑

=ρ η

πλ

πλ

1

2

2

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However, use of this equation leads to nonzero total pressure at the surface at the crest or trough of the wave.Thus, Equation 14–419 is modified to be:

(14–420)P g

coshZd

d

coshd

od

w ii

N i s

i

w= ∑

+

=ρ η

πλ η

πλ

1

2

2

which does result in a total pressure of zero at all points of the free surface. This dynamic pressure, which is cal-culated at the integration points during the stiffness pass, is extrapolated to the nodes for the stress pass. Thehydrodynamic pressure for Stokes fifth order wave theory is:

(14–421)P g

coshZ

coshd

od

w ii

i

i

= ∑

=ρ η

πλ

πλ

1

52

2

Other aspects of the Stokes fifth order wave theory are discussed by Skjelbreia et al. (31). The modification assuggested by Nishimura et al.(143) has been included. The stream function wave theory is described by Dean(59).

If both waves and current are present, the question of wave-current interaction must be dealt with. Three optionsare made available thru Kcr (input as KCRC with TB,WATER):

For Kcr = 0, the current velocity at all points above the mean sea level is simply set equal to Wo, where Wo is the

input current velocity at Z = 0.0. All points below the mean sea level have velocities selected as though therewere no wave.

For Kcr = 1, the current velocity profile is “stretched” or “compressed” to fit the wave. In equation form, the Z co-

ordinate location of current measurement is adjusted by

(14–422)Z j Z j

dd

s′ = +( ) ( )

η

where:

Z(j) = Z coordinate location of current measurement (input as Z(j))Z(j) = adjusted value of Z(j)

For Kcr = 2, the same adjustment as for Kcr = 1 is used, as well as a second change that accounts for “continuity.”

That is,

(14–423)W j W j

dd s

′ =+

( ) ( )η

where:

W(j) = velocity of current at this location (input as W(j))W(j) = adjusted value of W(j)

These three options are shown pictorially in Figure 14.38: “Velocity Profiles for Wave-Current Interactions”.

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Figure 14.38 Velocity Profiles for Wave-Current Interactions

"! $#%'&)(*

+ ,-"! $#%'&/.0*

0 12"! $#%'&43*

5 6 0 798: ; < # = = >%7

To compute the relative velocities ( &un , &ut ), both the fluid particle velocity and the structure velocity mustbe available so that one can be subtracted from the other. The fluid particle velocity is computed using relation-ships such as Equation 14–415 and Equation 14–416 as well as current effects. The structure velocity is availablethrough the Newmark time integration logic (see Section 17.2: Transient Analysis).

Finally, a generalized Morison's equation is used to compute a distributed load on the element to account forthe hydrodynamic effects:

(14–424)

/

F L CD

u u C D v

CD

u u

d D we

n n M w e n

T we

t t

= +

+

ρ ρ π

ρ

2 4

2

2& & &

& &

where:

F/Ld = vector of loads per unit length due to hydrodynamic effects

CD = coefficient of normal drag (see below)

ρw = water density (mass/length3) (input as DENSW on TB,WATER)

De = outside diameter of the pipe with insulation (length)

&un = normal relative particle velocity vector (length/time)CM = coefficient of inertia (input as CM on R command)

&vn = normal particle acceleration vector (length/time2)CT = coefficient of tangential drag (see below)

&ut = tangential relative particle velocity vector (length/time)

Two integration points along the length of the element are used to generate the load vector. Integration pointsbelow the mud line are simply bypassed. For elements intersecting the free surface, the integration points aredistributed along the wet length only. If the reduced load vector option is requested (KEYOPT(2) = 2), the momentterms are set equal to zero.

The coefficients of drag (CD,CT) may be defined in one of two ways:

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Section 14.59: PIPE59 - Immersed Pipe or Cable

1. They may be input as fixed numbers using the real constant table (input as CD and CT on R and RMOREcommands), or

2. They may be input as functions of Reynolds number using the water motion table (input as RE, CD, andCT).

The dependency on Reynolds number (Re) may be expressed as:

(14–425)C f ReD D= ( )

where:

fD = functional relationship (input on the water motion table as RE and CD with TB,WATER)

Re uD

ne w= & ρµ

µ = viscosity (input as VISC on MP command)

and

(14–426)C f ReT T=

where:

fT = functional relationship (input on the water motion table as RE and CT with TB,WATER)

Re = &uD

te wρµ

Temperature-dependent quantity may be input as µ, where the temperatures used are those given by inputquantities T(i) of the water motion table.

14.59.8. Stress Output

The below two equations are specialized either to end I or to end J.

The stress output for the pipe format (KEYOPT(1) ≠ 1), is similar to PIPE16 (Section 14.16: PIPE16 - Elastic StraightPipe). The average axial stress is:

(14–427)σx

n i i o o

o i

FA

D P D P

D D= + −

−

2 2

2 2

where:

σx = average axial stress (output as SAXL)

Fn = axial element reaction force (output as FX, adjusted for sign)

Pi = internal pressure (output as the first term of ELEMENT PRESSURES)

Po = external pressure = P Pos

od+ (output as the fifth term of the ELEMENT PRESSURES)

and the hoop stress is:

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(14–428)σh

i i o o i

o i

P D P D D

D D=

− +−

2 2 2 2

2 2( )

where:

σh = hoop stress at the outside surface of the pipe (output as SH)

Equation 14–428 is a specialization of Equation 14–408. The outside surface is chosen as the bending stressesusually dominate over pressure induced stresses.

All stress results are given at the nodes of the element. However, the hydrodynamic pressure had been computedonly at the two integration points. These two values are then used to compute hydrodynamic pressures at thetwo nodes of the element by extrapolation.

The stress output for the cable format (KEYOPT(1) = 1 with Di = 0.0) is similar to that for LINK8 (Section 14.8: LINK8

- 3-D Spar (or Truss)), except that the stress is given with and without the external pressure applied:

(14–429)σxI o

FA

P= +l

(14–430)σeI

FA

= l

(14–431)F Aa xI= σ

where:

σxI = axial stress (output as SAXL)

σeI = equivalent stress (output as SEQV)

Fl = axial force on node (output as FX)Fa = axial force in the element (output as FAXL)

14.60. PIPE60 - Plastic Curved Pipe (Elbow)

Integration PointsShape FunctionsMatrix or Vector

NoneNo shape functions are explicitly used. Rather, a flexibilitymatrix similar to that developed by Chen (4) is invertedand used.

Stiffness Matrix

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Section 14.60: PIPE60 - Plastic Curved Pipe (Elbow)

Integration PointsShape FunctionsMatrix or Vector

NoneNo shape functions are used. Rather a lumped massmatrix using only translational DOF is used.

Mass Matrix

8 around circumference ateach end of the element. Thepoints are located midwaybetween the inside and out-side surfaces

No shape functions are explicitly used. See developmentbelow.

Pressure, Thermal, andNewton-Raphson LoadVector

DistributionLoad Type

Bilinear across cross-section, linear along lengthElement Temperature

Constant across cross-section, linear along lengthNodal Temperature

Internal and External: constant along length and around circumference. Lateral: variestrigonometrically along length

Pressure

14.60.1. Assumptions and Restrictions

The radius/thickness ratio is assumed to be large.

14.60.2. Other Applicable Sections

The stiffness and mass matrices are identical to those derived for Section 14.18: PIPE18 - Elastic Curved Pipe (Elbow).Section 14.16: PIPE16 - Elastic Straight Pipe discusses some aspects of the elastic stress printout.

14.60.3. Load Vector

The element load vector is computed in a linear analysis by:

(14–432) [ ] F K uFl l+

and in a nonlinear (Newton-Raphson) analysis by:

(14–433) [ ]( )F K u uFnl l+ − −1

where:

Fl = element load vector (in element coordinates) (applied loads minus Newton-Raphson restoring force)from previous iteration

[ ]Kl = element stiffness matrix (in element coordinates)

uF = induced nodal displacements in the element (see Equation 14–434)un-1 = displacements of the previous iteration

The element coordinate system is a cylindrical system as shown in Figure 14.39: “3-D Plastic Curved Pipe ElementGeometry”.

The induced nodal displacement vector uF is defined by:

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(14–434)

sin cos

sin

( )

( )

( )

u

R

R

RD

F

jj

jj

m j

+

− ∑

− ∑

=

=

4 4 4

0

4 4

4

1

1

8

2 1

1

8

1

θ θ ε

θ ε

θ γjj

m

j

jj

m

j

jj

RD

RD

R

=

=

=

∑

∑

∑

1

8

1

1

8

1

1

8

6

6

4 4 4

θ ε

β

θ ε

β

θ θ

( )

( )

cos

sin

sin cos εε

θ ε

θ γ

θ ε

jj

jj

m jj

m

j

R

RD

RD

( )

( )

( )

( )

sin

2

1

8

2 2

1

8

2

1

8

2

0

4 4

4

6

=

=

=

∑

− ∑

− ∑

−ccos

sin

( )

β

θ ε

β

jj

m

j

jj

RD

=

=

∑

− ∑

1

8

2

1

8

6

≠ ≠

≠ ≠

≠ ≠

j j

j j

j j

j

2 6

4 8

2 6

,

,

,

≠≠ ≠4 8, j

where:

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Section 14.60: PIPE60 - Plastic Curved Pipe (Elbow)

ε ε ε ε ε εjth

xpr

xpl

xcr

xsw( )1 = + + ++ at end I

ε ε ε ε ε εjth

xpr

xpl

xcr

xsw( )2 = + + ++ at end J

γ γ γj xhpr

xhcr( )1 = + at end I

γ γ γj xhpr

xhcr( )2 = + at end J

εth = α(Tj - TREF) (= thermal strain)

α = thermal coefficient of expansion (input as ALPX on MP command)Tj = temperature at integration point j

εxpr

= axial strain due to pressure (see Equation 14–104)

εxpl

= plastic axial strain (see Section 4.1: Rate-Independent Plasticity)

εxcr

= axial creep strain (see Section 4.2: Rate-Dependent Plasticity)

εxsw

= swelling strain (see Section 4.4: Nonlinear Elasticity)

γxhpl

= plastic shear strain (see Section 4.1: Rate-Independent Plasticity)

γxhcr

= creep shear strain (see Section 4.2: Rate-Dependent Plasticity)R = radius of curvature (input as RADCUR on R command)Dm = 1/2 (Do + Di) (= average diameter)

Do = outside diameter (input as OD on R command)

Di = Do - 2t ( = inside diameter)

t = thickness (input as TKWALL on R command)θ = subtended angle of the elbowβj = angular position of integration point j on the circumference Figure 14.40: “Integration Point Locations

at End J” (output as ANGLE)

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Figure 14.39 3-D Plastic Curved Pipe Element Geometry

There are eight integration points around the circumference at each end of the element, as shown in Fig-ure 14.40: “Integration Point Locations at End J”. The assumption has been made that the elbow has a large radius-to-thickness ratio so that the integration points are located at the midsurface of the shell. Since there are integ-ration points only at each end of the element, the subtended angle of the element should not be too large. Forexample, if there are effects other than internal pressure and in-plane bending, the elements should have asubtended angle no larger than 45°.

Figure 14.40 Integration Point Locations at End J

!

"#$&%('

)+*# ,

!

14.60.4. Stress Calculations

The stress calculations take place at each integration point, and have a different basis than for PIPE18, the elasticelbow element. The calculations have three phases:

1. Computing the modified total strains (ε').

2. Using the modified total strains and the material properties, computing the change in plastic strains andthen the current elastic strains.

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Section 14.60: PIPE60 - Plastic Curved Pipe (Elbow)

3. Computing the current stresses based on the current elastic strains.

Phase 2 is discussed in Section 4.1: Rate-Independent Plasticity. Phase 1 and 3 are discussed below. Phase 1: Themodified total strains at an integration point are computed as:

(14–435) [ ] ′ = −ε σD b1

where:

′ =

′

′

′

′

ε

ε

ε

ε

γ

xd

hd

xh

r

[ ]

( )

D

E E E

E E E

E E E

E

− =

− −

− −

− −

+

1

10

10

10

0 0 02 1

ν ν

ν ν

ν ν

ν

x, h, r = subscripts representing axial, hoop, and radial directions, respectivelyE = Young's modulus (input as EX on MP command)ν = Poisson's ratio (input as PRXY or NUXY on MP command)

σb, the integration point stress vector before plasticity computations, is defined as:

(14–436) σ

σ

σ

σ

τ

b

x

h

r

xh

=

These terms are defined by:

(14–437)σx

xw y z z

i i o oF

AS M S M

DP D Pty= + = = −

4

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(14–438)σ ν νφ

φh y z zo

j

jS M S M

Dt

R r

R rPy= + + −

+

+

2

25

12

sin

sin( ii oP− )

(14–439)σr

i oP P= − +2

(14–440)τ β βxh w y j z j

x x

AF F

S M= − + −22

( cos sin )

where:

Fy, Fz, Mx = forces on element at node by integration point (see Equation 14–441 below)

A D Dwo i= −π

42 2( )

SD

D Dx

o i

o=−

324 4π( )

S S C Cy x j j j= − + − +(sin (( . . )sin . sin ))φ φ φ2 11 5 18 75 3 11 25 5

S S C Cz x j j j= + − +(cos (( . . )cos . cos ))φ φ φ2 11 5 18 75 3 11 25 5

φ β πj j= −

2

rD Do i= +

4

Pi = internal pressure (input on SFE command)

Po = external pressure (input on SFE command)

C CPRErt1 3

22

17 600 480= + +

CC C C

2 24 1

1

1 6 25 4 51

=− − −( ( . . ))ν

CRt

r3

2 21=

− ν

C CPRErt4 3

22

5 6 24= + +

P = Pi - Po

Note that Sy and Sz are expressed in three-term Fourier series around the circumference of the pipe cross-section.

These terms have been developed from the ASME Code(60). Note also that φj is the same angle from the element

y axis as βj is for PIPE20. The forces on both ends of the element (Fy, Mx, etc.) are computed from:

(14–441) [ ]([ ] )F T K u Fe R ep

e= ∆ − l

where:

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Section 14.60: PIPE60 - Plastic Curved Pipe (Elbow)

F F Me xI zJT= =… forces on element in element coordinate ssystem

[TR] = global to local conversion matrix (note that the local x axis is not straight but rather is curved along

the centerline of the element)[Ke] = element stiffness matrix (global Cartesian coordinates)

∆ue = element incremental displacement vector

Phase 3: Performed after the plasticity calculations, Phase 3 is done simply by:

(14–442) [ ] σ ε= D e

where:

εe = elastic strain after the plasticity calculations

The σ vector, which is used for output, is defined with the same terms as in Equation 14–436. But lastly, σr is

redefined by Equation 14–439 as this stress value must be maintained, regardless of the amount of plastic strain.

As long as the element remains elastic, additional printout is given during the solution phase. The stress intens-ification factors (Cσ) of PIPE18 are used in this printout, but are not used in the printout associated with the plastic

stresses and strains. The maximum principal stresses, the stress intensity, and equivalent stresses are compared(and replaced if necessary) to the values of the plastic printout at the eight positions around the circumferenceat each end. Also, the elastic printout is based on outer-fiber stresses, but the plastic printout is based on mid-thickness stresses. Further, other thin-walled approximations in Equation 14–437 and Equation 14–438 are notused by the elastic printout. Hence some apparent inconsistency appears in the printout.

14.61. SHELL61 - Axisymmetric-Harmonic Structural Shell

Integration PointsShape FunctionsMatrix or Vector

3 along lengthEquation 12–35, Equation 12–36, and Equation 12–37. Ifextra shape functions are not included (KEYOPT(3) = 1):Equation 12–32, Equation 12–33, and Equation 12–34

Stiffness Matrix; and Thermaland Pressure Load Vectors

Same as stiffness matrixEquation 12–26, Equation 12–27, and Equation 12–28Mass and Stress StiffnessMatrices

DistributionLoad Type

Linear through thickness and along length, harmonic around circumferenceElement Temperature

Constant through thickness, linear along length, harmonic around circumferenceNodal Temperature

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DistributionLoad Type

Linear along length, harmonic around circumferencePressure

Reference: Zienkiewicz(39)

14.61.1. Other Applicable Sections

Chapter 2, “Structures” discusses fundamentals of linear elements. Section 14.25: PLANE25 - Axisymmetric-Har-monic 4-Node Structural Solid has a discussion on temperature, applicable to this element.

14.61.2. Assumptions and Restrictions

The material properties are assumed to be constant around the entire circumference, regardless of temperaturedependent material properties or loading.

14.61.3. Stress, Force, and Moment Calculations

Element output comes in two forms:

1. Stresses as well as forces and moments per unit length: This printout is controlled by the KEYOPT(6). Thethru-the-thickness stress locations are shown in Figure 14.41: “Stress Locations”. The stresses are computedusing standard procedures as given in Section 2.3: Structural Strain and Stress Evaluations. The stressesmay then be integrated thru the thickness to give forces per unit length and moments per unit lengthat requested points along the length:

(14–443)T tx x c

= σ

(14–444)T tz z c

= σ

(14–445)T txz xz c

= σ

(14–446)Mt

x x c x c= −( )σ σ2

12

(14–447)Mt

z z c z c= −( )σ σ

2

12

(14–448)Mt

xz xz c xz c= −( )σ σ

2

12

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Section 14.61: SHELL61 - Axisymmetric-Harmonic Structural Shell

Figure 14.41 Stress Locations

φ

σ

σ

σ

where:

Tx, Tz, Txz, Mx, Mz, Mxz = resultant forces and moments (output as TX, TZ, TXZ, MX, MZ, MXZ, respectively)

t = thickness (input as TK(I), TK(J) on R command)σx, σy, σz, σxz = stresses (output as SX, SY, SZ, and SXZ, respectively)

σ σ σx c x t x b= + =( ) 2 x stress at centerplane (also nodal locationns)

σx t= x stress at top

σx b= x stress at bottom

2. Forces and moments on a circumference basis: This printout is controlled by KEYOPT(4). The values arecomputed using:

(14–449) [ ] ([ ] )F T K u F FRT

e e eth

epr

l = − −

where:

F F F F M F F F Mxr

yr

zr

zr

xr

yr

zr

zr T

l =

, , , , , , , , (1 1 1 1 2 2 2 2 output as MFOR and MMOM)

[TR] = local to global transformation matrix

[Ke] = element stiffness matrix

ue = nodal displacements

Feth = element thermal load vector

Fepr = element pressure load vector

Another difference between the two types of output are the nomenclature conventions. Since the first groupof output uses a shell nomenclature convention and the second group of output uses a nodal nomenclature

convention, Mz and Mzr

represent moments in different directions.

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The rest of this subsection will describe some of the expected relationships between these two methods ofoutput at the ends of the element. This is done to give a better understanding of the terms, and possibly detectpoor internal consistency, suggesting that a finer mesh is in order. It is advised to concentrate on the primaryload carrying mechanisms. In order to relate these two types of output in the printout, they have to be requestedwith both KEYOPT(6) > 1 and KEYOPT(4) = 1. Further, care must be taken to ensure that the same end of theelement is being considered.

The axial reaction force based on the stress over an angle ∆β is:

(14–450)F

y

tR y dyx

r x t x b x t x bc

t

t

=+

+−

−

−∫

( ) ( )( sin )

σ σ σ σβ φ

22

2

∆

or

(14–451)F R t sin

txr x t x b

c x t x b=

+− −

∆β

σ σσ σ φ

( )( )

2 12

2

where:

Rc = radius at midplane

t = thickness

The reaction moment based on the stress over an angle ∆β is:

(14–452)M

y

ty R y dyx

r x t x b x t x bc

t

t

=+

+−

−

−∫

( ) ( )( sin )

σ σ σ σβ φ

22

2

∆

or

(14–453)M

t sinR

txr x t x b

x t x b c= −+

+ −

∆β

σ σ φ σ σ( )

( )2 12 12

3 2

Since SHELL61 computes stiffness matrices and load vectors using the entire circumference for axisymmetric

structures, ∆β = 2π. Using this fact, the definition of σx c , and Equation 14–443 and Equation 14–446, Equa-

tion 14–451 and Equation 14–453 become:

(14–454)F R T sin Mxr

c x x= −2π φ( )

(14–455)M

t sinT R Mz

rx c x= − +

2

12

2π φ

As the definition of φ is critical for these equations, Figure 14.42: “Element Orientations” is provided to show φin all four quadrants.

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Section 14.61: SHELL61 - Axisymmetric-Harmonic Structural Shell

Figure 14.42 Element Orientations

φ

φ

φ

φ

In a uniform stress (σx) environment, a reaction moment will be generated to account for the greater material

on the outside side. This is equivalent to moving the reaction point outward a distance yf. yf is computed by:

(14–456)y

M

Ff

zr

xr

=

Using Equation 14–454 and Equation 14–455 and setting Mx to zero gives:

(14–457)y

tf

Rc= −

2

12

sinφ

14.62. SOLID62 - 3-D Magneto-Structural Solid

Chapter 14: Element Library

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Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–194, Equation 12–195, and Equa-tion 12–196

Magnetic Vector PotentialCoefficient, and Damping (EddyCurrent) Matrices; and Perman-ent Magnet and Applied Cur-rent Load Vector

2 x 2 x 2

Equation 12–191, Equation 12–192, and Equa-tion 12–193 or, if modified extra shape functions areincluded (KEYOPT(1) = 0) and element has 8 uniquenodes Equation 12–206, Equation 12–207, and Equa-tion 12–208

Stiffness Matrix and ThermalLoad Vector

2 x 2 x 2Equation 12–191, Equation 12–192 and Equa-tion 12–193

Mass and Stress StiffnessMatrices

2 x 2 x 2Same as damping matrixMagnetic Force Load Vector

2 x 2Equation 12–57 and Equation 12–58QuadPressure Load Vector

3Equation 12–38 and Equation 12–39Triangle

DistributionLoad Type

Trilinear thru elementCurrent Density and Phase Angle

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

References: Wilson(38), Taylor et al.(49), Coulomb(76), Biro et al.(120)

14.62.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 5.2: Derivation of Electromagnetic Matrices and Section 5.3: Electromagnetic Field Evaluationscontain a discussion of the 2-D magnetic vector potential formulation which is similar to the 3-D formulation ofthis element.

14.63. SHELL63 - Elastic Shell

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Section 14.63: SHELL63 - Elastic Shell

Integration PointsShape FunctionsMatrix or Vector

2 x 2

Equation 12–75 and Equa-tion 12–76 (and, if modified extrashape functions are included(KEYOPT(3) = 0) and element has4 unique nodes, Equation 12–78,Equation 12–79, and Equa-tion 12–80

Membrane / Quad

Stiffness Matrix and ThermalLoad Vector

1Equation 12–51, Equation 12–52,and Equation 12–53

Membrane / Triangle

3 (for each triangle)Four triangles that are overlaidare used. These subtrianglesrefer to Equation 12–53

Bending

2 x 2Equation 12–57, Equation 12–58,and Equation 12–59

Membrane / Quad

Mass, Foundation Stiffnessand Stress Stiffness Matrices

1Equation 12–38, Equation 12–39,and Equation 12–40

Membrane / Triangle

3 (for each triangle)

Four triangles that are overlaidare used. These triangles connectnodes IJK, IJL, KLI, and KLJ. w isdefined as given in Zien-kiewicz(39)

Bending

None

One-sixth (one- third for tri-angles) of the total pressuretimes the area is applied to eachnode normal of each subtriangleof the element

Reduced shell pressureloading (KEYOPT(6) =0) (Load vector ex-cludes moments)Transverse Pressure Load

Vector

Same as mass matrixSame as mass matrix

Consistent shell pres-sure loading (KEY-OPT(6) = 2) (Load vec-tor includes moments)

2Equation 12–57 and Equa-tion 12–58 specialized to theedge

Quad

Edge Pressure Load Vector

2Equation 12–38 and Equa-tion 12–39 specialized to theedge

Triangle

DistributionLoad Type

Bilinear in plane of element, linear thru thicknessElement Temperature

Bilinear in plane of element, constant thru thicknessNodal Temperature

Bilinear in plane of element, linear along each edgePressure

14.63.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

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14.63.2. Foundation Stiffness

If Kf, the foundation stiffness, is input, the out-of-plane stiffness matrix is augmented by three or four springs to

ground. The number of springs is equal to the number of distinct nodes, and their direction is normal to theplane of the element. The value of each spring is:

(14–458)K

KNf i

f

d, = ∆

where:

Kf,i = normal stiffness at node i

∆ = element areaKf = foundation stiffness (input as EFS on R command)

Nd = number of distinct nodes

The output includes the foundation pressure, computed as:

(14–459)σp

fI J K L

Kw w w w= + + +

4( )

where:

σp = foundation pressure (output as FOUND, PRESS)

wI, etc. = lateral deflection at node I, etc.

14.63.3. In-Plane Rotational Stiffness

The in-plane rotational (drilling) DOF has no stiffness associated with it, based on the shape functions. A smallstiffness is added to prevent a numerical instability following the approach presented by Kanok-Nukulchai(26)for nonwarped elements if KEYOPT(1) = 0. KEYOPT(3) = 2 is used to include the Allman-type rotational DOFs (asdescribed with SHELL43).

14.63.4. Warping

If all four nodes are not defined to be in the same flat plane (or if an initially flat element loses its flatness due tolarge displacements (using NLGEOM,ON)), additional calculations are performed in SHELL63. The purpose ofthe additional calculations is to convert the matrices and load vectors of the element from the points on the flatplane in which the element is derived to the actual nodes. Physically, this may be thought of as adding short rigidoffsets between the flat plane of the element and the actual nodes. (For the membrane stiffness only case(KEYOPT(1) = 1), the limits given with SHELL41 are used). When these offsets are required, it implies that theelement is not flat, but rather it is “warped”. To account for the warping, the following procedure is used: First,the normal to element is computed by taking the vector cross-product (the common normal) between the vectorfrom node I to node K and the vector from node J to node L. Then, the check can be made to see if extra calculationsare needed to account for warped elements. This check consists of comparing the normal to each of the fourelement corners with the element normal as defined above. The corner normals are computed by taking thevector cross-product of vectors representing the two adjacent edges. All vectors are normalized to 1.0. If any ofthe three global Cartesian components of each corner normal differs from the equivalent component of theelement normal by more than .00001, then the element is considered to be warped.

A warping factor is computed as:

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Section 14.63: SHELL63 - Elastic Shell

(14–460)φ = D

t

where:

D = component of the vector from the first node to the fourth node parallel to the element normalt = average thickness of the element

If:

φ ≤ 0.1 no warning message is printed.10 ≤ φ ≤ 1.0 a warning message is printed1.0 < φ a message suggesting the use of triangles is printed and the run terminates

To account for the warping, the following matrix is developed to adjust the output matrices and load vector:

(14–461)[ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

W

w

w

w

w

=

1

2

3

4

0 0 0

0 0 0

0 0 0

0 0 0

(14–462)[ ]w

Z

Z

i

io

io

=

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

where:

Z io = offset from average plane at node i

and the DOF are in the usual order of UX, UY, UZ, ROTX, ROTY, and ROTZ. To ensure the location of the averageplane goes through the middle of the element, the following condition is met:

(14–463)Z Z Z Zo o

10

20

3 4 0+ + + =

14.63.5. Options for Non-Uniform Material

SHELL63 can be adjusted for nonuniform materials, using an approach similar to that of Takemoto and Cook(107).Considering effects in the element x direction only, the loads are related to the displacement by:

(14–464)T tEx x x= ε

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(14–465)

Mt E

E

E

xx

xyy

x

x= −

−

3

212 1 ν

κ

where:

Tx = force per unit length

t = thickness (input as TK(I), TK(J), TK(K), TK(L) on R command)Ex = Young's modulus in x direction (input as EX on MP command)

Ey = Young's modulus in y direction (input as EY on MP command)

εx = strain of middle fiber in x direction

Mx = moment per unit length

νxy = Poisson's ratio (input as PRXY on MP command)

κx = curvature in x direction

A nonuniform material may be represented with Equation 14–465 as:

(14–466)

M Ct E

E

E

x rx

xyy

x

x= −

−

3

212 1 ν

κ

where:

Cr = bending moment multiplier (input as RMI on RMORE command)

The above discussion relates only to the formulation of the stiffness matrix.

Similarly, stresses for uniform materials are determined by:

(14–467)σ ε κx

topx xE

t= +

2

(14–468)σ ε κx

botx xE

t= −

2

where:

σxtop = x direction stress at top fiber

σxbot = x direction stress at bottom fiber

For nonuniform materials, the stresses are determined by:

(14–469)σ ε κxtop

x t xE c= +( )

(14–470)σ ε κxbot

x b xE c= −( )

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Section 14.63: SHELL63 - Elastic Shell

where:

ct = top bending stress multiplier (input as CTOP, RMORE command)

cb = bottom bending stress multiplier (input as CBOT, RMORE command)

The resultant moments (output as MX, MY, MXY) are determined from the output stresses rather than fromEquation 14–466.

14.63.6. Extrapolation of Results to the Nodes

Integration point results can be requested to be copied to the nodes (ERESX,NO command). For the case ofquadrilateral shaped elements, the bending results of each subtriangle are averaged and copied to the node ofthe quadrilateral which shares two edges with that subtriangle.

14.64. SOLID64 - 3-D Anisotropic Structural Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2

Equation 12–191, Equation 12–192, and Equation 12–193or if modified extra shape functions are included (KEY-OPT(1) = 0) and element has 8 unique nodes: Equa-tion 12–206, Equation 12–207, and Equation 12–208

Stiffness Matrix and ThermalLoad Vector

2 x 2 x 2Equation 12–191, Equation 12–192, and Equation 12–193Mass and Stress StiffnessMatrices

2 x 2Equation 12–57 and Equation 12–58QuadPressure Load Vector

3Equation 12–38 and Equation 12–39Triangle

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

References: Wilson(38), Taylor(49)

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14.64.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 13.5: Positive Definite Matrices defines positive definite matrices.

14.64.2. Stress-Strain Matrix

As referred to in Section 2.1: Structural Fundamentals, the stresses and strains are related by:

(14–471) [ ]( )σ ε ε= −D th

(14–472) [ ] ε ε σ= + −th D 1

where:

[D] = stress-strain matrix, stiffness form (input with TB,ANEL,,,,0)

[D]-1 = stress-strain matrix, flexibility form (input with TB,ANEL,,,,1)

The input must use the same order of components as given in Section 2.1: Structural Fundamentals, i.e. εx, εy, εz,

εxy, εyz, εxz. While εth is restricted to orthotropic input, [D] may be input as a full anisotropic matrix because 21

independent values are used in its makeup for this element. Symmetry of the [D] matrix is ensured, but it is upto the user to provide values so that the matrix is positive definite. If it is not, the program will terminate.

14.65. SOLID65 - 3-D Reinforced Concrete Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2

Equation 12–191, Equation 12–192, and Equation 12–193,or if modified extra shape functions are included (KEY-OPT(1) = 0) and element has 8 unique nodes Equa-tion 12–206, Equation 12–207, and Equation 12–208

Stiffness Matrix and ThermalLoad Vector

2 x 2 x 2Equation 12–191, Equation 12–192, and Equation 12–193Mass Matrix

2 x 2Equation 12–57 and Equation 12–58QuadPressure Load Vector

3Equation 12–38 and Equation 12–39Triangle

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Section 14.65: SOLID65 - 3-D Reinforced Concrete Solid

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

References: Willam and Warnke(37), Wilson(38), Taylor(49)

14.65.1. Assumptions and Restrictions

1. Cracking is permitted in three orthogonal directions at each integration point.

2. If cracking occurs at an integration point, the cracking is modeled through an adjustment of materialproperties which effectively treats the cracking as a “smeared band” of cracks, rather than discrete cracks.

3. The concrete material is assumed to be initially isotropic.

4. Whenever the reinforcement capability of the element is used, the reinforcement is assumed to be“smeared” throughout the element.

5. In addition to cracking and crushing, the concrete may also undergo plasticity, with the Drucker-Pragerfailure surface being most commonly used. In this case, the plasticity is done before the cracking andcrushing checks.

14.65.2. Description

SOLID65 allows the presence of four different materials within each element; one matrix material (e.g. concrete)and a maximum of three independent reinforcing materials. The concrete material is capable of directional in-tegration point cracking and crushing besides incorporating plastic and creep behavior. The reinforcement(which also incorporates creep and plasticity) has uniaxial stiffness only and is assumed to be smearedthroughout the element. Directional orientation is accomplished through user specified angles.

14.65.3. Linear Behavior - General

The stress-strain matrix [D] used for this element is defined as:

(14–473)[ ] [ ] [ ]D V D V Di

R

i

Nc

iR

i

Nr

ir r

= −

+= =∑ ∑1

1 1

where:

Nr = number of reinforcing materials (maximum of three, all reinforcement is ignored if M1 is zero. Also, if M1,

M2, or M3 equals the concrete material number, the reinforcement with that material number is ignored)

V iR = ratio of volume of reinforcing material i to total voluume of element (input as VRi on command)R

[Dc] = stress-strain matrix for concrete, defined by Equation 14–474

[Dr]i = stress-strain matrix for reinforcement i, defined by Equation 14–475

M1, M2, M3 = material numbers associated of reinforcement (input as MAT1, MAT2, and MAT3 on R command)

14.65.4. Linear Behavior - Concrete

The matrix [Dc] is derived by specializing and inverting the orthotropic stress-strain relations defined by Equa-tion 2–4 to the case of an isotropic material or

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(14–474)[ ]

( )( )

( )

( )

( )

( )D

Ec =+ −

−−

−−

1 1 2

1 0 0 0

1 0 0 0

1 0 0 0

0 0 01 2

20

ν ν

ν ν νν ν νν ν ν

ν00

0 0 0 01 2

20

0 0 0 0 01 2

2

( )

( )

−

−

ν

ν

where:

E = Young's modulus for concrete (input as EX on MP command)ν = Poisson's ratio for concrete (input as PRXY or NUXY on MP command)

14.65.5. Linear Behavior - Reinforcement

The orientation of the reinforcement i within an element is depicted in Figure 14.43: “Reinforcement Orientation”.

The element coordinate system is denoted by (X, Y, Z) and ( , , )x y zir

ir

ir

describes the coordinate system for rein-

forcement type i. The stress-strain matrix with respect to each coordinate system ( , , )x y zir

ir

ir

has the form

(14–475)

σ

σ

σ

σ

σ

σ

xxr

yyr

zzr

xyr

yzr

xzr

irE

=

0 0 0 0 00

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

ε

εxxr

yyyr

zzr

xyr

yzr

xzr

ri

xxr

yy

Dε

ε

ε

ε

ε

ε

= [ ]

rr

zzr

xyr

yzr

xzr

ε

ε

ε

ε

where:

E ir = Young’s modulus of reinforcement type i (input as EX onn command)MP

It may be seen that the only nonzero stress component is σxxr

, the axial stress in the x ir

direction of reinforcement

type i. The reinforcement direction x ir

is related to element coordinates X, Y, Z through

(14–476)

X

Y

Z

xi i

i i

i

ir

r

=

=cos cos

sin cos

sin

θ φθ φ

θ

l

l

1

2rr

rirx

l3

where:

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Section 14.65: SOLID65 - 3-D Reinforced Concrete Solid

θi = angle between the projection of the x ir

axis on XY plane and the X axis (input as THETA1, THETA2, and

THETA3 on R command)

φi = angle between the x ir

axis and the XY plane (input as PHI1, PHI2, and PHI3 on R command)

l ir

= direction cosines between x ir

axis and element X, Y, Z axes

Figure 14.43 Reinforcement Orientation

Since the reinforcement material matrix is defined in coordinates aligned in the direction of reinforcement ori-entation, it is necessary to construct a transformation of the form

(14–477)[ ] [ ] [ ] [ ]D T D TRi

r T ri

r=

in order to express the material behavior of the reinforcement in global coordinates. The form of this transform-ation by Schnobrich(29) is

(14–478)[ ]T

a a a a a a a a a

a a a a a a a

r =

112

122

132

11 12 12 13 11 13

212

222

232

21 22 22 233 21 23

312

322

332

31 32 32 33 31 33

11 21 12 22 13 22 2 2

a a

a a a a a a a a a

a a a a a a 3311 22

12 21

12 23

13 32

11 23

13 21

21 31 22 32 22 2 2

a a

a a

a a

a a

a a

a a

a a a a a

+ + +

33 3321 32

22 31

22 33

23 32

21 33

13 21

11 31 12 322 2

aa a

a a

a a

a a

a a

a a

a a a a

+ + +

22 13 3311 32

12 31

12 33

13 32

11 33

13 31a a

a a

a a

a a

a a

a a

a a

+ + +

where the coefficients aij are defined as

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(14–479)

a a a

a a a

a a a

m m m

n

r r r

r r r11 12 13

21 22 23

31 32 33

1 2 3

1 2 3

1

=

l l l

rr r rn n2 3

The vector l l l1 2 3r r r T

is defined by Equation 14–476 while

m m mr r r T1 2 3

and

n n nr r r T1 2 3

are unit

vectors mutually orthogonal to l l l1 2 3r r r T

thus defining a Cartesian coordinate referring to reinforcement

directions. If the operations presented by Equation 14–477 are performed substituting Equation 14–475 andEquation 14–478, the resulting reinforcement material matrix in element coordinates takes the form

(14–480)[ ] D E A Ari i

rd d

T=

where:

A a a a adT

=

11

2212

112

132L

Therefore, the only direction cosines used in [DR]i involve the uniquely defined unit vectorl l l1 2 3r r r T

.

14.65.6. Nonlinear Behavior - Concrete

As mentioned previously, the matrix material (e.g. concrete) is capable of plasticity, creep, cracking and crushing.The plasticity and creep formulations are the same as those implemented in SOLID45 (see Section 4.1: Rate-In-dependent Plasticity). The concrete material model with its cracking and crushing capabilities is discussed inSection 4.8: Concrete. This material model predicts either elastic behavior, cracking behavior or crushing beha-vior. If elastic behavior is predicted, the concrete is treated as a linear elastic material (discussed above). Ifcracking or crushing behavior is predicted, the elastic, stress-strain matrix is adjusted as discussed below for eachfailure mode.

14.65.7. Modeling of a Crack

The presence of a crack at an integration point is represented through modification of the stress-strain relationsby introducing a plane of weakness in a direction normal to the crack face. Also, a shear transfer coefficient βt

(constant C1 with TB,CONCR) is introduced which represents a shear strength reduction factor for those subsequent

loads which induce sliding (shear) across the crack face. The stress-strain relations for a material that has crackedin one direction only become:

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Section 14.65: SOLID65 - 3-D Reinforced Concrete Solid

(14–481)[ ]

( )

( )

DE

RE

cck

t

t

=+

+

− −

− −1

10 0 0 0 0

01

1 10 0 0

01

11

0 0 0

0 0 02

0 0ν

ν

νν

νν

ν νβ

00 0 0 012

0

0 0 0 0 02βt

where the superscript ck signifies that the stress strain relations refer to a coordinate system parallel to principal

stress directions with the xck axis perpendicular to the crack face. If KEYOPT(7) = 0, Rt = 0.0. If KEYOPT(7) = 1, Rt

is the slope (secant modulus) as defined in the figure below. Rt works with adaptive descent and diminishes to0.0 as the solution converges.

Figure 14.44 Strength of Cracked Condition

E

1 1

6

R

f

T

ck

tc

ε ckεε

f

t

t

where:

ft = uniaxial tensile cracking stress (input as C3 with TB,CONCR)

Tc = multiplier for amount of tensile stress relaxation (input as C9 with TB,CONCR, defaults to 0.6)

If the crack closes, then all compressive stresses normal to the crack plane are transmitted across the crack and

only a shear transfer coefficient βc (constant C2 with TB,CONCR) for a closed crack is introduced. Then [ ]Dcck

can

be expressed as

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(14–482)[ ]

( )( )

( )

( )D

Ecck c=

+ −

−−

−−

1 1 2

1 0 0 0

1 0 0 0

1 0 0 0

0 0 01 2

20

ν ν

ν ν νν ν νν ν ν

β ν00

0 0 0 01 2

20

0 0 0 0 01 2

2

( )

( )

−

−

ν

β νc

The stress-strain relations for concrete that has cracked in two directions are:

(14–483)[ ]

( )

( )

D E

RE

RE

cck

t

t

t

t

=+

+

0 0 0 0 0

0 0 0 0 0

0 0 1 0 0 0

0 0 02 1

0 0

0 0 0 02 1

0

0 0

βν

βν

00 0 02 1

βν

t

( )+

If both directions reclose,

(14–484)[ ]

( )( )

( )

( )D

Ecck c=

+ −

−−

−−

1 1 2

1 0 0 0

1 0 0 0

1 0 0 0

0 0 01 2

20

ν ν

ν ν νν ν νν ν ν

β ν00

0 0 0 01 2

20

0 0 0 0 01 2

2

( )

( )

−

−

ν

β νc

The stress-strain relations for concrete that has cracked in all three directions are:

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Section 14.65: SOLID65 - 3-D Reinforced Concrete Solid

(14–485)[ ]

( )

( )

D E

RE

RE

cck

t

t

t

t

=+

+

0 0 0 0 0

0 0 0 0 0

0 0 1 0 0 0

0 0 02 1

0 0

0 0 0 02 1

0

0 0

βν

βν

00 0 02 1

βν

t( )+

If all three cracks reclose, Equation 14–484 is followed. In total there are 16 possible combinations of crack ar-rangement and appropriate changes in stress-strain relationships incorporated in SOLID65. A note is output if1 >βc >βt >0 are not true.

The transformation of [ ]Dcck

to element coordinates has the form

(14–486)[ ] [ ] [ ][ ]D T D Tcck T

cck ck=

where [Tck] has a form identical to Equation 14–478 and the three columns of [A] in Equation 14–479 are nowthe principal direction vectors.

The open or closed status of integration point cracking is based on a strain value εckck

called the crack strain. Forthe case of a possible crack in the x direction, this strain is evaluated as

(14–487)ε

ε νν

ε ε

ε νεckck

xck

yck

zck

xck=

+−

+

+1

if no cracking has occurred

zzck

xck

if y direction has cracked

if y and z direction havε ee cracked

where:

ε ε εxck

yck

zck, and three normal component strains in crack= orientation

The vector εck is computed by:

(14–488) [ ] ε εck ckT= ′

where:

ε' = modified total strain (in element coordinates)

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ε', in turn, is defined as:

(14–489) ′ = + − −−ε ε ε ε εn n

eln n

thnpl

1 ∆ ∆ ∆

where:

n = substep number

εnel

− =1 elastic strain from previous substep

∆εn = total strain increment (based on ∆un, the displacement increment over the substep)

∆εnth = thermal strain increment

∆εnpl = plastic strain increment

If εckck

is less than zero, the associated crack is assumed to be closed.

If εckck

is greater than or equal to zero, the associated crack is assumed to be open. When cracking first occurs atan integration point, the crack is assumed to be open for the next iteration.

14.65.8. Modeling of Crushing

If the material at an integration point fails in uniaxial, biaxial, or triaxial compression, the material is assumed tocrush at that point. In SOLID65, crushing is defined as the complete deterioration of the structural integrity ofthe material (e.g. material spalling). Under conditions where crushing has occurred, material strength is assumedto have degraded to an extent such that the contribution to the stiffness of an element at the integration pointin question can be ignored.

14.65.9. Nonlinear Behavior - Reinforcement

The one-dimensional creep and plasticity behavior for SOLID65 reinforcement is modeled in the same manneras for LINK8.

14.66. Not Documented

No detail or element available at this time.

14.67. PLANE67 - 2-D Coupled Thermal-Electric Solid

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Section 14.67: PLANE67 - 2-D Coupled Thermal-Electric Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–112QuadElectrical Conductivity Matrix

3Equation 12–93Triangle

2 x 2Equation 12–111QuadThermal Conductivity Matrix and HeatGeneration Load Vector 3Equation 12–92Triangle

Same as conductivitymatrices

Same as for thermal conductivity matrix. Matrixis diagonalized as described in Section 13.2:Lumped Matrices

Specific Heat Matrix

2Same as thermal conductivity matrix evaluatedat the face

Convection Surface Matrix and LoadVector

Reference: Kohnke and Swanson(19)

14.67.1. Other Applicable Sections

Chapter 11, “Coupling” discusses coupled effects.

14.68. LINK68 - Coupled Thermal-Electric Line

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–14Electrical Conductivity Matrices

NoneEquation 12–13Thermal Conductivity and Specific HeatMatrices; and Heat Generation Load Vector

Reference: Kohnke and Swanson(19)

14.68.1. Other Applicable Sections

Chapter 11, “Coupling” discusses coupled effects.

Chapter 14: Element Library

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14.69. SOLID69 - 3-D Coupled Thermal-Electric Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–202Electrical Conductivity Matrix

2 x 2 x 2Equation 12–201Thermal Conductivity Matrix and HeatGeneration Load Vector

2 x 2 x 2Equation 12–201. Matrix is diagonalized asdescribed in Section 13.2: Lumped Matrices

Specific Heat Matrix

NoneEquation 12–201, specialized to the faceConvection Surface Matrix and Load Vector

Reference: Kohnke and Swanson(19)

14.69.1. Other Applicable Sections

Chapter 11, “Coupling” discusses coupled effects.

14.70. SOLID70 - 3-D Thermal Solid

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Section 14.70: SOLID70 - 3-D Thermal Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–201Conductivity Matrix and Heat GenerationLoad Vector

Same as conductivitymatrix

Equation 12–201. Matrix is diagonalized asdescribed in Section 13.2: Lumped Matrices

Specific Heat Matrix

2 x 2Equation 12–201 specialized to the faceConvection Surface Matrix and Load Vector

14.70.1. Other Applicable Sections

Section 6.2: Derivation of Heat Flow Matrices has a complete derivation of the matrices and load vectors of ageneral thermal analysis element. Mass transport is discussed in Section 14.55: PLANE55 - 2-D Thermal Solid.

14.70.2. Fluid Flow in a Porous Medium

An option (KEYOPT(7) = 1) is available to convert SOLID70 to a nonlinear steady-state fluid flow element. Pressureis the variable rather than temperature. From Equation 6–21, the element conductivity matrix is:

(14–490)[ ] [ ] [ ][ ] ( )K B D B d vole

tb T

vol

= ∫

[B] is defined by Equation 6–21 and for this option, [D] is defined as:

(14–491)[ ]D

K

K E

K

K E

K

K E

x

x

y

y

z

z

=

+

+

+

∞

∞

∞

∞

∞

∞

ρµ

ρ

µ

ρµ

0 0

0 0

0 0

where:

Kx∞

= absolute permeability of the porous medium in the x direction (input as KXX on MP command)ρ = mass density of the fluid (input as DENS on MP command)µ = viscosity of the fluid (input as VISC on MP command)

E S= ρβ α

β = visco-inertial parameter of the fluid (input as C on MP command)S = seepage velocity (at centroid from previous iteration, defined below)α = empirical exponent on S (input as MU on MP command)

For this option, no “specific heat” matrix or “heat generation” load vector is computed.

The pressure gradient components are computed by:

(14–492)

g

g

g

B T

xp

yp

zp

e

= [ ]

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where:

gxp

= pressure gradient in the x-direction (output as PRESSURE GRADIENT (X))Te = vector of element temperatures (pressures)

The pressure gradient is computed from:

(14–493)g g g gpxp

yp

zp= + +( ) ( ) ( )2 2 2

where:

gp = total pressure gradient (output as PRESSURE GRADIENT (TOTAL))

The mass flux components are:

(14–494)

f

f

f

D

g

g

g

x

y

z

xp

yp

zp

= −

[ ]

The vector sum of the mass flux components is:

(14–495)f f f fx y z= + +2 2 2

where:

f = mass flux (output as MASS FLUX)

The fluid velocity components are:

(14–496)

S

S

S

f

f

f

x

y

z

x

y

z

=

1ρ

where:

Sx = fluid velocity in the x-direction (output as FLUID VELOCITY (X))

and the maximum fluid velocity is:

(14–497)S

f=

ρ

where:

S = total fluid velocity (output as FLUID VELOCITY (TOTAL))

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Section 14.70: SOLID70 - 3-D Thermal Solid

14.71. MASS71 - Thermal Mass

Integration PointsShape FunctionsMatrix or Vector

NoneNoneSpecific Heat Matrix and Heat GenerationLoad Vector

14.71.1. Specific Heat Matrix

The specific heat matrix for this element is simply:

(14–498)[ ] [ ]C Cet o=

Co is defined as:

(14–499)C

C vol

Co p

a=

ρ ( ) if KEYOPT(3) = 0

if KEYOPT(3) = 1

where:

ρ = density (input as DENS on MP command)Cp = specific heat (input as C on MP command)

vol = volume (input as CON1 on R command)Ca = capacitance (input as CON1 on R command)

14.71.2. Heat Generation Load Vector

The heat generation load vector is:

(14–500) Q Ae

gq=

where:

AQ

A T A T A Tq

RA A=

+ + +

if A thru A are not provided

if A

1 6

1A1 2 3 54 6 thru A are provided6

QR = heat rate (input as QRATE on MP command)

A1, A2, etc. = constants (input as A1, A2, etc. on R command)

T T To= + =l absolute temperature

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TT

Tunif

ll

=′

for first iteration

for second and subsequent iiterations

Tunif = uniform temperature (input on BFUNIF command)

′Tl = temperature from previous iterationTo = offset temperature (input on TOFFST command)

14.72. Not Documented

No detail or element available at this time.

14.73. Not Documented

No detail or element available at this time.

14.74. HYPER74 - 2-D 8-Node Mixed u-P Hyperelastic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

3 x 3Equation 12–117, Equation 12–118, andEquation 12–119

QuadStiffness and Mass Matrices;and Thermal and Newton-Raphson Load Vectors 3

Equation 12–96, Equation 12–97, andEquation 12–98

Triangle

2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functions across element, constant thru thickness or aroundcircumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

References: Oden(123), Sussman(124)

14.74.1. Other Applicable Sections

For the basic formulation refer to Section 14.58: HYPER58 - 3-D 8-Node Mixed u-P Hyperelastic Solid. The hyper-elastic material model (Mooney-Rivlin) is described in Section 4.6: Hyperelasticity.

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Section 14.74: HYPER74 - 2-D 8-Node Mixed u-P Hyperelastic Solid

14.74.2. Assumptions and Restrictions

A dropped midside node implies that the edge is and remains straight.

14.75. PLANE75 - Axisymmetric-Harmonic 4-Node Thermal Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–144QuadConductivity Matrix and HeatGeneration Load Vector 3Equation 12–136Triangle

Same as conductivitymatrix

Same as conductivity matrix. Matrix is diagonalized asdescribed in Section 13.2: Lumped Matrices

Specific Heat Matrix

2Same as conductivity matrix specialized to the faceConvection Surface Matrix andLoad Vector

14.75.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of the element matrices and load vectors as well as heat fluxevaluations.

14.76. Not Documented

No detail or element available at this time.

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14.77. PLANE77 - 2-D 8-Node Thermal Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

3 x 3Equation 12–121QuadConductivity Matrix and HeatGeneration Load Vector 6Equation 12–101Triangle

Same as conductivitymatrix

Same as conductivity matrix. If KEYOPT(1) = 1, matrix isdiagonalized as described in Section 13.2: LumpedMatrices

Specific Heat Matrix

2Same as conductivity matrix, specialized to the faceConvection Surface Matrix andLoad Vector

14.77.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of the thermal element matrices and load vectors as well as heatflux evaluations. If KEYOPT(1) = 1, the specific heat matrix is diagonalized as described in Section 13.2: LumpedMatrices.

14.77.2. Assumptions and Restrictions

A dropped midside node implies that the edge is straight and that the temperature varies linearly along thatedge.

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Section 14.77: PLANE77 - 2-D 8-Node Thermal Solid

14.78. PLANE78 - Axisymmetric-Harmonic 8-Node Thermal Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

3 x 3Equation 12–151QuadConductivity Matrix and HeatGeneration Load Vector 6Equation 12–140Triangle

Same as conductivitymatrix

Same as conductivity matrix. If KEYOPT(1) = 1, matrix isdiagonalized as described in Section 13.2: LumpedMatrices

Specific Heat Matrix

2Same as stiffness matrix, specialized to the faceConvection Surface Matrix andLoad Vector

14.78.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of the thermal element matrices and load vectors as well as heatflux evaluations.

14.78.2. Assumptions and Restrictions

A dropped midside node implies that the edge is straight and that the temperature varies linearly along thatedge.

14.79. FLUID79 - 2-D Contained Fluid

! "

# $

%

&

')( *+,+ - ./ -0 1

2)( *+-43 / -0 1

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Integration PointsShape FunctionsGeometryMatrix or Vector

1 x 1 for bulk strain effects2 x 2 for shear and rotationalresistance effects

Equation 12–103 and Equation 12–104QuadStiffness and DampingMatrices; and Thermal LoadVector 1 x 1 for bulk strain effects

3 for shear and rotationalresistance effects

Equation 12–84 and Equation 12–85Triangle

Same as for shear effectsSame as stiffness matrix. Matrix is diagonalized as inSection 13.2: Lumped Matrices.

Mass Matrix

2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Average of the four nodal temperatures is used throughout the elementElement Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

14.79.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of element matrices and load vectors. The fluid aspects of thiselement are the same as described for FLUID80.

14.80. FLUID80 - 3-D Contained Fluid

Integration PointsShape FunctionsMatrix or Vector

1 x 1 x 1 for bulk strain effects2 x 2 x 2 for shear and rotational resistance effects

Equation 12–191, Equation 12–192, and Equa-tion 12–193

Stiffness and DampingMatrices; and ThermalLoad Vector

2 x 2 x 2Same as stiffness matrix. Matrix is diagonalized asdescribed in Section 13.2: Lumped Matrices

Mass Matrix

2 x 2Same as stiffness matrix, specialized to the facePressure Load Vector

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Section 14.80: FLUID80 - 3-D Contained Fluid

DistributionLoad Type

Average of the 8 nodal temperatures is used throughout elementElement Temperature

Average of the 8 nodal temperatures is used throughout elementNodal Temperature

Bilinear across each facePressure

14.80.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of element matrices and load vectors.

14.80.2. Assumptions and Restrictions

This element does not generate a consistent mass matrix; only the lumped mass matrix is available.

14.80.3. Material Properties

Rather than Equation 2–3, the stress-strain relationships used to develop the stiffness matrix and thermal loadvector are:

(14–501)

εγ

γ

γ

αbulk

xy

yz

xz

x

y

z

R

R

R

T

=

3

0

0

0

0

0

0

∆

+

10 0 0 0 0 0

01

0 0 0 0 0

0 01

0 0 0 0

0 0 01

0 0 0

0 0 0

K

S

S

S

001

0 0

0 0 0 0 01

0

0 0 0 0 0 01

B

B

B

P

M

M

M

xy

yz

xz

x

y

z

τ

τ

τ

where:

εbulkux

vy

wz

= = ∂∂

+ ∂∂

+ ∂∂

bulk strain

α = thermal coefficient of expansion (input as ALPX on MP command)∆T = change of temperature from reference temperatureK = fluid elastic (bulk) modulus (input as EX on MP command)P = pressureγ = shear strain

S = K x 10-9 (arbitrarily small number to give element some shear stability)τ = shear stressRi = rotation about axis i

B = K x 10-9 (arbitrarily small number to give element some rotational stability)Mi = twisting force about axis i

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A damping matrix is also developed based on:

(14–502)

&&

&

&&&

&

εγ

γ

γ

bulk

xy

yz

xz

x

y

z

R

R

R

=

0 0 0 0 00 0 0

01

0 0 0 0 0

0 01

0 0 0 0

0 0 01

0 0 0

0 0 0 01

0 0

0 0 0 0 01

0

0 0 0 0 0 01

η

η

η

c

c

c

P

M

M

M

xy

yz

xz

x

y

z

τ

τ

τ

where:

η = viscosity (input as VISC on MP command)c = .00001*η

and the (⋅) represents differentiation with respect to time.

A lumped mass matrix is developed, based on the density (input as DENS on MP command).

14.80.4. Free Surface Effects

The free surface is handled with an additional special spring effect. The necessity of these springs can be seenby studying a U-Tube, as shown in Figure 14.45: “U-Tube with Fluid”.

