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SOFiSTiK Verification Manual 1

Verification Manual Version 12.2 2009

prepared checked approved Date: 01.08.2003 Date: 19.01.2009 Date: 22.01.2009 S. Fahrendholz/R.Geiger Dr.-Ing. Jürgen Bellmann Dr.-Ing. Casimir Katz Document name: q:\dok\qs\verification.doc


Table of Content

1 Quality Certification ............................................................................................................... 3 1.1 Developement and Production ...................................................................................... 3 1.2 Reporting of Bugs and Software Enhancements ........................................................... 3

2 Validation .............................................................................................................................. 4 2.1 Test No IC1 - Tapered Membrane End Load ................................................................ 5 2.2 Test No IC2 - Tapered Membrane Gravity Load............................................................ 6 2.3 Test NO IC3 - Tapered Membrane Edge Shear ............................................................ 7 2.4 Test No IC4 - Tapered Membrane Gravity Load............................................................ 8 2.5 Test No IC5 - Circular Membrane Edge Pressure ......................................................... 9 2.6 Test No IC6 - Circular Membrane Point Load .............................................................. 10 2.7 Test No IC7 - Cicular Membrane Parabolic Temperature............................................ 11 2.8 Test No IC8 - Shear Diffusion ...................................................................................... 12 2.9 Test No IC9 - Elliptic Membrane .................................................................................. 13 2.10 Test No IC10 - Tapered Plate Edge Shear .............................................................. 14 2.11 Test No IC11 - Tapered Plate Gravity ...................................................................... 15 2.12 Test No IC12 - Elliptic Plate Normal Pressure ......................................................... 16 2.13 Test No IC13 - Skew Plate Normal Pressure ........................................................... 17 2.14 Test No IC14 - Tapered Thick Shell Pressure Load ................................................ 18 2.15 Test No IC15 - Tapered Thick Shell Self-weight ...................................................... 19 2.16 Test No IC16 - Cylinder-Taper-Sphere Temperature ............................................... 20 2.17 Test No IC17 - Hemisphere External Pressure ........................................................ 21 2.18 Test No IC18 - Hemisphere Point Load ................................................................... 22 2.19 Test No IC19 - Cylindrical Shell Edge Moment ........................................................ 23 2.20 Test No IC24 - Catenoidal Shell Internal Pressure .................................................. 24 2.21 Test No IC27 - Cylinder/Sphere Internal Pressure ................................................... 25 2.22 Test No IC28 - Circular Paraboloid Gravity .............................................................. 26 2.23 Test No IC29 - Z-Section Cantilever Torsion Bending ............................................. 27 2.24 Test No IC30 - Z-Section Cantilever Beam Bending ................................................ 28 2.25 Test No IC31 - Axisymmetric Hyperbolic Shell - Edge Loading ............................... 29 2.26 Test No IC32 - Axisymmetric Hyperbolic Shell - Pressure ....................................... 30 2.27 Test No IC34 - Axisymmetric Catenoidal Shell - Pressure ....................................... 31 2.28 Test No IC37 - Axisymmetric Stiffened Cylinder - Pressure ..................................... 32 2.29 Test No IC38 - Axisymmetric Cylinder/Sphere - Pressurre ...................................... 33 2.30 Test No IC39 - Axisymmetric Cylinder/Sphere - Pressure ....................................... 34 2.31 Test 5 Fundamental 2D Plasticity Benchmark ......................................................... 35 2.32 Test 5 "Dynamic for Deep Simply-Supported Beam" ............................................... 39 2.33 Test 5H "Harmonic Forced Vibration Response" ..................................................... 40 2.34 Test 5P "Periodic Forced Vibration Response ......................................................... 41 2.35 Test 5T "Transient Forced Vibration Response" ...................................................... 42

3 Literature ............................................................................................................................. 43


1 Quality Certification SOFiSTiK- Software is continuously developed since 1981 and used by over 10000 customers. To assure the highest quality for our customers we have installed a quality assurance system with the following steps.

1.1 Developement and Production

Each new software feature is thoroughly validated by a team of developers, supporters and external customers. A set of reference examples is thus created and documented (partly in German) During the life time of the software questions arising are treated by an intense discussion with customers, authorities and scientists to find the best interpretation.

For each minor release of the software (or at least once per month) an automatic comparison of the current results with the reference examples is performed to detect any deviations introduced by other bug fixes. These so called "current versions" are available for all customers with an automatic procedure via Internet. (SONAR) This assures that most bugs will be detected at an early stage.

Fast fixes of the software are published as separate beta-versions.

Once a year a QS-Release is published on CD/DVD. A period of approximately three months is foreseen to assure the actuality of the manuals, online help and to validate the overall consistency of the total software environment. The reference examples are tested on all major targets (i.e. Linux, Windows). These Versions are shipped to all customers. At most every two years we allow for a general new release with changes in the basic structure of the software (e.g. 16 to 32 Bit, or DOS to Windows or a change of the used compiler versions)

1.2 Reporting of Bugs and Software Enhancements Each request from our customers is traced with an helpdesk system assuring that no problem will be lost. All bug fixes or enhancements of the software are documented with version number and date in html files associated to every program module. Serious bugs will be announced to our customers via E-mail if the have registered to our news letter. There is also a forum on the internet www.sofistik.de / or ww.sofistik.com to discuss latest developments and analysis techniques. Although this procedure has minimized the number of errors in the Software, SOFiSTiK can not assure that their software is bug-free or that it will solve a particular problem in a way which is concluding with the opinion of the user in all details. We strongly recommend therefore to use the engineering skill when evaluating the results of any software. 1.3 Insurance SOFiSTiK is Member of the German Association of Consulting Engineers (VBI) and has an professional indemnity insurance.


2 Validation The tasks covered by SOFiSTiK- Software are so large, that it is not possible to validate all specific features with known reference solutions.

Thus there are variant sources of validation:

Internal verification examples for white box testing maintained by the programmer and not generally available to the public.

The examples in the manuals wiIl show the general behaviour of the program and will give expected or approved results for comparison.

The lections given at the annual user meeting of SOFiSTiK (since 1988) show the practical usage of the software within a wide range of applications and the scientific background.

Externally established examples. As SOFiSTiK is a member of the NAFEMS (www.nafems.org) we have taken most of the examples of the NAFEMS Benchmarks, which follow on the next pages. These are:

P07 Linear Static Benchmarks Vol. 1 Hitchings P08 Linear Static Benchmarks Vol. 2 Hitchings R0016 Selected Benchmarks for forced vibration Maguire et. al. R0026 Selected Benchmarks for Material Linken nonlinearity


2.1 Test No IC1 - Tapered Membrane End Load

Classification: Disk, Steel

Index: NAFEMS, Benchmark, Disk, Edge compression, Line load

Short description: In this test the simple disk system, depicted above, is analysed with an end load of 10 MN/m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 6, which then has 64x64 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx in point B. Benchmark value 61.3 MPa (8x8 elements), for an 8-node second order and a 16-node third order quad-element. The results of the benchmark and the program ASE can be found in the following table:

Benchmark (8-node quad) ASE Mesh Point A (MPa) Point B (MPa) Point A (MPa) Point B (MPa)

Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy 2x2 31.30 -3.10 -0.73 61.90 -10.80 0 48.37 -2.19 -3.56 65.91 -1.92 0 4x4 23.10 -0.91 -0.69 60.90 -9.36 0 37.29 -1.75 -1.76 62.37 -7.73 0 8x8 16.70 -0.61 -0.56 61.30 -8.97 0 27.97 -1.13 -0.81 61.66 -8.54 0 16x16 20.80 -0.73 -0.33 61.42 -8.75 0 32x32 15.32 -0.49 -0.11 61.36 -8.80 0 64x64 11.20 -0.34 -0.25 61.34 -8.81 0

further results in the DAT-file

Input file: ic1_e.dat

Last changed: 05.08.2003

Essential programs: ASE


2.2 Test No IC2 - Tapered Membrane Gravity Load


Index: NAFEMS, Benchmark, Disk, Self-weight, Gravitational load

Short description: In this test the simple disk system, depicted above, is analysed with a self-weight of p=70 kN/m² in the x-direction and a gravitational acceleration of 9.81 m/s², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 6, which then has 64x64 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx in point B. Benchmark value 0.247 MPa (8x8 elements), for an 8-node second order and a 16-node third order quad-element. The results of the benchmark and the program ASE can be found in the following table:


Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy

2x2 0.142 0 -0.00693 0.258 -0.0147 0 0.1404-

0.0120-

0.0033 0.1866 -

0.00113 0

4x4 0.100 -0.00193 -

0.00457 0.247 -0.0958 0 0.1270-

0.0054-

0.0007 0.2153 -

0.01850 0

8x8 0.0702 -0.00225 -

0.00317 0.247 -

0.00978 0 0.1035-

0.0031 0.0006 0.2305 -

0.01450 0

16x16 0.0799 -0.0021 0.0009 0.2385 -

0.01220 0

32x32 0.0600 -0.0015 0.0008 0.2427 -

0.01100 0

64x64 0.0443 -0.0011 0.0006 0.2448 -

0.01040 0






2.3 Test NO IC3 - Tapered Membrane Edge Shear


Index: NAFEMS, Benchmark, Disk, Shear loading

Short description: In this test the simple disk system, depicted above, is analysed for a uniform shear load of 100 MPa = 100000 kN/m²*0.1 m = 10000 kN/m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 6, which then has 64x64 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxy in point B. Benchmark value -26.9 MPa (8x8 elements), for a 16-node third order quad-element. The results of the benchmark and the program ASE can be found in the following table:



2x2 313.0 93.8 -68.6 0 0 -28.7 261.41 46.87 -71.65 0 0 -

42.63

4x4 299.0 89.8 -91.0 0 0 -27.9 285.95 70.38 -74.24 0 0 -

40.00

8x8 310.0 93.1 -110.0 0 0 -

27.3 294.13 79.93 -82.03 0 0 -

35.40

16x16 307.84 85.79 -93.72 0 0 -31.79

32x32 331.15 92.24 -107.03 0 0 -

29.51

64x64 363.07 100.52 -120.97 0 0 -

28.23






2.4 Test No IC4 - Tapered Membrane Gravity Load


Index: NAFEMS, Benchmark, Disk, Self-weight, Gravitational load

Short description: In this test the simple disk system, depicted above, is analysed with a self-weight of p=70 kN/m² in the y-direction and a gravitational acceleration of 9.81 m/s², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 6, which then has 64x64 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxy in point B. Benchmark value -0.200 MPa (8x8 elements), for an 8-node second order and a 16-node third order quad-element. The results of the benchmark and the program ASE can be found in the following table:


Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy 2x2 0.579 0.174 -0.234 0 0 -0.198 0.370 0.066 -0.183 0 0 -0.143 4x4 0.663 0.199 -0.251 0 0 -0.200 0.495 0.113 -0.205 0 0 -0.183 8x8 0.739 0.222 -0.276 0 0 -0.200 0.598 0.151 -0.222 0 0 -0.194 16x16 0.688 0.182 -0.241 0 0 -0.197 32x32 0.774 0.209 -0.265 0 0 -0.198 64x64 0.863 0.235 -0.293 0 0 -0.199






2.5 Test No IC5 - Circular Membrane Edge Pressure


Index: NAFEMS, Benchmark, Disk, Edge compression, Line load

Short description: In this test the disk system, depicted above, is analysed with an edge compression of 100 MPa, which is equivalent to a line load of 10000 kN/m for a thickness of 0.1 m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. Initially the system is generated with an element mesh consisting of 8x1 quadrilateral elements. Subsequently the number of elements are doubled, up to an element mesh having 64x8 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Syy in point D. Benchmark value -1150 MPa (analytic). The results of the benchmark and the program ASE can be found in the following table:

Benchmark ASE

Mesh Point D (MPa) radius 10m Point C (MPa)

radius 11m Point D (MPa)

radius 10m Point C (MPa)

radius 11m Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy

8x1 -3.54 -1150.0 0 -

103.0 -

1050.0 0 -

56.08-

1162.10 54.47-

55.77-

1032.86 48.12

16x2 -0.949 -

1150.0 0 -

101.0 -

1050.0 0 -

28.40-

1159.07 27.77-

78.14-

1044.31 23.73

32x4 -14.30-

1156.21 14.02-

89.13-

1048.74 11.78

64x8 -7.18 -1154.42 7.04-

94.57-

1050.64 5.87






2.6 Test No IC6 - Circular Membrane Point Load


Index: NAFEMS, Benchmark, Disk, Point load

Short description: In this test the disk system, depicted above, is analysed with a point load of 5000 N in radial direction at point B, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. Initially the system is generated with an element mesh consisting of 8x1 quadrilateral elements. Subsequently the number of elements are doubled, up to an element mesh having 64x8 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Syy in point D. Benchmark value -532 MPa (16x2 elements). The results of the benchmark and the program ASE can be found in the following table:

Benchmark ASE

Mesh Point D (MPa) radius 10m Point C (MPa)

radius 11m Point D (MPa)

radius 10m Point C (MPa)

radius 11m Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy

8x1 -9.74 -

523.0 -3.09 -8.54 377.0 0 -8.1-

510.60 21.66-

5.97 368.80 -

21.94

16x2 -3.26 -

532.0 -

0.405 -

0.794 368.0-

0.138-

7.35-

528.06 10.99-

6.81 362.31 -

10.74

32x4 -4.98-

533.20 5.90-

3.90 362.67 -5.01

64x8 -2.89-

534.15 3.10-

2.02 363.15 -2.38






2.7 Test No IC7 - Cicular Membrane Parabolic Temperature


Index: NAFEMS, Benchmark, Disk, Temperature load

Short description: In this test the disk system, depicted above, is analysed for a fluctuating temperature load, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. The temperature expansion coefficient is taken as 2.3E-4 /°C. The linearly fluctuating temperature load, which depends on the radius, was modeled with the program HYDRA. In order to be able to define the maximum temperature load, it is necessary to change the provided element arrangement, so that now the elements are defined having the same size and they are aligned in the radial direction (HYDRA and TALPA can only analyse temperature loads in nodes). The system is generated with element meshes of 4x2, 8x4, 16x8, 32x16 and 64x32 quad-elements. The calculation is made with the program TALPA. The GRAF-plot shows the temperature loads from HYDRA by means of a coloured diagram. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Syy in point A. Theoretical value -115 MPa (analytical), benchmark value -104 MPa (for 8x4 elements according to benchmark method 1). The results of the program TALPA are compared against the benchmark results from method 1. They can be found in the following table:

Benchmark TALPA Mesh Point A (MPa) Point A (MPa)

Sxx Syy Sxy Sxx Syy Sxy 4x2 -21.0 -70.0 -0.100 -29.4 -87.3 -12.008x4 -31.3 -104.0 -0.100 -34.2 -110.3 -7.5716x8 -34.8 -115.1 -3.9532x16 -34.8 -115.6 -1.9964x32 -34.7 -115.5 -0.99Theory -34.5 -115.0 0




Essential programs: HYDRA, TALPA


2.8 Test No IC8 - Shear Diffusion


Index: NAFEMS, Benchmark, Disk, Truss elements, Point load

Short description: In this test the disk system, depicted above, is analysed with edge beams and a point load of 10 kN in horizontal direction at point B, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh consisting of 24x2 quadrilateral elements and the respective beam elements (type TRUS). In the subsequent cases the number of elements are doubled, up to case 3 having an element mesh of 96x8 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Shear stress Sxy in point E. Theoretical value 27.8 MPa (analytical), benchmark value 34.3 MPa (24x2 element mesh with 8-node elements). The results of the benchmark and the program ASE can be found in the following table:

Benchmark ASE Mesh Point E (MPa) Point E (MPa)

