Finite Element Analysis David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA  February   Introduction FiniteelementanalysisFEAhasbecomecom
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Finite Element Analysis David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA February Introduction FiniteelementanalysisFEAhasbecomecom

Numericalsolutionstoevenverycomplicatedstress problemscannowbeobtainedroutinelyusingFEAandthemethodissoimportantthateven introductorytreatmentsofMechanicsofMaterialssuchasthesemodulesshouldoutlineits principalfeatures InspiteofthegreatpowerofFEAthed

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Finite Element Analysis David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA February Introduction FiniteelementanalysisFEAhasbecomecom




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Presentation on theme: "Finite Element Analysis David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA February Introduction FiniteelementanalysisFEAhasbecomecom"— Presentation transcript:


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Finite Element Analysis David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 February 28, 2001 Introduction Finiteelementanalysis(FEA)hasbecomecommonplaceinrecentyears,andisnowthebasis ofamultibilliondollarperyearindustry. Numericalsolutionstoevenverycomplicatedstress problemscannowbeobtainedroutinelyusingFEA,andthemethodissoimportantthateven introductorytreatmentsofMechanicsofMaterials–suchasthesemodules–shouldoutlineits principalfeatures.

InspiteofthegreatpowerofFEA,thedisadvantagesofcomputersolutionsmustbekeptin mindwhenusingthisandsimilarmethods:theydonotnecessarilyrevealhowthestressesare influencedbyimportantproblemvariablessuchasmaterialspropertiesandgeometricalfeatures, anderrorsininputdatacanproducewildlyincorrectresultsthatmaybeoverlookedbythe analyst.Perhapsthemostimportantfunctionoftheoreticalmodelingisthatofsharpeningthe designer’sintuition;usersoffiniteelementcodesshouldplantheirstrategytowardthisend, supplementingthecomputersimulationwithasmuchclosed-formandexperimentalanalysisas possible.

Finiteelementcodesarelesscomplicatedthanmanyofthewordprocessingandspreadsheet packagesfoundonmodernmicrocomputers. Nevertheless,theyarecomplexenoughthatmost usersdonotfinditeffectivetoprogramtheirowncode. Anumberofprewrittencommercial codesareavailable,representingabroadpricerangeandcompatiblewithmachinesfrommi- crocomputerstosupercomputers .However,userswithspecializedneedsshouldnotnecessarily shyawayfromcodedevelopment,andmayfindthecodesourcesavailableinsuchtextsasthat byZienkiewicz tobeausefulstartingpoint.MostfiniteelementsoftwareiswritteninFortran,

butsomenewercodessuchas felt areinCorothermoremodernprogramminglanguages. Inpractice,afiniteelementanalysisusuallyconsistsofthreeprincipalsteps: 1. Preprocessing: Theuserconstructsa model oftheparttobeanalyzedinwhichthegeom- etryisdividedintoanumberofdiscretesubregions,or“elements,”connectedatdiscrete pointscalled“nodes.” Certainofthesenodeswillhavefixeddisplacements,andothers willhaveprescribedloads. Thesemodelscanbeextremelytimeconsumingtoprepare, andcommercialcodesviewithoneanothertohavethemostuser-friendlygraphical“pre- processor”toassistinthisrathertediouschore.

Someofthesepreprocessorscanoverlay ameshonapreexistingCADfile,sothatfiniteelementanalysiscanbedoneconveniently aspartofthecomputerizeddrafting-and-designprocess. C.A. Brebbia, ed., Finite Element Systems, A Handbook, Springer-Verlag, Berlin, 1982. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, McGraw-Hill Co., London, 1989.
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2. Analysis: Thedatasetpreparedbythepreprocessorisusedasinputtothefiniteelement codeitself,whichconstructsandsolvesasystemoflinearornonlinearalgebraicequations ij where and