Note that if the left side is pushed down a distance of ∆h, the displaced fluid mass is:

(14–503)M h AD = ∆ ρ

where:

MD = mass of displaced fluid

∆h = distance fluid surface has movedA = cross-sectional area of U-Tubeρ = fluid density

Then, the force required to hold the fluid in place is

(14–504)F M gD D=

where:

FD = force required to hold the fluid in place

g = acceleration due to gravity (input on ACEL command)

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Section 14.80: FLUID80 - 3-D Contained Fluid

Figure 14.45 U-Tube with Fluid

Finally, the stiffness at the surface is the force divided by the distance, or

(14–505)K

Fh

AgsD= =

∆ρ

This expression is generalized to be:

(14–506)K A g C g C g Cs F x x y y z z= + +ρ ( )

where:

AF = area of the face of the element

gi = acceleration in the i direction

Ci = ith component of the normal to the face of the element

This results in adding springs from each node to ground, with the spring constants being positive on the top ofthe element, and negative on the bottom. For an interior node, positive and negative effects cancel out and, atthe bottom where the boundary must be fixed to keep the fluid from leaking out, the negative spring has noeffect. If KEYOPT(2) = 1, positive springs are added only to faces located at z = 0.0.

14.80.5. Other Assumptions and Limitations

The surface springs tend to retard the hydrostatic motions of the element from their correct values. The hydro-dynamic motions are not changed. From the definition of bulk modulus,

(14–507)u

PK

dzso

H

= ∫

where:

us = vertical motion of a static column of fluid (unit cross-sectional area)

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H = height of fluid columnP = pressure at any pointz = distance from free surface

The pressure is normally defined as:

(14–508)P gz= ρ

But this pressure effect is reduced by the presence of the surface springs, so that

(14–509)P gz K u g z us s s= − = −ρ ρ ( )

Combining Equation 14–507 and Equation 14–509 and integrating,

(14–510)u

gK

Hu Hs s= −

ρ 2

2

or

(14–511)u

H gK

gK

Hs =

+

1

1 2

2

ρρ

If there were no surface springs,

(14–512)ug

KH

s = ρ 2

2

Thus the error for hydrostatic effects is the departure from 1.0 of the factor (1 / (1+Hρg/K)), which is normallyquite small.

The 1 x 1 x 1 integration rule is used to permit the element to “bend” without the bulk modulus resistance beingmobilized, i.e.

Figure 14.46 Bending Without Resistance

While this motion is permitted, other motions in a static problem often result, which can be thought of as energy-free eddy currents. For this reason, small shear and rotational resistances are built in, as indicated in Equa-tion 14–501.

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Section 14.80: FLUID80 - 3-D Contained Fluid

14.81. FLUID81 - Axisymmetric-Harmonic Contained Fluid

Integration PointsShape FunctionsGeometryMatrix or Vector

1 for bulk strain effects2 x 2 for shear and rotational resistance effects

Equation 12–141, Equation 12–142, andEquation 12–143

QuadStiffness and DampingMatrices; and Thermal LoadVector 1 for bulk strain effects

3 for shear and rotational resistance effects

Equation 12–133, Equation 12–134, andEquation 12–135

Triangle

2 x 2Equation 12–103, Equation 12–104, andEquation 12–105

Quad

Mass Matrix

3Equation 12–84, Equation 12–86, andEquation 12–87

Triangle

2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Average of the four nodal temperatures is used throughout the elementElement Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

14.81.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of element matrices and load vectors. The fluid aspects of thiselement are the same as described for Section 14.80: FLUID80 - 3-D Contained Fluid except that a consistentmass matrix is also available (LUMPM,OFF).

14.81.2. Assumptions and Restrictions

The material properties are assumed to be constant around the entire circumference, regardless of temperaturedependent material properties or loading.

14.81.3. Load Vector Correction

When l (input as MODE on MODE command) > 0, the gravity that is required to be input for use as a gravityspring (input as ACELY on ACEL command) also is erroneously multiplied by the mass matrix for a gravity forceeffect. This erroneous effect is cancelled out by an element load vector that is automatically generated duringthe element stiffness pass.

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14.82. PLANE82 - 2-D 8-Node Structural Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–117 and Equation 12–118QuadMass, Stiffness and StressStiffness Matrices; andThermal Load Vector 3Equation 12–96 and Equation 12–97Triangle

2 along faceSame as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functions across element, constant thru thickness or aroundcircumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

Reference: Zienkiewicz(39)

14.82.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.82.2. Assumptions and Restrictions

A dropped midside node implies that the face is and remains straight.

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Section 14.82: PLANE82 - 2-D 8-Node Structural Solid

14.83. PLANE83 - Axisymmetric-Harmonic 8-Node Structural Solid

!

" # $ %

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–148, Equation 12–149, andEquation 12–150

QuadStiffness, Mass, and StressStiffness Matrices; andThermal Load Vector 3

Equation 12–137, Equation 12–138, andEquation 12–139

Triangle

2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functions across element, harmonic around circumferenceElement Temperature

Same as element temperature distributionNodal Temperature

Linear along each face, harmonic around circumferencePressure

Reference: Zienkiewicz(39)

14.83.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 14.25: PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid has a discussion of tem-perature applicable to this element.

14.83.2. Assumptions and Restrictions

A dropped midside node implies that the edge is and remains straight.

The material properties are assumed to be constant around the entire circumference, regardless of temperature-

dependent material properties or loading. For l (input as MODE on MODE command) > 0, extreme values for

combined stresses are obtained by computing these stresses at every 10/ l degrees and selecting the extremevalues.

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14.84. HYPER84 - 2-D Hyperelastic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2 (if KEYOPT(6) = 1, use 1 x1 for volumetric terms)

Equation 12–103, Equa-tion 12–104, and Equation 12–105

Quad 4 node (KEY-OPT(1) = 0)

Stiffness Matrix

1 (plane strain) 3 (axisymmet-ric) (if KEYOPT(6) = 1, use 1 x 1for volumetric terms)

Equation 12–84, Equation 12–86,and Equation 12–87

Tri. 4 node (KEYOPT(1)= 0)

3 x 3 (if KEYOPT(6) = 1, use 2 x2 for volumetric terms)

Equation 12–117, Equa-tion 12–118, and Equation 12–119

Quad 8 node (KEY-OPT(1) = 1)

3 (if KEYOPT(6) = 1, use 1 x 1for volumetric terms)

Equation 12–96, Equation 12–97,and Equation 12–98

Tri. 8 node (KEYOPT(1)= 1)

2 x 2 (KEYOPT(1) = 0)3 x 3 (KEYOPT(1) = 1)Same as stiffness matrixMass Matrix

2 (KEYOPT(1) = 0)3 (KEYOPT(1) = 1)Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functions across element, constant thru thickness or aroundcircumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

Reference: Oden(27), Zienkiewicz(39), Rivlin(89), Kao(90), Mooney(91), and Blatz(92)

14.84.1. Assumptions and Restrictions

A dropped midside node implies that the edge is and remains straight.

14.84.2. Other Applicable Sections

For the basic element formulation refer to Section 14.86: HYPER86 - 3-D Hyperelastic Solid. The hyperelasticmaterial models (Mooney-Rivlin and Blatz-Ko) are described in Section 4.6: Hyperelasticity.

14.85. Not Documented

No detail or element available at this time.

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Section 14.85: Not Documented

14.86. HYPER86 - 3-D Hyperelastic Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2 (if KEYOPT(6) = 1, use1 x 1 x 1 for volumetric terms)

Equation 12–191, Equation 12–192, and Equa-tion 12–193

Stiffness Matrix

2 x 2 x 2Same as stiffness matrixMass Matrix

2 x 2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

References: Oden(27), Zienkiewicz(39), Rivlin(89), Kao(90), Mooney(91), and Blatz(92)

14.86.1. Other Applicable Sections

The hyperelastic material models (Mooney-Rivlin and Blatz-Ko) are described in Section 4.6: Hyperelasticity.

14.86.2. Virtual Work Statement

The variational principle employed to derive the incremental stiffness matrix of the hyperelastic finite elementsdescribed in this section is the incremental principle of virtual work. Internal and external work as well as theirincrements are expressed in an equilibrium statement for an element.

(14–513)δ δ δ δU U V V+ = +& &

where:

δU = internal virtual work

δ &U = increment of internal virtual work

δV = external virtual work

δ &V = increment of external virtual work

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The internal virtual work is expressed as the integral over the volume of the current strain energy density function,W. The external virtual work is the work of the surface pressures over the current surfaces, as well as the work ofthe nodal point loads. Equation 14–513 can be expressed as follows:

(14–514)

δ δ δ

δ

Wd vol Wd vol Pn u dS

F

vol voli i

Selement

in

in

( ) ( ) ^

( ) ( )

∫ ∫ ∫∑+ =

+

&

∆eelement

∑

where:

vol = current element volumeW = strain energy density function per unit current volumeP = scalar pressure magnitude

n i^ = components of unit normal of current deformed surface

Fin( ) = applied nodal forces in i direction at node n

δui = displacement field variations of the i coordinate

S = current deformed surface area of element

δ∆ in = variation of nodal displacement in i direction at nodee n

14.86.3. Element Matrix Derivation

Equation 14–514 is the basic equilibrium relationship used to derive the element stiffness matrix and load vectors.Details of the strain energy density function and its variation with respect to current strain are outlined here.

The strain energy density is a function of the current strain components,

(14–515)W W Cij= ( )

where:

Cij = components of the right Cauchy-Green deformation tensor (defined below)

Many forms of this functional dependence are possible. The strain energy density functions available are givenin Section 4.6: Hyperelasticity.

Without selecting any particular form of W, expressions for the variation of W and an increment of the variationof W are given as follows:

(14–516)δ δW

WC

Cij

ij= ∂∂

(14–517)δ δ δ& & &W

WC C

C CWC

Cij kl

ij klij

ij= ∂∂ ∂

+ ∂∂

2

The deformation tensor [C] is comprised of the products of the deformation gradients [f]

(14–518)C f fij ki kj= = component of the Cauchy-Green strain tensor

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Section 14.86: HYPER86 - 3-D Hyperelastic Solid

(14–519)δ δ δC f f f fij ki kj ki kj= +

(14–520)δ δ δ& & &C f f f fij ki kj kj ki= +

(14–521)& & &C f f f fij ki kj ki kj= +

where:

fxXij

i

j= ∂

∂

Xi = undeformed position of a particle in direction i

xi = Xi + ui = deformed position of a particle in direction i

ui = displacement of particle in direction i

Substitution of Equation 14–516 through Equation 14–521 into Equation 14–513 yields an element equilibriumequation in terms of the external loads and internal strains, as shown by:

(14–522)

2 22∂

∂+ ∂

∂+ ∂

∂ ∂

WC

f fWC

f fW

C Cf f f f

ijki kj

ijki kj

ij klmi nj mk nlδ δ δ& &

= +

∫

∫∑ −

d vol

Pdet f f N u dS F

vol

ij j i oSelement

in

io

( )

( )^

%1 δ δ∆nn

element∑

where:

∂∂ ∂

=2W

C Cij klincremental moduli (fourth order tensor)

Nj^

= components of normal to original undeformed surface

δui = variation of the i coordinate displacement field

So = undeformed surface area over which P acts

It can be shown that there is a common virtual factor to all terms in Equation 14–522. Converting from tensorto matrix form, Equation 14–522 becomes:

(14–523)[ ( )] ( )K u u F F R uepr nd& = + −

where:

[Ke(u)] = current element stiffness matrix

Fpr = total current applied pressures (normal to current surface)

Fnd = total current applied nodal point loadsR(u) = current Newton-Raphson restoring force vector

&u = unknown nodal displacement incrementsu = current total nodal displacements before this solution

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Equation 14–523 gives the element stiffness equation. The unknown quantity is &u while the stiffness matrixand restoring force vector are functions of the current value of displacements u, through the deformation

gradient [f] and derivations of the strain energy density ∂ ∂W Cij and

∂ ∂ ∂2W C Cij kl .

14.86.4. Reduced Integration on Volumetric Term in Stiffness Matrix

This formulation may produce numerical instability in the nearly incompressible range (Poisson's ratio (ν) → 0.5).This can be viewed from the following energy relation:

(14–524)W d volT

vol= ∫

12

( )ε σ

where:

ε = strain vector = ε ε ε ε ε εxx yy zz xy yz xz

T

σ = stress vector = σ σ σ σ σ σxx yy zz xy yz xz

T

In the case of isotropic materials, the stress-strain relation in terms of shear and bulk modulus for 3-D stress stateis:

(14–525) ( [ ] [ ]) σ ε= +G D K Ds v

where:

GE=+

=2 1( )ν

shear modulus

KE=−

=3 1 2( )ν

bulk modulus

[ ]Ds =

− −

− −

− −

43

23

23

0 0 0

23

43

23

0 0 0

23

23

43

0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

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Section 14.86: HYPER86 - 3-D Hyperelastic Solid

[ ]Dv =

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

subscript s = deviatoricsubscript v = volumetric

The bulk modulus used in Equation 14–525 becomes unbounded as Poisson's ratio, ν, approaches 0.5. Now usingEquation 14–525 in Equation 14–524):

(14–526)W W Ws v= +

where:

W G D d vols svol

= =∫12

ε ε[ ] ( ) deviatoric (shear) strain enerrgy

W K D d volv vvol

= =∫12

ε ε[ ] ( ) volumetric strain energy

Now using the derivative given in Section 2.2: Derivation of Structural Matrices, the discretized finite elementrelationship becomes:

(14–527)( [ ] [ ]) α αG K K u Fs vnd+ =

where:

α = 1K

[Ks] = stiffness associated with shear energy

[Kv] = stiffness associated with volumetric energy

u = nodal displacements

Fnd = nodal load vector

As Poisson's ratio (ν) approaches 0.5, α approaches 0.0 so that Equation 14–527 reduces to:

(14–528)[ ] K uv = 0

In Equation 14–528 if [Kv] is non-singular, only the trivial solution is possible (i.e., u = 0). To enforce a nontrivial

solution, [Kv] has to be singular and is achieved by a reduced order integration scheme.

14.86.5. Description of Additional Output Strain Measures

The geometric strain measures output for the hyperelastic element are (1) unit extension and (2) angle changewith respect to the global Cartesian system.

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The stretch ratio provides the basis for interpretation of the finite strain tensor. The change of length per unit oforiginal length (unit extension) is defined as:

(14–529)Ξ Λp ds dS

dSdsdS

= − = − = −1 1

where:

ds = current lengthdS = original lengthΛ = stretch ratio

Ξp = unit extension

Defining the physical components of Lagrangian strain as follows:

(14–530)EE

G G

C

G Gijp ij

ii jj

ij ij

ii jj= =

−12

( )δ

where:

Eijp = Lagrangian strain tensor

Cij = right Cauchy-Green tensor

Gij = metric tensor of reference curvilinear system

The output strains in the directions of the global axes are the unit extensions and are defined as follows:

(14–531)Ξiip

iipE= − −1 2 1

where:

Ξiip =unit extension (output as UNEXTN (X, Y, Z)

1 2− =Eiip stretch ratio measure

The shear rotations (output as ROTANG (XY, YZ, XZ)) are defined as the angle change from the reference config-uration. The equation for the angle change ∆φij is:

(14–532)∆φ π θij ij

ijp

iip

jjp

sinE

E E= − =

+

+

−2

2

1 2 1 2

1

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Section 14.86: HYPER86 - 3-D Hyperelastic Solid

14.87. SOLID87 - 3-D 10-Node Tetrahedral Thermal Solid

Integration PointsShape FunctionsMatrix or Vector

4Equation 12–167Conductivity Matrix and HeatGeneration Load Vector

11Same as conductivity matrix. If KEYOPT(1) = 1, the matrixis diagonalized as described in Section 13.2: LumpedMatrices

Specific Heat Matrix

6

Equation 12–167 specialized to the face. Diagonalized surface matrix if KEYOPT(5) = 0,consistent surface matrix if KEYOPT(5) = 1

Convection Surface Matrix andLoad Vector

14.87.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors as well as heatflux evaluations. If KEYOPT(1) = 1, the specific heat matrix is diagonalized as described in Section 13.2: LumpedMatrices.

14.88. VISCO88 - 2-D 8-Node Viscoelastic Solid

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Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–117 and Equation 12–118QuadStiffness, Mass, and StressStiffness Matrices; andThermal and Newton-RaphsonLoad Vector

3Equation 12–96 and Equation 12–97Triangle

2 along faceSame as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functions across element, constant thru thickness or aroundcircumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

References: Zienkiewicz(39), Markovsky et al.(108), Scherer and Rekhson(109), Narayanaswamy(110), Zienkiewiczet al.(111), Taylor et al.(112)

14.88.1. Other Applicable Sections

Section 4.7: Viscoelasticity describes the basic theory regarding viscoelasticity.

14.89. VISCO89 - 3-D 20-Node Viscoelastic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

14Equation 12–209, Equation 12–210, andEquation 12–211

Brick

Stiffness, Mass, and StressStiffness Matrices; andThermal and Newton-RaphsonLoad Vectors

3 x 3Equation 12–186, Equation 12–187, andEquation 12–188

Wedge

2 x 2 x 2Equation 12–171, Equation 12–172, andEquation 12–173

Pyramid

4Equation 12–164, Equation 12–165, andEquation 12–166

Tet

3 x 3Equation 12–69 and Equation 12–70QuadPressure Load Vector

6Equation 12–46 and Equation 12–47Triangle

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Section 14.89: VISCO89 - 3-D 20-Node Viscoelastic Solid

DistributionLoad Type

Same as shape functions across elementElement Temperature

Same as shape functions across elementNodal Temperature

Bilinear across each facePressure

References: Zienkiewicz(39), Markovsky et al.(108), Scherer and Rekhson(109), Narayanaswamy(110), Zienkiewiczet al.(111), Taylor et al.(112)

14.89.1. Other Applicable Sections

Section 4.7: Viscoelasticity describes the basic theory regarding viscoelasticity. If KEYOPT(3) = 1, the matrix is di-agonalized as described in Section 13.2: Lumped Matrices.

14.90. SOLID90 - 3-D 20-Node Thermal Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

14Equation 12–212Brick

Conductivity Matrix and HeatGeneration Load Vector

3 x 3Equation 12–189Wedge

2 x 2 x 2Equation 12–174Pyramid

4Equation 12–167Tet

Same as conductivitymatrix

Same as conductivity matrix. If KEYOPT(1) = 1, the matrixis diagonalized as described in Section 13.2: LumpedMatrices.

Specific Heat Matrix

3 x 3Equation 12–73QuadConvection Surface Matrix andLoad Vector 6Equation 12–49Triangle

14.90.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors as well as heatflux evaluations.

Chapter 14: Element Library

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14.91. SHELL91 - Nonlinear Layered Structural Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

Thru-the-thickness: 3 for each layerIn-plane: 2 x 2

Equation 12–83QuadStiffness Matrix and ThermalLoad Vector

Thru-the-thickness: 3 for each layerIn-plane: 3

Equation 12–56TriangleStiffness Matrix and ThermalLoad Vector

Same as stiffness matrixEquation 12–69, Equation 12–70, andEquation 12–71

QuadMass and Stress StiffnessMatrices

Same as stiffness matrixEquation 12–46, Equation 12–47, andEquation 12–48

TriangleMass and Stress StiffnessMatrices

2 x 2Equation 12–71QuadTransverse Pressure LoadVector

3Equation 12–48TriangleTransverse Pressure LoadVector

2Same as in-plane mass matrix specialized to theedge.

Edge Pressure Load Vector

DistributionLoad Type

Linear thru each layer, bilinear in plane of elementElement Temperature

Constant thru thickness, bilinear in plane of elementNodal Temperature

Bilinear in plane of element, linear along each edgePressure

Reference: Ahmad(1), Cook(5)

14.91.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. The mass matrix is diagonalized as described in Section 13.2: Lumped Matrices.

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Section 14.91: SHELL91 - Nonlinear Layered Structural Shell

14.91.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to thecenterplane.

Each triad of integration points (in the r direction) is assumed to have the same element (material) orientation.

There is no significant stiffness associated with rotation about the element r axis. A nominal value of stiffness ispresent using the approach of Zienkiewicz(39), however, to prevent free rotation at the node.

14.91.3. Stress-Strain Relationship

The material property matrix [D]j for the layer j is:

(14–533)[ ]D

BE B E

B E BE

G

G

f

x xy x

xy x y

xy

yz

=

ν

ν

0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0GG

fxz

where:

BE

E E

y j

y j xy j x j

=−

,

, , ,( )ν 2

Ex,j = Young's modulus in layer x direction of layer j (input as EX on MP command)

νxy,j = Poisson's ratio in layer x-y plane of layer j (input as NUXY on MP command)

Gxy,j = shear modulus in layer x-y plane of layer j (input as GXY on MP command)

f A

t

=+

1 2

1 0 225 2

.

. ., whichever is greater

A = element area (in s-t plane)t = average total thickness

The above definition of f is designed to avoid shear locking. Unlike most other elements, the temperature-de-pendent material properties are evaluated at each of the in-plane integration points, rather than only at thecentroid.

14.91.4. Stress, Force and Moment Calculations

The shape functions assume that the transverse shear strains are constant thru the thickness. However, thesestrains must be zero at the free surface. Therefore, unless nonlinear materials are used or the sandwich optionis used (KEYOPT(9) = 1), they are adjusted by:

(14–534)σ σxzj xzjr′ = −32

1 2( )

Chapter 14: Element Library

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(14–535)σ σyzj yzjr′ = −3

21 2( )

where typically:

σxzj′ = adjusted value of transverse shear stress

σxz,j = transverse shear stress as computed from strain-displacement relationships

r = normal coordinate, varying from -1.0 (bottom) to +1.0 (top)

Even with this adjustment, these strains will not be exact due to the variable nature of the material propertiesthru the thickness. However, for thin shell environments, these strains and their resulting stresses are small incomparison to the x, y, and xy components. The interlaminar shear stresses are equivalent to the transverse shearstresses at the layer boundaries and are computed using equilibrium considerations, and hence are more accuratefor most applications.

14.91.5. Force and Moment Summations

The in-plane forces are computed as:

(14–536)T tx j

x jt

x jb

j

N=

+

=

∑σ σ, ,

21

l

(14–537)T ty j

y jt

y jb

j

N=

+

=

∑σ σ, ,

21

l

(14–538)T txy j

xy jt

xy jb

j

N=

+

=

∑σ σ, ,

21

l

where typically:

Tx = in-plane x force per unit length (output as TX)

Nl = numbers of layers

σx jt, = stress at top of layer j in element x direction

σx jb

, = stress at bottom of layer j in element x direction

tj = thickness of layer j

The out-of-plane moments are computed as:

(14–539)M t z z z zx j x j

bjb

jt

x jt

jt

jb

j

N= + + +

=∑1

62 2

1( ( ) ( )), ,σ σ

l

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Section 14.91: SHELL91 - Nonlinear Layered Structural Shell

(14–540)M t z z z zy j y j

bjb

jt

y jt

jt

jb

j

N= + + +

=∑1

62 2

1( ( ) ( )), ,σ σ

l

(14–541)M t z z z zxy j xy j

bjb

jt

xy jt

jt

jb

j

N= + + +

=∑1

62 2

1( ( ) ( )), ,σ σ

l

where, typically:

Mx = x-moment per unit length (output as MX)

z jb = z coordinate of bottom layer j

z jt = z coordinate of top layer j

z = coordinate normal to shell, with z = 0 being at shell midsurface

The transverse shear forces are computed as:

(14–542)N tx j xz j

j

N=

=∑ σ ,

1

l

(14–543)N ty j yz j

j

N=

=∑ σ ,

1

l

where, typically:

Nx = transverse x-shear force per unit length (output as NX)

σxz,j = average transverse shear stress in layer j in element x-z plane

For this computation of transverse shear forces, the shear stresses have not been adjusted as shown in Equa-tion 14–534 and Equation 14–535.

14.91.6. Interlaminar Shear Stress Calculation

In the absence of body forces, the in-plane equilibrium equations of infinitesimally small volume are:

(14–544)∂∂

+∂

∂+ ∂

∂=σ σ σx xy xz

x y z0

(14–545)

∂∂

+∂∂

+∂

∂=

σ σ σyx y yz

x y z0

Rewriting these in incremental form,

(14–546)∆ ∆ ∆

∆∆

∆σ σ σ

xzx xyz

x y= − +

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(14–547)∆ ∆

∆∆

∆∆

σσ σ

yzyx yzx y

= − +

Setting these equations in terms of layer j,

(14–548)∆

∆∆

∆∆

σσ σ

xz j jx j xy jtx y,

, ,= − +

(14–549)∆

∆∆

∆∆

σσ σ

yz j jyx j y jtx y,

, ,= − +

where:

∆σ σ σ σ σx j x j x j x j x j, , , , ,( ) .= + − −2 3 1 4 2 0

∆σ σ σ σ σxy j xy j xy j xy j xy j, , , , ,( ) .= + − −3 4 1 2 2 0

∆σ σ σ σ σyx j xy j xy j xy j xy j, , , , ,( ) .= + − −2 3 1 4 2 0

∆σ σ σ σ σy j y j y j y j y j, , , , ,( ) / .= + − −3 4 1 2 2 0

σx j,3 = stress in element x direction in layer j at integratiion point 3

∆x and ∆y are shown in Figure 14.47: “Integration Point Locations”.

Figure 14.47 Integration Point Locations

! "

∆#∆ $

Thus, the interlaminar shear stress is:

(14–550)τ σxk

xz jj

k

x jj

kS t= −

= =∑ ∑∆ ,

1 1

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Section 14.91: SHELL91 - Nonlinear Layered Structural Shell

(14–551)τ σyk

yz jj

k

y jj

kS t= −

= =∑ ∑∆ ,

1 1

where, typically,

τxk = interlaminar shear stress between layers k and k+1 (outtput as ILSXZ)

Stx

xz jj

N

= ==∑ ∆σ ,

( )1correction term

t = total thickness

14.91.7. Sandwich Option

If KEYOPT(9) = 1, SHELL91 uses “sandwich” logic. This causes:

• The term f in Equation 14–533 to be set to 1.0 for the middle layer (core).

• The transverse shear moduli (Gyz and Gxz) are set to zero for the top and bottom layers.

• The transverse shear strains and stresses in the face plate (non-core) layers are set to 0.0.

• As mentioned earlier, the adjustment to the transverse shear strains and stresses in the core as suggestedby Equation 14–534 and Equation 14–535 is not done.

14.92. SOLID92 - 3-D 10-Node Tetrahedral Structural Solid

Integration PointsShape FunctionsMatrix or Vector

4Equation 12–164, Equation 12–165, and Equation 12–166Stiffness, Mass, and StressStiffness Matrices; andThermal Load Vector

6Equation 12–164, Equation 12–165, and Equation 12–166specialized to the face

Pressure Load Vector

DistributionLoad Type

Same as shape functionsElement Temperature

Chapter 14: Element Library

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DistributionLoad Type

Same as shape functionsNodal Temperature

Linear over each facePressure

Reference: Zienkiewicz(39)

14.92.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.93. SHELL93 - 8-Node Structural Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

Thru-the-thickness:2 (linear material)5 (nonlinear material)

In-plane: 2 x 2

Equation 12–83Quad

Stiffness Matrix and ThermalLoad Vector Thru-the-thickness:

2 (linear material)5 (nonlinear material)

In-plane: 3

Equation 12–56Triangle

Same as stiffness matrixEquation 12–69, Equation 12–70, andEquation 12–71

QuadMass and Stress StiffnessMatrices

Same as stiffness matrixEquation 12–46, Equation 12–47, andEquation 12–48

Triangle

2 x 2Equation 12–71QuadTransverse Pressure LoadVector 3Equation 12–48Triangle

2Same as in-plane mass matrix, specialized to the edgeEdge Pressure Load Vector

DistributionLoad Type

Linear thru thickness, bilinear in plane of elementElement Temperature

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Section 14.93: SHELL93 - 8-Node Structural Shell

DistributionLoad Type

Constant thru thickness, bilinear in plane of elementNodal Temperature

Bilinear in plane of element, linear along each edgePressure

Reference: Ahmad(1) Cook(5)

14.93.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. The mass matrix is diagonalized as described in Section 13.2: Lumped Matrices.

14.93.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to thecenterplane.

Each pair of integration points (in the r direction) is assumed to have the same element (material) orientation.

There is no significant stiffness associated with rotation about the element r axis. A nominal value of stiffness ispresent (as described with SHELL63), however, to prevent free rotation at the node.

14.93.3. Stress-Strain Relationships

The material property matrix [D] for the element is:

(14–552)[ ]D

BE B E

B E BE

G

G

f

x xy x

xy x y

xy

yz

=

ν

ν

0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0GG

fxz

where:

BE

E E

y

y xy x

=− ( )ν 2

Ex = Young's modulus in element x direction (input as EX on MP command)

νxy = Poisson's ratio in element x-y plane (input as NUXY on MP command)

Gxy = shear modulus in element x-y plane (input as GXY on MP command)

f A

t

=+

1 2

1 0 225 2

.

. ., whichever is greater

A = element area (in s-t plane)t = average thickness

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The above definition of f is designed to avoid shear locking.

14.93.4. Stress Output

The stresses at the center of the element are computed by taking the average of the four integration points onthat plane. See Section 13.6: Nodal and Centroidal Data Evaluation for more details.

The output forces and moments are computed as described in Section 2.3: Structural Strain and Stress Evaluations.

14.94. CIRCU94 - Piezoelectric Circuit

I

J

K

Integration PointsShape FunctionsMatrix or Vector

NoneNone (lumped)Stiffness Matrix

NoneNone (lumped, harmonic analysis only)Damping Matrix

NoneNone (lumped)Load Vector

The piezoelectric circuit element, CIRCU94, simulates basic linear electric circuit components that can be directlyconnected to the piezoelectric FEA domain. For details about the underlying theory, see Wang and Ostergaard(323).It is suitable for the simulation of circuit-fed piezoelectric transducers, piezoelectric dampers for vibration control,crystal filters and oscillators etc.

14.94.1. Electric Circuit Elements

CIRCU94 contains 5 linear electric circuit element options:

(KEYOPT(1) = 0)a. Resistor

(KEYOPT(1) = 1)b. Inductor

(KEYOPT(1) = 2)c. Capacitor

(KEYOPT(1) = 3)d. Current Source

(KEYOPT(1) = 4)e. Voltage Source

Options a, b, c, d are defined by two nodes I and J (see figure above), each node having a VOLT DOF. The voltagesource is also characterized by a third node K with CURR DOF to represent an auxiliary charge variable.

14.94.2. Piezoelectric Circuit Element Matrices and Load Vectors

The finite element equations for the resistor, inductor, capacitor and current source of CIRCU94 are derived usingthe nodal analysis method (McCalla(188)) that enforces Kirchhoff's Current Law (KCL) at each circuit node. To be

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Section 14.94: CIRCU94 - Piezoelectric Circuit

compatible with the system of piezoelectric finite element equations (see Section 11.2: Piezoelectrics), the nodalanalysis method has been adapted to maintain the charge balance at each node:

(14–553)[ ] K V Q =

where:

[K] = stiffness (capacitance) matrixV = vector of nodal voltages (to be determined)Q = load vector of nodal charges

The voltage source is modeled using the modified nodal analysis method (McCalla(188)) in which the set of un-knowns is extended to include electric charge at the auxiliary node K, while the corresponding entry of the loadvector is augmented by the voltage source amplitude. In a transient analysis, different integration schemes areemployed to determine the vector of nodal voltages.

For a resistor, the generalized trapezoidal rule is used to approximate the charge at time step n+1 thus yielding:

(14–554)[ ]K

tR

=−

−

=θ∆ 1 1

1 1stiffness matrix

(14–555) V

V

V

In

Jn

=

=

+

+

1

1nodal voltages

(14–556) Q

Q

Q

Rn

Rn

=−

=

+

+

1

1element vector charge

where:

θ = first order time integration parameter (input on TINTP command)∆t = time increment (input on DELTIM command)R = resistance

Q i t qRn

Rn

Rn+ = − +1 1( )θ ∆

q i t i t qRn

Rn

Rn

Rn+ += + − +1 1 1θ θ∆ ∆( )

iV V

RRn I

nJn

++ +

= −11 1

The constitutive equation for an inductor is of second order with respect to the charge time-derivative, andtherefore the Newmark integration scheme is used to derive its finite element equation:

(14–557)[ ]K

tL

=−

−

=α∆ 2 1 1

1 1stiffness matrix

(14–558) Q

Q

Q

Ln

Ln

=−

=

+

+

1

1vector charge

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where:

L = inductance

QtL

V V i t qLn

In

Jn

Ln

Ln+ = −

− + +121

2α ∆ ∆( )

qtL

V VtL

V V i t qLn

In

Jn

In

Jn

Ln

Ln+ + += − + −

− + +12

1 121

2α α∆ ∆ ∆( ) ( )

it

LV V

tL

V V iLn

In

Jn

In

Jn

Ln+ + += − + − − +1 1 1 1δ δ∆ ∆

( ) ( ) ( )

α, δ = Newmark integration parameters (input on TINTP command

A capacitor with nodes I and J is represented by

(14–559)[ ]K C=

−−

=

1 1

1 1stiffness matrix

(14–560) Q

Q

Q

Cn

Cn

=−

=

+

+

1

1charge vector

where:

C = capacitance

Q C V V qCn

In

Jn

Cn+ = − − +1 ( )

q C V V C V V qCn

In

Jn

In

Jn

Cn+ + += − − − +1 1 1( ) ( )

For a current source, the [K] matrix is a null matrix, while the charge vector is updated at each time step as

(14–561) Q =

−

+

+

Q

Q

Sn

Sn

1

1

where:

Q tI tI QSn

Sn

Sn

Sn+ += + − +1 1 1θ θ∆ ∆( )

ISn+ =1 source current at time tn+1

Note that for the first substep of the first load step in a transient analysis, as well as on the transient analysis restart,all the integration parameters (θ, α, δ) are set to 1. For every subsequent substep/load step, ANSYS uses eitherthe default integration parameters or their values input using the TINTP command.

In a harmonic analysis, the time-derivative is replaced by jω, which produces

(14–562)[ ]K j

R= −

−−

ω

ω1 1 1

1 12

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Section 14.94: CIRCU94 - Piezoelectric Circuit

for a resistor,

(14–563)[ ]K

L= −

−−

1 1 1

1 12ω

for an inductor, and

(14–564)[ ]K C=

−−

=

1 1

1 1capacitor

where:

j = imaginary unitω = angular frequency (input on HARFRQ command)

The element charge vector Q is a null vector for all of the above components.

For a current source, the [K] matrix is a null matrix and the charge vector is calculated as

(14–565) Q =

−

Q

QS

S

where:

Qj

I eS Sj= 1

ωφ

IS = source current amplitude

φ = source current phase angle (in radians)

Note that all of the above matrices and load vectors are premultiplied by -1 before being assembled with thepiezoelectric finite element equations that use negative electric charge as a through variable (reaction “force”)for the VOLT degree of freedom.

14.95. SOLID95 - 3-D 20-Node Structural Solid

Chapter 14: Element Library

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Integration PointsShape FunctionsGeometryMatrix or Vector

14 if KEYOPT(11) = 02 x 2 x 2 if KEYOPT(11) = 1

Equation 12–209 , Equation 12–210,and Equation 12–211

Brick

Stiffness, Mass, and StressStiffness Matrices; andThermal Load Vector

3 x 3Equation 12–186, Equation 12–187, andEquation 12–188

Wedge

2 x 2 x 2Equation 12–171, Equation 12–172, andEquation 12–173

Pyramid

4Equation 12–164, Equation 12–165, andEquation 12–166

Tet

3 x 3Equation 12–69 and Equation 12–70QuadPressure Load Vector

6Equation 12–46 and Equation 12–47Triangle

DistributionLoad Type

Same as shape functions thru elementElement Temperature

Same as shape functions thru elementNodal Temperature

Bilinear across each facePressure

Reference: Zienkiewicz(39)

14.95.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. If KEYOPT(3) = 1, the mass matrix is diagonalized as described in Section 13.2: Lumped Matrices.

14.96. SOLID96 - 3-D Magnetic Scalar Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–203Magnetic Scalar Potential Coefficient Matrix; andLoad Vector of Magnetism due to Permanent Mag-nets, and Source Currents

References: Coulomb(76), Mayergoyz(119), Gyimesi(141,149)

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Section 14.96: SOLID96 - 3-D Magnetic Scalar Solid

14.96.1. Other Applicable Sections

Section 5.2: Derivation of Electromagnetic Matrices discusses the magnetic scalar potential method used by thiselement.

14.97. SOLID97 - 3-D Magnetic Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–194, Equation 12–195, and Equa-tion 12–196

Magnetic Vector Potential CoefficientMatrix and Load Vector of Magnetismdue to Source Currents, PermanentMagnets, and Applied Currents

2 x 2 x 2Equation 12–202Electric Potential Coefficient Matrix

DistributionLoad Type

Trilinearly thru elementCurrent Density, Voltage Load and Phase Angle Distribution

References: Coulomb(76), Mohammed(118), Biro et al.(120)

14.97.1. Other Applicable Sections

Section 5.2: Derivation of Electromagnetic Matrices and Section 5.3: Electromagnetic Field Evaluations containa discussion of the 2-D magnetic vector potential formulation which is similar to the 3-D formulation of thiselement.

Chapter 14: Element Library

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14.98. SOLID98 - Tetrahedral Coupled-Field Solid

Integration PointsShape FunctionsMatrix or Vector

4Equation 12–169Magnetic Potential CoefficientMatrix

4Equation 12–168Electric Conductivity Matrix

4Equation 12–167Thermal Conductivity Matrix

4Equation 12–164, Equation 12–165, and Equa-tion 12–166

Stiffness and Mass Matrices; andThermal Expansion Load Vector

4Same as combination of stiffness matrix and conduct-ivity matrix

Piezoelectric Coupling Matrix

11Same as conductivity matrix. If KEYOPT(3) = 1, matrixis diagonalized as described in Section 13.2: LumpedMatrices

Specific Heat Matrix

4Same as coefficient or conductivity matrix

Load Vector due to ImposedThermal and Electric Gradients,Heat Generation, Joule Heating,Magnetic Forces, Permanent Mag-net and Magnetism due to SourceCurrents

6Same as stiffness or conductivity matrix, specializedto the face

Load Vector due to Convectionand Pressures

References: Zienkiewicz(39), Coulomb(76), Mayergoyz(119), Gyimesi(141)

14.98.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Chapter 6, “Heat Flow” describes the derivation of thermal element matrices and load vectors aswell as heat flux evaluations. Section 5.2: Derivation of Electromagnetic Matrices describes the scalar potentialmethod, which is used by this element. Section 11.2: Piezoelectrics discusses the piezoelectric capability usedby the element. If KEYOPT(3) = 1, the specific heat matrix is diagonalized as described in Section 13.2: LumpedMatrices. Also, Section 14.69: SOLID69 - 3-D Coupled Thermal-Electric Solid discusses the thermoelectric capab-ility.

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Section 14.98: SOLID98 - Tetrahedral Coupled-Field Solid

14.99. SHELL99 - Linear Layered Structural Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

Thru the thickness: 2In-plane: 2 x 2 Equation 12–83Quad

Stiffness Matrix and ThermalLoad Vector Thru the thickness: 2

In-plane: 3Equation 12–56Triangle

Same as stiffness matrixEquation 12–69, Equation 12–70, andEquation 12–71

QuadMass and Stress StiffnessMatrices

Same as stiffness matrixEquation 12–46, Equation 12–47, andEquation 12–48

Triangle

2 x 2Equation 12–71QuadTransverse Pressure LoadVector 3Equation 12–48Triangle

2Same as in-plane mass matrix, specialized to theedge

Edge Pressure Load Vector

DistributionLoad Type

Linear thru thickness, bilinear in plane of elementElement Temperature

Constant thru thickness, bilinear in plane of elementNodal Temperature

Bilinear in plane of element, linear along each edgePressure

References: Ahmad(1), Cook(5), Yunus et al.(139)

14.99.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. The mass matrix is diagonalized as described in Section 13.2: Lumped Matrices.

14.99.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to thecenterplane.

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Each pair of integration points (in the r direction) is assumed to have the same material orientation.

There is no significant stiffness associated with rotation about the element r axis. A nominal value of stiffness ispresent using the approach of Zienkiewicz(39), however, to prevent free rotation at the node.

14.99.3. Direct Matrix Input

SHELL99 has two options for the direct input of the matrices that account for the stiffness and mass effects aswell as one thermal load distribution. This permits the user to incorporate the results of their own compositematerial programs, as well as lifting any restriction as to the number of layers.

If KEYOPT(2) = 3, the matrices [E0], [E1], [E2], [E3], and [E4] are input directly (input as A, B, D, E, and F, respectively

on the R and RMORE commands). For the thermal load, the vectors S0, S1, and S2 are also input directly (input

as MT, BT, and QT on the R and RMORE commands). [E3], [E4], and S2 are used only if KEYOPT(2) = 2 and if the

shell is curved. Further, for both cases, the average density is input directly (input as AVDENS on the RMOREcommand).

Considering the KEYOPT(2) = 2 case for a flat shell, the thru thickness accumulated effects can be derived followingthe theoretical formulation given in reference (139) as:

(14–566)[ ] [ ] [ ] [ ]E T D T drm j

Tj m j

r

r

j

N

jbt

jtp

01

= ∫=∑

l

(14–567)[ ] [ ] [ ] [ ]E r T D T drm j

Tj m j

r

r

j

N

jbt

jtp

11

= ∫=∑

l

(14–568)[ ] [ ] [ ] [ ]E r T D T drm j

Tj m j

r

r

j

N

jbt

jtp

22

1= ∫

=∑

l

(14–569)[ ] [ ] [ ] [ ]E r T D T drm j

Tj m j

r

r

j

N

jbt

jtp

33

1= ∫

=∑

l

(14–570)[ ] [ ] [ ] [ ]E r T D T drm j

Tj m j

r

r

j

N

jbt

jtp

44

1= ∫

=∑

l

(14–571) [ ] [ ] S T D drm j

Tj

thj

r

r

j

N

jbt

jtp

01

= ∫=∑ ε

l

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Section 14.99: SHELL99 - Linear Layered Structural Shell

(14–572) [ ] [ ] S r T D drm j

Tj

thj

r

r

j

N

jbt

jtp

11

= ∫=∑ ε

l

(14–573) [ ] [ ] S r T D drm j

Tj

thj

r

r

j

N

jbt

jtp

22

1= ∫

=∑ ε

l

where:

Nl = numbers of layers

[D]j = stress-strain relationships at point of interest within layer j

[Tm] = layer to element transformation matrix

[E0], [E1], [E2], S0, and S1 can be used to define the forces and moments on a unit square out of the flat shell:

(14–574)

[ ] [ ]

[ ] [ ]

N

M

E E

E E

S

S

=

−

0 1

1 2

0

1

εκ

where:

N = forces per unit lengthM = moments per unit lengthε = strainsk = curvatures

Each of the above matrices and load vectors are of sizes 6 x 6 and 6 x 1, as opposed to the 3 x 3 and 3 x 1 sizescommonly used in thin shell analysis. Thus, if only 3 x 3 matrix information is available, it is recommended to useKEYOPT(2) = 4, which transforms the 3 x 3 matrices to a 6 x 6 matrices (using [E0] as an example):

(14–575)[ ]E

G G G

G G G

L

G G G

H

o =

11 12 13

12 22 23

13 23 33

0 0 0

0 0 0

0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 HH

where:

G = 3 x 3 matrix of terms available from outside of the ANSYS program

L = G1110-8

H = G33C

C = transverse shear multiplier (input as TRSHEAR on RMORE command)

As discussed earlier, the values in [E0] (as well as other matrices) used by the ANSYS program for either the layer

or matrix input may be printed with KEYOPT(10) = 1 in order to verify the input.

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For matrix input, the required stress vector Nc needed for stress stiffening is computed as:

(14–576) ([ ][ ] [ ][ ]) N E B E B Sc = + +0 0 1 1 0δ

where:

δ = ue from the previous iteration

14.99.4. Stress Calculations

Strains and stresses are computed at the top and bottom of each layer (KEYOPT(9) = 0) or at the midthickness(KEYOPT(9) = 1). The strains within layer j are:

(14–577) [ ] [ ] ε j m j eT B u=

where:

ue = element displacement vector

The stresses within layer j are:

(14–578) [ ] ( )σ ε ε

jD j j

thj= −

where:

εthj = thermal strain in layer j

14.99.5. Force and Moment Summations

First, all stresses are converted from the layer orientation to the element orientation:

(14–579) [ ] σ σe j m j

TjT=

where:

σej = stresses in element orientation

To simplify the below descriptions, the subscript e is dropped. The in-plane forces are computed as:

(14–580)T tx j

x jt

x jb

j

N=

+

=

∑σ σ, ,

21

l

(14–581)T ty j

y jt

y jb

j

N=

+

=

∑σ σ, ,

21

l

(14–582)T txy jxy jt

xy jb

j

N=

+

=

∑σ σ, ,

21

l

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Section 14.99: SHELL99 - Linear Layered Structural Shell

where, typically,

Tx = in-plane force per unit length (output as TX)

σx jt, = stress at top of layer j in element x direction

σx jb

, = stress at bottom of layer j in element x direction

tj = thickness of layer j

The out-of-plane moments are computed as:

(14–583)M t z z z zx j x j

bjb

jt

x jt

jt

jb

j

N= + + +

=∑1

62 2

1( ( ) ( )), ,σ σ

l

(14–584)M t z z z zy j y j

bjb

jt

y jt

jt

jb

j

N= + + +

=∑1

62 2

1( ( ) ( )), ,σ σ

l

(14–585)M t z z z zxy j xy j

bjb

jt

xy jt

jt

jb

j

N= + + +

=∑1

62 2

1( ( ) ( )), ,σ σ

l

where, typically,

Mx = x-moment per unit length (output as MX)

z jb = z coordinate of bottom layer j

z jt = z coordinate of top layer j

z = coordinate normal to shell, with z = 0 being at shell midsurface

The transverse shear forces are computed as:

(14–586)N tx j xz j

j

N=

=∑ σ ,

1

l

(14–587)N ty j yz j

j

N=

=∑ σ ,

1

l

where, typically,

Nx = transverse x-shear force per unit length (output as NX)

σxz,j = average transverse shear stress in layer j in element x-z plane

For this computation of transverse shear forces, the shear stresses have not been adjusted as shown in the nextsubsection.

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14.99.6. Shear Strain Adjustment

The shape functions assume that the transverse shear strains are constant thru the thickness. However, thesestrains must be zero at the free surface. Therefore, they are adjusted by:

(14–588)′ = −ε εxz j xz jr, ,( )

32

1 2

(14–589)′ = −ε εyz j yz jr, ,( )

32

1 2

where typically,

′ =εxz j, adjusted value of transverse shear strain

εxz,j = transverse shear strain as computed from strain-displacement relationships

r = normal coordinate, varying from -1.0 (bottom) to +1.0 (top)

Even with this adjustment, these strains will not be exact due to the variable nature of the material propertiesthru the thickness. However, for thin shell environments, these strains and their resulting stresses are small incomparison to the x, y, and xy components. The interlaminar shear stresses are equivalent to the transverse shearstresses at the layer boundaries and are computed using equilibrium considerations, and hence are more accuratefor most applications.

14.99.7. Interlaminar Shear Stress Calculations

In the absence of body forces, the in-plane equilibrium equations of infinitesimally small volume are

(14–590)∂∂

+∂

∂+ ∂

∂=σ σ σx xy xz

x y z0

(14–591)

∂∂

+∂∂

+∂

∂=

σ σ σyx y yz

x y z0

Rewriting these in incremental form,

(14–592)∆ ∆ ∆

∆∆

∆σ σ σ

xzx xyz

x y= − +

(14–593)∆ ∆

∆∆

∆∆

σσ σ

yzyx yzx y

= − +

Setting these equations in terms of layer j,

(14–594)∆

∆∆

∆∆

σσ σ

xz j jx j xy jtx y,

, ,= − +

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Section 14.99: SHELL99 - Linear Layered Structural Shell

(14–595)∆

∆∆

∆∆

σσ σ

yz j jyx j y jtx y,

, ,= − +

where:

∆σ σ σ σ σx j x j x j x j x j, , , , ,( ) .= + − −2 3 1 4 2 0

∆σ σ σ σ σxy j xy j xy j xy j xy j, , , , ,( ) .= + − −3 4 1 2 2 0

∆σ σ σ σ σyx j xy j xy j xy j xy j, , , , ,( ) .= + − −2 3 1 4 2 0

∆σ σ σ σ σy j y j y j y j y j, , , , ,( ) / .= + − −3 4 1 2 2 0

σx j,3 = stress in element x direction in layer j at integratiion point 3

∆x and ∆y are shown in Figure 14.47: “Integration Point Locations” in Section 14.91: SHELL91 - Nonlinear LayeredStructural Shell.

The interlaminar shear stress components between layer k and layer k+1 may now be written as:

(14–596)τ σxk

xz jj

k

x jj

kS t= −

= =∑ ∑∆ ,

1 1

(14–597)τ σyk

yz jj

k

y jj

kS t= −

= =∑ ∑∆ ,

1 1

where, typically,

τxk = interlaminar shear stress between layers k and k+1 (outtput as ILSXZ)

Atx

xz jj

N

= ==∑ ∆σ ,

( )1correction term

t = total thickness

Finally,

(14–598)τ τ τkxk

yk= +( ) ( )2 2

where:

τk = maximum interlaminar shear stress between layers K and K + 1 (output as ILSUM)

The maximum of all values τk is τmaxk

(output as ILMAX). If τmaxk

is less than a small number β, the interlaminarshear stress printout is suppressed and the post data values are set to zero. β is determined by:

(14–599)β σ σ τ= + +− ∑10 8 ( )x y xy

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where the summation is over all integration points in the top and bottom layers (or in layers LP1 and LP2, if re-quested).

Finally, a check is made on the validity of the interlaminar shear stresses. R is defined as:

(14–600)Rt A Ax y

maxk

=+2 2

τ

where:

R = error measure (output as Max. Adjustment / Max. Stress)

R is output if it is greater than 0.1.

14.100. Not Documented

No detail or element available at this time.

14.101. Not Documented

No detail or element available at this time.

14.102. Not Documented

No detail or element available at this time.

14.103. Not Documented

No detail or element available at this time.

14.104. Not Documented

No detail or element available at this time.

14.105. Not Documented

No detail or element available at this time.

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Section 14.105: Not Documented

14.106. VISCO106 - 2-D 4-Node Viscoplastic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–103, Equation 12–104, andEquation 12–105

QuadStiffness and Mass Matrices;and Thermal Load Vector 3 if axisymmetric

1 if planeEquation 12–84, Equation 12–85, andEquation 12–86

Triangle

2Same as stiffness matrix, specialized to facePressure Load Vector

DistributionLoad Type

Bilinear across element, constant thru thickness or around circumferenceElement Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

References: Oden(123), Weber et al.(127), Anand(159) and Brown et al.(147)

14.106.1. Other Applicable Sections

For the basic element formulation refer to Section 14.107: VISCO107 - 3-D 8-Node Viscoplastic Solid. Rate-de-pendent plasticity (Anand's model) is described in Section 4.2: Rate-Dependent Plasticity.

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14.107. VISCO107 - 3-D 8-Node Viscoplastic Solid

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–191, Equation 12–192, and Equation 12–193Stiffness and Mass Matrices;and Thermal Load Vector

2 x 2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

References: Oden(123), Weber et al.(127)

14.107.1. Basic Assumptions

This section discusses the basic theory of the large strain viscoplastic elements. The elements developed use theupdated Lagrangian concept along with logarithmic (Hencky) strain and Cauchy (true) stress measures. Thematerial is limited to be isotropic in nature and elastic strains are assumed to be small relative to plastic strains.Further the plastic flow is assumed to be isochoric (i.e. volume preserving) and both the rate-independent andrate-dependent elastic-plastic constitutive relationship is considered.

The strain energy calculation is based on integration of loading rates and a large time increment may produceinaccurate energy values, even though stress-strain solutions are quite accurate.

14.107.2. Element Tangent Matrices and Newton-Raphson Restoring Force

The formulation considered is highly nonlinear in nature both from the point of view of kinematic or geometricconsideration as well as constitutive behavior. The full Newton-Raphson solution option is utilized when theNewton-Raphson restoring force is given by (see Equation 15–106):

(14–601) [ ] ( )F B d volnr T

vol= ∫ σ

where:

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Section 14.107: VISCO107 - 3-D 8-Node Viscoplastic Solid

[B] = strain-displacement matrixσ = Cauchy stress

Equation 14–601 is modified by assuming a decomposition of the Cauchy stress into the deviatoric part plus thepressure part:

(14–602) σ σ= ′ − q P

where:

σ' = Cauchy stress deviator

q = 1 1 1 0 0 0

P = hydrostatic stress- hydrostatic stress = (σx + σy + σz) / 3 (output as HPRES)

The pressure is separately interpolated to conveniently allow for enforcement of the incompressibility constraintassociated with large plastic strains (Oden and Kikuchi(123)). The restoring force can now be rewritten as:

(14–603) [ ] ( ) [ ] ( )F B d vol B q P d volnr T

vol

T= ′ −∫ ∫σ

The incompressibility constraint during plastic flow is enforced through the augmentation of the momentumequations with the additional equation:

(14–604)[ ] ( ( )) ( )

^N J J P d volP T

vol

∆ ∆ ∆− =∫ 0

where:

[NP] = shape function associated with the independently interpolated pressure DOF∆J = determinant of the relative deformation gradient (the relative volume change)

∆ J^

= constitutively prescribed function expressing the presssure-volume change relationship and expressed as:

(14–605)∆ ∆J

PK

^exp= −

where:

K = elastic bulk modulus for the material (= E / (3(1-2ν)))E = Young's modulus (input as EX on MP command)ν = Poisson's ratio (input as NUXY on MP command)

The total Cauchy stress is calculated by finding the deviatoric part from the constitutive equations using thestrains calculated from nodal displacements and subtracting the separately interpolated pressure, i.e.:

(14–606) σ σ= ′ − −q Po

where:

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Po = interpolated from the pressure field

The stiffness matrix for the static analysis is constructed by evaluating the exact Jacobian of the discretized system.This yields an equation of the form:

(14–607)

K K

K K

u

P

F F

F

uu up

pu pp

u

p

=

−

∆∆ 0

where:

F = external nodal forces∆u, ∆P = increments of displacement and pressure, respectively

[ ] ( )F B d volu T

vol= ∫ σ

[ ] ( ( )) ( )^

F N J J P d volp p T

vol

= −∫ ∆ ∆ ∆

[ ] [ [ ] ( )]Ku

B d voluu T

vol

= ∂∂ ∫ σ

[ ] [ ] [ [ ] ( )]K Kp

B d volup pu T T

vol= = ∂

∂ ∫ σ

[ ] [ [ ]( ( ) ( )]^

Kp

N J J P d volpp p

vol

= ∂∂

−∫ ∆ ∆ ∆

The tangent matrix developed in Equation 14–607 has two parts, namely the constitutive part and geometricpart. From the requirement of full tangent matrices, both the constitutive and geometric parts are essential, butthe numerical efficiency and stability considerations can prove to be different. Thus, it is left on the user to controlthe inclusion of stress (geometric) stiffness (using the SSTIF command). Symmetry of the stiffness matrix isachieved by assuming small strain increments for the constitutive part and negligible volume change duringthe step for the geometric part. The assumptions generally result in good convergence characteristic for theseelements even when these assumptions of small strain increments and negligible volume change are violated.Additional detail of the stiffness matrix can be found in Weber et al.(127).

For the constitutive part of the rate-dependent plasticity (Anand's model), see Section 4.2: Rate-DependentPlasticity. Section 13.1: Integration Point Locations describes the integration point locations.

