Moving Into the Nonlinear World With FEA by Desktop Engineering

7/28/13 Moving into the Nonlinear World with FEA by Desktop Engineering

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Sunday,7.28.2013

TechnologyforDesignEngineering

MovingintotheNonlinearWorldwithFEAAprimeronthemaintypesofnonlinearity,whenandhowtousethem.

byTonyAbbey|PublishedNovember1,2012

Mostengineerstraditionallymigratedfromthelinearfiniteelementanalysis(FEA)worldintothenonlinearworldwhenfacedwithstresslevelsabovetheelasticlimitinotherwords,plasticity.Thesedays,itismorelikelythatcomponentsinanassemblymakecontactwhenloadedorunloaded,andyouneedtomodelthatsituation.Nowyouareoffdownthenonlinearpath!

Ihadacolleaguewhosteadfastlydeclared,Theworldisnonlinear.Hewasquiteright,butwewouldmuchpreferwhendoingFEAtokeepthingsassimpleaspossibleandignorethatfact.Alotofthetimewecangetawaywiththis,butsometimeswecant.

AfteraquickdabbleinnonlinearFEA,alotofengineersunderstandthereasonforthecoyness:Nonlinearanalysisistoughtodoeffectivelyandefficientlyitisasteeplearningcurve.

TheNonlinearStrategy

Whenwecarryoutanonlinearanalysis,wearetakingajourneyintotheunknown.Fig.1showsatypicalnonlinearhistory.Noticehowtheloadisnowbrokendownintosmallersteps.Wecantjustapplythetotalandhopewegetagoodresult.Wehavetotiptoeuptheloadscale.

Hereweseetwoloadincrements:upto10%andupto20%.Inpractice,thesemaybeaslowas1%incrementsforahighlynonlinearproblem.

Ateachloadincrement,thesolverhastoiteratewithinthesolutiontofindaloadbalance.Thefirstapproximationisalinearanalysis,soitwillnotbeinbalanceiftheresponseisnonlinear.Thesolvertransformsthesolutionintoaonedimensionalsearchpath,lookingforthisbalancepoint.

Fig.1:Nonlinearstrategy.

Oncethefirstbalancepointisfound,wehavethefirstknownpointinourjourney.Wecanthencarryoutanotherexploratorylinearanalysisandseewherethattakesus.Again,itisanapproximationandwehavetoiteratetofindthenonlinearbalancepoint.Thejourneycontinues,findingeachbalancepointuptothefull100%loading.Wehavethenestablishedthenonlinearresponseateachpointinthestructure.

Thecostofdoingthesesuccessiveiterationscanbequitesignificant.Ifittakes10steps,witheachoneneedingafullstiffnessmatrixupdateandsolution,thatmeanstheanalysisis10timesmoreexpensive

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needingafullstiffnessmatrixupdateandsolution,thatmeanstheanalysisis10timesmoreexpensivethananequivalentsizedstaticsolution.Therearemanystrategiestomakethisupdatingasparsimoniousaspossible,soitisnotquitesosavageascalingeffectoncost.Butforahighlynonlinearstructure,thereisoftennootheroptionthantoacceptthecost.

Theconvergencetoabalanceateachstepcanbethecauseofmuchheartache.Ifweusetoolargeastep,orhavehighlynonlineareventssuchaslargecontactchanges,bigchangesinthematerialslopeorabruptcollapseduetononlinearbucklingorsnapthrough,thesolvercanhaveatoughtimeseekingabalance.Wemayevenfindthatthereisnophysicallyrealizablesolutionasinthecasewithacollapseload.

Thisleadsustooneofthemostimportanttips:Simplifythenonlinearityasmuchaspossible.Ifyouhavecontact,friction,geometricnonlinearityandplasticityallgoingonatonce,takeastepbackwardandintroducetheseeffectsoneatatime.Youmaybondthecontactsurfacestogetherforsimplicity,andmakethemateriallinear.Nowyoucanshakedownthegeometricnonlinearityandgetthattowork.Evenhere,youmaywanttostartwithasimplermodeltoestablishthephysicsofthestructuralbehaviorandworkupfromthere.

