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Summary of the Class before Exam1

Building a FEA Model Ingredients of a FEA software package

Steps in building a FEA model

Modeling considerations

1D Spring/Truss Elements Single 1D spring/truss element

Global stiffness matrix; properties

Boundary conditions

Shape functions and their properties

2D and 3D truss elements; rotation matrix

Beam elements 1D beam elements

2D beam elements

Distributed load

Frames Beam elements with bending and axial loads

3D beam elements with bending, axial loads, and torsion

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Building A FEA Model

Six basic ingredients of a FEA software package

1. Type of analysis

2. Geometry (defined through nodes)

3. Elements

4. Material properties5. Boundary conditions

6. Time functions

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Steps in building a FEA model

1. Discretization of the geometry into an equivalent system of finite elements with

associated nodes and choosing an appropriate element type to closely model the

actual physical behavior.2. Selection of an appropriate displacement (interpolation) function within each

element.

3. Defining the strain/displacement and stress/ strain relationships for deriving the

equations for each finite element

4. Derivation of the element stiffness matrix and equations. The nodal forces arerelated to the nodal displacements using the direct equilibrium method, or energy

methods or the weighted residual method.

5. Assembling the global stiffness matrix from the element stiffness matrices

6. The introduction of the boundary conditions.

7. Solving for the unknown degrees of freedom (or the generalized displacements)8. Solving for the element stresses and strains

9. Interpretation of the results for use in the design/analysis process

Building A FEA Model

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Building An FEA Model

Considerations in building a physical model

1. The nature of real-world problem2. The cost of conducting FEA

Considerations in building an FEA model

1. The accuracy of the FEA model

2. The cost of conducting FEA

The differences between a physical model and an FEA modelDeveloping a physical model is a process that simplifies a real-world problem into a

problem that is suitable for FEA.

Factors that affect the cost:

1. Degree of Freedoms (DOF): p

2. Number of Nodes : N

3. How the nodes are numbered

4. Number of Integration Points in Each Elements

5. Nonlinear Analysis

How to estimate the size (the memory need) of an FEA model?

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Element Stiffness Matrix and Global Stiffness Matrix

Shape funct ions, or interpolation functions:

approximation of displacement field

Nodal force-nodal displacement relationship

Matrix form of above relationship

Element stiffness matrix

Coordinate Transformation

[ ] ][][][ TKTK eTe =Global Stiffness Matrix

Direct stiffness matrix method

Write down global displacement vector

Use this vector as an index for DOFs

Airline tickets

Boundary Conditions

Coordinate rotation for skewed support

Cross out some terms

Solve global stif fness matrix

Solve nodal reaction force (external force)

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Spring/Truss Elements

1 2x

xd1 xf1 xd2 xf2

For the spring to be in equilibrium,

xxx dkdkf 212 +=

xxx dkdkf 211 =

=

x

x

x

x

dd

kkkk

ff

2

1

2

1

[ ] xe

x dkf = [ ]e

k stiffness matrix

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Spring/Truss Elements

1 2 x

1 2

x

xd1 xd2

T T

dx

du= ?=u

xL

dddu xxx

121

+= xx d

L

xd

L

xu 21

1 +

=

[ ]

=

x

x

d

dNNu

2

121

L

x

N

11 = L

x

N

2= Shape functions

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Beam Elements

Mechanics of Beams

2

2

xd

dvEIm=

EI

m

xd

dv=

2

2

Vxd

dm=

( )xV

xd

dvEI

3

3

=

)(

xwxd

dV= ( )xw

xd

dvEI

4

4

=

If ( ) 0 =xw

04

4

=xd

dvEI

m

V

+m

VIn beam theory

+ m

f

+m

f

+

In FEA

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Beam element

yd1

1

yd2

2

Beam Elements

Beam Elements ( )xVxddv

EI 3

3

= In general, ( ) 0 xV

432

23

1)( axaxaxaxv +++=

( ) 410 adxv y ===

( ) 310

axxd

dv===

( ) 322

12 23

aLaLaLxxd

dv++===

( ) 432

2

3

12 aLaLaLadLxv y +++===

Boundary conditions

Why we use a third order polynomial?

