Banded Solver method

1. _____tt~--- MassachusettsInstituteofTechnologyMITVideoCOurseVideoCourseStudyGuideFiniteElementProceduresforSolidsandStructuresLinearAnalys isKlaus-JOrgenBatheProfessorofMechanicalEngineering,MITPublishedbyMITCenterforAdvancedEngineeringstudyReorderNo672-2100

2. PREFACETheanalysisofcomplexstaticanddynamicproblemsinvolvesinessencethreestages: selectionofamathematicalmodel,analysisofthemodel,andinterpretationoftheresults. Duringrecentyearsthefiniteelementmethodimplementedonthedigitalcomputerhasbeenusedsuccessfullyinmodelingverycomplexproblemsinvariousareasofengineeringandhassignificantlyincreasedthepossibilitiesforsafeandcosteffectivedesign. However,theefficientuseofthemethodisonlypossibleifthebasicassumptionsoftheproceduresemployedareknown,andthemethodcanbeexercisedconfidentlyonthecomputer. Theobjectiveinthiscourseistosummarizemodernandeffectivefiniteelementproceduresforthelinearanalysesofstaticanddynamicproblems.Thematerialdiscussedinthelecturesincludesthebasicfiniteelementformulationsemployed, theeffectiveimplementationoftheseformulationsincomputerprograms,andrecommendationsontheactualuseofthemethodsinengineeringpractice.Thecourseisintendedforpracticingengineersandscientistswhowanttosolveproblemsusingmodemandefficientfiniteelementmethods. Finiteelementproceduresforthenonlinearanalysisofstructuresarepresentedinthefollow-upcourse,FiniteElementProceduresforSolidsandStructures-NonlinearAnalysis. Inthisstudyguideshortdescriptionsofthelecturesandtheviewgraphsusedinthelecturepresentationsaregiven. Belowtheshortdescriptionofeachlecture,referenceismadetotheaccompanyingtextbookforthecourse:FiniteElementProceduresinEngineeringAnalysis,byK.J.Bathe,PrenticeHall, Inc.,1982. Thetextbooksectionsandexamples,listedbelowtheshortdescriptionofeachlecture,provideimportantreadingandstudymaterialtothecourse.

3. ContentsLecturesl.Somebasicconceptsofengineeringanalysis1-12.Analysisofcontinuoussystems;differentialandvariationalformulations2-13.Formulationofthedisplacement-basedfiniteelementmethod_3-14.Generalizedcoordinatefiniteelementmodels4-15.Implementationofmethodsincomputerprograms; examplesSAp,ADINA5-16.Formulationandcalculationofisoparametricmodels6-17.Formulationofstructuralelements7-18.Numericalintegrations,modelingconsiderations8-19.Solutionoffiniteelementequilibriumequationsinstaticanalysis9-110.Solutionoffiniteelementequilibriumequationsindynamicanalysis10-11l.Modesuperpositionanalysis;timehistory11-112.Solutionmethodsforcalculationsoffrequenciesandmodeshapes12-1

4. SOMEBASICCONCEPTSOFENGINEERINGANALYSISLECTURE146MINUTESI-I

5. SolIebasicccnaceplsofeugiDeeriDgualysisLECTURE1Introductiontothecourse.objectiveoflecturesSomebasicconceptsofengineeringanalysis. discreteandcontinuoussystems.problemtypes:steady-state.propagationandeigenvalueproblemsAnalysisofdiscretesystems: exampleanalysisofaspringsystemBasicsolutionrequirementsUseandexplanationofthemoderndirectstiffnessmethodVariationalformulation TEXTBOOK:Sections:3.1and3.2.1.3.2.2.3.2.3.3.2.4Examples:3.1.3.2.3.3.3.4.3.5.3.6.3.7.3.8.3.9.3.10.3.11.3.12.3.13.3.141-2

6. Somebasicconcepts01engineeringaulysisINTRODUCTIONTOLINEARANALYSISOFSOLIDSANDSTRUCTURES•Thefiniteelementmethodisnowwidelyusedforanalysisofstructuralengineeringproblems. •'ncivil,aeronautical,mechanical, ocean,mining,nuclear,biomechanical,... engineering•Sincethefirstapplicationstwodecadesago, -wenowseeapplicationsinlinear,nonlinear,staticanddynamicanalysis. -variouscomputerprogramsareavailableandinsignificantuseMyobjectiveinthissetoflecturesis: •tointroducetoyoufiniteelementmethodsforthelinearanalysisofsolidsandstructures. ["Iinear"meaninginfinitesimallysmalldisplacementsandlinearelasticmaterialproeerties( Hooke'slawapplies)j•toconsider-theformulationofthefiniteelementequilibriumequations-thecalculationoffiniteelementmatrices-methodsforsolutionofthegoverningequations-computerimplementations.todiscussmodernandeffectivetechniques,andtheirpracticalusage. 1·3

7. SomebasicconceptsofengineeringanalysisREMARKS•Emphasisisgiventophysicalexplanationsratherthanmathematicalderivations• TechniquesdiscussedarethoseemployedinthecomputerprogramsSAPandADINASAP== StructuralAnalysisProgramADINA=AutomaticDynamicIncrementalNonlinearAnalysis•Thesefewlecturesrepresentaverybriefandcompactintroductiontothefieldoffiniteelementanalysis•WeshallfollowquitecloselycertainsectionsinthebookFiniteElementProceduresinEngineeringAnalysis, Prentice-Hall,Inc. (byK.J.Bathe). FiniteElementSolutionProcessPhysicalproblemEstablishfiniteelement...--~modelofphysicalproblem1I:I,-__S_ol_v_e_th_e_m_o_d_el__ I~ ~---iL-_I_n_te_r.;..p_re_t_t_h_e_re_s_u_lt_S_....JRevise(refine) themodel? 1-4

8. SolIebasicconceptsofengiDeeringanalysis10ft15ftI12at15°,. Analysisofcoolingtower. K~~~~~-~,-Fault(norestraintassumed) Altered'gritE=toEc., Analysisofdam. 1·5

9. SomebasicconceptsofengineeringanalysisB. t Wo E~~;;C=-------_........ FFiniteelementmeshfortireinflationanalysis. 1·6

10. SolDebasicconceptsofengineeringanalysisSegmentofasphericalcoverofalaservacuumtargetchamber. l,WppPINCHEDCYLINDRICALSHELLOD;:...,...----.---~~~~~~CEtW-50P-100-150•16x16MESH-200- DISPLACEMENTDISTRIBUTIONALONGDCOFPINCHEDCYLINDRICALSHELL•16x16MESH-0.2Mil""=0.1~CBENDINGMOMENTDISTRIBUTIONALONGDCOFPINCHEDCYLINDRICALSHELL1-7

11. SoBlebasicconcepts01engineeringanalysisIFiniteelementidealizationofwindtunnelfordynamicanalysisSOMEBASICCONCEPTSOFENGINEERINGANALYSISTheanalysisofanengineeringsystemrequires: -idealizationofsystem-formulationofequilibriumequations- solutionofequations-interpretationofresults1·8

12. SYSTEMSSomebasicconceptsofengineeringanalysisDISCRETEresponseisdescribedbyvariablesatafinitenumberofpointssetofalgebraic-- equationsCONTINUOUSresponseisdescribedbyvariablesataninfinitenumberofpointssetofdifferentialequationsPROBLEMTYPESARE• STEADY-STATE(statics) •PROPAGATION(dynamics) •EIGENVALUEFordiscreteandcontinuoussystemsAnalysisofcomplexcontinuoussystemrequiressolutionofdifferentialequationsusingnumerica lproceduresreductionofcontinuoussystemtodiscreteformpowerfulmechanism: thefiniteelementmethods, implementedondigitalcomputersANALYSISOFDISCRETESYSTEMSStepsinvolved: -systemidealizationintoelements-evaluationofelementequilibriumrequirements-elementassemblage-solutionofresponse1·9

13. SomebasicconceptsofengineeringanalysisExample: steady-stateanalysisofsystemofrigidcartsinterconnectedbyspringsPhysicallayoutELEMENTSU1U3I~:~l)..F(4)..31F(4) k,u1-F(')1-1]["1].[F14']-, '4[1u2-11UF(4) 33-F(2) F(2)---2, '2[1-1]["I]fF}]1uF(2)--122F(S)F(S) 23u,u2-t][F(5l]'5[1k3F(3)1u2=F1S)-1-----233F(3),-r1]fPl]'3[] -11uF(3) 221·10

14. SolIebasiccOIceplsofengineeringanalysisElementinterconnectionrequirements: F(4)+F(S)=R333TheseequationscanbewrittenintheformKU=E. EquilibriumequationsKU=R(a) +k4k1+k2+k3~-k2-k3UT=[u-1RT=[R-1 ·· ···: -k4....'"...............................·.·.. K=-k2-k3~k2+k3+kS~-kS·...................•................•.....•........·.·.1·11

15. Somebasicconceptsofengineeringanalysisandwenotethat~=t~(i) i=1where::] o0etc... ThisassemblageprocessiscalledthedirectstiffnessmethodThesteady-stateanalysisiscompletedbysolvingtheequationsin(a) 1·12

16. Somebasicconcepls01engineeringanalysisu,· .................::............... ·. ·.~....·.·.·.·.·.·.·u, K1....::............... ·. K= ·.....·.·.·: U1...............................:. K~•••~~.~•••••••••~~•••••••••••:...............~1---.JI.lfl--r/A~,111~~r/A1·13

17. SOlDebasicconceptsofengineeringanalysis ·....::.·.·u, ·.•'O••••••••••••••••••••••••••••••••••••••••••••••·.·.·. K= u, +K4; K1+K2+K3;-K2-K3·.'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O:'O'O'O'O'O'O'O'O'O'O'O'O'O'O:'O'O'O'O'O'O'O'O'O'O'O'O'O'O•·.·. K= ·.'O'O••••••••••••••••••••••••••••••••••••••••••••••·.··. . +K4; K1+K2+K3~-K2-K3-K4'O'O••••••••••••••••••••••••••••••••••••••••••••• K=-K2-K3~K2+K3+K5-K5'O'O•••••••••••••••:••••••••••••••••••••••••••••• u, IK1·14

18. SomebasicconceptsofengineeringanalysisInthisexampleweusedthedirectapproach;alternativelywecouldhaveusedavariationalapproach. Inthevariationalapproachweoperateonanextremumformulation: u=strainenergyofsystemW=totalpotentialoftheloadsEquilibriumequationsareobtainedfroman-0(b)~- 1IntheaboveanalysiswehaveU=~UT!!! W=UTRInvoking(b)weobtainKU=RNote:toobtainUandWweagainaddthecontributionsfromallelements1·15

19. SOlDebasicconceptsofengineeringanalysisPROPAGATIONPROBLEMSmaincharacteristic:theresponsechangeswithtime~needtoincludethed'Alembertforces: Fortheexample: m,aaM=am2aaam3EIGENVALUEPROBLEMSweareconcernedwiththegeneralizedeigenvalueproblem(EVP) Av=ABv!l,.!laresymmetricmatricesofordernvisavectorofordernAisascalarEVPsariseindynamicandbucklinganalysis1·16

20. SomebasicconceptsofengineeringanalysisExample:systemofrigidcarts~lU+KU=OLetU=<psinW(t-T) Thenweobtain_w2~~sinW(t-T) +K<psinW(t-T)=0--- HenceweobtaintheequationThereare3solutionsw,,~, (l)2'~2eigenpairsw3'~3Ingeneralwehavensolutions1·17

21. ANALYSISOFCONTINUOUSSYSTEMS; DIFFEBENTIALANDVABIATIONALFOBMULATIONSLECTURE259MINUTES2-1

22. Analysis01continnoussystems;differentialandvariationallonnDlationsLECTURE2BasicconceptsintheanalysisofcontinuoussystemsDifferentialandvariationalformulationsEssentialandnaturalboundaryconditionsDefinitionofem-IvariationalproblemPrincipleofvirtualdisplacementsRelationbetweenstationarityoftotalpotential,theprincipleofvirtualdisplacements,andthedifferentialformulationWeightedresidualmethods, Galerkin,leastsquaresmethodsRitzanalysismethodPropertiesoftheweightedresidualandRitzmethodsExampleanalysisofanonuniformbar,solutionaccuracy,introductiontothefiniteelementmethodTEXTBOOK:Sections:3.3.1,3.3.2,3.3.3Examples:3.15,3.16,3.17,3.18,3.19,3.20,3.21,3.22,3.23,3.24,3.252·2

