a picture album of the finite element method

1

A Picture Album of FEM

The Finite Element Method

Rahul Leslie

Deputy Director Buildings Design DRIQ, PWD Trivandrum

2

“The limitations of the human mind are such that it cannot grasp the behavior of its complex surroundings and creations in one operation.

Thus the process of subdividing all systems into their individual components…, whose behavior is readily understood, and then rebuilding the original system from such components to study its behavior is a natural way in which the engineer, the scientist, or even and economist proceeds.”

– O. C. Zienkiewicz

in “The Finite Element Method”

A Picture Album of FEM presented by Rahul Leslie

3

FEM

The Practical Side

Skip


4


5

Element Library

Elements

•One Dimensional Elements

•Two Dimensional Elements

•Three Dimensional Elements


6


7

Element LibraryOne Dimensional (1D) Elements

Two Dimensional (2D) Elements

Three Dimensional (3D) Elements


8

A Simple Example

For a simple nib, in the Finite Element Model…

It has supports

It is acted upon by forces

It has a body that resists force


9

Elements… …are connected together… …at the nodes…


10


11


12

…element property provided,

Truss, Beam

Axi-Symmetric

Plane stress, Plate Bending

Plane Strain, Axi-Symmetric

(No Specification

usually Required)

Element Specification (Type of Element )


13

1 Dimensional

Truss element

Frame element

2 Dimensional Ax-symmetric surface

2 Dimensional

Plane stress / Membrane element

Shell / plate element

Layered shell element

3 Dimensional Axi-symmetric solid

3 Dimensional

Solid element

Layered Solid element

Element Library


14

0 DimensionalMass

Spring Support

1 Dimensional

Gap/ Hook elements

Tension only / Compression only elements

Spring / Damper element

Rigid Link

2/3 Dimensional Glue

2 Dimensional

Panel Zone element

3 Dimensional

Element Library (Continued)


15


Beam: IYY, IZZ, A, J, SY, SZ

2D elem: Thickness

3D elem: (No Property Required)

Cross Sectional Properties

Truss: A


16


…material property provided,

For Linear Stress Analysis

Young’s Modulus

Poissons Ratio

E

υ (or μ)


17



…support conditions defined,


18

…and analysed.

…element property provived,


…support conditions defined,

…loading provided,


Coordinate System

Global system

X

Y

Z Rotational directions (RX, RY and RZ) are defined as:

When looking through the axis to the origin, anticlockwise is +ve

19


Coordinate System

20


Coordinate System

Local system for beams

GX

GY

GZ

XY

Z

X

Y

Z

Rotational directions (RX, RY and RZ) are defined as:


21


Coordinate System

Local system for beams – orientation node

Rotational directions (RX, RY and RZ) are defined as:


22


Coordinate System

23


Coordinate System

Local system for plates

Rotational directions RX and RY are along local X and Y

XY

Z

Direction Z is towards that side from which the nodes i, j, k, l in order appear anti-clockwise

k

j

i

l

Direction X is parallel to i-j, and directed from i end to j end.

Direction Y is perpendicular to X direction, and directed from j end to k end.

24


Coordinate System

25


Global & Local Coordinate Systems

26


Global & Local Coordinate Systems

27


28

Element FormulationsOne Dimensional (1D) Elements

Two Dimensional (2D) Elements

Three Dimensional (3D) Elements


29

Element FormulationsBased on:

Barycentric local coordinate system

Normalized-Cartesian local coordinate system

(l1, l2)

(a1, a2, a3)

Note:

l1 = L1/L

a1 = A1/A

v1 = V1/V

(v1, v2, v3, v4)

(r)

(r, s)

(r, s, t)


30

Element Library1D Elements

•Truss Elements

•Beam Elements (incomplete/complete)

2D Elements

•Plane Stress Elements (incomplete/complete)

•Plate Elements

•Shell Elements (incomplete/complete)

3D Elements

•Solid elements


31

Truss element

Truss element with DoF in local directions

Truss element with DoF in global directions


32

Beam element

Beam element with DoF in global directions

Beam element (text book derivation)

Beam element with DoF in local directions


33

Plane elements

Plane stress element (incomplete)Plane stress element (with drilling DoF) Plate element

Shell element


34

Solid element


35

Element Library (Continued…)1D Elements used for 2D structures

•1D Axi-symmetric elements

2D Elements used for 3D structures

•2D Axi-symmetric elements

•Plane Strain Elements


36

Examples with 1D Element

(1) Truss Elements

(2) Beam Elements


37

Examples with 1D Element

(1) Truss Elements:

A roof truss…


38

A Transmission Line Tower


39

A Truss Girder Crane


40

(2) Beam Elements:

An RCC frame…


41

A car frame…


42

Hyperbolic paraboloid shell roof…


43



44


St. Matthew’s Catholic Church, Michgan

St Francis Assisi Cathedral, Kerala


45

A roof(2) Plate Bending Elements:


46

A swimming pool and tank…


47

A Coca~Cola can…


48

Examples with 3D Elements


Hexahedral Elements Tetrahedral Elements

49


50

…a piston…

(using Hexahedral Elements)


51

…more machine parts…

(using Tetrahedral Elements)


52

…using Hexahedrons & Tetrahedrons together…


53

A bridge.


54

And a hip bone.


