Mindlin Plate Theory and Abaqus UEL Implementation

97 Pages·2017·4.1 MB·English

NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
Mindlin Plate Theory and Abaqus UEL
Implementation
Page1of97
NikhilPadhye
[email protected]
SubodhKalia
[email protected]
2
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
Contents
1 MindlinPlateTheory 7
1.1 Stressandstraincomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Shapefunctionsandisoparametricelementformulation . . . . . . . . . . . . . 13
1.3 ComputationofJacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Calculationofstressandstrainatintegrationpoints . . . . . . . . . . . . . . . . 18
1.5 Principleofvirtualworkandderivationofelementstiffnessmatrix . . . . . . . 24
2 SampleHandCalculations 29
2.1 BendingStiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 ShearStiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 SomeConsequencesofMindlinPlateTheory 37
4 ABAQUSUELImplementation 40
4.1 UELinputvariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 UELoutputvariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 PseudocodefortheUELsubroutine . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 BenchmarkingofUELImplementationforMindlinPlateElement 50
5.1 Singleelementtest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Testingmultipleelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Acknowledgments 59
Page3of97
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
List of Figures
1 W,Landt representthewidth,lengthandthicknessoftheplate,respectively. 7
2 Rotation of a material line (normal to the neutral plane in the undeformed
configuration),aboutY-axisinanti-clockwisedirectionwhenviewedfromthe
positiveY-axis,anddenotedbyθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
y
3 Isoparametricformulationfor4-nodedelementusing2x2quadintegration. . . 14
4 Four-nodedelementwithintegrationpointsinisoparametricformulation.. . . 30
5 UELsubroutineheaderfromAbaqus2016documentation,AbaqusUserSub-
routinesReferenceGuide,Section1.1.28. . . . . . . . . . . . . . . . . . . . . . . . 40
6 Input-OutputblockdiagramfortheUELsubroutine. . . . . . . . . . . . . . . . . 41
7 Boundaryconditionsforasingleelementtest. . . . . . . . . . . . . . . . . . . . . 50
8 ReactionforcesinZ-directionwithAbaqusS4element. . . . . . . . . . . . . . . 51
9 ReactionmomentsinX-directionwithAbaqusS4element. . . . . . . . . . . . . 52
10 ReactionmomentsinY-directionwithAbaqusS4element. . . . . . . . . . . . . 52
11 PlotsforreactionforcesinZ-directionatthenodeswithrespecttonodaldis-
placement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
12 PlotsforreactionmomentsinX-directionatthenodeswithrespecttonodal
displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
13 Plots of reaction moments in Y-direction at the nodes with respect to nodal
displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
14 PlatedeflectionfromMatlabcode. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
15 PlatedeflectionfromAbaqusS4elementanalysiswithascalefactorof106. . . 57
Page4of97
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
16 PlatedeflectionfromUELwithascalefactorof106.. . . . . . . . . . . . . . . . . 58
17 Comparisonofdeflectionsforalaterallyloadedplateclampedattheedges. . . 58
Page5of97
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
ABSTRACT
Inthisreport,wepresentamodelforplatebendingbasedontheMindlinplatetheoryfor
smallelasticdeformations. Therelatedfieldandconstitutiveequations,andfiniteelement
discretization (FEM) at an element level are also presented. The implementation of the
Mindlin plate element is carried out for the UEL module of ABAQUS. The overall goal of
thistechnicalreportistofacilitatetheunderstandingandimplementationoffiniteelement
discretizationfortheUELmoduleofAbaqus,sothatuserscandefinetheirownelements.The
overallproceduredescribedinthistechnicalreportisexpectedtoaidtheimplementation
ofcustomdefineduserelementsforABAQUS,atopicthatiscurrentlynotdiscussedwell
intheopinionoftheauthors. Forexample,otherthanofficialAbaqusdocumentation[1],
only one comprehensive documentation for UEL in Abaqus is available online [9]. The
currentdemonstrationiscarriedoutforsmalldeformations,forthesakeofsimplicity,and
applicationsinvolvingotherplate/shelltheories(alongwithlargedeformationorrotation
effects)willbedoneinthesubsequentreportsandpublications1. Althoughbenchmarking
onsimpletestproblemsisperformedinthisstudy,detailedanalysesontheperformanceof
currentplateformulationsuchaspatchtestperformance,shearlockingissues,etc. arenot
discussed. Readerisreferredto[3],[7],[4],[2]and[6]fordetaileddiscussionsonthesetopics,
includingmoresophisticatedfiniteelementmethods.
1Thereadersareencouragedtocontacttheauthorsincasetheyneedanyfurtherclarification.Useofthis
documentoranycontent/codepresentedhereforanycommercialpurposesisstrictlyprohibited.Readersare
welcometocitethisreportifitisfoundusefultotheminanyway.