23. Analysisofcontinuoussystems;differentialandvariationalformulationsBASICCONCEPTSOFFINITEELEMENTANALYSISCONTINUOUSSYSTEMS• Wediscussedsomebasicconceptsofanalysisofdiscretesystems•SomeadditionalbasicconceptsareusedinanalysisofcontinuoussystemsCONTINUOUSSYSTEMSdifferentialformulationtWeightedresidualmethods Galerkin_.._-----41~_ leastsquaresvariationalformulationRitzMethod.... finiteelementmethod2·3

24. AnalysisofcontinuoussysteDlS;differentialand,arialionalIOl'llulali. Example-DifferentialformulationaAI+A~aIdx-aAIxxoXX/ Young'smodulus,E~Lt:)massdensity, cross-sectionalarea,AR..~------- TheproblemgoverningdifferentialequationisDerivationofdifferentialequationTheelementforceequilibriumrequirementofatypicaldiffer entialelementisusingd'Alembert'sprincipler~.-;+~~dxI~ AreaA,massdensityp2=pAau~ Theconstitutiverelationisaua=EaxCombiningthetwoequationsaboveweobtain2 ·4

25. baIysis01COitiDlOUsystems;differatialaDdvariationaliOl'lDDlatiODSTheboundaryconditionsareu(O,t}=° EA~~(L,t)=ROwithinitialconditionsu(x,O}=° ~(xO)=°at' 9essential(displ.)B.C. 9natural(force)B.C. Ingeneral,wehavehighestorderof(spatial)derivativesinproblem- governingdifferentialequationis2m. highestorderof(spatial)derivativesinessentialb. c.is(m-1) highestorderofspatialderivativesinnaturalb. c.is(2m-1) Definition: WecallthisproblemaCm-1variationalproblem. 2·5

26. Analysis01continuoussystems;differentialandvariatioD,a1fOl'llolatiODSExample-VariationalformulationWehaveingeneralII=U-WFortherodfLII=J}EAoandiLau2B(--)dx-ufdx-uRaxLou=0oandwehave0II=0Thestationarycondition6II=0givesrLauaurL.BJO(EAax)(6ax)dx-)06ut-dx-6uLR=0Thisistheprincipleofvirtualdisplacementsgoverningtheproblem.Ingeneral,wewritethisprincipleasor(seealsoLecture3) 2·6

27. lIiIysisof..IiDIGUSsystems;differentialandvariatiooallormulatioDSHowever,wecannowderivethedifferentialequationofequilibriumandtheb.c.atx=l. Writinga8ufor8au,re-axaxcallingthatEAisconstantandusingintegrationbypartsyieldsdx+[EA~Iaxx=L-EA~dXx=oSinceQUOiszerobutQUisarbitraryatallotherpoints,wemusthaveandauIEAax-x=L=RBa2uAlsof=-Ap-and,at2hencewehave2·7

28. AnalysisofcODtiDaoassyst_diIIereatialandvariatioulfOlllalatiODSTheimportantpointisthatinvokingoIT=0andusingtheessentialb.c.onlywegenerate•theprincipleofvirtualdisplacements•theproblem-governingdifferentialaquatio!) •thenaturalb.c.(theseareinessence"containedin"IT, i.e.,inW). Inthederivationoftheproblemgoverningdifferentialequationweusedintegrationbyparts• thehighestspatialderivativeinITisoforderm. •Weuseintegrationbypartsm-times. TotalPotentialITIUseoIT=0andessential"b.c. ~ 2·8PrincipleofVirtualDisplacementsIIntegrationbyparts~ DifferentialEquationofEquilibriumandnaturalb.c. _solveproblem_solveproblem

29. balysisofaDa.syst-:diBerentialandvariatiouallnaiatiOlSWeightedResidualMethodsConsiderthesteady-stateproblem(3.6) withtheB.C. B.[</>]=q.,i=1,2,••• 11atboundary(3.7) Thebasicstepintheweightedresidual(andtheRitzanalysis) istoassumeasolutionoftheform(3.10) wherethefiarelinearlyindependenttrialfunctionsandtheaiaremultipliersthataredeterminedintheanalysis. Usingtheweightedresidualmethods, wechoosethefunctionsfiin(3.10) soastosatisfyallboundaryconditionsin(3.7)andwethencalculatetheresidual, nR=r-L2mCLa·f.](3.11) 1=111ThevariousweightedresidualmethodsdifferinthecriterionthattheyemploytocalculatetheaisuchthatRissmall. InalltechniqueswedeterminetheaisoastomakeaweightedaverageofRvanish. 2·9

30. Analysis01C.tinnoDSsystems;differentialandvariational10000nlationsGalerkinmethodInthistechniquetheparametersaiaredeterminedfromthenequationsff.RdD=O;=1,2,•••,nD1Leastsquaresmethod(3.12) Inthistechniquetheintegralofthesquareoftheresidualisminimizedwithrespecttotheparametersai'aaa. 1;=1,2,•••,n[Themethodscanbeextendedtooperatealsoonthenaturalboundaryconditions,ifthesearenotsatisfiedbythetrialfunctions.] RITZANALYSISMETHODLetnbethefunctionaloftheem-1variationalproblemthatisequivalenttothedifferentialformulationgivenin(3.6)and(3.7). IntheRitzmethodwesubstitutethetrialfunctions<pgivenin(3.10) intonandgeneratensimultaneousequationsfortheparametersaiusingthestationaryconditiononn, 2·10an0aa.= 1;=1,2,•••,n(3.14)

31. Analysisofcontinuoussystems;differentialandvariationalformulationsProperties•ThetrialfunctionsusedintheRitzanalysisneedonlysatisfytheessentialb.c. •SincetheapplicationofoIl=0generatestheprincipleofvirtualdisplacements,weineffectusethisprincipleintheRitzanalysis. •Byinvoking0II=0weminimizetheviolationoftheinternalequilibriumrequirementsandtheviolationofthenaturalb.c. •Asymmetriccoefficientmatrixisgenerated,offormKU=RExampleR=100N2Area=1em( ........_-x,u----------~---r;;;-==-e- .F---.;;B;",.,.CI-...--~~---·-I-..--------·-I100em80emFig.3.19.Barsubjectedtoconcentratedendforce. 2·11

32. AnalysisofCOitiDlOISsystems;differeatialad,ariali"fOllDaialiODSHerewehave1180IT=1EA(~)2dx2axo-100uIx=180andtheessentialboundaryconditionisuIx=O=0LetusassumethedisplacementsCase1u=a1x+a2iCase2~ u=I1JO0<x<100100<x<180WenotethatinvokingoIT=0weobtain1180oIT=(EA~~)o(~~)dx-100OUIx=180o=0ortheprincipleofvirtualdisplacements£180(~~u)(EA~~)dx=100OUIx=180oJETTdV=IT.F. --11V2·12

33. Analysisofcontinuoussystems;differentialandvariationalformulationsExactSolutionUsingintegrationbypartsweobtain~(EA~)=0axaxEA~=100axx=180Thesolutionisu=1~Ox;0<x<100100<x<180Thestressesinthebararea=100;0<x<100a=100;100<x<180(l+x-l00)2402·13

34. Analysisofcontinuoussystems;differentialandvariationalformulationsPerformingnowtheRitzanalysis: Case1f180dx+I(1+x-l00)2240100Invokingthatorr=0weobtainE[0.4467116and11634076128.6a1=---=E=---a-0.3412--EHence,wehavetheapproximatesolutionu=12C.60.341Ex-E2x2·14a=128.6-0.682x

35. Analysisofcontinuoussystems;differentialandvariationalformulationsCase2Herewehave100EJ12n=2(100uB) af180dx+I(1+x-l00)2240100Invokingagainon=0weobtainE[15.4-13][~:]=[~oo]240-1313Hence,wenowhave1000011846.2U=EUcEBando=1000<x<1001846.2=23.08x>100o=802-15

36. AulysisofCOilinDmassystems;diUerenliaiandvarialiOlla1I01'1BDlaIiGlSuEXACT~------::.:--~~~.-.-. "Solution2---..I~..,-__--r-__--.,r---~X15000E10000E5000-E- 100180CALCULATEDDISPLACEMENTS(J50100-I=:::==-==_==_:=os:=_=_=,==_=_== ""EXACT"I~ ~SOLUTION1I-<,JSOLUTION2L._._.~._._ -+---,~--------r-------~X100180CALCULATEDSTRESSES2·18

37. balysisofcoatiDloassystms;diBerenlialudvariationalfonnllatioasWenotethatinthislastanalysiseweusedtrialfunctionsthatdonotsatisfythenaturalb.c. ethetrialfunctionsthemselvesarecontinuous,butthederivativesarediscontinuousatpointB. 1foraem-variationalproblemweonlyneedcontinuityinthe(m-1)stderivativesofthefunctions; inthisproblemm=1. edomainsA-BandB-earefiniteelementsandWEPERFORMEDAFINITEELEMENTANALYSIS. 2·17

38. FORMULATIONOFTHEDISPLACEMENT-BASEDFINITEELEMENTMETHODLECTURE358MINUTES3·1

39. Formulationofthedisplacement-basedfiniteelementmethodLECTURE3Generaleffectiveformulationofthedisplacement- basedfiniteelementmethodPrincipleofvirtualdisplacementsDiscussionofvariousinterpolationandelementmatricesPhysicalexplanationofderivationsandequationsDirectstiffnessmethodStaticanddynamicconditionsImpositionofboundaryconditionsExampleanalysisofanonunifor mbar.detaileddiscussionofelementmatricesTEXTBOOK:Sections:4.1.4.2.1.4.2.2Examples:4.1.4.2.4.3.4.43·2

40. Formulationofthedisplacement-basedfiniteelementmethodFORMULATIONOFTHEDISPLACEMENTBASEDFINITEELEMENTMETHOD- Averygeneralformulation-Providesthebasisofalmostallfiniteelementanalysesperformedinpractice- TheformulationisreallyamodernapplicationoftheRitz/ Gelerkinproceduresdiscussedinlecture2-Considerstaticanddynamicconditions,butlinearanalysisFig.4.2.Generalthree-dimensionalbody. 3·3

41. FOl'DlulationofthedisplaceDlent·basedfinitee1mnentlDethodTheexternalforcesarefBf~FiXXfB=fBfS=fSFi=Fi(4.1)yyyfBfSFiZZZThedisplacementsofthebodyfromtheunloadedconfigurationaredenotedbyU,whereuT=[uVw] ThestrainscorrespondingtoUare, (4.2) ~T=[EXXEyyEZZYXyYyZYZX](4.3) andthestressescorrespondingto€ are3·4

42. Formulationofthedisplacement-basedfiniteelementmethodPrincipleofvirtualdisplacementswhereITT=[ITIfw](4.6) Fig.4.2.Generalthree-dimensionalbody. 3·5

43. Formulationofthedisplaceaenl-basedfililee1eDlenl.ethodx,u,, "" FiniteelementForelement(m)weuse: !!(m)(x,y,z)=!:!.(m)(x,y,z)0(4.8) "T!!=[U,V,W,U2V2W2•••UNVNWN] "T!!=[U,U2U3...Un](4.9) §.(m)(x,y,z)=~(m)(x,y,z)!!(4.'0) !.(m)=f(m)~(m)+-rI(m)(4.'1) 3·&

44. 'OI'IIalationofthedisplaceDlenl-basedfilileeleDlenlmethodRewrite(4.5)asasumofintegrationsovertheelements(4.12) Substituteinto(4.12)fortheelementdisplacements,strains,andstresses, using(4.8),to(4.10),____---..ll.c=~~-------(m)Tj-I--£ 'iTl~1B(m)Tc(m)B(m)dv(m)jU=If~v(m)-l--£1----~(m)=f.(m)~(m) j[IT(m)j(-£)(m)=B(m)l..u·) TLl(m)!!(m)1.BdV(m)--- I1mVI()T_",._~m:,.Lf. m)!!sCm)Tim)dScmljy:(m)=!!(m)~ _m_JV...:........<,==~I______(m)TElB(m)TTI(m)dv(m)j-USm;rm)---(m)T-___.r__.........1------....~ "<I::: (4.13) 3·7