arethedisplacementsandexternallyappliedforcesatthenodalpoints.The formationofthe matrixisdependentonthetypeofproblembeingattacked,andthis modulewilloutlinetheapproachfortrussandlinearelasticstressanalyses. Commercial codesmayhaveverylargeelementlibraries,withelementsappropriatetoawiderange ofproblemtypes.OneofFEA’sprincipaladvantagesisthatmanyproblemtypescanbe addressedwiththesamecode,merelybyspecifyingtheappropriateelementtypesfrom thelibrary. 3. Postprocessing: Intheearlierdaysoffiniteelementanalysis,theuserwouldporethrough

reamsofnumbersgeneratedbythecode,listingdisplacementsandstressesatdiscrete positionswithinthemodel. Itiseasytomissimportanttrendsandhotspotsthisway, andmoderncodesusegraphicaldisplaystoassistinvisualizingtheresults. Atypical postprocessordisplayoverlayscoloredcontoursrepresentingstresslevelsonthemodel, showingafull-fieldpicturesimilartothatofphotoelasticormoireexperimentalresults. Theoperationofaspecificcodeisusuallydetailedinthedocumentationaccompanyingthe software,andvendorsofthemoreexpensivecodeswilloftenofferworkshopsortrainingsessions

aswelltohelpuserslearntheintricaciesofcodeoperation.Oneproblemusersmayhaveeven afterthistrainingisthatthecodetendstobea“blackbox”whoseinnerworkingsarenot understood.Inthismodulewewilloutlinetheprinciplesunderlyingmostcurrentfiniteelement stressanalysiscodes,limitingthediscussiontolinearelasticanalysisfornow. Understanding thistheoryhelpsdissipatetheblack-boxsyndrome,andalsoservestosummarizetheanalytical foundationsofsolidmechanics. Matrixanalysisoftrusses Pin-jointedtrusses,discussedmorefullyinModule5,provideagoodwaytointroduceFEA concepts.

Thestaticanalysisoftrussescanbecarriedoutexactly,andtheequationsofeven complicatedtrussescanbeassembledinamatrixformamenabletonumericalsolution. This approach,sometimescalled“matrixanalysis,”providedthefoundationofearlyFEAdevelop- ment. Matrixanalysisoftrussesoperatesbyconsideringthestiffnessofeachtrusselementone atatime,andthenusingthesestiffnessestodeterminetheforcesthataresetupinthetruss elementsbythedisplacementsofthejoints,usuallycalled“nodes”infiniteelementanalysis. Thennotingthatthesumoftheforcescontributedbyeachelementtoanodemustequalthe

forcethatisexternallyappliedtothatnode,wecanassembleasequenceoflinearalgebraic equationsinwhichthenodaldisplacementsaretheunknownsandtheappliednodalforcesare knownquantities. Theseequationsareconvenientlywritteninmatrixform,whichgivesthe methoditsname: 11 12 ··· 21 22 ··· ··· nn
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Here and indicatethedeflectionatthe th nodeandtheforceatthe th node(these wouldactuallybevectorquantities,withsubcomponentsalongeachcoordinateaxis).The ij coefficientarrayiscalledthe globalstiffnessmatrix ,withthe ij componentbeingphysicallythe influenceofthe th displacementonthe

th force.Thematrixequationscanbeabbreviatedas ij or Ku (1) usingeithersubscriptsorboldfacetoindicatevectorandmatrixquantities. Eithertheforceexternallyappliedorthedisplacementisknownattheoutsetforeachnode, anditisimpossibletospecifysimultaneouslybothanarbitrarydisplacement and aforceona givennode. Theseprescribednodalforcesanddisplacementsaretheboundaryconditionsof theproblem. Itisthetaskofanalysistodeterminetheforcesthataccompanytheimposed displacements,andthedisplacementsatthenodeswhereknownexternalforcesareapplied. Stiffnessmatrixforasingletrusselement