Page6of97
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
1 MINDLIN PLATE THEORY
Top surface
W
z L
t
y
Mid surface
x
Figure1:W,Landt representthewidth,lengthandthicknessoftheplate,respectively.
A plate is commonly identified as a thin flat structure as shown in Figure 1. In order
to qualify as a plate, we require that the thickness (t) of the structure is small compared
toitslength(L)andwidth(W),i.e. t <<L,W. Thisassumptionsimplifiestheunderlying
governingequilibriumequationsandalsoallowsforseveralsimplificationsonthekinematics
ofdeformationduetothe“thin"natureofthestructureathand. TheMindlinplatetheory
canaccountforhomogeneousthroughthicknesssheardeformations(asopposedtoanother
elementarytheoryofKirchhoff-Loveplates).
Thetwokeyfeaturesofthistheoryare: (a)weassumebyconstructionthatσ =0,and
z
(b)sincethisisafirstordersheardeformationtheory;therefore,allowingσ andσ tobe
xz yz
non-zero. Weestimateσ ,σ fromγ ,γ whichthemselvesaretakentobenon-zeroand
xz yz xz yz
constantthroughthethicknessoftheplate. Membraneeffects(i.e. inplanestretchingofthe
platemid-surface)arenotincludedinthistheory. Someotherconsequencesofsuchad-hoc
Page7of97
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
assumptionsmadeinthistheoryarediscussedlaterinSection3.
1.1 Stressandstraincomponents
ThedeformationintheMindlinplatecanbecharacterizedbythedisplacementofthemid-
surface of the plate in the Z-direction (denoted by w), and rotation of the normal to the
mid-surface(denotedbyθ andθ )aboutX-axisandY-axis,respectively. Thuswerequire
x y
threedegreesoffreedomw,θ ,θ atanypointonthemid-surfaceoftheplatetodescribethe
x y
completedeformation.
LetusconsideramateriallineL L ,acrossthicknessoftheplate,perpendiculartothe
1 2
neutral plane in the undeformed configuration as shown in Figure 2. Upon deformation
we assume that points on the material line L L still remain on a straight line l l in the
1 2 1 2
deformedconfiguration.ThemateriallineL L isrotatedbyanangleθ intheanti-clockwise
1 2 y
directionwithrespecttotheY-axis. Forsmalldeformations,thedisplacementofanymaterial
pointlyingonthelineL L ,relativetothepoint‘O’onthemid-surface,intheX-direction,
1 2
canbeapproximatedas2:
u =zθ , (1)
x y
TheextensionalstrainintheX-directionisgivenas:
(cid:178) = ∂ux =z∂θy . (2)
x ∂x ∂x
2Wehaveassumednomembraneactionontheplate,i.e.,noin-planestrainsonthemid-surfaceofthe
plate,thereforethemid-surfacedisplacementdoesnotvaryspatially,nordoesitplaysanyroleinstrainsacross
thicknessasitshallmerelyenterasaconstantsumtou inequation 2.
x
Page8of97
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
In the above equation, z indicates the coordinate of a material point along Z-axis in
theundeformedconfiguration,andforthesakeofclaritywerepeatthatθ ismeasuredin
y
anti-clockwisedirectionaboutY-axiswhenobservingfrompositivedirectiontothenegative
direction.
Itisworthhighlightingthatarathertacitassumptionhasbeenmadeinthisformulation
(while locating a material point along the thickness direction) - regarding no change in
thickness(implying(cid:178) =0),whichseemsinconsistentwiththepriorassumptionofσ =0. In
z z
opinionoftheauthors,itisquiteimportanttounderstandthelimitationsofsuchclassical
plate/shelltheories(evenwiththeirmoreelegantextensionsasmadeintheliterature),since
theycanleadtoratherspuriouspredictionsinscenarioswheretheunderlyingassumptions
arequiteanoverreach.
Followingasimilarprocedureonecanestimateu =−zθ . Pleasenotetheminussignin
y x
theestimationofu ,whichwasnotpresentinthecaseofu . Thus,(cid:178) isgivenas
y x y
∂u ∂θ
(cid:178) = y =−z x (3)
y ∂y ∂y
andaccordinglytheshearstrainsintheXY-planearegivenas
(cid:183) ∂θ ∂θ (cid:184)
γ =z − x + y . (4)
xy ∂x ∂y
Theshearstrainsacrossthethickness(γ andγ )canbecalculatedas
xz yz
∂u ∂u
γ = x + z , (5)
xz ∂z ∂x
Page9of97
NikhilPadhye,SubodhKalia ImplementationofMindlinplateelement
N
θ
y
L
1
‘O’
L
2 z
Top surface
N
z L
1
y
u = z θ
x y
‘O’
x
L
2
Center line
Bottom surface
Figure2:Rotationofamaterialline(normaltotheneutralplaneintheundeformedconfiguration),
aboutY-axisinanti-clockwisedirectionwhenviewedfromthepositiveY-axis,anddenotedbyθ .
y
Page10of97