45. Formulationofthedisplacement-basedfiniteelement.ethodWeobtainKU=Rwhere(4.14) R=.Ba+Rs-R1+~(4.15) K=~fB(m)Tc(m)B(m)dV(m) -mJV(m)----(4.16) R="'1.H(m)TfB(m)dV(m)(4.17) ~~lm)-- R="'1HS(m)Tfs(m)dS(m)(4.18) -S~~m)-- R="'1B(m)TT1(m)dV(m)(4.19) -1~V(m)-- R=F~- Indynamicanalysiswehavef(m)T-B(m) ~B=~V(m).!:!.[1. _p(m).!:!.(m)~]dV(m) MD+KU=R(4.21) (4.22) (4.20) B(m)-B(m)••(m) 1.=1.-p!! 3·8

46. Formulationofthedisplacement-basedfiniteelementmethodToimposetheboundaryconditions, weuse~a~b~a~b+ ~at!t>b-~ =(4.38) .... ~a~+~a~=~-~b~-~b~ (4.39) ~=~a~+~b~+~a~+~b~ (4.40) ransformedegreesofeedomiA!•VTId!-fGlobaldegreesoffreedom;-VC,-:~:e)~I(restrained~ rl'fuL[ COSaT= sina-sina] cosa'/..U=TITFig.4.10.Transformationtoskewboundaryconditions3·9

47. Formulationofthedisplacement-basedfiniteelementmethodForthetransformationonthetotaldegreesoffreedomweusesothat.. Mu+Ku=Rwhere.th.th1Jcolumn!1.••j(4.41)ithrow1cosa.-sina. T= }h1(4.42)sina.cosa. 1LFig.4.11.Skewboundaryconditionimposedusingspringelement. Wecannowalsousethisprocedure(penaltymethod) SayUi=b,thentheconstraintequationis___3·10kU.=kb, wherek»k.. " (4.44)

48. FormDlationofthedisplacement·basedfiniteelementmethodExampleanalysis80xzy100Finiteelementsarea=1element® 100area=9J~ I"100-I80~I3·11

49. Formulationofthedisplacement·basedfiniteelementmethodElementinterpolationfunctions1.0I... L--IDisplacementandstraininterpolationmatrices: H(l}=[(l-L)ya]-100100v(m}=H(m}U!:!.(2}=[a(1-L):0]80!!(l)=[11a] 100100av=B(m}U!!(2)=[11ay-- a8080] 3·12

50. FOI'IDDlationofthedisplacement·basedfiniteelementmethodstiffnessmatrix-11005.=(1HEllOl~O[-l~Ol~Oo}YaaU1-80180HenceE[2.4-2.4=240-2.415.4a-13SimilarlyforM'.!!B'andsoon. Boundaryconditionsmuststillbeimposed. 3·13

51. GENERALIZEDCOORDINATEFINITEELEMENTMODELSLECTURE457MINUTES4·1

52. GeneralizedcoordinatefiniteelementmodelsLECTURE4Classificationofproblems:truss,planestress,planestrain,axisymmetric,beam,plateandshellconditions: correspondingdisplacement,strain,andstressvariablesDerivationofgeneralizedcoordinatemodelsOne-,two-,three-dimensionalelements,plateandshellelementsExampleanalysisofacantileverplate,detailedderivationofelementmatricesLumpedandconsistentloadingExampleresultsSummaryofthefiniteelementsolutionprocessSolutionerrorsConvergencerequirements,physicalexplanations, thepatchtestTEXTBOOK:Sections:4.2.3,4.2.4,4.2.5,4.2.6Examples:4.5,4.6,4.7,4.8,4.11,4.12,4.13,4.14,4.15,4.16,4.17,4.184-2

53. GeneralizedcoordinatefiniteeleDlentmodelsDERIVATIONOFSPECIFICFINITEELEMENTS•Generalizedcoordinatefiniteelementmodels~(m)=iB(m)TC(m)B(m)dV(m) V(m) aW)=JH(m)TLB(m)dV(m) V(m) R(m)=fHS(m)TfS(m)dS(m) !!S(m)-- Setc. Inessence,weneedH(m)B(m)C(m)-,-'- •ConvergenceofanalysisresultsAAcrosssectionA-A: TXXisuniform. Allotherstresscomponentsarezero. Fig.4.14.Variousstressandstrainconditionswithillustrativeexamples. (a)Uniaxialstresscondition:frameunderconcentratedloads. 4·3

54. Ge.raJizedcoordiDalefiniteelementlDOIIeIsHale I6I --1ZI TXX'Tyy,TXYareuniformacrossthethickness. Allotherstresscomponentsarezero. Fig.4.14.(b)Planestressconditions: membraneandbeamunderin-planeactions. u(x,y),v(x,y) arenon-zerow=0,Ezz=0Fig.4.14.(e)Planestraincondition: longdamsubjectedtowaterpressure. 4·4

55. GeneralizedcoordinatefiniteelementmodelsStructureandloadingareaxisymmetric. j( I III,I I-- Allotherstresscomponentsarenon-zero. Fig.4.14.(d)Axisymmetriccondition: cylinderunderinternalpressure. (beforedeformation) (afterdeformation) / SHELLFig.4.14.(e)Plateandshellstructures. 4·5

56. GeneralizedcoordinatefiniteelementmodelsProblemBarBeamPlanestressPlanestrainAxisymmetricThree-dimensionalPlateBendingDisplacementComponents uw u,vu,vu,vu,v,wwTable4.2(a)CorrespondingKinematicandStaticVariablesinVariousProblems. ProblemBarBeamPlanestressPlanestrainAxisymmetricThree-dimensionalPlateBendingStrainVector~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'a1wa1wa1w•••,IC...,=-dxZ'IC77=-OyZ,IC..,=20xoyTable4.2(b)CorrespondingKinematicandStaticVariablesinVariousProblems. 4·&

57. ProblemBarBeamPlanestressPlanestrainAxisymmetricThree-dimensionalPlateBendingGeneralizedcoordinatefiniteelementmodelsStressVector1:T[T;u,] [Mn] [TnTJIJIT"'JI] [TnTJIJIT"'JI] [TnTJIJIT"'JITn] [TnTYJITnT"'JITJI'Tu] [MnMJIJIM"'JI] Table4.2(e)CorrespondingKinematicandStaticVariablesinVariousProblems. ProblemMaterialMatrix.£ BarBeamPlaneStressEEl[ 1vEv11-1':&o01~.] Table4.3GeneralizedStress-StrainMatricesforIsotropicMaterialsandtheProblemsinTable4.2.4·7

58. GeneralizedcoordinatefiniteelementmodelsELEMENTDISPLACEMENTEXPANSIONS: Forone-dimensionalbarelementsFortwo-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) 4·8

59. Hence,ingeneralu=~exGeneralizedcoordinatefiniteelementmodels(4.50) (4.51/52) (4.53/54) Example(4.55) Y.VX.Vla)CantileverplaterNodalpoint6lp9Element0058CD@ Y.VV7147X.VV7(blFiniteelementidealizationFig.4.5.Finiteelementplanestressanalysis;i.e.TZZ=TZy=TZX=04·9

60. Generalizedcoordinatefiniteelementmodels2LJ2.=US--II--.......---------....~ element® Elementnodalpointno.4=structurenodalpointno.5. Fig.4.6.Typicaltwo-dimensionalfour-nodeelementdefinedinlocalcoordinatesystem. Forelement2wehave[ U{X,y)](2) =H(2)uv{x,y)-- whereuT=[U-14·10

61. GeneralizedcoordinateliniteelementmodelsToestablishH(2)weuse: or[ U(X,y)]=_~l!. v(x,y) where!=[~~}!=[1xyxy] andDefiningwehaveQ=Aa. HenceH=iPA-14·11

62. GeneralizedcoordinatefiniteelementmodelsHenceH=fl-l4and(1+x)(Hy)::aI•••IIa::(1+x)(1+y): H'ZJ=[0-0Ull:HII:HZIUJVJUzt':u.v. U2U3U4UsU6U7UsU9U1aI0:HIJH17:HI.H16:00:HI.Hu: o:HZJH21:H::H:6:00:H..Ha: VI-elementdegreesoffreedomU12U13U14UIS-assemblagedegreesHIs:00zerosOJoffreedomHzs:00zerosO2x18(a)Elementlayout(b)Local-globaldegreesoffreedomFig.4.7.Pressureloadingonelement(m) 4·12

63. GeneralizedcoordinatefiniteelementmodelsInplane-stressconditionstheelementstrainsarewhereE-au.E_av._au+avxx-ax'yy-ay,Yxy-ayaxHencewhereI=[~10Iy'OI00010I01IX10I4·13

64. GeneralizedcoordinatefiniteelementmodelsACTUALPHYSICALPROBLEMGEOMETRICDOMAINMATERIALLOADINGBOUNDARYCONDITIONS1MECHANICALIDEALIZATIONKINEMATICS,e.g.trussplanestressthree-dimensionalKirchhoffplateetc. MATERIAL,e.g.isotropiclinearelasticMooney-Rivlinrubberetc. LOADING,e.g.concentratedcentrifugaletc. BOUNDARYCONDITIONS,e.g.prescribed1displacementsetc. FINITEELEMENTSOLUTIONCHOICEOFELEMENTSANDSOLUTIONPROCEDURESYIELDS: GOVERNINGDIFFERENTIALEQUATIONSOFMOTIONe.g. ..!..(EA.!!!)=-p(x)axaxYIELDS: APPROXIMATERESPONSESOLUTIONOFMECHANICALIDEALIZATIONFig.4.23.FiniteElementSolutionProcess4·14

65. GeneralizedcoordinatefiniteelementmodelsSECTIONERRORERROROCCURRENCEINdiscussingerrorDISCRETIZATIONuseoffiniteelement4.2.5interpolationsNUMERICALevaluationoffinite5.8.1INTEGRATIONelementmatricesusing6.5.3INSPACEnumericalintegrationEVALUATIONOFuseofnonlinearmaterial6.4.2CONSTITUTIVEmodelsRELATIONSSOLUTIONOFdirecttimeintegration,9.2DYNAMICEQUILI-.modesuperposition9.4BRIUMEQUATIONSSOLUTIONOFGauss-Seidel,Newton-8.4FINITEELEr1ENTRaphson,Quasi-Newton8.6EQUATIONSBYmethods,eigenso1utions9.5ITERATION10.4ROUND-OFFsetting-upequationsand8.5theirsolutionTable4.4FiniteElementSolutionErrors4·15

66. GeneralizedcoordinatefiniteelementmodelsCONVERGENCEAssumeacompatibleelementlayoutisused, thenwehavemonotonicconvergencetothesolutionoftheproblemgoverningdifferentialequation, providedtheelementscontain: 1)allrequiredrigidbodymodes2)allrequiredconstantstrainstates~compatibleLWlayoutCDincompatiblelayout~ t:=no.ofelementsIfanincompatibleelementlayoutisused,theninadditioneverypatchofelementsmustbeabletorepresenttheconstantstrainstates.Thenwehaveconvergencebutnon-monotonicconvergence. 4·16

67. Geuralizedcoordinatefinitee1eJDeDtmodels7" /"'r->,; / ( ""1----- ,I IIII IIiI (a)Rigidbodymodesofaplanestresselement......~_QIIII(b)AnalysistoillustratetherigidbodymodeconditionRigidbodytranslationandrotation; elementmustbestressfree. Fig.4.24.Useofplanestresselementinanalysisofcantilever4·17

68. Generalizedcoordinatefiliteelellent.adels------- RigidbodymodeA2=0Poisson'sratio"0.30Young'sr------, modulus=1.0II 10-l II I II _1RigidbodymodeAl=0ItIII 01I II •I1. ...--,.".-~-- .J--- (' --- ,........... _-I--IIIIIIfRigidbodymodeA3=0.....I'JFlexuralmodeA4=0.57692Fig.4.25(a)Eigenvectorsandeigenvaluesoffour-nodeplanestresselement~- ......" ...~-, ........~ .----"'"="""- --~ I'- FlexuralmodeAs=0.57692ShearmodeA.=0.76923r--------1IIIIIII: IIIIIIL.J,-----, IIIIIIIIIIIIIIIIIStretchingmodeA7=0.76923UniformextensionmodeAs=1.92308Fig.4.25(b)Eigenvectorsandeigenvaluesoffour-nodeplanestresselement4·18