MaterialNonlinearity

Whenthestresslevelinacomponentexceedstheyieldpoint,thematerialintheaffectedzonestartstogoplastic.Thepresenceofplasticitymeansthematerialisfollowinganonlinearstressstraincurve,typifiedbytestresultsshowninFig.2.Somematerialsshowadistincttransitionfromlinearelastictononlinearplasticataclearlydefinedyieldpointothersshowaslowdriftoffthelinearcurve.Fig.3showshowtodealwiththedriftanarbitrarylineat0.1%or0.2%strainisdrawnparalleltotheinitialslope,andtheyieldpointistakentobewherethiscrossesthematerialcurve.

Fig.2:Variousmetalstressstraincurves.

WeusuallyhavetosimplifytheactualstressstraincurveinanFEAmaterialmodel.TheinputtoFEAcanberationalizedbyanelasticlinearsection,whoseslopematchesthelinearmaterialstiffnessE,andaplasticnonlinearpart,whichcanbeaconstantslope(thetwoslopesaredescribedasabilinearfit),oravaryingslopedefinedbyadatatable.BothtypesareshowninFig.3.

Fig.3:FEAmaterialfittodata.

TheFig.3insetshowstheloadingandunloadingalongtheelasticandplasticcurves.Theunloadingoccursparalleltotheelasticcurve,andleavesalockedinstrainwhenfullyunloaded.Thisisthe

residualplasticstrainleftinthestructure,withassociatedpermanentset.

Ifanonlinearsolutionisused,allelementsinthemodelaremonitoredtocheckforvaluesaboveyield.Ifthisoccurs,theelementsintheaffectedregionshaveamodifiedmaterialstiffnessterminvoked,whichusesthenewslopeoftheelasticplasticcurve.

Ingeneral,weneedtotiptoealongthiscurvesothatstiffnessupdatesarecarriedoutslowly,andequilibriumenforcedaswego.

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equilibriumenforcedaswego.

Fig.4:Progressiveaxialstressdistribution.

Ifweadoptthisgradualincrementapproach,wecancapturethegrowthofaplasticzone.Itisimportanttorealize,however,thatastheplasticzonegrows,thestressflowinthecomponentandaroundtheplasticzonewillchange.Often,aplasticlobetypeshapeappears,andthengrows.Fig.4showsasequenceyieldstressfrontgrowtharoundaholeinaplateasloadisincreased.Fig.5showsthecorrespondingplasticstraingrowth.

Fig.5:Progressiveplasticstrains.

Wemayhaveasituationwhereonlyverylocalplasticityisseen,suchasatalocalnotchorotherstressraiser,orattheextremefiberpositionsofabeaminbending.Theanalysistendstobestableastheoverallstiffnesschangesaresmall.Conversely,astheplasticzonesspread,suchasinthefinalstagesoftheloadedholeinFig.5,largesectionsofthemodelareaffectedandtheoverallstiffness

changescanbesignificant.Thismakesitdifficultforthenonlinearsolvertohandlethesolution.

GeometricNonlinearity

LookingatthetentinFig.6,wecanseethewallshavebeendeflectedinwardunderthewindpressure.Internalbalancingloadsinthetentwalldevelopasitmovesintothisshape,andthecorrespondinginternalstressesaredominatedbymembraneorinplanestresses.Itwouldntmakesensetothinkofthetentinitsundeformed,flat,initialstate,tryingtoresistthewindpressure.Whenastructurehastodeformandtheloadscanonlybebalancedinthatconfiguration,thephenomenaiscalledgeometricnonlinearity.Noticethereisnomaterialnonlinearityhereitisalldowntotheloadinganddeflection.

Fig.6:Nonlineardeflections.

Contrastthiswithastiffbridgedeck,carryingitsownweightplustraffic,windloading,etc.Theloadswillbebeamedfromthebridgedecktothefoundationstructure,withverylittledeformationofthebridgedeck.Wecanignoretheinfluenceofdeformationstogetaloadbalancethisisalinearsolution.Manyyearsago,Ihadaclientwhotriedtoanalyzeatentwallwithalinearsolution.Itcantworkbecauseitneedstodeflecttotransmittheloads.