Asafirststepindevelopingasetofmatrixequationsthatdescribetrusssystems,weneeda relationshipbetweentheforcesanddisplacementsateachendofasingletrusselement.Consider suchanelementinthe planeasshowninFig.1,attachedtonodesnumbered and and inclinedatanangle fromthehorizontal. Figure1:Individualtrusselement. Consideringtheelongationvector toberesolvedindirectionsalongandtransversetothe element,theelongationinthetrusselementcanbewrittenintermsofthedifferencesinthe displacementsofitsendpoints: =( cos sin cos sin where and

arethehorizontalandverticalcomponentsofthedeflections,respectively.(The displacementsatnode drawninFig.1arenegative.) Thisrelationcanbewritteninmatrix formas: scs Here =cos and =sin Theaxialforce thataccompaniestheelongation isgivenbyHooke’slawforlinearelastic bodiesas =( AE/L .ThehorizontalandverticalnodalforcesareshowninFig.2;thesecan bewrittenintermsofthetotalaxialforceas:
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Figure2:Componentsofnodalforce. xi yi xj yj AE AE scs Carryingoutthematrixmultiplication: xi yi xj yj AE cs cs cs s cs cs c cs cs cs s (2) Thequantity in brackets, multiplied by AE/L , is known as

the“element stiffness matrix ij . Eachofitstermshasaphysicalsignificance,representingthecontributionofoneofthe displacementstooneoftheforces.Theglobalsystemofequationsisformedbycombiningthe elementstiffnessmatricesfromeachtrusselementinturn,sotheircomputationiscentraltothe methodofmatrixstructuralanalysis.Theprincipaldifferencebetweenthematrixtrussmethod andthegeneralfiniteelementmethodisinhowtheelementstiffnessmatricesareformed;most oftheothercomputeroperationsarethesame. Assemblyofmultipleelementcontributions Figure3:Elementcontributionstototalnodalforce.

Thenextstepistoconsideranassemblageofmanytrusselementsconnectedbypinjoints. Each element meeting at ajoint, ornode, will contributea forcethereas dictated bythe displacementsofboththatelement’snodes(seeFig.3). Tomaintainstaticequilibrium,all
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elementforcecontributions elem atagivennodemustsumtotheforce ext thatisexternally appliedatthatnode: ext elem elem =( elem elem ij )=( elem elem ij ij Eachelementstiffnessmatrix elem ij isaddedtotheappropriatelocationoftheoverall,or“global stiffnessmatrix ij thatrelatesallofthetrussdisplacementsandforces.Thisprocessiscalled

“assembly.”Theindexnumbersintheaboverelationmustbethe“global”numbersassigned tothetrussstructureasawhole.However,itisgenerallyconvenienttocomputetheindividual elementstiffnessmatricesusingalocalscheme,andthentohavethecomputerconverttoglobal numberswhenassemblingtheindividualmatrices. Example1 Theassemblyprocessisattheheartofthefiniteelementmethod,anditisworthwhiletodoasimple casebyhandtoseehowitreallyworks. Considerthetwo-elementtrussproblemofFig.4,withthe nodesbeingassignedarbitrary“global”numbersfrom1to3. Sinceeachnodecaningeneralmovein twodirections,thereare3

2=6totaldegreesoffreedomintheproblem. Theglobalstiffnessmatrix willthenbea6 6arrayrelatingthesixdisplacementstothesixexternallyappliedforces. Onlyone ofthedisplacementsisunknowninthiscase,sinceallbuttheverticaldisplacementofnode2(degreeof freedomnumber4)isconstrainedtobezero. Figure4showsaworkablelistingoftheglobalnumbers, andalso“local”numbersforeachindividualelement. Figure4:Globalandlocalnumberingforthetwo-elementtruss. Usingthelocalnumbers,the4 4elementstiffnessmatrixofeachofthetwoelementscanbeevaluated accordingtoEqn.2.Theinclinationangleiscalculatedfromthenodalcoordinatesas