69. (0®G) ®®-.® /@) @@ GeneralizedcoordiDatefiniteelementlDodels ·11~17'c. IT,I>~.f: 20ISa)compatibleelementmesh;2constantstressa=1000N/cmineachelement.YYb)incompatibleelementmesh; node17belongstoelement4, nodes19and20belongtoelement5,andnode18belongstoelement6. Fig.4.30(a)Effectofdisplacementincompatibilityinstressprediction0yystresspredictedbytheincompatibleelementmesh: PointOyy(N/m2) A1066B716C359D1303E1303Fig.4.30(b)Effectofdisplacementincompatibilityinstressprediction4·19

70. IMPLEMENTATIONorMETHODSINCOMPUTERPROGRAMS; EXAMPLESSAP,ADINALECTURE556MINUTES5·1

71. "pi_entationofmetllodsincomputerprograDlS;examplesSIP,ADlRALECTURE5ImplementationofthefiniteelementmethodThecomputerprogramsSAPandADINADetailsofallocationofnodalpointdegreesoffreedom.calculationofmatrices.theassemblageprocessExampleanalysisofacantileverplateOut- of-coresolutionEff&ctivenodal-pointnumberingFlowchartoftotalsolutionprocessIntroductiontodifferenteffectivefiniteelementsusedinone.two.three-dimensional.beam. plateandshellanalysesTEXTBOOK:AppendixA,Sections:1.3.8.2.3Examples:A.I.A.2.A.3.A.4.ExampleProgramSTAP5·2

72. l_pIg_talioaof_ethodsinCODIpDterprogram;mDlplesSAP,ADINAN=no.ofd.o.f. oftotalstructureTK(m)=1.B(m)C(m)B(m)dV(m) -V(m)--- R(m)=1.H(m)TfB(m)dV(m) -Bv(m)-- H(m)B(m)-- kxN.hNIMPLEMENTATIONOFTHEFINITEELEMENTMETHODWederivedtheequilibriumequationswhereInpractice, wecalculatecompactedelementmatrices. K=~K(m);R=~R(m) -m--Bm!..!B~,~B'nxnnxln=no.ofelementd.o.f. tl~ kxnR,xnThestressanalysisprocesscanbeunderstoodtoconsistofessentiallythreephases: 1.CalculationofstructurematricesK,M,C,andR,whicheverareapplicable. 2.Solutionofequilibriumequations. 3.Evaluationofelementstresses. 5·3

73. IDlpl_81taliolofDlethodsintoDlpulerprogrw;mDlplesSAP,ADIlIThecalculationofthestructurematricesisperformedasfollows: 1.Thenodalpointandelementinformationarereadand/ orgenerated. 2.Theelementstiffnessmatrices, massanddampingmatrices,andequivalentnodalloadsarecalculated. 3.ThestructurematricesK,M, C,andR,whicheverareapplicable,areassembled. lSz::6tW::3Zx/U::1/Sx::4r-----yV::2Sy::5Fig.A.1.Possibledegreesoffreedomatanodalpoint. I-nodalpoint_.-... -....iID(I,J)= Degreeoffreedom5·4

74. ....taIiOiofIIeIWsincoaplerP....UUlplesSIP,ABilATemperatureattopfaceal00"CaOem<D® E=l(Jl1N/cm22E=2xl(Jl1NIcm2t.,-0.15t"=0.2084_17-7~TemperatureatDegreeofbottomface=70'Cfreedomnumberat659t~l1@4ElementE.2xl(Jl1Nlem2numberE·lOSN/em2t4II'"0.20t10.,-0.153825__-9NodeFig.A.2.Finiteelementcantileveridealization. 1nthiscasethe10arrayisgivenby11100000011100000010=1111111111111111111111111111111111]15·5

75. 0.040.080.0] 0.00.00.0] 70.085.0100.0] IJDpleDIeDtatiODofmethodsinCODIpater.programs;examplesSAP,ADIIAandthen00013579110002468101210=000000000000000000000000000000000000AlsoXT=[0.00.00.060.060.060.0120.0120.0120.0] TY=[0.040.080.00.040.080.0TZ=[0.00.00.00.00.00.0TT=[70.085.0100.070.085.0100.05·6

76. Implementationofmethodsincomputerprograms;examplesSAP,ADINAFortheelementswehaveElement1:nodenumbers:5,2,1,4; materialpropertyset:1Element2:nodenumbers:6325·I,,, materialpropertyset:1Element3:nodenumbers:8547·,,,, materialpropertyset:2Element4:nodenumbers:9658-,,,, materialpropertyset:2CORRESPONDINGCOLUMNANDROWNUMBERSForcompactedImatrix12345678For!1..,4000012oJLMT=[34000012] 5·7

77. Implementationofmethodsincomputerprograms;examplesSAP,ADINASimilarly,wecanobtaintheLMarraysthatcorrespondtotheelements2,3,and4.Wehaveforelement2, LMT=[56000034] forelement3, LMT=[910341278] andforelement4, LMT=[11125634910] JSkYline.~000"o000"------m=3ok36'006'- k45k460"0(a)ActualstiffnessmatrixksskS6k6612461012161822A(21)storeskS8Fig.A.3.Storageschemeusedforatypicalstiffnessmatrix. A(17) A(16) A(15tA(14) A(13) A(12) A(91A(8) A(7) A(6)A(lllA(lO) Symmetric(b)ArrayAstoringelementsofK. A(l)A(3) A(2)A(S) A(4) ,.mK=3·1"kllk120k14knk230k33k34K= k44A= 5·8

78. _pl••taiioooflDeIJaodsinCOIDpaterprograJDS;eDIIIplesSAP,ADINAx=NONZEROELEMENT0=ZEROELEMENT.--,COLUMNHEIGHTSIIIX00010010o000:00:0xixx0100xXIX001000XIX00X00X0X1000IxxlxXIOxixXiXSYMMETRICIXXlXXIXIXELEMENTSINORIGINALSTIFFNESSMATRIXFig.10.Typicalelementpatterninastiffnessmatrixusingblockstorage. BLOCK1BLOCK2~---~ IX0X0XIXXiXXiXIX~_BLOCK4ELEMENTSINDECOMPOSEDSTIFFNESSMATRIXFig.10.Typicalelementpatterninastiffnessmatrixusingblockstorage. 5·9

79. IIIlpl••tationofmethodsincomputerprograms;examplesSAP,ABilA203~1234567891011121,141516171819~21222324252627282930313233322283033(b)Goodnodalpointnumbering, mk+1=16. Fig.A.4.Badandgoodnodalpointnumberingforfiniteelementassemblage. (a)Badnodalpointnumbering, mk+1=46.911141619212426293164512712172227~, ,3810131518202355·10

80. "pI••tationofIlethodsincOIlpulerprogram;exallplesSAP,ADINASTARTREADNEXTDATACASEReadnodalpointdata(coordinates,boundaryconditions)andestablishequationnumbersinthe10array. Calculateandstoreloadvectonforallloadcases. Read.generate.andstoreelementdata.Loopoverallelementgroups. Readelementgroupdata,andassembleglobalstructurestiffnessmatrix.Loopoverallelementgroups. Calculate.b..Q..!:.Tfactorizationofglobalstiffnessmatrix(·) FOREACHLOADCASEReadloadvectorandcalculatenodalpointdisplacements.~---1Readelementgroupdataandcalculateelementstresses. Loopoverallelementgroups. ENDFig.A.5.FlowchartofprogramSTAP.*SeeSection8.2.2.5-11

81. Implementationofmethodsincomputerprograms;examplesSAP,ADINAzONE-DIMENSIONALELEMENTI-'-------4!RINGELEMENTx;--.-------------...yFig.12.Trusselementp.A.42.5-12z23Fig.13.Two-dimensionalplanestress,planestrainandaxisymmetricelements. p..A.43. y

82. yImplementationofmetbodsincomputerprograms;examplesSAP,ADINA2---5x~------Fig.14.Three-dimensionalsolid-------....~ andthickshellelementp.A.44. zyFig.15.Three-dimensionalbeamelementpA.45.5·13

83. Implementationofmethodsincomputerprograms;examplesSAP,ADINA3-16NODESTRANSITIONELEMENT• • ----.__e_ ---L~-----• yxFig.16.Thinshellelement(variable-number-nodes) p.A.46.5·14

84. FOBMULATIONANDCALCULATIONOFISOPABAMETBICMODELSLECTURE657MINUTES6·1

85. FOl'DlolationandcalculationofisoparmetricmodelsLECTURE6FormulationandcalculationofisoparametriccontinuumelementsTruss.plane-stress.plane-strain.axisymmetricandthree-dimensionalelementsVariable-number-nodeselements.curvedelementsDerivationofinterpolations. displacementandstraininterpolationmatrices.theJacobiantransformationVariousexamples:shiftingofinternalnodestoachievestresssingularitiesforfracturemechanicsanalysisTEXTBOOK: Sections:5.1.5.2.5.3.1.5.3.3.5.5.1Examples:5.1.5.2.5.3.5.4.5.5.5.6.5.7.5.8.5.9.5.10.5.11,5.12.5.13.5.14.5.15.5.16.5.176·2

86. FOI'DlDlatiOludcalculationofisopariUHbicmodelsFORMULATIONANDCALCULATIONOFISOPARAMETRICFINITEELEMENTSinterpolationmatricesandel ementmatrices-Weconsideredearlier(lecture4)generalizedcoordinatefiniteelementmodels-WenowwanttodiscussamoregeneralapproachtoderivingtherequiredisoparametricelementslsoparametricElementsBasicConcept:(ContinuumElements) InterpolateGeometryNx=Li=lh.x.; IINy=Li=1h.y.; IINz=Li=lh.z. IIInterpolateDisplacementsNu=1: i=1h.u. IINv=Li==1h.v. IINw=Li=1h.w. IIN=numberofnodes&·3

87. Formulationandcalculationofisoparametricmodels1/0ElementTruss2/0ElementsPlanestressContinuumPlanestrainElementsAxisymmetricAnalysis3/0ElementsThree-dimensionalThickShell(a)Trussandcableelements(b)Two-dimensionalelementsFig.5.2.Sometypicalcontinuumelements6·4

88. FOI'IlalationandcalcalatioD01isoparametricmodels(c)Three-dimensionalelementsFig.5.2.SometypicalcontinuumelementsConsiderspecialgeometriesfirst: ~~==-=l======~I=-==r======r==~1Truss,2unitslong6·5

89. F..utioaandcalculationofisoparalDebiclDodelsSIll( 11~J~ll(- 11- r1-DElement2Nodes: 2/Delement,2x2unitsSimilarly3/Delement2x2x2units(r-s-taxes) -11.0~~-+-.._h1=%(1+r) 2-r1-r

90. Formulationudcalculation01isoparUletriclIodeis1.0----...-e_----......:::::...::::=---..:...::-:::;.h2=Y.z(1-r)-Y.z(1-r2) 2312-0Element4Nodes: 3Similarlyh2=%(1-r)(1+5) h3=%(1-r)(1-5) h4=%(1+r)(1-s) /-r-r----+~~-rh1=~(1+r)(1+5) 46-7

91. Formulationandcalculationofisoparametricmodels6·833ConstructionofSnodeelement(2dimensional) firstobtainhS: ....--+-+--+~_.....1-+--+-I--I--I---I--------I--.._rThenobtainh1andh2: tfF--.-~-_-_-~-r~~_------..1L...4.!1.01h1=%(1+r)(1+s) -%hSSim.h2=%(1-r)(1+s) -%hS4

92. Formulationandcalculationofisoparametricmodelsr=+1y63 r=-1 ---;q----- r=0s=o---.. 8rx(a)Fourto9variable-number-nodestwo-dimensionalelementFig.5.5.Interpolationfunctionsoffourtoninevariable-number-nodestwo-dimensionalelement. -~hs·1·.-~hs-~hs-;h6...-~h6-~h7Includeonlyifnodeiisdefinedh,=~(l+r)(l+s) h2=~(l-r)(l+s) h3=~(l-r)(1-s) h.=~(1+r)(l-s) hs=~(1-r2)(1+s) 'h6=~(1-S2)(1-r) h7=~(1-r2)(1-s) hs=i(1-s2)(1+r) h~::(1-r"")(1-S'") i=5i=6i=7i=8I::r-~her-~h<j-ihq-~hq-ihq-1h<j-th" -th<f(b)InterpolationfunctionsFig.5.5.Interpolationfunctionsoffourtoninevariable-number-nodestwo-dimensionalelement: 6·9

93. FonnulationandcalculationofisoparametricmodelsHavingobtainedthehiwecanconstructthematricesHand!!: -TheelementsofHarethe·h· -I(orzero) -TheelementsofBarethederivativesofthehi(orzero), Becauseforthe2x2x2elements~wecanuse1:;'=~ x==ry==sz==tEXAMPLE4node2dim.element6·10

