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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)
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