=tan Theresultingmatrixforelement1is: (1) 25 00 43 30 25 00 43 30 43 30 75 00 43 30 75 00 25 00 43 30 25 00 43 30 43 30 75 00 43 30 75 00 10 andforelement2: (2) 25 00 43 30 25 00 43 30 43 30 75 00 43 30 75 00 25 00 43 30 25 00 43 30 43 30 75 00 43 30 75 00 10 (Itisimportanttheunitsbeconsistent;herelengthsareininches,forcesinpounds,andmoduliinpsi. Themodulusofbothelementsis =10Mpsiandbothhavearea =0 1in .) Thesematriceshave rowsandcolumnsnumberedfrom1to4,correspondingtothelocaldegreesoffreedomoftheelement.
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However,eachofthelocaldegreesoffreedomcanbematchedtooneoftheglobaldegreesoftheoverall problem.ByinspectionofFig.4,wecanformthefollowingtablethatmapslocaltoglobalnumbers: local global, global, element1 element2 11 3 22 4 33 5 44 6 Usingthistable,weseeforinstancethattheseconddegreeoffreedomforelement2isthefourthdegree offreedomintheglobalnumberingsystem,andthethirdlocaldegreeoffreedomcorrespondstothefifth globaldegreeoffreedom. Hencethevalueinthesecondrowandthirdcolumnoftheelementstiffness matrixofelement2,denoted (2) 23 ,shouldbeaddedintothepositioninthefourthrowandfifthcolumn

ofthe6 6globalstiffnessmatrix.Wewritethisas (2) 23 Eachofthesixteenpositionsinthestiffnessmatrixofeachofthetwoelementsmustbeaddedintothe globalmatrixaccordingtothemappinggivenbythetable.Thisgivestheresult (1) 11 (1) 12 (1) 13 (1) 14 00 (1) 21 (1) 22 (1) 23 (1) 24 00 (1) 31 (1) 32 (1) 33 (2) 11 (1) 34 (2) 12 (2) 13 (2) 14 (1) 41 (1) 42 (1) 43 (2) 21 (1) 44 (2) 22 (2) 23 (2) 24 00 (2) 31 (2) 32 (2) 33 (2) 34 00 (2) 41 (2) 42 (2) 43 (2) 44 ThismatrixpremultipliesthevectorofnodaldisplacementsaccordingtoEqn.1toyieldthevectorof externallyappliednodalforces.

Thefullsystemequations,takingintoaccounttheknownforcesand displacements,arethen 10 25 43 25 043 30 00 00 43 375 043 75 00 00 00 25 043 350 00 25 43 30 43 75 00 0 150 43 75 00 00 25 43 325 043 30 00 43 75 043 375 00 1732 Notethateithertheforceorthedisplacementforeachdegreeoffreedomisknown,withtheaccompanying displacementorforcebeingunknown.Hereonlyoneofthedisplacements( )isunknown,butinmost problemstheunknowndisplacementsfaroutnumbertheunknownforces. Notealsothatonlythose elementsthatarephysicallyconnectedtoagivennodecancontributeaforcetothatnode. Inmost

cases,thisresultsintheglobalstiffnessmatrixcontainingmanyzeroescorrespondingtonodalpairsthat arenotspannedbyanelement. Effectivecomputerimplementationswilltakeadvantageofthematrix sparsenesstoconservememoryandreduceexecutiontime. Inlargerproblemsthematrixequationsaresolvedfortheunknowndisplacementsandforcesby Gaussianreductionorothertechniques.Inthistwo-elementproblem,thesolutionforthesingleunknown displacementcanbewrittendownalmostfrominspection.Multiplyingoutthefourthrowofthesystem, wehave 0+0+0+150 10 +0+0= 1732 1732 150 10 01155in

Nowanyoftheunknownforcescanbeobtaineddirectly.Multiplyingoutthefirstrowforinstancegives
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0+0+0+(43 4)( 0115) 10 +0+0= 500lb Thenegativesignhereindicatesthehorizontalforceonglobalnode#1istotheleft,oppositethedirection assumedinFig.4. Theprocessofcyclingthrougheachelementtoformtheelementstiffnessmatrix,assembling theelementmatrixintothecorrectpositionsintheglobalmatrix,solvingtheequationsfor displacementsandthenback-multiplyingtocomputetheforces,andprintingtheresultscanbe automatedtomakeaveryversatilecomputercode.