94. Formulationandcalculationofisoparametricmodelsah10ah40araru1[ Erlah1ah4v100ESSasasu2Yrsah1ah13h4ah4asat'asarv4..Iv- BWenoteagainr==xs=yGENERALELEMENTSY,vsr=+1s=+1r---t---4_r• 6·11

95. FormulationandcalculationofisoparametricmodelsDisplacementandgeometryinter- polationasbefore,but[:]=[:::]l~]Aside:asasasaycannotuseoraaar---ax+...axaraa-=Jax(ingeneral)ar- a_J-1a(5.25)a-x-arUsing(5.25)wecanfindthematrix.!!.ofgeneralelementsThe!:!andJ!matricesareafunctionofr,s,t;fortheintegrationthususedv=detJdrdsdt6·12

96. FOI'Dlalationudcalculationofisoparmebic.odelsFig.5.9.Sometwo-dimensionalelementsElement1z.._----+----......-"·r+-3,'4- """1--1-----------t..~16em. XElement22. ... J= +--_1<'0'II=> 1+3.....-------.....; cDG-W o112132&-13

97. FormulationandcalculationofisoparametricmodelsElement3c.1V (1+5)] (3+r) 2cl"l12. I• -+---~~--,c:'3,.~'t'.....,.. .I. ...L...3-"' 1------14--, 1c.W'I3r=-INaturalspace3 •I I,-.-I'L/4ActualphysicalspaceFig.5.23.Quarter-pointonedimensionalelement. 6·14

98. FormulationandcalculationofisoparametricmodelsHerewehave3x=L: i=1henceL2h.x.9x=-4(1+r) 11J=[!:..+!'-LJ-22andorSincer=2.Jf-1Wenote1singularityatX=0! /x6·15

99. FormulationandcalculationofisoparaDlebicDlodelsNumericalIntegrationGaussIntegrationNewton-CotesFormulasK='"a··kF··k-!:JIJ-IJI,J,kxx6·16

100. FORMULATIONOFSTRUCTURALELEMENTSLECTURE752MINUTES7·1

101. FormulationofstructuralelementsLECTURE7FormulationandcalculationofisoparametricstructuralelementsBeam,plateandshellelementsFormulationusingMindlinplatetheoryandunifiedgeneJ," alcontinuumformulationAssumptionsusedincludingsheardeformationsDemonstrativeexamples:two-dimensionalbeam, plateelementsDiscussionofgeneralvariable-number-nodeselementsTransitionelementsbetweenstructuralandcontinuumelementsLow- versushigh-orderelementsTEXTBOOK:Sections:5.4.1,5.4.2,5.5.2,5.6.1Examples:5.20,5.21,5.22,5.23,5.24,5.25,5.26,5.277·2

102. FORMULATIONOFSTRUCTURALELEMENTS•beam,plateandshellelements•isoparametricapproachforinterpolationsContinuumApproachFOI'IIDlati....slnclDraie1U11DIsStrengthofMaterialsApproach•straightbeamelementsusebeamtheoryincludingsheareffects•plateelementsuseplatetheoryincludingsheareffects(ReissnerIMindlin) "particlesremainonastraightlineduringdeformation" Usethegeneralprincipleofvirtlialdisplacements,but--excludethestresscomponentsnotapplicable--usekinematicconstraintsforparticlesonsectionsoriginallvnormaltothemidsurfacee. g. beame.g. shell7·3

103. Formulationofstructuralelements.. xNeutralaxisBeamsectionBoundaryconditionsbetweenbeamelementsDeformationofcross-sectionwi=wi;-0+0xxdw_dwdx-0-dx+0xxa)BeamdeformationsexcludingsheareffectFig.5.29.BeamdeformationmechanismsNeutralaxisBeamsectionDeformationofcross-sectionWI-Wix-Ox+OBoundaryconditionsbetweenbeamelements./ b)BeamdeformationsincludingsheareffectFig.5.29.Beamdeformationmechanisms7·4

104. FormulationofstructuralelementsWeusedwS=--ydx(5.48) (5.49) _(LJpwdxoL-LmSdxo(5.50) L+GAkJ(~~-S)o(~~-S)dxoL-ipoWdxoL-imoSdx=0o(5.51) 7·5

105. Formulationofstructuralelements(a)BeamwithappliedloadingE=Young'smodulus,G=shearmodulus3k=§..A=abI=ab6',12Fig.5.30.Formulationoftwodimensionalbeamelement( b)Two,three-andfour-nodemodels; 0i={3i'i=1,...,q(InterpolationfunctionsaregiveninFig.5.4) Fig.5.30.Formulationoftwodimensionalbeamelement7 ·6

106. FormulationofstructuralelementsTheinterpolationsarenowqW=~h.w.L..J11i=, qB=~h.e.L..J11i=, (5.52) w=HU'B=HU1-/-'.:...:.s- dW=BU'~=BUdX1-/-'dX~- WhereTQ.=[w,Wq8,8qJ~=[h,hq0OJ~=[00h,hqJ(5.53) (5.54) and!!w=J-1[:~l...:>0...0] __,f,dh,dhq] ~-JLO...adr'...ar(5.55) 7·7

107. FormulationofstructuralelementsSothatK=E1f1T~~detJdr-1and+GAkt-1T(~-tla)(~-~)detJdr(5.56) R=f~pdetJdr-1+/~mdetJdr(5.57) -1Consideringtheorderofinterpolationsrequired, westudyGAk(5.60) ex.=ITHence-useparabolic(orhigher-order) elements.discreteKirchhofftheory-reducednumericalintegration7-8

108. FormulationofstructuralelementsFig.5.33.Three-dimensionalmoregeneralbeamelementHereweuse(5.61) qQ,z(r,s,t)=Lk=lq+~'bhQ,Vk2L.-kksxk=lqqQ,y(r,s,t)=LhkQ,Yk+iLakhkQ,V~yk=lk=lq+~'"bhQ,Vk2LJkksyk=lqhkQ,Zk+iLakhkQ,V~Zk=lq+~2'"bh£VkLJkkszk=l7·9

109. FormulationofstructuralelementsSothat10u(r,s,t)=x-xv(r,s,t)=ly_0y(5.62) 10w(r,s,t)=z-zqv(r,s,t)=L: k=landqtqku(r,s,t)=L:hkuk+"2L:akhkVtxk=lk=lq+t.EbkhkV~xk=ltqhkvk+2Lk=lq+tL: k=lqw(r,s,t)=L: k=l(5.63) 7·10

110. FormulationofstructuralelementsFinally,weexpressthevectorsV~ andV~intermsofrotationsabouttheCartesianaxesx,y,z, kakv=ex'is...:..s~ whereekxe=ek~yekz(5.65) (5.66) Wecannowfind£nnqYni;=~!4~(5.67) k=lYnl;; whereuT=[Ukvkwkekekek](5.68)~xyzandthenalsohaveTnnEaa£nnTn~=aGkaYn~ TnI';;0aGkYnl;; (5.77) 7-11

111. Formulationofstructuralelementsandw=w(x,y) ....------ (5.78) HenceFig.5.36.DeformationmechanismsinanalysisofplateincludingsheardeformationsEXXdl dXdSEyy=z_-.1.(5.79)dyYxydSX_dSydydXdWSyYyzdy- =(5.80) dWYzx-+SdXx7·12

112. FormulationofstructuralelementsandLXX1vaLyy=z_E_v1a2l-vaal-vLxy2(5.81) awLyzay-ByE(5.82)=2(1+v) LZXaw+BaxxThetotalpotentialfortheelementis: 1II=- 2LxydzdA+~ 2ffh/yyZYzx]~yzJdxdAA-h/2~zx-fwPdAA(5.83) 7·13

113. FormulationofstructuralelementsorperformingtheintegrationthroughthethicknessIT=tiT.<q,.<dA+t//f,;ydAAA-I:PdA(5.84) AwhereK= as_.-J.- ayasx_~ ayax;y= aw+saxx(5.86) 1v0Eh310C=.v~12(l-v2)1-v0027·14[ 1Ehkf.s=2{1+v)0(5.87)

114. FormulationofstructuralelementsUsingtheconditionc5TI=0weobtaintheprincipleofvirtualdisplacementsfortheplateelement. -fwpdA=0A(5.88) Weusetheinterpolationsqw=~h.w.LJ11i=lqS=~h.eiyLJ1xi=landqx=~h.x.LJ11i=l(5.89) qY=~h.y.LJ11;=17·15

115. FormulationofstructuralelementssMid-surfacer....-~-----t~ Fig.5.38.9-nodeshellelementForshellelementsweproceedasintheformulationofthegeneralbeamelements, (5.90) 7·16

116. FormulationofstructuralelementsTherefore, whereToexpressY~intermsofrotationsatthenodal-pointkwedefine(5.91) (5.92) °V1k=(exOvk)/Iex°Vkl(5.93a)--y-n-y-nthenVk°Vk°VkS..:...n=-~O',k+-1k(5.94) 7·17

117. Finally,weneedtorecognizetheuseofthefollowingstress-strainlawl=~h~(5.100) 1vaaaa1aaaaJlaaaT(1_~2)!2sh~h=~h1-vaa-2- 1-va-2- symmetric1-v2(5.101) 16·nodeparentelementwithcubicinterpolationI-2-I5•• 2•• Somederivedelements: 64£>-[> 000o'.'. Variable-number-nodesshellelement7·18

118. Formnlalionofstructuralelementsa)Shellintersections• b)SolidtoshellintersectionFig.5.39.Useofshelltransitionelements7·19

119. NUMERICALINTEGRATIONS, MODELINGCONSIDERATIONSLECTURE847MINUTES8·1

120. Numericalintegrations,modelingconsiderationsLECTURE8EvaluationofisoparametricelementmatricesNumercialintegrations.Gauss.Newton-CotesformulasBasicconceptsusedandactualnumericaloperationsperformedPracticalconsiderationsRequiredorderofintegration. simpleexamplesCalculationofstressesRecommendedelementsandintegrationordersforone-,two-.three-dimensionalanalysis.andplateandshellstructuresModelingconsiderationsusingtheelements. TEXTBOOK:Sections:5.7.1.5.7.2.5.7.3.5.7.4.5.8.1.5.8.2.5.8.3Examples:5.28.5.29.5.30.5.31.5.32.5.33.5.34.5.35.5.36.5.37.5.38.5.398·2

121. Numericalintegrations.modelingconsiderationsNUMERICALINTEGRATION. SOMEMODELINGCONSIDERATIONS•Newton-Cotesformulas•Gaussintegration•Practicalconsiderations•ChoiceofelementsWehadK=fBTCBdV(4.29) -V--- M=JpHTHdV(4.30) -V-- R=fHTfBdV(4.31) ~V-- TR=fHSfSdS(4.32)-sS-- Rr=f~T!.rdV(4..33) V8·3

122. Numericalintegrations,modelingconsiderationsInisoparametricfiniteelementanalysiswehave: -thedisplacementinterpolationmatrixt:!(r,s,t) -thestrain-displacementinterpolationmatrix~(r,s,t) Wherer,s,tvaryfrom-1to+1. Henceweneedtouse: dV=det.4drdsdtHence,wenowhave,forexampleintwo-dimensionalanalysis: +1+1!$=ff~T~~detAdrds-1-1+1+1M=ffptlTttdetJdrds-1-1etc... 8·4

123. Numericalintegrations,modelingconsiderationsTheevaluationoftheintegralsiscarriedouteffectivelyusingnumericalintegration,e.g.: K=L~a.·.F..-.4JlJ-lJ1Jwherea... IJF··-IJi,jdenotetheintegrationpoints=weightcoefficients=B··TCB··detJ··-IJ--IJ~J-r-r=±O.5775=±O.577r=±O.7755=±0.775r=05=0, 2x2-pointintegration8·5

124. Numericalintegrations.modelingcoDSideratiODSzL.-----.~Y3x3-pointintegrationConsiderone-dimensionalintegrationandtheconceptofaninterpolatingpolynomial. 1storderinterpolating---"'--polynomialinx. .-8a II a+b-2- xb

125. Numericalintegrations,modelingconsiderationsIactualfunctionF2ndorderinterpolating~~~~polynomialinx. aa+b2betc.... InNewton-Cotesintegrationweusesamplingpointsatequaldistances, andbn{F(r)dr=(b-a)~C.nF.+RJLJ11na;=0(5.123) n=numberofintervalsCin=Newton-Cotesconstantsinterpolatingpolynomialisofordern. 8·7