14.107.3. Plastic Energy Output

(14–608)E vole

pl T pli

j

N

i

N cs=

==∑∑

intσ ε∆

11

where:

Nint = number of integration points

Ncs = total number of converged substeps

∆εpl = plastic strain increment

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Section 14.107: VISCO107 - 3-D 8-Node Viscoplastic Solid

14.108. VISCO108 - 2-D 8-Node Viscoplastic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

3 x 3Equation 12–117, Equation 12–118, andEquation 12–119

Quad

Stiffness and Mass Matrices;and Thermal Load Vector 3Equation 12–96, Equation 12–97, and

Equation 12–98Triangle

2Same as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functions across element, constant thru thickness or aroundcircumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

References: Oden(123), Weber et al.(127), Anand(159) and Brown et al.(147)

14.108.1. Other Applicable Sections

For the basic element formulation refer to Section 14.107: VISCO107 - 3-D 8-Node Viscoplastic Solid. Rate-de-pendent plasticity (Anand's model) is described in Section 4.2: Rate-Dependent Plasticity.

14.108.2. Assumptions and Restrictions

A dropped midside node implies that the edge is and remains straight.

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14.109. TRANS109 - 2-D Electromechanical Transducer

TRANS109 realizes strong electromechanical coupling between distributed and lumped mechanical and electro-static systems. TRANS109 is especially suitable for the analysis of Micro Electromechanical Systems (MEMS): ac-celerometers, pressure sensors, microactuators, gyroscopes, torsional actuators, filters, HF and optical switches,etc.

TRANS109 (Gyimesi and Ostergaard(329) and Gyimesi et al.(346)) is the 2-D extension of strongly coupled linetransducer TRANS126 (Gyimesi and Ostergaard(248)), (Section 11.5: Review of Coupled ElectromechanicalMethods, and Section 14.126: TRANS126 - Electromechanical Transducer). TRANS109 is a 2-D 3-node elementwith triangle geometry. It supports three degrees of freedom at its nodes: mechanical displacement, UX and UY,as well as electrical scalar potential, VOLT. Its reaction solutions are mechanical forces, FX and FY, and electricalcharge, CHRG.

The element potential energy is stored in the electrostatic domain. The energy change is associated with thechange of potential distribution in the system, which produces mechanical reaction forces. The finite elementformulation of the TRANS109 transducer follows standard Ritz-Galerkin variational principles which ensure thatit is compatible with regular finite elements. The electrostatic energy definition is

(14–609)W V C VT= 1

2 [ ]

where:

V = vector of nodal voltagessuperscript T = denotes matrix transpose[C] = element capacitance matrix

The vector of nodal electrostatic charges, q, can be obtained as

(14–610) [ ] q C v=

where:

q = vector of nodal charges

The capacitance matrix, [C], depends on the element geometry:

(14–611)[ ] [ ]( )C C u=

where:

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Section 14.109: TRANS109 - 2-D Electromechanical Transducer

u = vector of nodal displacements

According to the principle of virtual work

(14–612) f

dWd

u=

where:

f = vector of nodal mechanical reaction forces

At equilibrium, the electrostatic forces between each transducer elements as well as transducers and mechanicalelements balance each other. The mesh, including the air region, deforms so that the force equilibrium be obtained.

During solution, TRANS109 automatically morphs the mesh based on equilibrium considerations. This meansthat users need to create an initial mesh using usual meshing tools, then during solution TRANS109 automaticallychanges the mesh according to the force equilibrium criteria. No new nodes or elements are created duringmorphing, but the displacements of the original nodes are constantly updated according to the electromechan-ical force balance. The morph supports large displacements, even of uneven meshes.

14.110. INFIN110 - 2-D Infinite Solid

Integration PointsMapping and Shape FunctionsMatrix or Vector

2 x 2Equation 12–123, Equation 12–126, and Equa-tion 12–127

Magnetic Potential CoefficientMatrix

2 x 2Equation 12–124, Equation 12–126, and Equa-tion 12–127

Thermal Conductivity and SpecificHeat Matrices

2 x 2Equation 12–125, Equation 12–126, and Equa-tion 12–127

Electrical Potential CoefficientMatrix

References: Zienkiewicz et al.(169), Damjanic' and Owen(170), Marques and Owen(171), Li et al.(172)

14.110.1. Mapping Functions

The theory for the infinite mapping functions is briefly summarized here. Consider the 1-D situation shown below:

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Figure 14.48 Global to Local Mapping of a 1-D Infinite Element

J K M

x

xk

x j

xo

O J Ka a

(in global coordinates)

M(at infinity)

MAP

(in local coordinates)t=-1 t=1t=0

r

The 1-D element may be thought of as one edge of the infinite element of Figure 14.49: “Mapping of 2-D SolidInfinite Element”. It extends from node J, through node K to the point M at infinity and is mapped onto the parent

element defined by the local coordinate system in the range -1 ≤ t ≤ 1.

Figure 14.49 Mapping of 2-D Solid Infinite Element

**

t

s

M

J

IK

N 8

L

Y

N

X,R

Poles of Mapping

Map

8

M

I

J

The position of the "pole", xo, is arbitrary, and once chosen, the location of node K is defined by

(14–613)x x xK J o= 2

The interpolation from local to global positions is performed as

(14–614)x t M t x M t xJ J K K( ) ( ) ( )= +

where:

MJ(t) = -2t/(1 - t)

MK(t) = 1 - MJ(t)

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Section 14.110: INFIN110 - 2-D Infinite Solid

Examining the above mapping, it can be seen that t = -1, 0, 1 correspond respectively to the global positions x= xJ, xK, ∞ , respectively.

The basic field variable, A (Az for KEYOPT(1) = 0, VOLT for KEYOPT(1) = 1 or TEMP for KEYOPT(1) = 2) can be inter-

polated using standard shape functions, which when written in polynomial form becomes

(14–615)A t b b t b t b t( ) = + + + + − − − −0 1 22

33

Solving Equation 14–614 for t yields

(14–616)t

ar

= −12

where:

r = distance from the pole, O, to a general point within the elementa = xK - xJ as shown in Figure 14.49: “Mapping of 2-D Solid Infinite Element”

Substituting Equation 14–616 into Equation 14–615 gives

(14–617)A t c

cr

c

r

c

r( ) = + + + + − − − − −0

1 22

33

Where c0 = 0 is implied since the variable A is assumed to vanish at infinity.

Equation 14–617 is truncated at the quadratic (r2) term in the present implementation. Equation 14–617 alsoshows the role of the pole position, O.

In 2-D (Figure 14.49: “Mapping of 2-D Solid Infinite Element”) mapping is achieved by the shape function products.The mapping functions and the Lagrangian isoparametric shape functions for 2-D and axisymmetric 4 nodequadrilaterals are given in Section 12.6.6: 2-D and Axisymmetric 4 Node Quadrilateral Infinite Solids. The shapefunctions for the nodes M and N are not needed as the field variable, A, is assumed to vanish at infinity.

14.110.2. Matrices

The matrices can be written as:

(14–618)[ ] [ ] [ ][ ]K B D B dvole

vol

T= ∫

(14–619)[ ] C C N N dvole c

vol

T= ∫

where:

N = shape functions given in Section 12.6.6: 2-D and Axisymmetric 4 Node Quadrilateral Infinite Solids[B] = defined later

1. AZ DOF (KEYOPT(1) = 0)

[Ke] = magnetic potential coefficient matrix

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[ ]Do

=

11 0

0 1µ

µo = magnetic permeability of free space (input on EMUNIT command)

The infinite elements can be used in magnetodynamic analysis even though these elements do notcompute mass matrices. This is because air has negligible conductivity.

2. VOLT DOF (KEYOPT(1) = 1)

[Ke] = electrical potential coefficient matrix

[ ]Dx

y

=

ε

ε

0

0

εx, εy = electrical permittivity (input as PERX and PERY on MP command)

3. TEMP DOF (KEYOPT(1) = 2)

[Ke] = thermal conductivity matrix

[Ce] = specific heat matrix

[ ]D

k

k

x

y

=

0

0

kx, ky = thermal conductivities in the x and y direction (input as KXX and KYY on MP command)

Cc = ρ Cp

ρ = density of the fluid (input as DENS on MP command)Cp = specific heat of the fluid (input as C on MP command)

Although it is assumed that the nodal DOFs are zero at infinity, it is possible to solve thermal problems in whichthe nodal temperatures tend to some constant value, To, rather than zero. In that case, the temperature differential,

θ (= T - To), may be thought to be posed as the nodal DOF. The actual temperature can then be easily found from

T = θ + To. For transient analysis, θ must be zero at infinity t > 0, where t is time. Neumann boundary condition is

automatically satisfied at infinity. Note that with these infinite elements, meaningful steady-state thermal ana-lysis can be performed only when the problem is driven by heat sources or sinks (BF or BFE command with Lab= HGEN).

The Bi vectors of the [B] matrix in Equation 14–618 contain the derivatives of Ni with respect to the global co-

ordinates which are evaluated according to

(14–620) [ ]B

Nx

Ny

J

Ns

Nt

i

i

i

i

i

=

∂∂

∂∂

=

∂∂

∂∂

1

where:

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Section 14.110: INFIN110 - 2-D Infinite Solid

[J] = Jacobian matrix which defines the geometric mapping

[J] is given by

(14–621)[ ]J

Ms

xMs

y

Mt

xMt

yi

ii

ii

ii

ii

= ∑

∂∂

∂∂

∂∂

∂∂

=1

4

The mapping functions [M] in terms of s and t are given in Section 12.6.6: 2-D and Axisymmetric 4 Node Quadri-lateral Infinite Solids. The domain differential d(vol) must also be written in terms of the local coordinates, sothat

(14–622)d vol dx dy J dsdt( ) | |= =

Subject to the evaluation of Bi and d(vol), which involves the mapping functions, the element matrices [Ke] and

[Ce] may now be computed in the standard manner using Gaussian quadrature.

14.111. INFIN111 - 3-D Infinite Solid

Integration PointsMapping and Shape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–129, Equation 12–130, Equation 12–131,Equation 12–132, Equation 12–133, and Equation 12–134

Magnetic or Electrical ScalarPotential Coefficient Matrix orThermal Conductivity Matrix

2 x 2 x 2Equation 12–129, Equation 12–132, Equation 12–133,and Equation 12–134

Specific Heat Matrix

2 x 2 x 2Equation 12–126, Equation 12–127, Equation 12–128,Equation 12–132, Equation 12–133, and Equation 12–134

Magnetic Vector PotentialCoefficient Matrix

14.111.1. Other Applicable Sections

See Section 14.110: INFIN110 - 2-D Infinite Solid for the theoretical development of infinite solid elements. Thederivation presented in Section 14.110: INFIN110 - 2-D Infinite Solid for 2-D can be extended to 3-D in astraightforward manner.

Chapter 14: Element Library

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14.112. Not Documented

No detail or element available at this time.

14.113. Not Documented

No detail or element available at this time.

14.114. Not Documented

No detail or element available at this time.

14.115. INTER115 - 3-D Magnetic Interface

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–60, Equation 12–61,Equation 12–64, and Equation 12–66

QuadCoefficient Matrix and Load Vector

1Equation 12–41, Equation 12–42,Equation 12–43, and Equation 12–45

TriangleCoefficient Matrix and Load Vector

14.115.1. Element Matrix Derivation

A general 3-D electromagnetics problem is schematically shown in Figure 14.50: “A General ElectromagneticsAnalysis Field and Its Component Regions”. The analysis region of the problem may be divided into three parts.Ω1 is the region of conduction, in which the conductivity, σ, is not zero so that eddy currents may be induced.

Ω1 may also be a ferromagnetic region so that the permeability µ is much larger than that of the free space, µo.

However, no source currents exist in Ω1. Both Ω2 and Ω3 are regions free of eddy currents. There may be source

currents present in these regions. A distinction is made between Ω2 and Ω3 to ensure that the scalar potential

region, Ω3, is single-connected and to provide an option to place the source currents in either the vector potential

or the scalar potential region. ΓB and ΓH represent boundaries on which fluxes are parallel and normal respectively.

In Ω1, due to the nonzero conductivity and/or high permeability, the magnetic vector potential together with

the electric scalar potential are employed to model the influence of eddy currents. In Ω2, only the magnetic

vector potential is used. In Ω3, the total magnetic field is composed of a reduced field which is derived from the

magnetic reduced scalar potential, φ, and the field, Hs, which is computed using the Biot-Savart law.

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Section 14.115: INTER115 - 3-D Magnetic Interface

Figure 14.50 A General Electromagnetics Analysis Field and Its Component Regions

14.115.2. Formulation

The A, V-A-θ Formulation

The equations relating the various field quantities are constituted by the following subset of Maxwell's equationswith the displacement currents neglected.

(14–623)

∇ × − − =

∇ × + ∂∂

=

∇ ⋅ =

H J J

EBt

B

s e 0

0

0

inn Ω1

(14–624)

∇ × =

∇ ⋅ =

H J

B

s

02 3in Ω Ω∪

The constitutive relationships are:

(14–625) [ ] B H= µ

The boundary and interface conditions, respectively, are:

(14–626) B nTB⋅ = 0 on Γ

(14–627) H n H× = 0 on Γ

Chapter 14: Element Library

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(14–628)

,

B n B n

H n H n

T1 1 2 2

1 1 2 2

12

0

0

⋅ + ⋅ =

× + × =

on Γ ΓΓ Γ13 23,

Variables are defined in Section Section 5.1: Electromagnetic Field Fundamentals.

By introducing the magnetic vector potential, A (AX, AY, AZ), both in Ω1 and Ω2; the electric scalar potential V

(VOLT) in Ω1; and the generalized scalar potential φg (MAG) in Ω3, the field quantities can be written in terms of

various potentials as:

(14–629) B A= ∇ × in andΩ Ω1 2

(14–630) E

At

V= − ∂∂

− ∇ in Ω1

(14–631) H Hs g= ∇φ in Ω3

In order to make the solution of potential A unique, the Coulomb gauge condition is applied to define the di-vergence of A in addition to its curl.

Substituting Equation 14–629 through Equation 14–631 into the field equations and the boundary conditionsEquation 14–623 through Equation 14–628 and using the Galerkin form of the method of weighted residualequations, the weak form of the differential equations in terms of the potentials A, V and φg can be obtained.

Through some algebraic manipulations and by applying the boundary as well as interface conditions, respectively,the finite element equations may be written as:

(14–632)

Ω Ω1 2+∫ ∇ × ∇ × + ∇⋅ ∇⋅ + ⋅ ∂( [ ] ) [ ]( ) [ ]( [ ] ) ( ) [ ][ ]N A N A NA

T TA

T TA

Tν ν σ AAt

Nvt

d N n dAT

AT

g

∂

+ ⋅∇ ∂∂

− ⋅∇ ×+∫[ ][ ] [ ] σ φΩ ΓΓ Γ13 23 3

== ⋅ ×( ) + ⋅+∫ ∫Γ Γ ΩΓ Ω13 23 22 2[ ] [ ] N Hs n d N J dA

TA

T

(14–633)Ω Ω1

0∫ ∇ ⋅ ∂∂

+ ∇ ⋅∇ ∂∂

=[ ] [ ] σ σN

At

Nvt

d

(14–634)

− ∇ ⋅∇ + ⋅ ∇ ×

+ ⋅ ∇ ×

∫ ∫

∫

Ω Γ

Γ

Ω Γ3 23

13

2

1

[ ]( ) ( )

µ φN d N n A d

N n

Tg

AA d N H dTs [ ] ( ) = − ∇( ) ⋅∫Γ ΩΩ2

µ

where:

[NA] = matrix of element shape functions for A

N = vector of element shape function for both V and φv = related to the potential V as:

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Section 14.115: INTER115 - 3-D Magnetic Interface

(14–635)V

vt

= ∂∂

A number of interface terms arise in the above equations because of the coupling of vector potential and scalarpotential formulations across different regions. These are the terms that involve integration over the surfaceshared by two adjoining subregions and are given as:

(14–636)I N n dA g1 313 23

= − ⋅ ∇ ×+∫ [ ] ( )Γ Γ Γφ

(14–637)I N n A d2 313 23

= − ⋅ ∇ ×+∫ ( )Γ Γ Γ

(14–638)I NA H n ds3 313 23

= − ⋅ ×+∫ [ ] ( )Γ Γ Γ

where:

Γij = surface at the interface of subregions Ωi and Ωj, respectively.

The term, I3, contributes to the load vector while the terms, I1 and I2, contribute to the coefficient matrix. The

asymmetric contributions of I1 and I2 to the coefficient matrix may be made symmetric following the procedure

by Emson and Simkin(176). After some algebraic manipulations including applying the Stokes' theorem, we get

(14–639)I I I2 21 22= +

(14–640)I N n A d21 313 23

= − ∇ × ⋅+∫ ( ) Γ Γ Γ

(14–641)I N A d22

13 23= ⋅

+∫ Γ ΓÑ l

It is observed from Equation 14–639 that the integrals represented by I1 and I2 are symmetric if the condition I22

= 0 is satisfied. The integral given by I22 is evaluated along a closed path lying on the interface. If the interface

lies completely inside the region of the problem, the integrals over the internal edges will cancel each other; ifthe integral path is on a plane of symmetry, the tangential component of A will be zero, so the integral will bevanish; and if the integral path is on the part of the boundary where the scalar potential is prescribed, the termscontaining N will be omitted and the symmetry of the matrix will be ensured. Therefore, the condition that ensuressymmetry can usually be satisfied. Even if, as in some special cases, the condition can not be directly satisfied,the region may be remeshed to make the interface of the vector and scalar potential regions lie completely insidethe problem domain. Thus, the symmetry condition can be assumed to hold without any loss of generality.

Replacing the vector and scalar potentials by the shape functions and nodal degrees of freedom as describedby Equation 14–642 through Equation 14–645,

(14–642) [ ] A N AAT

e=

(14–643)∂∂

=At

N AAT

e[ ]

(14–644)φ φgT

eN=

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(14–645)V

vt

N VTe= ∂

∂=

the above manipulations finally result in the following set of finite element equations:

(14–646)

( [ ] ) [ ]( [ ] ) [ ] [ ] [ ]

[ ]

∇ × ∇ × + ∇⋅ ∇ ⋅

+

∫ N N N N d AAT T

AT

AT

AT

eν ν

σ

Ω

Ω

Ω1

11 1

13

∫ ∫⋅ + ⋅∇

−∫ ⋅ ∇

[ ] [ ] [ ][ ]

[ ]

N N d A N N d V

N N

AT

A e AT

e

AT

Ω ΩΩ

Γ

& σ

××( ) = −∫ ⋅ × [ ] [ ] n d N N n d He AT

A s3 313Γ ΓΓφ

(14–647)[ ] [ ] [ ] σ σΩ ΩΩ Ω

1 10∫ ∫∇ ⋅ + ∇ ⋅∇ =N N d A N N d VT

A eT

e&

(14–648)

− ∇ ⋅∇ + ⋅ ∇ ×

+ ⋅ ∇ ×

∫ ∫

∫

[ ]( ) ( )

(

µ φΩ Γ

Γ

Ω Γ3 23

13

2

1

N d N n A d

N n

Tg

) ( ) [ ] A d N H dTsΓ ΩΩ= − ∇ ⋅∫ 2

µ

(14–649)

− ∇ ⋅∇ − ∇ × ⋅

= −

∫ ∫ +[ ] [ ]

[ ]

µ φ

µ

Ω Γ ΓΩ Γ3 13 23 3N N d N n N d AT T

A ee

ΩΩ Ω2

∫ ∇ [ ] N N d HTA s

Equation 14–646 through Equation 14–649 represent a symmetric system of equations for the entire problem.

The interface elements couple the vector potential and scalar potential regions, and therefore have AX, AY, AZand MAG degrees of freedom at each node. The coefficient matrix and the load vector terms in Equation 14–646through Equation 14–649 are computed in the magnetic vector potential elements (SOLID97), the scalar potentialelements SOLID96, SOLID98 with KEYOPT(1) = 10, or SOLID5 with KEYOPT(1) = 10) and the interface elements(INTER115). The only terms in these equations that are computed in the interface elements are given by:

Coefficient Matrix:

(14–650)

[ ] [ ] ( )

( ) [ ]

K N N n d

N n N

AT

TA

= − ⋅ ∇ ×

− ∇ × ⋅

+

+

∫

∫

Γ Γ

Γ Γ

Γ13 23

13 23

3

3 ddΓ

Load Vector:

(14–651) [ ] ([ ] )F N N n dA

TA= − ⋅ ×+∫Γ Γ Γ

13 23 3

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Section 14.115: INTER115 - 3-D Magnetic Interface

14.116. FLUID116 - Coupled Thermal-Fluid Pipe

Integration PointsShape FunctionsGeometryMatrix or Vector

NoneEquation 12–13Between nodes I and J

Thermal Conductivity MatrixNoneNone

Convection betweennodes I and K andbetween nodes J and L(optional)

NoneEquation 12–12Between nodes I and JPressure Conductivity Matrix

NoneEquation 12–13Specific Heat Matrix and HeatGeneration Load Vector

14.116.1. Assumptions and Restrictions

Transient pressure and compressibility effects are also not included.

14.116.2. Combined Equations

The thermal and pressure aspects of the problem have been combined into one element having two differenttypes of working variables: temperatures and pressures. The equilibrium equations for one element have theform of:

(14–652)N C T

NK

Kc

t

c

t

p

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

0

0 0 0

0

0

+

&

=

+

T

P

Q

wN Q

Hc

g

where:

[Ct] = specific heat matrix for one channelT = nodal temperature vector

&T = vector of variations of nodal temperature with respectt to time

Chapter 14: Element Library

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P = nodal pressure vector

[Kt] = thermal conductivity matrix for one channel (includes effects of convection and mass transport)

[Kp] = pressure conductivity matrix for one channelQ = nodal heat flow vector (input as HEAT on F command)w = nodal fluid flow vector (input as FLOW on F command)

Qg = internal heat generation vector for one channelH = gravity and pumping effects vector for one channelNc = number of parallel flow channels (input as Nc on R command)

14.116.3. Thermal Matrix Definitions

Specific Heat Matrix

The specific heat matrix is a diagonal matrix with each term being the sum of the corresponding row of a con-sistent specific heat matrix:

(14–653)[ ]C At

c=

1 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

where:

AC AL

cu p o=

ρ2

ρ

ρ

u

gas abs

or

PR T

= =

=

effective density

(ideal gas law

if Rgas 0 0.

)) if Rgas ≠

0 0.

ρ = mass density (input as DENS on MP commandP = pressure (average of first two nodes)Tabs = T + TOFFST = absolute temperature

T = temperature (average of first two nodes)TOFFST = offset temperature (input on TOFFST command)Cp = specific heat (input as C on MP command)

A = flow cross-sectional area (input as A on R command)Lo = length of member (distance between nodes I and J)

Rgas = gas constant (input as Rgas on R command)

Thermal Conductivity Matrix

The thermal conductivity matrix is given by:

(14–654)[ ]K

B B B B B B

B B B B B B

B B

B B

t =

+ − − + −− − + + −

−−

1 2 4 1 4 2

1 5 1 3 5 3

2 2

3 3

0

0

0 0

0 0

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Section 14.116: FLUID116 - Coupled Thermal-Fluid Pipe

where:

BAKs

1 =l

Ks = thermal conductivity (input as KXX on MP command)

B2 = h AI

h = film coefficient (defined below)

AI = lateral area of pipe associated with end I (input as (Ann I) on command)

(defaults to if KEYOPT(2) = 2, defa

RπDL

2uults to DL if KEYOPT(2) = 3)π

B3 = h AJ

AJ = lateral area of pipe associated with end I (input as (Ann J) on command)

(defaults to if KEYOPT(2) = 2, defa

RπDL

2uults to DL if KEYOPT(2) = 4)π

D = hydraulic diameter (input as D on R command)

BwCp

4 =if flow is from node J to node I

if flow is from no0 dde I to node J

BwCp

5 =

0

if flow is from node I to node J

if flow is from noode J to node I

w = mass fluid flow rate in the element

w may be determined by the program or may be input by the user:

(14–655)W =

computed from previous iteration if pressure is a degreee of freedom

or

input (VAL1 on SFE,,, command) if pressHFLUX uure is not a degree of freedom

The above definitions of B4 and B5, as used by Equation 14–654, cause the energy change due to mass transport

to be lumped at the outlet node.

The film coefficient h is defined as:

(14–656)h =

material property input (HF on MP command) if KEYOPT(4) = 0

or

NuK

Dif KEYOPT(4) = 1

or

table input (TB, HFLM table)

s

iif KEYOPT(4) = 2,3, or 4

or

defined by user programmable

featture User116Hfif KEYOPT(4) = 5

Nu, the Nusselt number, is defined for KEYOPT(4) = 1 as:

(14–657)Nu N N Re PrN N

= +1 23 4

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where:

N1 to N4 = input constants (input on R commands)

Re = =wDAµ

Reynolds number

µ = viscosity (input as VISC on MP command)

Pr = =C

Kp

s

µPrandtl number

A common usage of Equation 14–657 is the Dittus-Boelter correlation for fully developed turbulent flow insmooth tubes (Holman(55)):

(14–658)Nu Re Pra= 0 023 0 8. .

where:

a = 0.4 for heating

0.3 for cooling

Heat Generation Load Vector

The internal heat generation load vector is due to both average heating effects and viscous damping:

(14–659) Q

Q

Qg

n

n=

0

0

where:

QL

Aq V C F vno

DF ver= +2

2( )&&& π µ

&&&q = internal heat generation rate per unit volume (input oon or command)BF BFE

VDF = viscous damping multiplier (input on RMORE command)

Cver = units conversion factor (input on RMORE command)

F = flow type factor = 8.0 if Re 2500.0

0.21420 if Re > 2500.0

≤

v = average velocity

The expression for the viscous damping part of Qn is based on fully developed laminar flow.

14.116.4. Fluid Equations

Bernoulli's equation is:

(14–660)ZP v

gP

ZP v

gC

vgI

I I PMPJ

J JL

a+ + + = + + +γ γ γ

2 2 2

2 2 2

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Section 14.116: FLUID116 - Coupled Thermal-Fluid Pipe

where:

Z = coordinate in the negative acceleration directionP = pressureγ = ρgg = acceleration of gravityv = velocityPPMP = pump pressure (input as Pp on R command)

CL = loss coefficient

The loss coefficient is defined as:

(14–661)C

fDL = +l

lβ

where:

β = =extra flow loss factor

if KEYOPT(8) = 0

or

if KEY

fD

k

all

lOOPT(8) = 1

la = additional length to account for extra flow losses (input as La on R command)

k = loss coefficient for typical fittings (input as K on R command)f = Moody friction factor, defined below:

For the first iteration of the first load step,

(14–662)f

f f

fm m

m=

≠=

if

if

0 0

1 0 0 0

.

. .

where:

fm = input as MU on MP command

For all subsequent iterations

(14–663)f

f

fx

m=if KEYOPT(7) = 0

if KEYOPT(7) = 1

table input(defined bby TB, FLOW) if KEYOPT(7) = 2,3

The smooth pipe empirical correlation is:

(14–664)fRe

Re

ReRe

x =

< ≤

<

640 2500

0 3162500

1 4

or

.

( ) /

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Bernoulli's Equation 14–660 may be simplified for this element, since the cross-sectional area of the pipe doesnot change. Therefore, continuity requires all velocities not to vary along the length. Hence v1 = v2 = va, so that

Bernoulli's Equation 14–660 reduces to:

(14–665)Z Z

P P PC

vgI J

I J PMPL− +

−+ =

γ γ

2

2

Writing Equation 14–665 in terms of mass flow rate (w = ρAv), and rearranging terms to match the second halfof Equation 14–652,

(14–666)2 22

22 2ρ ρ

γA

CP P w

g AC

Z ZP

LI J

LI J

PMP( )− = + − + −

Since the pressure drop (PI - PJ) is not linearly related to the flow (w), a nonlinear solution will be required. As the

w term may not be squared in the solution, the square root of all terms is taken in a heuristic way:

(14–667)A

CP P w A

CZ Z g P

LI J

LI J PMP

2 2ρ ρ ρ− = + + −(( ) )

Defining:

(14–668)B A

CcL

= 2ρ

and

(14–669)P Z Z g PL I J PMP= − + −( )ρ

Equation 14–667 reduces to:

(14–670)B P P w B Pc I J c L− = +

Hence, the pressure conductivity matrix is based on the term

B

P Pc

I J− and the pressure (gravity and pumping)

load vector is based on the term Bc PL.

Two further points:

1. Bc is generalized as:

(14–671)B

AC

c

L

=

2ρif KEYOPT(6) = 0

input constant (input as C on R commmand) if KEYOPT(6) = 1

table input (defined by TB,FCON) if KKEYOPT(6) = 2 or 3

defined by user programmablefeature, Useer116Cond

if KEYOPT(6) = 4

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Section 14.116: FLUID116 - Coupled Thermal-Fluid Pipe

1. (-ZI + ZJ)g is generalized as:

(14–672)( ) − + =Z Z g x aI JT

t∆

where:

∆x = vector from node I to node Jat = translational acceleration vector which includes effects of angular velocities (see Section 15.1:

Acceleration Effect)

14.117. SOLID117 - 3-D 20-Node Magnetic Edge

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–194, Equation 12–195, and Equa-tion 12–196 for magnetic vector potential; Equa-tion 12–258 thru Equation 12–269 for edge-flux

Edge Formulation of MagneticVector Potential Coefficient Matrixand Load Vector of Magnetism dueto Source Currents, PermanentMagnets, and Applied Currents

2 x 2 x 2Equation 12–250 thru Equation 12–257Electric Potential Coefficient Matrix

DistributionLoad Type

Trilinearly varying over the thru elementCurrent Density, Voltage Load and Phase Angle Distribution

References: Biro et al.(120), Gyimesi and Ostergaard(201), Gyimesi and Ostergaard (221), Ostergaard and Gy-imesi(222), Ostergaard and Gyimesi(223), Preis et al.(203), Nedelec(204), Kameari(206), Jin(207)

14.117.1. Other Applicable Sections

The following sections describe the theorem of the magnetic edge element using edge flux DOF:

• Section 5.1.3: Magnetic Vector Potential

• Section 5.1.4: Edge Flux Degrees of Freedom

• Section 5.1.6: Harmonic Analysis Using Complex Formalism

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• Section 5.2.2: Magnetic Vector Potential

• Section 12.9: Electromagnetic Edge Elements

• Section 13.1: Integration Point Locations

SOLID117 of the ANSYS Elements Reference serves as a reference user guide. 3-D Magnetostatics and Fundamentalsof Edge-based Analysis, 3-D Harmonic Magnetic Analysis (Edge-Based), and 3-D Transient Magnetic Analysis(Edge-Based) of the ANSYS Low-Frequency Electromagnetic Analysis Guide describe respectively static, harmonicand transient analyses by magnetic element SOLID117.

14.117.2. Matrix Formulation of Low Frequency Edge Element and TreeGauging

This low frequency electromagnetic element eliminates the shortcomings of nodal vector potential formulationdiscussed in Section 5.1.6: Harmonic Analysis Using Complex Formalism. The pertinent shape functions arepresented in Section 12.9: Electromagnetic Edge Elements.

The column vector of nodal vector potential components in SOLID97 is denoted by Ae, that of time integrated

scalar potentials by νe. (See definitions in Section 5.2.2: Magnetic Vector Potential.) The vector potential, A,

can be expressed by linear combinations of both corner node vector potential DOFs, Ae, as in SOLID97, and

side node edge-flux DOFs, AZ. For this reason there is a linear relationship between Ae and AZ.

(14–673) [ ] A T AeR Z=

where:

[TR] = transformation matrix. Relationship Equation 14–673 allows to compute the stiffness and dampingmatrices as well as load vectors of SOLID117 in terms of SOLID97.

Substituting Equation 14–673 into Equation 5–109 and Equation 5–110 provides

(14–674) ([ ] [ ] [ ] [ ] )A K A K C d dt A C d dt JZ T ZZz

ZVe

ZZz

ZVe

Z+ + + − =ν ν 00

(14–675) ([ ] [ ] [ ] [ ] )ν ν νeT VZ

zVV

eVZ

zVV

etK A K C d dt A C d dt l+ + + − = 00

where:

[ ] [ ] [ ][ ]K T K TZZ R T AA R=

[ ] [ ] [ ][ ]C T C TZZ R T AA R=

[ ] [ ] [ ][ ]K T K TZV R T AA R=

[ ] [ ] [ ][ ]C T C TZV R T AV R=

[ ] [ ] [ ]J T JZ R T A=

[ ] [ ][ ]K K TVZ VA R=

[ ] [ ][ ]C C TVZ VA R=

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Section 14.117: SOLID117 - 3-D 20-Node Magnetic Edge

Equation 14–674 and Equation 14–675 need to be properly gauged to obtain uniqueness. For more on thistopic see for example Preiss et al.(203). SOLID117 applies a tree gauging algorithm. It considers the relationshipbetween nodes and edges by a topological graph. A fundamental tree of a graph is an assembly of edges consti-tuting a path over which there is one and only one way between different nodes. It can be shown that the edge-flux DOFs over the fundamental tree can be set to zero providing uniqueness without violating generality.

The tree gauging applied is transparent to most users. At the solution phase the extra constraints are automat-ically supplied over the tree edges on top of the set of constraints provided by users. After equation solution,the extra constraints are removed. This method is good for most of the practical problems. However, expertusers may apply their own gauging for specific problems by turning the tree gauging off by the command,GAUGE,OFF.

14.118. Not Documented

No detail or element available at this time.

14.119. HF119 - 3-D High-Frequency Magnetic Tetrahedral Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

VariablePolynomial variable in or-der of 1

Equation 12–164, Equa-tion 12–165, and Equa-tion 12–166

Stiffness, Mass and DampingMatrices

VariablePolynomial variable in or-der of 1

Equation 12–46 andEquation 12–47

Surface PORT, INF, IMPD, SHLDLoad Vectors

DistributionLoad Type

Linear across each faceSurface Loads

14.119.1. Other Applicable Sections

Section 5.6: Inductance, Flux and Energy Computation by LMATRIX and SENERGY Macros describes the derivationof element matrices and load vectors as well as results evaluations.

14.119.2. Solution Shape Functions - H (curl) Conforming Elements

HF119, along with HF120, uses a set of vector solution functions, which belong to the finite element functionalspace, H(curl), introduced by Nedelec(158). These vector functions have, among others, a very useful property,

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i.e., they possess tangential continuity on the boundary between two adjacent elements. This property fits nat-urally the need of HF119 to solve the electric field E based on the Maxwell's equations, since E is only tangentiallycontinuous across material interfaces.

Similar to HF120 as discussed in Section 14.120.2: Solution Shape Functions - H(curl) Conforming Element, theelectric field E is approximated by:

(14–676)E r E W ri i

i

Nvur r u ru r( ) ( )=

=∑

1

where:

rr

= position vector within the element

Nv = number of vector functions

Ei = covariant components of E at proper locations (AX DOFs)

Wi = vector shape functions defined in the tetrahedral element

Refer to the tetrahedral element shown at the beginning of this subsection. The geometry of the element isrepresented by the following mapping:

(14–677)r N L L L L rj j

j

r r=

=∑ ( , , , )1 2 3 4

1

10

where:

Nj = nodal shape functions

Lj = volume coordinates

rj = nodal coordinates

Consider the local oblique coordinate system (s, t, r) based on node K. A set of unitary vectors can be defined as:

(14–678)a

rL

rL

ar

Lr

La

rL

rL

rr r

rr r

rr r

11 3

22 3

34 3

= ∂∂

− ∂∂

= ∂∂

− ∂∂

= ∂∂

− ∂∂

These defines subsequently the gradients of the four volume coordinates:

(14–679)

∇ = × ∇ = ×

∇ = × ∇ = −∇ − ∇ − ∇

=

La a

JL

a aJ

La a

JL L L L

J

t t

t

t

12 3

23 1

41 2

3 1 2 4

r r r r

r r

aa a ar r r1 2 3⋅ ×

The vector shape functions for the first order tetrahedral element can be conveniently defined as

(14–680)W L L L L i j I J K L i jij i j j iu ru

= ∇ − ∇ = ≠, , , ,

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Section 14.119: HF119 - 3-D High-Frequency Magnetic Tetrahedral Solid

The first order element is often referred to as the Whitney element (Whitney(208)).

14.120. HF120 - High-Frequency Magnetic Brick Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

VariablePolynomial variable in orderfrom 1 to 2

Brick, Equation 12–209,Equation 12–210, andEquation 12–211Stiffness, Mass and Damping

Matrices

VariablePolynomial variable in orderfrom 1 to 2

Wedge, Equation 12–186,Equation 12–187, andEquation 12–188

VariablePolynomial variable in orderfrom 1 to 2

Quad, Equation 12–69and Equation 12–70Surface PORT, INF, IMPD, SHLD

Load VectorsVariable

Polynomial variable in orderfrom 1 to 2

Triangle, Equation 12–46and Equation 12–47

DistributionLoad Type

Bilinear across each faceSurface Loads

14.120.1. Other Applicable Sections

Section 5.6: Inductance, Flux and Energy Computation by LMATRIX and SENERGY Macros describes the derivationof element matrices and load vectors as well as result evaluations.

14.120.2. Solution Shape Functions - H(curl) Conforming Element

HF120 uses a set of vector solution functions, which belong to the finite element functional space, H(curl), intro-duced by Nedelec(158). These vector functions have, among others, a very useful property, i.e., they possesstangential continuity on the boundary between two adjacent elements. This property fits naturally the need ofHF120 to solve the electric field E based on the Maxwell's equations, since E is only tangentially continuous acrossmaterial interfaces.

The electric field E is approximated by:

(14–681)E r E W ri ii

Nvur r u ru r( ) ( )=

=∑

1

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where:

rr

= position vector within the element

Nv = number of vector shape functions

Wi = vector shape functions defined in the brick element

Ei = covariant components of E

In the following, three aspects in Equation 14–681 are explained, i.e., how to define the Wi functions, how to

choose the number of functions Nv, and what are the physical meanings of the associated expansion coefficients

Ei. Recall that coefficients Ei are represented by the AX degrees of freedom (DOF) in HF120.

To proceed, a few geometric definitions associated with an oblique coordinate system are necessary. Refer tothe brick element shown at the beginning of this subsection. The geometry of the element is determined by thefollowing mapping:

(14–682)r N s t r ri i

j

r r=

=∑ ( , , )

1

20

where:

Ni = standard isoparametric shape functions

rir

= global coordinates for the 20 nodes

Based on the mapping, a set of unitary basis vectors can be defined (Stratton(209)):

(14–683)ars

art

arr

rr

rr

rr

1 2 3= ∂∂

= ∂∂

= ∂∂

These are simply tangent vectors in the local oblique coordinate system (s, t, r). Alternatively, a set of reciprocalunitary basis vectors can also be defined:

(14–684)

aa a

Ja

a aJ

aa a

JJ a a a

rr r

rr r

rr r

r r r

1 2 3 2 3 1

3 1 21 2 3

= × = ×

= × = ⋅ ×

A vector F may be represented using either set of basis vectors:

(14–685)F f a f ai

ii

jj

j

r r r= =

= =∑ ∑

1

3

1

3

where:

fj = covariant components

fi = contravariant components.

Given the covariant components of a vector F, its curl is found to be

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Section 14.120: HF120 - High-Frequency Magnetic Brick Solid

(14–686)∇ × = ∂

∂∂∂

∂∂

FJ

a a a

s t rf f f

r

r r r

11 2 3

1 2 3

Having introduced the above geometric concepts, appropriate vector shape functions for the brick element aredefined next. For the first order element (KEYOPT(1) = 1), there is one function associated with each edge:

(14–687)w

a

a

a

i

i

i

i

uru

r

r

r=

φ

φ

φ

1

2

3

,

,

,

i=Q,S,U,W

i=R,T,V,X

i=Y,Z,A,B

where:

φi = scalar functions.

Therefore, Nv = 12.

Now consider the second order brick (KEYOPT(1) = 2). There are two functions defined for each edge. For examplefor node Q:

(14–688)w a w ai i i i

uru r uru r( ) ( ) ( ) ( ),1 1 1 2 2 1

= =φ φ

In addition, there are two functions defined associated with each face of the brick. For example, for the faceMNOP (r = 1):

(14–689)w a w af f f f

uru r uru r( ) ( ) ( ) ( ),1 1 1 2 2 1

= =φ φ

The total number of functions are Nv = 36.

Since each vector functions Wi has only one covariant component, it becomes clear that each expansion coeffi-

cients Ei in (1), i.e., the AX DOF, represents a covariant component of the electric field E at a proper location, aside

from a scale factor that may apply. The curl of E can be readily computed by using Equation 14–686.

Similarly, we can define vector shape functions for the wedge shape by combining functions from the brick andtetrahedral shapes. See Section 14.119: HF119 - 3-D High-Frequency Magnetic Tetrahedral Solid for tetrahedralfunctions.

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14.121. PLANE121 - 2-D 8-Node Electrostatic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

3 x 3Equation 12–122QuadDielectric Permittivity and Electrical Con-ductivity Coefficient Matrices, ChargeDensity Load Vector 3Equation 12–102Triangle

2Same as coefficient matrix, specialized tothe face

Surface Charge Density and Load Vector

14.121.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of the electrostatic element matrices and load vectors aswell as electric field evaluations.

14.121.2. Assumptions and Restrictions

A dropped midside node implies that the edge is straight and that the potential varies linearly along that edge.

14.122. SOLID122 - 3-D 20-Node Electrostatic Solid

!

"#

$

%

&

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Section 14.122: SOLID122 - 3-D 20-Node Electrostatic Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

14Equation 12–213BrickDielectric Permittivity and ElectricalConductivity Coefficient Matrices,Charge Density Load Vector

3 x 3Equation 12–190Wedge

8Equation 12–175Pyramid

4Equation 12–168Tet

3 x 3Equation 12–74QuadSurface Charge Density Load Vector

6Equation 12–50Triangle

14.122.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of electrostatic element matrices and load vectors as wellas electric field evaluations.

14.123. SOLID123 - 3-D 10-Node Tetrahedral Electrostatic Solid

Integration PointsShape FunctionsMatrix or Vector

4Equation 12–168Dielectric Permittivity and ElectricalConductivity Coefficient Matrices,Charge Density Load Vector

6Equation 12–168 specialized to the faceCharge Density Surface Load Vector

14.123.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of electrostatic element matrices and load vectors as wellas electric field evaluations.

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14.124. CIRCU124 - Electric Circuit

"!#%$

&

'

(

Integration PointsShape FunctionsMatrix or Vector

NoneNone (lumped)Stiffness Matrix

NoneNone (lumped, harmonic analysis only)Damping Matrix

NoneNone (lumped)Load Vector

14.124.1. Electric Circuit Elements

CIRCU124 contains 13 linear electric circuit element options. These may be classified into two groups:

1. Independent Circuit Element options, defined by 2 or 3 nodes:

Resistor (KEYOPT(1) = 0)

Inductor (KEYOPT(1) = 1)

Capacitor (KEYOPT(1) = 2)

Current Source (KEYOPT(1) = 3)

Voltage Source (KEYOPT(1) = 4)

2. Dependent Circuit Element options, defined by 3, 4, 5, or 6 nodes:

Stranded coil current source (KEYOPT(1) = 5)

2-D massive conductor voltage source (KEYOPT(1) = 6)

3-D massive conductor voltage source (KEYOPT(1) = 7)

Mutual inductor (KEYOPT(1) = 8)

Voltage-controlled current source (KEYOPT(1) = 9)

Voltage-controlled voltage source (KEYOPT(1) = 10)

Current-controlled voltage source (KEYOPT(1) = 11)

Current-controlled current source (KEYOPT(1) = 12)

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Section 14.124: CIRCU124 - Electric Circuit

14.124.2. Electric Circuit Element Matrices

All circuit options in CIRCU124 are based on Kirchhoff's Current Law. These options use stiffness matrices basedon a simple lumped circuit model.

For transient analysis, an inductor with nodes I and J can be presented by:

(14–690)θ∆tL

V

V

I

I

n

Jn

Ln

Ln

1 1

1 11

1

1

1

1− −

=

+

+

+

+

where:

L = inductanceVI = voltage at node I

VJ = voltage at node J

∆t = time incrementθ = time integration parametern = time step n

It

LV V iL

nIn

Jn

Ln+ = − − +1 1( )

( )θ ∆

it

LV V IL

nIn

Jn

Ln+ + + += − +1 1 1 1θ∆

( )

A capacitor with nodes I and J is represented by:

(14–691)C

t

V

V

I

I

In

Jn

cn

cnθ∆

1 1

1 1

1

1

1

1

−−

=

−

+

+

+

+

where:

C = capacitance

Ic

tV V ic

nIn

Jn

cn+ = − − − −1 1

θθ

θ∆( )

ic

tV V Ic

nIn

Jn

cn+ + + += +1 1 1 1

θ∆( )

Similarly, a mutual inductor with nodes I, J, K and L has the following matrix:

(14–692)θ∆t

L L M

L L M M

L L M M

M M L L

M M L L

VI

1 22

2 2

2 2

1 1

1 1

−

− −− −− −

− −

VV

V

V

I

I

I

I

J

K

L

n

n

n

n

=

−

−

+

+

+

+

11

11

21

21

where:

L1 = input side inductance

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L2 = output side inductanceM = mutual inductance

It

L L ML V V M V V in

In

Jn

Kn

Ln n

11

1 22 2 1

1+ = −−

− − − +( )[ ( ) ( )]

θ ∆

It

L L MM V V L V V in

In

Jn

Kn

Ln n

21

1 22 1 2

1+ = −−

− − + − +( )[ ( ) ( )]

θ ∆

it

L L ML V V M V V In

In

Jn

Kn

Ln n

11

1 22 2

1 1 1 11

1+ + + + + +=−

− − − +θ∆[ ( ) ( )]

it

L L MM V V L V V In

In

Jn

Kn

Ln n

21

1 22

1 11

1 11

1+ + + + + +=−

− − + − +θ∆[ ( ) ( )]

For harmonic analysis, the above three circuit element options have only a damping matrix. For an inductor:

(14–693)−

−−

1 1 1

1 12ω L

for a capacitor:

(14–694)j Cω

1 1

1 1

−−

and for a mutual inductor:

(14–695)−

−

− −− −− −

− −

12

1 22

2 2

2 2

1 1

1 1

ω ( )L L M

L L M M

L L M M

M M L L

M M L L

14.125. CIRCU125 - Diode

! #"$% &! #"$

' '

((

Integration PointsShape FunctionsMatrix or Vector

NoneNone (lumped)Stiffness Matrix

NoneNoneDamping Matrix

NoneNone (lumped)Load Vector

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Section 14.125: CIRCU125 - Diode

14.125.1. Diode Elements

CIRCU125 has two highly nonlinear electric circuit element options:

• Common Diode (KEYOPT(1) = 0)

• Zener Diode (KEYOPT(1) = 1)

The I-V characteristics of common and Zener Diodes are plotted in Figure 14.51: “I-V (Current-Voltage) Character-istics of CIRCU125”.

As can be seen, the characteristics of the diodes are approximated by a piece-wise linear curve. The commondiode has two sections corresponding to open and close states. The Zener diode has three sections correspondingto open, block, and Zener states. The parameters of the piece-wise linear curves are described by real constantsdepending on KEYOPT(1) selection.

Figure 14.51 I-V (Current-Voltage) Characteristics of CIRCU125

I I

VZ

RZ

RB

V VV

F

RF

VF

RB

RF

Legend: = Forward voltage

= Zener voltage

= Slope of forward resistance

= Slope of blocking resistance

= Slope of Zener resistance

FV

ZVRFRBRZ

(a) Common Diode (b) Zener Diode

14.125.2. Norton Equivalents

The behavior of a diode in a given state is described by the Norton equivalent circuit representation (see Fig-ure 14.52: “Norton Current Definition”).

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The Norton equivalent conductance, G, is the derivative (steepness) of the I-V curve to a pertinent diode state.The Norton equivalent current generator, I, is the current where the extension of the linear section of the I-Vcurve intersects the I-axis.

Figure 14.52 Norton Current Definition

dynamic resitance

I Norton CurrentN

V

R

I

14.125.3. Element Matrix and Load Vector

The element matrix and load vectors are obtained by using the nodal potential formulation, a circuit analysistechnique which suits perfectly for coupling lumped circuit elements to distributed finite element models.

The stiffness matrix is:

(14–696)K G=

−−

1 1

1 1

The load vector is:

(14–697)F I=

−

1

1

where:

G and I = Norton equivalents of the diode in the pertinent state of operation.

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Section 14.125: CIRCU125 - Diode

14.126. TRANS126 - Electromechanical Transducer

The line electromechanical transducer element, TRANS126, realizes strong coupling between distributed andlumped mechanical and electrostatic systems. For details about its theory see Gyimesi and Ostergaard(248). Formore general geometries and selection between various transducers, see Section 14.109: TRANS109 - 2-D Elec-tromechanical Transducer and Section 11.5: Review of Coupled Electromechanical Methods. TRANS126 is especiallysuitable for the analysis of Micro Electromechanical Systems (MEMS): accelerometers, pressure sensors, microactuators, gyroscopes, torsional actuators, filters, HF switches, etc.

Figure 14.53 Electromechanical Transducer

Physical representation

Finite element representation

V

EMT

K

m

m

D

D

+ -

I+

K

See, for example, Figure 14.53: “Electromechanical Transducer” with a damped spring mass resonator driven bya parallel plate capacitor fed by a voltage generator constituting an electromechanical system. The left sideshows the physical layout of the transducer connected to the mechanical system, the right side shows theequivalent electromechanical transducer element connected to the mechanical system.

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TRANS126 is a 2 node element each node having a structural (UX, UY or UZ) and an electrical (VOLT) DOFs. Theforce between the plates is attractive:

(14–698)F

dCdx

V= 12

2

where:

F = forceC = capacitancex = gap sizeV = voltage between capacitor electrodes

The capacitance can be obtained by using the CMATRIX macro for which the theory is given in Section 5.10:Capacitance Computation.

The current is

(14–699)I C

dVdt

dCdx

vV= +

where:

I = currentt = time

vdxdt

= =

velocity of gap opening

The first term is the usual capacitive current due to voltage change; the second term is the motion induced current.

For small signal analysis:

(14–700)F F D v D

dVdt

K x K Vxv xv xx xv= + + + +0 ∆ ∆

(14–701)I I D v D

dVdt

K x K Vvx vv vx vv= + + + +0 ∆ ∆

where:

F0 = force at the operating point

I0 = current at the operating point

[D] = linearized damping matrices[K] = linearized stiffness matrices∆x = gap change between the operating point and the actual solution∆V = voltage change between the operating point and the actual solution

The stiffness and damping matrices characterize the transducer for small signal prestressed harmonic, modaland transient analyses.

For large signal static and transient analyses, the Newton-Raphson algorithm is applied with F0 and I0 constituting

the Newton-Raphson restoring force and [K] and [D] the tangent stiffness and damping matrices.

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Section 14.126: TRANS126 - Electromechanical Transducer

(14–702)K

dFdx

C Vxx = = ′′12

2

where:

Kxx = electrostatic stiffness (output as ESTIF)

F = electrostatic force between capacitor platesV = voltage between capacitor electrodesC'' = second derivative of capacitance with respect to gap displacement

(14–703)K

dIdV

C vvv = = ′

where:

Kvv = motion conductivity (output as CONDUCT)

I = currentC' = first derivative of capacitance with respect to gap displacementv = velocity of gap opening

Definitions of additional post items for the electromechanical transducer are as follows:

(14–704)P Fvm =

where:

Pm = mechanical power (output as MECHPOWER)

F = force between capacitor platesv = velocity of gap opening

(14–705)P VIe =

where:

Pe = electrical power (output as ELECPOWER)

V = voltage between capacitor electrodesI = current

(14–706)W CVc = 1

22

where:

Wc = electrostatic energy of capacitor (output as CENERGY)

V = voltage between capacitor electrodesC = capacitance

(14–707)F

dCdx

V= 12

2

where:

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F = electrostatic force between capacitor plates (output as EFORCE)C = capacitancex = gap size

dCdx

= first derivative of capacitance with regard to gap

V = voltage between capacitor electrodes

dVdt

= voltage rate (output as DVDT)

14.127. SOLID127 - 3-D Tetrahedral Electrostatic Solid p-Element

Integration PointsShape FunctionsGeometric Shape FunctionsMatrix or Vector

VariablePolynomial variable in orderfrom 2 to 8

Equation 12–164, Equa-tion 12–165, and Equa-tion 12–166

Coefficient Matrix andCharge Density LoadVector

VariablePolynomial variable in orderfrom 2 to 8

Same as coefficient matrixspecialized to face

Surface Charge DensityLoad Vector

14.127.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of electrostatic element matrices and load vectors as wellas electric field evaluations.

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Section 14.127: SOLID127 - 3-D Tetrahedral Electrostatic Solid p-Element

14.128. SOLID128 - 3-D Brick Electrostatic Solid p-Element

Integration PointsShape FunctionsGeometryMatrix or Vector

VariablePolynomial variable in orderfrom 2 to 8

Brick, Equation 12–209, Equa-tion 12–210, and Equa-tion 12–211Coefficient Matrix and

Charge Density LoadVector

VariablePolynomial variable in orderfrom 2 to 8

Wedge, Equation 12–186,Equation 12–187, and Equa-tion 12–188

VariablePolynomial variable in orderfrom 2 to 8

Quad, Equation 12–69 andEquation 12–70Surface Charge Density

Load VectorVariable

Polynomial variable in orderfrom 2 to 8

Triangle, Equation 12–46 andEquation 12–47

14.128.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of electrostatic element matrices and load vectors as wellas electric field evaluations.

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14.129. FLUID129 - 2-D Infinite Acoustic

Integration PointsShape FunctionsMatrix or Vector

2Equation 12–12Fluid Stiffness and Damping Matrices

14.129.1. Other Applicable Sections

The mathematical formulation and finite element discretization are presented in Section 14.130: FLUID130 - 3-D Infinite Acoustic.

14.130. FLUID130 - 3-D Infinite Acoustic

Integration PointsShape FunctionsMatrix or Vector

2 x 2Equation 12–110Fluid Stiffness and Damping Matrices

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Section 14.130: FLUID130 - 3-D Infinite Acoustic

14.130.1. Mathematical Formulation and F.E. Discretization

The exterior structural acoustics problem typically involves a structure submerged in an infinite, homogeneous,inviscid fluid. The fluid is considered linear, meaning that there is a linear relationship between pressure fluctu-ations and changes in density. Equation 14–708 is the linearized, lossless wave equation for the propagation ofsound in fluids.

(14–708)∇ = +2

21

Pc

P in&& Ω

where:

P = pressurec = speed of sound in the fluid (input as SONC on MP command)&&P = second derivative of pressure with respect to time

Ω+ = unbounded region occupied by the fluid

In addition to Equation 14–708), the following Sommerfeld radiation condition (which simply states that thewaves generated within the fluid are outgoing) needs to be satisfied at infinity:

(14–709)lim

r rrd

Pc

P→∞

− +

=12

10&

where:

r = distance from the originPr = pressure derivative along the radial direction

d = dimensionality of the problem (i.e., d =3 or d =2 if Ω+ is 3-D or 2-D respectively

A primary difficulty associated with the use of finite elements for the modeling of the infinite medium stemsprecisely from the need to satisfy the Sommerfeld radiation condition, Equation 14–709. A typical approach for

tackling the difficulty consists of truncating the unbounded domain Ω+ by the introduction of an absorbing(artificial) boundary Γa at some distance from the structure.

Figure 14.54 Absorbing Boundary

R

IL

KJ

Γa

x

y

z

nΓa

n

J

R

x

I

y

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The equation of motion Equation 14–708 is then solved in the annular region Ωf which is bounded by the fluid-

structure interface Γ and the absorbing boundary Γa. In order, however, for the resulting problem in Ωf to be

well-posed, an appropriate condition needs to be specified on Γa. Towards this end, the following second-order

conditions are used (Kallivokas et al.(218)) on Γa:

In two dimensions:

(14–710)P P

cP

cP cP c Pn n+ = − + −

+ + +

γ κ γ κ κγλλ1 1

212

18

12

2&&

where:

n = outward normal to Γa

Pn = pressure derivative in the normal direction

Pλλ = pressure derivative along Γa

k = curvature of Γa

γ = stability parameter

In three dimensions:

(14–711)

& && &P Pc

P Hc

P

H Pc

EG

GE

PGE

P

n n

uu

v

+ = − + −

+ +

+

γ γ

γ

1

2

+ −v

cH K P

22( )

where:

n = outward normalu and v = orthogonal curvilinear surface coordinates (e.g., the meridional and polar angles in spherical co-ordinates)Pu, Pv = pressure derivatives in the Γa surface directions

H and K = mean and Gaussian curvature, respectivelyE and G = usual coefficients of the first fundamental form

14.130.2. Finite Element Discretization

Following a Galerkin based procedure, Equation 14–708 is multiplied by a virtual quantity δP and integrated over

the annular domain Ωf. By using the divergence theorem on the resulting equation it can be shown that:

(14–712)

12c

PPd P Pd PP d PP df fn a n

f f a

δ δ δ δ&& Ω Ω Γ ΓΩ Ω Γ Γ∫ ∫ ∫ ∫+ ∇ ⋅ ∇ − = −

Upon discretization of Equation 14–712, the first term on the left hand side will yield the mass matrix of the fluidwhile the second term will yield the stiffness matrix.

