Complete Study Guide - Finite Element Analysis …

1 _____tt~---MassachusettsInstituteofTechn ologyMITV ideoCOurseVideoCourseStudyGuideFiniteEle mentProceduresforSolidsandStructures LinearAnalysisKlaus-JOrgenBatheProfessor ofMechanicalEngineering,MITP ublishedbyMITC enterforAdvancedEngineeringstudyReorderN o672-2100 PREFACET heanalysisof complexstaticanddynamicproblemsin volvesinessencethreestages:selectionof a mathematicalmodel,analysisofthemodel, ,theefficientuseofthemethodisonlypossibl eifthebasicassumptionsoftheproceduresemp loyedareknown, to thelecturesincludesthebasicfiniteelement formulationsem ployed,theeffectiveimplementationof theseformulationsincomputerprograms, solveprob ,FiniteElementProceduresforSolidsandStru ctures- ,referenceismadetotheaccompanyingtextboo kforthecourse:FiniteElementProceduresinE ngineeringAnalysis, ,Prentice Hall,Inc., ,listedbelowtheshortdescriptionofeachlec ture, ; ;examplesSAp, , ; valueproblemsAnalysisofdiscretesystems:e xampleanalysisofaspringsystemBasicsoluti onrequirementsUseandexplanationofthemode rndirectstiff nessmethodVariationalformulationTEXTBOOK : Thefiniteelementmethodisnowwidelyusedfor analysisofstructuralengineeringproblems.

2 'ncivil,aeronautical,mechanical,ocean,mi ning,nuclear,biomechani cal,..engineering Sincethefirstapplicationstwodecadesago,- wenowseeapplicationsinlinear,nonlinear, variouscomputerprogramsareavailableandin significantuseMyobjectiveinthissetoflect uresis: tointroducetoyoufiniteelementmethodsfort helinearanalysisofsolidsandstructures.[" Iinear"meaninginfinitesi mallysmalldisplacementsandlinearelasticm aterialproeer ties(Hooke'slawapplies)j toconsider-theformulationofthefiniteelem entequilibriumequations-thecalculationof finiteelementmatrices- methodsforsolutionofthegoverningequation s- , 3 SomebasicconceptsofengineeringanalysisRE MARKS Emphasisisgiventophysicalexplanationsrat herthanmathe maticalderivations Techniquesdiscussedarethoseemployedinthe computerpro gramsSAPandADINASAP==StructuralAnalysisP rogramADINA=AutomaticDynamicIncrementalN onlinearAnalysis Thesefewlecturesrepresenta verybriefandcompactintroductiontothefiel doffiniteelementanalysis Weshallfollowquitecloselycertainsections inthebookFiniteElementProceduresinEngine eringAnalysis,Prentice-Hall,Inc.]

3 ( ). -~modelofphysicalproblem1I:I,-__S_ol_v_e _th_e_m_o_d_el__I~~- - -iL-_I_n_te_r.;.. (refine)themodel?1-4 SolIebasicconceptsofengiDeeringanalysis1 0ft15ftI12at15 ,. ~~~~~-~,-Fault\\(norestraintassumed)Alte red'gritE=toEc., WoE~~;;C= 6 SolDebasicconceptsofengineeringanalysisS egmentofa ,WppPINCHEDCYLINDRICALSHELLOD;:..,.. ~~~~~~CEtW-50P-100-150 16x16 MESH-200-DISPLACEMENTDISTRIBUTIONALONGDC OFPINCHEDCYLINDRICALSHELL ""= ~CBENDINGMOMENTDISTRIBUTIONALONGDCOFPINC HEDCYLINDRICALSHELL1-7 SoBlebasicconcepts01engineeringanalysisI Finiteelementidealizationofwindtunnelfor dynamicanalysisSOMEBASICCONCEPTSOFENGINE ERINGANALYSIST heanalysisofanengineeringsystemrequires: - idealizationofsystem- formulationofequili briumequations- solutionofequations- interpretationofresults1 8 SYSTEMSS omebasicconceptsofengineeringanalysisDIS CRETE responseisdescribedbyvariablesatafiniten umberofpointssetofalge braic--equationsCONTINUOUS responseisdescribedbyvariablesataninfini tenumberofpointssetofdiffer entialequationsPROBLEMTYPESARE STEADY-STATE(statics) PROPAGATION(dynamics) EIGENVALUEF ordiscreteandcontinuoussystemsAnalysisof complexcontinu oussystemrequiressolutionofdifferentiale quationsusingnumericalproceduresreductio nofcontinuoussystemtodiscreteformpowerfu lmechanism:thefiniteelementmethods,imple mentedondigitalcomputersANALYSISOFDISCRE TESYSTEMSS tepsinvolved:- systemidealizationintoelements- evaluationofelementequilibriumrequiremen ts- elementassemblage- solutionofresponse1 9 SomebasicconceptsofengineeringanalysisEx ample.

4 Steady-stateanalysisofsystemofrigidcarts interconnectedbyspringsPhysicallayoutELE MENTSU1U3I~:~\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 SolIebasiccOIceplsofengineeringanalysisE lementinterconnectionrequirements:F(4)+F (S)=R333 TheseequationscanbewrittenintheformK U= U=R(a)+k4k1+k2+k3~-k2- k3UT=[u-1RT=[R-1 '"..K=-k2- k3~k2+k3+kS~-kS ..1 11 Somebasicconceptsofengineeringanalysisan dwenotethat~=t~(i)i=1where::] (a)1 12 Somebasicconcepls01engineeringanalysisu, ..::..~..u, ::..K= .. :.K~ ~~.~ ~~ :..~ \fl--r/A~,1\1\1\~~r/A1 13 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.