126. Numericalintegrations,modelingconsiderationsUpperBoundonErrorR.asNumberofaFunctionofIntervalsnqqCncnqC·CntheDerivativeofF23561110-I(b-a}lF"(r)"2T214110-3(b-a)5PV(r)6"6"6" 313311O-3(b-a)5F'V(r)"8"8"8"847321232710-6(b-a)7FVI(r)9090909090519755050751910-6(b-a)7Fv'(r)288288US288ill288641216272722721641lO-'(b-a)'FVIU(r)840840840840840840840Table5.1.Newton-Cotesnumbersanderrorestimates. InGaussnumericalintegrationweusebfF(r)dr"U1F(r1)+u2F(r2)+••. a+0.F(r)+Rnnn(5.124) whereboththeweightsa1•...•anandthesamplingpointsr1•...•~ arevariables. Theinterp(llatingpolynomialisnowoforder2n-1.8·8

127. Numericalintegrations,modelingconsiderationsnrj/X, 1O.(I5zeros)2.(I5zeros) 2±0.5773502691896261.0ססoo0ססoo0ססoo3±0.7745966692414830.5555555555555560.0ססoo0ססoo0ססoo0.8888888888888894±0.8611363115940530.347854845137454±0.3399810435848560.6521451548625465±0.9061798459386640.236926885056189±0.5384693101056830.4786286704993660.0ססoo0ססoo0ססoo0.5688888888888896±0.9324695142031520.171324492379170±0.6612093864662650.360761573048139±0.2386191860831970.467913934572691Table5.2.SamplingpointsandweightsinGauss-Legendrenumeri- calintegration. Nowlet, ribeasamplingpointandelibethecorrespondingweightfortheinterval-1to+1. Thentheactualsamplingpointandweightfortheintervalatobarea+b+b-ar.andb-ael. -2-212IandtheriandelicanbetabulatedasinTable5.2.8·9

128. Numericalintegrations,modelingconsiderationsIntwo-andthree-dimensionalanalysisweuse+1+1ffF(r,s)drds=I:"1-1-11or+1fF(ri's)ds-1(5.131) +1+1ffF(r,s)drds=I:,,;,,/(ri'sj) -1-1i,j(5.132) andcorrespondingto(5.113), a·IJ•=a.a.,wherea.anda. IJIJaretheintegrationweightsforone-dimensionalintegration. Similarly, +1+1+1ff1F(r,s,t}drdsdt-1-1-1=~a.·a.·a.kF(r.,s.,tk)LJ1J1Ji,j,k(5.133) anda··k=a.Q.Qk.IJIJ8·10

129. Numericalintegrations,modelingconsiderationsPracticaluseofnumericalintegration.Theintegrationorderrequiredtoevaluateaspecificelementmatrixexactlycanbeevaluatedbystudyingthefunctionftobeintegrated. •Inpractice,theintegrationisfrequentlynotperformedexactly, butthe·integrationordermustbehighenough. Consideringtheevaluationoftheelementmatrices,wenotethefollowingrequirements: a)stiffnessmatrixevaluation: (1)theelementmatrixdoesnotcontainanyspuriouszeroenergymodes(i.e.,therankoftheelementstiffnessmatrixisnotsmallerthanevaluatedexactly);and(2)theelementcontainstherequiredconstantstrainstates. b)massmatrixevaluation: thetotalelementmassmustbeincluded. c)forcevectorevaluations: thetotalloadsmustbeincluded. 8·11

130. Numericalintegrations,modelingconsiderationsDemonstrativeexample2x2Gaussintegration"absurd"results3x3GaussintegrationcorrectresultsFig.5.46.8-nodeplanestresselementsupportedatBbyaspring. Stresscalculations(5.136) •stressescanbecalculatedatanypointoftheelement. •stressesare,ingeneral,discontinuousacrosselementboundaries. 8-12

131. Numericalintegrations.modelingconsiderationsthickness=1cmA-p3xl0721~[E=N/cm<3>e.CD)=0.3I1>p300N=c· :.....,--of3c.m.3Coft'1. A8... '100N!Crrt'l. / (a)Cantileversubjectedtobendingmomentandfiniteelementsolutions. Fig.5.47.Predictedlongitudinalstressdistributionsinanalysisofcantilever. =a. 8·13

132. NumericalintegratiODS.modelingcoDSideratioDS'?,A, ~@B<D4l, ,C. I,s_a~"? v=0.3P=lOON"" 8&<DCo174+/lA/e-t'- CoAA ®B8<DCoc~I".00"'Ie-." (b)Cantileversubjectedtotip-shearforceandfiniteelementsolutionsFig.5.47.Predictedlongitudinalstressdistributionsinanalysisofcantilever. SomemodelingconsiderationsWeneed•aqualitativeknowledgeoftheresponsetobepredicted•athoroughknowledgeoftheprinciplesofmechanicsandthefiniteelementproceduresavailable•parabolic/undistortedelementsusuallymosteffective8-14

133. Numericalintegrations,modelingconsiderationsTable5.6Elementsusuallyeffectiveinanalysis. TYPEOFPROBLEMTRUSSORCABLETWO-DIMENSIONALPLANESTRESSPLANESTRAINAXISYMMETRICTHREE-DIMENSIONALELEMENT2-node8-nodeor9-node20-node DD 3-DBEAM-=~ 3-nodeor4-node-/..... PLATESHELL9-node9-nodeor16-nodeL7~~ 8·15

134. Numericalintegrations,modelingconsiderations4/'1odeIelEJmerrt1SnodegI'oole..~kl'7ll'"t~ e1er1l(1'It.iJII a)4-nodeto8-nodeelementtransitionregion84-Ioc:(e4nodeele,"~"t. eIemtnt"" A4-nodeel....~I"tc119BU.sVAA'Ve-llA~ CU, ConstraintuA=(uC+uB)/2equations: vA=(vC+vB)/2b)4-nodeto4-nodeelementtransition/. ! c)8-nodetofiner8-nodeelementlayouttransitionregionFig.5.49.Sometransitionswithcompatibleelementlayouts8·16

135. SOLUTIONOFFINITEELEMENTEQUILIBRIUMEQUATIONSINSTATICANALYSISLECTURE960MINUTES9·1

136. SolutionofIiDilee1eDleulequilihrilllequationsiuslaticaaalysisLECTURE9SolutionoffiniteelementequationsinstaticanalysisBasicGausseliminationStaticcondensationSubstructuringMulti-levelsubstructuringFrontalsolutiontl>tT-factorization(columnreductionscheme) asusedinSAPandADINACholeskyfactorizationOut-of-coresolutionoflargesystemsDemonstrationofbasictechniquesusingsimpleexamplesPhysicalinterpretationofthebasicoperationsusedTEXTBOOK:Sections:8.1.8.2.1.8.2.2.8.2.3.8.2.4. Examples:8.1.8.2.8.3.8.4.8.5.8.6.8.7.8.8.8.9.8.109·2

137. SoJutiODoffililee1emenlequilihrillDequationsinslaticanalysisSOLUTIONOFEQUILIBRIUMEQUATIONSINSTATICANALYSIS•Iterativemethods, e.g.Gauss-8eidel•DireetmethodsthesearebasicallyvariationsofGausselimination-staticcondensation-substructuring-frontalsolution-.LQ..bTfactorization-Choleskydecomposition-Crout-columnreduction(skyline)solverTHEBASICGAUSSELIMINATIONPROCEDUREConsidertheGausseliminationsolutionof5-4,0U,0-46-4,U2, =(8.2),-46-4U300,-45U409·3

138. SolationoffiniteelementeqailihriUlequationsinstaticanalysisSTEP1:Subtractamultipleofequation1fromequations2and3toobtainzeroelementsinthefirstcolumnofK. r------------ oll!16I5-5 II oI_~29:55Io:-45-41o1-45(8.3) 5-4oo9·4ooo14165-5r-------- 0:~_20I77I0:_2065I714I=(8.4)

139. SolationoffiniteelementeqailillriUlequationsinstaticanalysisSTEP3: 5-410U10014161U21S-s1520.-8(8.5) 007-TU3"7r--- 7000I5U4I"6"6II Nowsolvefortheunknownsu4, U3'U2andU,: 12=5(8.6) 1-(-156)U3-(1)U4_13U=--------:;-;;-----214-S519367o-(-4)35-(1)15-(0)"5_8U=----~----15-"59·5

140. SolutionoffiniteelementeqDilihriDlDequationsinslaticanalysisSTATICCONDENSATIONPartitionmatricesinto[~a~-ac][!!a][Ba] .!Sea~-ec!!c=BeHenceand(8.28) (-1)-1~a-~ilC.!Sec.!Sea!!a=Ba-~c.!Sec~---------KaaExampleteer:~ aIU105I-410I---+------------ -46-41U21= 1-46-4U3001-45U40~c'---y----' ~aHence(8.30)gives~-,-- 6-4-4[1/5][-41Kaa=-46-411-450'---1....- 9·8sothat141615-5K=1629-4-a.a-550]1-45~- andwehaveobtainedthe3x3unreducedmatrixin(8.3)

141. SoIltiOloffiniteelemelteqlilihrilDlequationsinstaticaualysis5-40VI:1-46-4U21-46-4U3:101-45U414-!§U2"55-!§29-4U305-5-45V40Fig.8.1PhysicalsystemsconsideredintheGausseliminationsolutionofthesimplysupportedbeam. 9-7

142. SolutiouoffiniteelementeqDilihriomequationsinstaticanalysisSUBSTRUCTURING•Weusestaticcondensationontheinternaldegreesoffreedomofasubstructure•theresultisanewstiffnessmatrixofthesubstructureinvolvingboundarydegreesoffreedomonly-?-? -~-o--oe--c>---nl-650x50Example......--.-L32x32Fig.8.3.Trusselementwithlinearlyvaryingarea. Wehavefortheelement. 9·8[ 17~~-206L3-2048-28

143. SoIali.oIliDilee1emealeqailihrilDleqaaliODSiastalicaaalysisFirstrearrangetheequationsEA,['76"L3-20StaticcondensationofU2givesEA,Ir76L33][-20]-[lJ[-2025-2848orll.EA,[19L-1and9·9

144. SolutionoffiDileelemeulequilibrilllequati.inslaticaDalysisMulti-levelSubstructuringI'L'I'L~,L,I.L.1A2A4A,ISA,II16A, ,,'~-&-o=2:E~f'·'n-~-UUr;,U6U7UsUgIU2U3u.RsBarwithlinearlyvaryingarea-II1- U,-u3u2---I•.-U,u3(a)First-levelsubstructure---IIII1- U,-Usu3_II•I1- U,Us(b)Second-levelsubstructure_IIIIIIII1- U,-ugUs.Rr;,-.IIIIIII1- U,ug(c)Third-levelsubstructureandactualstructure. Fig.8.5.Analysisofbarusingsubstructuring. '-10

145. SolutionoffiDilee1eDleulequilihrioequti.illstaticanalysisFrontalSolutionElementqElementq+1Elementq+2Elementq+3-------- mm+3~IN:" Element1Element44WavefrontWavefrontfornode1fornode2Fig.8.6.Frontalsolutionofplanestressfiniteelementidealization. •Thefrontalsolutionconsistsofsuccessivestaticcondensationofnodaldegreesoffreedom. •Solutionisperformedintheorderoftheelementnumbering. •Samenumberofoperationsareperformedinthefrontalsolutionasintheskylinesolution,iftheelementnumberinginthewavefrontsolutioncorrespondstothenodalpointnumberingintheskylinesolution. 9·11

146. SolutionoffiniteelementequilibriumequationsinstaticanalysisLDLTFACTORIZATION-isthebasisoftheskylinesolu- tion(columnreductionscheme) -BasicStepL-1K=K--1--1Example: 5-4a5-4a4-46-4a~416555= 1a-46-4a_1629-4-555aaaa-45a-45Wenote44-1-5-5L=1~11-1aa5S- oaaaaa9·12

147. SolutionoffiniteelementequilibriumequationsinstaticanalysisProceedinginthesameway-1-11.21.1K:=SxxxxxxxxxSx.......xupper:=triangularxxmatrixxxHenceorAlso,because~issymmetricwhere0:=diagona1rnatrixd..:=s.. 11119·13