Next, the following finite element approximations for quantities on the absorbing boundary Γa placed at a radius

R and their virtual counterparts are introduced:

(14–713)P x t x P t q x t x q t q x t x qT T( , ) ( ) ( ), ( , ) ( ) ( ), ( , ) ( )( ) ( ) ( )= = =N N N211 1 2

3(( )( )2 t

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Section 14.130: FLUID130 - 3-D Infinite Acoustic

(14–714)δ δ δ δ δ δP x P x q x q x q x q xT T T( ) ( ), ( ) ( ), ( ) ( )( ) ( ) ( ) ( )= = =N N N1 2 31 1 2 2

where:

N1, N2, N3 = vectors of shape functions ( = N1, N2, N3)

P, q(1), q(2) = unknown nodal values (P is output as degree of freedom PRES. q(1) and q(2) are solved for but notoutput).

Furthermore, the shape functions in Equation 14–713 and Equation 14–714 are set to:

(14–715)N N N N1 2 3= = =

The element stiffness and damping matrices reduce to:

For two dimensional case:

(14–716)[ ]K

R

d R d d

R daD

Te

Te

Te

Te

ae

ae

ae

ae

2

2

218

4 4

4=

−∫ ∫ ∫

∫

NN NN NN

NN

λ λ λ

λ

Γ Γ Γ

Γ

−−

−

∫

∫ ∫

4 0

0

2R d

d d

Te

Te

Te

ae

ae

ae

NN

NN NN

λ

λ λ

Γ

Γ Γ

(14–717)[ ]C

c

d

R d

d

aD

Te

Te

Te

ae

ae

ae

2 218

8 0 0

0 4 0

0 0

=

∫

∫

∫

NN

NN

NN

λ

λ

λ

Γ

Γ

Γ

where:

dλe = arc-length differential

These matrices are 6 x 6 in size, having 2 nodes per element with 3 degrees of freedom per node (P, q(1), q(2)).

For three dimensional case:

(14–718)[ ]K

R

dA R dA

R dA RaD

Te

s s Te

s T se

ae

ae

ae

3

2

2 2

12

2

=

∇ ⋅ ∇

∇ ⋅ ∇ −

∫ ∫

∫

NN N N

N N

Γ Γ

Γ

∇∇ ⋅ ∇

∫ s s TedA

ae

N NΓ

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(14–719)[ ]C

C

dA

R dAaD

Te

s s Te

ae

ae

32

12

2 0

0=

− ∇ ⋅ ∇

∫

∫

NN

N N

Γ

Γ

where:

dAe = area differential

These matrices are 8 x 8 in size, having 4 nodes per element with 2 degrees of freedom per node (P, q) (Barry etal.(217)).

For axisymmetric case:

(14–720)[ ]K

R

xd R xd

R xd R xdaDa

Te

Te

Te

Te

ae

ae

ae

2

2

2 2

2

=−

∫ ∫

∫π

λ λ

λ λ

NN NN

N N NN

Γ Γ

Γ ΓΓae∫

(14–721)[ ]C

C

xd

R xdaDa

Te

Te

ae

ae

22

2 0

0=

−

∫

∫π

λ

λ

NN

NN

Γ

Γ

where:

x = radius

These matrices are 4 x 4 in size having 2 nodes per element with 2 degrees of freedom per node (P, q).

14.131. SHELL131 - 4-Node Layered Thermal Shell

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Section 14.131: SHELL131 - 4-Node Layered Thermal Shell

Layer IntegrationPoints

Layer Shape FunctionsGeometryMatrix or Vector

In-Plane: 2 x 2In-Plane: Equation 12–643 unknowns per node per lay-er (KEYOPT(3) = 0)

Conductivity Matrix, HeatGeneration Load Vector,and Convection SurfaceMatrix and Load Vector

Thru Thickness: 2Thru Thickness: Equation 12–25

In-Plane: 2 x 2In-Plane: Equation 12–642 unknowns per node per lay-er (KEYOPT(3) = 1) Thru Thickness: 1Thru Thickness: Equation 12–13

In-Plane: 2 x 2In-Plane: Equation 12–641 unknown per node per layer(KEYOPT(3) = 2) Thru Thickness: 1Thru Thickness: Constant

Same as conductivitymatrix

Same as conductivity matrix. Matrix is diagonalized as describedin Section 13.2: Lumped Matrices

Specific Heat Matrix

14.131.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of the thermal element matrices and load vectors as well as heatflux evaluations.

14.132. SHELL132 - 8-Node Layered Thermal Shell

Layer IntegrationPoints

Layer Shape FunctionsGeometryMatrix or Vector

Quad: 3 x 3Triangle: 3Equation 12–73In-Plane3 unknowns per node per

layer (KEYOPT(3) = 0)Conductivity Matrix,Heat GenerationLoad Vector, Specif-ic Heat Matrix andConvection SurfaceMatrix and LoadVector

2Equation 12–25Thru Thickness

Quad: 3 x 3Triangle: 3Equation 12–73In-Plane2 unknowns per node per

layer (KEYOPT(3) = 1)1Equation 12–13Thru Thickness

Quad: 3 x 3Triangle: 3Equation 12–73In-Plane1 unknown per node per

layer (KEYOPT(3) = 2)1ConstantThru Thickness

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14.132.1. Other Applicable Sections

Chapter 6, “Heat Flow” describes the derivation of the thermal element matrices and load vectors as well as heatflux evaluations.

14.133. Not Documented

No detail or element available at this time.

14.134. Not Documented

No detail or element available at this time.

14.135. Not Documented

No detail or element available at this time.

14.136. FLUID136 - 3-D Squeeze Film Fluid Element

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2 (4-node)Equation 12–63Quad, if KEYOPT(2) = 0Conductivity Matrix and VelocityLoad Vector 3 x 3 (8-node)Equation 12–91Quad, if KEYOPT(2) = 1

Same as conductivitymatrix

Same as conductivity matrix. If KEYOPT(1) = 1, matrix isdiagonalized as described in Section 13.2: LumpedMatrices

Damping Matrix

14.136.1. Other Applicable Sections

Section 7.8: Squeeze Film Theory describes the governing squeeze film equations used as a basis for formingthe element matrices.

14.136.2. Assumptions and Restrictions

A dropped midside node implies that the edge is straight and that the pressure varies linearly along that edge.

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Section 14.136: FLUID136 - 3-D Squeeze Film Fluid Element

14.137. Not Documented

No detail or element available at this time.

14.138. FLUID138 - 3-D Viscous Fluid Link Element

Integration PointsShape FunctionsMatrix or Vector

NoneEquation 12–12Pressure and Damping Matrices

14.138.1. Other Applicable Sections

Section 7.8: Squeeze Film Theory describes the governing squeeze film equations used as a basis for formingthe element matrices.

Chapter 14: Element Library

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14.139. FLUID139 - 3-D Slide Film Fluid Element

!#"%$ & &

!'"%$ & &

!'"%$ & &

)(*

)(*)(

+-, /.10

+-, /.10

+-, .20

+-, /.3

+4, /.3

+-, /.3

5 ./6 , $ & &

5 ./6 , $ & &

0

0

0

3

0 7)( 0 7% 0 7 0 7/ 8 0 7/ 0 7/9( 0 7%

:%;

5 ./6 , $ & &

0 7% 3

0 79( 0 7/ 0 7/ 0 7/ 8 0 7/ 0 7/9( 0 7/%

<>=

<>=

<>=

Integration PointsShape FunctionsMatrix or Vector

NoneAnalytical FormulaFluid, Stiffness, Mass, and DampingMatrices

14.139.1. Other Applicable Sections

Section 7.9: Slide Film Theory describes the governing slide film equations used as a basis for forming the elementmatrices.

14.140. Not Documented

No detail or element available at this time.

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Section 14.140: Not Documented

14.141. FLUID141 - 2-D Fluid-Thermal

t

Y

X,R

J

KS

L

I

Integration PointsShape FunctionsGeometryMatrix or Vector

if 2-D 1 (default) or 2 x 2; ifaxisymmetric 1 or 2 x 2 (de-fault) (adjustable with theFLDATA,QUAD,MOMD com-mand)

Equation 12–107, Equation 12–108, andEquation 12–109

QuadAdvection-Diffusion Matricesfor Momentum Equations (X,Y and Z)

1Equation 12–107, Equation 12–108, andEquation 12–109

Triangle

Same as for momentumequation, but adjustable(with theFLDATA,QUAD,PRSD com-mand)

Equation 12–110Quad

Advection-Diffusion Matrix forPressure Equation 12–110Triangle

Same as for momentum,equations but adjustable(with theFLDATA,QUAD,THRD com-mand)

Equation 12–111Quad

Advection-Diffusion Matrix forEnergy (Temperature) Equation 12–111Triangle

Same as for momentum,equations but adjustable(with theFLDATA,QUAD,TRBD com-mand)

Equation 12–113 and Equation 12–114QuadAdvection-Diffusion Matricesfor Turbulent Kinetic Energyand Dissipation Rate Equation 12–113 and Equation 12–114Triangle

Same as momentum equa-tions, but adjustable (withthe FLDATA,QUAD,MOMScommand)

Same as momentum equation matrixMomentum Equation SourceVector

Same as pressure equations,but adjustable (with theFLDATA,QUAD,PRSS com-mand)

Same as pressure matrixPressure Equation SourceVector

Same as temperature equa-tions, but adjustable (withthe FLDATA,QUAD,THRScommand)

Same as temperature matrixHeat Generation Vector

Chapter 14: Element Library

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Integration PointsShape FunctionsGeometryMatrix or Vector

Same as kinetic energy anddissipation rate equations,but adjustable (with theFLDATA,QUAD,TRBS com-mand)

Same as kinetic energy and dissipation rate matricesTurbulent Kinetic Energy andDissipation Rate Source TermVectors

1Same as momentum equation matrixDistributed Resistance SourceTerm Vector

NoneOne-half of the element face length times the heatflow rate is applied at each edge node

Convection Surface Matrix andLoad Vector and Heat FluxLoad Vector

14.141.1. Other Applicable Sections

Chapter 7, “Fluid Flow” describes the derivation of the applicable matrices, vectors, and output quantities.Chapter 6, “Heat Flow” describes the derivation of the heat transfer logic, including the film coefficient treatment.

14.142. FLUID142 - 3-D Fluid-Thermal

Integration PointsShape FunctionsGeometryMatrix or Vector

1 (default) or 2 x 2 x 2 (adjustablewith the FLDATA,QUAD,MOMDcommand)

Equation 12–197, Equa-tion 12–198, and Equation 12–199

Brick, Pyramid,and WedgeAdvection-Diffusion Mat-

rix for Momentum Equa-tions (X, Y and Z)

1Equation 12–197, Equa-tion 12–198, and Equation 12–199

Tet

Same as for equation momentum,but adjustable (with theFLDATA,QUAD,PRSD command)

Equation 12–200Brick, Pyramid,and WedgeAdvection-Diffusion Mat-

rix for Pressure

1Equation 12–200Tet

Same as for momentum, equationsbut adjustable (with theFLDATA,QUAD,THRD command)

Equation 12–201Brick, Pyramid,and Wedge

Advection-Diffusion Mat-rix for Energy (Temperat-ure)

1Equation 12–201Tet

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Section 14.142: FLUID142 - 3-D Fluid-Thermal

Integration PointsShape FunctionsGeometryMatrix or Vector

Same as for momentum, equationsbut adjustable (with theFLDATA,QUAD,TRBD command)

Equation 12–204 and Equa-tion 12–205

Brick, Pyramid,and Wedge

Advection-DiffusionMatrices for TurbulentKinetic Energy and Dissip-ation Rate 1

Equation 12–204 and Equa-tion 12–205

Tet

1 (default) or 2 x 2 x 2 but ad-justable (with theFLDATA,QUAD,MOMS command)

Equation 12–197, Equa-tion 12–198, and Equation 12–199

Brick, Pyramid,and WedgeMomentum Equation

Source Vector

1Equation 12–197, Equa-tion 12–198, and Equation 12–199

Tet

Same as for equation momentum,but adjustable (with theFLDATA,QUAD,PRSS command)

Equation 12–200Brick, Pyramid,and WedgePressure Equation Source

Vector

1Equation 12–200Tet

Same as for momentum, equationsbut adjustable (with theFLDATA,QUAD,THRS command)

Equation 12–201Brick, Pyramid,and WedgeHeat Generation Vector

1Equation 12–201Tet

Same as for momentum, equationsbut adjustable (with theFLDATA,QUAD,TRBS command)

Equation 12–204 and Equa-tion 12–205

Brick, Pyramid,and WedgeTurbulent Kinetic Energy

and Dissipation RateSource Term Vectors

1Equation 12–204 and Equa-tion 12–205

Tet

Same as momentum equationsource vector

Same as momentum equation source vectorDistributed ResistanceSource Term Vector

None

One-fourth of the element surfacearea times the heat flow rate isapplied at each face node

Brick, Pyramid,and WedgeConvection Surface Mat-

rix and Load Vector andHeat Flux Load Vector One-third of the element surface

area times the heat flow rate isapplied at each face node

Tet

14.142.1. Other Applicable Sections

Chapter 7, “Fluid Flow” describes the derivation of the applicable matrices, vectors, and output quantities.Chapter 6, “Heat Flow” describes the derivation of the heat transfer logic, including the film coefficient treatment.

14.142.2. Distributed Resistance Main Diagonal Modification

Suppose the matrix equation representation for the momentum equation in the X direction written withoutdistributed resistance may be represented by the expression:

(14–722)A V bxm

x xm=

The source terms for the distributed resistances are summed:

(14–723)D K V

f V

DCRx

xx

hxx= + +

ρρ

µ

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where:

DRx = distributed resistance in the x directionKx = loss coefficient in the X direction

ρ = densityfx = friction factor for the X direction

µ = viscosityCx = permeability in the X direction

| V | = velocity magnitudeDhx = hydraulic diameter in the X direction

Consider the ith node algebraic equation. The main diagonal of the A matrix and the source terms are modifiedas follows:

(14–724)A A Diimx

iimx

iRx= +

(14–725)b b D Vimx

imx

iRx

x= + 2

14.142.3. Turbulent Kinetic Energy Source Term Linearization

The source terms are modified for the turbulent kinetic energy k and the turbulent kinetic energy dissipationrate ε to prevent negative values of kinetic energy.

The source terms for the kinetic energy combine as follows:

(14–726)S

VX

VX

V

Xk ti

j

i

j

j

i= ∂

∂∂∂

+∂∂

−µ ρε

where the velocity spatial derivatives are written in index notation and µt is the turbulent viscosity:

(14–727)µ ρεµt C

k=2

where:

ρ = densityCµ = constant

The source term may thus be rewritten:

(14–728)S

VX

VX

V

XC

kk t

i

j

i

j

j

i t= ∂

∂∂∂

+∂∂

−µ ρ

µµ2

2

A truncated Taylor series expansion of the kinetic energy term around the previous (old) value is expressed:

(14–729)S S

Sk

k kk kk

koldold

old

= + ∂∂

−( )

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Section 14.142: FLUID142 - 3-D Fluid-Thermal

The partial derivative of the source term with respect to the kinetic energy is:

(14–730)

∂∂

= −Sk

Ckk

t2 2

µρµ

The source term is thus expressed

(14–731)S

VX

VX

V

XC

kC

kkk t

i

j

i

j

j

i

old

t

old

t= ∂

∂∂∂

+∂∂

+ −µ ρ

µρ

µµ µ2

222

The first two terms are the source term, and the final term is moved to the coefficient matrix. Denote by Ak thecoefficient matrix of the turbulent kinetic energy equation before the linearization. The main diagonal of the ithrow of the equation becomes:

(14–732)A A C

kiik

iik old

t= + 2 2

µρµ

and the source term is:

(14–733)S

VX

VX

V

XC

kk t

i

j

i

j

j

i

old

t= ∂

∂∂∂

+∂∂

+µ ρ

µµ2

2

14.142.4. Turbulent Kinetic Energy Dissipation Rate

Source Term Linearization

The source term for the dissipation rate is handled in a similar fashion.

(14–734)S C

kVX

VX

V

XC

kti

j

i

j

j

iε µ ε ρ ε= ∂

∂∂∂

+∂∂

−1 2

2

Replace ε using the expression for the turbulent viscosity to yield

(14–735)S C C k

VX

VX

V

XC

ki

j

i

j

j

iε µρ ρ ε= ∂

∂∂∂

+∂∂

−1 2

2

A truncated Taylor series expansion of the dissipation source term around the previous (old) value is expressed

(14–736)S S

Sold

old

oldε εε

εεε ε= +

∂∂

−( )

The partial derivative of the dissipation rate source term with respect to ε is:

(14–737)∂∂

= −SC

kε

ερ ε

2 2

The dissipation source term is thus expressed

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(14–738)S C C k

VX

VX

V

XC

kC

ki

j

i

j

j

i

old oldε µρ ρ

ερ

εε= ∂

∂∂∂

+∂∂

+ −1 2

2

22

The first two terms are the source term, and the final term is moved to the coefficient matrix. Denote by Aε thecoefficient matrix of the turbulent kinetic energy dissipation rate equation before the linearization. The maindiagonal of the ith row of the equation becomes:

(14–739)A A C

kii iioldε ε ρ

ε= + 2 2

and the source term is:

(14–740)S C C k

VX

VX

V

XC

ki

j

i

j

j

i

oldε µ µρ ρ

ε= ∂

∂∂∂

+∂∂

+1

2

14.143. SHELL143 - 4-Node Plastic Small Strain Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

In-plane: 2 x 2Thru-the-thickness:2 (linear material)5 (nonlinear material)

Equation 12–81Quad

Stiffness Matrix and ThermalLoad Vector In-plane: 1

Thru-the-thickness:2 (linear material)5 (nonlinear material)

Equation 12–54Triangle

Same as stiffness matrixEquation 12–57, Equation 12–58, andEquation 12–59

QuadMass and Stress StiffnessMatrices

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Section 14.143: SHELL143 - 4-Node Plastic Small Strain Shell

Equation 12–38, Equation 12–39, andEquation 12–40

Triangle

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–59QuadTransverse Pressure LoadVector 1Equation 12–40Triangle

2Equation 12–57 and Equation 12–58specialized to the edge

Quad

Edge Pressure Load Vector

2Equation 12–38 and Equation 12–39specialized to the edge

Triangle

DistributionLoad Type

Bilinear in plane of element, linear thru thicknessElement Temperature

Bilinear in plane of element, constant thru thicknessNodal Temperature

Bilinear in plane of element and linear along each edgePressure

References: Ahmad(1), Cook(5), Dvorkin(96), Dvorkin(97), Bathe and Dvorkin(98), Allman(113), Cook(114), MacNealand Harder(115)

14.143.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.143.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to thecenterplane.

Each pair of integration points (in the r direction) is assumed to have the same element (material) orientation.

This element does not generate a consistent mass matrix; only the lumped mass matrix is available.

14.143.3. Assumed Displacement Shape Functions

The assumed displacement and transverse shear strain shape functions are given in Chapter 12, “Shape Functions”.The basic shape functions are essentially a condensation of those used for SHELL93 . The basic functions for thetransverse shear strain have been changed to avoid shear locking (Dvorkin(96), Dvorkin(97), Bathe and Dvorkin(98))and are pictured in Figure 14.30: “Shape Functions for the Transverse Strains”. One result of the use of thesedisplacement and strain shapes is that elastic rectangular elements give constant curvature results for flat elements,and also, in the absence of membrane loads, for curved elements. Thus, for these cases, nodal stresses are thesame as centroidal stresses. Both SHELL63 and SHELL93 can have linearly varying curvatures.

14.143.4. Stress-Strain Relationships

The material property matrix [D] for the element is described in Section 14.43.4: Stress-Strain Relationships. It isthe same as SHELL43 .

14.143.5. In-Plane Rotational DOF

If KEYOPT(3) is 0 or 1, there is no significant stiffness associated with the in-plane rotation DOF (rotation aboutthe element r axis). A nominal value of stiffness is present (as described with SHELL63 ), however, to prevent free

Chapter 14: Element Library

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rotation at the node. KEYOPT(3) = 2 is used to include the Allman-type rotational DOFs (as described by Allman(113)and Cook(114)). Such rotations improve the in-plane and general 3-D shell performance of the element. However,one of the outcomes of using the Allman rotation is that the element stiffness matrix contains up to two spuriouszero energy modes.

14.143.6. Spurious Mode Control with Allman Rotation

This procedure is described in Section 14.43.6: Spurious Mode Control with Allman Rotation. The same procedureas implemented in SHELL43 is used here.

14.143.7. Natural Space Extra Shape Functions with Allman Rotation

One of the outcomes of the Allman rotation is the dissimilar displacement variation along the normal and tan-gential directions of the element edges. The result of such variation is that the in-plane bending stiffness of the

elements is too large by a factor 1/(1-ν2) and sometimes termed as Poisson's ratio locking. To overcome thisdifficulty, two natural space (s and t) nodeless in-plane displacement shape functions are added in the elementstiffness matrix formulation and then condensed out at the element level. The element thus generated is freeof Poisson's ratio locking. For details of a similar implementation, refer to Yunus et al.(117).

14.143.8. Warping

A warping factor is computed as:

(14–741)φ = D

t

where:

D = component of the vector from the first node to the fourth node parallel to the element normalt = average thickness of the element

If φ > 1.0, a warning message is printed.

14.143.9. Consistent Tangent

A consistent tangent matrix implemented with the finite rotation capability is available by using KEYOPT(2) = 1.The theory is described in Section 3.2.8: Consistent Tangent Stiffness Matrix and Finite Rotation.

14.143.10. Stress Output

The stresses at the center of the element are computed by taking the average of the four integration points onthat plane.

The output forces and moments are computed as described in Section 2.3: Structural Strain and Stress Evaluations.

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Section 14.143: SHELL143 - 4-Node Plastic Small Strain Shell

14.144. ROM144 - Reduced Order Electrostatic-Structuralqi

qj

qk

ql

qm

qn

qo

qp

qq

Vs Vt Vu Vv Vw

UcUd

Ue

Uf

Ug

Uh

Uii

Ujj

UkkUll

Integration PointsShape FunctionsMatrix or Vector

NoneNone (lumped)Stiffness Matrix

NoneNone (lumped)Damping Matrix

NoneNone (lumped)Mass Matrix

NoneNone (lumped)Load Vector

ROM144 represents a reduced order model of distributed electostatic-structural systems. The element is derivedfrom a series of uncoupled static FEM analyses using electrostatic and structural elements (Section 15.10: ReducedOrder Modeling of Coupled Domains). The element fully couples the electrostatic-structural domains and issuitable for simulating the electromechanical response of micro-electromechanical systems (MEMS) such asclamped beams, micromirror actuators, and RF switches.

ROM144 is defined by either 20 (KEYOPT(1) = 0) or 30 nodes (KEYOPT(1) = 1). The first 10 nodes are associatedwith modal amplitudes, and represented by the EMF DOF labels. Nodes 11 to 20 have electric potential (VOLT)DOFs, of which only the first five are used. The last 10 optional nodes (21 to 30) have structural (UX) DOF to rep-resent master node displacements in the operating direction of the device. For each master node, ROM144 in-ternally uses additional structural DOFs (UY) to account for Lagrange multipliers used to represent internalnodal forces.

14.144.1. Element Matrices and Load Vectors

The FE equations of the 20-node option of ROM144 are derived from the system of governing equations of acoupled electrostatic-structural system in modal coordinates (Equation 15–102 and Equation 15–103)

(14–742)

K K

K K

q

V

D

D D

q

V

qq qV

Vq VV

qq

Vq VV

+

+

0 &&

MM q

V

F

I

qq 0

0 0

=

&&&&

where:

K = stiffness matrix

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D = damping matrixM = mass matrixq q q, ,& && = modal amplitude and its first and second derivativves with respect to time

V V V, ,& && = electrode voltage and its first and second derivattives with respect to time

F = forceI = electric current

The system of Equation 14–742 is similar to that of the Section 14.126: TRANS126 - Electromechanical Transducerelement with the difference that the structural DOFs are generalized coordinates (modal amplitudes) and theelectrical DOFs are the electrode voltages of the multiple conductors of the electromechanical device.

The contribution to the ROM144 FE matrices and load vectors from the electrostatic domain is calculated basedon the electrostatic co-energy Wel (Section 15.10: Reduced Order Modeling of Coupled Domains).

The electrostatic forces are the first derivative of the co-energy with respect to the modal coordinates:

(14–743)F

Wqk

el

k= − ∂

∂

where:

Fk = electrostatic force

Wel = co-energy

qk = modal coordinate

k = index of modal coordinate

Electrode charges are the first derivatives of the co-energy with respect to the conductor voltage:

(14–744)Q

WVi

el

i= ∂

∂

where:

Qi = electrode charge

Vi = conductor voltage

i = index of conductor

The corresponding electrode current Ii is calculated as a time-derivative of the electrode charge Qi. Both, electro-

static forces and the electrode currents are stored in the Newton-Raphson restoring force vector.

The stiffness matrix terms for the electrostatic domain are computed as follows:

(14–745)K

Fqkl

qq k

l= ∂

∂

(14–746)K

FVki

qV k

i= ∂

∂

(14–747)K

Iqik

Vq i

k= ∂

∂

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Section 14.144: ROM144 - Reduced Order Electrostatic-Structural

(14–748)K

IVij

VV i

j= ∂

∂

where:

l = index of modal coordinatej = index of conductor

The damping matrix terms for the electrostatic domain are calculated as follows:

(14–749)D Dqq qV= = 0

(14–750)D

Iqik

Vq i

k= ∂

∂ &

(14–751)D

I

VijVV i

j= ∂

∂ &

There is no contribution to the mass matrix from the electrostatic domain.

The contribution to the FE matrices and load vectors from the structural domain is calculated based on the strainenergy WSENE (Section 15.10: Reduced Order Modeling of Coupled Domains). The Newton-Raphson restoring

force F, stiffness K, mass M, and damping matrix D are computed according to Equation 14–752 to Equation 14–755.

(14–752)F

Wqi

SENE

i= ∂

∂

(14–753)K

Wq qij

qq SENE

j i= ∂

∂ ∂

2

(14–754)M

W

qii

i

SENE

i

=∂

∂12

2

2ω

(14–755)D Mii i i ii= 2ξ ω

where:

i, j = indices of modal coordinatesωi = angular frequency of ith eigenmode

ξi = modal damping factor (input as Damp on the RMMRANGE command

14.144.2. Combination of Modal Coordinates and Nodal Displacement atMaster Nodes

For the 30-node option of ROM144, it is necessary to establish a self-consistent description of both modal co-ordinates and nodal displacements at master nodes (defined on the RMASTER command defining the generation

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pass) in order to connect ROM144 to other structural elements UX DOF or to apply nonzero structural displacementconstraints or forces.

Modal coordinates qi describe the amplitude of a global deflection state that affects the entire structure. On the

other hand, a nodal displacement ui is related to a special point of the structure and represents the true local

deflection state.

Both modal and nodal descriptions can be transformed into each other. The relationship between modal coordin-ates qj and nodal displacements ui is given by:

(14–756)u qi ij j

j

m= ∑

=φ

1

where:

φij = jth eigenmode shape at node i

m = number of eigenmodes considered

Similarly, nodal forces Fi can be transformed into modal forces fj by:

(14–757)f Fj ij í

i

n= ∑

=φ

1

where:

n = number of master nodes

Both the displacement boundary conditions at master nodes ui and attached elements create internal nodal

forces Fi in the operating direction. The latter are additional unknowns in the total equation system, and can be

viewed as Lagrange multipliers λi mapped to the UY DOF. Hence each master UX DOF requires two equations

in the system FE equations in order to obtain a unique solution. This is illustrated on the example of a FE equation(stiffness matrix only) with 3 modal amplitude DOFs (q1, q2, q3), 2 conductors (V1, V2), and 2 master UX DOFs (u1,

u2):

(14–758)L+

K K K K K

K K K K K

qq qq qq qV qV

qq qq qq qV11 12 13 11 12 11 21

21 22 23 21 22

0 0φ φqqV

qq qq qq qV qV

Vq Vq V

K K K K K

K K K

φ φ

φ φ

12 22

31 32 33 13 32 13 23

11 12 13

0 0

0 0

qq VV VV

Vq Vq Vq VV VV

K K

K K K K K

11 12

21 22 23 21 22

11 12 13

0 0 0 0

0 0 0 0

0 0 0 0φ φ φ −−−

−

−

1 00 0 0 0 0 1

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

21 22 23

11

22

φ φ φ

K

K

uu

uu

∗−−

qqqVV

uu

1

2

3

1

2

1

2

1

2

λλ

=

fffII

F

F

a

a

1

2

3

1

2

1

2

00

Modal amplitude 1 (EMF)

Modal amplitude 2 ((EMF)

Modal amplitude 3 (EMF)

Electrode voltage 1 (VOLT)

Elecctrode voltage 2 (VOLT)

Lagrange multiplier 1 (UY)

Lagrange multiplier 2 (UY)

Master displacement 1 (UX)

Master displaccement 2 (UX)

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Section 14.144: ROM144 - Reduced Order Electrostatic-Structural

Rows 6 and 7 of Equation 14–758 correspond to the modal and nodal displacement relationship of Equa-tion 14–756, while column 6 and 7 - to nodal and modal force relationship (Equation 14–757). Rows and columns(8) and (9) correspond to the force-displacement relationship for the UX DOF at master nodes:

(14–759)K u Fij i i

ai= − λ

(14–760)λ i iF=

where K iiuu

is set to zero by the ROM144 element. These matrix coefficients represent the stiffness caused byother elements attached to the master node UX DOF of ROM144.

14.144.3. Element Loads

In the generation pass of the ROM tool, the ith mode contribution factors e i

j for each element load case j (Sec-

tion 15.10: Reduced Order Modeling of Coupled Domains) are calculated and stored in the ROM database file.In the Use Pass, the element loads can be scaled and superimposed in order to define special load situationssuch as acting gravity, external acceleration or a pressure difference. The corresponding modal forces for the jth

load casef j

E

(Equation 15–102) is:

(14–761)f e KjE

ij

iiqq= ( )0

where:

K iiqq( )0 = modal stiffness of the ith eigenmode at the initall position ( for all modes)qi = 0

14.145. PLANE145 - 2-D Quadrilateral Structural Solid p-Element

Integration PointsSolution Shape Func-

tionsGeometry / Geometric Shape

FunctionsMatrix or Vector

VariablePolynomial variable inorder from 2 to 8

Quad, Equation 12–117 andEquation 12–118Stiffness Matrix; and Thermal

and Inertial Load VectorsVariable

Polynomial variable inorder from 2 to 8

Triangle, Equation 12–96 andEquation 12–97

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Integration PointsSolution Shape Func-

tionsGeometry / Geometric Shape

FunctionsMatrix or Vector

VariablePolynomial variable inorder from 2 to 8

Same as stiffness matrix, special-ized to the edge

Pressure Load Vector

DistributionLoad Type

Same as geometric shape functions across element, constant thru thicknessor around circumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear across each facePressure

Reference: Szabo and Babuska(192)

14.145.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.146. PLANE146 - 2-D Triangular Structural Solid p-Element

Integration PointsSolution Shape Func-

tionsGeometric Shape FunctionsMatrix or Vector

VariablePolynomial variable inorder from 2 to 8

Equation 12–96 and Equa-tion 12–97

Stiffness Matrix; and Thermaland Inertial Load Vectors

VariablePolynomial variable inorder from 2 to 8

Same as stiffness matrix, special-ized to the edge

Pressure Load Vector

DistributionLoad Type

Same as geometric shape functions across element, constant thru thicknessor around circumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear across each facePressure

Reference: Szabo and Babuska(192)

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Section 14.146: PLANE146 - 2-D Triangular Structural Solid p-Element

14.146.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.147. SOLID147 - 3-D Brick Structural Solid p-Element

Integration PointsSolution Shape Func-

tionsGeometry / Geometric Shape

FunctionsMatrix or Vector

VariablePolynomial variable inorder from 2 to 8

Equation 12–209, Equation 12–210,and Equation 12–211Stiffness Matrix; and Thermal

and Inertial Load VectorsVariable

Polynomial variable inorder from 2 to 8

Wedge, Equation 12–186, Equa-tion 12–187, and Equation 12–188

VariablePolynomial variable inorder from 2 to 8

Quad, Equation 12–69 and Equa-tion 12–70

Pressure Load Vector

VariablePolynomial variable inorder from 2 to 8

Triangle, Equation 12–46 andEquation 12–47

DistributionLoad Type

Same as geometric shape functions thru elementElement Temperature

Same as geometric shape functions thru elementNodal Temperature

Bilinear across each facePressure

Reference: Szabo and Babuska(192)

14.147.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

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14.148. SOLID148 - 3-D Tetrahedral Structural Solid p-Element

Integration PointsSolution Shape Func-

tionsGeometric Shape FunctionsMatrix or Vector

VariablePolynomial variable inorder from 2 to 8

Equation 12–164, Equation 12–165,and Equation 12–166

Stiffness Matrix; and Thermaland Inertial Load Vectors

VariablePolynomial variable inorder from 2 to 8

Same as stiffness matrix specializedto face

Pressure Load Vector

DistributionLoad Type

Same as geometric shape functionsElement Temperature

Same as geometric shape functionsNodal Temperature

Linear across each facePressure

Reference: Szabo and Babuska(192)

14.148.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.149. Not Documented

No detail or element available at this time.

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Section 14.149: Not Documented

14.150. SHELL150 - 8-Node Structural Shell p-Element

Integration PointsSolution Shape FunctionsGeometry / Geometric Shape

FunctionsMatrix or Vector

Thru-the-thickness: 2In-plane: Variable

Polynomial variable in orderfrom 2 to 8

Quad, Equation 12–83Stiffness Matrix; andThermal and InertialLoad Vectors Thru-the-thickness: 2

In-plane: VariablePolynomial variable in orderfrom 2 to 8

Triangle, Equation 12–56

VariablePolynomial variable in orderfrom 2 to 8

Quad, Equation 12–71Transverse PressureLoad Vector

VariablePolynomial variable in orderfrom 2 to 8

Triangle, Equation 12–48

VariablePolynomial variable in orderfrom 2 to 8

Same as in-plane stiffness mat-rix, specialized to the edge

Edge Pressure LoadVector

DistributionLoad Type

Linear thru thickness, bilinear in plane of elementElement Temperature

Constant thru thickness, bilinear in plane of elementNodal Temperature

Bilinear across plane of element, linear along each edgePressure

Reference: Ahmad(1), Cook(5), Szabo and Babuska(192)

14.150.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.150.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to thecenterplane.

Each pair of integration points (in the r direction) is assumed to have the same element (material) orientation.

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There is no significant stiffness associated with rotation about the element r axis.

This element uses a lumped (translation only) inertial load vector.

14.150.3. Stress-Strain Relationships

The material property matrix [D] for the element is:

(14–762)[ ]D

BE B E

B E BE

G

G

f

x x

x y

xy

xy

xy

yz

=

ν

ν

0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0GG

fxz

where:

BE

E E

y

y xy x

=− ( )ν 2

Ex = Young's modulus in element x direction (input as EX on MP command)

νxy = Poisson's ratio in element x-y plane (input as PRXY on MP command)

Gxy = shear modulus in element x-y plane (input as GXY on MP command)

f A

t

= +

1 2

1 0 225

2

.

. . , whichever is greater

A = element area (in s-t plane)t = average thickness

The above definition of f is designed to avoid shear locking.

14.151. SURF151 - 2-D Thermal Surface Effect

! "

# #

$

$

%& ! "

'& ""

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Section 14.151: SURF151 - 2-D Thermal Surface Effect

Integration PointsShape FunctionsMatrix or Vector

2

2

w = C1 + C2x with no midside node

w = C1 + C2x + C3x2 with midside nodeAll

DistributionLoad Type

Same as shape functionsAll Loads

The logic is very similar to that given for Section 14.152: SURF152 - 3-D Thermal Surface Effect.

14.152. SURF152 - 3-D Thermal Surface Effect

!

"

#

$

%

&'

!

"

#

$ ()

*

+,

Integration PointsShape FunctionsGeometry / Midside NodesMatrix or Vector

3 x 3Equation 12–73Quad, if KEYOPT(4) = 0 (has midsidenodes)

Convection Surface Mat-rix and Load Vector; andHeat Generation LoadVector

2 x 2Equation 12–64Quad, if KEYOPT(4) = 1 (has no mid-side nodes)

6Equation 12–49Triangle, if KEYOPT(4) = 0 (has mid-side nodes)

3Equation 12–90Triangle, if KEYOPT(4) = 0 (has nomidside nodes)

DistributionLoad Type

Same as shape functionsAll Loads

14.152.1. Matrices and Load Vectors

When the extra node is not present, the logic is the same as given and as described in Section 6.2: Derivation ofHeat Flow Matrices. The discussion below relates to theory that uses the extra node.

The conductivity matrix is based on one-dimensional flow to and away from the surface. The form is conceptuallythe same as for LINK33 (Equation 14–252) except that the surface has four or eight nodes instead of only onenode. Using the example of convection and no midside nodes are requested (KEYOPT(4) = 1) (resulting in a 5 x5 matrix), the first four terms of the main diagonal are:

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(14–763)h N d areaf

area ( )∫

where:

hf =

film coefficient (input on command with KVAL=1)

h (u

SFE

IIf KEYOPT(5) = 1 and user programmable

feature USRSURF116 ooutput argument KEY(1) = 1,

this definition supercedes the other.)

hu = output argument for film coefficient of USRSURF116

N = vector of shape functions

which represents the main diagonal of the upper-left corner of the conductivity matrix. The remaining terms ofthis corner are all zero. The last main diagonal term is simply the sum of all four terms of Equation 14–763 andthe off-diagonal terms in the fifth column and row are the negative of the main diagonal of each row and column,respectively.

If midside nodes are present (KEYOPT(4) = 0) (resulting in a 9 x 9 matrix) Equation 14–763 is replaced by:

(14–764)h N N d areaf

T

area ( )∫

which represents the upper-left corner of the conductivity matrix. The last main diagonal is simply the sum ofall 64 terms of Equation 14–764 and the off-diagonal terms in the ninth column and row are the negative of thesum of each row and column respectively.

Radiation is handled similarly, except that the approach discussed for LINK31 in Section 14.31: LINK31 - RadiationLink is used. A load vector is also generated. The area used is the area of the element. The form factor is discussedin a subsequent section.

An additional load vector is formed when using the extra node by:

(14–765) [ ] Q K Tc tc ve=

where:

Qc = load vector to be formed

[Ktc] = element conductivity matrix due to convection

T TvevG T

=

0 0 0L

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Section 14.152: SURF152 - 3-D Thermal Surface Effect

TvG =

output argument TEMVEL if the user

programmable feature USRSURF116

is used.

T if KEYOPT(6) = 1

(see next section)

0

v

..0 for all other cases

TEMVEL from USRSURF116 is the difference between the bulk temperature and the temperature of the extranode.

14.152.2. Adiabatic Wall Temperature as Bulk Temperature

There is special logic that accesses FLUID116 information where FLUID116 has had KEYOPT(2) set equal to 1.This logic uses SURF151 or SURF152 with the extra node present (KEYOPT(5) = 1) and computes an adiabaticwall temperature (KEYOPT(6) = 1). For this case, Tv, as used above, is defined as:

(14–766)T

F V V

g J C

V Fv

R rel abs

c c pf

rel ref=

−

− Ω

( )

( ) (

2 2

2

2if KEYOPT(1) = 0

FR ss

c c pf

Rc c p

f

R

g J C

Fg J C

)

)

2

2

2

2

if KEYOPT(1) = 1

if KEYOPT(1)(V116 = 2

where:

FR = recovery factor (see Equation 14–767)

VV R

F R Rrelabs

ref s=

−−

ΩΩ Ω

if KEYOPT(1) = 0

if KEYOPT(1) = 1

Vabs = absolute value of fluid velocity (input as VABS on R command)

Ω = angular velocity of moving wall (input as OMEGA on R command)R = average radius of this elementΩref = reference angular velocity (input as (An)I and (An)J on R command of FLUID116)

Fs = slip factor (input as SLIPFAI, SLIPFAJ on R command of FLUID116)

V116 = velocity of fluid at extra node from FLUID116

gc = gravitational constant used for units consistency (input as GC on R command)

Jc = Joule constant used to convert work units to heat units (input as JC on R command)

Cpf = specific heat of fluid (from FLUID116)

The recovery factor is computed as follows:

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(14–767)F

C

R

nCn

n=

if KEYOPT(2) = 0

if KEYOPT(2) = 1

if KEYOPT(2)

Pr

Pr == 2

where:

Cn = constant used for recovery factor calculation (input as NRF on R command)

Pr = =C

K

pf f

xf

µPrandtl number

n =

0 5000. if Re < 2500.0

0.3333 if Re > 2500.0

µf = viscosity of fluid (from FLUID116)

Kxf = conductivity of fluid (from FLUID116)

Re = =ρµ

f

fVD

Reynold’s number

ρf = density of fluid (from FLUID116)D = diameter of fluid pipe (from FLUID116)

(14–768)V

V

Vl=

Re if KEYOPT(1) = 0,1

if KEYOPT(1) = 2116

where:

V = velocity used to compute Reynold's number

The adiabatic wall temperature is reported as:

(14–769)T T Taw ex v= +

where:

Taw = adiabatic wall temperature

Tex = temperature of extra node

KEYOPT(1) = 0 or 1 is ordinarily used for turbomachinry analysis, whereas KEYOPT(1) = 2 is ordinarily used forflow past stationary objects. For turbomachinery analyses Tex is assumed to be the total temperature, but for

flow past stationary objects Tex is assumed to be the static temperature.

14.152.3. Film Coefficient Adjustment

After the first coefficient has been determined, it is adjusted if KEYOPT(7) = 1:

(14–770)′ = −h h T Tf f S B

n( )

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Section 14.152: SURF152 - 3-D Thermal Surface Effect

where:

′ =hf adjusted film coefficient

hf = unadjusted film coefficient

TS = surface temperature

TB = bulk temperature (Taw, if defined)

n = real constant (input as ENN on RMORE command)

14.152.4. Radiation Form Factor Calculation

The form factor is computed as:

(14–771)F =

input (FORMF on command) if KEYOPT(9) = 1

B if KEYOPT(9)

R

== 2 or 3

also,

F = form factor (output as FORM FACTOR)

Developing B further

B =

≤

− >

>

cos if

cos if and KEYOPT(9) = 2

if and K

0

α α

α α

α

90

90

90

o

o

o EEYOPT(9) = 3

α = angle between element z axis at integration point being processed and the line connecting the integrationpoint and the extra node (see Figure 14.55: “Form Factor Calculation”)

Figure 14.55 Form Factor Calculation

Extra node (Q)

αL

I J

K

F is then used in the two-surface radiation equation:

(14–772)Q AF T Te

rQ= −σε ( )4 4

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where:

σ = Stefan-Boltzmann constant (input as SBCONST on R command)ε = emissivity (input as EMIS on MP command)A = element area

Note that this “form factor” does not have any distance affects. Thus, if distances are to be included, they mustall be similar in size, as in an object on or near the earth being warmed by the sun. For this case, distance affectscan be included by an adjusted value of σ.

14.153. SURF153 - 2-D Structural Surface Effect

Integration PointsShape FunctionsMidside NodesMatrix or Vector

3w = C1 + C2x + C3x2If KEYOPT(4) = 0 (has midside nodes)All

2w = C1 + C2xIf KEYOPT(4) = 1 (has no midsidenodes)

All

DistributionLoad Type

Same as shape functionsAll Loads

The logic is very similar to that given for SURF154 in Section 14.154: SURF154 - 3-D Structural Surface Effect withthe differences noted below:

1. For surface tension (input as SURT on R command)) on axisymmetric models (KEYOPT(3) = 1), an averageforce is used on both end nodes.

2. For surface tension with midside nodes, no load is applied at the middle node, and only the componentdirected towards the other end node is used.

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Section 14.153: SURF153 - 2-D Structural Surface Effect

14.154. SURF154 - 3-D Structural Surface Effect

Integration PointsShape FunctionsGeometry / Midside NodesMatrix or Vector

3 x 3Equation 12–71Quad, if KEYOPT(4) = 0 (has midsidenodes)

Stiffness and DampingMatrices, and PressureLoad Vector

2 x 2Equation 12–59Quad, if KEYOPT(4) = 1 (has no mid-side nodes)

6Equation 12–56 (op-tion w)

Triangle, if KEYOPT(4) = 0 (has mid-side nodes)

3Equation 12–53Triangle, if KEYOPT(4) = 0 (has nomidside nodes)

3 x 3Equation 12–69, Equa-tion 12–70 and Equa-tion 12–71

Quad, if KEYOPT(4) = 0 (has midsidenodes)

Mass and Stress StiffnessMatrices

2 x 2Equation 12–57, Equa-tion 12–58 and Equa-tion 12–59

Quad, if KEYOPT(4) = 1 (has no mid-side nodes)

6Equation 12–56Triangle, if KEYOPT(4) = 0 (has mid-side nodes)

3Equation 12–51, Equa-tion 12–52 and Equa-tion 12–53

Triangle, if KEYOPT(4) = 0 (has nomidside nodes)

3 x 3Equation 12–69 andEquation 12–70

Quad, if KEYOPT(4) = 0 (has midsidenodes)

Surface Tension LoadVector

2 x 2Equation 12–57 andEquation 12–58

Quad, if KEYOPT(4) = 1 (has no mid-side nodes)

6Equation 12–56 (op-tions u and v)

Triangle, if KEYOPT(4) = 0 (has mid-side nodes)

3Equation 12–51 andEquation 12–52

Triangle, if KEYOPT(4) = 0 (has nomidside nodes)

DistributionLoad Type

Same as shape functionsAll Loads

The stiffness matrix is:

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(14–773)

[ ]

K

k N N dA

ef

fz z

T

A

=

= ∫element foundation stiffness matrix

where:

kf = foundation stiffness (input as EFS on R command)A = area of elementNz = vector of shape functions representing motions normal to the surface

The mass matrix is:

(14–774)

[ ]

M

N N dA A N N dA

eT

Ad

T

A

=

= +∫ ∫element mass matrix

thρ

where:

th = thickness (input as TKI, TKJ, TKK, TKL on RMORE command)

ρ = density (input as DENS on MP command)N = vector of shape functionsAd = added mass per unit area (input as ADMSUA on R command)

If the command LUMPM,ON is used, [Me] is diagonalized as described in Section 13.2: Lumped Matrices.

The element damping matrix is:

(14–775)[ ] C N N dAe

T

A= =∫µ element damping matrix

where:

µ = dissipation (input as VISC on MP command)

The element stress stiffness matrix is:

(14–776)[ ] [ ] [ ][ ]S S S S dAe g

Tm g

A

= =∫ element mass matrix

where:

[Sg] = derivatives of shape functions of normal motions

[ ]S

s

sm =

0 0

0 0

0 0 0

s = in-plane force per unit length (input as SURT on R command)

If pressure is applied to face 1, the pressure load stiffness matrix is computed as described in Section 3.3.4:Pressure Load Stiffness.

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Section 14.154: SURF154 - 3-D Structural Surface Effect

The element load vector is:

(14–777) F F Fe est

epr= +

where:

F s N dEest

pE

= =∫ surface tension force vector

Np = vector of shape functions representing in-plane motions normal to the edge

E = edge of element

( ( F N P N P N P P Z N N Nepr

xP

x yP

y zP

zA

v f x X y Y z Z= + + + + +∫ τ τ τ ))dA

= pressure load vector

N

N

NxP x

xe=

if KEYOPT(2) = 0

if KEYOPT(2) = 1

N

N

NyP y

ye

=

if KEYOPT(2) = 0

if KEYOPT(2) = 1

N

N

NzP z

ze=

if KEYOPT(2) = 0

if KEYOPT(2) = 1

Nx = vector of shape functions representing motion in element x direction

Ny = vector of shape functions representing motion in element y direction

Nxe = vector of shape functions representing motion in the local coordinate x direction

Nye = vector of shape functions representing motion in the local coordinate y direction

Nze = vector of shape functions representing motion in the local coordinate z direction

P P Px y z, , =

distributed pressures over element in element x, yy, and z directions (input as VAL1 thru VAL4

with LKEY = 2,,3,1, respectively, on SFE command, if KEYOPT(2) = 0

distriibuted pressures over element in local x, y, and z directiions (input as VAL1 thru VAL4

with LKEY = 1,2,3, respectiveely, on SFE command, if KEYOPT(2) = 1

Pv = uniform pressure magnitude

PP

v =

1cosθ if KEYOPT(11) = 0 or 1

if KEYOPT(11) = 2P1

P1 = input (VAL1 with LKEY = 5 on SFE command)

θ = angle between element normal and applied load direction

Zf =≤1 0

0 0

0 0.

.

.if KEYOPT(12) = 0 or cos

if KEYOPT(12) = 1 an

θ

dd cosθ > 0 0.

τxx x y zD D D D= + +

≠2 2 2 if KEYOPT(11)

if KEYOPT(11) = 1

1

0.0

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τyy x y zD D D D= + +

≠2 2 2 if KEYOPT(11) 1

if KEYOPT(11) = 10.0

τz z x y zD D D D= + +2 2 2

Dx, Dy, Dz = vector directions (input as VAL2 thru VAL4 with LKEY = 5 on SFE command)

NX, NY, NZ = vectors of shape functions in global Cartesian coordinates

The integration used to arrive at Fepr

is the usual numerical integration, even if KEYOPT(6) ≠ 0. The outputquantities “average face pressures” are the average of the pressure values at the integration points.

14.155. Not Documented

No detail or element available at this time.

14.156. Not Documented

No detail or element available at this time.

14.157. SHELL157 - Thermal-Electric Shell

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–65. No variation thruthickness

Quad

Electrical Conductivity Matrix

1Equation 12–65. No variation thruthickness

Triangle

2 x 2Equation 12–64 and Equation 12–65.No variation thru thickness

QuadThermal Conductivity Matrix;Heat Generation Load andConvection Surface Matrix andLoad Vectors 1

Equation 12–90 and Equation 12–65.No variation thru thickness

Triangle

Same as conductivitymatrix

Same as conductivity matrix. Matrix is diagonalized asdescribed in Section 13.2: Lumped Matrices

Specific Heat Matrix

14.157.1. Other Applicable Sections

Chapter 11, “Coupling” discusses coupled effects.

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Section 14.157: SHELL157 - Thermal-Electric Shell

14.158. HYPER158 - 3-D 10-Node Tetrahedral Mixed u-P HyperelasticSolid

Integration PointsShape FunctionsMatrix or Vector

4Equation 12–164, Equation 12–165, and Equation 12–166Stiffness and Mass Matrices;and Thermal Load Vector

6Equation 12–164, Equation 12–165, and Equation 12–166specialized to the face

Pressure Load Vector

DistributionLoad Type

Same as shape functionsElement Temperature

Same as shape functionsNodal Temperature

Linear over each facePressure

Reference: Oden and Kikuchi(123), Sussman and Bathe(124)

14.158.1. Other Applicable Sections

For the basic element formulation, refer to Section 14.58: HYPER58 - 3-D 8-Node Mixed u-P Hyperelastic Solid.The hyperelastic material model (Mooney-Rivlin) is described in Section 4.6: Hyperelasticity.

14.159. Not Documented

No detail or element available at this time.

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14.160. LINK160 - Explicit 3-D Spar (or Truss)

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

14.161. BEAM161 - Explicit 3-D Beam

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

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Section 14.161: BEAM161 - Explicit 3-D Beam

14.162. PLANE162 - Explicit 2-D Structural Solid

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

14.163. SHELL163 - Explicit Thin Structural Shell

!

"$#%&

')(*,+-/. 10243 05 -76 2 +984-;:< 02 -/*4=>+?84-/-< -@A- 2 +9B

C

D

E

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

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14.164. SOLID164 - Explicit 3-D Structural Solid

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

14.165. COMBI165 - Explicit Spring-Damper

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

14–319ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.165: COMBI165 - Explicit Spring-Damper

14.166. MASS166 - Explicit 3-D Structural Mass

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

14.167. LINK167 - Explicit Tension-Only Spar

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

14.168. SOLID168 - Explicit 3-D 10-Node Tetrahedral Structural Solid

Chapter 14: Element Library

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.14–320

For all theoretical information about this element, see the LS-DYNA Theoretical Manual(199).

14.169. TARGE169 - 2-D Target Segment

!" #

$%&

14.169.1. Other Applicable Sections

Section 14.170: TARGE170 - 3-D Target Segment discusses Target Elements.

14.169.2. Segment Types

TARGE169 supports six 2-D segment types:

Figure 14.56 2-D Segment Types

Line

Arc, clockwise

Arc, counterclockwise

Parabola

Circle

Pilot Node

I J

I

I

I

I

I

K

K

J

K

J

J

14–321ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 14.169: TARGE169 - 2-D Target Segment

14.170. TARGE170 - 3-D Target Segment

!

"

#

$

14.170.1. Introduction

In studying the contact between two bodies, the surface of one body is conventionally taken as a contact surfaceand the surface of the other body as a target surface. The “contact-target” pair concept has been widely used infinite element simulations. For rigid-flexible contact, the contact surface is associated with the deformable body;and the target surface must be the rigid surface. For flexible-flexible contact, both contact and target surfacesare associated with deformable bodies. The contact and target surfaces constitute a “Contact Pair”.

TARGE170 is used to represent various 3-D target surfaces for the associated contact elements (CONTA173 andCONTA174 ). The contact elements themselves overlay the solid elements describing the boundary of a deformablebody that is potentially in contact with the rigid target surface, defined by TARGE170. Hence, a “target” is simplya geometric entity in space that senses and responds when one or more contact elements move into a targetsegment element.

14.170.2. Segment Types

The target surface is modelled through a set of target segments; typically several target segments comprise onetarget surface. Each target segment is a single element with a specific shape or segment type. TARGE170 supportseight 3-D segment types; see Figure 14.57: “3-D Segment Types”

Chapter 14: Element Library

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Figure 14.57 3-D Segment Types

K

I

J3 Node Triangle( , TRIA)TSHAP

Cylinder( , CYLI)R1=RadiusTSHAP

TSHAP4 Node Quadrilateral( ,Quad)

Cone( ,CONE)R1=Radius (I)R2=Radius (J)

TSHAP

TSHAPPilot node( ,PILO)

Sphere( ,SPHE)R1=RadiusTSHAP

6 Node Triangle( ,TRI6)TSHAP

8 Node Quadrilateral( ,QUA8)TSHAP

I

I

I

I

JJ

J J

J

O

K

L

L

L

I

K

I

I

P

M

N

NM

K

14.170.3. Reaction Forces

The reaction forces on the entire rigid target surface are obtained by summing all the nodal forces of the associatedcontact elements. The reaction forces are accumulated on the pilot node. If the pilot node has not been explicitlydefined by the user, one of the target nodes (generally the one with the smallest number) will be used to accu-mulate the reaction forces.

14.171. CONTA171 - 2-D 2-Node Surface-to-Surface Contact

"!$# %&

'

( )* +,# -./# # -10%!$# 2%324

Integration PointsShape FunctionsMatrix or Vector

2W = C1 + C2 xStiffness Matrix

14.171.1. Other Applicable Sections

The CONTA171 description is the same as for Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contactexcept that it is 2-D and there are no midside nodes.