Interpolation function for deflection.

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Beam Elements: displacement function

( ) ( )212213112

++= LddLa yy

432

23

1)( axaxaxaxv +++=

( ) ( )212122 213

+= LddLa yy

13 =a yda 13=

[ ]{ }dNv= [ ] [ ]4321 NNNNN = { } [ ]Tyy ddd 2211 =In matrix form

( )32331 321 LLxxLN += ( )3223

32 21

LxLxLxLN +=

( )LxxL

N23

3332

1+= ( )223

34

1LxLx

LN =

Note: ( ) 101 ==xN ( ) 01 ==LxN ( ) 102 ==x

xddN ( ) 0

2 ==Lx

xddN

Beam Elements

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Beam Elements: Stress and Strain

( )2

2

xd

dvEIxm =

( )3

3

xd

dvEIxV =

Beam Elements

[ ]

=

2

2

1

1

2

2

1

1

y

y

e

y

y

d

d

K

m

fm

f

[ ]

=

22

22

3

4626612612

2646

612612

LLLL

LL

LLLL

LL

L

EIK

e

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Torsion in FEA

Consider torsion only

1 2

xm

1

xm

2

x1 x2

L

JL

dGM

=

( )xxxxL

GJmm 1221

==

=

x

x

x

x

L

GJmm

2

1

2

1

1111

x

Frames

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Consider shear, bending, and torsion

1 2

xm1 xm2

x1 x2

L

xzm1 z1zm2 z2

yf1

yd1 y

f2

yd2

=

x

x

x

x

L

GJ

m

m

2

1

2

1

11

11

=

z

y

z

y

z

y

z

y

d

d

LLLLLL

LLLL

LL

L

EI

m

f

m

f

2

2

1

1

22

22

3

2

2

1

1

4626612612

2646

612612

Frames

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Beam In 3D Space

Start from Local Coordinate

x

Consider bending in y zand , and torsion in x

Consider shear in y zand , and axial inx

yy

df22

,

zz df 22,

xx df 22 ,

zzm 22,

xxm 22,

yym 22,

Frames

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Frames

=

z

y

x

z

y

x

z

y

x

z

y

x

z

y

yy

zz

zzz

yyy

yyyy

zzzz

z

y

x

z

y

x

z

y

x

z

y

x

d

d

d

d

d

d

L

EIL

EIL

GJL

EI

L

EIL

EI

L

EIL

AEL

EI

L

EI

L

EIL

EI

L

EI

L

EIL

GJ

L

GJL

EI

L

EI

L

EI

L

EIL

EI

L

EI

L

EI

L

EIL

AE

L

AE

m

m

m

f

f

f

m

m

m

f

f

f

2

2

2

2

2

2

1

1

1

1

1

1

23

23

2

2

2323

2323

2

2

2

2

2

2

1

1

1

1

1

1

4

04

00

06

012

6000

12

00000

2

000

6

0

4

02

06

004

0000000

0

6

0

12

000

6

0

12

6000

120

6000

12

0000000000

Beam In 3D Space, Local Coordinate

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Shape Functions

Shape functions describe the shape of displacement field and are one of

the determinant factors in governing the efficiency and accuracy of FEA.

A shape function takes the value of ONE at the node associated with it,

and takes the value of ZERO at other nodes.

The choice of shape functions depends on:

The number of nodes in the element.

The function should be able to describe rigid body motion, and

constant strain.

L

xN

11 =

L

xN

2=Two nodes truss:

( )32331

321

LLxxL

N += ( )322332 21

LxLxLxL

N +=

( )LxxL

N23

3332

1+= ( )223

34

1LxLx

L

N =

Two nodes

beam bending:

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Rotation Matrix

=

y

x

y

x

f

f

f

f

1

1

1

1

cossin

sincos

or

=

=

y

x

y

x

y

x

f

f

f

f

f

f

1

1

1

1

1

1

cossin

sincos

cossin

sincos

[ ]{ }fTf = { } [ ] fTf 1=

[ ] [ ]TTT =1

x

y

x

xf1

yf1

xf1

yf1

y

Measured from tocounterclockwisex x

Measured from tocounterclockwise

xx

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=

y

x

y

x

y

x

y

x

f

f

ff

CS

SC

CSSC

f

f

ff

2

2

1

1

2

2

1

1

00

00

0000

=

y

x

y

x

y

x

y

x

d

d

dd

CS

SC

CSSC

d

d

dd

2

2

1

1

2

2

1

1

00

00

0000

xyzzyx

=

y

x

y

x

y

x

y

x

f

f

f

f

CS

SC

CS

SC

f

f

f

f

2

2

1

1

2

2

1

1

00

00

00

00

=

y

x

y

x

y

x

y

x

d

d

d

d

CS

SC

CS

SC

d

d

d

d

2

2

1

1

2

2

1

1

00

00

00

00

zyxxyz

Rotation Matrix


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Coordinate transformation matrix

x

y

z

xy

z

=

x

y

x

x

y

x

ff

f

ff

f

1

1

1

1

1

1

x

y

x y z

z

( )xx,cos ( )yx,cos ( )zx,cos

( )xy,cos ( )yy,cos ( )zy,cos( )xz,cos ( )yz,cos ( )zz,cos


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2D and 3D elements: Element stiffness matrix in global coordinate

{ } [ ] { }dTKTfe

T ][][=

[ ] ][][][ TKTK eT=


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Spring Elements

Global Stiffness matrix: direct stiffness matrix method

i j

Element I

)()(

)()(

Ijj

Iij

IijIii

kk

kk

[ ]

=K

1 i j N

1

i

j

N

)(I

ijk

)(Iijk

)(Iiik

)(I

jjk

N by N matrix

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Global Stiffness Matrix

Global Stiffness matrix: Properties

1. It is symmetric jiij KK =

2. The element on the diagonal of the matrix is always positive. 0>

iiK

3. The product of the i-th row of the global stiffness matrix and the

global displacement matrix gives the external force on the i-th DOF

of the system.

4. Kij is equal to the reaction force on the i-th DOF due to a unit displacement

on thej-th DOF whereas all the other DOFs are fixed.

5. The global stiffness matrix is singular.

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Boundary Conditions

Boundary Conditions

1 3 x

01 =xd

xF1

xd3

2

xd2

2

1k 2k

3P 2P

+

=

x

x

x

x

x

d

d

kkkk

kk

kk

F

F

F

3

2

2121

22

11

3

2

1

0

0

0

+

=

x

x

x

d

d

kkkk

kk

kk

P

P

F

3

2

2121

22

11

3

2

1

0

0

0X X X

X

X

How to find ?xF1 This method is applicable to other elements.

Why we need BC?

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Inclined or Skewed Supports

Sometimes, a boundary condition can be more conveniently applied in a local

coordinate.

In the local coordinate system: 03 =yd

In the global coordinate system:

x

y

x

y

1

2 3

=

y

x

y

x

dd

dd

3

3

3

3

cossinsincos

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[ ] [ ][ ] [ ] [ ] [ ]

[ ][ ] [ ][ ] [ ]

=

3

2

1

3333233

3232212

1211

3

2

1

d

d

d

TKTKT

TKKK

KK

F

F

F

T

T

Frames Inclined or Skewed Supports

=

3

3

3

2

2

2

1

1

1

3

3

3

2

2

2

1

1

1

y

x

y

x

y

x

y

x

y

x

y

x

d

d

d

d

d

d

XXXXXX

XXXXXX

XXXXXXXXXXXXXXX

XXXXXXXXX

XXXXXXXXXXXXXXX

XXXXXX

XXXXXX

M

F

FM

F

FM

F

F

=0

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( ) ( )= L

ddistribute xdxvxwW0

Beam Elements

Distributed loads: Work Equivalence Method

22221111 mdfmdfW yyyydiscrete +++=

( )= L

xdNxwm0

21( )=

L

y xdNxwf0

11

( )= L

y xdNxwf0

32 ( )=

L xdNxwm0

42

Note, x refers to local coordinate, it takes values between 0 and L.

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