148. SolutionoffiniteeleJDentequilihriDIIequationsinstaticanalysisIntheCholeskyfactorization,weusewheret=LD~ SOLUTIONOFEQUATIONSUsing9·14K=L0LTwehaveLV=RoLTU=Vwhere-IV:=L--n-land(8.16) (8.17) (8.18) (8.19) (8.20)

149. SolutionoffiniteelementequilibrimnequationsinstaticanalysisCOLUMNREDUCTIONSCHEME5-416-416-45~ 4545-5514-414-4-556-46-455~ 541541-55-551481148575-715-415-4TT559·15

150. Solationoffiniteelementeqailihriameqaati.instaticanalysisX=NONZEROELEMENT0=ZEROELEMENT_~COLUMNHEIGHTSSYMMETRICo0000o0000'-----, X000Xo0000o0x00oX000XXXX0XXXXXXXXXELEMENTSINORIGINALSTIFFNESSMATRIXTypicalelementpatterninastiffnessmatrixSKYLINEo0000o0000L...-_ X000XX000XX0X0XXXX0XXXXXXXXXXXXXXXXELEMENTSINDECOMPOSEDSTIFFNESSMATRIXTypicalelementpatterninastiffnessmatrix9-16

151. SYMMETRICSolutionoffiniteelementequilibriumequationsinstaticanalysisx=NONZEROELEMENT0=ZEROELEMENTCOLUMNHEIGHTSIII-x000100:0o000:0010xixx0100xXlX001000xIx00x00x0XO00xxixXIOxixxixIxXlXxIxIxELEMENTSINORIGINALSTIFFNESSMATRIXTypicalelementpatterninastiffnessmatrixusingblockstorage. 9·17

152. SOLUTIONOFFINITEELEMENTEQUILIBRIUMEQUATIONSINDYNAMICANALYSISLECTURE1056MINUTES10·1

153. Solotionoffinitee1mnenteqoiIihrioequationsindynaDlicanalysisLECTURE10SolutionofdynamicresponsebydirectintegrationBasicconceptsusedExplicitandimplicittechniquesImplementationofmethodsDetaileddiscussionofcentraldifferenceandNewmarkmethodsStabilityandaccuracyconsiderationsIntegrationerrorsModelingofstructuralvibrationandwavepropagationproblemsSelectionofelementandtimestepsizesIRec ommendationsontheuseofthemethodsinpracticeTEXTBOOK:Sections:9.1.9.2.1.9.2.2.9.2.3.9.2.4.9.2.5.9.4.1.9.4.2.9.4.3.9.4.4Examples:9.1.9.2.9.3.9.4.9.5.9.1210·2

154. SolutionoffiniteelementequilihriDlequationsindyDalDicualysisDIRECTINTEGRATIONSOLUTIONOFEQUILIBRIUMEQUATIONSINDYNAMICANALYSISMU+CU+KU=R------- •explicit,implicitintegration•computationalconsiderations•selectionofsolutiontimestep(b.t) •somemodelingconsiderationsEquilibriumequationsindynamicanalysisMU+CU+KU=R(9.1) or10·3

155. SolutionoffiniteelelleulequilihrilllequatiolSindynaJDicanalysisLoaddescriptiontimetime-- Fig.1.EvaluationofexternallyappliednodalpointloadvectortRattimet. THECENTRALDIFFERENCEMETHOD(COM) to=_l_(_t-tltu+t+tltU)(9.4) -2tlt-- anexplicitintegrationscheme10·4

156. SolationoffiniteeleDlenteqailibrimnequationsindynanaicanalysisCombining(9.3)to(9.5)weobtain(-'-M+-'-c)t+~tu=tR_~K__2_M)tu2-2~t----2-- ~t~t-(-'-M_-'-c)t-~tu2-2~t--~t(9.6) wherewenote!t!!=(~!(mT!! =~(l5-(m)tlL)=~t£(m) Computationalconsiderations•tostartthesolution.use(9.7) •inpractice.mostlyusedwithlumpedmassmatrixandlow-orderelements. 10·5

157. SolutionoffiniteelementequilibriumequationsindynamicanalysisStabilityandAccuracyofCOM-l'Itmustbesmallerthanl'IterTnl'Iter=TI;Tn=smallestnaturalperiodinthesystemhencemethodisconditionallystable_inpractice,useforcontinuumelements, l'It<l'IL-ee=~ forlower-orderelementsL'lL=smallestdistancebetweennodesforhigh-orderelementsl'IL=(smallestdistancebetweennodes)/(rel.stiffnessfactor) •methodusedmainlyforwavepropagationanalysis•numberofoperationsexno.ofelementsandno.oftimesteps10·6

158. SolutionoffiniteelelDenteqoiIibriDIIeqoatiouindynandcanalysisTHENEWMARKMETHOD(9.28) {9.29Janimplicitintegrationschemesolutionisobtainedusing.Inpractice,weusemostlya.=la,0=~ whichistheconstant-average-accelerationmethod(Newmark'smethod) •methodisunconditionallystable•methodisusedprimarilyforanalysisofstructuraldynamicsproblems•numberofoperations==~nm2+2nmt10·7

159. SolutionoffiniteelementequilibriDIIequationsindynmicanalysisAccuracyconsiderations•timestep!'1tischosenbasedonaccuracyconsiderationsonly•Considertheequations~1U+KU=RandwhereK¢. --1Using¢"1K¢=0.22::w·~1<p. 1--1whereweobtainnequationsfromwhichtosolveforxi(t)(seelecture11) 10·8..2Tx.+w.x.=~.R111~1-i=l,...,n

160. Solution01finiteeleDlentequilibriDllequationsindynaDlicanalysisHence,thedirectstep-by-stepsolutionofr~O+KU=Rcorrespondstothedirectstep-bystepsolutionof.. 2x·+w.x·111withi=l,...,nnU=~<I>.x. -~-l1i=1Therefore,tostudytheaccuracyoftheNewmarkmethod,wecanstudythesolutionofthesingledegreeoffreedomequation..2x+wx=rConsiderthecase..2x+wx=ao·ax=0··2x=-w10·9

161. SolotionoffiniteelementeqoiIihriDlequationsindynandcanalysis19.019.015.0Houbolt15.0method § 11.011.0.. le....5-wE!:...Cle.g7.007.0'"~C/IC0> "iii'"u"85.0'"5.0"0.;:'"'""0Co~ '".~ C/IQ.:! 3.0E3.0c'"'"8.l:! tf:! c'.01~4t€'"1.~l:! '"Q" 1.01.0~PE0.060.100.140.180.060.100.140.18Fig.9.8(a)Percentageperiodelonga-Fig.9.8(b)Percentageperiodelonga- tionsandamplitudedecays.tionsandamplitudedecays. 4t-----r----:--r--r----r-----,...-----,-----, equation..2.2.tx+~wx+wx=S1npstaticresponse2131----+--f-+-+----t-----t-----.,t----'-1...o'0~ "t:JCtIoCJ'ECtIc:: >o123Fig. 9.4.Thedynamicloadfactor10·10

162. SoIIIi.offilile81••1eqailihrillDeqaaliOlSindJllillicanalysisD:.r•1.05-nYNAMICRESPONSE_..-STATICRESPONSE~=0.05g7T+gir.itz~~.:.::::7'--!2C' ~1-.j.'",fs! ,;1•1!T 8174-._--+---t-"---....__..--t----+-._--+_..-........-..._-.-1'c.oeo.."~fl.'JOn.7~I.00II,.,~. Responseofasingledegreeoffreedomsystem. DLF..0.50-DYNAMICRESPONSE---STATIc.:RESPONSE.f...=3.0w.... ....,-------.//' --~-----=~---':....;-,,---=__==_7'~---_.~~.:.--==---/-/"7--_____...... + g, ::i-+-~--+---+---"..-------t-----+---+I--t-__---+--+1~--+--+I::-:---+----,+1:::---+----,+1::-:---+------::+-'::-:---+-----:<' c.':;:C.25~.I)C:."L:.00:..?~I.SO1.752.002.252.502.753.00TIllEResponseofasingledegreeoffreedomsystem. 10·11

163. SolutionoffiniteelementequilibriumequationsindynamicanalysisModelingofastructuralvibrationproblem1)Identifythefrequenciescontainedintheloading, usingaFourieranalysisifnecessary. 2)Chooseafiniteelementmeshthataccuratelyrepresentsallfrequenciesuptoaboutfourtimesthehighestfrequencywcontainedintheloading. u3)Performthedirectintegrationanalysis.Thetimestep/':,tforthissolutionshouldequalabout120Tu,whereTu=2n/wu'orbesmallerforstabilityreasons. ModelingofawavepropagationproblemIfweassumethatthewavelengthisLw'thetotaltimeforthewavetotravelpastapointis(9.100) wherecisthewavespeed.Assumingthatntimestepsarenecessarytorepresentthewave,weuse(9.101) andthe"effectivelength"ofafiniteelementshouldbe10·12c/':,t(9.102)

164. SoIaliOi..filile81••1eqailihriDleqaali_indJUlDicualysisSUMMARYOFSTEP-BY-STEPINTEGRATIONS-INITIALCALCULATIONS--- 1.FormlinearstiffnessmatrixK, massmatrixManddampingmatrix~,whicheverapplicable; Calculatethefollowingconstants: Newmarkmethod:0>0.50,ex.2:.0.25(0.5+0)22aO=,/(aAt) a4=0/ex.-, as=-a3a,=O/(aAt) as=I1t(O/ex.-2)/2ag=I1t('-0) a3=,/(2ex.)-, a7=-a2Centraldifferencemethod: a,='/2I1t...0O·0·· 2.Inltlahze!!.,!!.,!!.; Forcentraldifferencemethodonly,calculateI1tufrominitialconditions:- 3.Formeffectivelinearcoefficientmatrix; inimplicittimeintegration: inexplicittimeintegration: M=a~+a,f. 10·13

165. Solutionoffiniteelementequilibriumequationsindynamicanalysis4.IndynamicanalysisusingimplicittimeintegrationtriangularizeR:. ---FOREACHSTEP--- (j)Formeffectiveloadvector; inimplicittimeintegration: inexplicittimeintegration: (ii)Solvefordisplacementincrements; inimplicittimeintegration: inexplicittimeintegration: 10·14

166. SoI.ti.offilileelOl.1equilihriDlequationsindynamicanalysisNewmarkMethod: CentralDifferenceMethod: 10·15

167. MODESUPERPOSITIONANALYSIS;TIMEBISTORYLECTURE1148MINUTES11·1

168. ModeslperpClilionanalysis;lilliebistoryLECTURE11SolutionofdynamicresponsebymodesuperpositionThebasicideaofmodesuperpositionDerivationofdecoupledequations SolutionwithandwithoutdampingCaugheyandRayleighdampingCalculationofdampingmatrixforgivendampingratiosSelectionofnumberofmodalcoordinatesErrorsanduseofstaticcorrectionPracticalconsiderationsTEXTBOOK:Sections:9.3.1.9.3.2.9.3.3Examples:9.6.9.7.9.8.9.9.9.10.9.1111·2

169. Modesuperpositionanalysis;timehistoryModeSuperpositionAnalysisBasicideais: transformdynamicequilibriumequationsintoamoreeffectiveformforsolution, using!L=1:.!(t) nxlnxnnxlP=transformationmatrix!(t)=generalizeddisplacementsUsing!L(t)=1:.!(t) onMU+c0+KU=Rweobtain(9.30) (9.1) ~R(t)+fi(t)+R!(t)~(t) (9.31) whereCfT~f; R=PTR(9.32) 11·3

170. (9.34) ModesDperJMlilionualysis;tiDlehistoryAneffectivetransformationmatrixfisestablishedusingthedisplacementsolutionsofthefreevibrationequilibriumequationswi thdampingneglected, M0+KU=0Usingweobtainthegeneralizedeigenproblem, (9.36) withtheneigensolutions(w~,p..,), 22(ul2'~),...,(wn'.P.n),and11·4T1==0'<P1"M'""-_.:t:..Ji=ji.,j2<W-n(9.37) (9.38)

171. Modesuperpositionanalysis;timehistoryDefining(9.39) wecanwriteandhave(9.40) Nowusing!L(t)=!~Jt) ¢TM¢=I(9.41) (9.42) weobtainequilibriumequationsthatcorrespondtothemodalgeneralizeddisplacements!(t)+!T~!!(t)+r;i~(t)=!T!S.(t) (9.43) Theinitialconditionson~(t)areobtainedusing(9.42)andtheM-orthonormalityof¢;i.e., attime0wehave(9.44) 11·5