Sxy Sxy 24x2 34.3 25.463 48x4 28.804 96x8 28.939 Theory 27.8



Last changed : 05.08.2003



2.9 Test No IC9 - Elliptic Membrane


Index: NAFEMS, Benchmark, Disk, Line load

Short description: In this test the disk system, depicted above, is analysed with a line load of 1000 kN/m = 10MPa*0.1m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh consisting of 3x2 quadrilateral elements. In case 2 the 3x2 element mesh is subdivided by triangular elements. For case 3 the system consists of 6x4 quad-elements, and is respectively subdivided by triangular elements in case 4. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Syy in point D. Theoretical value 92.7 MPa (analytical), benchmark value 90.5 MPa (6x4 quad-elements). The results of the benchmark and the program ASE can be found in the following table:

Benchmark ASE Mesh Point D (MPa) Point D (MPa)

Syy Syy 3x2 82.0 62.27 3x2 quad-elements subdivided by tri-elements 53.30

6x4 90.5 82.35 6x4 quad-elements subdivided by tri-elements 74.96

Theory 92.7






2.10 Test No IC10 - Tapered Plate Edge Shear

Classification: Plate, Steel

Index: NAFEMS, Benchmark, Plate, Shear loading

Short description: In this test the simple plate system, depicted above, is analysed with a line load of 10 kN/m acting along the edge DB, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 6, which then has 64x64 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx at the top side of the plate in point B. Benchmark value 14.7 MPa (8x8 elements), for an 8-node second order and a 16-node third order quad-element. The results of the benchmark and the program ASE can be found in the following table:


Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy 2x2 9.21 2.89 0.993 14.3 4.29 0 -12.114 -3.536 0.688 -13.970 -4.475 0 4x4 6.27 1.73 0.324 14.5 4.36 0 -10.190 -2.915 0.850 -14.113 -4.329 0 8x8 3.89 0.821 -0.184 14.6 4.37 0 -6.579 -2.016 0.910 -14.653 -4.442 0 16x16 -3.924 -1.120 0.917 -14.663 -4.410 0 32x32 -2.023 -0.549 0.951 -14.636 -4.393 0 64x64 -1.416 -0.358 1.045 -14.628 -4.389 0






2.11 Test No IC11 - Tapered Plate Gravity


Index: NAFEMS, Benchmark, Plate, Self-weight, Gravitational load

Short description: In this test the simple plate system, depicted above, is analysed with a self-weight of p=70 kN/m² in the z-direction and a gravitational acceleration of 9.81 m/s², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 6, which then has 64x64 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx at the top side of the plate in point B. Benchmark value 26 MPa (8x8 elements), for an 8-node second order and a 16-node third order quad-element. The results of the benchmark and the program ASE can be found in the following table:


Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy Sxx Syy Sxy 2x2 17.7 5.85 2.37 24.0 7.21 0 -21.769 -5.880 1.095 -24.412 -8.201 0 4x4 13.2 3.92 1.10 25.5 7.64 0 -19.516 -5.238 1.402 -25.010 -7.767 0 8x8 9.0 2.20 0.03 25.7 7.71 0 -14.439 -4.049 1.554 -25.909 -7.864 0 16x16 -9.172 -2.602 1.648 -25.916 -7.795 0 32x32 -5.549 -1.567 1.803 -25.869 -7.764 0 64x64 -4.128 -1.154 2.079 -25.855 -7.757 0






2.12 Test No IC12 - Elliptic Plate Normal Pressure


Index: NAFEMS, Benchmark, Plate, distributed load

Short description: In this test the plate system, depicted above, is analysed with a distributed load of 1000 kN/m², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 3x2 quadrilateral elements. In the subsequent case 2 the number of elements are doubled (6x4 quadrilateral elements). The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Syy at the top side of the plate in point D. Analytical value 118 MPa. Benchmark value 158 MPa (6x4 elements) for an 8-node element and 177 MPa (6x4 elements) for a 16-node element. The results of the benchmark and the program ASE can be found in the following table:

Benchmark (8-node quad) ASE Mesh Point A (MPa) Point D (MPa) Point A (MPa) Point D (MPa)


3x2 184.0 8.24 -2.38 -3.20 142.0 -5.87 215.98 32.7 36.41 14.05 209.76 5.19

6x4 178.0 3.69 -0.178 4.93 158.0-

0.490 200.06 17.08 18.03 6.20 158.79 1.95






2.13 Test No IC13 - Skew Plate Normal Pressure


Index: NAFEMS, Benchmark, Plate, Distributed load

Short description: In this test the simple plate system, depicted above, is analysed with a distributed load in the z-direction of p= -0.7 kN/m² = -0.7 kPa, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 4, which then has 16x16 quadrilateral elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: max. principal stress at the under side of the plate for point E. Theoretical value 0.802 MPa (analytical), benchmark value 0.795 MPa (8-node quad-element). The results of the program ASE are compared against the benchmark results, they can be found in the following table:

Benchmark ASE Mesh Point E (MPa) Point E (MPa)

P1 P2 P3 P1 P2 P3 2x2 0.757 0.252 -0.227 0.457 0.289 -0.04204x4 0.795 0.335 -0.130 0.734 0.416 -0.07958x8 0.776 0.446 -0.0825616x16 0.791 0.452 -0.0849






2.14 Test No IC14 - Tapered Thick Shell Pressure Load

Classification: Shell, Steel

Index: NAFEMS, Benchmark, Shell, Edge compression, axisymmetrical state

Short description: In this test the simple shell system, depicted above, is analysed for an axisymmetrical state, with an edge compression of 100 MPa, which is equivalent to a line load of 10000 kN/m for a thickness of 0.1 m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 5, which then has 32x32 quadrilateral elements. The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Szz (Stt) in point C. Benchmark value 237 MPa (8x8 elements). The results of the benchmark and the program TALPA can be found in the following table:

Benchmark TALPA Mesh Point B (MPa) Point C (MPa) Point B (MPa) Point C (MPa)

Sxx Syy Szz Sxy Sxx Syy Szz Sxy Sxx Syy Szz Sxy Sxx Syy Szz Sxy (Syy) (Srr) (Stt) (Sry) (Syy) (Srr) (Stt) (Sry) (Syy) (Srr) (Stt) (Sry) (Syy) (Srr) (Stt) (Sry)

2x2 60.3 105 215 0.813 -4.08 40.9 228 -0.250 41.2 86.9 200 6.45 8.98 54.4 232 9.29

4x4 69.9 102 217 0.156 -2.40 38.3 227 -0.933 52.2 95.0 208 3.51 4.21 46.6 230 5.35

8x8 77.8 101 219 -0.124 -

0.593 30.9 237-

0.306 62.8 97.8 212 1.82 1.09 41.8 229 2.19

16x16 71.8 99.1 216 0.91 0.21 40.5 228 0.7932x32 78.9 99.6 218 0.43 0.04 38.0 230 0.27




Essential programs: TALPA


2.15 Test No IC15 - Tapered Thick Shell Self-weight


Index: NAFEMS, Benchmark, Shell, Self-weight, axisymmetrical state

Short description: In this test the simple shell system, depicted above, is analysed for an axisymmetrical state, with a self-weight of p=70 kN/m² in the x-direction (benchmark y-direction) and a gravitational acceleration of 9.81 m/s², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 2x2 quadrilateral elements. In the subsequent cases the number of elements are doubled, up to case 5, which then has 32x32 quadrilateral elements. The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Szz (Stt) in point C. Benchmark value 22.8 KPa (8x8 elements). The results of the benchmark and the program TALPA can be found in the following table:

Benchmark TALPA Mesh Point B (KPa) Point C (KPa) Point B (KPa) Point C (KPa)


