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oundary & 'nitial!onditions

inear elastic material Three partial dierential equations in

displacements

Second order in each coordinate Second order in time

*n a free surface in each direction Specify stress or displacement but not both

'nitial conditions for each directionspecify %isplacement and velocity

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Surface +orces

Specify pressure (,,) shears (,. ,y)or

Specify displacements (uv$)

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/igid ody 0otion 1 2%

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/igid ody %isplacements 12%

Strains vanish

'ntegrating normal strains

'ntegrating shear strain

ence (U, V are constants)

%isplacement solution

,0,0,0 =

+

==

==

=

x

v

y

u

y

v

x

uxyyyxx

( ) ( )xgvyfu == ,

( ) ( ) ( ) ( ) constant&or ====+ xgxgxgyf 00

( ) ( ) ( ) xVxgyUyfyf +=== & ,,

( ) ( ) xVyxvyUyxu +== ,,,

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/eactions $ith E.cessive!onstraints

E.tra constraint in hori,ontal direction $ill adde.cessive stress4ertical constraints and loads produce point loadin5nite stresses

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/igid ody %isplacements 13%

6ll strains vanish

'n terms of displacements

'ntegrating yields displacements

7here

6nd

0,0 ===== zxyzxyzzyyxx

kwjviuu ++=

rUu +=

vectorconstant=++= kWjViUU

vectorconstant=++= kji zyx

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Self-Equilibrating +orces

E.amples include8 "niform pressure (submarine or bathysphere)

Thermal e.pansion

!s remove rigid body translations &rotations

!onstrain si. degrees offreedom (3 dofs at one

point 2 dofs at a second

and 9 dof at a third)

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:late & eam %ofs at Each;ode

eam 1 3translations

3 rotations

:late 1 3translations

2 rotations

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;astran +E !ode 1 :lateElements

6ll nodes for all elements types have si.dofs

3 for translation

3 for rotation +lat plate models need dofs perpendicular

to plane of model constrained (set to ,ero)

Shells made of plate elements do not Solid elements need all three rotations at

each node set to ,ero

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Simply Supported eamE.ample

+i. si. dofs 1 < translation and 9 rotation

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Simply Supported eamE.ample

+i. si. dofs 1 < translation and 9 rotation

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!antilever eam

+i. si. dofs 1 3 translation and 3 rotation

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'nternal Surfaces & !rac#s

!rac#s 'nternal Surfaces

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ert, !ontact - Gaps &+riction

ert, !ontact Gap & +rictionElements

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Transformations

ZZ

YXYYXX

=

+=+=

'

cossin'sincos'

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"se of Symmetry

0a#es a large problem smaller

6.isymmetry reduces a 3% problemto 2%

/ecall stress & strain symmetric

E.amples8

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:eriodic oundary!onditions

Stress & Strain are periodic

0ean displacements can varylinearly $ith coordinates due toe.pansion and rotation

+or

01

011

vv

uCu

=

+=

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0ulti-:oint !onstraints -Tying

=

=

mmnmnn

m

m

d

d

d

x

x

x

ccc

ccc

ccc

DCX

......

...

............

...

...

2

1

2

1

21

22221

11211

$here .iare speci5ed degrees of freedom

ci=d=are #no$n constants>

or

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%istant oundary !onditions

uild a su?ciently large model

6t least 2@ times length of largestdimension of interest

Substructure a large coarse model

"se output from large model as input toa re5ned local model

"se super-elements or substructuring

"se in5nite elements ($hen available)