Larger-scaletruss(andother)finiteelementanalysisarebestdonewithadedicatedcomputer code,andanexcellentoneforlearningthemethodisavailablefromthewebat http://felt.sourceforge.net/ .Thiscode,named felt ,wasauthoredbyJasonGobatand DarrenAtkinsonforeducationaluse,andincorporatesanumberofnovelfeaturestopromote user-friendliness. Completeinformationdescribingthiscode,aswellastheC-languagesource andanumberoftrialrunsandauxiliarycodemodulesisavailableviatheirwebpages. Ifyou haveaccesstoX-windowworkstations,agraphicalshellnamed velvet isavailableaswell. Example2

Figure5:Thesix-elementtruss,asdevelopedinthe velvet/felt FEAgraphicalinterface. Toillustratehowthiscodeoperatesforasomewhatlargerproblem,considerthesix-elementtrussof Fig.5,whichwasanalyzedinModule5bothbythejoint-at-a-timefreebodyanalysisapproachandby Castigliano’smethod. Theinputdataset, whichcanbewrittenmanuallyordevelopedgraphicallyin velvet ,employs parsingtechniquestosimplifywhatcanbeaverytediousanderror-pronestepinfiniteelementanalysis. Thedatasetforthis6-elementtrussis: problem description nodes=5 elements=6 nodes 1 x=0 y=100 z=0 constraint=pin
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2 x=100 y=100 z=0

constraint=planar 3 x=200 y=100 z=0 force=P 4 x=0 y=0 z=0 constraint=pin 5 x=100 y=0 z=0 constraint=planar truss elements 1 nodes=[1,2] material=steel 2 nodes=[2,3] 3 nodes=[4,2] 4 nodes=[2,5] 5 nodes=[5,3] 6 nodes=[4,5] material properties steel E=3e+07 A=0.5 distributed loads constraints free Tx=u Ty=u Tz=u Rx=u Ry=u Rz=u pin Tx=c Ty=c Tz=c Rx=u Ry=u Rz=u planar Tx=u Ty=u Tz=c Rx=u Ry=u Rz=u forces P Fy=-1000 end Themeaningoftheselinesshouldbefairlyevidentoninspection,althoughthe felt documentation shouldbeconsultedformoredetail.Theoutputproducedby felt forthesedatais: ** ** Nodal

Displacements ----------------------------------------------------------------------------- Node # DOF 1 DOF 2 DOF 3 DOF 4 DOF 5 DOF 6 ----------------------------------------------------------------------------- 1 000000 2 0.013333 -0.03219 0000 3 0.02 -0.084379 0000 4 000000 5 -0.0066667 -0.038856 0000 Element Stresses ------------------------------------------------------------------------------- 1: 4000 2: 2000 3: -2828.4 4: 2000 5: -2828.4 6: -2000 Reaction Forces ----------------------------------- Node # DOF Reaction Force -----------------------------------
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1 Tx -2000

1Ty 0 1Tz 0 2Tz 0 3Tz 0 4 Tx 2000 4 Ty 1000 4Tz 0 5Tz 0 Material Usage Summary -------------------------- Material: steel Number: 6 Length: 682.8427 Mass: 0.0000 Total mass: 0.0000 Theverticaldisplacementofnode3(the DOF 2 value)is-0.0844,thesamevalueobtainedbythe closed-formmethodsofModule5.Figure6showsthe velvet graphicaloutputforthetrussdeflections (greatlymagnified). Figure6:The6-elementtrussinitsoriginalanddeformedshape. GeneralStressAnalysis Theelementstiffnessmatrixcouldbeformedexactlyfortrusselements,butthisisnotthecase forgeneralstressanalysissituations.