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Section 14.171: CONTA171 - 2-D 2-Node Surface-to-Surface Contact

14.172. CONTA172 - 2-D 3-Node Surface-to-Surface Contact

!#" $%&' ()*+,#" !-" $%

Integration PointsShape FunctionsMatrix or Vector

2W = C1 + C2 x + C3x2Stiffness Matrix

14.172.1. Other Applicable Sections

The CONTA172 description is the same as for Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contactexcept that it is 2-D.

14.173. CONTA173 - 3-D 4-Node Surface-to-Surface Contact

.

/

0

1

01

/324.

566789 :;<=>:?@<;AB ?C:8<6

D 7E;:8 ;F-G <IHJ<IE;6

AB ?C):8<*7+C A 7G 9 =K AL <G GF#G <IHJ<IE;

M

N

O

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–57, Equation 12–58, andEquation 12–59

QuadStiffness and Stress Stiff-ness Matrices

3Equation 12–51, Equation 12–52, andEquation 12–53

Triangle

14.173.1. Other Applicable Sections

The CONTA173 description is the same as for Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contactexcept there are no midside nodes.

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14.174. CONTA174 - 3-D 8-Node Surface-to-Surface Contact

!

" $# % &(' $)*# +

,.- !- /

/

0

12,

0 1

2

$' 3 4 ' '&(' )#

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2If KEYOPT(4) = 0 (has midside nodes) Equa-tion 12–69, Equation 12–70, and Equa-tion 12–71

QuadStiffness and Stress Stiff-ness Matrices

3If KEYOPT(4) = 0 (has midside nodes) Equa-tion 12–56

Triangle

14.174.1. Introduction

CONTA174 is an 8-node element that is intended for general rigid-flexible and flexible-flexible contact analysis.In a general contact analysis, the area of contact between two (or more) bodies is generally not known in advance.CONTA174 is applicable to 3-D geometries. It may be applied for contact between solid bodies or shells.

14.174.2. Contact Kinematics

Contact Pair

In studying the contact between two bodies, the surface of one body is conventionally taken as a contact surfaceand the surface of the other body as a target surface. For rigid-flexible contact, the contact surface is associatedwith the deformable body; and the target surface must be the rigid surface. For flexible-flexible contact, bothcontact and target surfaces are associated with deformable bodies. The contact and target surfaces constitutea “Contact Pair”.

The CONTA174 contact element is associated with the 3-D target segment elements (TARGE170) using a sharedreal constant set number. This element is located on the surface of 3-D solid, shell elements (called underlyingelement). It has the same geometric characteristics as the underlying elements. The contact surface can be eitherside or both sides of the shell or beam elements.

Location of Contact Detection

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Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contact

Figure 14.58 Contact Detection Point Location at Gauss Point

Rigidbody

Gauss integrationpoint

Deformable solid

Contact segmentTarget segment

CONTA174 is surface-to-surface contact element. The contact detection points are the integration point and arelocated either at nodal points or Gauss points. The contact elements is constrained against penetration intotarget surface at its integration points. However, the target surface can, in principle, penetrate through into thecontact surface. See Figure 14.58: “Contact Detection Point Location at Gauss Point”. CONTA174 uses Gauss in-tegration points as a default (Cescotto and Charlier(213), Cescotto and Zhu(214)), which generally provides moreaccurate results than those using the nodes themselves as the integration points. A disadvantage with the useof nodal contact points is that: when for a uniform pressure, the kinematically equivalent forces at the nodes areunrepresentative and indicate release at corners.

Penetration Distance

The penetration distance is measured along the normal direction of contact surface located at integration pointsto the target surface (Cescotto and Charlier(214)). See Figure 14.59: “Penetration Distance”. It is uniquely definedeven the geometry of the target surface is not smooth. Such discontinuities may be due to physical corners onthe target surface, or may be introduced by a numerical discretization process (e.g. finite elements). Based onthe present way of calculating penetration distance there is no restriction on the shape of the rigid target surface.Smoothing is not always necessary typically for the concave corner. For the convex corner, it is still recommendedto smooth out the region of abrupt curvature changes (see Figure 14.60: “Smoothing Convex Corner”).

Figure 14.59 Penetration Distance

Integration point

Target surface

Contact element

Penetration distance

Chapter 14: Element Library

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Figure 14.60 Smoothing Convex Corner

Smoothing Radius

Outward normal

Pinball Algorithm

The position and the motion of a contact element relative to its associated target surface determine the contactelement status. The program monitors each contact element and assigns a status:

STAT = 0 Open far-field contactSTAT = 1 Open near-field contactSTAT = 2 Sliding contactSTAT = 3 Sticking contact

A contact element is considered to be in near-field contact when the element enters a pinball region, which iscentered on the integration point of the contact element. The computational cost of searching for contact dependson the size of the pinball region. Far-field contact element calculations are simple and add few computationaldemands. The near-field calculations (for contact elements that are nearly or actually in contact) are slower andmore complex. The most complex calculations occur the elements are in actual contact.

Setting a proper pinball region is useful to overcome spurious contact definitions if the target surface has severalconvex regions. The current default setting should be appropriate for most contact problems.

14.174.3. Frictional Model

Coulomb's Law

In the basic Coulomb friction model, two contacting surfaces can carry shear stresses up to a certain magnitudeacross their interface before they start sliding relative to each other. The state is known as sticking. The Coulombfriction model is defined as:

(14–778)τ µlim P b= +

(14–779)τ τ≤ lim

where:

τlim = limit shear stress

τ = equivalent shear stressµ = frictional coefficient (input using MU on MP command)

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Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contact

P = contact normal pressureb = contact cohesion (input as COHE on R command)

Once the equivalent shear stress exceeds τlim, the contact and target surfaces will slide relative to each other.

This state is known as sliding. The sticking/sliding calculations determine when a point transitions from stickingto sliding or vice versa. The contact cohesion provides sliding resistance even with zero normal pressure,

CONTA174 provides an option for defining a maximum equivalent shear stress τmax (input as TAUMAX on RMOREcommand) so that, regardless of the magnitude of the contact pressure, sliding will occur if the magnitude ofthe equivalent shear stress reaches this value.

Figure 14.61 Friction Model

| |t

t

Sliding

lim

Sticking

max

b

m

t

p

Static and Dynamic Friction

CONTA174 provides the exponential friction model, which is used to smooth the transition between the staticcoefficient of friction and the dynamic coefficient of friction according to the formula (Benson and Hallquist(317)):

(14–780)µ υ µ µ µ υ( ) ( )= + − −

d s dc

e

where:

υ = slip rateµd = dynamic friction coefficient (input as MU on MP command)

µs= Rf µd = static friction coefficient

Rf = ratio of static and dynamic friction (input as FACT on RMORE command)

c = decay coefficient (input as DC on RMORE command)

Integration of Frictional Law

The integration of the frictional mode is similar to that of nonassociated theory of plasticity (see Section 4.1:Rate-Independent Plasticity). In each substep that sliding friction occurs, an elastic predictor is computed incontact traction space. The predictor is modified with a radial return mapping function, providing both a smallelastic deformation along sliding response as developed by Giannakopoulos(135).

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Algorithmic Symmetrization

Contact problems involving friction produce non-symmetric stiffness. Using an unsymmetric solver(NROPT,UNSYM) is more computationally expensive than a symmetric solver for each iteration. For this reason,a symmetrization algorithm developed by Laursen and Simo(216) is used by which most frictional contactproblems can be solved using solvers for symmetric systems. If frictional stresses have a substantial influenceon the overall displacement field and the magnitude of the frictional stresses is highly solution dependent, anysymmetric approximation to the stiffness matrix may provide a low rate of convergence. In such cases, the useof an unsymmetric stiffness matrix is more computationally efficient.

14.174.4. Contact Algorithm

Four different contact algorithms are implemented in this element (selected by KEYOPT(2)).

• Pure penalty method

• Augmented Lagrangian method (Simo and Laursens(215))

• Pure Lagrange multiplier method (Bathe(2))

• Lagrange multiplier on contact normal and penalty on frictional direction

Pure Penalty Method

This method requires both contact normal and tangential stiffness. The main drawback is that the amount pen-etration between the two surfaces depends on this stiffness. Higher stiffness values decrease the amount ofpenetration but can lead to ill-conditioning of the global stiffness matrix and to convergence difficulties. Ideally,you want a high enough stiffness that contact penetration is acceptably small, but a low enough stiffness thatthe problem will be well-behaved in terms of convergence or matrix ill-conditioning.

The contact traction vector is:

(14–781)

P

y

z

τ

τ

where:

P = normal contact pressureτy = tangential contact stress in y direction

τz = tangential contact stress in z direction

The contact pressure is:

(14–782)P

u

K u un

n n n=

>≤

0 0

0

if

if

where:

Kn = contact normal stiffness

un = contact gap size

The frictional stress is obtained by Coulomb's law:

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Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contact

(14–783)τ

τ τ τ µ

µ τ τ τ µy

s y y z

n n y z

K u P

K u P=

= + − <

= + − =

if (sticking)

if (s

2 2

2 2

0

0 lliding)

where:

Ks = tangential contact stiffness (input as FKS on R command)

uy = contact slip distance in y direction

µ = frictional coefficient (input as MU on MP command)

Augmented Lagrangian Method

The augmented Lagrangian method is an iterative series of penalty updates to find the Lagrange multipliers (i.e.,contact tractions). Compared to the penalty method, the augmented Lagrangian method usually leads to betterconditioning and is less sensitive to the magnitude of the contact stiffness coefficient. However, in some analyses,the augmented Lagrangian method may require additional iterations, especially if the deformed mesh becomesexcessively distorted.

The contact pressure is defined by:

(14–784)P

Kn

n n i n=

>+ ≤

+

0 0

01

if

if

µµ λ µ

where:

λλ ελ εi

i n n n

i n

K u u

u= =+ >

<

1

if

if

ε = compatibility tolerance (input as FTOLN on R command)λi = Lagrange multiplier component at iteration i

The Lagrange multiplier component λi is computed locally (for each element) and iteratively.

Pure Lagrange Multiplier Method

The pure Lagrange multiplier method does not require contact stiffness. Instead it requires chattering controlparameters. Theoretically, the pure Lagrange multiplier method enforces zero penetration when contact is closedand “zero slip” when sticking contact occurs. However the pure Lagrange multiplier method adds additionaldegrees of freedom to the model and requires additional iterations to stabilize contact conditions. This will increasethe computational cost. This algorithm has chattering problems due to contact status changes between openand closed or between sliding and sticking. The other main drawback of the Lagrange multiplier method is theoverconstraint occuring in the model. The model is overconstrained when a contact constraint condition at anode conflicts with a prescribed boundary condition on that degree of freedom (e.g., D command) at the samenode. Overconstraints can lead to convergence difficulties and/or inaccurate results. The Lagrange multipliermethod also introduces zero diagonal terms in the stiffness matrix, so that iterative solvers (e.g., PCG) can notbe used.

The contact traction components (i.e., Lagrange multiplier parameters) become unknown DOFs for each element.The associated Newton-Raphson load vector is:

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(14–785) , , , , ,F P u u unr y z n y zT

= τ τ

Lagrange Multiplier on Contact Normal and Penalty on Frictional Direction

In this method only the contact normal pressure is treated as a Lagrange multiplier. The tangential contactstresses are calculated based on the penalty method (see Equation 14–783).

This method allows only a very small amount of slip for a sticking contact condition. It overcomes chatteringproblems due to contact status change between sliding and sticking which often occurs in the pure LagrangeMultiplier method. Therefore this algorithm treats frictional sliding contact problems much better than the pureLagrange method.

14.174.5. Thermal/Structural Contact

Combined structural and thermal contact is specified if KEYOPT(1) = 1, which indicates that structural and thermalDOFs are active. Pure thermal contact is specified if KEYOPT(1) = 2. The thermal contact features (Zhu andCescotto(280)) are:

Thermal Contact Conduction

(14–786)q K T Tc T C= − ≥( ) if STAT 2

where:

q = heat flux (heat flow rate per area)Kc = thermal contact conductance coefficient (input as TCC on R command)

TT = temperature on target surface

TC = temperature on contact surface

Heat Convection

(14–787)q h T Tf e C= − ≤( ) if STAT 1

where:

hf = convection coefficient (input on SFE command with Lab = CONV and KVAL = 1)

T

T

e

T

=

if STAT = 1

environmental temperature (input on

com

SFE

mmand with Lab = CONV and KVAL = 2)

if STAT = 0

Heat Radiation

(14–788)q F T T T Te o C o= + − +

≤σε ( ) ( )4 4 1if STAT

where:

σ = Stefan-Boltzmann constant (input as SBCT on R command)ε = emissivity (input using EMIS on MP command)

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Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contact

F = radiation view factor (input as RDVF on R command)To = temperature offset (input as VALUE on TOFFST command)

Heat Generation Due to Frictional Sliding

(14–789)

q F F t v

q F F t vc w f

T w f

== −

>( )1

0if STAT = 2 and µ

where:

qc = amount of frictional dissipation on contact surface

qT = amount of frictional dissipation on target surface

Fw = weight factor for the distribution of heat between two contact and target surfaces (input as FWGT on

R command)Ff = fractional dissipated energy converted into heat (input on FHTG on R command)

t = equivalent frictional stressv = sliding rate

Note — When KEYOPT(1) = 2, heat generation due to friction is ignored.

14.174.6. Electric Contact

Combined structural, thermal, and electric contact is specified if KEYOPT(1) = 3. Combined thermal and electriccontact is specified if KEYOPT(1) = 4. Combined structural and electric contact is specified if KEYOPT(1) = 5. Pureelectric contact is specified if KEYOPT(1) = 6. The electric contact features are:

Electric Current Conduction (KEYOPT(1) = 3 or 4)

(14–790)J

LV VT C= −σ

( )

where:

J = current densityσ/L = electric conductivity per unit length (input as ECC on R command)VT = voltage on target surface

VC = voltage on contact surface

Electrostatic (KEYOPT(1) = 5 or 6)

(14–791)QA

CA

V VT C= −( )

where:

Q

A= charge per unit area

C

A= capacitance per unit area (input as ECC on command)R

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14.174.7. Magnetic Contact

The magnetic contact is specified if KEYOPT(1) = 7. Using the magnetic scalar potential approach, the 3-D mag-netic flux across the contacting interface is defined by:

(14–792)ψ φ φ µn

M t c o gnC AH= − −( )

where:

ψn = magnetic fluxφt = magnetic potential at target surface (MAG degree of freedom)

φc = magnetic potential at contact surface (MAG degree of freedom)

CM = magnetic contact permeance coefficient

µo = free space permeability

A = contact area

Hgn

= normal component of the “guess” magnetic field (See Equation 5–16)

The gap permeance is defined as the ratio of the magnetic flux in the gap to the total magnetic potential differenceacross the gap. The equation for gap permeance is:

(14–793)P A to= µ /

where:

t = gap thickness

The magnetic contact permeance coefficient is defined as:

(14–794)C tM o= µ /

The above equations are only valid for 3-D analysis using the Magnetic Scalar Potential approach.

14.175. CONTA175 - 2-D/3-D Node-to-Surface Contact

! #"%$&('*)+!,

-/.0 ! 1'324-/.0 ! 1'324

5678 9":$;&('32<!,

>=@?;A >=B?CDA

E

F G

HE

No detail or element available at this time.

Integration PointsShape FunctionsGeometryMatrix or Vector

NoneNoneNormal DirectionStiffness Matrix

NoneNoneSliding Direction

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Section 14.175: CONTA175 - 2-D/3-D Node-to-Surface Contact

14.175.1. Other Applicable Sections

The CONTA175 description is the same as for Section 14.174: CONTA174 - 3-D 8-Node Surface-to-Surface Contactexcept that it is a one node contact element.

14.175.2. Contact Models

The contact model can be either contact force based (KEYOPT(3) = 0, default) or contact traction based (KEYOPT(3)= 1). For a contact traction based model, ANSYS can determine the area associated with the contact node. Forthe single point contact case, a unit area will be used which is equivalent to the contact force based model.

14.175.3. Contact Forces

In order to satisfy contact compatibility, forces are developed in a direction normal (n-direction) to the targetthat will tend to reduce the penetration to an acceptable numerical level. In addition to normal contact forces,friction forces are developed in directions that are tangent to the target plane.

(14–795)F

K unn n

=

>

≤

0 0

0

if u

if un

n

where:

Fn = normal contact force

Kn = contact normal stiffness (input FKN on R command)

un = contact gap size

(14–796)F

K u F F

K u F Fr

T r s n

n n s n

=+ − <

+ − =

if (sticking)

if (sli

F

F

r2

r2

2

2

0

0

µ

µ µ dding)

where:

Fr = tangential contact force in r direction

Fs = tangential contact force in z direction

KT = tangential contact stiffness (input on FKT on R command)

ur = contact slip distance in y direction

M = frictional coefficient (input as MU on MP command)

14.176. Not Documented

No detail or element available at this time.

14.177. Not Documented

No detail or element available at this time.

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14.178. CONTA178 - 3-D Node-to-Node Contact

Integration PointsShape FunctionsGeometryMatrix or Vector

NoneNoneNormal DirectionStiffness Matrix

NoneNoneSliding Direction

DistributionLoad Type

None - average used for material property evaluationElement Temperature

None - average used for material property evaluationNodal Temperature

14.178.1. Introduction

CONTA178 represents contact and sliding between any two nodes of any types of elements. This node-to-nodecontact element can handle cases when the contact location is known beforehand.

CONTA178 is applicable to 3-D geometries. It can also be used in 2-D and axisymmetric models by constrainingthe UZ degrees of freedom. The element is capable of supporting compression in the contact normal directionand Coulomb friction in the tangential direction.

14.178.2. Contact Algorithms

Four different contact algorithms are implemented in this element.

• Pure penalty method

• Augmented Lagrange method

• Pure Lagrange multiplier method

• Lagrange multiplier on contact normal penalty on frictional direction

Pure Penalty Method

The Newton-Raphson load vector is:

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Section 14.178: CONTA178 - 3-D Node-to-Node Contact

(14–797) F

F

F

F

F

F

F

nr

n

sy

sz

n

sy

sz

l =−−

−

where:

Fn = normal contact force

Fsy = tangential contact force in y direction

Fsz = tangential contact force in z direction

(14–798)F

U

K U Unn

n n n

if=

>≤

0 0

0

if

where:

Kn = contact normal stiffness (input FKN on R command)

un = contact gap size

(14–799)F

K u F F F

K u F F Fsy

s y sy sz n

n n sy sz n

sticking=

+ − <

+ − =

if

if

2 2

2 2

0µ

µ µ

( )

00 ( )sliding

where:

Ks = tangential contact stiffness (input as FKS on R command)

uy = contact slip distance in y direction

µ = frictional coefficient (input as MU on MP command)

Augmented Lagrange Method

(14–800)F

K u u

unn n n

n=

≤>

if

if

0

0 0

where:

λτ

ii n nk u u

+ = =+

1 Lagrange multiplier force at iteration i+1if nn

i nu

>≤

ετ εif

ε = user-defined compatibility tolerance (input as TOLN on R command)

The Lagrange multiplier component of force λ is computed locally (for each element) and iteratively.

Pure Lagrange Multiplier Method

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The contact forces (i.e., Lagrange multiplier components of forces) become unknown DOFs for each element.The associated Newton-Raphson load vector is:

(14–801) F

F

F

F

F

F

F

u

u

u

nr

n

sy

sz

n

sy

sz

n

y

z

=−−

−

Lagrange Multiplier on Contact Normal Penalty on Frictional Direction

In this method only the contact normal face is treated as a Lagrange multiplier. The tangential forces are calculatedbased on penalty method:

(14–802)F

K u F F F

F F F Fsy

s y sy sz n

n sy sz n

=+ − ≤

+ − >

if

if

2 2

2 2

0

0

µ

µ µ

14.178.3. Element Damper

The damping capability is only used for modal and transient analyses. Damping is only active in the contactnormal direction when contact is closed. The damping force is computed as:

(14–803)F C VD v= −

where:

V = relative velocity between two contact nodes in contact normal direction

C C C Vv v v= +1 2

Cv1 = constant damping coefficient (input as CV1 on R command)

Cv2 = linear damping coefficient (input as CV2 on R command)

14.179. PRETS179 - Pretension

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Section 14.179: PRETS179 - Pretension

Integration PointsShape FunctionsMatrix or Vector

NoneNoneStiffness Matrix

DistributionLoad Type

Applied on pretension node K across entire pretension sectionPretension Force

14.179.1. Introduction

The element is used to represent a two or three dimensional section for a bolted structure. The pretension sectioncan carry a pretension load. The pretension node (K) on each section is used to control and monitor the totaltension load.

14.179.2. Assumptions and Restrictions

The pretension element is not capable of carrying bending or torsion loads.

14.180. LINK180 - 3-D Finite Strain Spar (or Truss)

Integration PointsShape FunctionsMatrix or Vector

1Equation 12–6Stiffness Matrix; and Thermaland Newton Raphson LoadVectors

1Equation 12–6, Equation 12–7, and Equation 12–8Mass and Stress StiffeningMatrices

DistributionLoad Type

Linear along lengthElement Temperature

Linear along lengthNodal Temperature

Reference: Cook et al.(117)

14.180.1. Assumptions and Restrictions

The theory for this element is a reduction of the theory for Section 14.189: BEAM189 - 3-D Quadratic Finite StrainBeam. The reductions include only 2 nodes, no bending or shear effects, no pressures, and the entire elementas only one integration point.

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The element is not capable of carrying bending loads. The stress is assumed to be uniform over the entire element.

14.180.2. Element Mass Matrix

All element matrices and load vectors described below are generated in the element coordinate system and arethen converted to the global coordinate system. The element stiffness matrix is:

The element mass matrix is:

(14–804)[ ]M

ALl =

ρ2

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

where:A = element cross-sectional area (input as AREA on R command)L = element lengthρ = density (input as DENS on MP command)

14.181. SHELL181 - 4-Node Finite Strain Shell

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Section 14.181: SHELL181 - 4-Node Finite Strain Shell

Integration PointsShape FunctionsMatrix or Vector

In-plane: 1 x 1 (KEYOPT(3) = 0)2 x 2 (KEYOPT(3) = 2)

Thru-the-thickness:

5 for real constant input1, 3, 5, 7, or 9 per layer for sec-tion data input for general shelloption (KEYOPT(1) = 0)1 per layer for section data in-put for membrane shell option(KEYOPT(1) = 1)

Equation 12–81Stiffness Matrix; and ThermalLoad Vector

Closed form integrationEquation 12–57, Equation 12–58, and Equa-tion 12–59

Mass and Stress StiffnessMatrices

2 x 2Equation 12–59Transverse Pressure LoadVector

2Equation 12–57 and Equation 12–58 specializedto the edge

Edge Pressure Load Vector

DistributionLoad Type

Bilinear in plane of element, linear thru each layerElement Temperature

Bilinear in plane of element, constant thru thicknessNodal Temperature

Bilinear in plane of element and linear along each edgePressure

References: Ahmad(1), Cook(5), Dvorkin(96), Dvorkin(97), Bathe and Dvorkin(98), Allman(113), Cook(114), MacNealand Harder(115)

14.181.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.181.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to thecenterplane.

Each pair of integration points (in the r direction) is assumed to have the same element (material) orientation.

14.181.3. Assumed Displacement Shape Functions

The assumed displacement and transverse shear strain shape functions are given in Chapter 12, “Shape Functions”.The basic functions for the transverse shear strain have been changed to avoid shear locking (Dvorkin(96),Dvorkin(97), Bathe and Dvorkin(98)) and are pictured in Figure 14.30: “Shape Functions for the Transverse Strains”in Section 14.43: SHELL43 - 4-Node Plastic Large Strain Shell.

14.181.4. Membrane Option

A membrane option is available for SHELL181 if KEYOPT(1) = 1. For this option, there is no bending stiffness orrotational degrees of freedom. There is only one integration point per layer, regardless of other input.

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14.181.5. Warping

A warping factor is computed as:

(14–805)φ = D

t

where:

D = component of the vector from the first node to the fourth node parallel to the element normalt = average thickness of the element

If φ > 1.0, a warning message is printed.

14.182. PLANE182 - 2-D 4-Node Structural Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2 if KEYOPT(1) = 0, 2, or 31 if KEYOPT(1) = 1

Equation 12–103 and Equa-tion 12–104

QuadStiffness and Stress StiffnessMatrices; and Thermal LoadVector

1Equation 12–84 and Equa-tion 12–85

Triangle

2 x 2Same as stiffness matrix

QuadMass Matrix

1Triangle

2Same as stiffness matrix, specialized to facePressure Load Vector

DistributionLoad Type

Bilinear across element, constant thru thickness or around circumferenceElement Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

14.182.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement.

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Section 14.182: PLANE182 - 2-D 4-Node Structural Solid

14.182.2. Theory

If KEYOPT(1) = 0, this element uses B method (selective reduced integration technique for volumetric terms)(Hughes(219), Nagtegaal et al.(220)).

If KEYOPT(1) = 1, the uniform reduced integration technique (Flanagan and Belytschko(232)) is used.

If KEYOPT(1) = 2 or 3, the enhanced strain formulations from the work of Simo and Rifai(318), Simo andArmero(319), Simo et al.(320), Andelfinger and Ramm(321), and Nagtegaal and Fox(322) are used. It introduces5 internal degrees of freedom to prevent shear and volumetric locking for KEYOPT(1) = 2, and 4 internal degreesof freedom to prevent shear locking for KEYOPT(1) = 3. If mixed u-P formulation is employed with the enhancedstrain formulations, only 4 degrees of freedom for overcoming shear locking are activated.

14.183. PLANE183 - 2-D 8-Node Structural Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–117 and Equation 12–118QuadStiffness and Stress StiffnessMatrices; and Thermal LoadVector 3Equation 12–96 and Equation 12–97Triangle

3 x 3Same as stiffness matrix

QuadMass Matrix

3Triangle

2 along faceSame as stiffness matrix, specialized to the facePressure Load Vector

DistributionLoad Type

Same as shape functions across element, constant thru thickness or aroundcircumference

Element Temperature

Same as element temperature distributionNodal Temperature

Linear along each facePressure

Reference: Zienkiewicz(39)

14.183.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement.

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14.183.2. Assumptions and Restrictions

A dropped midside node implies that the face is and remains straight.

14.184. MPC184 - Multipoint Constraint Rigid Link and Rigid Beam Ele-ment

MPC184 comprises a general class of multipoint constraint elements that implement kinematic constraints usingLagrange multipliers. The elements are loosely classified here as “constraint elements” and “joint elements”. Allof these elements are used in situations that require you to impose some kind of constraint to meet certain re-quirements. Since these elements are implemented using Lagrange multipliers, the constraint forces and momentsare available for output purposes. The different constraint elements and joint elements are identified by KEYOPT(1).

14.184.1. Slider Element

The slider element (KEYOPT(1) = 3) is a 3-node, 2-D or 3-D element that allows a “slave” node to slide on a linejoining two “master” nodes. KEYOPT(2) = 0 identifies a 3-D slider element, while KEYOPT(2) = 1 identifies a 2-Dslider element.

Figure 14.62 184.2 Slider Constraint Geometry

The constraints required to maintain the “slave” node on the line joining the two “master” nodes are as follows:

Define a unit vector n as:

(14–806)n

x x= −

−

J I

J Ix x

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Section 14.184: MPC184 - Multipoint Constraint Rigid Link and Rigid Beam Element

where:

xI, xJ = position vectors of nodes I and J in the current configuration

Identify unit vectors l and m such that l, m, and n form an orthonormal set.

The constraints are then defined as:

(14–807)( )x x LK I− ⋅ = 0

(14–808)( )x x MK I− ⋅ = 0

where:

xk = position vector of the node K in the current configuration

Let i, j, and k be the global base vectors. Then we can define the unit vector l as:

(14–809)l

n in i

i= ××

≠if n

If n = l, then:

(14–810)l

n kn k

= ××

Finally, the unit vector m is defined as:

(14–811)m n l= ×

The virtual work contributions are obtained from taking the variations of the above equations.

14.184.2. Spherical Element

The spherical element (KEYOPT(1) = 5) is a 2-node, 2-D or 3-D element that allows the two nodes to haveidentical displacements. Rotational components at the nodes, if any, are left free. KEYOPT(2) = 0 identifies a 3-Dspherical element, while KEYOPT(2) = 1 identifies a 2-D spherical element.

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Figure 14.63 184.3 Spherical Constraint Geometry

The constraints imposed in a spherical element is simply:

(14–812)u uI J=

where:

uI and uJ = displacement vectors of nodes I and J, respectively.

That is, the displacements at the two nodes are made identical. The virtual work contributions are obtained fromtaking the variations of the above equation.

14.184.3. Revolute Joint Element

The revolute joint element (KEYOPT(1) = 6) is a 2-node 3-D element. The two nodes that form the element arecoincident and kinematic constraints are imposed such that only the rotation about the revolute axis is free. Thecapabilities of this element include certain control features such as stops, locks, and actuating loads/boundaryconditions that can be imposed on the available component of relative motion between the two nodes of theelement. For example, stops can be specified for the rotation about the revolute axis. This limits the rotationaround the revolute axis to be within a certain range. Displacement or force boundary conditions may be imposedon the component of relative motion between the two nodes allowing for “actuation” of the joint. The drivingforce or displacements arise from the actuating mechanisms like an electric or hydraulic system that drives thesejoints. Material behavior in the form of linear stiffness and damping may also be specified on available componentof relative motion.

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Section 14.184: MPC184 - Multipoint Constraint Rigid Link and Rigid Beam Element

Figure 14.64 184.4 Revolute Joint Geometry

!

"#

Local coordinate systems at the nodes are required to define the kinematic constraints for the element. Theselocal coordinate systems evolve with the rotation at the underlying node. The coordinate system at node I ismandatory, while the coordinate system at node J is optional.

The constraints imposed in a revolute joint element are described in MPC184. The virtual work contributions areobtained by taking the variations of these equations (see Section 3.6: Constraints and Lagrange MultiplierMethod.

14.184.4. Universal Joint Element

The universal joint element (KEYOPT(1) = 7) is a 2-node 3-D element. The two nodes that form the element arecoincident and kinematic constraints are imposed such that only two rotational components of relative motionbetween the two nodes are free. The capabilities of this element include certain control features such as stops,locks, and actuating loads/boundary conditions that can be imposed on the available components of relativemotion between the two nodes of the element. Linear stiffness and damping behavior may also be associatedwith the available components of relative motion in the element.

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Figure 14.65 184.5 Universal Joint Geometry

!

Local coordinate systems at the nodes are required to define the kinematic constraints for the element. Theselocal coordinate systems evolve with the rotation at the underlying node. The coordinate system at node I ismandatory, while the coordinate system at node J is optional.

The constraints imposed in a universal joint element are described in MPC184. The virtual work contributionsare obtained by taking the variations of the constraint equations (see Section 3.6: Constraints and LagrangeMultiplier Method

14.185. SOLID185 - 3-D 8-Node Structural Solid

"

#

$

%

&

'(

)

*

+

,

-/. 0

12. 345.76

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2 if KEYOPT(2) = 0, 2, or 31 if KEYOPT(2) = 1

Equation 12–191, Equation 12–192, andEquation 12–193

Stiffness and Stress StiffnessMatrices; and Thermal LoadVector

2 x 2 x 2Same as stiffness matrixMass Matrix

2 x 2Equation 12–57 and Equa-tion 12–58

Quad

Pressure Load Vector

3Equation 12–38 and Equa-tion 12–39

Triangle

DistributionLoad Type

Trilinear thru elementElement Temperature

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Section 14.185: SOLID185 - 3-D 8-Node Structural Solid

DistributionLoad Type

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

14.185.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement.

14.185.2. Theory

If KEYOPT(2) = 0, this element uses B method (selective reduced integration technique for volumetric terms)(Hughes(219), Nagtegaal et al.(220)).

If KEYOPT(2) = 1, the uniform reduced integration technique (Flanagan and Belytschko(232)) is used.

If KEYOPT(2) = 2 or 3, the enhanced strain formulations from the work of Simo and Rifai(318), Simo andArmero(319), Simo et al.(320), Andelfinger and Ramm(321), and Nagtegaal and Fox(322) are used. It introduces13 internal degrees of freedom to prevent shear and volumetric locking for KEYOPT(2) = 2, and 9 degrees offreedom to prevent shear locking only for KEYOPT(2) = 3. If mixed u-P formulation is employed with the enhancedstrain formulations, only 9 degrees of freedom for overcoming shear locking are activated.

14.186. SOLID186 - 3-D 20-Node Structural Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

14 if KEYOPT(2) = 12 x 2 x 2 if KEYOPT(2) = 0

Equation 12–209, Equation 12–210, andEquation 12–211

Brick

Stiffness and Stress StiffnessMatrices; and Thermal LoadVector

3 x 3Equation 12–186, Equation 12–187, andEquation 12–188

Wedge

2 x 2 x 2Equation 12–171, Equation 12–172, andEquation 12–173

Pyramid

4Equation 12–164, Equation 12–165, andEquation 12–166

Tet

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Integration PointsShape FunctionsGeometryMatrix or Vector

3 x 3 x 3 if brick. If othershapes, same as stiffness mat-rix

Same as stiffness matrix.Mass Matrix

3 x 3Equation 12–69 and Equation 12–70QuadPressure Load Vector

6Equation 12–46 and Equation 12–47Triangle

DistributionLoad Type

Same as shape functions thru elementElement Temperature

Same as shape functions thru elementNodal Temperature

Bilinear across each facePressure

Reference: Zienkiewicz(39)

14.186.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement.

14.187. SOLID187 - 3-D 10-Node Tetrahedral Structural Solid

Integration PointsShape FunctionsMatrix or Vector

4Equation 12–164, Equation 12–165, and Equation 12–166Stiffness, Mass, and StressStiffness Matrices; andThermal Load Vector

6Equation 12–164, Equation 12–165, and Equation 12–166specialized to the face

Pressure Load Vector

DistributionLoad Type

Same as shape functionsElement Temperature

Same as shape functionsNodal Temperature

Linear over each facePressure

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Section 14.187: SOLID187 - 3-D 10-Node Tetrahedral Structural Solid

Reference: Zienkiewicz(39)

14.187.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement.

14.188. BEAM188 - 3-D Linear Finite Strain Beam

Integration PointsShape FunctionsMatrix or Vector

Along the length: 1Across the section: see

Section 14.189: BEAM189 - 3-D Quadratic Finite StrainBeam

Equation 12–6, Equation 12–7, Equation 12–8, Equa-tion 12–9, Equation 12–10, and Equation 12–11

Stiffness and Stress StiffnessMatrices; and Thermal andNewton-Raphson Load Vec-tors

Along the length: 2Across the section: 1 Same as stiffness matrix

Mass Matrix and PressureLoad Vector

DistributionLoad Type

Bilinear across cross-section and linear along length (see Section 14.24: BEAM24- 3-D Thin-walled Beam for details)

Element Temperature

Constant across cross-section, linear along lengthNodal Temperature

Linear along length. The pressure is assumed to act along the element x-axis.Pressure

References: Simo and Vu-Quoc(237), Ibrahimbegovic(238).

The theory for this element is the same as Section 14.189: BEAM189 - 3-D Quadratic Finite Strain Beam, exceptthat it is a linear, 2-node beam element.

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14.189. BEAM189 - 3-D Quadratic Finite Strain Beam

Integration PointsShape FunctionsMatrix or Vector

Along the length: 2Across the section: see text below

Equation 12–19, Equation 12–20, Equation 12–21, Equa-tion 12–22, Equation 12–23, and Equation 12–24

Stiffness and Stress StiffnessMatrices; and Thermal andNewton-Raphson Load Vec-tors

Along the length: 3Across the section: 1 Same as stiffness matrix

Mass Matrix and PressureLoad Vector

DistributionLoad Type

Bilinear across cross-section and linear along length (see Section 14.24: BEAM24- 3-D Thin-walled Beam for details)

Element Temperature

Constant across cross-section, linear along lengthNodal Temperature

Linear along length. The pressure is assumed to act along the element x-axis.Pressure

References: Simo and Vu-Quoc(237), Ibrahimbegovic(238).

14.189.1. Assumptions and Restrictions

The elements are based on Timoshenko beam theory, and hence shear deformation effects are included. Theelement is a quadratic (3-node) beam element in 3-D with six degrees of freedom at each node. The DOF at eachnode includes translations in x, y, and z directions, and rotations about the x, y, and z directions. Warping of crosssections is considered optionally (KEYOPT(1)).

This element is well-suited for linear, large rotation, and/or large strain nonlinear applications. If KEYOPT(2) = 0,the cross sectional dimensions are scaled uniformly as a function of axial strain in nonlinear analysis such thatthe volume of the element is preserved.

The element includes stress stiffness terms, by default, in any analysis using large deformation (NLGEOM,ON).The stress stiffness terms provided enable the elements to analyze flexural, lateral and torsional stability problems(using eigenvalue buckling or collapse studies with arc length methods). Pressure load stiffness (Section 3.3.4:Pressure Load Stiffness) is included.

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Section 14.189: BEAM189 - 3-D Quadratic Finite Strain Beam

Transverse shear strain is constant through cross-section, i.e., cross sections remain plane and undistorted afterdeformation. The element can be used for slender or stout beams. Due to the limitations of first order shear de-formation theory, only moderately “thick” beams may be analyzed. Slenderness ratio of a beam structure maybe used in judging the applicability of the element. It is important to note that this ratio should be calculatedusing some global distance measures, and not based on individual element dimensions. A slenderness ratiogreater than 30 is recommended.

Currently these elements support only elastic relation between transverse shear forces and transverse shearstrains. Orthotropic elastic material properties with bilinear and multilinear isotropic hardening plasticity options(BISO, MISO) may be used. User may specify transverse shear stiffnesses using real constants.

The St. Venant warping functions for torsional behavior is determined in the undeformed state, and is used todefine shear strain even after yielding. The element does not provide options to recalculate the torsional sheardistribution on cross sections during the analysis and possible partial plastic yielding of cross section. As such,large inelastic deformation due to torsional loading should be treated with caution and carefully verified.

The elements are provided with section relevant quantities (area of integration, position, Poisson function,function derivatives, etc.) automatically at a number of section points by the use of section commands. Eachsection is assumed to be an assembly of predetermined number of 9 node cells which illustrates a sectionmodel of a rectangular section. Each cell has 4 integration points.

Figure 14.66 Section Model

Rectangular Section

Section NodesSection Integration Points

When the material has inelastic behavior or the temperature varies across the section, constitutive calculationsare performed at each of the section integration points. For all other cases, the element uses the precalculatedproperties of the section at each element integration point along the length. The restrained warping formulationused may be found in Timoshenko and Gere(246) and Schulz and Fillippou(247).

14.189.2. Stress Evaluation

Several stress evaluation options exist. The section strains and generalized stresses are evaluated at element in-tegration points and then linearly extrapolated to the nodes of the element.

If the material is elastic, stresses and strains are available after extrapolation in cross-section at the nodes ofsection mesh. If the material is plastic, stresses and strains are moved without extrapolation to the section nodes(from section integration points).

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14.190. SOLSH190 - 3-D 8-Node Solid Shell

!

"

#

$

% &&'

$ '

% '

Integration PointsShape FunctionsMatrix or Vector

2 x 2 x 2Equation 12–191, Equation 12–192, and Equation 12–193Stiffness and Stress StiffnessMatrices; and Thermal LoadVector

2 x 2 x 2Same as stiffness matrixMass Matrix

2 x 2Equation 12–57 and Equation 12–58QuadPressure Load Vector

3Equation 12–38 and Equation 12–39Triangle

DistributionLoad Type

Trilinear thru elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

14.190.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement.

14.190.2. Theory

SOLSH190 is a 3-D solid element free of locking in bending-dominant situations. Unlike shell elements, SOLSH190is compatible with general 3-D constitutive relations and can be connected directly with other continuum ele-ments.

SOLSH190 utilizes a suite of special kinematic formulations, including assumed strain method (Bathe and Dvor-kin(98)) to overcome locking when the shell thickness becomes extremely small.

SOLSH190 employs enhanced strain formulations (Simo and Rifai(318), Simo et al.(320)) to improve the accuracyin in-plane bending situations. The satisfaction of the in-plane patch test is ensured. Incompatible shape functionsare used to overcome the thickness locking.

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Section 14.190: SOLSH190 - 3-D 8-Node Solid Shell

14.191. SOLID191 - 3-D 20-Node Layered Structural Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

Thru-the-thickness: 3 for each layerIn-plane: 2 x 2

Equation 12–209, Equation 12–210,and Equation 12–211

Brick

Stiffness Matrix

Thru-the-thickness: 3 for each layerIn-plane: 3

Equation 12–186 , Equation 12–187,and Equation 12–188

Wedge

Thru-the-thickness: 3 for each layerIn-plane: 2 x 2

Equation 12–171, Equation 12–172,and Equation 12–173

Pyramid

Thru-the-thickness: 3 for each layerIn-plane: 3

Equation 12–164 , Equation 12–165,and Equation 12–166

Tet

Same as stiffness matrixMass Matrix

Same as stiffness matrixStress Stiffness Matrix

Same as stiffness matrixThermal Load Vector

3 x 3Equation 12–69 and Equation 12–70QuadPressure Load Vector

6Equation 12–46 and Equation 12–47Triangle

DistributionLoad Type

Linear thru each layer, bilinear in plane of elementElement Temperature

Trilinear thru elementNodal Temperature

Bilinear across each facePressure

Reference: Zienkiewicz(39)

14.191.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

Section 14.46: SOLID46 - 3-D 8-Node Layered Structural Solid includes the description of the effective materialproperties and the interlaminar shear stress calculation which also applies to SOLID191.

Chapter 14: Element Library

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14.192. INTER192 - 2-D 4-Node Gasket

Integration PointsShape FunctionsMatrix or Vector

2Linear in x and y directionsStiffness Matrix

Same as stiffness matrixSame as stiffness matrixThermal Load Vector

DistributionLoad Type

Based on element shape function, constant through the direction perpendicularto element plane

Element temperature

Same as element temperature distributionNodal temperature

14.192.1. Other Applicable Sections

The theory for this element is described in Section 14.194: INTER194 - 3-D 16-Node Gasket.

14.193. INTER193 - 2-D 6-Node Gasket

Integration PointsShape FunctionsMatrix or Vector

2Linear in x, quadratic in y directionStiffness Matrix

Same as stiffness matrixSame as stiffness matrixThermal Load Vector

DistributionLoad Type

Based on element shape function, constant through the direction perpendicularto element plane

Element temperature

Same as element temperature distributionNodal temperature

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Section 14.193: INTER193 - 2-D 6-Node Gasket

14.193.1. Other Applicable Sections

The theory for this element is described in Section 14.194: INTER194 - 3-D 16-Node Gasket.

14.194. INTER194 - 3-D 16-Node Gasket

Integration PointsShape FunctionsMatrix or Vector

2 x 2Linear in x, quadratic in y and z directionsStiffness Matrix

Same as stiffness matrixSame as stiffness matrixThermal Load Vector

DistributionLoad Type

Based on element shape function, constant through the direction perpendicularto element plane

Element temperature

Same as element temperature distributionNodal temperature

14.194.1. Element Technology

The element is designed specially for simulation of gasket joints, where the primary deformation is confined tothe gasket through-thickness direction. The through-thickness deformation of gasket is decoupled from theother deformations and the membrane (in-plane) stiffness contribution is neglected. The element offers a directmeans to quantify the through-thickness behavior of the gasket joints. The pressure-deformation behavior ob-tained from experimental measurement can be applied to the gasket material. See Section 4.3: Gasket Materialfor detailed description of gasket material options.

The element is composed of bottom and top surfaces. An element midplane is created by averaging the coordin-ates of node pairs from the bottom and top surfaces of the elements. The numerical integration of interfaceelements is performed in the element midplane. The element formulation is based on a corotational procedure.The virtual work in an element is written as:

(14–813)δ δW T ddS

Sint

int

= ∫

where:

t = traction force across the elementd = closure across the element

Chapter 14: Element Library

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Sint = midplane of the interface surfaces

The integration is performed in the corotational equilibrium configuration and the Gauss integration procedureis used.

The relative deformation between top and bottom surfaces is defined as:

(14–814)d u u= −TOP BOTTOM

where, uTOP and uBOTTOM are the displacement of top and bottom surfaces of interface elements in the local elementcoordinate system based on the midplane of element.

The thickness direction is defined as the normal direction of the mid plane of the element at the integrationpoint.

14.195. INTER195 - 3-D 8-Node Gasket

Integration PointsShape FunctionsMatrix or Vector

2 x 2Linear in x, bilinear in y and z directionsStiffness Matrix

Same as stiffness matrixSame as stiffness matrixThermal Load Vector

DistributionLoad Type

Based on element shape function, constant through the direction perpendicularto element plane

Element temperature

Same as element temperature distributionNodal temperature

14.195.1. Other Applicable Sections

The theory for this element is described in Section 14.194: INTER194 - 3-D 16-Node Gasket.

14.196. Not Documented

No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

Chapter 14: Element Library

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14.208. SHELL208 - 2-Node Finite Strain Axisymmetric Shell

! #"%$ ! #"'&

()!*+,(-.(0/*

Integration PointsShape FunctionsMatrix or Vector

Along-the-length:1 (KEYOPT(3) = 0)2 (KEYOPT(3) = 2)

Thru-the-thickness:

1, 3, 5, 7, or 9 per layer

KEYOPT(3) = 0: Equation 12–6, Equation 12–7,and Equation 12–11

KEYOPT(3) = 2: Equation 12–19, Equation 12–20,and Equation 12–24

Stiffness and Stress Stiffness Mat-rix; and Thermal and Newton-Raphson Load Vectors

Along-the-length:2 (KEYOPT(3) = 0)3 (KEYOPT(3) = 2)

Thru-the-thickness:

1, 3, 5, 7, or 9 per layer

Same as stiffness matrixMass Matrix and Pressure LoadVector

DistributionLoad Type

Linear along length and linear thru thicknessElement Temperature

Linear along length and constant thru thicknessNodal Temperature

Linear along lengthPressure

References: Ahmad(1), Cook(5)

14.208.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.208.2. Assumptions and Restrictions

Normals to the centerline are assumed to remain straight after deformation, but not necessarily normal to thecenterline.

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Section 14.208: SHELL208 - 2-Node Finite Strain Axisymmetric Shell

14.209. SHELL209 - 2-Node Finite Strain Axisymmetric Shell

Integration PointsShape FunctionsMatrix or Vector

Along-the-length: 2

Thru-the-thickness:

1, 3, 5, 7, or 9 per layer

Equation 12–19, Equation 12–20, andEquation 12–24Stiffness and Stress Stiffness Mat-

rix; and Thermal and Newton-Raphson Load Vectors

Along-the-length: 3

Thru-the-thickness:

1, 3, 5, 7, or 9 per layer

Same as stiffness matrixMass Matrix and Pressure LoadVector

DistributionLoad Type

Linear along length and linear thru thicknessElement Temperature

Linear along length and constant thru thicknessNodal Temperature

Linear along lengthPressure

References: Ahmad(1), Cook(5)

14.209.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations.

14.209.2. Assumptions and Restrictions

Normals to the centerline are assumed to remain straight after deformation, but not necessarily normal to thecenterline.

14.210. Not Documented

No detail or element available at this time.

Chapter 14: Element Library

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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Section 14.222: Not Documented

14.223. PLANE223 - 2-D 8-Node Coupled-Field Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

2 x 2Equation 12–117 and Equation 12–118QuadStiffness and Stress StiffnessMatrices; and Thermal Expan-sion Load Vector 3Equation 12–96 and Equation 12–97Triangle

3 x 3Same as stiffness matrix

QuadMass Matrix

3Triangle

2 along faceSame as stiffness matrix, specialized to the facePressure Load Vector

2 x 2Equation 12–121QuadThermal Conductivity Matrixand Heat Generation LoadVector 3Equation 12–101Triangle

Same as thermal conductivity matrixSpecific Heat Matrix

2Same as thermal conductivity matrix, specialized to theface

Convection Surface Matrix andLoad Vector

2 x 2Equation 12–122QuadDielectric Permittivity andElectrical ConductivityMatrices; Charge Density LoadVector; Joule Heating or Pelti-er Heat Flux Load Vectors

3Equation 12–102Triangle

Same as combination of stiffness matrix and dielectric matrixPiezoelectric Coupling Matrix

Same as combination of electrical conductivity and thermal conductivity matricesSeebeck Coefficient CouplingMatrix

2 along faceSame as dielectric matrix, specialized to the faceSurface Charge Density LoadVector

14.223.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement. Chapter 5, “Electromagnetics” describes the derivation of dielectric and electric conduction matrices.Section 11.2: Piezoelectrics discusses the piezoelectric capability used by the element. Section 11.3: Piezoresistivitydiscusses the piezoresistive effect. Section 11.4: Thermoelectrics discusses the thermoelectric effects.

Chapter 14: Element Library

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No detail or element available at this time.

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No detail or element available at this time.

14.226. SOLID226 - 3-D 20-Node Coupled-Field Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

14 if KEYOPT(2) = 12 x 2 x 2 if KEYOPT(2) = 0

Equation 12–209, Equation 12–210, andEquation 12–211

Brick

Stiffness and Stress StiffnessMatrices; and Thermal Expan-sion Load Vector

3 x 3Equation 12–186, Equation 12–187, andEquation 12–188

Wedge

2 x 2 x 2Equation 12–171, Equation 12–172, andEquation 12–173

Pyramid

4Equation 12–164, Equation 12–165, andEquation 12–166

Tet

3 x 3 x 3 if brick. If othershapes, same as stiffness mat-rix

Same as stiffness matrix.Mass Matrix

3 x 3Equation 12–69 and Equation 12–70QuadPressure Load Vector

6Equation 12–46 and Equation 12–47Triangle

14Equation 12–212BrickThermal Conductivity Matrixand Heat Generation LoadVector

3 x 3Equation 12–189Wedge

2 x 2 x 2Equation 12–174Pyramid

4Equation 12–167Tet

Same as thermal conductivity matrixSpecific Heat Matrix

3 x 3Equation 12–73QuadConvection Surface Matrix andLoad Vector 6Equation 12–49Triangle

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Section 14.226: SOLID226 - 3-D 20-Node Coupled-Field Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

14Equation 12–213BrickDielectric Permittivity andElectrical ConductivityMatrices; Charge Density LoadVector; Joule Heating or Pelti-er Heat Flux Load Vectors

3 x 3Equation 12–190Wedge

2 x 2 x 2Equation 12–175Pyramid

4Equation 12–168Tet

Same as combination of stiffness matrix and dielectric matrix.Piezoelectric Coupling Matrix

Same as combination of electrical conductivity and thermal conductivity matricesSeebeck Coefficient CouplingMatrix

3 x 3Equation 12–175QuadSurface Charge Density LoadVector 6Equation 12–50Triangle

14.226.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement. Chapter 5, “Electromagnetics” describes the derivation of dielectric and electric conduction matrices.Section 11.2: Piezoelectrics discusses the piezoelectric capability used by the element. Section 11.3: Piezoresistivitydiscusses the piezoresistive effect. Section 11.4: Thermoelectrics discusses the thermoelectric effects.

14.227. SOLID227 - 3-D 10-Node Coupled-Field Solid

Integration PointsShape FunctionsMatrix or Vector

4Equation 12–164, Equation 12–165, and Equation 12–166

Stiffness, Mass, and StressStiffness Matrices; andThermal Expansion Load Vec-tor

6Equation 12–164, Equation 12–165, and Equation 12–166specialized to the face

Pressure Load Vector

2 x 2Equation 12–167Thermal Conductivity Matrixand Heat Generation LoadVector

11Same as thermal conductivity matrixSpecific Heat Matrix

6Equation 12–167 specialized to the face. Consistent sur-face matrix.

Convection Surface Matrix andLoad Vector

Chapter 14: Element Library

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Integration PointsShape FunctionsMatrix or Vector

2 x 2Equation 12–168

Dielectric Permittivity andElectrical ConductivityMatrices; Charge Density LoadVector; Joule Heating or Pelti-er Heat Flux Load Vectors

Same as combination of stiffness matrix and dielectric matrixPiezoelectric Coupling Matrix

Same as combination of electrical conductivity and thermal conductivity matricesSeebeck Coefficient CouplingMatrix

6Equation 12–168 specialized to the faceSurface Charge Density LoadVector

14.227.1. Other Applicable Sections

Chapter 2, “Structures” describes the derivation of structural element matrices and load vectors as well as stressevaluations. Section 3.5: General Element Formulations gives the general element formulations used by thiselement. Chapter 5, “Electromagnetics” describes the derivation of dielectric and electric conduction matrices.Section 11.2: Piezoelectrics discusses the piezoelectric capability used by the element. Section 11.3: Piezoresistivitydiscusses the piezoresistive effect. Section 11.4: Thermoelectrics discusses the thermoelectric effects.

14.228. Not Documented

No detail or element available at this time.

14.229. Not Documented

No detail or element available at this time.

14.230. PLANE230 - 2-D 8-Node Electric Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

3 x 3Equation 12–122QuadElectrical Conductivity and DielectricPermittivity Coefficient Matrices 3Equation 12–102Triangle

14.230.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of the electric element matrices and load vectors as wellas electric field evaluations.

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Section 14.230: PLANE230 - 2-D 8-Node Electric Solid

14.230.2. Assumptions and Restrictions

A dropped midside node implies that the edge is straight and that the potential varies linearly along that edge.

14.231. SOLID231 - 3-D 20-Node Electric Solid

Integration PointsShape FunctionsGeometryMatrix or Vector

14Equation 12–213Brick

Electrical Conductivity and DielectricPermittivity Coefficient Matrices

3 x 3Equation 12–190Wedge

8Equation 12–175Pyramid

4Equation 12–168Tet

14.231.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of electric element matrices and load vectors as well aselectric field evaluations.

14.232. SOLID232 - 3-D 10-Node Tetrahedral Electric Solid

!

"#

Chapter 14: Element Library

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Integration PointsShape FunctionsMatrix or Vector

4Equation 12–168Dielectric Permittivity and Electrical Con-ductivity Coefficient Matrices, ChargeDensity Load Vector

14.232.1. Other Applicable Sections

Chapter 5, “Electromagnetics” describes the derivation of electric element matrices and load vectors as well aselectric field evaluations.

14.233. Not Documented

No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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Section 14.242: Not Documented

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

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No detail or element available at this time.

14.249. Not Documented

No detail or element available at this time.

14.250. Not Documented

No detail or element available at this time.

14.251. SURF251 - 2-D Radiosity Surface

SURF251 is used only for postprocessing of radiation quantities, such as radiation heat flux. See SURF251 in theANSYS Elements Reference for details.

Chapter 14: Element Library

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14.252. SURF252 - 3-D Thermal Radiosity Surface

SURF252 is used only for postprocessing of radiation quantities, such as radiation heat flux. See SURF252 in theANSYS Elements Reference for details.

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Section 14.252: SURF252 - 3-D Thermal Radiosity Surface

14–370

Chapter 15: Analysis Tools

15.1. Acceleration Effect

Accelerations are applicable only to elements with displacement degrees of freedom (DOFs).

The acceleration vector ac which causes applied loads consists of a vector with a term for every degree of freedom

in the model. In the description below, a typical node having a specific location and accelerations associatedwith the three translations and three rotations will be considered:

(15–1)

a

a

act

r=

where:

a a a at td

tI

tr= + + = translational acceleration vector

a a ar rI

rr= + = rotational acceleration vector

where:

atd

= accelerations in global Cartesian coordinates (input on ACEL command)

atI

= translational acceleration vector due to inertia relief (see Section 15.2: Inertia Relief)

arI

= rotational acceleration vector due to inertia relief (see Section 15.2: Inertia Relief)

atr

= translational acceleration vector due to rotations (defined below)

arr

= angular acceleration vector due to input rotational accelerations (defined below)

ANSYS defines three types of rotations:

Rotation 1: The whole structure rotates about each of the global Cartesian axes (input on OMEGA and DO-MEGA commands)Rotation 2: The element component rotates about an axis defined by user (input on CMOMEGA and CMDO-MEGA commands).Rotation 3: The global origin rotates about the axis by user if Rotation 1 appears or the rotational axis rotatesabout the axis defined by user if Rotation 2 appears (input on CGOMGA, DCGOMG, and CGLOC commands)

Up to two out of the three types of rotations may be applied on a structure at the same time.

