lecture 5 schauwecker
TRANSCRIPT
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Pages561578Finites
Element
Procedure,
Klaus
Jrgen
Bathe
1. Example6.18
2. Displacement/PressureFormulationsforLarge
Deformations
3. StructuralElements:
BeamandAxisymmetricShellElements
4. Example6.20
5. StructuralElements:PlateandGeneralShellElements
16.10.2009 1
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Example6.18
Fournodeplanestrainelementin
largedisplacement/large
strain
conditions
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Quest:Establishthematrices
CorrespondingtoTLusingTable6.5
16.10.2009 2Example6.18
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Reminder
TL
Total
Lagrangian
formulation
Toget:
Weneedtoknow:
16.10.2009 3Ex
ample6.18
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Table6.5
Onlyfor3Daxisymmetricelements
16.10.2009 4Example6.18
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QuadrilateralElements
16.10.2009 5Example6.18
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16.10.2009 Example6.18 6
r=1 r=+1
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Table6.5
tu11=1
tu12=0.5
16.10.2009 7Example6.18
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16.10.2009 8Example6.18
...
...
...
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Table6.5
16.10.2009 9Example6.18
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6.4Displacement/Pressure
Formulationfor
Large
Deformations
16.10.2009 1
(Page
561
567)
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WhyDisplacement/PressureFormulationsfor
LargeDeformations?
Apuredisplacementbasedprocedureisnoteffectivefor(almost)incompressibleanalysis
Materialinlargedeformations(likerubberlike
materialsor
metals
under
large
strain)
behave
almostimcompressible
Thereforetotal
and
updated
Lagrangian
formulationsshouldbeextendedtoincompressibleanalysis
16.10.2009Displacement/Pressure Formulations for
LargeDeformations 2
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TotalLagrangianFormulation
puredisplacement
based
16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 3
Equationofmotion(based
on
the
principle
of
virtualdisplacements):
Wedefine
the
incremental
potential*:
*Theoverbar denotesthatthatonlydisplacementfieldswhereconsidered(notpressurefields)
Nowtheprincipleofvirtual
displacement
can
be
rewritten
as:
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Displacement/Pressure
TotalLagrangian
Formulation
16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 4
TotalPotential
Potentialofthedisplacements
Potentialofthepressure
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16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 5
Pressureofthedisplacementfield
Totalelementpressureattimet
bulkmodulus
PotentialQ
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Bulk
Modulus
Thebulkmodulus(Kompressionsmodul)isdefined
as
the
pressure
increase
needed
to
causeagivenrelativedecreaseinvolume
16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 6
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Linearization
16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 7
VectorsFU, FPcontainstheseElements
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16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 8
MatricesKUU, KUP,KPP,KPUcontainstheseElements
Linearization
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16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 9
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16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 10
Displacement/Pressure
UpdatedLagrangian
Formulation
TotalLagrange:
UpdatedLagrange:
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16.10.2009Displacement/PressureFormulationsfor
LargeDeformations 11
InnovationinULformulation
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6.5.1StructuralElements:
Beam
and
Axisymmetric
Shell
Elements
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(Page568 572)
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16.10.2009 BeamandAxisymmetricShellElements 2
ThevectorsVs andVt definetheOrientationofthecrosssectionofthebeam:theyarenormaltotheaxisofthebeamandtoeachotherThe
values
aand
bdefine
the
size
of
the
cross
section
of
the
beam
DimensionsoftheBeam
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16.10.2009 BeamandAxisymmetricShellElements 3
Displacement u/IncrementalDisplacementu
t
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16.10.2009 BeamandAxisymmetricShellElements 4
VectorVt andVsFEMI:
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Problem:LargeRotation
16.10.2009 BeamandAxisymmetricShellElements 5
>thereforesecondorderhastobetakenintoaccount(higherordertermsareneglected)
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Result
16.10.2009 BeamandAxisymmetricShellElements 6
NowtheEquationcanbesolved
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Example6.20
TwoNodeBeamElementinLargeDisplacementandRotation
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Evaluatethecoordinateanddisplacementinterpolationsandderivatesthatarerequiredforthe
calculationof
the
strain
displacement
matrices
of
theULandTLformulations
16.10.2009 Example 6.20 2
Firstwehavetocalculate:
q = 2
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16.10.2009 Example6.20 3
q 2
TwoNodeElement:
=0 =0 =0
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16.10.2009 Example6.20 4
Coordinates
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16.10.2009 Example6.20 5
Displacement
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16.10.2009 Example6.20 6
IncrementalDisplacement
second
order
terms
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16.10.2009 Example6.20 7
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16.10.2009 Example6.20 8
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16.10.2009 Example6.20 9
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16.10.2009 Example6.20 10
Derivatives
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16.10.2009 Example6.20 11
Derivatives
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6.5.2StructuralElements:
StructuralElements:PlateandGeneralShellElements
16.10.2009 1
(Page575 578)
PlateandGeneralShellElements
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16.10.2009 PlateandGeneralShellElements 2
Wenowextendthetheoryforbeamsforgeneralshellelements
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Rotation
16.10.2009 PlateandGeneralShellElements 3
NormalVector
and aretheRotationoftheVector about and
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6 Pl d G l Sh ll El
Note:theprocedureforcomputingthedisplacements,etc.isnowthesameasalways.ButasseeninFEMItherecanbeproblemswithshearlocking.ThereforeMixedInterpolationshouldbeconsidered.