Therelationbetweennodalforcesanddisplacementsare notknowninadvanceforgeneraltwo-orthree-dimensionalstressanalysisproblems,andan approximaterelationmustbeusedinordertowriteoutanelementstiffnessmatrix. Intheusual“displacementformulation”ofthefiniteelementmethod,thegoverningequa- tionsarecombinedsoastohaveonlydisplacementsappearingasunknowns;thiscanbedoneby usingtheHookeanconstitutiveequationstoreplacethestressesintheequilibriumequationsby thestrains,andthenusingthekinematicequationstoreplacethestrainsbythedisplacements. Thisgives DLu (3)
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Ofcourse,itisoftenimpossibletosolvetheseequationsinclosedformfortheirregularbound- aryconditions encounteredinpractical problems. However, theequationsareamenableto discretizationandsolutionbynumericaltechniquessuchasfinitedifferencesorfiniteelements. Finiteelementmethodsareoneofseveralapproximatenumericaltechniquesavailablefor thesolutionofengineeringboundaryvalueproblems. Problemsinthemechanicsofmaterials oftenleadtoequationsofthistype,andfiniteelementmethodshaveanumberofadvantages inhandlingthem.Themethodisparticularlywellsuitedtoproblemswithirregulargeometries

andboundaryconditions,anditcanbeimplementedingeneralcomputercodesthatcanbe usedformanydifferentproblems. Toobtainanumericalsolutionforthestressanalysisproblem,letuspostulateafunction x,y )asanapproximationto x,y x,y )(4) Manydifferentformsmightbeadoptedfortheapproximation . Thefiniteelementmethod discretizesthesolutiondomainintoanassemblageofsubregions,or“elements,”eachofwhichhas itsownapproximatingfunctions. Specifically,theapproximationforthedisplacement x,y withinanelementiswrittenasacombinationofthe(asyetunknown)displacementsatthe nodesbelongingtothatelement: x,y )=

x,y (5) Heretheindex rangesovertheelement’snodes, arethenodaldisplacements,andthe are “interpolationfunctions.” Theseinterpolationfunctionsareusuallysimplepolynomials(gen- erallylinear,quadratic,oroccasionallycubicpolynomials)thatarechosentobecomeunityat node andzeroattheotherelementnodes.Theinterpolationfunctionscanbeevaluatedatany positionwithintheelementbymeansofstandardsubroutines,sotheapproximatedisplacement atanypositionwithintheelementcanbeobtainedintermsofthenodaldisplacementsdirectly fromEqn.5. Figure7:Interpolationinonedimension.

Theinterpolationconceptcanbeillustratedbyaskinghowwemightguessthevalueofa function )atanarbitrarypoint locatedbetweentwonodesat =0and =1,assuming weknowsomehowthenodalvalues (0)and (1). Wemightassumethatasareasonable approximation )simplyvarieslinearlybetweenthesetwovaluesasshowninFig.7,and write )= (1 )+ or 10
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)= )+ )=(1 )= Herethe and arethelinearinterpolationfunctionsforthisone-dimensionalapproxima- tion. Finiteelementcodeshavesubroutinesthatextendthisinterpolationconcepttotwoand threedimensions. Approximationsforthestrainandstressfollowdirectlyfromthedisplacements: (6) DB

(7) where x,y )= x,y )isanarrayofderivativesoftheinterpolationfunctions: j,x j,y j,y j,x (8) A“virtualwork”argumentcannowbeinvokedtorelatethenodaldisplacement appearing atnode totheforcesappliedexternallyatnode :ifasmall,or“virtual,”displacement is superimposedonnode ,theincreaseinstrainenergy δU withinanelementconnectedtothat nodeisgivenby: δU dV (9) where isthevolumeoftheelement. Usingtheapproximatestrainobtainedfromtheinter- polateddisplacements, istheapproximatevirtualincreaseinstraininducedbythe virtualnodaldisplacement.UsingEqn.7andthematrixidentity( AB wehave: δU DB dV