The angular acceleration vector due to rotations is:

(15–2) arr = + + ×& &ω ωΩ Ω

The translational acceleration vector due to rotations is:

(15–3) ( ) ( ) ( ) a r r r Rtr = × × + × + ⋅ × × + × × +ω ω ω ω& 2 Ω Ω Ω &&Ω × R

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

where:

x = vector cross product

In the case where the rotations are the combination of Rotation 1 and Rotation 3:

ω = angular velocity vector defined about the global Cartesian origin (input on OMEGA command)Ω = angular velocity vector of the overall structure about the point CG (input on CGOMGA command) &ω = angular acceleration vector defined about the global Cartesian origin (input on DOMEGA command)

&Ω = angular acceleration vector of the overall structure about the point CG (input on DCGOMG command)r = position vector (see Figure 15.1: “Rotational Coordinate System (Rotations 1 and 3)”)R = vector from CG to the global Cartesian origin (computed from input on CGLOC command, with directionopposite as shown in Figure 15.1: “Rotational Coordinate System (Rotations 1 and 3)”.

In the case where the rotations are Rotation 1 and Rotation 2:

ω = angular velocity vector defined about the rotational axis of the element component (input on CMOMEGAcommand)Ω = angular velocity vector defined about the global Cartesian origin (input on OMEGA command) &ω = angular acceleration vector defined about the rotational axis of the element component (input onCMDOMEGA command)

&Ω = angular acceleration vector defined about the global Cartesian origin (input on DOMEGA command)r = position vector (see Figure 15.2: “Rotational Coordinate System (Rotations 1 and 2)”)R = vector from about the global Cartesian origin to the point on the rotational axis of the component (seeFigure 15.2: “Rotational Coordinate System (Rotations 1 and 2)”).

In the case where the rotations are Rotation 2 and Rotation 3:

ω = angular velocity vector defined about the rotational axis of the element component (input on CMOMEGAcommand)Ω = angular velocity vector of the overall structure about the point CG (input on CGOMGA command) &ω = angular acceleration vector defined about the rotational axis of the element component (input onCMDOMEGA command)

&Ω = angular acceleration vector of the overall structure about the point CG (input on DCGOMG command)r = position vector (see Figure 15.3: “Rotational Coordinate System (Rotations 2 and 3)”)R = vector from CG to the point on the rotational axis of the component (see Figure 15.3: “Rotational Co-ordinate System (Rotations 2 and 3)”)

Chapter 15: Analysis Tools

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Figure 15.1 Rotational Coordinate System (Rotations 1 and 3)

Overall system

CG

Ω,Ω.

ω,ω.

R X

Y

Z

rOrigin of globalCartesiancoordinatesystem

ModelPoint beingstudied

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Section 15.1: Acceleration Effect

Figure 15.2 Rotational Coordinate System (Rotations 1 and 2)

Overall system

Ω,Ω.

ω,ω.

R

X

Y

Z

r

elementcomponent

ModelPoint beingstudied

Point on rotational axis of the component

Origin of globalCartesiancoordinatesystem

Chapter 15: Analysis Tools

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Figure 15.3 Rotational Coordinate System (Rotations 2 and 3)

Overall system

Ω,Ω.

ω,ω.

R

r

elementcomponent

ModelPoint beingstudied

Point on rotational axis of the component

CG

For MASS21 with KEYOPT(3) = 0 and MATRIX27 with KEYOPT(3) = 2, additional Euler's equation terms are con-sidered:

(15–4) [ ] M IT T= ×ω ω

where:

M = additional moments generated by the angular velocity[I] = matrix of input moments of inertiaωT = total applied angular velocities: = ω + Ω

15.2. Inertia Relief

Inertia relief is applicable only to the structural parts of linear analyses.

An equivalent free-body analysis is performed if a static analysis (ANTYPE,STATIC) and inertia relief (IRLF,1) areused. This is a technique in which the applied forces and torques are balanced by inertial forces induced by anacceleration field. Consider the application of an acceleration field (to be determined) that precisely cancels orbalances the applied loads:

(15–5) ( )

( ) ( )

F a d vol

F r a r d vol

ta

tI

vol

ra

rI

vol

+ =

+ × ×

∫∫

ρ

ρ

0

== 0

15–5ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 15.2: Inertia Relief

where:

Fta

= force components of the applied load vector

atI

= translational acceleration vector due to inertia relief (to be determined)ρ = densityvol = volume of model

Fra

= moment components of the applied load vector

r X Y Z T= = position vector

arI

= rotational acceleration vector due to inertia relief (to be determined)x = vector cross product

In the finite element implementation, the position vector r and the moment in the applied load vector Fra

are taken with respect to the origin. Considering further specialization for finite elements, Equation 15–5 is re-written in equivalent form as:

(15–6)

[ ]

[ ]

F M a

F M a

ta

t tI

ra

t rI

+ =

+ =

0

0

where:

[Mt] = mass tensor for the entire finite element model (developed below)

[Mr] = mass moments and mass products of the inertia tensor for the entire finite element model (developed

below)

Once [Mt] and [Mr] are developed, then atI

and arI

in Equation 15–6 can be solved. The output inertia relief

summary includes atI

(output as TRANSLATIONAL ACCELERATIONS) and arI

(output as ROTATIONAL ACCEL-ERATIONS).

The computation for [Mt] and [Mr] proceeds on an element-by-element basis:

(15–7)[ ] [ ] ( )M m d volt e

vol= =

∑ ∫1 0 0

0 1 0

0 0 1

ρ

(15–8)[ ] [ ]M I

y z xy xz

xy x z yz

xz yz x y

dr e= =

+ − −

− + −

− − +

∑

2 2

2 2

2 2

ρ (( )volvol∫

in which [me] and [Ie] relate to individual elements, and the summations are for all elements in the model. The

output `precision mass summary' includes components of [Mt] (labeled as TOTAL MASS) and [Mr] (MOMENTS

AND PRODUCTS OF INERTIA TENSOR ABOUT ORIGIN).

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The evaluation for components of [me] are simply obtained from a row-by-row summation applied to the ele-

mental mass matrix over translational (x, y, z) degrees of freedom. It should be noted that [me] is a diagonal

matrix (mxy = 0, mxz = 0, etc.). The computation for [Ie] is somewhat more involved, but can be summarized in

the following form:

(15–9)[ ] [ ] [ ][ ]I b M beT

e=

where:

[Me] = elemental mass matrix (which may be either lumped or consistent)

[b] = matrix which consists of nodal positions and unity components

The forms of [b] and, of course, [Me] are dependent on the type of element under consideration. The description

of element mass matrices [Me] is given in Section 2.2: Derivation of Structural Matrices. The derivation for [b]

comes about by comparing Equation 15–5 and Equation 15–6 on a per element basis, and eliminating Fra

toyield

(15–10)[ ] ( )M a r a r d volr r

IrI

vol= × ×∫ ρ

where:

vol = element volume

After a little manipulation, the acceleration field in Equation 15–10 can be dropped, leaving the definition of [Ie]

in Equation 15–9.

It can be shown that if the mass matrix in Equation 15–9 is derived in a consistent manner, then the componentsin [Ie] are quite precise. This is demonstrated as follows. Consider the inertia tensor in standard form:

(15–11)[ ] ( )I

y z xy xz

xy x z yz

xz yz x y

d vole v=

+ − −

− + −

− − +

2 2

2 2

2 2

ρool∫

which can be rewritten in product form:

(15–12)[ ] [ ] [ ] ( )I Q Q d vole

Tvol= ∫ ρ

The matrix [Q] is a skew-symmetric matrix.

(15–13)[ ]Q

z y

z x

y x

=−

−−

0

0

0

Next, shape functions are introduced by way of their basic form,

(15–14) [ ]r XYZ N x y z x y zT T= = … 1 1 1 2 2 2

15–7ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

Section 15.2: Inertia Relief

where:

[N] = usual matrix containing individual shape functions

Omitting the tedious algebra, Equation 15–13 and Equation 15–14 are combined to obtain

(15–15)[ ] [ ][ ]Q N b=

where:

(15–16)[ ]b

z y z y

z x z x

y x y x

T =− −

− −− −

0 0

0 0

0 0

2 1 2 2

1 1 2 2

1 1 2 2

………

Inserting Equation 15–16 into Equation 15–12 leads to

(15–17)[ ] [ ] [ ] [ ] ( )[ ]I b N N d vol be

T Tvol= ∫ ρ

Noting that the integral in Equation 15–17 is the consistent mass matrix for a solid element,

(15–18)[ ] [ ] [ ] ( )M N N d vole

Tvol= ∫ ρ

So it follows that Equation 15–9 is recovered from the combination of Equation 15–17 and Equation 15–18.

As stated above, the exact form of [b] and [Me] used in Equation 15–9 varies depending on the type of element

under consideration. Equation 15–16 and Equation 15–18 apply to all solid elements (in 2-D, z = 0). For discreteelements, such as beams and shells, certain adjustments are made to [b] in order to account for moments of in-ertia corresponding to individual rotational degrees of freedom. For 3-D beams, for example, [b] takes the form:

(15–19)[ ]b T

z y z y

z x z x

y x y x

=− −

− −

− −

0 1 0 0 0 1 0 0

0 0 1 0 0 0 1 0

0 0 0 1 0

2 1 2 2

1 1 2 2

1 1 2 2

…

…

00 0 1…

In any case, it is worth repeating that precise [Ie] and [Mr] matrices result when consistent mass matrices are used

in Equation 15–9.

If inertia relief is requested (IRLF,1), then the mx, my, and mz diagonal components in [Mt] as well as all tensor

components in [Mr] are calculated. Then the acceleration fields atI

and arI

are computed by the inversion of

Equation 15–6. The body forces that correspond to these accelerations are added to the user-imposed loadvector, thereby making the net or resultant reaction forces null. The user may request only a mass summary for[Mt] and [Mr] (IRLF,-1).

The calculations for [Mt], [Mr], at

I and ar

I are made at every substep of every load step where they are requested,

reflecting changes in material density and applied loads.

Several limitations apply, and it is useful to list them below.

1. Element mass and/or density must be defined in the model.

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2. Substructures are not allowed.

3. In a model containing both 2-D and 3-D elements, only Mt(1,1) and Mt(2,2) in [Mt] and Mr(3,3) in [Mr] are

correct in the precise mass summary. All other terms in [Mt] and [Mr] should be ignored. The acceleration

balance is, however, correct.

4. Axisymmetric elements are not allowed.

5. If grounded gap elements are in the model, their status should not change from their original status.Otherwise the exact kinematic constraints stated above might be violated.

6. The “CENTER OF MASS” output does not include the effects of offsets or tapering on beam elements(BEAM23, BEAM24, BEAM44, BEAM54, BEAM188, and BEAM189) , as well as the layered elements (SHELL91,SHELL99, SOLID46, and SOLID191). Breaking up each tapered element into several elements will give amore accurate solution.

15.3. Damping Matrices

The damping matrix ([C]) may be used in harmonic, damped modal and transient analyses as well as substructuregeneration. In its most general form, it is:

(15–20)[ ] [ ] ( )[ ] [ ] [ ] [C M K K C Cc j

mj j

j

N

km

= + + + +

+ +

=∑α β β β βξ2

1 Ω ξξ ]k

Ne

=∑

1

where:

[C] = structure damping matrixα = mass matrix multiplier (input on ALPHAD command)[M] = structure mass matrixβ = stiffness matrix multiplier (input on BETAD command)βc = variable stiffness matrix multiplier (see Equation 15–23)

[K] = structure stiffness matrixNm = number of materials with DAMP or DMPR input

β jm

= stiffness matrix multiplier for material j (input as DAMP on MP command)

βξj = constant (frequency-independent) stiffness matrix coefficient for material j (input as DMPR on MP

command)Ω = circular excitation frequencyKj = portion of structure stiffness matrix based on material j

Ne = number of elements with specified damping

Ck = element damping matrix

Cξ = frequency-dependent damping matrix (see Equation 15–21)

Element damping matrices are available for:

Dynamic Fluid CouplingFLUID383-D Elastic BeamBEAM4

CombinationCOMBIN40Revolute JointCOMBIN7

2-D Contained FluidFLUID79Linear ActuatorLINK11

3-D Contained FluidFLUID80Spring-DamperCOMBIN14

Axisymmetric-Harmonic Contained FluidFLUID81Elastic Straight PipePIPE16

2-D Structural Surface EffectSURF153Stiffness, Damping, or Mass MatrixMATRIX27

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Section 15.3: Damping Matrices

3-D Structural Surface EffectSURF154ControlCOMBIN37

Note that [K], the structure stiffness matrix, may include plasticity and/or large deflection effects (i.e., may be thetangent matrix).

For the special case of thin-film fluid behavior, damping parameters may be computed for structures and usedin a subsequent structural analysis (see Section 15.19: Modal Projection Method).

The frequency-dependent damping matrix Cξ is specified indirectly by defining a damping ratio, ξd. This effect

is available only in the Spectrum (ANTYPE,SPECTR), the Harmonic Response with mode superposition (AN-TYPE,HARM with HROPT,MSUP) Analyses, as well as the Transient Analysis with mode superposition (AN-TYPE,TRANS with TRNOPT,MSUP).

Cξ may be calculated from the specified ξd as follows:

(15–21) [ ] Φ Φi

Ti i

diCξ ξ ω= 2

where:

ξid

= damping ratio for mode shape i (defined below)Φi = shape of mode i

ωi = circular natural frequency associated with mode shape i = 2πfi

fi = natural frequency associated with mode shape i

The damping ratio ξi

d is the combination of:

(15–22)ξ ξ ξi

dim= +

where:

ξ = constant damping ratio (input on DMPRAT command)

ξim

= modal damping ratio for mode shape i (input on MDAMP command)

Actually ξi

d is used directly. Cξ is never explicitly computed.

βc , available for the Harmonic Response Analyses (ANTYPE,HARM with HROPT,FULL or HROPT,REDUC), is used

to give a constant damping ratio, regardless of frequency. The damping ratio is the ratio between actual dampingand critical damping. The stiffness matrix multiplier is related to the damping ratio by:

(15–23)β ξ

πξc f

= = 2Ω

where:

ξ = constant damping ratio (input on DMPRAT command)Ω = excitation circular frequency in the range between ΩB and ΩE

ΩB = 2πFB

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ΩE = 2πFE

fB = beginning frequency (input as FREQB,HARFRQ command)

fE = end frequency (input as FREQE,HARFRQ command)

15.4. Element Reordering

The ANSYS program provides a capability for reordering the elements to reduce the wavefront. Since thewavefront solver processes the elements sequentially, the order of the elements greatly affects the size of thewavefront. To minimize the wavefront is to minimize the number of DOFs that are active at the same time.

Each element has a location, or order, number which represents its sequence in the solution process. Initially,this order number is equal to the identification number of the element. Reordering changes the order numberfor each element. (The element identification numbers are not changed during reordering and are used in pre-processing and postprocessing.) The new order is used only during the solution phase and is transparent to theuser but can be displayed (using the /PNUM,ELEM command). Reordering can be accomplished in one of threeways:

15.4.1. Reordering Based on Topology with a Program-Defined Starting Surface

This sorting algorithm is used by default, requiring no explicit action by the user. The sorting may also be accessedby initiating the reordering (WAVES command), but without a wave starting list (WSTART command). Thestarting surface is defined by the program using a graph theory algorithm (Hoit and Wilson(99), Cuthill andMcKee(100), Georges and McIntyre(101)). The automatic algorithm defines a set of accumulated nodal and elementweights as suggested by Hoit and Wilson(99). These accumulated nodal and element weights are then used todevelop the element ordering scheme.

15.4.2. Reordering Based on Topology with a User- Defined Starting Surface

This sorting algorithm is initiated (using the WAVES command) and uses a starting surface (input on the WSTARTcommand), and then possibly is guided by other surfaces (also input on the WSTART command). These surfaces,as required by the algorithm, consist of lists of nodes (wave lists) which are used to start and stop the orderingprocess. The steps taken by the program are:

1. Define each coupled node set and constraint equation as an element.

2. Bring in wave list (defined on WSTART command).

3. Define candidate elements (elements having nodes in present wave list, but not in any other wave list).

4. If no candidate elements were found, go to step 2 and start again for next wave list. If no more wavelists, then stop.

5. Find the best candidate based on:

a. element that brings in the least number of new nodes (nodes not in present wave list) - Subset Aof candidate elements.

b. if Subset A has more than one element, then element from Subset A on the surface of the model -Subset B of candidate elements.

c. if Subset B has more than one element, then element from Subset B with the lowest elementnumber.

6. Remove processed nodes from wave list and include new nodes from best candidate.

7. If best candidate element is not a coupled node set or constraint equation, then save element.

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Section 15.4: Element Reordering

8. Repeat steps 3 to 7 until all elements have been processed.

Restrictions on the use of reordering based on topology are:

1. Master DOFs and imposed displacement conditions are not considered.

2. Any discontinuous models must have at least one node from each part included in a list.

15.4.3. Reordering Based on Geometry

This sorting algorithm (accessed with the WSORT command) is performed by a sweep through the elementcentroids along one of the three global or local axes, either in the positive or negative direction.

15.4.4. Automatic Reordering

If no reordering was explicitly requested (accessed with the NOORDER command), models are automaticallyreordered before solution. Both methods outlined in Section 15.4.1: Reordering Based on Topology with a Program-Defined Starting Surface and Section 15.4.3: Reordering Based on Geometry (in three positive directions) areused and the ordering resulting in the smallest wavefront is used.

15.5. Automatic Master DOF Selection

The program permits the user to select the master degrees of freedom (MDOF) (input on M command), theprogram to select them (input on TOTAL command), or any combination of these two options. Any user selectedMDOF are always retained DOFs during the Guyan reduction. Consider the case where the program selects allof the MDOF. (This method is described by Henshell and Ong(9)). Define:

NS = Number of MDOFS to be selected

NA = Number of total active DOFs in the structure

The procedure then goes through the following steps:

1. The first NS completed DOFs that are encountered by the wavefront solver are initially presumed to be

MDOF. (An option is available to exclude the rotational DOFs (NRMDF = 1, TOTAL command)).

2. The next DOF is brought into the solver. All of the NS + 1 DOFs then have the quantity (Qi) computed:

(15–24)Q

KMi

ii

ii=

where:

Kii = ith main diagonal term of the current stiffness matrix

Mii = ith main diagonal term of the current mass matrix (or stress stiffness matrix for buckling)

If Kii or Mii is zero or negative, row i is eliminated. This removes tension DOFs in buckling.

1. The largest of the Qi terms is identified and then eliminated.

2. All remaining DOFs are thus processed in the same manner. Therefore, NA - NS DOFs are eliminated.

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It may be seen that there sometimes is a path dependency on the resulting selection of MDOF. Specifically, oneselection would result if the elements are read in from left to right, and a different one might result if the elementsare read in from right to left. However, this difference usually yields insignificant differences in the results.

The use of this algorithm presumes a reasonably regular structure. If the structure has an irregular mass distribution,the automatically selected MDOF may be concentrated totally in the high mass regions, in which case themanual selection of some MDOF should be used.

15.6. Automatic Time Stepping

The method of automatic time stepping (or automatic loading) is one in which the time step size and/or theapplied loads are automatically determined in response to the current state of the analysis under consideration.This method (accessed with AUTOTS,ON) may be applied to structural, thermal, electric, and magnetic analysesthat are performed in the time domain (using the TIME command), and includes static (or steady state) (AN-TYPE,STATIC) and dynamic (or transient) (ANTYPE,TRANS) situations.

An important point to be made here is that automatic loading always works through the adjustment of the timestep size; and that the loads that are applied are automatically adjusted if ramped boundary conditions are ac-tivated (using KBC,0). In other words the time step size is always subjected to possible adjustment when auto-matic loading is engaged. Applied loads and boundary conditions, however, will vary according to how they areapplied and whether the boundary conditions are stepped or ramped. That is why this method may also bethought of as automatic loading.

There are two important features of the automatic time stepping algorithm. The first feature concerns the abilityto estimate the next time step size, based on current and past analysis conditions, and make proper load adjust-ments. In other words, given conditions at the current time, tn, and the previous time increment, ∆tn, the primary

aim is to determine the next time increment, ∆tn+1. Since the determination of ∆tn+1 is largely predictive, this

part of the automatic time stepping algorithm is referred to as the time step prediction component.

The second feature of automatic time stepping is referred to as the time step bisection component. Its purposeis to decide whether or not to reduce the present time step size, ∆tn, and redo the substep with a smaller step

size. For example, working from the last converged solution at time point tn-1, the present solution begins with

a predicted time step, ∆tn. Equilibrium iterations are performed; and if proper convergence is either not achieved

or not anticipated, this time step is reduce to ∆tn/2 (i.e., it is bisected), and the analysis begins again from time

tn-1. Multiple bisections can occur per substep for various reasons (discussed later).

15.6.1. Time Step Prediction

At a given converged solution at time, tn, and with the previous time increment, ∆tn, the goal is to predict the

appropriate time step size to use as the next substep. This step size is derived from the results of several unrelatedcomputations and is most easily expressed as the minimization statement:

(15–25)∆ ∆ ∆ ∆ ∆ ∆ ∆t min t t t t t tn eq g c p+ =1 1 2( , , , , , )

where:

∆teq = time increment which is limited by the number of equilibrium iterations needed for convergence at

the last converged time point. The more iterations required for convergence, the smaller the predicted timestep. This is a general measure of all active nonlinearities. Increasing the maximum number of equilibriumiterations (using the NEQIT command) will tend to promote larger time step sizes.

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Section 15.6: Automatic Time Stepping

∆t1 = time increment which is limited by the response eigenvalue computation for 1st order systems (e.g.,

thermal transients) (input on the TINTP command).∆t2 = time increment which is limited by the response frequency computation for 2nd order systems (e.g.,

structural dynamics). The aim is to maintain 20 points per cycle (described below).∆tg = time increment that represents the time point at which a gap or a nonlinear (multi-status) element will

change abruptly from one condition to another (status change). KEYOPT(7) allows further control for theCONTAC elements.∆tc = time increment based on the allowable creep strain increment (described below).

∆tp = time increment based on the allowable plastic strain increment. The limit is set at 5% per time step

(described below).

Several trial step sizes are calculated, and the minimum one is selected for the next time step. This predictedvalue is further restricted to a range of values expressed by

(15–26)∆ ∆ ∆t min F t tn n max+ ≤1 ( , )

and

(15–27)∆ ∆ ∆t max t F tn n min+ ≥1 ( / , )

where:

F = increase/decrease factor. F = 2, if static analysis; F = 3, if dynamic (see the ANTYPE and TIMINT commands)∆tmax = maximum time step size (DTMAX from the DELTIM command or the equivalent quantity calculated

from the NSUBST command)∆tmin = minimum time step size (DTMIN from the DELTIM command or the equivalent quantity calculated

from the NSUBST command)

In other words, the current time step is increased or decreased by at most a factor of 2 (or 3 if dynamic), and itmay not be less than ∆tmin or greater than ∆tmax.

15.6.2. Time Step Bisection

When bisection occurs, the current substep solution (∆tn) is removed, and the time step size is reduced by 50%.

If applied loads are ramped (KBC,0), then the current load increment is also reduced by the same amount. Oneor more bisections can take place for several reasons, namely:

1. The number of equilibrium iterations used for this substep exceeds the number allowed (NEQIT com-mand).

2. It appears likely that all equilibrium iterations will be used.

3. A negative pivot message was encountered in the solution, suggesting instability.

4. The largest calculated displacement DOF exceeds the limit (DLIM on the NCNV command).

5. An illegal element distortion is detected (e.g., negative radius in an axisymmetric analysis)

More than one bisection may be performed per substep. However, bisection of the time-step size is limited bythe minimum size (defined by DTMIN input on the DELTIM command or the equivalent NSUBST input).

15.6.3. The Response Eigenvalue for 1st Order Transients

The response eigenvalue is used in the computation of ∆t1 and is defined as:

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(15–28)λr

T T

Tu K u

u C u= [ ]

[ ]

∆ ∆∆ ∆

where:

λr = response eigenvalue (item RESEIG for POST26 SOLU command and *GET command)

∆u = substep solution vector (tn-1 to tn)

[KT] = the Dirichlet matrix. In a heat transfer or an electrical conduction analysis this matrix is referred to asthe conductivity matrix; in magnetics this is called the magnetic “stiffness”. The superscript T denotes theuse of a tangent matrix in nonlinear situations[C] = the damping matrix. In heat transfer this is called the specific heat matrix.

The product of the response eigenvalue and the previous time step (∆tn) has been employed by Hughes(145)

for the evaluation of 1st order explicit/implicit systems. In Hughes(145) the quantity ∆tnλ is referred to as the

“oscillation limit”, where λ is the maximum eigenvalue. For unconditionally stable systems, the primary restrictionon time-step size is that the inequality ∆tnλ >> 1 should be avoided. Hence it is very conservative to propose

that ∆tnλ = 1.

Since the time integration used employs the trapezoidal rule (Equation 17–31), all analyses of 1st order systemsare unconditionally stable. The response eigenvalue supplied by means of Equation 15–28 represents the dom-inate eigenvalue and not the maximum; and the time-step restriction above is restated as:

(15–29)∆t f fn rλ ≅ <( )1

This equation expresses the primary aim of automatic time stepping for 1st order transient analyses. Thequantity ∆tnλr appears as the oscillation limit output during automatic loading. The default is f = 1/2, and can

be changed (using OSLIM and TOL on the TINTP command). The quantity ∆t1 is approximated as:

(15–30)∆∆ ∆

tt

ftn r n

1 =λ

15.6.4. The Response Frequency for Structural Dynamics

The response frequency is used in the computation of ∆t2 and is defined as (Bergan(105)):

(15–31)f

u K u

u M ur

T T

T2

22= [ ]

( ) [ ]

∆ ∆∆ ∆π

where:

fr = response frequency (item RESFRQ for POST26 SOLU command and *GET command)

∆u = substep solution vector (tn-1 to tn)

[KT] = tangent stiffness matrix[M] = mass matrix

This equation is a nonlinear form of Rayleigh's quotient. The related response period is:

(15–32)T fr r= 1/

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Section 15.6: Automatic Time Stepping

Using Tr, the time increment limited by the response frequency is:

(15–33)∆t Tr2 20= /

15.6.5. Creep Time Increment

The time step size may be increased or decreased by comparing the value of the creep ratio Cmax (Section 4.2:

Rate-Dependent Plasticity) to the creep criterion Ccr. Ccr is equal to .10 unless it is redefined (using the CRPLIMcommand). The time step estimate is computed as:

(15–34)∆ ∆t t

CCc n

cr

max=

∆tc is used in Equation 15–25 only if it differs from ∆tn by more than 10%.

15.6.6. Plasticity Time Increment

The time step size is increased or decreased by comparing the value of the effective plastic strain increment ∆%εnpl

(Equation 4–26) to 0.05 (5%). The time step estimate is computed as:

(15–35)∆ ∆

∆t tp n

npl

= .05

%ε

∆tp is used in Equation 15–25 only if it differs from ∆tn by more than 10%.

15.7. Solving for Unknowns and Reactions

In general, the equations that are solved for static linear analyses are:

(15–36)[ ] K u F=

or

(15–37)[ ] K u F Fa r= +

where:

[K] = total stiffness or conductivity matrix = [ ]Ke

m

N

=∑

1

u = nodal degree of freedom (DOF) vectorN = number of elements[Ke] = element stiffness or conductivity matrix

Fr = nodal reaction load vector

Fa, the total applied load vector, is defined by:

(15–38) F F Fa nd e= +

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where:

Fnd = applied nodal load vector

Fe = total of all element load vector effects (pressure, acceleration, thermal, gravity)

Equation 15–36 thru Equation 15–38 are similar to Equation 17–1 thru Equation 17–4.

If sufficient boundary conditions are specified on u to guarantee a unique solution, Equation 15–36 can besolved to obtain the node DOF values at each node in the model.

Rewriting Equation 15–37 for linear analyses by separating out the matrix and vectors into those DOFs with andwithout imposed values,

(15–39)

[ ] [ ]

[ ] [ ]

K K

K K

u

u

F

F

cc cs

csT

ss

c

s

ca

sa

=

+

F

F

cr

sr

where:

s = subscript representing DOFs with imposed values (specified DOFs)c = subscript representing DOFs without imposed values (computed DOFs)

Note that us is known, but not necessarily equal to 0. Since the reactions at DOFs without imposed values

must be zero, Equation 15–39 can be written as:

(15–40)

[ ] [ ]

[ ] [ ]

K K

K K

u

u

F

F

cc cs

csT

ss

c

s

ca

sa

=

+

0

Fsr

The top part of Equation 15–40 may be solved for uc:

(15–41) [ ] ( [ ] )u K K u Fc cc cs s ca= − +−1

The actual numerical solution process is not as indicated here but is done more efficiently using one of the variousequation solvers discussed in Section 15.8: Equation Solvers.

15.7.1. Reaction Forces

The reaction vector Fsr

, may be developed for linear models from the bottom part of Equation 15–40:

(15–42) [ ] [ ] F K u K u Fsr

csT

c ss s sa= + −

where:

Fsr

= reaction forces (output using either OUTPR,RSOL or PRRSOL command)

Alternatively, the nodal reaction load vector may be considered over all DOFs by combining Equation 15–37 andEquation 15–38 to get:

(15–43) [ ] F K u F Fr nd e= − −

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Section 15.7: Solving for Unknowns and Reactions

where only the loads at imposed DOF are output. Where applicable, the transient/dynamic effects are added:

(15–44) [ ] [ ] [ ] F M u C u K u F Fr nd e= + + − −&& &

where:

[M] = total mass matrix[C] = total damping or conductivity matrix

&u , &&u = defined below

The element static nodal loads are:

(15–45) [ ] F K u Fek

e e ee= − +

where:

Fek

= element nodal loads (output using OUTPR,NLOAD, or PRESOL commands)e = subscript for element matrices and load vectors

The element damping and inertial loads are:

(15–46) [ ] F C ueD

e= − &

(15–47) [ ] F M ueI

e= &&

where:

FeD

= element damping nodal load (output using OUTPR,NLOAD, or PRESOL commands)

FeI

= element inertial nodal load (output using OUTPR,NLOAD, or PRESOL commands)

Thus,

(15–48) ( ) F F F F Fr

eK

eD

eI nd

m

N= − + + −

=∑

1

The derivatives of the nodal DOF with respect to time are:

&u = first derivative of the nodal DOF with respect to time, e.g., velocity

&&u = second derivative of the nodal DOF with respect to time, e.g., acceleration

Section 17.2: Transient Analysis and Section 17.4: Harmonic Response Analyses discuss the transient and harmonicdamping and inertia loads.

If an imposed DOF value is part of a constraint equation, the nodal reaction load vector is further modified usingthe appropriate terms of the right hand side of Equation 15–143; that is, the forces on the non-unique DOFs aresummed into the unique DOF (the one with the imposed DOF value) to give the total reaction force acting onthat DOF.

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15.7.2. Disequilibrium

The following circumstances could cause a disequilibrium, usually a moment disequilibrium:

Explanation of Possible DifficultyProgram Option

If the 4 nodes do not lie in a flat plane moment equilibrium may not bepreserved, as no internal corrections are done. However, the program re-quires such elements to be input very close to flat.

non-planar, 4-node membrane shellelementsSHELL41SHELL63 with KEYOPT(1) = 1

The user can write any form of relationship between the displacements,and these may include fictitious forces or moments. Thus, the reactionforces printout can be used to detect input errors.

nodal coupling constraint equations(CP, CE commands)

The user has the option to input almost any type of erroneous input, sothat such input should be checked carefully. For example, all terms repres-enting UX degrees of freedom of one UX row of the matrix should sum tozero to preserve equilibrium.

MATRIX27User generated super- element matrix

Noncoincident nodes can cause a moment disequilibrium. (This is usuallynot a problem if one of the nodes is attached to a non-rotating ground).

COMBIN7CONTAC12COMBIN37FLUID38COMBIN39COMBIN40

Elements with one node having a different nodal coordinate system fromthe other are inconsistent.

COMBIN14 (with KEYOPT(2) > 0)MATRIX27COMBIN37FLUID38COMBIN39COMBIN40

The following circumstances could cause an apparent disequilibrium:

1. All nodal coordinate systems are not parallel to the global Cartesian coordinate system. However, if allnodal forces are rotated to the global Cartesian coordinate system, equilibrium should be seen to besatisfied.

2. The solution is not converged. This applies to the potential discrepancy between applied and internalelement forces in a nonlinear analysis.

3. The mesh is too coarse. This may manifest itself for elements where there is an element force printoutat the nodes, such as SHELL61 (axisymmetric-harmonic structural shell).

4. Stress stiffening only (SSTIF,ON), (discussed in Section 3.3: Stress Stiffening) is used. Note that momentequilibrium seems not to be preserved in equation (3.6). However, if the implicit updating of the coordin-ates is also considered (NLGEOM,ON), equilibrium will be seen to be preserved.

5. The “TOTAL” of the moments (MX, MY, MZ) given with the reaction forces does not necessarily representequilibrium. It only represents the sum of all applicable moments. Moment equilibrium would also needthe effects of forces taken about an arbitrary point.

6. Axisymmetric models are used with forces or pressures with a radial component. These loads will oftenbe partially equilibrated by hoop stresses, which do not show up in the reaction forces.

7. Shell elements have an elastic foundation described. The load carried by the elastic foundation is notseen in the reaction forces.

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Section 15.7: Solving for Unknowns and Reactions

8. In substructure expansion pass with the resolve method used, the reaction forces at the master degreeof freedom are different from that given by the backsubstitution method (see Section 17.6: SubstructuringAnalysis).

15.8. Equation Solvers

The system of simultaneous linear equations generated by the finite element procedure is solved either usinga direct elimination process or an iterative method. A direct elimination process is primarily a Gaussian eliminationapproach which involves solving for the unknown vector of variables u in Equation 15–49:

(15–49)[ ] K u F=

where:

[K] = global stiffness/conductivity matrixu = global vector of nodal unknownF = global applied load vector

The direct elimination process involves decomposition (factorization) of the matrix [K] into lower and upper tri-angular matrices, [K] = [L][U]. Then forward and back substitutions using [L] and [U] are made to compute thesolution vector u.

A typical iterative method involves an initial guess, u1, of the solution vector u and then a successive steps of

iteration leading to a sequence of vectors u2, u3, . . . such that, in the limit, un = u as n tends to infinity. The

calculation of un + 1 involves [K], F, and the u vectors from one or two of the previous iterations. Typically the

solution converges to within a specified tolerance after a finite number of iterations.

There are two direct solvers available, the Sparse Direct Solver and the Frontal Solver, and a wider choice of iter-ative solvers. In the following sections all of the solvers are described under two major subsections: Direct Solversand Iterative Solvers.

15.8.1. Direct Solvers

The two direct solvers that are available are the Sparse Direct Solver, and the Frontal (Wavefront) Solver. TheSparse Direct Solver makes use of the fact that the finite element matrices are normally sparsely populated. Thissparseness allows the system of simultaneous equations to be solved efficiently by minimizing the operationcounts. The Frontal Solver, on the other hand, is designed to minimize the memory used in the solution processalthough the operation count is generally more than that of the Sparse Direct Solver.

15.8.2. Sparse Direct Solver

As described in the introductory section, the linear matrix equation, (Equation 15–49) is solved by triangulardecomposition of matrix [K] to yield the following equation:

(15–50)[ ][ ] L U u F=

where:

[L] = lower triangular matrix[U] = upper triangular matrix

By substituting:

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(15–51) [ ] w U u=

we can obtain u by first solving the triangular matrix system for w by using the forward pass operation givenby:

(15–52)[ ] L w F=

and then computing u using the back substitution operation on a triangular matrix given by:

(15–53)[ ] U u w=

When [K] is symmetric, the above procedure could use the substitution:

(15–54)[ ] [ ][ ]K L L T=

However, it is modified as:

(15–55)[ ] [ ][ ][ ]K L D L T= ′ ′

where:

[D] = a diagonal matrix

The diagonal terms of [D] may be negative in the case of some nonlinear finite element analysis. This allows thegeneration of [L'] without the consideration of a square root of negative number. Therefore, Equation 15–50through Equation 15–53 become:

(15–56)[ ][ ][ ] L D L u FT′ ′ =

(15–57) [ ][ ] w D L uT= ′

(15–58)[ ] L w F′ =

and

(15–59)[ ][ ] D L u FT′ =

Since [K] is normally sparsely populated with coefficients dominantly located around the main diagonal, theSparse Direct Solver (accessed with EQSLV,SPARSE) is designed to handle only the nonzero entries in [K]. Ingeneral, during the Cholesky decomposition of [K] shown in Equation 15–50 or Equation 15–56, nonzero coeffi-cients appear in [L] or [L'] at coefficient locations where [K] matrix had zero entries. The Sparse Direct Solver al-gorithm minimizes this fill-in by judiciously reordering the equation numbers in [K].

The performance of a direct solution method is greatly optimized through the equations reordering procedurewhich involves relabeling of the variables in the vector u. This simply amounts to permuting the rows andcolumns of [K] and the rows of F with the objective of minimizing fill-in. So, when the decomposition step inEquation 15–50 or Equation 15–56 is performed on the reordered [K] matrix, the fill-in that occurs in [L] or [L']matrix is kept to a minimum. This enormously contributes to optimizing the performance of the Sparse DirectSolver.

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Section 15.8: Equation Solvers

To achieve minimum fill-in, different matrix coefficient reordering algorithms are available in the literature(George and Liu(302)). The Sparse Direct Solver uses two different reordering schemes. They are the MinimumDegree ordering and the METIS ordering. The choice of which reordering method to use is automated in thesolver algorithm in order to yield the least fill-in.

15.8.3. Frontal Solver

The frontal (or wavefront) solution procedure is discussed by Irons(17) and Melosh and Bamford(25). The numberof equations which are active after any element has been processed during the solution procedure is called thewavefront at that point. The method used places a wavefront restriction on the problem definition, which dependsupon the amount of memory available for a given problem. Many thousand DOFs (degrees of freedom) on thewavefront can be handled in memory on some currently available computers. Wavefront limits tend to be re-strictive only for the analysis of arbitrary 3-D solids. In the wavefront procedure, the sequence in which the elementsare processed in the solver (the element “order”) is crucial to minimize the size of the wavefront.

The computer time required for the solution procedure is proportional to the square of the mean wavefront size.Therefore, it is advantageous to be able to estimate and minimize the wavefront size. The wavefront size is de-termined by the sequence in which the elements are arranged. The node numbers of all elements are scannedto determine which element is the last to use each node. As the total system of equations is assembled from theelement matrices, the equations for a node which occurs for the last time are algebraically solved in terms of theremaining unknowns and eliminated from the assembled matrix by Gauss elimination. The active equations arerepresented by:

(15–60)K u Fkj j

j

L

k=∑ =

1

where:

Kkj = stiffness term relating the force at DOF k to the displacement at DOF j

uj = nodal displacement of DOF j

Fk = nodal force of DOF k

k = equation (row) numberj = column numberL = number of equations

To eliminate a typical equation i = k, the equation is first normalized to

(15–61)

K

Ku

FK

ij

iij

j

Li

ii=∑ =

1

This is rewritten as:

(15–62)K u Fij j

j

L

i∗

=

∗∑ =1

where:

KK

Kijij

ii

∗ =

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FF

Kii

ii

∗ =

Here Kii is known as the “pivot”. If the absolute value of Kii is sufficiently small, it is numerically indistinguishable

from zero. This usually means the structure is insufficiently constrained (or needs more master DOFs for reducedanalyses).

Pivots are categorized as shown in the figure below.

Figure 15.4 Ranges of Pivot Values

!"# $ %&

' ($ %&

)+* *

, ).-

/ ).-

In Figure 15.4: “ Ranges of Pivot Values”,

(15–63)K

K

o

max

=

×

−

−

10

7 888 10

13

31

( )

.

or whichever is greatter

where:

Kmax = max (Kii) encountered up to this point in the wavefront

The number of small positive and negative pivots (N) are reported with the message:

There are (N) small equation solver pivot terms. This may occur during a Newton-Raphson iterationprocedure, is so noted, and usually can be ignored. Otherwise, it usually represents an uncon-strained structure or a reduced analysis with insufficient master DOFs and generates an errormessage.

Large negative pivots will cause the error message:

Large negative pivot value (value) at node (node, DOF).

Variations of this message exist for problems with buckling, stress stiffening or spin softening. Large or smallnegative pivots for piezoelectric, acoustic, coupled fluid-thermal, circuits, interface elements, as well as for arclength usages are not counted or flagged as they are commonly expected.

Equation 15–62 is written to a file for later backsubstitution. The remaining equations are modified as:

(15–64)K K K Kkj kj ki ij∗ ∗= −

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Section 15.8: Equation Solvers

(15–65)F F K Fk k ki i= − ∗

where:

k ≠ i

so that

(15–66)K u Fkj j

j

L

k∗

=

− ∗∑ =1

1

where k varies from 1 to L-1. Having eliminated row i from Equation 15–66, the process is repeated for all otherrows eligible to be eliminated.

The equations for a node which occurs for the first time are added to the assembled matrix as the solution pro-gresses. Thus, the assembled matrix expands and contracts as node make their first and last appearance in theelement definitions. The varying size of the active matrix is the instantaneous wavefront size.

When several elements are connected to the same node point, the DOFs associated with these elements remainactive in memory until the wavefront “passes” all elements connected to the node. DOFs related by constraintequations or coupled nodes remain active until the wavefront “passes” all elements connected to the relatedDOFs. Master DOFs remain active in memory and are not deleted from the wavefront. This procedure is shownby the flow chart shown in the following figure.

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Figure 15.5 Wavefront Flow Chart

"!#$#%&'(*)+," *) '& - .,$/+"0 - +

%( (

12+3'.45768!

9;:<

9;:=9 >?

9 9(@ 5A68!B C ,

5+

C ,

5D+

E

F)4"+ ((B

9 G G!# 9 $HJI ,K G!B

+,G G!# 9 $H'', " +L,ML/'+&, B

D+,G G!# 9 $H',' 3&+.', + 3L,*)H),N4+ GF)L/),+L',' 4 N*)"0 - "MLB

D+,G G!# 9 $H',' 3&+F (,O.L +N*)LH)N4+ "*)H)+'', 4 PF) Q - GMB

12+ J G!# 9 $ +R*)0 - +(/S(MGT0.N L +

UVXWYKZ[W\

]]]

U=^`_La^(^bdcfegh(hai^(^(j(gLkY7l(kQm`^aLghU=VXW"n o2pq`rf^haigLotsqhdsd^akXq`saFn ut^c0vq`r^Nm`^(^LoXqcwcf^LkQm(x ^(j\;oj(^y2hgaUVXWXpzv^ai^|NG\ YHZ~W

C ,

C ,

C

C ,

5+

5D+

5+

5+

To reduce the maximum wavefront size, the elements must be ordered so that the element for which each nodeis first mentioned is as close as possible in sequence to the element for which it is mentioned last. In geometricterms, the elements should be ordered so that the wavefront sweeps through the model continuously from oneend to the other in the direction which has the largest number of nodes. For example, consider a rectangularmodel having 6 nodes in one direction and 20 nodes in the other direction, as shown below.

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Section 15.8: Equation Solvers

Figure 15.6 Sample Mesh

!" #$&%')("*+#

,-$"#". !" #/$&%'0(1*+#

The elements should be ordered along one 6 node edge and progress toward the other 6 node edge. In thisway, equations will be deleted from the assembled matrix as soon as possible after they are added, thus minim-izing the wavefront size. All elements, including those of different types, should be included in the “one sweep”definition. See Section 15.4: Element Reordering for element reordering to reduce wavefront size.

See Section 13.3: Reuse of Matrices for when matrices can be reused.

The sparse direct solver is the default solver for all analyses, except for electromagnetic analyses with CIRCU124elements present, analyses that include both p-elements and constraint equations, spectrum analyses, andsubstructuring analyses (which each use the frontal direct solver by default). For nonlinear problems, the sparsedirect solver provides robust solution with good CPU performance, usually faster than the frontal solver.

15.8.4. Iterative Solver

The ANSYS program offers a large number of iterative solvers as alternatives to the direct solvers (sparse orfrontal solvers). These alternatives in many cases can result in less I/O or disk usage, less total elapsed time, andmore scalable parallel performance. However, in general, iterative solvers are not as robust as the direct solvers.For numerical challenges such as a nearly-singular matrix (matrix with small pivots) or a matrix that includesLagrangian multipliers, the direct solver is an effective solution tool, while an iterative solver is less effective ormay even fail.

The first three iterative solvers are based on the conjugate gradient (CG) method. The first of these three CGsolvers is the Jacobi Conjugate Gradient (JCG) solver (Mahinthakumar and Hoole(144)) (chosen with the EQSLV,JCGcommand) which is suitable for well-conditioned problems. Well-conditioned problems often arise from heattransfer, acoustics, magnetics and solid 2-D / 3-D structural analyses. The JCG solver is available for real andcomplex symmetric and unsymmetric matrices. The second solver is the Preconditioned Conjugate Gradient(PCG) solver (chosen with the EQSLV,PCG command) which is efficient and reliable for all types of analyses in-cluding the ill-conditioned beam/shell structural analysis. The PCG solver is made available through a licensefrom Computational Applications and System Integration, Inc. of Champaign, Illinois (USA). The PCG solver isonly valid for real symmetric stiffness matrices. The third solver is the Incomplete Cholesky Conjugate Gradient(ICCG) solver (internally developed, unpublished work) (chosen with the EQSLV,ICCG command). The ICCGsolver is more robust than the JCG solver for handling ill-conditioned matrices. The ICCG solver is available forreal and complex, symmetric and unsymmetric matrices.

The typical system of equations to be solved iteratively is given as :

(15–67)[ ] K u F=

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where:

[K] = global coefficient matrixu = unknown vectorF = global load vector

In the CG method, the solution is found as a series of vectors pi:

(15–68) u p p pm m= + + … +α α α1 1 2 2

where m is no larger than the matrix size n. The scheme is guaranteed to converge in n or fewer iterations on aninfinite precision machine. However, since the scheme is implemented on a machine with finite precision, itsometimes requires more than n iterations to converge. The solvers allow up to a maximum of 2n iterations. Ifit still does not converge after the 2n iterations, the solution will be abandoned with an error message. The un-converged situation is often due to an inadequate number of boundary constraints being used (rigid body motion).The rate of convergence of the CG algorithm is proportional to the square root of the conditioning number of[K] where the condition number of [K] is equal to the ratio of the maximum eigenvalue of [K] to the minimumeigenvalue of [K] . A preconditioning procedure is used to reduce the condition number of linear Equation 15–67.In the JCG algorithm, the diagonal terms of [K] are used as the preconditioner [Q], while in the ICCG and PCG al-gorithms, a more sophisticated preconditioner [Q] is used. The CG algorithm with preconditioning is showncollectively as Equation 15–69.

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Section 15.8: Equation Solvers

(15–69)

[ ]

)

u

R F

z Q F

0

0

01

2

0==

=

≤

−

Do i=1, n

If (Norm(R) then

se

εtt

quit loop

Else

If(i=1)then

u u

p R

z

i

T

=

==

=

−1

1

1 0

10

0β

α RR

p K p

R R K p

T0

1 1

1 0 1 1

[ ]

[ ] = − αElse

Applying preconditiooning: [ ]

z Q R

z R

z R

p

i i

ii

Ti

iT

i

i

−−

−

− −

− −

=

=

11

1

1 1

2 2

β

[ ]

z p

z R

p K p

R R

i i i

ii

Ti

iT

i

i i i

− −

− −

−

+

=

= −

1 1

1 1

1

β

α

α [[ ] K pi

Endif

Endif

End loop

Convergence is achieved when:

(15–70)

R R

F Fi

Ti

T≤ ε2

where:

ε = user supplied tolerance (TOLER on the EQSLV command; output as SPECIFIED TOLERANCE)Ri = F - [K] ui

ui = solution vector at iteration i

also, for the JCG and ICCG solvers:

(15–71) R RiT

i = output as CALCULATED NORM

(15–72) F FT ε2 = output as TARGET NORM

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It is assumed that the initial starting vector u0 is a zero vector.

Another two iterative solvers are provided by the ANSYS program to achieve a more scalable parallel performance.These are the algebraic multigrid (AMG) solver and the distributed domain solver (DDS).

The AMG solver (accessed with EQSLV,AMG), is made available through a license from Solvers International, Inc.of Colorado (USA), and is written for shared-memory architecture machines. AMG solver works on the incomingtotal equation matrix and automatically creates a few levels of coarser equation matrices. Iterative convergenceis accomplished by iterating between a coarse and a fine matrix. The maximum scalability that can be achievedusing 8 CPU processors is about a 5 times speedup in total elapsed time. For the ill-conditioned problems wherethe ill-conditioning is caused by high aspect ratio elements, a large amount of constraint equations, or shell/beamattached to solid elements, the AMG solver with one CPU processor is more efficient than any of the three CGsolvers. The AMG solver is also valid with constraint equations and coupling.

The DDS solver (accessed with EQSLV,DDS) is applicable for the distributed memory as well as the shared memorymachines. A distributed memory system can be typically constructed by linking different machines with Ethernet,Myrinet or similar cables. The DDS solver automatically decomposes the mesh into a number of small subdomains(like substructures) and then sends different subdomains to different processors. The subdomain (substructure)is solved by a direct solver such as the sparse direct solver, and then subsequently, the interface degrees offreedom (DOF) (like master DOF of a substructure) is solved by an iterative solver. During the DDS solution, thereis a continual exchange of information between machines. After the interface DOF are solved, the DDS solverautomatically calculates the complete solutions (like substructure expansion pass) in the parallel mode. Thescalability of the DDS solver is superior to other solvers and is only limited to the number of processors available.However, the current version of the DDS solver does not support applications with constraint equations orcoupling.

15.9. Mode Superposition Method

Mode superposition method is a method of using the natural frequencies and mode shapes from the modalanalysis (ANTYPE,MODAL) to characterize the dynamic response of a structure to transient (ANTYPE,TRANSwith TRNOPT,MSUP, Section 17.2: Transient Analysis), or steady harmonic (ANTYPE,HARM with HROPT,MSUP,Section 17.4: Harmonic Response Analyses) excitations.

The equations of motion may be expressed as in Equation 17–5:

(15–73)[ ] [ ] [ ] M u C u K u F&& &+ + =

F is the time-varying load vector, given by

(15–74) F F s Fnd s= +

where:

Fnd = time varying nodal forcess = load vector scale factor (input on LVSCALE command)

Fs = load vector from the modal analysis (see below)

The load vector Fs is computed when doing a modal analysis and its generation is the same as for a substructureload vector, described in Section 17.6: Substructuring Analysis.

The following development is similar to that given by Bathe(2):

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Section 15.9: Mode Superposition Method

Define a set of modal coordinates yi such that

(15–75) u yi i

i

n=

=∑ φ

1

where:

φi = the mode shape of mode i

n = the number of modes to be used (input as MAXMODE on TRNOPT or HROPT commands)

Note that Equation 15–75 hinders the use of nonzero displacement input, since defining yi in terms of u is not

straight forward. The inverse relationship does exist (Equation 15–75) for the case where all the displacementsare known, but not when only some are known. Substituting Equation 15–75 into Equation 15–73,

(15–76)[ ] [ ] [ ] M y C y K y Fi i

i

n

i ii

n

i ii

nφ φ φ&& &

= = =∑ ∑ ∑+ + =

1 1 1

Premultiply by a typical mode shape φiT :

(15–77)

[ ] [ ]

[ ]

φ φ φ φ

φ φ

jT

i ii

n

jT

i ii

n

jT

i ii

M y C y

K y

&& &= =∑ ∑+

+

1 1

==∑ =

1

n

jT F φ

The orthogonal condition of the natural modes states that

(15–78) [ ] φ φj

TiM i j= ≠0

(15–79) [ ] φ φj

TiK i j= ≠0

In the mode superposition method using Lanczos and subspace extraction methods, only Rayleigh or constantdamping is allowed so that:

(15–80) [ ] φ φj

TiC i j= ≠0

Applying these conditions to Equation 15–77, only the i = j terms remain:

(15–81) [ ] [ ] [ ] [ ]φ φ φ φ φ φ φj

Tj j j

Tj j j

Tj j j

TM y C y K y F&& &+ + =

The coefficients of &&y j ,

&y j , and yj, are derived as follows:

1.Coefficient of

&&y j :

By the normality condition (Equation 17–42),

(15–82) [ ] φ φj

TjM = 1

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2.Coefficient of

&y j :

The damping term is based on treating the modal coordinate as a single DOF system (shown in Equa-tion 15–73) for which:

(15–83) [ ] φ φ φj

Tj j jC C= 2

and

(15–84) [ ] φ φ φj

Tj j jM M= =2 1

Figure 15.7 Single Degree of Freedom Oscillator

Equation 15–84 can give a definition of φj:

(15–85)φ j

jM= 1

From (Tse(68)),

(15–86)C K Mj j j j= 2ξ

where:

ξj = fraction of critical damping for mode j

and,

(15–87)ωj j jK M= ( )

where:

ωj = natural circular frequency of mode j

Combining Equation 15–85 thru Equation 15–82 with Equation 15–83,

(15–88) [ ] φ φ ξ

ξ ω

jT

j j j jj

j j

C K MM

=

=

21

2

2

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Section 15.9: Mode Superposition Method

3. Coefficient of yj:

From Equation 17–39,

(15–89)[ ] [ ] K Mj j jφ ω φ= 2

Premultiply by φjT,

(15–90) [ ] [ ] φ φ ω φ φj

Tj j j

TjK M= 2

Substituting Equation 15–82 for the mass term,

(15–91) [ ] φ φ ωj

Tj jK = 2

For convenient notation, let

(15–92)f Fj j

T= φ

represent the right-hand side of Equation 15–81. Substituting Equation 15–82, Equation 15–88, Equa-tion 15–91 and Equation 15–92 into Equation 15–81, the equation of motion of the modal coordinatesis obtained:

(15–93)&& &y y y fj j j j j j j+ + =2 2ω ξ ω

Since j represents any mode, Equation 15–93 represents n uncoupled equations in the n unknowns yj.

The advantage of the uncoupled system (ANTYPE,TRAN with TRNOPT,MSUP) is that all the computa-tionally expensive matrix algebra has been done in the eigensolver, and long transients may be analyzedinexpensively in modal coordinates with Equation 15–75. In harmonic analysis (ANTYPE,HARM withHROPT,MSUP), frequencies may be scanned faster than by the reduced harmonic response (AN-TYPE,HARM with HROPT,REDUC) method.

The yj are converted back into geometric displacements u (the system response to the loading) by using

Equation 15–75. That is, the individual modal responses yj are superimposed to obtain the actual response,

and hence the name “mode superposition”.

If the modal analysis was performed using the reduced method (MODOPT,REDUC), then the matrices

and load vectors in the above equations would be in terms of the master DOFs (i.e., ^u ).

For the QR damped mode extraction method, we can write the differential equations of motion inmodal coordinate as follows:

(15–94)[ ] [ ] [ ][ ] [ ] [ ] I y C y y FT T&& &+ + =Φ Φ Λ Φ2

15.9.1. Modal Damping

The modal damping, ξj, is the combination of several ANSYS damping inputs:

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(15–95)ξ α ω βω ξ ξj jj j

m= + + +( ) ( )2 2

where:

α = uniform mass damping multiplier (input on ALPHAD command)β = uniform stiffness damping multiplier (input on BETAD command)ξ = constant damping ratio (input on DMPRAT command)

ξ jm

= modal damping ratio (input on MDAMP command)

Because of the assumption in Equation 15–80, explicit damping in such elements as COMBIN14 is not allowed

by the mode superposition procedure. In addition constant stiffness matrix multiplier βm (input as DAMP on MP

command) and constant material damping coefficients βξ (input as DMPR on MP command) are not applicablein modal damping since the resulting modal damping matrices are not uncoupled in the modal subspace (seeEquation 15–80 and Equation 15–176).