2x2 -111 -2.36 14.6 -5.46 -11.9 -17.3 16.6 -1.93 -122 11.3 13.0 -2.37 -50.5 -

25.7 4.40 -1.43

4x4 -76.1 0.871 26.7 -3.06 -2.68 -17.5 19.4-

0.634 -106 2.43 17.0 -3.46 -29.0 -

20.4 11.0 -2.38

8x8 -51.8 1.72 34.5 -1.86 -0.933 -

14.5 22.8-

0.398 -83.8 0.39 24.0 -3.20 -15.7 -

19.8 14.8 -1.63

16x16 -63.0 0.02 30.5 -2.44 -8.21 -18.5 17.4 -0.94

32x32 -46.3 0.031 35.6 -1.69 -4.20 -17.0 19.5 -0.64






2.16 Test No IC16 - Cylinder-Taper-Sphere Temperature


Index: NAFEMS, Benchmark, cylindrical shell, axisymmetric state

Short description: In this test the cylindrical shell system, depicted above, is analysed for a fluctuating temperature load, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. The temperature expansion coefficient is taken as 2.3E-4 /°C. The linearly fluctuating temperature load, which depends on the coordinates, was modeled with the program HYDRA. The system is generated with element meshes of 5x1, 10x2, 20x4 and 40x8 quad-elements. The calculation is made with the program TALPA, because only this program is able to analyse an axisymmetrical state. The GRAF-plot shows the temperature loads from HYDRA by means of a coloured diagram. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx (Syy) in point A. Benchmark value -105 MPa (10x2 elements). The results of the benchmark and the program TALPA can be found in the following table:

Benchmark TALPA Mesh Point A (MPa) Point F (MPa) Point A (MPa) Point F (MPa)


5x1 -104 -1.83 55.5 -1.10 -183 -

17.2-

61.8 30.3 97.1 10.7-

53.4 -11.2 124 25.7 49.7 -29.8

10x2 -105 -1.02 55.5 -0.49 -203 -

17.6-

68.5 33.0 100.9 1.29-

56.0 -5.16 114 10.8 38.9 -33.1

20x4 103.2 -0.005-

56.2 -2.40 111 8.40 36.7 -33.6

40x8 103.5 -0.054-

56.2 -1.35 110 9.81 36.3 -34.0






2.17 Test No IC17 - Hemisphere External Pressure

Classification: Shell, Glas

Index: NAFEMS, Benchmark, Shell, Distributed load, Symmetry conditions

Short description: In this test the shell system, depicted above, is analysed with a distributed load, which acts from the outside, of 1MPa = 1000 kN/m² (print error: see DAT-file). For this an isotropic material with an E-modulus of 68.25*10³ MPa and a poisson's ratio of 0.3 is used. The system is generated, so that different element meshes, consisting of quad-elements, can be analysed. The following element meshes are analysed: 4x4 elements, 8x8 elements and 16x16 elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Radial displacement in point G. Theoretical value -1.282 mm analytically. Benchmark value -1.272 mm (coarse mesh, 16-node quad-element). The results of the benchmark and the program ASE can be found in the following table:

Mesh Deflection in points (mm) A B C D E F G Benchmark

3 element -1.280 -1.209 -1.280 -1.273 -1.269 -1.273 -1.272 ASE

4x4 -1.2130 -1.4757 -1.2130 -1.5072 -1.1692 -1.5072 -0.58678x8 -1.2714 -1.4085 -1.2714 -1.3878 -1.2703 -1.3878 -1.015716x16 -1.2771 -1.3194 -1.2771 -1.3129 -1.2770 -1.3129 -1.2072






2.18 Test No IC18 - Hemisphere Point Load

Classification: Shell, Glas

Index: NAFEMS, Benchmark, Shell, Point load, Symmetry conditions

Short description: In this test the shell system, depicted above, is analysed with two point loads, each of 2 kN (acting towards the inside and outside respectively), whereby an isotropic material with an E-modulus of 68.25*10³ MPa and a poisson's ratio of 0.3 are used. The system is generated in such a manner, so as to allow the calculation of several different element meshes consisting of quad-elements. The following element meshes are investigated: 4x4 elements, 8x8 elements and 16x16 elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: x-displacement for point A. Theoretical value 0.185 m analytically. Benchmark value 0.1838 m (fine mesh, 8-node quad-element). The results of the benchmark and the program ASE can be found in the following table:

Mesh Benchmark ASE Deflection in points (m) A B A B ux uz ux uy ux uz ux uy

4x4 0.12006 0.06356 0.03953 -0.039538x8 0.1838 0.09 0.0628 -0.0628 0.17978 0.08965 0.06036 -0.0603616x16 0.18563 0.09102 0.06265 -0.06255Theory 0.185






2.19 Test No IC19 - Cylindrical Shell Edge Moment


Index: NAFEMS, Benchmark, Shell, Point load, Symmetry conditions

Short descriptions: In this test the shell system, depicted above, is analysed with a moment loading, which is applied to the edge CD, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. The system is generated, so that an element mesh consisting of 2x2 quadrilateral elements can be calculated. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress at the outer top surface for point E. Theoretical value 60.0 MPa analytically. Benchmark value 60.0 MPa . The results of the benchmark and the program ASE can be found in the following table:

Mesh Benchmark ASE Point E top Point E bottom Point E top Point E bottom Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa)

2x2 60.0 18.0 -60.0 -18.0 60.0 18.0 -60.0 -18.0 Theory 60.0 18.0 -60.0 -18.0






2.20 Test No IC24 - Catenoidal Shell Internal Pressure


Index: NAFEMS, Benchmark, Shell, Distributed load

Short description: In this test the shell system, depicted above, is analysed with a distributed load of 1 MPa = 1000 kN/m², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. The system is generated in such a manner, so as to allow the calculation of several different element meshes consisting of quad-elements. The following element meshes are investigated: 3x3 elements, 6x6 elements and 12x12 elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Syy (Saa) at point A. Theoretical value -69.1 MPa analytically. Benchmark value -68.89 MPa (6x6 element mesh with 8-node quad-elements). The results of the benchmark and the program ASE can be found in the following table:

Mesh Benchmark ASE Point A Point A Sxx (MPa) Syy (MPa) Sxx (MPa) Syy (MPa) (Stt) (Saa) (Stt) (Saa)

3x3 31.63 -62.01 30.89 -63.56 6x6 26.55 -68.89 27.07 -66.41 12x12 23.64 -66.22 Theory 30.95 -69.06






2.21 Test No IC27 - Cylinder/Sphere Internal Pressure


Index: NAFEMS, Benchmark, Shell, Distributed load

Short description: In this test the shell system, depicted above, is analysed with a distributed load of 1 MPa = 1000 kN/m², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. The system is generated in such a manner, so as to allow the calculation of several different element meshes consisting of quad-elements. The following element meshes are investigated: 4,3,2,4,6 elements in the z-direction x 4 elements between the x- and y-axis and double the number of elements, 8,6,4,8,12 elements in the z-direction x 8 elements between the x- and y-axis. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Stt at the outer surface of point D. Theoretical value 38.5 MPa analytically. Benchmark value 38.6 MPa . The results of the benchmark and the program ASE can be found in the following table:

Mesh Benchmark ASE Point D outside Point D inside Point D outside Point D inside Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa)

4,3,2,4,6 x4 38.6 25.7 35.0 13.6 38.103 25.348 34.664 13.833 8,6,4,8,12x8 38.442 25.696 34.967 14.112 Theory 38.5 25.9 35.0 14.2






2.22 Test No IC28 - Circular Paraboloid Gravity


Index: NAFEMS, Benchmark, Shell, Self-weight

Short description: In this test the shell system, depicted above, is analysed for its self-weight of p=70 kN/m³, in the z-direction, and a gravitational acceleration of 10 m/s², whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. The system is generated in such a manner, so as to allow the calculation of several different element meshes consisting of quad-elements. The following element meshes are investigated: 2x2 elements, 4x4 elements and 6x6 elements. The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx at the bottom surface of point D. Theoretical value -0.613 MPa analytically. Benchmark value -0.620 MPa (6x6 elements). The results of the benchmark and the program ASE can be found in the following table:

Mesh Benchmark ASE Point D top Point D bottom Point D top Point D bottom Sxx (MPa) Syy (MPa) Sxx (MPa) Syy (MPa) Sxx (MPa) Syy (MPa) Sxx (MPa) Syy (MPa)

2x2 -1.78 -1.78 -1.69 -1.69 -0.487 -0.490 -0.175 -0.178 4x4 -0.495 -0.495 -0.572 -0.572 -0.560 -0.560 -0.619 -0.619 6x6 -0.608 -0.608 -0.620 -0.620 -0.596 -0.596 -0.601 -0.601