Ofteninpractice,wegettoamiddlegroundwhenwecometodeflectionsofthinwalledstructures,likeplates.ThelineartheoryweuseinFEAassumesthedeflectionsnormaltotheplateare,atmost,

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likeplates.ThelineartheoryweuseinFEAassumesthedeflectionsnormaltotheplateare,atmost,around25%to50%oftheplatethickness.Underthatassumption,pressureloadisbeamedfromthecentertotheedges,likethebridge.Ifdeflectiongoesbeyondthis,theloadtransferstartstopickupthetentlikemembraneorinplaneloading.Thestresssystemchangesfromallbendingandsheartobending,shearandmembrane.WeneedtomovetoanonlinearFEAtopickthisup.

Wecanseethegeometricnonlineartermcomingintoplayinasimpleexample.Fig.7showsarigidbarconnectedtoapivotatoneend.Alinearrotationalspringresistsmotionatthatend.Weapplyasmallforceatthefreeendofthebar,anditrotatesslightlyaboutthepivot.Themovementappliedtothebarbalancesthetorqueproducedintherotaryspring.Thebarmovement(forcetimeslength)equalsthetorquereactioninthespring.Wecanalsoestimatetherotationofthebaraboutthepivotbyknowingthetorqueandthespringrate.

Fig.7:Rigidrodandtorsionalspring.

Theimportantpointhereisthatwecarryouttheforcebalanceintheundeformedpositionasalinearapproximation.Thisisthebasisoflinearanalysis.

Asthebarrotates,themovementgetssmaller,becausethelineofactionoftheforcecreepsclosertothepivot.Ifweincludethiseffect,wearebalancingtherodandspringinthedeformedposition.Thisisanonlinearsolution.

Thelinearsolution...juststayslinear!Itdoesntcarehowabsurdthesolutiongets.Thinkaboutaforceof2E5Nappliedatthetip.Therotationisofftheclock.Wecouldextrapolateandcalculatearotationthatimpliestheroddoesfourorfivecompletepirouettesaroundthepivot.Itisanabsurdanswer,butweoftenseeexactlythatinalinearanalysisthatistryingtohandlenonlineargeometricresponse.Wecantmapthisbacktoaphysicallymeaningfulsituation.Wehaveovershotthelimitsofalinearanalysis,andneedtomovetononlinearanalysis.

Thenonlinearsolutionshowsadivergenceawayfromthelinearataround30ofrotation.Ifweapply2E5Nnow,wegetarotationofaround82clearlybalancedonlyinanearlyverticaldeformedstate.

Sometimes,analysiswithgeometricnonlinearityiscalledalargedisplacementanalysis.Idontlikethistermbecauseitbegsthequestionhowlargeislarge?Everyanalysisismaterial,loadingandconfigurationdependent,sothereisnorealanswer.

GeometricNonlinearLoadTypes

Thepressureforceinatentwallwillalwaysfollowthetentwalldeformation.Aninflatingtoyballoonchangesshapedramatically,butpressureisalwaysnormaltothesurface.Therodforcewasappliedandstayedintheverticalsense,whatevertherodrotation.Thisisanonfollowerload.

Itisimportanttoestablishwhattypeofloadingispresent.Gravityloadswillalwaysbenonfollower,butabearingloadfromanadjacentstructurecanbeeither.

Thesetupofafollowerforceisstraightforwardifitispressureloading.AllFEAsolversshouldbeabletoautomaticallyadjustthenormaldirectionunderdeformation.

Aloadappliedasapointforceistrickier.Thevectorassociatedwiththeforcedirectionhastobeupdated.Typically,asetofanchornodesisused.Asthesedeform,theywillupdatetheforcevector.However,ifnodesarebadlychosen,theforcevectorisslavedtotheseandcanresultinbizarrechanges.Itisbettertoconvertanypointforcestoapressuredistributedoverasmallpadofsurfacearea.Thisisgoodpractice,eveninalinearanalysis,toavoidspuriouslocalizedstresses.

ContactImplementationTheearlyimplementationsofcontactwerenothingmorethanasetofnonlinearsprings,asshowninFig.8.Thesewerereferredtoasgapelements.Theyarelittleusedthesedays,butillustratesomeoftheprinciplesstillused.Whenthegapisopen,thespringstiffnessisweak,likechewinggum.Whenthegapisshut,thestiffnessisthesameasthesurroundingmaterial.