15.10. Reduced Order Modeling of Coupled Domains

A direct finite element solution of coupled-physics problems is computationally very expensive. The goal of thereduced-order modeling is to generate a fast and accurate description of the coupled-physics systems to char-acterize their static or dynamic responses. The method presented here is based on a modal representation ofcoupled domains and can be viewed as an extension of the Section 15.9: Mode Superposition Method to nonlinearstructural and coupled-physics systems (Gabbay, et al.(230), Mehner, et al.(250), Mehner, et al.(335), and Mehner,et al.(336)).

In the mode superposition method, the deformation state u of the structural domain is described by a factoredsum of mode shapes:

(15–96)u x y z t u q t x y zeq i i

i

m( , , , ) ( ) ( , , )= +

=∑ φ

1

where:

qi = modal amplitude of mode i

φi = mode shape

ueq = deformation in equilibrium state in the initial prestress position

m = number of considered modes

By substituting Equation 15–96 into the governing equations of motion, we obtain m constitutive equationsthat describe nonlinear structural systems in modal coordinates qi:

(15–97)m q m q

Wq

f S fi i i i i iSENE

iiN

kl i

S

l&& &+ + ∂

∂= +∑ ∑2 ξ ω

where:

mi = modal mass

ξi = modal damping factor

ωi = angular frequency

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Section 15.10: Reduced Order Modeling of Coupled Domains

WSENE = strain energy

f iN

= modal node force

f iE

= modal element forceSl = element load scale factor (input on RMLVSCALE command)

In a general case, Equation 15–97 are coupled since the strain energy WSENE depends on the generalized coordin-

ates qi. For linear structural systems, Equation 15–97 reduces to Equation 15–93.

Reduced Order Modeling (ROM) substantially reduces running time since the dynamic behavior of most structurescan be accurately represented by a few eigenmodes. The ROM method presented here is a three step procedurestarting with a Generation Pass, followed by a Use Pass Section 14.144: ROM144 - Reduced Order Electrostatic-Structural, which can either be performed within ANSYS or externally in system simulator environment, and finallyan optional Expansion Pass to extract the full DOF set solution according to Equation 15–96.

The entire algorithm can be outlined as follows:

• Determine the linear elastic modes from the modal analysis (ANTYPE,MODAL) of the structural problem.

• Select the most important modes based on their contribution to the test load displacement (RMMSELECTcommand).

• Displace the structure to various linear combinations of eigenmodes and compute energy functions forsingle physics domains at each deflection state (RMSMPLE command).

• Fit strain energy function to polynomial functions (RMRGENERATE command).

• Derive the ROM finite element equations from the polynomial representations of the energy functions.

15.10.1. Selection of Modal Basis Functions

Modes used for ROM can either be determined from the results of the test load application or based on theirmodal stiffness at the initial position.

Case 1: Test Load is Available (TMOD option on RMMSELECT command)

The test load drives the structure to a typical deformation state, which is representative for most load situationsin the Use Pass. The mode contribution factors ai are determined from

(15–98)

φ φ φ

φ φ φ

φ φ φ

φ φ φ φ

11

12

1

21

22

2

31

32

3

1 2 3

L

L

LM M O M

m

m

m

n n n nm

=

a

a

a

u

u

u

umn

1

2

1

2

3MM

where:

φi = mode shapes at the neutral plane nodes obtained from the results of the modal analysis (RMNEVECcommand)

ui = displacements at the neutral plane nodes obtained from the results of the test load (TLOAD option onRMNDISP command).

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Mode contribution factors ai are necessary to determine what modes are used and their amplitude range. Note

that only those modes are considered in Equation 15–98, which actually act in the operating direction (specifiedon the RMANL command). Criterion is that the maximum of the modal displacement in operating direction isat least 50% of the maximum displacement amplitude. The solution vector ai indicates how much each mode

contributes to the deflection state. A specified number of modes (Nmode of the RMMSELECT command) areconsidered unless the mode contribution factors are less than 0.1%.

Equation 15–98 solved by the least squares method and the results are scaled in such a way that the sum of allm mode contribution factors ai is equal to one. Modes with highest ai are suggested as basis functions.

Usually the first two modes are declared as dominant. The second mode is not dominant if either its eigenfrequencyis higher than five times the frequency of the first mode, or its mode contribution factor is smaller than 10%.

The operating range of each mode is proportional to their mode contribution factors taking into account thetotal deflection range (Dmax and Dmin input on the RMMSELECT command). Modal amplitudes smaller than2.5% of Dmax are increased automatically in order to prevent numerical round-off errors.

Case 2: Test Load is not Available (NMOD option on RMMSELECT command)

The first Nmode eigenmodes in the operating direction are chosen as basis functions. Likewise, a consideredmode must have a modal displacement maximum in operating direction of 50% with respect to the modalamplitude.

The minimum and maximum operating range of each mode is determined by:

(15–99)qD

iMax Min

ij

j

m=

−

=

−

∑/

ωω

22

1

1

where:

DMax/Min = total deflection range of the structure (input on RMMSELECT command)

15.10.2. Element Loads

Up to 5 element loads such as acting gravity, external acceleration or a pressure difference may be specified inthe Generation Pass and then scaled and superimposed in the Use Pass. In the same way as mode contribution

factors ai are determined for the test load, the mode contribution factors e i

j for each element load case are de-

termined by a least squares fit:

(15–100)

φ φ φ

φ φ φ

φ φ φ

φ φ φ φ

11

12

1

21

22

2

31

32

3

1 2 3

L

L

LM M O M

k

k

k

n n n nk

=

e

e

e

u

u

u

u

j

j

nj

j

j

j

nj

1

2

1

2

3MM

where:

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u ij = displacements at the neutral plane nodes obtained from the results of the element load j (ELOAD option

on RMNDISP command).

Here index k represents the number of modes, which have been selected for the ROM. The coefficients e i

j are

used to calculate modal element forces (see Section 14.144.1: Element Matrices and Load Vectors).

15.10.3. Mode Combinations for Finite Element Data Acquisition and EnergyComputation

In a general case, the energy functions depend on all basis functions. In the case of m modes and k data points

in each mode direction one would need km sample points.

A large number of examples have shown that lower eigenmodes affect all modes strongly whereby interactionsamong higher eigenmodes are negligible. An explanation for this statement is that lower modes are characterizedby large amplitudes, which substantially change the operating point of the system. On the other hand, theamplitudes of higher modes are reasonably small, and they do not influence the operating point.

Taking advantage of those properties is a core step in reducing the computational effort. After the mode selectionprocedure, the lowest modes are classified into dominant and relevant. For the dominant modes, the numberof data points in the mode direction defaults to 11 and 5 respectively for the first and second dominant modesrespectively. The default number of steps for relevant modes is 3. Larger (than the default above) number ofsteps can be specified on the RMMRANGE command.

A very important advantage of the ROM approach is that all finite element data can be extracted from a seriesof single domain runs. First, the structure is displaced to the linear combinations of eigenmodes by imposingdisplacement constrains to the neutral plane nodes. Then a static analysis is performed at each data point todetermine the strain energy.

Both the sample point generation and the energy computation are controlled by the command RMSMPLE.

15.10.4. Function Fit Methods for Strain Energy

The objective of function fit is to represent the acquired FE data in a closed form so that the ROM FE elementmatrices (Section 14.144: ROM144 - Reduced Order Electrostatic-Structural) are easily derived from the analyticalrepresentations of energy functions.

The ROM tool uses polynomials to fit the energy functions. Polynomials are very convenient since they can capturesmooth functions with high accuracy, can be described by a few parameters and allow a simple computation oftheir local derivatives. Moreover, strain energy functions are inherent polynomials. In the case of linear systems,the strain energy can be exactly described by a polynomial of order two since the stiffness is constant. Stress-stiffened problems are captured by polynomials of order four.

The energy function fit procedure (RMRGENERATE command) calculates nc coefficients that fit a polynomial to

the n values of strain energy:

(15–101)[ ] A K WPOLY SENE=

where:

[A] = n x nc matrix of polynomial terms

KPOLY = vector of desired coefficients

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Note that the number of FE data (WSENE) points n for a mode must be larger than the polynomial order P for the

corresponding mode (input on RMPORDER command). Equation 15–101 is solved by means of a least squaresmethod since the number of FE data points n is usually much larger than the number polynomial coefficientsnc.

The ROM tool uses four polynomial types (input on RMROPTIONS command):

LagrangePascalReduced LagrangeReduced Pascal

Lagrange and Pascal coefficient terms that form matrix [A] in Equation 15–101 are shown in Figure 15.8: “Set forLagrange and Pascal Polynomials”.

Figure 15.8 Set for Lagrange and Pascal Polynomials

Polynomials for Order 3 for Three Modes (1-x, 2-y, 3-z)

Reduced Lagrange and Reduced Pascal polynomial types allow a further reduction of KPOLY by considering only

coefficients located on the surface of the brick and pyramid respectively .

15.10.5. Coupled Electrostatic-Structural Systems

The ROM method is applicable to electrostatic-structural systems.

The constitutive equations for a coupled electrostatic-structural system in modal coordinates are:

(15–102)m q m q

Wq

f S fWqi i i i i i

SENE

iiN

kl i

E

l

el

i

&& &+ +∂

∂= ∑ + ∑ −

∂∂

2 ξ ω

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for the modal amplitudes and

(15–103)I Q

WVi iel

i= = ∂

∂&

where:

Ii = current in conductor i

Qi = charge on the ith conductor

Vi = ith conductor voltage

The electrostatic co-energy is given by:

(15–104)WC

V Velijr

i jr

= −∑2

2( )

where:

Cij = lumped capacitance between conductors i and j (input on RMCAP command)

r = index of considered capacitance

15.10.6. Computation of Capacitance Data and Function Fit

The capacitances Cij, and the electrostatic co-energy respectively, are functions of the modal coordinates qi. As

the strain energy WSENE for the structural domain, the lumped capacitances are calculated for each k data points

in each mode direction, and then fitted to polynomials. Following each structural analysis to determine the strainenergy WSENE, (n-1) linear simulations are performed in the deformed electrostatic domain, where n is the number

of conductors, to calculate the lumped capacitances. The capacitance data fit is similar to the strain energy fitdescribed above (Section 15.10.4: Function Fit Methods for Strain Energy). It is sometimes necessary to fit theinverted capacitance function (using the Invert option on the RMROPTIONS command).

15.11. Newton-Raphson Procedure

15.11.1. Overview

The finite element discretization process yields a set of simultaneous equations:

(15–105)[ ] K u Fa=

where:

[K] = coefficient matrixu = vector of unknown DOF (degree of freedom) values

Fa = vector of applied loads

If the coefficient matrix [K] is itself a function of the unknown DOF values (or their derivatives) then Equation 15–105is a nonlinear equation. The Newton-Raphson method is an iterative process of solving the nonlinear equationsand can be written as (Bathe(2)):

(15–106)[ ] K u F Fi

Ti

ainr∆ = −

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(15–107) u u ui i i+ = +1 ∆

where:

[ ]KiT

= Jacobian matrix (tangent matrix)i = subscript representing the current equilibrium iteration

Finr

= vector of restoring loads corresponding to the element internal loads

Both [ ]KiT

and Fi

nr are evaluated based on the values given by ui. The right-hand side of Equation 15–106

is the residual or out-of-balance load vector; i.e., the amount the system is out of equilibrium. A single solutioniteration is depicted graphically in Figure 15.9: “Newton-Raphson Solution - One Iteration” for a one DOF model.

In a structural analysis, [ ]KiT

is the tangent stiffness matrix, ui is the displacement vector and Fi

nr is the

restoring force vector calculated from the element stresses. In a thermal analysis, [ ]KiT

is the conductivity matrix,

ui is the temperature vector and Fi

nr is the resisting load vector calculated from the element heat flows. In

an electromagnetic analysis, [ ]KiT

is the Dirichlet matrix, ui is the magnetic potential vector, and Fi

nr is the

resisting load vector calculated from element magnetic fluxes. In a transient analysis, [ ]KiT

is the effective coef-

ficient matrix and Fi

nr is the effective applied load vector which includes the inertia and damping effects.

As seen in the following figures, more than one Newton-Raphson iteration is needed to obtain a convergedsolution. The general algorithm proceeds as follows:

1. Assume u0. u0 is usually the converged solution from the previous time step. On the first time step,

u0 = 0.

2.Compute the updated tangent matrix [ ]Ki

T and the restoring load

Finr

from configuration ui.

3. Calculate ∆ui from Equation 15–106.

4. Add ∆ui to ui in order to obtain the next approximation ui + 1 (Equation 15–107).

5. Repeat steps 2 to 4 until convergence is obtained.

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Figure 15.9 Newton-Raphson Solution - One Iteration

Figure 15.10: “Newton-Raphson Solution - Next Iteration” shows the solution of the next iteration (i + 1) of theexample from Figure 15.9: “Newton-Raphson Solution - One Iteration”. The subsequent iterations would proceedin a similar manner.

The solution obtained at the end of the iteration process would correspond to load level Fa. The final converged

solution would be in equilibrium, such that the restoring load vector Fi

nr (computed from the current stress

state, heat flows, etc.) would equal the applied load vector Fa (or at least to within some tolerance). None ofthe intermediate solutions would be in equilibrium.

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Figure 15.10 Newton-Raphson Solution - Next Iteration

If the analysis included path-dependent nonlinearities (such as plasticity), then the solution process requiresthat some intermediate steps be in equilibrium in order to correctly follow the load path. This is accomplished

effectively by specifying a step-by-step incremental analysis; i.e., the final load vector Fa is reached by applyingthe load in increments and performing the Newton-Raphson iterations at each step:

(15–108)[ ] , ,K u F Fn i

Ti n

an inr∆ = −

where:

[Kn,i] = tangent matrix for time step n, iteration i

Fna

= total applied force vector at time step n

,Fn inr

= restoring force vector for time step n, iteration i

This process is the incremental Newton-Raphson procedure and is shown in Figure 15.11: “Incremental Newton-Raphson Procedure”. The Newton-Raphson procedure guarantees convergence if and only if the solution at anyiteration ui is “near” the exact solution. Therefore, even without a path-dependent nonlinearity, the incremental

approach (i.e., applying the loads in increments) is sometimes required in order to obtain a solution correspondingto the final load level.

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Figure 15.11 Incremental Newton-Raphson Procedure

When the stiffness matrix is updated every iteration (as indicated in Equation 15–106 and Equation 15–108) theprocess is termed a full Newton-Raphson solution procedure ( NROPT,FULL or NROPT,UNSYM). Alternatively,the stiffness matrix could be updated less frequently using the modified Newton-Raphson procedure(NROPT,MODI). Specifically, for static or transient analyses, it would be updated only during the first or seconditeration of each substep, respectively. Use of the initial-stiffness procedure (NROPT,INIT) prevents any updatingof the stiffness matrix, as shown in Figure 15.12: “Initial-Stiffness Newton-Raphson”. If a multistatus element isin the model, however, it would be updated at iteration in which it changes status, irrespective of the Newton-Raphson option. The modified and initial-stiffness Newton-Raphson procedures converge more slowly than thefull Newton-Raphson procedure, but they require fewer matrix reformulations and inversions. A few elementsform an approximate tangent matrix so that the convergence characteristics are somewhat different.

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Figure 15.12 Initial-Stiffness Newton-Raphson

15.11.2. Convergence

The iteration process described in the previous section continues until convergence is achieved. The maximumnumber of allowed equilibrium iterations (input on NEQIT command) are performed in order to obtain conver-gence.

Convergence is assumed when

(15–109) R RR ref< ε (out-of-balance convergence)

and/or

(15–110) ∆u ui u ref< ε (DOF increment convergence)

where R is the residual vector:

(15–111) R F Fa nr= −

which is the right-hand side of the Newton-Raphson Equation 15–106. ∆ui is the DOF increment vector, εR and

εu are tolerances (TOLER on the CNVTOL command) and Rref and uref are reference values (VALUE on the CNVTOL

command). ||⋅ || is a vector norm; that is, a scalar measure of the magnitude of the vector (defined below).

Convergence, therefore, is obtained when size of the residual (disequilibrium) is less than a tolerance times areference value and/or when the size of the DOF increment is less than a tolerance times a reference value. Thedefault is to use out-of-balance convergence checking only. The default tolerance are .001 (for both εu and εR).

There are three available norms (NORM on the CNVTOL command) to choose from:

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Section 15.11: Newton-Raphson Procedure

1.Infinite norm

R max Ri∞ =

2.L1 norm

R Ri1= ∑

3.

L2 norm ( )R Ri2

212= ∑

For DOF increment convergence, substitute ∆u for R in the above equations. The infinite norm is simply themaximum value in the vector (maximum residual or maximum DOF increment), the L1 norm is the sum of theabsolute value of the terms, and the L2 norm is the square root of the sum of the squares (SRSS) value of theterms, also called the Euclidean norm. The default is to use the L2 norm.

The default out-of-balance reference value Rref is ||Fa||. For DOFs with imposed displacement constraints, Fnr

at those DOFs are used in the computation of Rref. For structural DOFs, if ||Fa|| falls below 1.0, then Rref uses 1.0

as its value. This occurs most often in rigid body motion (e.g., stress-free rotation) analyses. For thermal DOFs, if

||Fa|| falls below 1.0E-6, then Rref uses 1.0E-6 as its value. For all other DOFs, Rref uses 0.0. The default reference

value uref is ||u||.

15.11.3. Predictor

The solution used for the start of each time step n un,0 is usually equal to the current DOF solution un -1. The

tangent matrix [Kn,0] and restoring load Fn,0 are based on this configuration. The predictor option (PRED com-

mand) extrapolates the DOF solution using the previous history in order to take a better guess at the next solution.

In static analyses, the prediction is based on the displacement increments accumulated over the previous timestep, factored by the time-step size:

(15–112) ,u u un n n0 1= +− β ∆

where:

∆un = displacement increment accumulated over the previous time step

n = current time step

(15–113) ∆ ∆u un i

i

NEQIT=

=∑

1

and β is defined as:

(15–114)β =

−

∆∆

tt

n

n 1

where:

∆tn = current time-step size

∆tn-1 = previous time-step size

β is not allowed to be greater than 5.

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In transient analyses, the prediction is based on the current velocities and accelerations using the Newmarkformulas for structural DOFs:

(15–115) ( ) ,u u u t u tn n n n n n0 1 1 1

212

= + + −− − −& &&∆ ∆α

where:

, , u u un n n− − −1 1 1& && = current displacements, velocities and accelerations

∆tn = current time-step size

α = Newmark parameter (input on TINTP command)

For thermal, magnetic and other first order systems, the prediction is based on the trapezoidal formula:

(15–116) ( ) ,u u u tn n n n0 1 11= + −− −θ & ∆

where:

un - 1 = current temperatures (or magnetic potentials)

&un−1 = current rates of these quantitiesθ = trapezoidal time integration parameter (input on TINTP command)

See Section 17.2: Transient Analysis for more details on the transient procedures.

The subsequent equilibrium iterations provide DOF increments ∆u with respect to the predicted DOF valueun,0, hence this is a predictor-corrector algorithm.

15.11.4. Adaptive Descent

Adaptive descent (Adptky on the NROPT command) is a technique which switches to a “stiffer” matrix if conver-gence difficulties are encountered, and switches back to the full tangent as the solution convergences, resultingin the desired rapid convergence rate (Eggert(152)).

The matrix used in the Newton-Raphson equation (Equation 15–106) is defined as the sum of two matrices:

(15–117)[ ] [ ] ( )[ ]K K KiT S T= + −ξ ξ1

where:

[KS] = secant (or most stable) matrix

[KT] = tangent matrixξ = descent parameter

The program adaptively adjusts the descent parameter (ξ) during the equilibrium iterations as follows:

1. Start each substep using the tangent matrix (ξ = 0).

2. Monitor the change in the residual ||R||2 over the equilibrium iterations:

If it increases (indicating possible divergence):

• remove the current solution if ξ < 1, reset ξ to 1 and redo the iteration using the secant matrix

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• if already at ξ = 1, continue iterating

If it decreases (indicating converging solution):

• If ξ = 1 (secant matrix) and the residual has decreased for three iterations in a row (or 2 if ξ was in-creased to 1 during the equilibrium iteration process by (a.) above), then reduce ξ by a factor of 1/4(set it to 0.25) and continue iterating.

• If the ξ < 1, decrease it again by a factor of 1/4 and continue iterating. Once ξ is below 0.0156, set itto 0.0 (use the tangent matrix).

3. If a negative pivot message is encountered (indicating an ill-conditioned matrix):

• If ξ < 1, remove the current solution, reset ξ = 1 and redo the iteration using the secant matrix.

• If ξ = 1, bisect the time step if automatic time stepping is active, otherwise terminate the execution.

The nonlinearities which make use of adaptive descent (that is, they form a secant matrix if ξ > 0) include: plasticity,contact, stress stiffness with large strain, nonlinear magnetics using the scalar potential formulation, the concreteelement SOLID65 with KEYOPT(7) = 1, and the membrane shell element SHELL41 with KEYOPT(1) = 2. Adaptivedescent is used by default in these cases unless the line search or arc-length options are on. It is only availablewith full Newton-Raphson, where the matrix is updated every iteration. Full Newton-Raphson is also the defaultfor plasticity, contact and large strain nonlinearities.

15.11.5. Line Search

The line search option (accessed with LNSRCH command) attempts to improve a Newton-Raphson solution∆ui by scaling the solution vector by a scalar value termed the line search parameter.

Consider Equation 15–107 again:

(15–118) u u ui i i+ = +1 ∆

In some solution situations, the use of the full ∆ui leads to solution instabilities. Hence, if the line search option

is used, Equation 15–118 is modified to be:

(15–119) u u s ui i i+ = +1 ∆

where:

s = line search parameter, 0.05 < s < 1.0

s is automatically determined by minimizing the energy of the system, which reduces to finding the zero of thenonlinear equation:

(15–120)g u F F s us iT a nr

i= − ( ( ))∆ ∆

where:

gs = gradient of the potential energy with respect to s

An iterative solution scheme based on regula falsi is used to solve Equation 15–120 (Schweizerhof and Wrig-gers(153)). Iterations are continued until either:

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1.gs is less than 0.5 go, where go is the value of Equation 15–120 at s = 0.0 (that is, using Fn

nr−1 for Fnr

(s∆u)).

2. gs is not changing significantly between iterations.

3. Six iterations have been performed.

If go > 0.0, no iterations are performed and s is set to 1.0. s is not allowed below 0.05.

The scaled solution ∆ui is used to update the current DOF values ui+1 in Equation 15–107 and the next equi-

librium iteration is performed.

15.11.6. Arc-Length Method

The arc-length method (accessed with ARCLEN,ON) is suitable for nonlinear static equilibrium solutions of unstableproblems. Applications of the arc-length method involves the tracing of a complex path in the load-displacementresponse into the buckling/post buckling regimes. The arc-length method uses the explicit spherical iterationsto maintain the orthogonality between the arc-length radius and orthogonal directions as described by Fordeand Stiemer(174). It is assumed that all load magnitudes are controlled by a single scalar parameter (i.e., the totalload factor). Unsmooth or discontinuous load-displacement response in the cases often seen in contact analysesand elastic-perfectly plastic analyses cannot be traced effectively by the arc-length solution method. Mathemat-ically, the arc-length method can be viewed as the trace of a single equilibrium curve in a space spanned by thenodal displacement variables and the total load factor. Therefore, all options of the Newton-Raphson methodare still the basic method for the arc-length solution. As the displacement vectors and the scalar load factor aretreated as unknowns, the arc-length method itself is an automatic load step method (AUTOTS,ON is not needed).For problems with sharp turns in the load-displacement curve or path dependent materials, it is necessary tolimit the arc-length radius (arc-length load step size) using the initial arc-length radius (using the NSUBST com-mand). During the solution, the arc-length method will vary the arc-length radius at each arc-length substepaccording to the degree of nonlinearities that is involved.

The range of variation of the arc-length radius is limited by the maximum and minimum multipliers (MAXARCand MINARC on the ARCLEN command).

In the arc-length procedure, nonlinear Equation 15–106 is recast associated with the total load factor λ:

(15–121)[ ] K u F Fi

Ti

ainr∆ = λ

where λ is normally within the range -1.0 ≥ l ≥ 1.0. Writing the proportional loading factor λ in an incrementalform yields at substep n and iteration i (see Figure 15.13: “Arc-Length Approach with Full Newton-RaphsonMethod”):

(15–122)[ ] ( ) K u F F F Ri

Ti

an i

ainr

i∆ ∆− = + − = −λ λ λ

where:

∆λ = incremental load factor (as shown in Figure 15.13: “Arc-Length Approach with Full Newton-RaphsonMethod”)

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Section 15.11: Newton-Raphson Procedure

Figure 15.13 Arc-Length Approach with Full Newton-Raphson Method

"!#! $! % &

! % &

'

'

( '

'

(

)

*

*+ +

( )

)

( *+( '

The incremental displacement ∆ui can be written into two parts following Equation 15–122:

(15–123) ∆ ∆ ∆ ∆u u ui iI

iII= +λ

where:

∆uiI

= displacement due to a unit load factor

∆uiII

= displacement increment from the conventional Newton-Raphson method

These are defined by:

(15–124) [ ] ∆u K FiI

iT a= −1

(15–125) [ ] ∆u K RiII

iT

i= − −1

In each arc-length iteration, it is necessary to use Equation 15–124 and Equation 15–125 to solve for ∆uiI

and

∆uiII

. The incremental load factor ∆λ in Equation 15–123 is determined by the arc-length equation which canbe written as, for instance, at iteration i (see Figure 15.13: “Arc-Length Approach with Full Newton-RaphsonMethod”):

(15–126)li i nT

nu u2 2 2= +λ β ∆ ∆

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where:

β = scaling factor (with units of displacement) used to ensure the correct scale in the equations∆un = sum of all the displacement increments ∆ui of this iteration

The arc-length radius l i is forced, during the iterations, to be identical to the radius iteration l 1 at the first iter-

ation, i.e.

(15–127)l l … li i= = =−1 1

While the arc-length radius l 1 at iteration 1 of a substep is determined by using the initial arc-length radius

(defined by the NSUBST command), the limit range (defined by the ARCLEN command) and some logic of theautomatic time (load) step method (Section 15.6: Automatic Time Stepping).

Equation 15–123 together with Equation 15–126 uniquely determines the solution vector (∆ui, ∆λ)T. However,

there are many ways to solve for ∆λ approximately. The explicit spherical iteration method is used to ensureorthogonality (Forde and Stiemer(174)). In this method, the required residual ri (a scalar) for explicit iteration on

a sphere is first calculated. Then the arc-length load increment factor is determined by formula:

(15–128)∆ ∆ ∆

∆ ∆λ

β λ= −

+r u u

u ui n

TiII

i nT

iI

2

The method works well even in the situation where the vicinity of the critical point has sharp solution changes.Finally, the solution vectors are updated according to (see Figure 15.13: “Arc-Length Approach with Full Newton-Raphson Method”):

(15–129) u u u ui n n i+ = + +1 ∆ ∆

and

(15–130)λ λ λ λi n i+ = + +1 ∆

where:

n = current substep number

Values of λn and ∆λ are available in POST26 (SOLU command) corresponding to labels ALLF and ALDLF, respect-

ively. The normalized arc-length radius label ARCL (SOLU) corresponds to value l li i

0, where

li0

is the initialarc-length radius defined (by the NSUBST command) through Equation 15–126 (an arc-length radius at the firstiteration of the first substep).

In the case where the applied loads are greater or smaller than the maximum or minimum critical loads, arc-length will continue the iterations in cycles because |λ| does not approach unity. It is recommended to terminatethe arc-length iterations (using the ARCTRM or NCNV commands).

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15.12. Constraint Equations

15.12.1. Derivation of Matrix and Load Vector Operations

Given the set of L linear simultaneous equations in unknowns uj (same as Equation 15–60):

(15–131)K u F k Lkj j

j

L

k=∑ = ≤ ≤

11( )

subject to the linear constraint equation (input on CE command)

(15–132)C u Cj j

j

L

o=∑ =

1

normalize Equation 15–132 with respect to the prime DOF ui by dividing by Ci to get:

(15–133)C u Cj j

j

L

o∗

=

∗∑ =1

where:

C C Cj j i∗ =

C C Co o i∗ =

which is written to a file for backsubstitution instead of Equation 15–62. Equation 15–133 is expanded (recall

Ci∗

= 1) as:

(15–134)u C u C j ii j j

j

L

o+ = ≠∗

=

∗∑1

( )

Equation 15–131 may be similarly expanded as:

(15–135)K u K u F j iki i kj j

j

L

k+ = ≠=∑

1( )

Multiply Equation 15–134 by Kki and subtract from Equation 15–135 to get:

(15–136)( ) ( )K C K u F C K j ikj j ki j

j

L

k o ki− = − ≠∗

=

∗∑1

Specializing Equation 15–136 for k = i allows it to be written as:

(15–137)( ) ( )K C K u F C K j iij j ii j

j

L

i o ii− = − ≠∗

=

∗∑1

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This may be considered to be a revised form of the constraint equation. Introducing a Lagrange multiplier λk,

Equation 15–136 and Equation 15–137 may be combined as:

(15–138)

( )

( )

K C K u F C K

K C K u F C

kj j ki jj

L

k o ki

k ij j ii jj

L

i o

− − +

+ − − +

∗

=

∗

∗

=

∗

∑

∑

1

1λ KK j iii

= ≠0( )

By the standard Lagrange multiplier procedure (see Denn(8)):

(15–139)λk

i

k

uu

= ∂∂

Solving Equation 15–134 for ui,

(15–140)u C C u j ii o j j

j

L= − ≠∗ ∗

=∑

1( )

so that

(15–141)λk kC= − ∗

Substituting Equation 15–141 into Equation 15–138 and rearranging terms,

(15–142)

( )K C K C K C C K u

F C K C F C C

kj j ki k ij k j ii jj

L

k o ki k i k o

− − +

= − − +

∗ ∗ ∗ ∗

=∗ ∗ ∗

∑1

∗∗ ≠K j iii ( )

or

(15–143)K u F k Lkj j

j

L

k∗

=

− ∗∑ = ≤ ≤ −1

11 1( )

where:

K K C K C K C C Kkj kj j ki k ij k j ii∗ ∗ ∗ ∗ ∗= − − +

= (replaces Equation 15–64 in Gauss elimination)

F F C K C F C C Kk k o ki k i k o ii∗ ∗ ∗ ∗ ∗= − − +

= (replaces Equation 15–65 in Gauss elimination)

15.13. This section intentionally omitted

This section intentionally omitted

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15.14. Eigenvalue and Eigenvector Extraction

The eigenvalue and eigenvector problem needs to be solved for mode-frequency and buckling analyses. It hasthe form of:

(15–144)[ ] [ ] K Mi i iφ λ φ=

where:

[K] = structure stiffness matrixφi = eigenvector

λi = eigenvalue

[M] = structure mass matrix

For prestressed modal analyses, the [K] matrix includes the stress stiffness matrix [S]. For eigenvalue bucklinganalyses, the [M] matrix is replaced with the stress stiffness matrix [S]. The discussions given in the rest of thissection assume a modal analysis (ANTYPE,MODAL) except as noted, but also generally applies to eigenvaluebuckling analyses.

The eigenvalue and eigenvector extraction procedures available include the reduced, subspace, block Lanczos,unsymmetric, damped, and QR damped methods (MODOPT and BUCOPT commands) outlined inTable 15.1: “Procedures Used for Eigenvalue and Eigenvector Extraction”. The Power Dynamics method usessubspace iterations, but employs the PCG solver. Each method is discussed subsequently. Shifting, applicableto all methods, is discussed at the end of this section.

Table 15.1 Procedures Used for Eigenvalue and Eigenvector Extraction

QR damped ei-gensolver

Damped eigen-solver

Unsymmetriceigensolver

Block LanczosSubspaceReducedProcedure

MODOPT, QR-DAMP

MODOPT,DAMP

MODOPT, UN-SYM

MODOPT, LANBMODOPT,SUBSP

MODOPT,REDUC

Input

Symmetric orunsymmetricdamped systems

Symmetric orunsymmetricdamped systems

Unsymmetricmatrices

Symmetric (notavailable forbuckling)

SymmetricAny (but notrecom- men-ded for buck-ling)

Usages

K*, C*, MK*, C*, M*K*, M*K, MK, MK, MApplicableMatrices++

ModalNoneNoneNoneNoneGuyanReduction

QR algorithm forreduced modaldamping matrix

Lanczos which internally uses QRiterations

Lanczos whichinternally usesQL algorithm

Subspacewhich intern-ally uses Jac-obi

HBIExtractionTechnique

++ K = stiffness matrix, C = damping matrix, M = mass or stress stiffening matrix, * = can be unsymmetric

The PowerDynamics method is the same as the subspace method, except it uses the iterative solver instead ofthe frontal direct equation solver to solve Equation 15–156.

15.14.1. Reduced Method

For the reduced procedure (accessed with MODOPT,REDUC), the system of equations is first condensed downto those DOFs associated with the master DOFs by Guyan reduction. This condensation procedure is discussed

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in Section 17.6: Substructuring Analysis (Equation 17–88 and Equation 17–100). The set of n master DOFs char-acterize the natural frequencies of interest in the system. The selection of the master DOFs is discussed in moredetail in Section 15.5: Automatic Master DOF Selection of this manual and in Modal Analysis of the ANSYS Struc-tural Analysis Guide. This technique preserves the potential energy of the system but modifies, to some extent,the kinetic energy. The kinetic energy of the low frequency modes is less sensitive to the condensation than thekinetic energy of the high frequency modes. The number of master DOFs selected should usually be at leastequal to twice the number of frequencies of interest. This reduced form may be expressed as:

(15–145)[ ] [ ] ^ ^ ^ ^K Mi i iφ λ φ=

where:

[ ]^K = reduced stiffness matrix (known)

^φi = eigenvector (unknown)

λi = eigenvalue (unknown)

[ ]^

M = reduced mass matrix (known)

Next, the actual eigenvalue extraction is performed. The extraction technique employed is the HBI (Householder-Bisection-Inverse iteration) extraction technique and consists of the following five steps:

15.14.1.1. Transformation of the Generalized Eigenproblem to a Standard Eigen-problem

Equation 15–145 must be transformed to the desired form which is the standard eigenproblem (with [A] beingsymmetric):

(15–146)[ ] A ψ λ ψ=

This is accomplished by the following steps:

Premultiply both sides of Equation 15–145 by [ ]^

M −1 :

(15–147)[ ] [ ] ^ ^ ^ ^

M K− =1 φ λ φ

Decompose [ ]^

M into [L][L]T by Cholesky decomposition, where [L] is a lower triangular matrix. Combining withEquation 15–147,

(15–148)[ ] [ ] [ ] ^ ^ ^

L L KT− − =1 φ λ φ

It is convenient to define:

(15–149) [ ] ^φ ψ= −L T

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Section 15.14: Eigenvalue and Eigenvector Extraction

Combining Equation 15–148 and Equation 15–149), and reducing yields:

(15–150)[ ] [ ][ ] ^

L K L T− − =1

ψ λ ψ

or

(15–151)[ ] A ψ λ ψ=

where:

[ ] [ ] [ ][ ]^

A L K L T= − −1

Note that the symmetry of [A] has been preserved by this procedure.

15.14.1.2. Reduce [A] to Tridiagonal Form

This step is performed by Householder's method through a series of similarity transformations yielding

(15–152)[ ] [ ] [ ][ ]B T A TT=

where:

[B] = tridiagonalized form of [A][T] = matrix constructed to tridiagonalize [A], solved for iteratively (Bathe(2))

The eigenproblem is reduced to:

(15–153)[ ] B ψ λ ψ=

Note that the eigenvalues (λ) have not changed through these transformations, but the eigenvectors are relatedby:

(15–154) [ ] [ ] ^φ ψi

TiL L= −

15.14.1.3. Eigenvalue Calculation

Use Sturm sequence checks with the bisection method to determine the eigenvalues.

15.14.1.4. Eigenvector Calculation

The eigenvectors are evaluated using inverse iteration with shifting. The eigenvectors associated with multipleeigenvalues are evaluated using initial vector deflation by Gram-Schmidt orthogonalization in the inverse iterationprocedure.

15.14.1.5. Eigenvector Transformation

After the eigenvectors Ψi are evaluated,

^φi mode shapes are recovered through Equation 15–154.

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In the expansion pass, the eigenvectors are expanded from the master DOFs to the total DOFs.

15.14.2. Subspace Method

The subspace iteration method (accessed with MODOPT,SUBSP or BUCOPT,SUBSP) is described in detail byBathe(2). Enhancements as suggested by Wilson and Itoh(166) are also included as outlined subsequently. Thebasic algorithm consists of the following steps:

1. Define the initial shift s:

• In a modal analysis (ANTYPE,MODAL), s = FREQB on the MODOPT command (defaults to -4π2).

• In a buckling analysis (ANTYPE,BUCKLE), s = SHIFT on the BUCOPT command (defaults to 0.0).

2. Initialize the starting vectors [X0] (described below).

3. Triangularize the shifted matrix

[ ] [ ] [ ]K K s M∗ = + (15–155)

where:

[K] = assembled stiffness matrix[M] = assembled mass (or stress stiffness) matrix

A Sturm sequence check (described below) is performed if this is a shift point other than the initial shiftand it is requested (Strmck = ALL (default) or PART on the SUBOPT command).

4. For each subspace iteration n (1 to NM), do steps 5 to 14:

where:

NM = maximum number of subspace iterations (input as NUMSSI on the SUBOPT command)

5. Form [F] = [M][Xn-1] and scale [F] by λn-1

where:

λn-1 = previously estimated eigenvalues

6.Solve for [ ]Xn :

(15–156)[ ][ ] [ ]K X Fn∗ =

These equations are solved using the frontal direct equation solver (EQSLV,FRONT) or the iterative PCGsolver (EQSLV,PCG).

7.Scale the vectors [ ]Xn by (λn-1 - s) / λn-1

8. M-orthogonalize the vectors to the previously converged vectors (Gram-Schmidt orthogonalization).

9.Define the subspace matrices [ ]K and [ ]M :

(15–157)[ ] [ ] [ ][ ]K X K XnT

n=

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(15–158)[ ] [ ] [ ][ ]M X M XnT

n=

10.Adjust for the shift, [ ] [ ] [ ]K K s M∗ = +

11. Compute the eigenvalues and vectors of the subspace using a generalized Jacobi iteration:

[ ][ ] [ ][ ] K Q M Q n∗ = λ (15–159)

where:

[Q] = subspace eigenvectorsλn = updated eigenvalues

12. Update the approximation to the eigenvectors:

[ ] [ ][ ]X X Qn n=(15–160)

13. If any negative or redundant modes are found, remove them and create a new random vector.

14. Check for convergence (described below):

• All requested modes converged? If yes, go to step 15.

• If a new shift is required (described below), go to step 3

• Go to the next iteration, step 4

15. Perform a final Sturm sequence check if requested (Strmck = ALL (default) on the SUBOPT command).

Steps 5 thru 12 are only done on the unconverged vectors: once an eigenvalue has converged, the associatedeigenvector is no longer iterated on. The Gram-Schmidt procedure (step 8) ensures that the unconverged eigen-vectors remain orthogonal to the converged vectors not being iterated on. The remainder of this section detailssome of the steps involved.

15.14.2.1. Convergence

The convergence check (step 14a) requires that all of the requested eigenvalues satisfy the convergence ratio:

(15–161)e

Btoli

i n i n= − <−( ) ( )λ λ 1

where:

(λi)n = value of ith eigenvalue as computed in iteration n

(λi)n-1 = value of ith eigenvalue as computed in iteration n-1

Bi n

=

1 0.

( )λwhichever is greater

tol = tolerance value, set to 1.0E-5

15.14.2.2. Starting Vectors

The number of starting (iteration) vectors used is determined from:

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(15–162)q p d= +

where:

p = requested number of modes to extract (NMODE on the MODOPT or BUCOPT commands)d = number of extra iteration vectors to use (NPAD on the SUBOPT command, defaults to 4)

The q starting vectors [X0] (step 2) are initialized as follows. For each predefined rigid-body motion (Dof on the

RIGID command), define a rigid-body vector:

1. If a translational rigid-body motion, set the DOF slot in X0 to 1.0 (X0 is a column of [X0]).

2. If a rotational rigid-body motion, set the DOF slot in X0 corresponding to a unit rotation about the

global origin corresponding to the Dof label.

The rigid-body vectors are M-orthogonalized (Gram-Schmidt orthogonalization). The remainder of the vectorsare initialized to random vectors.

15.14.2.3. Sturm Sequence Check

The Sturm sequence check computes the number of negative pivots encountered during the triangularizationof the shifted matrix [K*]. This number will match the number of converged eigenvalues unless some eigenvalueshave been missed. In that case, more iteration vectors must be used (NPAD on the SUBOPT command) or theinitial shift (see step 1) was past the first mode. For the final Sturm sequence check, the shift used is defined as:

(15–163)s p p p= + −+λ λ λ0 1 1. ( )

where:

λp = eigenvalue of the last requested mode

λp+1 = eigenvalue of the next computed mode

15.14.2.4. Shifting Strategy

In order to improve the rate of convergence during the iteration process, a shifting strategy is adopted as follows(step 14b):

1. If the current converged mode(s) is zero(s) and the next mode i+1 is nonzero, shift to just below thenonzero mode:

(15–164)s i

i i

i= −+

+ +

+λ

λ λλ1

1 1

1

05

5

.

.

if is close to being converged

if

nnot

2. If the number of iterations since the last shift exceeds NS, then shift to just below the next unconverged

mode i+1:

(15–165)s i

i i i= −−

++ +λ

λ λ λλ1

1 105

5

. ( )

. (

if is close to being converged

ii i+ − 1 λ ) if not

where:

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NS = minimum number of subspace iterations completed before a shift is performed (input as NSHIFT

on the SUBOPT command)

If the mode is part of a cluster, then the next lowest unique mode is used to define the shift. If this is thefirst shift, then use:

(15–166)s = .9 1λ

The more shifts that are allowed (smaller value of NS), the faster the convergence, but the more matrix triangu-

larizations that must be performed.

15.14.2.5. Sliding Window

To improve the efficiency of the iterations, a subset qw of the q iteration vectors may be used in the iteration

process for the subspace working size, (qw is defined with SUBSIZ on the SUBOPT command (qw defaults to q)).

Steps 2 through 14 are performed using these working vectors. When a vector converges, it is removed fromthe iteration process and it is replaced by a new random vector until all p requested vectors have been found.

15.14.3. Block Lanczos

The block Lanczos eigenvalue extraction method (accessed with MODOPT,LANB or BUCOPT,LANB) is availablefor large symmetric eigenvalue problems. Typically, this solver is applicable to the type of problems solved usingthe subspace eigenvalue method, however, at a faster convergence rate.

A block shifted Lanczos algorithm, as found in Grimes et al.(195) is the theoretical basis of the eigensolver. Themethod used by the modal analysis employs an automated shift strategy, combined with Sturm sequence checks,to extract the number of eigenvalues requested. The Sturm sequence check also ensures that the requestednumber of eigenfrequencies beyond the user supplied shift frequency (FREQE on the MODOPT command) isfound without missing any modes.

The block Lanczos algorithm is a variation of the classical Lanczos algorithm, where the Lanczos recursions areperformed using a block of vectors, as opposed to a single vector. Additional theoretical details on the classicalLanczos method can be found in Rajakumar and Rogers(196).

Use of the block Lanczos method for solving large models (100,000 DOF, for example) with many constraintequations (CE) can require a significant amount of computer memory. This occurs when certain constraintequations lead to a huge wavefront size. For this reason, the Lagrange Multiplier approach is implemented totreat constraint equations in the block Lanczos eigensolver, rather than explicitly eliminating them prior towriting matrices to file.full. For details about the Lagrange Multiplier formulation theory refer to Cook(5).

15.14.4. Unsymmetric Method

The unsymmetric eigensolver (accessed with MODOPT,UNSYM) is applicable whenever the system matrices areunsymmetric. For example, an acoustic fluid-structure interaction problem using FLUID30 elements results inunsymmetric matrices. Also, certain problems involving the input matrix element MATRIX27, such as in rotordynamics can give rise to unsymmetric system matrices. A generalized eigenvalue problem given by the followingequation

(15–167)[ ] [ ] K Mi i iφ λ φ=

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can be setup and solved using the mode-frequency analysis (ANTYPE,MODAL). The matrices [K] and [M] are thesystem stiffness and mass matrices, respectively. Either or both [K] and [M] can be unsymmetric. φi is the eigen-

vector.

The method employed to solve the unsymmetric eigenvalue problem is the Lanczos algorithm. Starting fromtwo random vectors v1 and w1, the system matrices [K] and [M] (size n) are transformed into a tridiagonal

matrix [B] (subspace size q, where q ≤ n), through the Lanczos biorthogonal transformation as discussed in Ra-jakumar and Rogers(16). Eigenvalues of the [B] matrix, µi, are computed as approximations of the original system

eigenvalues λi. The QR algorithm (Wilkinson(18)) is used to extract the eigenvalues of the [B] matrix. As the sub-

space size q is increased, the µ will converge to closely approximate the eigenvalues of the original problem.

The transformed problem is a standard eigenvalue problem given by:

(15–168)[ ] B y yi i i= µ

The eigenvalues and eigenvectors of Equation 15–167 and Equation 15–168 are related by:

(15–169)λ

µii

= 1

(15–170) [ ] φi iV y=

where:

[V] = matrix of Lanczos vectors (size n x q).

For the unsymmetric modal analysis, the real part (ωi) of the complex frequency is used to compute the element

kinetic energy.

This method does not perform a Sturm Sequence check for possible missing modes. At the lower end of thespectrum close to the shift (input as FREQB on MODOPT command), the frequencies usually converge withoutmissing modes.

15.14.5. Damped Method

The damped eigensolver (accessed with MODOPT,DAMP) is applicable only when the system damping matrixneeds to be included in Equation 15–144, where the eigenproblem becomes a quadratic eigenvalue problemgiven by:

(15–171)[ ] [ ] [ ] K C Mi i i i iφ λ φ λ φ+ = − 2

where:

λi = − λi (defined below)

[C] = damping matrix

Matrices may be symmetric or unsymmetric. For problems involving rotordynamic stability, spinning structureswith gyroscopic effects, and/or damped structural eigenfrequencies, the above equation needs to be solved to

get the complex eigenvalues λi given by:

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Section 15.14: Eigenvalue and Eigenvector Extraction

(15–172)λ σ ωi i ij= ±

where:

λi = complex eigenvalueσi = real part of the eigenvalue

ωi = imaginary part of the eigenvalue

j = −1

The dynamic response of the system is given by:

(15–173) ( )u ei ij ti i= ±φ σ ω

where:

t = time

For the ith eigenvalue, the system is stable if σi is negative and unstable if σi is positive.

The method employed to solve the damped eigenvalue problem is the Lanczos algorithm (Rajakumar andAli(142)). Starting from four random vectors v1, w1, p1, and q1, the system matrices [K], [M], and [C] are

transformed into a subspace tridiagonal matrix [B] of size q ≤ n), through the Lanczos generalized biorthogonaltransformation. Eigenvalues of the [B] matrix, µi, are computed as an approximation of the original system eigen-

values λi . The QR algorithm (Wilkinson(18)) is used to extract the eigenvalues of the [B] matrix. As the subspacesize q is increased, the eigenvalues µi will converge to closely approximate the eigenvalues of the original

problem. The transformed problem is given by Equation 15–168 and from there on, the eigenvalues and eigen-vectors computation follow along the same lines as for the unsymmetric eigensolver.

This method does not perform a Sturm Sequence check for possible missing modes. At the lower end of thespectrum close to the shift (input as FREQB on the MODOPT command), the frequencies usually converge withoutmissing modes.

For the damped modal analysis, the imaginary part (ωi) of the complex frequency is used to compute the element

kinetic energy.

15.14.6. QR Damped Method

The QR damped method (accessed with MODOPT,QRDAMP) is a procedure for determining the complex eigen-values and corresponding eigenvectors of damped linear systems. This solver allows for nonsymmetric [K] and[C] matrices. The solver is computationally efficient compared to damp eigensolver (MODOPT,DAMP). Thismethod employs the modal orthogonal coordinate transformation of system matrices to reduce the eigenprobleminto the modal subspace. QR algorithm is then used to calculate eigenvalues of the resulting quadratic eigenvalueproblem in the modal subspace.

The equations of elastic structural systems without external excitation can be written in the following form:

(15–174)[ ] [ ] [ ] M u C u K u&& &+ + = 0

(See Equation 17–5 for definitions).

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It has been recognized that performing computations in the modal subspace is more efficient than in the fulleigen space. The stiffness matrix [K] can be symmetrized by rearranging the unsymmetric contributions; that is,the original stiffness matrix [K] can be divided into symmetric and unsymmetric parts. By dropping the dampingmatrix [C] and the unsymmetric contributions of [K], the symmetric block Lanczos eigenvalue problem is firstsolved to find real eigenvalues and the coresponding eigenvectors. In the present implementation, the unsym-metric element stiffness matrix is zeroed out for block Lanczos eigenvalue extraction. Following is the coordinatetransformation (see Equation 15–75) used to transform the full eigen problem into modal subspace:

(15–175) [ ] u y= Φ

where:

[Φ] = eigenvector matrix normalized with respect to the mass matrix [M]y = vector of modal coordinates

By using Equation 15–175 in Equation 15–174, we can write the differential equations of motion in the modalsubspace as follows:

(15–176)[ ] [ ] [ ][ ] ([ ] [ ] [ ][ ]) I y C y K yT T

unsym&& &+ + + =Φ Φ Λ Φ Φ2 0

where:

[Λ2] = a diagonal matrix containing the first n eigen frequencies ωi

For classically damped systems, the modal damping matrix [Φ]T[C][Φ] is a diagonal matrix with the diagonalterms being 2ξiωi, where ξi is the damping ratio of the i-th mode. For non-classically damped systems, the

modal damping matrix is either symmetric or unsymmetric. Unsymmetric stiffness contributions of the originalstiffness are projected onto the modal subspace to compute the reduced unsymmetric modal stiffness matrix

[Φ]T [Kunsym] [Φ].

Introducing the 2n-dimensional state variable vector approach, Equation 15–176 can be written in reduced formas follows:

(15–177)[ ] [ ] I z D z& =

where:

z

y

y=

&

[ ][ ] [ ]

[ ] [ ] [ ][ ] [ ] [ ][ ]D

O I

K CTunsym

T=− − −

Λ Φ Φ Φ Φ2

The 2n eigenvalues of Equation 15–177 are calculated using the QR algorithm (Press et al.(254)). The inverse iter-ation method (Wilkinson and Reinsch(357)) is used to calculate the complex modal subspace eigenvectors. Thefull complex eigenvectors, ψ, of original system is recovered using the following equation:

(15–178) [ ] ψ = Φ z

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15.14.7. Shifting

In some cases it is desirable to shift the values of eigenvalues either up or down. These fall in two categories:

1. Shifting down, so that the solution of problems with rigid body modes does not require working with asingular matrix.

2. Shifting up, so that the bottom range of eigenvalues will not be computed, because they had effectivelybeen converted to negative eigenvalues. This will, in general, result in better accuracy for the highermodes. The shift introduced is:

(15–179)λ λ λ= +o i

where:

λ = desired eigenvalueλo = eigenvalue shift

λi = eigenvalue that is extracted

λo, the eigenvalue shift is computed as:

(15–180)λo

bs

=

if buckling analysis

(input as on commandSHIFT BUCOPT ))

where s = constantif modal analysis

(input as

or

sm( )2 2πFREEQB on command)MODOPT

Equation 15–179 is combined with Equation 15–144 to give:

(15–181)[ ] ( )[ ] K Mi o i iφ λ λ φ= +

Rearranging,

(15–182)([ ] [ ]) [ ] K M Mo i i i− =λ φ λ φ

or

(15–183)[ ] [ ] K Mi i i′ =φ λ φ

where:

[K]' = [K] - λo [M]

It may be seen that if [K] is singular, as in the case of rigid body motion, [K]' will not be singular if [M] is positivedefinite (which it normally is) and if λo is input as a negative number. A default shift of λo = -1.0 is used for a

modal analysis.

Once λi is computed, λ is computed from Equation 15–179 and reported. Large shifts with the subspace iteration

method are not recommended as they introduce some degradation of the convergence and this may affect ac-curacy of the final results.

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15.14.8. Repeated Eigenvalues

Repeated roots or eigenvalues are possible to compute. This occurs, for example, for a thin, axisymmetric pole.Two independent sets of orthogonal motions are possible.

In these cases, the eigenvectors are not unique, as there are an infinite number of correct solutions. However,in the special case of two or more identical but disconnected structures run as one analysis, eigenvectors mayinclude components from more than one structure. To reduce confusion in such cases, it is recommended torun a separate analysis for each structure.

15.15. Analysis of Cyclic Symmetric Structures

15.15.1. Modal Analysis

Given a cyclic symmetric (periodic) structure such as a fan wheel, a modal analysis can be performed for the entirestructure by modelling only one sector of it. A proper basic sector represents a pattern that, if repeated n timesin cylindrical coordinate space, would yield the complete structure.

Figure 15.14 Typical Cyclic Symmetric Structure

In a flat circular membrane, mode shapes are identified by harmonic indices. For more information, see CyclicSymmetry Analysis of the ANSYS Advanced Analysis Techniques Guide.

Constraint relationships (equations) can be defined to relate the lower (θ = 0) and higher (θ = α, where α = sectorangle) angle edges of the basic sector to allow calculation of natural frequencies related to a given number ofharmonic indices. The basic sector is duplicated in the modal analysis to satisfy the required constraint relationshipsand to obtain nodal displacements. This technique was adapted from Dickens(148).

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Section 15.15: Analysis of Cyclic Symmetric Structures

Figure 15.15 Basic Sector Definition

High Component Nodes

ZY

X

CSYS = 1

Low Component Nodes

Sector angle α

Constraint equations relating the lower and higher angle edges of the two sectors are written:

(15–184)

u

u

k k

k k

u

uA

B

A

B

′

′

=

−

cos sin

sin cos

α αα α

where:

uA, uB = calculated displacements on lower angle side of basic and duplicated sectors (A and B, respectively)

u uA B′ ′, = displacements on higher angle side of basic and duplicated sectors (A and B, respectively) determined

from constraint relationships

k = harmonic index 0,1,2

N/2 if N is even

N-1

2if N is odd

=

...

α = 2π/N = sector angleN = number of sectors in 360°

Three basic steps in the procedure are briefly:

1. The CYCLIC command in /PREP7 automatically detects the cyclic symmetry model information, such asedge components, the number of sectors, the sector angles, and the corresponding cyclic coordinatesystem.

2. The CYCOPT command in /SOLU generates a duplicated sector and applies cyclic symmetry constraints(Equation 15–184) between the basic and the duplicated sectors.

3. The /CYCEXPAND command in /POST1 expands a cyclically symmetry response by combining the basicand the duplicated sectors results (Equation 15–185) to the entire structure.

15.15.2. Complete Mode Shape Derivation

The mode shape in each sector is obtained from the eigenvector solution. The displacement components (x, y,or z) at any node in sector j for harmonic index k, in the full structure is given by:

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(15–185)u u j k u j kA B= − − −cos( ) sin( )1 1α α

where:

j = sector number, varies from 1 to NuA = basic sector displacement

uB = duplicate sector displacement

The complete procedure addressing static, modal, and prestressed modal analyses of cyclic symmetric structuresis contained in Cyclic Symmetry Analysis of the ANSYS Advanced Analysis Techniques Guide.

15.15.3. Cyclic Symmetry Transformations

The cyclic symmetric solution sequences consist of three basic steps. The first step transforms applied loads tocyclic symmetric components using finite Fourier theory and enforces cyclic symmetry constraint equations (seeEquation 15–184) for each harmonic index (nodal diameter) (k = 0, 1, . . ., N/2).

Any applied load on the full 360° model is treated through a Fourier transformation process and applied on tothe cyclic sector. For each value of harmonic index, k, the procedure solves the corresponding linear equation.The responses in each of the harmonic indices are calculated as separate load steps at the solution stage. Theresponses are expanded via the Fourier expansion (Equation 15–185). They are then combined to get the completeresponse of the full structure in postprocessing.