2.23 Test No IC29 - Z-Section Cantilever Torsion Bending


Index: NAFEMS, Benchmark, Folded plates, Torsion

Short description: In this test a folded plate, depicted above, is analysed for a torsional load of 1.2 MNm, which consists of two point loads, each of 0.6 MN. The analysis uses an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3. For case 1 the system is generated with an element mesh of 8x3 quadrilateral elements. In the subsequent case 2 the number of elements are doubled (16x6). The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx at point A. Theoretical value -108.8 MPa analytically. Benchmark value -110.1 MPa . The results of the benchmark and the program ASE can be found in the following table:

s ASE 16x6 elements Theory Benchmark

(m) Szz (MPa) Ssz (MPa) Szz (MPa) Ssz (MPa) Szz (MPa) Ssz (MPa) 0.0 Point A -111.07 -3.39 -108.8 0.0 -110.1 -2.0 0.5 -37.26 -4.86 -36.26 -5.877 -36.9 -6.15 1.0 36.83 -4.65 36.26 -5.877 36.2 -1.35 2.0 37.55 0.0 36.26 0.0 37.3 0.0 3.0 36.83 4.65 36.26 5.877 36.2 1.35 3.5 -37.26 4.86 -36.26 5.877 -36.9 6.15 4.0 -111.07 3.39 -108.8 0.0 -110.1 2.0






2.24 Test No IC30 - Z-Section Cantilever Beam Bending


Index: NAFEMS, Benchmark, folded plate, bending

Short description: In this test a folded plate, depicted above, is analysed for an individual load of 6.0 MN. The analysis uses an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3. For case 1 the system is generated with an element mesh of 8x3 quadrilateral elements. In the subsequent case 2 the number of elements are doubled (16x6). The calculation is made with the program ASE. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx at point A. Theoretical value -193.0 MPa analytically. Benchmark value -191.0 MPa . The results of the benchmark and the program ASE can be found in the following table:

s ASE 16x6 elements Theory Benchmark

(m) Szz (MPa) Ssz (MPa) Szz (MPa) Ssz (MPa) Szz (MPa) Ssz (MPa) 0.0 Point A -190.48 -3.16 -193 0 -191 9.4 0.5 96.76 0.08 96.5 -3.24 96.7 -4.2 1.0 383.80 3.31 386 12.8 383 16.2 2.0 -0.01 29.94 0 38.6 0 34.7 3.0 -383.83 3.31 -386 12.8 -383 16.2 3.5 -96.74 0.08 -96.5 -3.24 -96.7 -4.2 4.0 190.55 -3.16 193 0 191 9.4






2.25 Test No IC31 - Axisymmetric Hyperbolic Shell - Edge Loading


Index: NAFEMS, Benchmark, Shell, Point load, axisymmetric state

Short description: In this test a shell system, depicted above, is analysed for an axisymmetrical state, with a point load of 1 MN/radian, which is equivalent to a line load of 1000 kN/0.01m (element width=shell thickness) for a thickness of 1.0 m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 10x1 quadrilateral elements, whereby the element width is equivalent to the shell thickness of 0.01 m. In the subsequent case the number of elements are doubled for the shell axis, therefore 20x1 elements are used. The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Stt (Szz) in point B at the outer side of the shell. Theoretical value 81.65 MPa analytically. Benchmark value 85.01 MPa (20x1). The results of the benchmark and the program TALPA can be found in the following table:

Mesh Benchmark TALPA Point B Point B Sxx (MPa) Szz (MPa) Sxx (MPa) Szz (MPa) (Saa) (Stt) (Saa) (Stt)

10x1 43.52 92.67 79.60 81.31 20x1 71.71 85.01 79.47 80.95 Theory 81.65 81.65






2.26 Test No IC32 - Axisymmetric Hyperbolic Shell - Pressure


Index: NAFEMS, Benchmark, Shell, Linen load, axisymmetrical state

Short description: In this test a shell system, depicted above, is analysed for an axisymmetrical state, with an internal pressure of 1 MPa, which is equivalent to a line load of 1000 kN/m for a thickness of 1.0 m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 10x1 quadrilateral elements, whereby the element width is equivalent to the shell thickness of 0.01 m. In the subsequent case the number of elements are doubled for the shell axis, therefore 20x1 elements are used. The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx (Saa) in point B at the outer side of the shell. Theoretical value -50.0 MPa analytically. Benchmark value -56.12 MPa (20x1). The results of the benchmark and the program TALPA can be found in the following table:

Mesh Benchmark TALPA Point B Point B Sxx (MPa) Szz (MPa) Sxx (MPa) Szz (MPa) (Saa) (Stt) (Saa) (Stt)

10x1 -74.06 55.24 -45.96 51.19 20x1 -56.12 50.00 -46.31 51.36 Theory -50.00 50.00






2.27 Test No IC34 - Axisymmetric Catenoidal Shell - Pressure


Index: NAFEMS, Benchmark, Shell, Line load, axisymmetricalstate

:

Short description In this test a shell system, depicted above, is analysed for an axisymmetrical state, with an internal pressure of 1 MPa, which is equivalent to a line load of 1000 kN/m for a thickness of 1.0 m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 10x1 quadrilateral elements, whereby the element width is equivalent to the shell thickness of 0.01 m. In the subsequent case the number of elements are doubled for the shell axis, therefore 20x1 elements are used. The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Sxx (Saa) in point B at the outer side of the shell. Theoretical value -69.06 MPa analytically. Benchmark value -70.44 MPa (20x1). The results of the benchmark and the program TALPA can be found in the following table:

Mesh Benchmark TALPA Point B Point B Sxx (MPa) Szz (MPa) Sxx (MPa) Szz (MPa) (Saa) (Sbb) (Saa) (Sbb)

10x1 -74.54 33.33 -66.03 31.72 20x1 -70.44 30.60 -66.47 31.72 Theory -69.06 30.95






2.28 Test No IC37 - Axisymmetric Stiffened Cylinder - Pressure


Index: NAFEMS, Benchmark, Shell, Line load, axisymmetrical state

Short description: In this test a cylinder, depicted above, is analysed for an axisymmetrical state, with an internal pressure of 1 MPa, which is equivalent to a line load of 1000 kN/m for a thickness of 1.0 m. Additionally a single load of 125 kN/0.01 m, which is distributed over the cylinder thickness, is applied. For the cylinder an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 12x1 quadrilateral elements, whereby the element width is equivalent to the shell thickness of 0.01 m, and two additional quad-elements on the outer side of the cylinder, for the thicker cross section, are generated. In the subsequent case 2 the number of elements are doubled for the shell axis, therefore 24x1 elements are used, with four additional quad-elements on the outer side of the cylinder (for the thicker cross section). The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Stt at the inner side of point C. Theoretical value 49.11 MPa analytically. Benchmark value 1.62 MPa . The results of the benchmark and the program TALPA can be found in the following table:

Mesh Benchmark TALPA Point C inside Point C outside Point C inside Point C outside Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa)

12x1 1.81 30.0 48.3 43.9 21.5 45.4 -2.56 36.6 24x1 1.62 29.9 48.4 43.9 22.0 44.9 -3.65 35.8

Point B inside Point B outside Point B inside Point B outside 12x1 29.6 49.9 20.4 47.2 11.96 48.48 13.04 47.56 24x1 29.6 49.9 20.4 47.1 9.73 48.59 15.25 48.97






2.29 Test No IC38 - Axisymmetric Cylinder/Sphere - Pressurre



Short description: In this test a shell, depicted above, is analysed for an axisymmetrical state, with an internal pressure of 1 MPa, which is equivalent to a line load of 1000 kN/m for a thickness of 1.0 m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 1/3 are used. For case 1 the system is generated with an element mesh of 8 + 8 quadrilateral elements, whereby the element width is equivalent to the shell thickness of 0.025 m, and the width of two elements is equivalent to the cylinder thickness of 0.0625 m. In the subsequent case 2 the number of elements are doubled for the shell axis, therefore 16 + 16 elements are used. The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Stt at the outer side of point B. Theoretical value 16.0 MPa analytically. Benchmark value 15.86 MPa . The results of the benchmark and the program TALPA can be found in the following table:

Mesh Benchmark TALPA Point B inside Point B outside Point B inside Point B outside Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa)

8 + 8 9.786 16.86 6.466 15.76 19.74 13.82 20.37 13.82 16 + 16 8.660 16.30 7.395 15.86 20.73 15.72 20.35 12.64 Theory 8.0 16.0 8.0 16.0






2.30 Test No IC39 - Axisymmetric Cylinder/Sphere - Pressure



Short description: In this test a shell system, depicted above, is analysed for an axisymmetrical state, with a line load of 1 MPa = 1000 kN/m²*1.0 m, whereby an isotropic material with an E-modulus of 210*10³ MPa and a poisson's ratio of 0.3 are used. For case 1 the system is generated with an element mesh of 4,3,2,4,6x1 elements and for case 2 with 8,6,4,8,12x1 elements, whereby the element width is equivalent to the shell thickness of 0.025 m. The calculation was made with the program TALPA, because only this program can analyse an axisymmetrical state. Compare to Benchmark Test No IC27: Cylinder/Sphere Internal Pressure

More precise description: NAFEMS: Linear Statics Benchmarks, Vol. 1

Results: Stress Stt at the outer side of point D. Theoretical value 38.5 MPa analytically. Benchmark value 38.6 MPa . The results of the benchmark and the program TALPA can be found in the following table:

Mesh Benchmark TALPA Point D inside Point D outside Point D inside Point D outside Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa) Stt (MPa) Saa (MPa)

4,3,2,4,6x1 34.89 14.18 36.73 24.28 8,6,4,8,12x1 35.0 14.2 38.5 25.9 34.83 13.87 36.85 24.95

Point C inside Point C outside Point C inside Point C outside 4,3,2,4,6x1 29.66 19.53 28.52 18.96 8,6,4,8,12x1 30.0 20.0 30.0 20.0 29.70 19.55 28.55 18.95






2.31 Test 5 Fundamental 2D Plasticity Benchmark

Classification: 2D steel plate, plane strain conditions, elasto-plastic Von Mises material a) perfectly plastic b) with isotropic linear hardening

Search Terms: NAFEMS, benchmark, Von Mises plasticity, isotropic hardening, displacement control, residual stresses

Short Description: Compute the stress path for a defined sequence of loading/ unloading steps for a) a perfectly plastic Von Mises material b) Von Mises material with isotropic linear hardening Detailed Description:

A square steel plate of edge length L = 1 m is subjected to a sequence of eight imposed straining steps:

Stretching in x-direction until the plate just yields, followed by further stretching in x-direction causing plastic flow, i.e. post yield behaviour.

Stretching in y-direction in two steps.

Compression in x-direction in two steps.

Compression in y-direction in two steps.

At the end of the final load step, the plate is returned to its original dimensions. Comparison of results see tabs below.

Input File: 2Dplasticity.dat

Last Changed: 23.07.2003

Mainly affected programs: TALPA


2.32 Test 5 "Dynamic for Deep Simply-Supported Beam"

Classification: Beam, general material

Search Terms: NAFEMS, Benchmark,Dynamic,Forced Vibration,Eigenvalue

:

Short Description Calculate the main eigenvalues of a simply supported 3D beam. Related examples: Test 5H (Harmonic Forced Vibration Response) Test 5P (Periodic Forced Vibration Response) Test 5T (Transient Forced Vibration Response)

Detailed Description:

The beam length is 10 m and is divided in 5 elements. The cross section is a rectangle with 2m side length. All displacements are constrained, also the torsional rotation at the beginning. Shear deformation are considered but not as a Timoshenko beam, though classic beam with shear correction factors. The matrices include rotational masses.

Results:

Modes DYNA NAFEMS (Closed Form Solution)NAFEMS

(Exact 3-D beam elem.)Flexural Modes 1 & 2 42.531 Hz 42.650 Hz 42.710 Hz Torsional Mode 3 71.342 Hz 71.200 Hz 71.495 Hz Extensional Mode 4 125.51 Hz 125.00 Hz 125.51 Hz Flexural Modes 5 & 6 145.09 Hz 148.15 Hz 150.76 Hz Torsional Mode 7 221.10 Hz 213.61 Hz 221.57 Hz Flexural Modes 8 & 9 266.01 Hz 283.47 Hz 301.08 Hz

Input File: test5.dat


Mainly affected programs: DYNA


2.33 Test 5H "Harmonic Forced Vibration Response"

Classification: Beam, not-classified material

Search Terms: NAFEMS, Benchmark,Dynamic,Forced Vibration,Harmonic Forced Vibration

n:

Short Descriptio Calculate the response from a steady state harmonic function in the range from 0 to 60 Hz. Related examples: Test 5 (Deep Simply Supported Beam/Eigenvalues) Test 5P (Periodic Forced Vibration Response) Test 5T (Transient Forced Vibration Response)


The beam is taken from the "Test 5". Actually there are modifications. In the so called "Timoshenko Beam" examples (which is a classical beam with shear factors), there are 6 elements, because we need a result in the middle of the beam. In the "Engineer Beams" (without shear and rotary inertia) we use 10 elements according to the NAFEMS paper.

The load is a classical steady state harmonic function in the range from 0 to 60 Hz. The peak force F0=1E6 N/m is distributed over the whole beam. A modal damping of 2% is used in all 16 modes of the modal analysis. The direct solution has damping factors of a=5.36 and b=7.46E-05.

Results:

Solutions Peak Displacement[mm] Peak Stress

[N/mm²] at Frequency

[Hz] DYNA: Auto Modal Analysis (Timoshenko Beam) 13.23 242.0 42.55

DYNA: Loop-LC Modal Analysis (Timoshenko Beam) 13.23 242.1 42.55

DYNA: Auto Modal Analysis (Engineers Beam) 12.15 242.1 45.30

DYNA: Loop-LC Modal Analysis (Engineers Beam) 12.15 241.8 45.30

DYNA: Direct Solution (Engineers Beam) 12.15 241.8 45.30 Reference Solution 13.45 241.9 42.65 NAFEMS Modal Analysis (middle of Timoshenko/Eng. Beam) 13.44 240.3 42.60

NAFEMS Direct Solution (Engineers Beam) 13.39 242.0 42.58

Input Files: test5h-timo.dat test5h-eng.dat




2.34 Test 5P "Periodic Forced Vibration Response


Search Terms: NAFEMS, Benchmark,Dynamic,Forced Vibration,Periodic Forced Vibration

:

Short Description Calculate the response from two (superpositioned) steady state harmonic functions at 20Hz. Related examples: Test 5 (Deep Simply Supported Beam/Eigenvalues) Test 5H (Harmonic Forced Vibration Response) Test 5T (Transient Forced Vibration Response)


The beam is taken from the "Test 5". Actually there are modifications. In the so called "Timoshenko Beam" examples (which is a classical beam with shear factors), there are 6 elements, because we need a result in the middle of the beam. In the "Engineer Beams" (without shear and rotational inertia) we use 10 elements according to the NAFEMS paper.

The load is a superpositionend steady state harmonic function at 20 Hz. The function is

F = F0 (sin wt - sin 3wt) , where F0 is 1E6 N/m distributed over the whole beam. A modal damping of 2% is used in all 16 modes of the modal analysis. The direct solution has damping factors of a=5.36 and b=7.46E-05.

