GENERALIZEDCOORDINATEFINITEELEMENTMODELSLECTURE 457MINUTES4·1GeneralizedcoordinatefiniteelementmodelsLECTURE 4 Classificationofproblems:truss,planestress,planestrain,axisymmetric,beam,plateandshellcon-ditions:correspondingdisplacement,strain,andstressvariablesDerivationofgeneralizedcoordinatemodelsOne-,two-,three-dimensionalelements,plateandshellelementsExampleanalysisof acantileverplate,detailedderivationofelementmatricesLumpedandconsistentloadingExampleresultsSummaryofthefiniteelementsolutionprocessSolutionerrorsConvergencerequirements,physicalexplana-tions,thepatchtestTEXTBOOK:Sections:4.2.3, 4.2.4, 4.2.5, 4.2.6Examples:4.5, 4.6, 4.7, 4.8, 4.11, 4.12, 4.13, 4.14,4.15, 4.16, 4.17, 4.184-2GeneralizedcoordinatefiniteeleDlentmodelsDERIVATIONOF SPECIFICFINITEELEMENTS• Generalized coordinatefiniteelement models~(m)= iB(m)TC(m) B(m)dV(m)V(m)aW)= JH(m)TLB(m)dV(m)V(m)R(m)= f HS(m)T f S(m)dS(m)!!S (m) - -Setc.Inessence,weneedH(m) B(m) C (m)-,-'-• Convergenceofanalysis resultsAAcross sectionA-A:TXXisuniform.Allotherstresscomponentsare zero.Fig. 4.14. Variousstressandstrainconditionswithillustrative examples.(a)Uniaxialstresscondition: frameunder concentratedloads.4·3Ge.raJizedcoordiDalefiniteelementlDOIIeIsHale\I6I\\-\-1ZI\TXX' Tyy, TXYare uniformacross the thickness.Allotherstresscomponentsare zero.Fig. 4.14. (b) Planestressconditions:membrane andbeamunder in-planeactions.u(x,y),v(x,y)are non-zerow=0 , Ezz= 0Fig. 4.14.(e)Plane strain condition:long dam subjectedtowater pressure.4·4GeneralizedcoordinatefiniteelementmodelsStructureandloadingare axisymmetric.j(IIII,II\--Allotherstresscomponentsare non-zero.Fig.4.14. (d) Axisymmetric condition:cylinder under internal pressure.(before deformation)(afterdeformation)/SHELLFig. 4.14. (e) Plate and shell structures.4·5GeneralizedcoordinatefiniteelementmodelsProblemBarBeamPlane stressPlane strainAxisymmetricThree-dimensionalPlate BendingDisplacementComponentsuwu,vu,vu,vu,v,wwTable 4.2(a)Corresponding Kine-matic and Static Variables in VariousProblems.ProblemBarBeamPlane stressPlane strainAxisymmetricThree-dimensionalPlate BendingStrain Vector~T-(E"...,)[IC...,](E"...,El'l')'"7)(E...,EJ"7)'..7)[E...,E"77)'''7Eu)[E...,E"77Eu)'''7)'76)'...,)(IC...,1(771("7).auauauauNolallon:E..=ax'£7 = a/ )'''7 =ay+ax'a1w a1w a1w•••,IC...,=-dxZ'IC77= - OyZ,IC.., = 20xoyTable 4.2 (b) Corresponding Kine-matic and Static Variables in VariousProblems.4·&ProblemBarBeamPlane stressPlane strainAxisymmetricThree-dimensionalPlate BendingGeneralizedcoordinatefiniteelementmodelsStress Vector1:T[T;u,][Mn][TnTJIJIT"'JI][TnTJIJIT"'JI][TnTJIJIT"'JITn][TnTYJITnT"'JITJI'Tu][MnMJIJIM"'JI]Table4.2(e) Corresponding Kine-matic and Static VariablesinVariousProblems.ProblemMaterialMatrix.£BarBeamPlane StressEEl[1vE v 11-1':&o 01~.]Table4.3Generalized Stress-StrainMatrices for Isotropic MaterialsandtheProblemsinTable 4.2.4·7GeneralizedcoordinatefiniteelementmodelsELEMENT DISPLACEMENT EXPANSIONS:Forone-dimensionalbarelementsFor two-dimensionalelements(4.47)Forplatebendingelements2w(x,y)=Y,+Y2x+Y3Y+Y4xy+Y5x+•..(4.48)Forthree-dimensionalsolidelementsu(x,y,z)=a,+Ozx+~Y+Ci4Z +~xy+...w(x,y,z)=Y,+y2x+y3y+y4z+y5xy+...