(10) (Thenodaldisplacements and arenotfunctionsof and ,andsocanbebroughtfrom insidetheintegral.) Theincreaseinstrainenergy δU mustequaltheworkdonebythenodal forces;thisis: δW (11) EquatingEqns.10and11andcancelingthecommonfactor ,wehave: DB dV (12) Thisisofthedesiredform ij ,where ij DB dV istheelementstiffness. Finally,theintegralinEqn.12mustbereplacedbyanumericalequivalentacceptabletothe computer. Gauss-Legendrenumericalintegrationiscommonlyusedinfiniteelementcodesfor thispurpose,sincethattechniqueprovidesahighratioofaccuracytocomputingeffort.Stated briefly,

theintegration consistsofevaluatingtheintegrandatoptimallyselectedintegration pointswithintheelement,andformingaweightedsummationoftheintegrandvaluesatthese points.Inthecaseofintegrationovertwo-dimensionalelementareas,thiscanbewritten: 11
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x,y dA ,y (13) Thelocationofthesamplingpoints ,y andtheassociatedweights areprovidedby standardsubroutines.Inmostmoderncodes,theseroutinesmaptheelementintoaconvenient shape,determinetheintegrationpointsandweightsinthetransformedcoordinateframe,and thenmaptheresultsbacktotheoriginalframe.Thefunctions usedearlierforinterpolation

canbeusedforthemappingaswell,achievingasignificanteconomyincoding.Thisyieldswhat areknownas“numericallyintegratedisoparametricelements,”andtheseareamainstayofthe finiteelementindustry. Equation12,withtheintegralreplacedbynumericalintegrationsoftheforminEqn.13,is thefiniteelementcounterpartofEqn.3,thedifferentialgoverningequation.Thecomputerwill carryouttheanalysisbyloopingovereachelement,andwithineachelementloopingoverthe individualintegrationpoints.Ateachintegrationpointthecomponentsoftheelementstiffness matrix ij

arecomputedaccordingtoEqn.12,andaddedintotheappropriatepositionsofthe ij globalstiffnessmatrixaswasdoneintheassemblystepofmatrixtrussmethoddescribedin theprevioussection.Itcanbeappreciatedthatagooddealofcomputationisinvolvedjustin formingthetermsofthestiffnessmatrix,andthatthefiniteelementmethodcouldneverhave beendevelopedwithoutconvenientandinexpensiveaccesstoacomputer. Stressesaroundacircularhole Wehaveconsideredtheproblemofauniaxiallyloadedplatecontainingacircularholeinprevious modules,includingthetheoreticalKirschsolution(Module16)andexperimentaldeterminations

usingbothphotoelasticandmoiremethods(Module17).Thisproblemisofpracticalimportance —-forinstance,wehavenotedthedangerousstressconcentrationthatappearsnearrivetholes —anditisalsoquitedemandinginboththeoreticalandnumericalanalyses.Sincethestresses risesharplynearthehole, afiniteelement gridmustberefinedthereinordertoproduce acceptableresults. Figure8:Meshforcircular-holeproblem. Figure8showsameshofthree-nodedtriangularelementsdevelopedbythe felt-velvet 12
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graphicalFEApackagethatcanbeusedtoapproximatethedisplacementsandstressesaround

auniaxiallyloadedplatecontainingacircularhole. Sinceboththeoreticalandexperimental resultsforthisstressfieldareavailableasmentionedabove,thecircular-holeproblemisagood oneforbecomingfamiliarwithcodeoperation. Theusershouldtakeadvantageofsymmetrytoreduceproblemsizewheneverpossible,and inthiscaseonlyonequadrantoftheproblemneedbemeshed. Thecenteroftheholeiskept fixed,sothesymmetryrequiresthatnodesalongtheleftedgebeallowedtomovevertically butnothorizontally. Similarly,nodesalongtheloweredgeareconstrainedverticallybutleft