Results:

Solutions Peak Displacement[mm] Peak Stress

[N/mm²] DYNA: Modal Analysis (Timoshenko Beam) 1.066 19.51

DYNA: Modal Analysis (Engineers Beam) 1.292 25.72

DYNA: Direct Solution (Engineers Beam) 0.962 18.62

Reference Solution 0.951 17.10 NAFEMS Modal Analysis (middle of Timoshenko/Eng. Beam) 0.949 17.05

NAFEMS Direct Solution (Engineers Beam) 0.953 17.33

Input Files: test5p-timo.dat test5p-eng.dat




2.35 Test 5T "Transient Forced Vibration Response"


Search Terms: NAFEMS, Benchmark,Dynamic,Forced Vibration,Transient Forced Vibration

Short Description: Calculate the response from a suddenly applied step load with 1E6 N/m Related examples: Test 5 (Deep Simply Supported Beam/Eigenvalues) Test 5H (Harmonic Forced Vibration Response) Test 5P (Periodic Forced Vibration Response)


The beam is taken from the "Test 5". Actually there are modifications. In the so called "Timoshenko Beam" examples (which is a classical beam with shear factors), there are 6 elements, because we need a result in the middle of the beam. In the "Engineer Beams" (without shear and rotational inertia) we use 10 elements according to the NAFEMS paper.

The load is a suddenly applied step load with 1E6 N/m, distributed over the whole beam. The time step is 0.0001 s and the observation time is set to 1.0 s.

A modal damping of 2% is used in all 16 modes of the modal analysis. The direct solution has damping factors of a=5.36 and b=7.46E-05.

Results:

Solutions Peak Displacement[mm] at time[sec]

Peak Stress [N/mm²]

Static Displacement[mm]

DYNA: Modal Analysis (Timoshenko Beam) 1.025 0.0118 18.68 0.526

DYNA: Direct Solution (Timoshenko Beam) 1.023 0.0116 18.50 0.526

DYNA: Modal Analysis (Engineers Beam) 0.940 0.0111 18.28 0.484

DYNA: Direct Solutions (Engineers Beam) 0.941 0.0111 18.44 0.484

Reference Solution 1.043 0.0117 18.76 0.538 NAFEMS Modal Analysis (middle of Timoshenko/Eng. Beam) 1.040 0.0116 18.71 0.536

NAFEMS Direct Solution (middle of Timoshenko/Eng. Beam) 1.057 0.0117 18.77 0.536

Input Files: test5t-timo.dat test5t-eng.dat




3 Literature (1977) Katz, C.

Die Anwendung der Theorie der Finiten Elemente auf die ebene Sickerströmung mit freier Oberfläche, Diplomarbeit in Grundbau und Bodenmechanik, TU München

(1978) Werner, H., Katz, C., Stieda, J., Axhausen, K. TOP - Benutzer- und DV-Handbuch, CAD-Bericht Kfk-CAD 67, Kernforschungszentrum Karlsruhe

(1979) Werner, H., Axhausen, K., Katz, C. Programmaufbau und Datenstrukturen in entwurfsunterstützenden Programmketten in P.J. Pahl, E. Stein, W. Wunderlich, Finite Elemente in der Baupraxis, Springer, Berlin

(1979) Katz, C., Verschuer Th. v., Werner, H. Data Handling in a Design supporting program chain 5th intern. Seminar on Computational Aspects of the Finite Element Method CAFEM 5, Berlin

(1979) Gonzales, A., Katz, C., Werner, H. Programmbaustein Spannbetonnachweise CAD-Bericht Kfk-CAD 73, Kernforschungszentrum Karlsruhe

(1979) Werner, H., Katz, C., Verschuer Th. v., Axhausen, K. Programmkette Tunnelbau, CAD-Bericht Kfk-CAD 124 (CAD im Grundbau) Seminare im Haus der Technik Essen, Kernforschungszentrum Karlsruhe

(1979) Werner, H., Axhausen, K., Katz, C. Techniques in User Oriented Finite Element Programs for Geomechanical Design Practise. 3rd Intern. Conf. On Numerical Methods in Geomechanics, Aachen, A. A. Balkema Rotterdam

(1980) Axhausen, K., Fink, Th., Katz, C., Rank, E., Stieda, J., Verschuer, Th. v., Werner, H. SET - Berechnungen im Erd-, Grund- und Tunnelbau, Benutzerhandbuch, CAD-Bericht Kfk-CAD 173, Kernforschungszentrum Karlsruhe

(1980) Axhausen, K., Fink, Th., Katz, C., Rank, Verschuer Th. v., Werner, H. Die Programmkette SET - Berechnungen um konstruktiven Ingenieurbau, Benutzerhandbuch, CAD-Bericht KfK-CAD 174, Kernforschungszentrum Karlsruhe

(1980) Fink, Th., Katz, C., Rank, E., Unger, C., Verschuer, Th.v., Werner, H. Die Programmkette SET - Berechnungen um konstruktiven Ingenieurbau, Benutzerhandbuch und Beispielsammlung. CAD-Bericht Kfk-CAD 175, Kernforschungszentrum Karlsruhe

(1981) Katz, C. The use of Green‘s Functions in the Numerical Analysis of Potential, Elastic and Plate Bending Problems. C. A. Brebbia (Ed.) Boundary Element Methods, Springer, Berlin

(1982) Katz, C. Ein symmertrisches Verfahren zur Berechnung von Problemen der Potential, -Scheiben- oder Plattentheorie mit Greenschen Funktionen. Mitteilungen Institut Bauingenieurwesen I TU München Heft 7

(1982) Katz, C. Some Improvements in 2D Boundary Elements using Integration by parts. 4th Internat. Seminar on Boundary Element Methods in Engineering, Southampton, Springer, Berlin

(1982) Katz, C., Werner, H. Implementation of nonlinear Boundary Conditions in Finite Element Analysis Computers & Structures Vol. 15m pp 299-304

(1983) Werner, H. , Stieda, J., Katz, C., Verschuer, Th.v. Rechnereinsatz für Entwurfsaufgaben im konstruktiven Ingenieurbau. Bauingenieur Vol. 58, pp 361-368

(1983) Rank, E., Katz, C., Werner, H. On the importance of the discrete Maximum Principle in Transient Analysis Using Finite Element Methods, Int. Journ. for Numerical Methods in Engineering, Vol. 19, 1771-1782


(1984) Werner, H., Kühl, Ch., Stieda, J., Katz, C., Verschuer, Th.v. Das Softwarekonzept SYRAKUS - Eine Lösung zur Anpassung unterschiedlicher Entwurfs-Programme, Bauingenieur Vol. 59, pp 455-460

(1985) Hartmann, F., Katz, C., Protopsaltis, B. Boundary Elements and Symmetry, Ingenieurarchiv 55, pp 440-449

(1985) Groth, P., Hilber, H.M., Katz, C., Werner, H. FEDIS - Finite Element Data Interface Standard, Projektbericht Kfk-PFT 114, Kernforschungszentrum Karlsruhe

(1985) Katz, C. Finite Elemente auf Mikrorechnern Finite Elemente - Anwendungen in der Baupraxis, Ernst & Sohn, Berlin

(1985) Katz, C. Analytic Integration of isoparametric 2D-Boundary Elements Proceedings of the 7th Intern. Conference on Boundary Elements, Como (C.A. Brebbia, G. Maier ed) Springer, Berlin

(1986) Katz, C. Self-Adaptive Boundary Elements for the Shear-Stress in Beams Proceedings of the 2nd Boundary Element Technology Conference, MIT (J.J. Connor, C.A. Brebbia ed) Computational Mechanics Publications

(1986) Katz, C. Berechnungen von allgemeinen Pfahlwerken, Bauingenieur Vol. 61, Heft 12, 563-568

(1987) Bellmann, J.: Hierarchisch Finite-Element-Ansätze und adaptive Methoden für Scheiben- und Plattenprobleme Mitteilungen aus dem Institut für Bauingenieurwesen I Technische Universität München Dissertation (1987) Heft 21

(1987) Bellmann, J.: Convergence of Hierarchical Finite Elements Fachgebiet Elektronisches Rechnen im Konstruktiven Ingenieurbau, Technische Universität München, Bericht 87/4 April 1987

(1987) Bellmann J, Mackert M., Bauer H.: Finite Kluftelemente für felsmechanische Berechnungen Fachgebiet Elektronisches Rechnen im Konstruktiven Ingenieurbau, Technische Universität München, Bericht 87/1 September 1987