172. ModeSUperpClitiODaualysis;tilDebisloryAnalysiswithDampingNeglected(9.45) i.e.,nindividualequationsoftheform2.x.(t)+w.x.(t)=r.(t)1111wherewithTaX'I=lj).MU1-1--t=O•.TO'X'I=-'--cp.MU1-1--t=Oi=',2,...,n(9.46) (9.47) UsingtheDuhamelintegralwehave=-'jtr1·(T)sinw.(t-T)dTw.110(9.48) +a..sinw.t+8.cosw·t1111wherea.iand8iaredeterminedfromtheinitialconditionsin(9.47). Andthen11-& (9.49)

173. ModesDperp.itionanalysis;timehistory4f----..-----:--..--r----,..----~---_r_---..., equation••2.2.x+E;,wx+WX=S1nPtstaticresponse~=-031-__-+__+--+-+-__+-__-+-+-__--., 02.... 0..... uCtI..... -0CtI0uECtIr::::: >- 023Fig.9.4.ThedynamicloadfactorHenceweuseuP=~¢.x·(t) --~--l1i=1whereuP-UTheerrorcanbemeasuredusing(9.50) 11·7

174. Modesuperpositionanalysis;timehistoryStaticcorrectionAssumethatweusedpmodestoobtain~p,thenletn~_=LriUl~) i=1HenceTr.=¢.R1-1- ThenandKflUfiRAnalysiswithDampingIncludedRecall,wehave!(t)+!Tf!i(t)+fi!(t)=!T~(t) (9.43) IfthedampingisproportionalT¢.C(po=2w.E;,.cS.. -1---J111Jandwehave(9.51) x.(t)+2w.E;,.x.(t)+w~x.(t)=r1 ·(t) 111111i=l,...,n(9.52) 11·8

175. Modesuperpositionanalysis;timehistoryAdampingmatrixthatsatisfiestherelationin(9.51)isobtainedusingtheCaugheyseries, (9.56) wherethecoefficientsak'k=,,•••,p, arecalculatedfromthepsimultane- ousequationsAspecialcaseisRayleighdamping, C=a~1+BK----- example: Assume~,=0.02w,=2calculateaandBWeuse(9.55) or'/ a+Bw:-2w.~. --1112w.~. 1111·9

176. Modesuperpositionanalysis;timehistoryUsingthisrelationforwl'[,1andw2'[,2'weobtaintwoequationsforaand13: a+4ii=0.08a+913=0.60Thesolutionisa=-0.336and13=O.104.ThusthedampingmatrixtobeusedisC=-0.336M+0.104KNotethatsince2a+13w.=2w.[,. 111foranyi,wehave,onceaand13havebeenestablished, E,.= 12a+SW. 12w. 1a13=-+-w2w.2i111·10

177. ModesDperp.itionanalysis;timehistoryResponsesolutionAsinthecaseofnodamping. wesolvePequationsx.+2w.E,.x·+w~x.=r. 1111111withr·1ITOxit=0"--.!i!i.!:L•ITO'xit=0=!if1.!:LandthenPuP~¢.x.(t)LJ-11i=1Practicalconsiderationsmodesuperpositionanalysisiseffective-whentheresponseliesinafewmodesonly,P«n-whentheresponseistobeobtainedovermanytimeintervals( orthemodalresponsecanbeobtainedinclosedform). e.g.earthquakeengineeringvibrationexcitation-itmaybeimportanttocalculateEp(t)orthestaticcorrection. 11·11

178. SOLUTIONMETHODSFORCALCULATIONSOFFREQUENCIESANDMODESBAPESLECTURE1258MINUTES12·1

179. SolutionmethodsforcalculationsoffrequenciesandmodeshapesLECTURE12SolutionmethodsforfiniteelementeigenproblemsStandardandgeneralizedeigenproblemsBasicconceptsofvectoriterationmethods. polynomialiterationtechniques.Sturmsequencemethods.'transformationmethodsLargeeigenproblemsDetailsofthedeterminantsearchandsubspaceiterationmethodsSelectionofappropriatetechnique.practicalconsiderationsTEXTBOOK:Sections:12.1.12.2.1.12.2.2.12.2.3.12.3.1.12.3.2.12.3.3.12.3.4.12.3.6(thematerialinChapter11isalsoreferredto) Examples:12.1.12.2.12.3.12.412·2

180. SolatiumethodsforcalculationsoffrequenciesandmodeshapesSOLUTIONMETHODSFOREIGENPROBLEMSStandardEVP: r!=! nxnGeneralizedEVP: !sP.-=!i!-(=w2) QuadraticEVP: MostemphasisonthegeneralizedEVPe.g.earthquakeengineering"LargeEVP"n>500m>601p=l,...,3"nIndynamicanalysis,proportionaldampingrsP.-=w2!i! Ifzerofreq.arepresentwecanusethefollowingprocedurersP.-+)1IisP.-=(w2+~r)!isP.- or(r+)1!i)sP.-=!isP.= w2+)1or2W=-)112·3

181. SolationlIethodslorcalcalatiouoIlreqa.ciesandlIodeshapesp(A) p(A)=det(K-A~) Inbucklinganalysis.!$.!=A~! wherep(A)=det(~-A~) p(A) 12·4

182. SolutionmethodsforcalculationsoffrequenciesandmodeshapesRewriteproblemas: andsolveforlargestK: ....---~ (~-~~)!=n.K2£. TraditionalApproach:TransformthegeneralizedEVPorquadraticEVPintoastandardform, thensolveusingoneofthemanytechniquesavailablee.g. .Ki=;!iiM=I::I::Ti=hTjJhence~:t=;i;K=1::-1K[-TorM=W02WTetc... 12·5

183. SolotiOl.elhodslorcalcolationsoIlreqoeaciesud.odesllapesDirectsolutionismoreeffective. ConsidertheGen.EVP!!=AM! with1.3...1neigenpairs(Ai'1.i) arerequiredori=l,,pi=r,,sThesolutionproceduresinuseoperateonthebasicequationsthathavetobesatisfied. 1)VECTORITERATIONTECHNIQUESEquation: e.g.InverseIt. ~P_=A~~ !~+l=M~ ~+l•ForwardIteration•RayleighQuotientIterationcanbeemployedtocalculateoneeigenvalueandvector, deflatethentocalculateadditionaleigenpairConvergenceto"aneigenpair", whichoneisnotguaranteed(convergencemayalsobeslow) 12·6

184. Solutionmethodsforcalculationsoffrequenciesandmodeshapes2)POLYNOMIALITERATIONMETHODS!~=A~~~(K-AM)¢0Hencep(A)det(~-A!:1)=0,,, NewtonIterationp(A)2naO+alA+a2A+...+anAbO(A-Al)(A-A2)'"(A-An) Implicitpolynomialiteration: Explicitpolynomialiteration: eExpandthepolynomialanditerateforzeros. eTechniquenotsuitableforlargerproblems-muchworktoobtainai's-unstableprocessp(Pi)=det(IS.-Pi!y!) =detLDLT=IId.. ---.IIIeaccurate,providedwedonotencounterlargemultipliersewedirectlysolveforAl,... euseSECANTITERATION: Pi+l=Pi- edeflatepolynomialafterconvergencetoA112-7

185. Solutionmethodsforcalculationsoffrequenciesandmodeshapes]J.11- p(A)/(A-A,) II IIIConvergenceguaranteedtoA1'thenA2,etc.butcanbeslowwhenwecalculatemultipleroots. CareneedbetakeninLDLTfactorization. 12·8

186. SaI.liOi.6Jdsforcalculali.offreql8ciesiIIldoleshapes3)STURMSEQUENCEMETHODS1234t:::}·..... !<p=A!119·~;;.·....-....·..·...·.. NumberofnegativeelementsinDisequaltothenumberofeigenvaluessmallerthanJ.1S. 3rdassociatedconstraintproblem2ndassociatedconstraintproblem1stassociatedconstraintproblem12·9

187. SolutionlDethodslorcalculations01frequenciesudlDodeshapes3)STURMSEQUENCEMETHODSTCalculate~-].lS.~=hQh, CountnumberofnegativeelementsinQanduseastrategytoisolateeigenvalue(s). interval, ,, ,/ ].ls1].lS2Tf..•NeedtotakecareinLDLaetonzatlon--- •Convergencecanbeveryslow4)TRANSFORMATIONMETHODSj<PTK<P=A~!=A~!--T--< PM<P=I--- Construct<Piteratively: _ n=[Al...'n]<P=[~,...~J;HA-----...... " 12·10

188. 5oI1tiOi.elhodsforcalculations01frequenciesad.odeshapesTTT~...~~lff1~...~-~ TTT~...~f1!if1~...~-le.g.generalizedJacobimethod•Herewecalculatealleigenpairssimultaneously•Expensiveandineffective(impossible)orlargeproblems. Forlargeeigenproblemsitisbesttousecombinationsoftheabovebasictechniques: •Determinantsearchtogetneararoot•Vectoriterationtoobtaineigenvectorandeigenvalue•Transformationmethodfororthogonalizationofiterationvectors. •Sturmsequencemethodtoensurethatrequiredeigenvalue(s)has(orhave)beencalculated12·11

189. SolutionmethodsforcalCl1atiouoffrequenciesandmodesJlapesTHEDETERMINANTSEARCHMETHODp(A) A1)IterateonpolynomialtoobtainshiftsclosetoA1P(l1;)=det(~-11;~) T=detLDL=nd.. ---;1111;+1=].1;-nP(l1;)-P(11;_1) 11;-11;_1nisnormally=1.0n=2.,4.,8.,...whenconvergenceisslowSameprocedurecanbeemployedtoobtainshiftnearA;,providedP(A)isdeflatedofA1'...,A;_12)UseSturmsequencepropertytocheckwhether11;+1islargerthananunknowneigenvalue. 12·12~-.. /...

190. .... .... .... SolutionlOethodsforcalculationsoffreqoBciesudlOodeshapes3)OncelJi+1islargerthananunknowneigenvalue,useinverseiterationtocalculatetheeigenvectorandeigenvaluelJi+1k=1,2,... •~+l~+l=-T-~ (~+l!i~+l) -Tp(~+l)= ~+l!i~k-T~ ~+l!i~+l4)Iterationvectormustbedeflatedofthepreviouslycalculatedeigenvectorsusing,e.g.GramSchmidtorthogonalization. IfconvergenceisslowuseRayleighquotientiteration12·13

191. SolutionmethodslorcalculationsoIlrequenciesudmodeshapesAdvantage: Calculatesonlyeigenpairsactuallyrequired;nopriortransformationofeigenproblemDisadvantage: Manytriangularfactorizations•EffectiveonlyforsmallbandedsystemsWeneedanalgorithmwithlessfactorizationsandmorevectoriterationswhenthebandwidthofthesystemislarge. SUBSPACEITERATIONMETHODIteratewithqvectorswher:'thelowestpeigenvaluesandeigenvectorsarerequired. inverse{K4+1=',14k=1,2,... iteration-- ~+1-TK-~+1=4+1~+1-T~14+1=4+1~+1~+1=~+1~+1~+14+1=~+1~+112·14

192. Solutionmethodsforcalculationsoffrequenciesandmodeshapes"Underconditions"wehaveCONDITION: startingsubspacespannedbyX,mustnotbeorthogonaltoleastdominantsubspacerequired. UseSturmsequencecheckeigenvaluepeigenvaluesT!5.-flSt1=~Q~ no.of-veelementsinDmustbeequaltop. Convergencerate: flSconvergencereachedwhen<tal12·15

193. SolutionmethodsforcalculationsoffrequenciesandmodeshapesStartingVectorsTwochoices1)~lx.=e., ~~ j=2,...,q-l2. x=randomvector42)LanczosmethodHereweneedtouseqmuchlargerthanp. Checksoneigenpairs1.Sturmsequencechecks11~!~Q,+1)_A~Q,+l)~!~Q,+1)[12E:.= 1[IK¢~9,+l)II--12importantin!!!.solutions. Reference:AnAcceleratedSubspaceIterationMethod,J.ComputerMethodsinAppliedMechanicsandEngineering,Vol.23, pp.313-331,1980.12·16

194. MIT OpenCourseWare
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Resource: Finite Element Procedures for Solids and Structures
Klaus-Jürgen Bathe
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