freetomovehorizontally.Loadsareappliedtothenodesalongtheupperedge,witheachload beingtheresultantofthefar-fieldstressactingalonghalfoftheelementboundariesbetween thegivennodeanditsneighbors. (Thefar-fieldstressistakenasunity.) Portionsofthe felt inputdatasetforthisproblemare: problem description nodes=76 elements=116 nodes 1 x=1 y=-0 z=0 constraint=slide_x 2 x=1.19644 y=-0 z=0 3 x=0.984562 y=0.167939 z=0 constraint=free 4 x=0.940634 y=0.335841 z=0 5 x=1.07888 y=0.235833 z=0 72 x=3.99602 y=3.01892 z=0 73 x=3.99602 y=3.51942 z=0 74 x=3.33267 y=4 z=0 75 x=3.57706 y=3.65664 z=0 76 x=4

y=4 z=0 CSTPlaneStress elements 1 nodes=[13,12,23] material=steel 2 nodes=[67,58,55] 6 nodes=[50,41,40] 7 nodes=[68,67,69] load=load_case_1 8 nodes=[68,58,67] 9 nodes=[57,58,68] load=load_case_1 10 nodes=[57,51,58] 11 nodes=[52,51,57] load=load_case_1 12 nodes=[37,39,52] load=load_case_1 13 nodes=[39,51,52] 116 nodes=[2,3,1] material properties steel E=2.05e+11 nu=0.33 t=1 distributed loads load_case_1 color=red direction=GlobalY values=(1,1) (3,1) 13
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constraints free Tx=u Ty=u Tz=u Rx=u Ry=u Rz=u slide_x color=red Tx=u Ty=c Tz=c Rx=u Ry=u Rz=u slide_y color=red Tx=c Ty=u

Tz=c Rx=u Ry=u Rz=u end The -displacementsandverticalstresses arecontouredinFig.9(a)and(b)respectively; theseshouldbecomparedwiththephotoelasticandmoireanalysesgiveninModule17,Figs.8 and10(a). Thestressattheintegrationpointclosesttothe -axisattheholeiscomputed tobe y,max =3 26,9%largerthanthetheoreticalvalueof3.00. Indrawingthecontoursof Fig.9b,thepostprocessorextrapolatedthestressestothenodesbyfittingaleast-squaresplane throughthestressesatallfourintegrationpointswithintheelement. Thisproducesaneven highervalueforthestressconcentrationfactor,3.593. Theusermustbeawarethatgraphical

postprocessorssmoothresultsthatarethemselvesonlyapproximations,sonumericalinaccuracy isarealpossibility. Refiningthemesh,especiallyneartheregionofhigheststressgradientat theholemeridian,wouldreducethiserror. Figure9:Verticaldisplacements(a)andstresses(b)ascomputedforthemeshofFig.8. Problems 1. (a)–(h)UseFEAtodeterminetheforceineachelementofthetrussesdrawnbelow. 2. (a)–(c)Writeouttheglobalstiffnessmatricesforthetrusseslistedbelow,andsolve fortheunknownforcesanddisplacements. Foreachelementassume =30Mpsiand =0 1in 3.

Obtainaplane-stressfiniteelementsolutionforacantileveredbeamwithasingleloadat thefreeend.Usearbitrarilychosen(butreasonable)dimensionsandmaterialproperties. Plotthestresses and xy asfunctionsof atanarbitrarystationalongthespan;also plotthestressesgivenbytheelementarytheoryofbeambending(c.f. Module13)and assessthemagnitudeofthenumericalerror. 4. Repeatthepreviousproblem,butwithasymmetrically-loadedbeaminthree-pointbend- ing. 14
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Prob.1 Prob.2 5. Useaxisymmetricelementstoobtainafiniteelementsolutionfortheradialstressina

thick-walledpressurevessel(usingarbitrarygeometryandmaterialparameters).Compare theresultswiththetheoreticalsolution(c.f.Prob.2inModule16). 15
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Prob.3 Prob.4 16