(1987) Bellmann J, Mackert M.: Isotrope und anisotrope Fließ- und Quellgesetze für Finite-Element-Berechnungen im Erd- und Tunnelbau Fachgebiet Elektronisches Rechnen im Konstruktiven Ingenieurbau, Technische Universität München, Bericht 87/7 September 1987

(1987) Katz, C., Murphy‘s Law in Boundary Element Implementations Boundary Elements IX, C.A. Brebbia, W.L. Wendland, G. Kuhn ed) Computational Mechanics Publication, Southampton/Springer, Berlin

(1987) Katz, C. What makes boundary elements so difficult? Microsoftware for Engineers Vol. 3 No. 4 pp 216-219

(1989) Bellmann J., Rank E.: Die p- und hp- Version der Finite-Element-Methode oder: Lohnen sich höherwertige Element? Bauingenieur 64 (1989) Heft 2, S. 67-72

(1989) Katz, C. Fehlerabschätzungen Finiter Element Berechnungen 1. FEM - Tagung Kaiserslautern, Universität Kaiserslautern

(1992) Katz, C., Stieda, J. Praktische FE-Berechnungen mit Plattenbalken, Bauinformatik Vol. 3 Heft 1

(1992) Katz, C. Erfahrungen mit der nichtlinearen Berechnung von Stahlbetonrahmen nach EC2 2. FEM - Tagung Kaiserslautern, Universität Kaiserslautern


(1994) Katz, C. Bauwerk-Boden Wechselwirkung, 3. FEM -Tagung Darmstadt, TH Darmstadt

(1995) Katz, C. Kann die FE-Methode wirklich alles? Finite Elemente in der Baupraxis, (E. Ramm, E. Stein, W. Wunderlich ed) Stuttgart Ernst & Sohn, Berlin

(1996) Katz, C. Protopsaltis, B. ed. SOFiSTiK Software für Statik und Konstruktion, 7. und 8. Anwender Seminar, A.A.Balkema, Rotterdam

(1997) Katz, C. Fließzonentheorie mit Interaktion aller Stabschnittgrößen bei Stahltragwerken Stahlbau 66 (1997), 205 -213

(1997) Flesch, R. Baudynamik praxisgerecht, Band 2, Anwendungen + Beispiele, Bauverlag Wiesbaden

(1998) Katz, C. Protopsaltis, B. ed. SOFiSTiK Software für Statik und Konstruktion, 9. und 10. Anwender Seminar, A.A.Balkema, Rotterdam

(1998) Bellmann, J.: Membrantragwerke und Seifenhaut – Unterschiede in der Formfindung Bauingenieur 73 (1998) 118-123

(1998) Gebhard, S., Katz, C., Bellmann, J. FEM-Berechnungen im Workstation Cluster

(1999) Katz, C. Bellmann, J. From Geometry to FE-Analysis, European Conference on Computational Mechanics ECCM 99

(1999) Heidkamp, H. Die Endverankerung von externen Spanngliedern in Feldlisenen von Massivbau-Hohlkasten-brücken. Eine Finite-Element Parameterstudie zur Erfassung der lokalen Momentenwirkung im Krafteinleitungsbereich. Diplomarbeit, Institut f. Massivbau, RWTH-Aachen

(1999) Achatz S., Glück M., Halfmann A., Katz C., Numerische Berechnung der Fluid-Struktur-Wechselwirkung auf Vektor-Parallelrechnern mit verteiltem Speicher. Lehrstuhlbericht Lehrstuhl für Bauinformatik der TU-München.

(2000) Heidkamp, H. Solution of Nonlinear Equilibrium Equations in Finite Element Analyses of Concrete Structures Master-Thesis, Department of Structural Mechanics, Chalmers University of Technology

(2000) Katz, C. Protopsaltis, B. ed. SOFiSTiK Software für Statik und Konstruktion 3, 11. und 12. Anwender Seminar, A.A.Balkema, Rotterdam

(2000) Halfmann, A, Rank, E. Rücker, M, Katz, C. Gebhard, S. Integrierte Modellierungs- und Berechnungssoftware für den konstruktiven Ingeneieurbau, Systemarchitektur und Netzgenerierung, Bauingenieur 75, pp 60-66

(2001) Halfmann A., Rank E., Glück M., Breuer M., Durst F., Bellmann J., Katz C. Computational Engineering for Wind-Exposed Thin-Walled Structures, Lehrstuhlbericht Lehrstuhl für Bauinformatik der TU-München.

(2002) Hartmann, F., Katz, C. Statik mit Finiten Elementen, Springer, Berlin

(2002) Bellmann J., Köppl J.: Die Überdachung des Rundmischbettes für ein Zementwerk in Harburg bei Donauwörth Stahlbau 71 (2002) Heft 7, S. 484-489

(2002) Heidkamp, H., Katz, C. Soils with swelling potential - Proposal of a final state formulation within an implicit integration scheme and illustrative FE-calculations. 5th World Congress on Computational Mechanics (WCCM V), Wien 2002

(2003) Katz, C. Protopsaltis, B. ed. SOFiSTiK Software für Statik und Konstruktion 4, 13. und 14. Anwender Seminar, A.A.Balkema, Rotterdam


(2003) Bellmann, J.: Membranes - From Formfinding to Cutting Pattern Proceedings of the international conference: Textile Composites and Inflatable Structures, CIMNE, Barcelona (2003), ISBN 84-95999-29-3

Lectures given at variant seminars (incomplete list): (1983) Informationsseminar L. Scheck Ulm

“Die Methode der Finiten Elemente zur Berechnung von Plattentragwerken auf Mikrorechnern“ (1986) Seminar über Finite Elemente bei instationären Problemen

Mitteilung aus dem Forschungsschwerpunkt Simulation und Optimierung deterministischer stochastischer dynamischer Systeme, Universität der Bundeswehr Neubiberg „Instationäre Berechnung von Grundwasserproblemen“

(1988) Brückenbau: Einflußlinien / Spannbetonbemessung , SOFiSTiK-User-Meeting (1989) Bemessung von allgemeinen Stabetonquerschnitten , SOFiSTiK-User-Meeting

Fehlerschätzung von FE-Berechnungen , SOFiSTiK-User-Meeting (1990) Eigenspannungen, Bemessung nach EC2 , SOFiSTiK-User-Meeting

Grundlagen dynamischer Berechnungen , SOFiSTiK-User-Meeting Der Einfluß der Lagerung bei schiefen Plattenrändern , SOFiSTiK-User-Meeting

(1991) Nichtlineare Berechnungen von Betontragwerken , SOFiSTiK-User-Meeting Probleme mit Unterzügen , SOFiSTiK-User-Meeting Dynamische Berechnung einer Rohrbrücke , SOFiSTiK-User-Meeting

(1992) Über den Einfluß der Bauabfolge bei Grundbauproblemen , SOFiSTiK-User-Meeting (1993) VBI - Fortbildungsseminar Bayern: Neuerungen und Tendenzen im konstruktiven Ingenieurbau

„Bewertung von Ergebnissen nach der FEM“ (1993) Forum Bauinformatik, TU München „Innovationsfragen in der Praxis“ (1993) Schubbemessung in Stahlbeton , SOFiSTiK-User-Meeting (1994) Vorspannung und Finite Elemente / Unterzüge , SOFiSTiK-User-Meeting

Bauwerk-Boden Wechselwirkungen , SOFiSTiK-User-Meeting (1995) Vertrauen ist gut - Kontrolle ist besser , SOFiSTiK-User-Meeting

Bemessungsverfahren nach EC4 Teil 10 - SOFiSTiK-User-Meeting (1996) Massivbau-Seminar Fachhochschule Biberach

FE-Berechnungen im Hochbau, Praxis und Fehlerquellen im Alltag (2000) Massivbau-Seminar Fachhochschule Biberach

Anerkannte Regeln der Baukunst ? Interpretation von DIN-Vorschriften aus der Sicht eines Softwareherstellers.

(2002) Heidkamp, H., Filus, M.: FEM-Baugrubenberechnung, SOFiSTiK-User-Meeting

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