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Fig.8:Originalgapelement.

Tocreateaclosedgap,thenodesAandBmustpassthroughoneanother.Thiscreatesaprobleminthatwearenotnowmodelingtherealsituation,bothnodesjusttouchingandthenmovingtogether.Onlybyhavingpenetrationcanwegetaresistingforce.Thisisbasicallythepenaltystiffnessmethodusedinmanycontactalgorithms.

Tomodelreality,itisreallyrequiredtomakecontactsbejusttouching,anddevelopthecorrespondingreactionforcesinamorenaturalway.Thisisdonebyanauxiliarysetofforcesintroducedtomakethecontactandformthereactionforces.ThismethodisknownastheLagrangianmethod,andisalaterdevelopment.

Manysolversactuallyuseamixofthesetwomethods.Thesearchforgoodstablemethodsstillcontinues,andwecanexpectnewsolverdevelopmentsoverthenextfewyears.

Thetechnologyhasmovedbeyondsimplegapsandnowcoverswholeregionsofcontactmesh.Nowarbitraryzonesofamodelcanbeassessedtoseewhethertheywillmoveintocontactorseparateastheloadingisapplied.Theterminologyofamastersurfaceandslavesurfaceisoftenusedtodifferentiatethetwosurfacesthatcomeintocontact.AtypicalsetupisshowninFig.9.Insinglesidedcontact,theregionsformedbythemasterelementssearchforslavenodesthatwillpassthroughthenetandwillformconnections.Theregioncanbeshellelementsorthefacesofsolidelements.

Fig.9:Generalsurfacecontact.

Thecomputationalcostofthissearchcanbequitehighratherlikeraytracing,somethodsareusedtocutdownwhoseeswhominageneralsolution.Costequatestosolutiontimeandmemoryrequirements.Still,techniquesaregettingmoreefficientastechnologyimproves,sowecanexpecttoseethecostgoingdownandversatilityincreasing.Oneexampleisthatdoublesidedcontactwhereslavesurfaceslookformasternodesisnowverycommon.

ControllingJaggies

Otherissuesincludegettingridofthejaggies.TheFEAmeshisusuallyadiscontinuousdiscretization,andtheslaveregionintroducesintomasterregions,asshowninFig.10.Thisresultsinasetofartificialpointloads,whichdestroytheattempttohavecontinuousbearingforces,forexample.Thisisverydifficulttoavoid,evenbycarefulmeshing.

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Fig.10:Avoidingthejaggies.

Techniquesareavailablethatfitacommoninterpolatedsurfacetowhichthenodesmove,oreveninherittheactualCADgeometrysurface.Asmoothbearingloaddistributioncanthenbeachieved.

Anothercommonissuewithcontactsisthatofloosecomponentsnotproperlyabuttingeachother.AtypicalCADmodelwillplacecomponentsatnominalpositionsforexample,apinandlugdefinedconcentrically.InFEA,werequirethepintostartoffbeinginbearingcontactwiththelugtoestablishaloadpath.IfwecanmodifytheCADmodel,thatsagreathelp.However,itcanbedifficulttoestablishastableinitialloadpathatsmallinitialloadsteps.

Othermethodsincludeputtinginveryweakspringstohelpstabilizeorinanextremecase,runningatimebasedanalysis.Thetimescaledoesnotmatter,butwhatreallyhelpsisthateachnewloadstepconvergenceisnotjustworkingwithanupdatedstiffnesswearepassingforwardthedynamiceffectsandthekeyhereisinertia.

Therearemanytypesofnonlinearitywehavelookedatthemainareas.Itisimportanttoassesswhatlevelofnonlinearityisneededtoadequatelymodeltheproblem,thentakesmallbitesofthecherryandexplorethenatureofthenonlinearitycarefully.Wewanttostartsimplyandexploreandfarweneedtogo!

TonyAbbeyisaconsultantanalystwithhisowncompany,FETraining.HealsoworksastrainingmanagerforNAFEMS,responsiblefordevelopingandimplementingtrainingclasses,includingawiderangeofelearningclasses.SendemailaboutthisarticletoDEEditors@deskeng.com.

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