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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.