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1

Engineering Mechanics 425

Introduction to Finite Element Analysis

Instructor:

T. J. Rudolphi

Professor, Engineering Mechanics

epart!ent of "erospace Engineering

2### $o%e $all

E!ail: rudolphi&iastate.edu

Phone: 2'4())'5

mailto:rudolphi@iastate.edumailto:rudolphi@iastate.edu

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2

*hat is finite ele!ents an+%a+

-inite ele!ents, or the finite ele!ent !ethod, is a co!putationaltechniue to sol/e differential euations( 0oundar+ /alue pro0le!s,initial /alue pro0le!s, eigen/alue pro0le!s.

Elastic defor!ations, stress anal+sis, structural anal+sis

i0rations and /i0rational anal+sis of structures

Ther!al anal+sis and heat transfer

"coustics, %a/e propagation pro0le!s

Electro!agnetic fields, stead+ and transient

round %ater flo%, flo% through porous !edia

-luid flo%, aerod+na!ics

3atural s+ste!s, 0iological s+ste!s anal+sis, population d+na!ics Econo!ic s+ste!s anal+sis

*eather prediction

Etc., etc, etc.

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#

)

:0artheofpieceele!entalanof!Euili0riu

=++= + xx

xx dfxpxxpF

xxxxfxpxxp +=++ ,)

:gettointegralon thetheore!!ean /aluethe6se

.andofin ter!s!euili0riuofeuationaldifferentitheis%hich

)

)

:)asli!ittheta7eand0+i/ide

xfxp

fdx

dp

fx

xpxxp

xx

=+

=++

"n Introductor+ E8a!ple: 9ne i!ensional Elastic efor!ation

lengthunitperforceapplied

section(crosson theforceinternalresultant

!oduluselasticstress,

strain

atpointaofntdisplace!e

coordinateglo0al

:3otation

======

===

f

AAp

EE

dxdu

xu

x

xxp +

x

P

f AE

x xx +

xp

f

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4

./ar+ %ithcouldandthat3ote

.ntdisplace!etheofin ter!s!euili0riuofeuationaldifferentiordersecondtheis%hich

)

iseuationaldifferentitheand

so,is%hichla%,s$oo7euseThen

xEA

u

fdx

du

AEdx

d

dxduAEAp

dxduEE

=+

====

,,

:ends0othatntdisplace!ethee/aluateand

hold,sassu!ptiona0o/ethe%here,880ar,theofele!entpieceaE8a!ine

2211221

2

2)2

1

2

211

2

1)21

1

21

xuuxuuCxCxfAEu

CxCxfAEu

x

++=

++=

21

2

)2

1

1)

)

)

:Integrate

constant.isandconstantin %hichcase-or the

CxCxfAEu

Cxfdx

duAE

fdxduAE

dxd

AEff

++=

+=

=

=

2p

1xx= 2xx=

1p

)f

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5

+

=

2

2

2

1)

2

1

2

1

2

1

21

1

:aseuationsthesecan %riteonefor!,!atri8In

x

xf

u

uAE

C

C

x

x

lengthele!ent,2

11

2111

21

1

:andforsol/ed0ecan%hich

12

21

21)

2

1

12

2

2

2

1)

2

1

1212

2

2

2

1)

2

1

1

2

1

2

1

21

=

+

+

=

+

=

+

=

xxlxx

xxf

u

u

xxl

AE

xxf

uuAE

xxxx

x

xf

u

uAE

x

x

C

C

CC

[ ]

[ ]

+

+

+=

+=++=

21

21)

2

1

12

2

)21

2

12

)21

21

2

)21

21

2

111

1

:5foreuationtheinto0ac7andfor/aluesPut the

xx

xxf

u

u

xxl

AExxf

C

CxxfCxCxfAEu

xuCC

2p

1xx= 2xx=

1p

)f

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;

[ ] ( )

( )

+=

==

2

or

22

11

:isforceinternalthe"lso,

21)12

12)

2

1

xxxfuu

l

AExp

xxxfu

u

lAE

dxduAExp

p

( )

( )

+=

+=

==

2

2

:resultsthere,andate/aluatedisforceinternaltheIf

212)1222

211)1211

21

xxxfuu

l

AEpxp

xxxfuu

l

AEpxp

xxxxp

[ ] ( )

[ ] ( ) 2

1

or,

2

:isele!entin thentdisplace!efor thee8pressionthe-inall+,

21)

2

1

12

21)

2

1

12

xxxxAE

f

u

uxxxx

lxu

xxxxf

u

uxxxx

l

AExAEu

+=

+=

2

p

1xx= 2xx=

1p

)f

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