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7/28/13 Moving into the Nonlinear World with FEA by Desktop Engineering
www.deskeng.com/articles/aabhcg.htm 1/6
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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7/28/13 Moving into the Nonlinear World with FEA by Desktop Engineering
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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.
7/28/13 Moving into the Nonlinear World with FEA by Desktop Engineering
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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,
7/28/13 Moving into the Nonlinear World with FEA by Desktop Engineering
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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.
7/28/13 Moving into the Nonlinear World with FEA by Desktop Engineering
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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.
7/28/13 Moving into the Nonlinear World with FEA by Desktop Engineering
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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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