(4.49)4·8Hence, in generalu =~exGeneralizedcoordinatefiniteelementmodels(4.50)(4.51/52)(4.53/54)Example(4.55)Y.VX.Vla) Cantilever platerNodalpoint6lp9Element [email protected](bl Finite element idealizationFig. 4.5. Finite element planestress analysis; i.e.TZZ=TZy=TZX=04·9Generalizedcoordinatefiniteelementmodels2LJ2.=US--II--.......---------....~element ®Elementnodalpointno.4=structurenodalpointno.5 .Fig.4.6.Typicaltwo-dimensionalfour-node element defined in localcoordinatesystem.Forelement 2wehave[U{X,y)] (2)= H(2) uv{x,y)--whereuT=[U- 14·10GeneralizedcoordinateliniteelementmodelsToestablishH (2) weuse:or[U(X,y)]=_~l!.v(x,y)where! =[~~}!=[1x y xy]andDefiningwehaveQ =Aa.HenceH=iPA-14·11GeneralizedcoordinatefiniteelementmodelsHenceH =fl-l4and(1+x ) (Hy): : aI•••I Ia : :(1+x)(1+y):H'ZJ=[0- 0Ull:HII: HZIUJVJUzt':u. v.U2U3U4UsU6U7UsU9U1aI 0 : HIJH17:HI.H16: 00:HI.Hu:o :HZJH21:H::H:6:00:H..Ha:VI-elementdegreesoffreedomU12 U13 U14UIS-assemblagedegreesHIs:0 0 zerosOJoffreedomHzs: 0 0 zeros O2x18(a)Elementlayout(b) Local-global degrees offreedomFig. 4.7.Pressureloading onelement (m)4·12GeneralizedcoordinatefiniteelementmodelsInplane-stressconditionstheelementstrainsarewhereE -au. E _av._au+avxx- ax'yy-ay, Yxy-ayaxHencewhereI =[~10Iy'OI0 0010I0 1IX 10I4·13GeneralizedcoordinatefiniteelementmodelsACTUALPHYSICALPROBLEMGEOMETRICDOMAINMATERIALLOADINGBOUNDARYCONDITIONS1MECHANICALIDEALIZATIONKINEMATICS,e.g. trussplane stressthree-dimensionalKirchhoff plateetc.MATERIAL,e.g. isotropiclinearelasticMooney-Rivlinrubberetc.LOADING,e.g. concentratedcentrifugaletc.BOUNDARYCONDITIONS,e.g. prescribed1displacementsetc.FINITEELEMENTSOLUTIONCHOICEOFELEMENTSANDSOLUTIONPROCEDURESYIELDS:GOVERNINGDIFFERENTIALEQUATIONSOFMOTIONe.g...!..(EA.!!!)= -p(x)axaxYIELDS:APPROXIMATERESPONSESOLUTIONOFMECHANICALIDEALIZATIONFig.4.23. Finite Element SolutionProcess4·14GeneralizedcoordinatefiniteelementmodelsSECTIONERRORERROROCCURRENCEINdiscussingerrorDISCRETIZATIONuseoffiniteelement4.2.5interpolationsNUMERICALevaluationoffinite5.8.1INTEGRATIONelement matrices using6.5.3INSPACEnumericalintegrationEVALUATIONOFuseofnonlinear material6.4.2CONSTITUTIVEmodelsRELATIONSSOLUTIONOFdirecttimeintegration,9.2DYNAMICEQUILI-.modesuperposition9.4BRIUMEQUATIONSSOLUTIONOFGauss-Seidel,Newton-8.4FINITEELEr1ENTRaphson,Quasi-Newton8.6EQUATIONSBYmethods, eigenso1utions 9.5ITERATION10.4ROUND-OFFsetting-upequationsand8.5theirsolutionTable4.4Finite ElementSolution Errors4·15GeneralizedcoordinatefiniteelementmodelsCONVERGENCEAssume a compatibleelement layoutisused,then we have monotonicconvergencetothesolutionofthe problem-governing differentialequation, provided theelements contain:1)allrequiredrigidbody modes2)allrequired constantstrain states~compatibleLWlayoutCDincompatiblelayout~t:=no.ofelementsIf an incompatible elementlayoutisused, theninadditionevery patchofelements mustbe abletorepresent the constantstrain states. Then we haveconvergencebutnon-monotonicconvergence.4·16Geuralizedcoordinatefinitee1eJDeDtmodels7 "/ "'r->,;/(""1-----,IIIIIIIiI(a)Rigid body modesofa planestresselement......~_QIIII(b) Analysistoillustrate the rigidbody mode conditionRigidbodytranslationandrotation;elementmustbestress-free.Fig. 4.24.Useofplanestresselementin