55

Examples of 2D ElementsPosing as 3D Elements

1) Plane Strain Elements

2) Axi-Symmetric Elements

3) Fibre-section Model


56

1) Plane Strain Elements (& Generalised Plane Strain Elements)


57

A Dam …


58

…and a tunnel…


59

2) Axi-Symmetric Elements


60

Some machine part…


61


62


63

3) Fibre Section Model


64


65

Examples of Models with Elements of 1D, 2D, 3D together


66

Wright’s aero-plane…


67

An RCC frame (with trusses)…


68

Shearwalls using shells,

slabs using plates or shells &

Beams, columns and braces with frame elements…


69

The Burj-Khalifa (Dubai) building – one floor…


70

An overhead water tank…


71

FE Model of an RCC Structure –

Solid Elements for Concrete and Line Elements for Reinforcement


72

Steps in an FEM Analysis

Preprocessor/Modeling

Analysis run / Solve

Post-processing/View results


73

• Identification of type of analysis

• Idealization, ie., choice of element type/types

• Identification of the appropriateness of analysis by FEM

Preprocessor / Modeling:


74

A connecting plate…

… brick elements…

… shell elements.


75

Shear wall…

… plate elements…

… frame elements & rigid links.


76

• Identification of type of analysis

• Idealization, ie., choice of element type/types

• Discretization of the solution region (meshing)

• Identification of the appropriateness of analysis by FEM

• Creation of material behavior model

• Application of boundary conditions

Preprocessor / Modeling:


77

• Assembly of global stiffness matrix [K]

• formulation of load vector {F}

• Solution of [K]{u}={F} to get nodal displacements {u}

• Formulation of element stiffness matrices

Analysis run / Solve:


78


79


80


81


82


83


84


85


86


87


88


89


90

• Assembly of global stiffness matrix [K]

• formulation of load vector {F}

• Solution of [K]{u}={F} to get nodal displacements {u}

• Formulation of element stiffness matrices

• (Solution of [K] – λ[M] in case of dynamic analysis)

Analysis run / Solve:

• Calculation of elemental stresses

• (Solution of [K] – λ[Kg] in case of buckling analysis)


91

• Interpret and validate results

• If required, re-formulate, and re-analysis

• View results (displacements, stresses, mode shapes, etc.)

Postprocessing / View results:


92

Types of Analysis

Static

Dynamic

Buckling


93


94


95


96


97


98

End of Part I

99

FEM Packages

Rahul LeslieDeputy Director,

Buildings Design,DRIQ Board, PWD

Presented by

100

Where FEM started from…


101


1) The first paper in FEM:

• M. J. Turner, R. W. Clough, H. C. Martin & L. J. Topp, “Stiffness and Deflection Analysis of Complex Structures”, J. Aeronautical Science 23 (9), pp. 805-823, Sept. 1956.

2) The coining of the name ‘Finite Element Method’:

• R. W. Clough, “The Finite Element Method in Plane Stress Analysis”, Proc. 2nd ASCE Conf. On Electronic Computation, Pittsburg, Sept. 1960.


102


3) Introduction of Isoparametric elements that made FEM so very versatile:

• B. M. Irons & O. C. Zienkiewicz, “The Isoparametric Finite Element System – A New Concept in Finite Element Analysis”, Proc. Conf. Recent Advances in Stress Analysis, Royal Aeronautical Society, London, 1968.

4) SAP-IV -- the first FEM Package :

• K. J. Bathe, E. L. Wilson & F. E. Peterson, “SAP IV – A Structural Analysis Program for Static and Dynamic Response of Linear Systems”, Report No. 73/11, Earthquake Engineering Research Center, June 1973.


103

The Packages that model them..First FE Package

• SAP IV


104


105


106


107


108


109


110

http://nisee.berkeley.edu/elibrary/Software/SAP4ZIP


111


112


113


114


115


116


117


118


119


120

Civil Engg. Specific

• STAAD.Pro

• SAP2000, ETABS

• MIDAS/Gen

• SCADDS, STRUDS, RISA-3D,Orion,etc.

• NISA/Civil

First FE Package

• SAP IV

General FEM

• ANSYS, Civil FEM

• ABAQUS

• MSC-NASTRAN

• MARC, ADINA, etc.

• NISA, STARDYNE, etc.


121


122

End of Part

Rahul [email protected]

123

FEM - II

The Finite Element Method

Rahul Leslie

Deputy Director Buildings Design DRIQ, PWD Trivandrum

124

“…The laws of nature are written in the language of mathematics. These often take the form of ordinary or partial differential equations.

The electronic digital computer is an amazingly fast calculating tool; but it can handle only arithmetic.

– G. S. Ramaswamy

in “Design and Construction of Concrete Shell Roofs”

Finite Element techniques do precisely this.”

The differential equation governing the physical phenomena… have therefore to be reduced to a system of simultaneous equation, before the computer can solve then by a series of arithmetical operations.


125

Higher order elements


126

First Order Elements

Second Order Elements

Third Order Elements


127


128

Lagrangian

Seredipity

Transitional

Skip NASAA Picture Album of FEM presented by Rahul Leslie

129


130


131


132


133

Fineness of Meshing


134

The course… The medium… The fine…

Skip demo


135


136


137


138


139


140

The Good... The Bad.. The Ugly


141

Mapped Meshing, Free Meshing

Mapped meshing… Free meshing…A Picture Album of FEM presented by Rahul Leslie

142


143


144


145

Poor meshing… Better meshing…


146

Poor meshing… Better meshing…


147

Reducing the problem

Detail Suppression

Dimensional Reduction

Plane Strain


148

Planar Symmetry

Axi-Symmetry

Cyclic Symmetry

Repetitive Symmetry


149

The Cessna Plane… …analysed…

…by sub-structuring.


150

Domain Decomposition for Distributed Computing


151



152



153


FE Domain Decomposition

Distributed Matrix Solver


154

Thank you!

Rahul [email protected]