DrAlex FEM Introduction

7/27/2019 DrAlex FEM Introduction

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19 October, 2013

Introduction to

Finite Element Analysis

for Skeletal Structures

Dr. Nick A Alexander 2006

Department of Civil Engineering

University of Bristol


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Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol

General Structural Problem

For the following structure

we want to compute

Bending moments, shear

forces, axial forces

Bending, shear and axial

stresses and strains Deflections


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Finite Element Analysis (FEA),

The basic idea Complex structures systems are often too complicated to simply derive

relationships between applied loads, deflections and internal stresses.

Hence large structures are divided up into many individual f ini te elements;that have a much simpler structural form,

e.g. a beam or column

The relationship between load, displacement, stressesand strains in af ini te elementcan be determined

Thus, it is computationally possible for a complex structure to be modelledby assembling (aggregation) many individual f ini te elements. Theaggregation process must satisfy equilibrium and continuity.


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Restraints

Modelling Idealisation

Nodes

connection points(not necessarily

hinges)

Elements

Loads


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For afinite element, we need to derive

the relationship between

External Loads

Deflections/deformations

Internal stresses and strains


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Derivingforce-displacement

relationship for general finite element.

Five basic steps

(i ) Conjecture a displacement function

(ii) Use nodal boundary conditions

(ii i) Derive strain

displacement relationship(iv) Derive stressdisplacement relationship

(v) Use principle of Vi r tual Work


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Bar Element example

f1f2

u1 u2

x

Node

(a hinge)Element

Deformed shape


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Bar Element example

(i ) Conjecture a displacement function

)1(1

2

1

21 a

a

a

xxaaxu N

x

u(x)


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Bar Element example

(ii) Express u(x)in terms of nodal displacements by

using boundary conditions.

2

1

2

1

1

01

a

a

Lu

u

u(0) = u1 u(L) = u2

)2(au A

Deformed shape


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Bar Element example

Sub (2) into (1)

uL

xuxu

1

1

1

01,1

AN

)3(,1 uuLx

Lxxu C

Displacement polynomial that satisfies boundary conditions


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Bar Element example

(ii i) Derive strain-displacement relationship by using

mechanics theory

)4(11

uLLuudx

d

dx

du

x

BC

Axial Strain


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Bar Element example

(iv) Derive stress-displacement relationship by using

elasticity theory

)5(uExExB

Axial Stress

Elastic Modulus


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Bar Element example

(v) Use principle of Vi r tual Work

TT

udxBBuEAdxuBuBEA

dxxxA

dydzdxxxdxdydzxxWI

.

.

..

Internal work

fu

f

f

uuWT

E

2

1

21External work

Bar cross-sectional areaAWork = StressxStrainx Volume


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Bar Element example

Equate internal and external work

TTT udxBBuEAfu

WWIE

)6( dxBBEAufT

k,k

Stiffness matrix


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Bar Element example

Resul tant sti f fness matr ix

)7(11

11

0 11

11

11

0 1

1

22

22

L

EAdxEA

dxEA

L

LL

LL

LL

L

L

L

k


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Assembling issue (1); Element coords.

Element axes are not all the same.

So there is a need for a coordinate transformation

XY

Z

Global axes


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FX

FZ

q

v

u

Z

X

f

f

F

F

qq

qq

cossin

sincos

fF

f

f

f

f

F

F

F

F

v

u

v

u

z

x

z

x

R

2

2

1

1

2

2

1

1

cossin00

sincos00

00cossin

00sincos

qq

qq

qq

qq

fu1fv1

fv2

fu2

Fx1

Fz1 Fx2

Fz2

for forces

q (8)


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Similarly for displacement

u1v1

v2

u2

X1

Z1 X2

Z2

uU

v

u

v

u

Z

X

Z

X

R

2

2

1

1

2

2

1

1

cossin00

sincos00

00cossin

00sincos

qq

qq

qq

qq

q

(9)


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Element force-displacement in global coordinate

2

2

1

1

2

2

1

1

0000

0101

0000

0101

v

u

v

u

L

EA

f

f

f

f

v

u

v

u

uf k

1,, RRRkRKKUF

Local coordinates

Global coordinates

Element nodal forces and displacements in

local coordinates

Element stiffness matrix

in local coordinates (extended)