The Fourier transformation from physical components, X, to the different harmonic index components, X , isgiven by the following:

Harmonic Index, k = 0 (symmetric mode):

(15–186)X

NXk j

j

N

==

= ∑01

1

Harmonic Index, 0 < k < N/2 (degenerate mode)

Basic sector:

(15–187)( ) cos( )X

NX j kk A j

j

N= −

=∑2

11

α

Duplicate sector:

(15–188)( ) sin( )X

NX j kk B j

j

N= −

=∑2

11

α

For N even only, Harmonic Index, k = N/2 (antisymmetric mode):

(15–189)X

NXk N

j

j

N

j=−

== −∑/

( )( )21

1

11

where:

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Section 15.15: Analysis of Cyclic Symmetric Structures

X = any physical component, such as displacements, forces, pressure loads, temperatures, and inertial loads

X = cyclic symmetric component

The transformation to physical components, X, from the cyclic symmetry, X , components is recovered by thefollowing equation:

(15–190)X X X j k X j k Xj k

k

K

kA kBj

k N= + − + − + −==

−=∑0

1

121 1 1[ cos( ) sin( ) ] ( ) /α α

The last term ( ) /− −=1 1

2j

k NX exists only for N even.

15.16. Mass Moments of Inertia

The computation of the mass moments and products of inertia, as well as the model center of mass, is describedin this section. The model center of mass is computed as:

(15–191)X

AMc

x=

(15–192)YA

Mcy=

(15–193)Z

AMc

z=

where typical terms are:

Xc = X coordinate of model center of mass (output as XC)

A m Xx i ii

N=

=∑

1

N = number of elements

m i = =mass of element i

function of real constants, if applicaable

or

Viρ

ρ = element density, based on average element temperatureVi = volume of element i

X N Xi oT

i= =X coordinate of the centroid of element i

No = vector of element shape functions, evaluated at the origin of the element coordinate system

Xi = global X coordinates of the nodes of element i

M mii

N= =

=∑

1mass of model (output as TOTAL MASS)

The moments and products of inertia with respect to the origin are:

(15–194)I m Y Zxx i i ii

N= +

=∑ (( ) ( ) )2 2

1

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(15–195)I m X Zyy i i i

i

N= +

=∑ (( ) ( ) )2 2

1

(15–196)I m X Yzz i i i

i

N= +

=∑ (( ) ( ) )2 2

1

(15–197)I m X Yxy i i i

i

N= −

=∑ (( )( ))

1

(15–198)I m Y Zyz i i i

i

N= −

=∑ (( )( ))

1

(15–199)I m X Zxz i i i

i

N= −

=∑ (( )( ))

1

where typical terms are:

Ixx = mass moment of inertia about the X axis through the model center of mass (output as IXX)

Ixy = mass product of inertia with respect to the X and Y axes through the model center of mass (output as

IXY)

Equation 15–194 and Equation 15–196 are adjusted for axisymmetric elements.

The moments and products of inertia with respect to the model center of mass (the components of the inertiatensor) are:

(15–200)I I M Y Zxx xx c c′ = − +(( ) ( ) )2 2

(15–201)I I M X Zyy yy c c′ = − +(( ) ( ) )2 2

(15–202)I I M X Yzz zz c c′ = − +(( ) ( ) )2 2

(15–203)I I MX Yxy xy c c′ = +

(15–204)I I MY Zyz yz c c′ = +

(15–205)I I MX Zxz xz c c′ = +

where typical terms are:

Ixx′

= mass moment of inertia about the X axis through the model center of mass (output as IXX)

Ixy′

= mass product of inertia with respect to the X and Y axes through the model center of mass (output asIXY)

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Section 15.16: Mass Moments of Inertia

15.16.1. Accuracy of the Calculations

The above mass calculations are not intended to be precise for all situations, but rather have been programmedfor speed. It may be seen from the above development that only the mass (mi) and the center of mass (Xi, Yi, and

Zi) of each element are included. Effects that are not considered are:

1. The mass being different in different directions.

2. The presence of rotational inertia terms.

3. The mixture of axisymmetric elements with non-axisymmetric elements (can cause negative momentsof inertia).

4. Tapered thicknesses.

5. Offsets used with beams and shells.

6. Trapezoidal-shaped elements.

7. The generalized plane strain option of Section 14.182: PLANE182 - 2-D 4-Node Structural Solid and Sec-tion 14.183: PLANE183 - 2-D 8-Node Structural Solid. (When these are present, the center of mass andmoment calculations are completely bypassed.)

Thus, if these effects are important, a separate analysis can be performed using inertia relief to find more precisecenter of mass and moments of inertia (using IRLF,-1). Inertia relief logic uses the element mass matrices directly;however, its center of mass calculations also do not include the effects of offsets.

It should be emphasized that the computations for displacements, stresses, reactions, etc. are correct with noneof the above approximations.

15.16.2. Effect of KSUM, LSUM, ASUM, and VSUM Commands

The center of mass and mass moment of inertia calculations for keypoints, lines, areas, and volumes (accessedby KSUM, LSUM, ASUM, VSUM, and *GET commands) use equations similar to Equation 15–191 throughEquation 15–205 with the following changes:

1. Only selected solid model entities are included.

2. Lines, areas, and volumes are approximated by numerically integrating to account for rotary inertias.

3. Keypoints are assumed to be unit masses without rotary inertia.

4. Lines are assumed to have unit mass per unit length.

5. Each area uses the thickness as:

t

first real constant in the table assigned to the

area (by=

the or command)

1.0 if there is no such assign

AATT AMESH

mment or real constant table

(15–206)

where:

t = thickness

6. Each area or volume is assumed to have density as:

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ρ =

input density (DENS for the material assigned to the areaa

or volume (by the or command)

1.0 i

AATT/VATT AMESH/VMESH

ff there is no such assignment or material property

(15–207)

where:

ρ = density

Composite material elements presume the element material number (defined with the MAT command).

15.17. Energies

Energies are available in the solution printout (by setting Item = VENG on the OUTPR command) or in postpro-cessing (by choosing items SENE, TENE, KENE, and AENE on the ETABLE command). For each element,

(15–208)

Evol E E

epo

T eli

i

NINT

epl

s=

+ +=∑1

2 1 σ ε

if element allows only

ddisplacement and rotational

degree of freedom (DOF),

either is nonlinear or uses

integration points, and is not

a p-eleement

all other cases

12

([ ] [ ]) u K S ueT

e e e+

=ppotential energy (includes strain energy)

(accessed with SSENE or TENE on command)

ETABLE

(15–209)

E u M ueki

eT

e e=

=

12

[ ] & &

kinematic energy (accessed with KENEE on

(computed only for transient and mo

command) ETABLE

ddal analyses)

(15–210)

E Qeart t

j

NCS

=

=

=∫

121

[ ] γ γ

artificial energy associated withh hourglass control

command

(accessed with AENE on

ETABLE )) (SOLID45, SOLID182, SOLID185, SHELL181 only)

where:

NINT = number of integration pointsσ = stress vector

εel = elastic strain vectorvoli = volume of integration point i

Eepl

= plastic strain energyEs = stress stiffening energy

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Section 15.17: Energies

=12

[ ] u S ueT

e e if [S ] is available and ,OFF usede

0.

NLGEOM

00 all other cases

[Ke] = element stiffness/conductivity matrix

[Se] = element stress stiffness matrix

u = element DOF vector

&u = time derivative of element DOF vector[Me] = element mass matrix

NCS = total number of converged substepsγ = hourglass strain energy defined in Flanagan and Belytschko(242) due to one point integrations.[Q] = hourglass control stiffness defined in Flanagan and Belytschko(242).

As may be seen from the bottom part of Equation 15–208 as well as Equation 15–209, all types of DOFs arecombined, e.g., SOLID5 using both UX, UY, UZ, TEMP, VOLT, and MAG DOF. An exception to this is the piezoelectricelements, described in Section 11.2: Piezoelectrics, which do report energies by separate types of DOFs in theNMISC record of element results. See Section 15.14: Eigenvalue and Eigenvector Extraction when complex fre-quencies are used. Also, if the bottom part of Equation 15–208 is used, any nonlinearities are ignored. Elementswith other incomplete aspects with respect to energy are reported in Table 15.2: “Exceptions for Element Energies”.

Artificial energy has no physical meaning. It is used to control the hourglass mode introduced by reduced integ-

ration. The rule-of-thumb to check if the element is stable or not due to the use of reduced integration is if

AENE

SENE

< 5% is true. When this inequality is true, the element using reduced integration is considered stable (i.e., functionsthe same way as fully integrated element).

A discussion of error energy is given in Table 15.2: “Exceptions for Element Energies”.

Table 15.2 Exceptions for Element Energies

ExceptionElement

Warping[1] thermal gradient not includedBEAM4

Thru-wall thermal gradient not includedPIPE16

Thru-wall thermal gradient not includedPIPE17

Thru-wall thermal gradient not includedPIPE18

No potential energyFLUID29

No potential energyFLUID30

No potential energyLINK31

No potential energyLINK34

No potential energyCOMBIN39

Foundation stiffness effects not includedSHELL41

Warping[1] thermal gradient not includedBEAM44

Thru-wall thermal gradient not includedPIPE59

Nonlinear and thermal effects not includedPIPE60

Thermal effects not includedSHELL61

Foundation stiffness effects not includedSHELL63

Foundation stiffness effects not includedSHELL99

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ExceptionElement

No potential energyFLUID141

No potential energyFLUID142

Thermal effects not includedPLANE145

Thermal effects not includedPLANE146

Thermal effects not includedSOLID147

Thermal effects not includedSOLID148

Thermal effects not includedSHELL150

1. Warping implies for example that temperatures T1 + T3 ≠ T2 + T4, i.e., some thermal strain is locked in.

For VISCO106, VISCO107, and VISCO108, a plastic energy per unit volume is also available. See Section 14.107:VISCO107 - 3-D 8-Node Viscoplastic Solid.

15.18. ANSYS Workbench Product Adaptive Solutions

Nearly every ANSYS Workbench product result can be calculated to a user-specified accuracy. The specified ac-curacy is achieved by means of adaptive and iterative analysis, whereby h-adaptive methodology is employed.The h-adaptive method begins with an initial finite element model that is refined over various iterations by re-placing coarse elements with finer elements in selected regions of the model. This is effectively a selectiveremeshing procedure. The criterion for which elements are selected for adaptive refinement depends on geometryand on what ANSYS Workbench product results quantities are requested. The result quantity φ, the expectedaccuracy E (expressed as a percentage), and the region R on the geometry that is being subjected to adaptiveanalysis may be selected. The user-specified accuracy is achieved when convergence is satisfied as follows:

(15–211)100 1 2 31φ φ

φi i

iE i n in R+ −

< = …, , , , , ( )

where i denotes the iteration number. It should be clear that results are compared from iteration i to iterationi+1. Iteration in this context includes a full analysis in which h-adaptive meshing and solving are performed.

The ANSYS Workbench product uses two different criteria for its adaptive procedures. The first criterion merelyidentifies the largest elements (LE), which are deleted and replaced with a finer finite element representation.The second employs a Zienkiewicz-Zhu (ZZ) norm for stress in structural analysis and heat flux in thermal analysis(which is the same as discussed in Section 19.7: POST1 - Error Approximation Technique). The relationshipbetween the desired accurate result and the criterion is listed in Table 15.3: “ANSYS Workbench Product AdaptivityMethods”.

Table 15.3 ANSYS Workbench Product Adaptivity Methods

Adaptive CriterionResult

ZZ normStresses and strains

ZZ normStructural margins and factors of safety

ZZ normFatigue damage and life

ZZ normHeat flows

ZZ normTemperatures

ZZ normDeformations

LEMode frequencies

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Section 15.18: ANSYS Workbench Product Adaptive Solutions

As mentioned above, geometry plays a role in the ANSYS Workbench product adaptive method. In general, ac-curate results and solutions can be devised for the entire assembly, a part or a collection of parts, or a surface ora collection of surfaces. The user makes the decision as to which region of the geometry applies. If accurate resultson a certain surface are desired, the ANSYS Workbench product ignores the aforementioned criterion and simplyrefines all elements on the surfaces that comprise the defined region. The reasoning here is that the user restrictsthe region where accurate results are desired. In addition, there is nothing limiting the user from having multipleaccuracy specification. In other words, specified accuracy in a selected region and results with specified accuracyover the entire model can be achieved.

15.19. Modal Projection Method

15.19.1. Extraction of Modal Damping Parameter for Squeeze Film Problems

A constant damping ratio is often applied for harmonic response analysis. In practice this approach only leadsto satisfying results if all frequency steps can be represented by the same damping ratio or the operating rangeencloses just one eigenmode. Difficulties arise if the damping ratio depends strongly on the excitation frequencyas happens in case of viscous damping in gaseous environment.

A typical damping ratio verse frequency function is shown below. For this example, the damping ratio is almostconstant below the cut-off frequency. Harmonic oscillations at frequencies below cutoff are strongly damped.Above cut-off the damping ratio decreases. Close to the structural eigenfrequency the damping ratio droppeddown to about 0.25 and a clear resonance peak can be observed.

Figure 15.16 Damping and Amplitude Ratio vs. Frequency

Damping and stiffness coefficients in modal coordinates are defined based on their nodal coordinate values as:

(15–212)C Cii i

Ti= φ φ*

and

(15–213)K Kii i

Ti= φ φ*

where:

Cii = damping coefficient in modal coordinates

φi = eigenvectors (modal coordinates)

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C* = finite element damping matrix in modal coordinatesKii = stiffness coefficient in modal coordinates

K* = finite element stiffness matrix in nodal coordinates

Unfortunately, both matrices C* and K* are not directly available for the fluid part of the coupled domain problem(e.g., squeeze film elements FLUID136). Moreover eigenvectors are derived from the structural part of the coupleddomain problem and consequently neither the modal damping matrix nor the modal stiffness matrix of the flu-idic system are necessarily orthogonal. Essential off-diagonal elements occur in case of asymmetric film arrange-ments or asymmetric plate motion as shown below.

Figure 15.17 Fluid Pressure From Modal Excitation Distribution

The goal is to express the viscous damping in modal coordinates as follows:

(15–214)Cq Kq f& + =

where:

f = modal force vectorq = vector of modal amplitudesC = unknown modal damping matrixK = unknown modal squeeze stiffness matrix

The following algorithm is necessary to compute all coefficients of the modal damping and stiffness matrix:

1. Start with the first mode and excite the fluid elements by wall velocities which correspond to a unitmodal velocity. In fact the nodal velocities become equal to the eigenvector of the appropriate mode.

2. Compute the real and imaginary part of the pressure distribution in a harmonic response analyses.

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Section 15.19: Modal Projection Method

3. Compute modal forces with regard to all other modes. The ith modal force states how much the pressure

distribution of the first mode really acts on the ith mode.

4. The computed modal forces can be used to extract all damping and squeeze stiffness coefficients of thefirst column in the C and K matrix.

5. Repeat step 1 with the next eigenvector and compute the next column of C and K.

The theoretical background is given by the following equations. Each coefficient Cji and Kji is defined by:

(15–215)C q K q F qji i ji i j

Ti& &+ = φ ( )

and

(15–216)F q N p q dAiT

i( ) ( )= ∫ &

where:

F(qi) = complex nodal damping force vector caused by a unit modal velocity of the source mode i.

Note that the modal forces are complex numbers with a real and imaginary part. The real part represents thedamping force and the imaginary part the squeeze force, which is cause by the fluid compression. The dampingand squeeze coefficients are given by:

(15–217)CN p q dA

qjijT T

i

i=

∫φ Re ( )&

&

and

(15–218)KN p q dA

qjijT T

i

i=

∫φ Im ( )&

Assuming the structure is excited by a unit modal velocity we obtain:

(15–219)C N p dAji j

T Ti= ∫φ φRe ( )

and

(15–220)K N p dAji j

T Ti= ∫Ω φ φIm ( )

Modal damping ratios ξ or the squeeze stiffness to structural stiffness ratio KRatio are defined only for the main

diagonal elements. These numbers are computed by:

(15–221)ξ

ωiii

i i

Cm

=2

and

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(15–222)K

KRatio

ii

i

=ω2

where:

mi = modal mass and the eigenfrequency

The damping ratio is necessary to compute ALPHAD and BETAD parameters for Rayleigh damping models or tospecify constant or modal damping by means of DMPRAT or MDAMP.

The squeeze to stiffness ratio specifies how much the structural stiffness is affected by the squeeze film. It cannot directly be applied to structural elements but is helpful for user defined reduced order models.

Modal damping parameter are automatically extracted by the DMPEXT command for a given frequency or afrequency range. The real and imaginary part of the pressure distribution of the source mode (eigenvector whichwas used to stimulate the system) is saved in the first load case after executing DMPEXT and can be used forfurther postprocessing. The shape function of the target mode is available as PRES degree of freedom at thesecond load case.

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Section 15.19: Modal Projection Method

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Chapter 16: This chapter intentionally omitted.This chapter is reserved for future use.

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16–2

Chapter 17: Analysis ProceduresThis chapter of the manual is designed to give users an understanding of the theoretical basis of the overallanalysis procedures. The derivation of the individual element matrices and load vectors is discussed in Section 2.2:Derivation of Structural Matrices, Section 5.2: Derivation of Electromagnetic Matrices, Section 6.2: Derivation ofHeat Flow Matrices, Section 7.2: Derivation of Fluid Flow Matrices, and Section 8.2: Derivation of Acoustics FluidMatrices.

In the matrix displacement method of analysis based upon finite element idealization, the structure being analyzedmust be approximated as an assembly of discrete regions (called elements) connected at a finite number ofpoints (called nodes). If the “force-displacement” relationship for each of these discrete structural elements isknown (the element “stiffness” matrix) then the “force-displacement relationship” for the entire “structure” canbe assembled using standard matrix methods. These methods are well documented (see, for example, Zien-kiewicz(39)) and are also discussed in Chapter 15, “Analysis Tools”. Thermal, fluid flow, and electromagneticanalyses are done on an analogous basis by replacing the above words in quotes with the appropriate terms.However, the terms displacement, force, and stiffness will be frequently used throughout this chapter, eventhough it is understood that the concepts apply to all valid effects also.

All analysis types for iterative or transient problems automatically reuse the element matrices or the overallstructural matrix whenever it is applicable. See Section 13.3: Reuse of Matrices for more details.

17.1. Static Analysis

17.1.1. Assumptions and Restrictions

The static analysis (ANTYPE,STATIC) solution method is valid for all degrees of freedom (DOFs). Inertial anddamping effects are ignored, except for static acceleration fields.

17.1.2. Description of Structural Systems

The overall equilibrium equations for linear structural static analysis are:

(17–1)[ ] K u F=

or

(17–2)[ ] K u F Fa r= +

where:

[ ] [ ]K Kem

N= =

=∑total stiffness matrix

1

u = nodal displacement vectorN = number of elements[Ke] = element stiffness matrix (described in Chapter 14, “Element Library”) (may include the element stress

stiffness matrix (described in Section 3.3: Stress Stiffening))

Fr = reaction load vector

Fa, the total applied load vector, is defined by:

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.

(17–3) ( )F F F F Fa nd ac

eth

m

N

epr= + + +

=∑

1

where:

Fnd = applied nodal load vector

Fac = - [M] ac = acceleration load vector

[ ] [ ]M Mem

N= =

=∑total mass matrix

1

[Me] = element mass matrix (described in Section 2.2: Derivation of Structural Matrices)

ac = total acceleration vector (defined in Section 15.1: Acceleration Effect)

Feth

= element thermal load vector (described in Section 2.2: Derivation of Structural Matrices)

Fepr

= element pressure load vector (described in Section 2.2: Derivation of Structural Matrices)

To illustrate the load vectors in Equation 17–2, consider a one element column model, loaded only by its ownweight, as shown in Figure 17.1: “Applied and Reaction Load Vectors”. Note that the lower applied gravity loadis applied directly to the imposed displacement, and therefore causes no strain; nevertheless, it contributes tothe reaction load vector just as much as the upper applied gravity load.

Figure 17.1 Applied and Reaction Load Vectors

Warning: If the stiffness for a certain DOF is zero, any applied loads on that DOF are ignored.

Section 15.7: Solving for Unknowns and Reactions discusses the solution of Equation 17–2 and the computationof the reaction loads. Section 15.11: Newton-Raphson Procedure describes the global equation for a nonlinearanalysis. Inertia relief is discussed in Section 15.2: Inertia Relief.

17.1.3. Description of Thermal, Magnetic and Other First Order Systems

The overall equations for linear 1st order systems are the same as for a linear structural static analysis, Equation 17–1and Equation 17–2. [K], though, is the total coefficient matrix (e.g., the conductivity matrix in a thermal analysis)

and u is the nodal DOF values. Fa, the total applied load vector, is defined by:

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(17–4) Q Q Qa nd

em

N= +

=∑

1

Table 17.1: “Nomenclature” relates the nomenclature used in Section 6.2: Derivation of Heat Flow Matrices andSection 5.2: Derivation of Electromagnetic Matrices for thermal, magnetic and electrical analyses to Equation 17–2and Equation 17–4. See Table 11.3: “Nomenclature of Coefficient Matrices” for a more detailed nomenclaturedescription.

Table 17.1 Nomenclature

FeFndu

Q Q Qe eg

ec+ + heat flux heat

generation convection

Qnd heat flowT temperatureThermal

Fe coercive forceFnd fluxφ scalar potentialScalar Magnetic

Fe current density and coercive

forceFnd current seg-ment

A vector potentialVector Magnetic

-Ind currentV voltageElectrical

Section 15.7: Solving for Unknowns and Reactions discusses the solution of Equation 17–2 and Section 15.11:Newton-Raphson Procedure describes the global equation for a nonlinear analysis.

17.2. Transient Analysis

The transient analysis solution method (ANTYPE,TRANS) used depends on the DOFs involved. Structural,acoustic, and other second order systems (that is, the systems are second order in time) are solved using onemethod and the thermal, magnetic, electrical and other first order systems are solved using another. Eachmethod is described subsequently. If the analysis contains both first and second order DOFs (e.g. structural andmagnetic), then each DOF is solved using the appropriate method. For matrix coupling between first and secondorder effects such as for piezoelectric analysis, a combined procedure is used.

17.2.1. Assumptions and Restrictions

1. Initial conditions are known.

2. No gyroscopic or Coriolis effects are included in a structural analysis (except for the gyroscopic dampingin BEAM4 and PIPE16).

17.2.2. Description of Structural and Other Second Order Systems

The transient dynamic equilibrium equation of interest is as follows for a linear structure:

(17–5)[ ] [ ] [ ] M u C u K u Fa&& &+ + =

where:

[M] = structural mass matrix[C] = structural damping matrix[K] = structural stiffness matrix

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Section 17.2: Transient Analysis

&&u = nodal acceleration vector

&u = nodal velocity vectoru = nodal displacement vector

Fa = applied load vector

There are two methods in the ANSYS program which can be employed for the solution of the linear Equation 17–5:the forward difference time integration method and the Newmark time integration method (including an improvedalgorithm called HHT). The forward difference method is used for explicit transient analyses and is described inthe LS-DYNA Theoretical Manual(199). The Newmark method and HHT method are used for implicit transientanalyses and are described below.

The Newmark method uses finite difference expansions in the time interval ∆t, in which it is assumed that(Bathe(2)):

(17–6) ( ) & & && &&u u u u tn n n n+ += + − +[ ]1 11 δ δ ∆

(17–7) u u u t u u tn n n n n+ += + + −

+

1 1

212

& && &&∆ ∆α α

where:

α, δ = Newmark integration parameters∆t = tn+1 - tn

un = nodal displacement vector at time tn

&un = nodal velocity vector at time tn

&&u n = nodal acceleration vector at time tn

un + 1 = nodal displacement vector at time tn + 1

&un + 1 = nodal velocity vector at time tn + 1

&&u n + 1 = nodal acceleration vector at time tn + 1

Since the primary aim is the computation of displacements un + 1, the governing Equation 17–5 is evaluated at

time tn + 1 as:

(17–8)[ ] [ ] [ ] M u C u K u Fn n na&& &+ + ++ + =1 1 1

The solution for the displacement at time tn + 1 is obtained by first rearranging Equation 17–6 and Equation 17–7,

such that:

(17–9) ( ) && & &&u a u u a u a un n n n n+ += − − −1 0 1 2 3

(17–10) & & && &&u u a u a un n n n+ += + +1 6 7 1

where:

at1 = δ

α∆a

t0 2

1=α∆

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a31

21= −

αa

t21=

α∆

at

5 22= −

∆ δα

a4 1= −δα

a t7 = δ∆a t6 1= −∆ ( )δ

Noting that &&u n + 1 in Equation 17–9 can be substituted into Equation 17–10, equations for &&u n + 1 and &un + 1

can be expressed only in terms of the unknown un + 1. The equations for &&u n + 1 and &un + 1 are then combined

with Equation 17–8 to form:

(17–11)( [ ] [ ] [ ])

[ ]( )

a M a C K u F

M a u a u a un

a

n n n

0 1 1

0 2 3

+ + = ++ + +

+& && [[ ]( )C a u a u a un n n1 4 5+ +& &&

Once a solution is obtained for un + 1, velocities and accelerations are updated as described in Equation 17–9

and Equation 17–10.

As described by Zienkiewicz(39), the solution of Equation 17–8 by means of Newmark Equation 17–6 and Equa-tion 17–7 is unconditionally stable for:

(17–12)α δ δ δ α≥ +

≥ + + >14

12

12

12

02

, ,

The Newmark parameters are related to the input as follows:

(17–13)α γ δ γ= + = +1

41

12

2( ) ,

where:

γ = amplitude decay factor (input on TINTP command).

Alternatively, the α and δ parameters may be input directly (using the TINTP command). By inspection of Equa-

tion 17–12 and Equation 17–13, unconditional stability is achieved when δ γ α γ= + ≥ +1

214

1 2, ( ) and γ ≥ 0.

Thus all solutions of Equation 17–12 are stable if γ ≥ 0. For a piezoelectric analysis, the Crank-Nicholson andconstant average acceleration methods must both be requested, that is, α = 0.25, δ = 0.5, and θ (THETA) = 0.5(using the TINTP command).

Typically the amplitude decay factor (γ) in Equation 17–13 takes a small value (the default is 0.005). The Newmark

method becomes the constant average acceleration method when γ = 0, which in turns means α = 1

4 and δ = 1

2(Bathe(2)). Results from the constant average acceleration method do not show any numerical damping in termsof displacement amplitude errors. If other sources of damping are not present, the lack of numerical dampingcan be undesirable in that the higher frequencies of the structure can produce unacceptable levels of numericalnoise (Zienkiewicz(39)). A certain level of numerical damping is usually desired and is achieved by degradingthe Newmark approximation by setting γ > 0.

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Section 17.2: Transient Analysis

In particular, it is desirable to have a controllable numerical damping in the higher frequency modes, since usingfinite elements to discretize the spatial domain, the results of these higher frequency modes are less accurate.However, the addition of high frequency numerical damping should not incur a loss of accuracy nor introduceexcessive numerical damping in the important low frequency modes. In the full transient analysis, the HHT timeintegration method (Chung and Hulbert(351)) has the desired property for the numerical damping.

The basic form of the HHT method is given by:

(17–14)[ ] [ ] [ ] M u C u K u Fn n n n

am f f f

&& &+ − + − + − + −+ + =1 1 1 1α α α α

where:

( ) && && &&u u un m n m nm+ − += − +1 11α α α

( ) & & &u u un f n f nf+ − += − +1 11α α α

( ) u u un f n f nf+ − += − +1 11α α α

( ) F F Fna

f na

f na

f+ − += − +1 11α α α

Comparing Equation 17–14 with Equation 17–5, one can see that the transient dynamic equilibrium equationconsidered in the HHT method is a linear combination of two successive time steps of n and n+1. αm and αf are

two extra integration parameters for the interpolation of the acceleration and the displacement, velocity andloads.

Introducing the Newmark assumption as given in Equation 17–6 and Equation 17–17 into Equation 17–14, thedisplacement un+1 at the time step n+1 can be obtained:

(17–15)( [ ] [ ] ( )[ ]) ( ) ia M a C K u F F Ff n f n

af n

af n0 1 1 11 1+ + − = − + −+ +α α α α nnt

[ ]( ) [ ](

+

+ + + + +M a u a u a u C a u a u an n n n n0 2 3 1 4 5& && & &&uun )

where:

atm

0 21= − αα∆

atf

11= −( )α δ

α∆

atm

21= − α

α∆

a m3

12

1= − −αα

a f4

11= − −( )α δ

α

a tf5 12

1= − −( )( )α δα

∆

The four parameters α, δ, αf, and αm used in the HHT method are related to the input as follows (Hilber et al(352)),

Chapter 17: Analysis Procedures

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.17–6

(17–16)

α γ

δ γ

α γα

= +

= +

==

14

1

12

0

2( )

f

m

γ = amplitude decay factor (input on TINTP command)

Alternatively, α, δ, αf, and αm can be input directly (using the TINTP command). But for the unconditional stability

and the second order accuracy of the time integration, they should satisfy the following relationships:

(17–17)

δ

α δ

δ α α

α α

≥

≥

= − +

≤ ≤

121212

12

m f

m f

If both αm and αf are zero when using this alternative, the HHT method is same as Newmark method.

Using this alternative, two other methods of parameter determination are possible. Given an amplitude decayfactor γ, the four integration parameters can be chosen as follows (Wood et al(353)):

(17–18)

α γ

δ γ

αα γ

= +

= +

== −

14

1

120

2( )

f

m

or they can be chosen as follows (Chung and Hulbert(351)):

(17–19)

α γ

δ γ

α γ

α γ

= +

= +

= −

= −

14

1

121

21 3

2

2( )

f

m

The parameters chosen according to Equation 17–16, or Equation 17–18, Equation 17–19 all satisfy the conditionsset in Equation 17–17. They are unconditionally stable and the second order accurate. Equation 17–16 andEquation 17–18 have a similar amount of numerical damping. Equation 17–19 has the least numerical damping

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Section 17.2: Transient Analysis

for the lower frequency modes. In this way,

11

−+

γγ is approximately the percentage of numerical damping for

the highest frequency of the structure.

17.2.2.1. Solution

Three methods of solution for the Newmark method (Equation 17–11) are available: full, reduced and mode su-perposition (TRNOPT command) and each are described subsequently. Only the full solution method is availablefor HHT (Equation 17–14).

Full Solution Method

The full solution method (TRNOPT,FULL) solves Equation 17–11 directly and makes no additional assumptions.In a nonlinear analysis, the Newton-Raphson method (Section 15.11: Newton-Raphson Procedure) is employedalong with the Newmark assumptions. The inversion of Equation 17–11 (or its nonlinear equivalent) employsthe same wavefront solver used for a static analysis in Section 15.8: Equation Solvers. Section 15.6: AutomaticTime Stepping discusses the procedure for the program to automatically determine the time step size requiredfor each time step.

Inherent to the Newmark method is that the values of uo, &uo, and &&u o at the start of the transient must be

known. Nonzero initial conditions are input either directly (with the IC commands) or by performing a staticanalysis load step (or load steps) prior to the start of the transient itself. Static load steps are performed in atransient analysis by turning off the transient time integration effects (with the TIMINT,OFF command). Thetransient itself can then be started (by TIMINT,ON). The default with transient analysis (ANTYPE,TRANS) is for

the transient to be running (TIMINT,ON); that is, to start the transient immediately. (This implies u = &u = &&u = 0. The initial conditions are outlined in the subsequent paragraphs. Cases referring to “no previous load step”mean that the first load step is transient.

Initial Displacement -

The initial displacements are:

(17–20)

uo =

0 if no previous load step available and no initialcoonditions ( commands) are used.

if no previous load

IC

′us step available but initialconditions ( commands) are uIC ssed.

if previous load step available which was runas a

us static analysis ( OFF)TIMINT,

where:

uo = vector of initial displacements

′us = displacement vector specified by the initial conditions (IC command)us = displacement vector resulting from a static analysis (TIMINT,OFF) of the previous load step

Initial Velocity -

Chapter 17: Analysis Procedures

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.17–8

The initial velocities are:

(17–21)

&uo =

0 if no previous load step available and no initialcconditions ( commands) are used.

if no previous lo

IC

& ′us aad step available but initialconditions ( commands) areIC used.

if previous load step available which us − −ust

1∆

wwas runas a static analysis ( OFF)TIMINT,

where:

&uo = vector of initial velocities

’&us = vector of velocities specified by the initial conditions (IC commands)us = displacements from a static analysis (TIMINT,OFF) of the previous load step

us-1 = displacement corresponding to the time point before us solution. us-1 is 0 if us is the first solution

of the analysis (i.e. load step 1 substep 1).∆t = time increment between s and s-1

Initial Acceleration -

The initial acceleration is simply:

(17–22) &&uo = 0

where:

&&u o = vector of initial accelerations

If a nonzero initial acceleration is required as for a free fall problem, an extra load step at the beginning of thetransient can be used. This load step would have a small time span, step boundary conditions, and a few timesteps which would allow the acceleration to be well represented at the end of the load step.

Nodal and Reaction Load Computation -

Inertia, damping and static loads on the nodes of each element are computed.

The inertial load part of the element output is computed by:

(17–23) F M uem

e e= &&

where:

Fem = vector of element inertial forces

[Me] = element mass matrix

&&u e = element acceleration vector

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Section 17.2: Transient Analysis

The acceleration of a typical DOF is given by Equation 17–9 for time tn+1. The acceleration vector &&u e is the average

acceleration between time tn + 1 and time tn, since the Newmark assumptions (Equation 17–6 and Equation 17–7)

assume the average acceleration represents the true acceleration.

The damping load part of the element output is computed by:

(17–24) F C uec

e e= &

where:

Fec = vector of element damping forces

[Ce] = element damping matrix

&ue = element velocity vector

The velocity of a typical DOF is given by Equation 17–10.

The static load is part of the element output computed in the same way as in a static analysis (Section 15.7:Solving for Unknowns and Reactions). The nodal reaction loads are computed as the negative of the sum of allthree types of loads (inertia, damping, and static) over all elements connected to a given fixed displacementnode.

Reduced Solution Method

The reduced solution method (TRNOPT,REDUC) uses reduced structure matrices to solve the time-dependentequation of motion (Equation 17–5) for linear structures. The solution method imposes the following additionalassumptions and restrictions:

1. Constant [M], [C], and [K] matrices. (A gap condition is permitted as described below.) This implies nolarge deflections or change of stress stiffening, as well as no plasticity, creep, or swelling.

2. Constant time step size.

3. No element load vectors. This implies no pressures or thermal strains. Only nodal forces applied directlyat master DOF or acceleration effects acting on the reduced mass matrix are permitted.

4. Nonzero displacements may be applied only at master DOF.

Description of Analysis -

This method usually runs faster than the full transient dynamic analysis by several orders of magnitude, principallybecause the matrix on the left-hand side of Equation 17–11 needs to be inverted only once and the transientanalysis is then reduced to a series of matrix multiplications. Also, the technique of “matrix reduction” discussedin Section 17.6: Substructuring Analysis is used in this method, so that the matrix representing the system willbe reduced to the essential DOFs required to characterize the response of the system. These essential DOFs arereferred to as the “master degrees of freedom”. Their automatic selection is discussed in Section 15.5: AutomaticMaster DOF Selection and guidelines for their manual selection are given in Modal Analysis of the ANSYS Struc-tural Analysis Guide. The reduction of Equation 17–11 for the reduced transient method results in:

(17–25)

( [ ] [ ] [ ])

[ ](

^ ^ ^ ^ ^

^ ^ ^

a M a C K u F

M a u a u a

n

n n

0 1 1

0 2 3

+ + = +

+ +

+

&&& &&u C a u a u a un n n n^ ^ ^ ^ ^) [ ]( )+ + +1 4 5

Chapter 17: Analysis Procedures

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.17–10

where the coefficients (ai) are defined after Equation 17–10. The ^ symbol is used to denote reduced matrices

and vectors. [ ]^K may contain prestressed effects (PSTRES,ON) corresponding to a non-varying stress state as

described in Section 3.3: Stress Stiffening. These equations, which have been reduced to the master DOFs, arethen solved by inverting the left-hand side of Equation 17–25 and performing a matrix multiplication at eachtime step.

For the initial conditions, a static solution is done at time = 0 using the given loads to define ^uo , &uo , and

&&uo are assumed to be zero.

A “quasi-linear” analysis variation is also available with the reduced method. This variation allows interfaces(gaps) between any of the master DOFs and ground, or between any pair of master DOFs. If the gap is initiallyclosed, these interfaces are accounted for by including the stiffness of the interface in the stiffness matrix, but ifthe gap should later open, a force is applied in the load vector to nullify the effect to the stiffness. If the gap isinitially open, it causes no effect on the initial solution, but if it should later close, a force is again applied in theload vector.

The force associated with the gap is:

(17–26)F k ugp gp g=

where:

kgp = gap stiffness (input as STIF, GP command)

ug = uA - uB - ugp

uA, uB = displacement across gap (must be master degrees of freedom)

ugp= initial size of gap (input as GAP, GP command)

This procedure adds an explicit term to the implicit integration procedure. An alternate procedure is to use thefull method, modeling the linear portions of the structure as superelements and the gaps as gap elements. Thislatter procedure (implicit integration) normally allows larger time steps because it modifies both the stiffnessmatrix and load vector when the gaps change status.

Expansion Pass -

The expansion pass of the reduced transient analysis involves computing the displacements at slave DOFs (seeEquation 17–97) and computing element stresses.

Nodal load output consists of the static loads only as described for a static analysis (Section 15.7: Solving forUnknowns and Reactions). The reaction load values represent the negative of the sum of the above static loadsover all elements connected to a given fixed displacement node. Damping and inertia forces are not includedin the reaction loads.

Mode Superposition Method

The mode superposition method (TRNOPT,MSUP) uses the natural frequencies and mode shapes of a linearstructure to predict the response to transient forcing functions. This solution method imposes the followingadditional assumptions and restrictions:

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Section 17.2: Transient Analysis

1. Constant [K] and [M] matrices. (A gap condition is permitted as described under the reduced solutionmethod.) This implies no large deflections or change of stress stiffening, as well as no plasticity, creep,or swelling.

2. Constant time step size.

3. There are no element damping matrices. However, various types of system damping are available.

4. Time varying imposed displacements are not allowed.

The development of the general mode superposition procedure is described in Section 15.9: Mode SuperpositionMethod. Equation 15–93 and Equation 15–94 are integrated through time for each mode by the Newmarkmethod.

The initial value of the modal coordinates at time = 0.0 are computed by solving Equation 15–93 with &&yo and

&yo assumed to be zero.

(17–27)y Fj j

To j= /φ ω2

where:

Fo = the forces applied at time = 0.0

The load vector, which must be converted to modal coordinates (Equation 15–92) at each time step, is given by

(17–28)F F s F F Fnd s

gp ma = + + +

where:

Fnd = nodal force vectors = load vector scale factor (input as FACT, LVSCALE command)

Fs = load vector from the modal analysis (see Section 15.9: Mode Superposition Method).Fgp = gap force vector (Equation 17–26) (not available for QR damped eigensolver).

Fma = inertial force (Fma = [M] a)

a = acceleration vector ( input with ACEL command) (see Section 15.1: Acceleration Effect)

In the modal superposition method, the damping force associated with gap is added to Equation 17–26:

(17–29) [ ] F K u C ugp gp g gp g= + &

where:

Cgp = gap damping (input as DAMP, GP command)

&ug = &u

A - &uB

&uA - &u

B = velocity across gap

If the modal analysis was performed using the reduced method (MODOPT,REDUC), then the matrices and vectors

in the above equations would be in terms of the master DOFs (e.g. u^ ).

Expansion Pass -

Chapter 17: Analysis Procedures

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The expansion pass of the mode superposition transient analysis involves computing the displacements at slaveDOFs if the reduced modal analysis (MODOPT,REDUC) was used (see Equation 17–97) and computing elementstresses.

Nodal load output consists of the static loads only as described for a static analysis (Section 15.7: Solving forUnknowns and Reactions). The reaction load values represent the negative of the sum of the static loads overall elements connected to a given fixed displacement node. Damping and inertia forces are not included in thereaction loads.

17.2.3. Description of Thermal, Magnetic and Other First Order Systems

The governing equation of interest is as follows:

(17–30)[ ] [ ] C u K u Fa& + =

where:

[C] = damping matrix[K] = coefficient matrixu = vector of DOF values

&u = time rate of the DOF values

Fa = applied load vector

In a thermal analysis, [C] is the specific heat matrix, [K] the conductivity matrix, u the vector of nodal temperatures

and Fa the applied heat flows. Table 17.2: “Nomenclature” relates the nomenclature used in Section 6.2: Deriv-ation of Heat Flow Matrices and Section 5.2: Derivation of Electromagnetic Matrices for thermal, magnetic andelectrical analyses to Equation 17–30.

Table 17.2 Nomenclature

Fau

Qa heat flowT temperatureThermal

Fa fluxφ scalar potentialScalar Magnetic

Fa current segmentA vector potentialVector Magnetic

Ia currentV voltageElectrical

The reduced and the mode superposition procedures do not apply to first order systems.

The procedure employed for the solution of Equation 17–30 is the generalized trapezoidal rule (Hughes(165)):

(17–31) ( ) u u t u t un n n n+ += + − +1 11 θ θ∆ ∆& &

where:

θ = transient integration parameter (input on TINTP command)∆t = tn + 1 - tn

un = nodal DOF values at time tn

&un = time rate of the nodal DOF values at time tn (computed at previous time step)

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Section 17.2: Transient Analysis

Equation 17–30 can be written at time tn + 1 as:

(17–32)[ ] [ ] C u K u Fn na& + ++ =1 1

Substituting &un + 1 from Equation 17–31 into this equation yields:

(17–33)1 1 1

1θ θθ

θ∆ ∆tC K u F C

tu un

an n[ ] [ ] [ ] +

= + + −

+ &

The solution of Equation 17–33 employs the same solvers used for static analysis in Section 17.1: Static Analysis.

Once un+1 is obtained, &un + 1 is updated using Equation 17–31. In a nonlinear analysis, the Newton-Raphson

method (Section 15.11: Newton-Raphson Procedure) is employed along with the generalized trapezoidal assump-tion, Equation 17–31.

The transient integration parameter θ (input on TINTP command) defaults to 0.5 (Crank-Nicholson method) ifsolution control is not used (SOLCONTROL,OFF) and 1.0 (backward Euler method) if solution control is used(SOLCONTROL,ON). If θ = 1, the method is referred to as the backward Euler method. For all θ > 0, the system

equations that follow are said to be implicit. In addition, for the more limiting case of θ ≥ 1/2, the solution ofthese equations is said to be unconditionally stable; i.e., stability is not a factor in time step (∆t) selection. Theavailable range of θ (using TINTP command) is therefore limited to

(17–34)12

1≤ ≤θ

which corresponds to an unconditionally stable, implicit method. For a piezoelectric analysis, the Crank-Nicholsonand constant average acceleration methods must both be requested with α (ALPHA) = 0.25, δ (DELTA) = 0.5, and

θ = 0.5 (on the TINTP command). Since the &un influences un + 1, sudden changes in loading need to be handled

carefully for values of θ < 1.0. See the ANSYS Basic Analysis Guide for more details.

The generalized-trapezoidal method requires that the values of uo and &uo at the start of the transient must

be known. Nonzero initial conditions are input either directly (with the IC command) (for uo) or by performing

a static analysis load step (or load steps) prior to the start of the transient itself. Static load steps are performedin a transient analysis by turning off the transient time integration effects (with the TIMINT,OFF command). Thetransient itself can then started (TIMINT,ON). The default for transient analysis (ANTYPE,TRANS) is to start the

transient immediately (TIMINT,ON). This implies (u = &u = 0). The initial conditions are outlined in the sub-sequent paragraphs.

Initial DOF Values -

The initial DOF values for first order systems are:

Chapter 17: Analysis Procedures

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.17–14

(17–35)

u

a

o =

if no previous load step available and noinitial coonditions ( commands) are used

if no previous load

IC

′us sstep available but theinitial conditions ( commands) arIC ee used

if previous load step available run as astatic

us aanalysis ( ,OFF)TIMINT

where:

uo = vector of initial DOF values

a = vector of uniform DOF values

′us = DOF vector directly specified (IC command)us = DOF vector resulting from a static analysis (TIMINT,OFF) of the previous load step available

a is set to TEMP (BFUNIF command) and/or to the temperature specified by the initial conditions (IC commands)for thermal DOFs (temperatures) and zero for other DOFs.

Nodal and Reaction Load Computation -

Damping and static loads on the nodes of each element are computed.

The damping load part of the element output is computed by:

(17–36) [ ] F C uec

e e= &

where:

Fec = vector of element damping loads

[Ce] = element damping matrix

&ue = element velocity vector

The velocity of a typical DOF is given by Equation 17–31. The velocity vector &ue is the average velocity between

time tn and time tn + 1, since the general trapezoidal rule (Equation 17–31) assumes the average velocity represents

the true velocity.

The static load is part of the element output computed in the same way as in a static analysis (Section 15.7:Solving for Unknowns and Reactions). The nodal reaction loads are computed as the negative of the sum of bothtypes of loads (damping and static) over all elements connected to a given fixed DOF node.

17.3. Mode-Frequency Analysis

17.3.1. Assumptions and Restrictions

1. Valid for structural and fluid degrees of freedom (DOFs).

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Section 17.3: Mode-Frequency Analysis

2. The structure has constant stiffness and mass effects.

3. There is no damping, unless the damped eigensolver (MODOPT,DAMP or MODOPT,QRDAMP) is selected.

4. The structure has no time varying forces, displacements, pressures, or temperatures applied (free vibration).

17.3.2. Description of Analysis

This analysis type (accessed with ANTYPE,MODAL) is used for natural frequency and mode shape determination.The equation of motion for an undamped system, expressed in matrix notation using the above assumptions is:

(17–37)[ ] [ ] M u K u&& + = 0

Note that [K], the structure stiffness matrix, may include prestress effects (PSTRES,ON). For a discussion of thedamped eigensolver (MODOPT,DAMP or MODOPT,QRDAMP) see Section 15.14: Eigenvalue and EigenvectorExtraction.

For a linear system, free vibrations will be harmonic of the form:

(17–38) cosu ti i= φ ω

where:

φi = eigenvector representing the mode shape of the ith natural frequency

ωi = ith natural circular frequency (radians per unit time)

t = time

Thus, Equation 17–37 becomes:

(17–39)( [ ] [ ]) − + =ω φi M K

i2 0

This equality is satisfied if either φi = 0 or if the determinant of ([K] - ω2 [M]) is zero. The first option is the trivial

one and, therefore, is not of interest. Thus, the second one gives the solution:

(17–40)[ ] [ ]K M− =ω2 0

This is an eigenvalue problem which may be solved for up to n values of ω2 and n eigenvectors φi which satisfy

Equation 17–39 where n is the number of DOFs. The eigenvalue and eigenvector extraction techniques are dis-cussed in Section 15.14: Eigenvalue and Eigenvector Extraction.

Rather than outputting the natural circular frequencies ω , the natural frequencies (f) are output; where:

(17–41)f i

i= ωπ2

where:

fi = ith natural frequency (cycles per unit time)

If normalization of each eigenvector φi to the mass matrix is selected (MODOPT,,,,,,OFF):

(17–42) [ ] φ φiT

iM = 1

Chapter 17: Analysis Procedures

ANSYS, Inc. Theory Reference . ANSYS Release 9.0 . 002114 . © SAS IP, Inc.17–16

If normalization of each eigenvector φi to 1.0 is selected (MODOPT,,,,,,ON), φi is normalized such that its largest

component is 1.0 (unity).

If the reduced mode extraction method was selected (MODOPT,REDUC), the n eigenvectors can then be expandedin the expansion pass (using the MXPAND command) to the full set of structure modal displacement DOFs using:

(17–43) [ ] [ ] ^φ φs i ss sm iK K= − −1

where:

φsi = slave DOFs vector of mode i (slave degrees of freedom are those DOFs that had been condensed out)

[Kss], [Ksm] = submatrix parts as shown in Equation 17–82

^φ i = master DOF vector of mode i

A discussion of effective mass is given in Section 17.7: Spectrum Analysis.

17.4. Harmonic Response Analyses

The harmonic response analysis (ANTYPE,HARMIC) solves the time-dependent equations of motion (Equa-tion 17–5) for linear structures undergoing steady-state vibration.

17.4.1. Assumptions and Restrictions

1. Valid for structural, fluid and magnetic degrees of freedom (DOFs).

2. The entire structure has constant or frequency-dependent stiffness, damping, and mass effects.

3. All loads and displacements vary sinusoidally at the same known frequency (although not necessarilyin phase).

4. Element loads are assumed to be real (in-phase) only, except for:

a. current density

b. pressures in SURF153 and SURF154

17.4.2. Description of Analysis

Consider the general equation of motion for a structural system (Equation 17–5).

(17–44)[ ] [ ] [ ] M u C u K u Fa&& &+ + =

where:

[M] = structural mass matrix[C] = structural damping matrix[K] = structural stiffness matrix

&&u = nodal acceleration vector

&u = nodal velocity vectoru = nodal displacement vector

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Section 17.4: Harmonic Response Analyses

Fa = applied load vector

As stated above, all points in the structure are moving at the same known frequency, however, not necessarilyin phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements maybe defined as:

(17–45) maxu u e ei i t= φ Ω

where:

umax = maximum displacement

i = square root of -1Ω= imposed circular frequency (radians/time) = 2πff = imposed frequency (cycles/time) (input as FREQB and FREQE on the HARFRQ command)t = timeφ = displacement phase shift (radians)

Note that umax and φ may be different at each DOF. The use of complex notation allows a compact and efficient

description and solution of the problem. Equation 17–45 can be rewritten as:

(17–46) (cos sin )maxu u i ei t= +φ φ Ω

or as:

(17–47) ( )u u i u ei t= +1 2Ω

where:

u1 = umax cos φ = real displacement vector (input as VALUE on D command, when specified)

u2 = umax sin φ = imaginary displacement vector (input as VALUE2 on D command, when specified)

The force vector can be specified analogously to the displacement:

(17–48) maxF F e ei i t= ψ Ω

(17–49) (cos sin )maxF F i ei t= +ψ ψ Ω

(17–50) ( )F F i F ei t= +1 2Ω

where:

Fmax = force amplitude

ψ = force phase shift (radians)F1 = Fmax cos ψ = real force vector (input as VALUE on F command, when specified)

F1 = Fmax sin ψ = imaginary force vector (input as on VALUE2 on F command, when specified)

Substituting Equation 17–47 and Equation 17–50 into Equation 17–44 gives:

(17–51)( [ ] [ ] [ ])( ) ( )− + + + = +Ω Ω Ω Ω21 2 1 2M i C K u i u e F i F ei t i t

Chapter 17: Analysis Procedures

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The dependence on time (eiΩt) is the same on both sides of the equation and may therefore be removed:

(17–52)([ ] [ ] [ ])( ) K M i C u i u F i F− + + = +Ω Ω21 2 1 2

The solution of this equation is discussed later.

17.4.3. Complex Displacement Output

The complex displacement output at each DOF may be given in one of two forms:

1. The same form as u1 and u2 as defined in Equation 17–47 (selected with the command HROUT,ON).

2. The form umax and φ (amplitude and phase angle (in degrees)), as defined in Equation 17–46 (selected

with the command HROUT,OFF). These two terms are computed at each DOF as:

(17–53)u u uimax = +222

(17–54)φ = −tan 1 2

1

uu

Note that the phase angle φ is relative to the input forcing phase angle ψ.

17.4.4. Nodal and Reaction Load Computation

Inertia, damping and static loads on the nodes of each element are computed.

The real and imaginary inertia load parts of the element output are computed by:

(17–55) [ ] F M ume e e1

21= Ω

(17–56) [ ] F M ume e e2

22= Ω

where:

Fme1 = vector of element inertia forces (real part)

[Me] = element mass matrix

u1e = element real displacement vector

Fme2 = vector of element inertia (imaginary part)

u2e = element imaginary displacement vector

The real and imaginary damping loads part of the element output are computed by:

(17–57) [ ] F C uc

e e e1 2= −Ω

(17–58) [ ] F C uc

e e e2 1= Ω

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where:

Fce1 = vector of element damping forces (real part)

[Ce] = element damping matrix

Fce2 = vector of element damping forces (imaginary part)

The real static load is computed the same way as in a static analysis (Section 15.7: Solving for Unknowns andReactions) using the real part of the displacement solution u1e. The imaginary static load is computed also the

same way, using the imaginary part u2e. Note that the imaginary part of the element loads (e.g., Fpr) are normally

zero, except for current density loads.

The nodal reaction loads are computed as the sum of all three types of loads (inertia, damping, and static) overall elements connected to a given fixed displacement node.

17.4.5. Solution

Four methods of solution to Equation 17–52 are available: full, reduced, mode superposition, and VariationalTechnology and each are described subsequently.

17.4.5.1. Full Solution Method

The full solution method (HROPT,FULL) solves Equation 17–52 directly. Equation 17–52 may be expressed as:

(17–59)[ ] K u Fc c c=

where c denotes a complex matrix or vector. Equation 17–59 is solved using the same wavefront solver used fora static analysis in Section 15.8: Equation Solvers, except that it is done using complex arithmetic.

17.4.5.2. Reduced Solution Method

The reduced solution method (HROPT,REDUC) uses reduced structure matrices to solve the equation of motion(Equation 17–44). This solution method imposes the following additional assumptions and restrictions:

1. No element load vectors (e.g., pressures or thermal strains). Only nodal forces applied directly at masterDOF or acceleration effects acting on the reduced mass matrix are permitted.

2. Nonzero displacements may be applied only at master DOF.

This method usually runs faster than the full harmonic analysis by several orders of magnitude, principally becausethe technique of “matrix reduction” discussed in Section 17.6: Substructuring Analysis is used so that the matrixrepresenting the system will be reduced to the essential DOFs required to characterize the response of the system.These essential DOFs are referred to as the “master degrees of freedom”. Their automatic selection is discussedin Section 15.5: Automatic Master DOF Selection and guidelines for their manual selection are given in ModalAnalysis of the ANSYS Structural Analysis Guide. The reduction of Equation 17–52 for the reduced method resultsin:

(17–60)([ ] [ ] [ ])( ) ^ ^ ^ ^ ^ ^ ^K M i C u i u F i F− + + = +Ω Ω2

1 2 1 2

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where the ^ denotes reduced matrices and vectors. These equations, which have been reduced to the master

DOFs, are then solved in the same way as the full method. [ ]^K may contain prestressed effects (PSTRES,ON)

corresponding to a non-varying stress state, described in Section 3.3: Stress Stiffening.

17.4.5.2.1. Expansion Pass

The reduced harmonic response method produces a solution of complex displacements at the master DOFsonly. In order to complete the analysis, an expansion pass is performed (EXPASS,ON). As in the full method, both

a real and imaginary solution corresponding to u^

1) and u^2) can be expanded (see Equation 17–97) and element

stresses obtained (HREXP,ALL).

Alternatively, a solution at a certain phase angle may be obtained (HREXP,ANGLE). The solution is computed atthis phase angle for each master DOF by:

(17–61)u u^ ^max cos( )= −φ θ

where:

u^max = amplitude given by Equation 17–53

φ = computed phase angle given by Equation 17–54

θ θ π= ′ 2360

θ' = input as ANGLE (in degrees), HREXP Command

This solution is then expanded and stresses obtained for these displacements. In this case, only the real part ofthe nodal loads is computed.

17.4.5.3. Mode Superposition Method

The mode superposition method (HROPT,MSUP) uses the natural frequencies and mode shapes to compute theresponse to a sinusoidally varying forcing function. This solution method imposes the following additional as-sumptions and restrictions:

1. Nonzero imposed harmonic displacements are not allowed.

2. There are no element damping matrices. However, various types of system damping are available.

The development of the general mode superposition procedure is given in Section 15.9: Mode SuperpositionMethod. The equation of motion (Equation 17–44) is converted to modal form, as described in Section 15.9:Mode Superposition Method. Equation 15–93 is:

(17–62)&& &y y y fj j j j j j j+ + =2 2ω ξ ω

where:

yj = modal coordinate

ωj = natural circular frequency of mode j

ξi = fraction of critical damping for mode j

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