Element stiffness matrix

in global coordinatesElement nodal forces and displacements

in global coordinates


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Element sti f fness matrix in global coordinates

qqqqqq

qqqqqq

qqqqqq

qqqqqq

22

22

22

22

1

sinsincossinsincos

sincoscossincoscos

sinsincossinsincossincoscossincoscos

L

EARkRK

(10)


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Assembling issue (2); Structure matrix

Element and nodal number ing

Element

number

1

23

Node number

1

2

3

0000

0101

0000

0101

1 LEAK

1010

0000

1010

0000

3 LEAK

45

90

L m

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

2 LEAK

P [kN]


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Create Structure sti ffness matr ix

from element sti ffness matr ices

3

3

1

1

3

3

1

1

0000

0101

0000

0101

Z

X

Z

X

F

F

F

F

LEA

z

x

z

x

3

3

2

2

1

1

3

3

2

2

1

1

0000

0101

0000

0101

Z

X

Z

X

Z

X

F

F

F

F

F

F

LEA

z

x

z

x

z

x

1,31,1

3,1 3,3


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3

3

2

2

42424242

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

3

3

2

2

Z

X

Z

X

F

F

F

F

LEA

z

x

z

x


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3,2

3

3

2

2

1

1

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

3

3

2

2

1

1

00

101

0000

0101

Z

X

Z

XZ

X

F

F

F

FF

F

LEA

z

x

z

x

z

x

3,3


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3

3

2

2

1

1

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

3

3

2

2

1

1

00

101

110

00

001010

010001

Z

X

Z

X

Z

X

F

F

F

F

F

F

LEA

z

x

z

x

z

x

2

2

1

1

2

2

1

1

1010

0000

1010

0000

Z

X

Z

X

F

F

F

F

LEA

z

x

z

x


2,11,1

2,1 2,2


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bbba

abaa

kkb

kka

ba


00

0

00

bbba

abaa

kkb

kka

ba

ith Elements

Stiffness Matrix with

Node numbers a and b

a

b

i

ithElementith Element in Structure

Stiffness Matrix with

Node numbers a and b


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3

3

2

2

1

1

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

24

2

4

2

4

2

4

2

3

3

2

2

1

1

00

101

11000

001010

010001

Z

X

ZX

Z

X

F

F

FF

F

F

LEA

z

x

z

x

z

x


Structure stiffness matrix

vector of nodal displacementsvector of nodal forces

sss UF K


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Assembling issue (3); Supports

How to deal with the problem of supports (restraints) ?

These are nodes where the displacements are known, zero in

the perfectly rigid support case.

3

3

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

4

2

2

2

1

1

0

0

0

0

00

101

110

00

001010

010001

0

Z

X

P

F

F

F

F

LEA

z

x

z

x

known applied nodal loads P unknown nodal

Displacements D

unknown

support

Reactions

R

knownsupport

Displacements

Ds


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Assembling issue (3); Supports

This system of equations needs to be partitionedto determine

the unknown nodal displacements and support reactions.

D

D

sKK

KK

P

R

2221

1211 DD

DD

2221

1211

KKP

KKR

s

s

D

221

1

1

4

2

4

2

4

2

4

21

22

3

3 01EAPL

EAL

PPK

Z

X

D

P

P

P

K

F

F

F

F

EAPL

LEA

z

x

z

x

000

01

221

1

4

2

4

2

4

2

4

212

2

2

1

1

Determine

nodal

displacements

Determine

support

reactions

sKPK DD

21

1

22

Solve system of algebraic equations

key numerical process


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Finishing off; calculate element

internal stresses, strains and actions. Calc. nodal displacements in local coordinates, eqn (9)

Calc. element strains, equation (4)

EA

Pux

EAPLLL

011B

2

2

1

1

1000

01000010

0001

221

10

0

v

uv

u

EAPLFor

Element 1

EAPLu

u 0

2

1

Negative sign indicates

Compression

For

Element 1


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Finishing off; calculate element

stresses, strains and actions. Calc. element stresses, equation (5)

Calc. element axial forces

AP

xEx Negative sign indicatesCompression

PAxxF

For

Element 1

ForElement 1


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Summary of FEA

Divide complex structure into

many finite elements

connected at nodes

Compute

Element Stiffness

Matrices (global Coords.)

Assemble

Structure Stiffness Matrix

and applied load vector

Introduce Supports

(partitioning)

Solve partitioned system

determine reactions and

nodal displacements

Compute element

nodal displacements

in local coords

Compute element internal

stresses, strains and actions

*

* key numerical process

DrAlex FEM Introduction

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