5 K= .'O'O ..+K4;K1+K2+ K3~-K2-K3-K4'O'O K=-K2-K3~K2+K3+K5-K5'O'O : u,IK1 14 SomebasicconceptsofengineeringanalysisIn thisexampleweusedthedirectapproach;alter nativelywecouldhaveuseda :u=strainenergyofsystemW=totalpotentialo ftheloadsEquilibriumequationsareobtained froman-0(b)~-1 IntheaboveanalysiswehaveU=~UT!!!W=UTRI nvoking(b)weobtainK U=RNote:toobtainUandWweagainaddthecontri butionsfromallelements1 15 SOlDebasicconceptsofengineeringanalysisP ROPAGATIONPROBLEMS maincharacteristic:theresponsechangeswit htime~needtoincludethed'Alembertforces:F ortheexample:m,aaM=am2aaam3 EIGENVALUEPROBLEMS weareconcernedwiththegeneralizedeigenval ueproblem(EVP)Av=A Bv!l,.!laresymmetricmatricesofordernvisa vectorofordernAisascalarEVPsariseindynam icandbucklinganalysis1 16 SomebasicconceptsofengineeringanalysisEx ample:systemofrigidcarts~lU+KU=OLetU=<ps inW(t-T)Thenweobtain_w2~~sinW(t-T)+K<psi nW(t-T)=0---HenceweobtaintheequationTher eare3 solutionsw,,~,(l)2'~2eigenpairsw3'~3 Ingeneralwehavensolutions1 17 ANALYSISOFCONTINUOUSSYSTEMS;DIFFEBENTIAL ANDVABIATIONALFOBMULATIONSLECTURE259 MINUTES2-1 Analysis01continnoussystems.

6 DifferentialandvariationallonnDlationsLE CTURE2 Basicconceptsintheanalysisofcontinuoussy stemsDifferentialandvariationalformulati onsEssentialandnaturalboundaryconditions Definitionofem-IvariationalproblemPrinci pleofvirtualdisplacementsRelationbetween stationarityoftotalpotential,theprincipl eofvirtualdisplacements,andthediffer entialformulationWeightedresidualmethods ,Galerkin,leastsquaresmethodsRitzanalysi smethodPropertiesoftheweightedresidualan dRitzmethodsExampleanalysisofanonuniform bar,solutionaccuracy,introductiontothefi niteelementmethodTEXTBOOK: , , , , , , , , , , , , 2 Analysisofcontinuoussystems;differential andvariationalformulationsBASICCONCEPTSO FFINITEELEMENTANALYSIS CONTINUOUSSYSTEMS Wediscussedsomebasicconceptsofanalysisof discretesystems ~ 3 AnalysisofcontinuoussysteDlS;differentia land,arialionalIOl' DifferentialformulationaAI+A~aIdx-aAIxxo XX/Young'smodulus,E~Lt:)massdensity,cros s-sectionalarea, ~-------Theproblemgoverningdifferentiale quationisDerivationofdifferentialequatio nTheelementforceequilibriumrequire mentofa typicaldifferentialelementisusingd'Alemb ert'sprincipler~.

7 -;+~~dxI~AreaA,massdensityp2=pAau~Thecon stitutiverelationisaua=E axCombiningthetwoequationsaboveweobtain2 4baIysis01 COitiDlOUsystems;differatialaDdvariation aliOl'lDDlatiODST heboundaryconditionsareu(O,t}= EA~~(L,t)=ROwithinitialconditionsu(x,O}= ~(xO)= at'9essential(displ.) (force) ,wehavehighestorderof(spatial)deriva tivesinproblem-governingdif (spatial)deriva (m-1)highestorderofspatialderiva (2m-1)Definition:Wecallthisproblema 5 Analysis01continuoussystems;differential andvariatioD,a1fOl'llolatiODSE xample- VariationalformulationWehaveingeneralII= U-WFortherodfLII=J }EAoandiLau2B(--)dx-ufdx- uRaxLou=0oandwehave0II=0 Thestationarycondition6II= (EAax)(6ax)dx-)06ut-dx-6uLR= ,wewritethisprincipleasor(seealsoLecture 3)2 ;differentialandvariatiooallormulatioDSH owever, .Writinga8ufor8au,re-axaxcallingthatEAis constantandusingintegrationbypartsyields dx+[EA~Iaxx=L-EA~\dXx=oSinceQUOiszerobut QUisarbitraryatallotherpoints,wemusthave andauIEAax-x=L=RBa2uAlsof=-Ap -and,at2hencewehave2 7 AnalysisofcODtiDaoassyst_diIIereatialand variatioulfOlllalatiODST heimportantpointisthatinvokingoIT=0 theprincipleofvirtualdisplacements theproblem-governingdiffer entialaquatio!)]

8 (theseareinessence"containedin"IT, ,inW).In thederivationoftheproblem governingdifferentialequationweusedinteg rationbyparts thehighestspatialderivativeinITisoforder m. " ~2 8 PrincipleofVirtualDisplacementsIIntegrat ionbyparts~ :diBerentialandvariatiouallnaiatiOlSWeig htedResidualMethodsConsiderthesteady-sta teproblem( ) [</>]=q.,i=1,2, 11atboundary( )Thebasicstepintheweightedresidual(andth eRitzanalysis)istoassumea solutionoftheform( )wherethefiarelinearlyindepen denttrialfunctionsandtheaiaremultipliers thataredeter ,wechoosethefunctionsfiin( )soastosatisfyallboundaryconditionsin( )andwethencalculatetheresidual,nR=r -L2mCLa f.]( )1= ; dD=O;=1,2, ,nD1 Leastsquaresmethod( )Inthistechniquetheintegralofthesquareof theresidualisminimizedwithrespecttothepa rametersai' ;=1,2, ,n[Themethodscanbeextendedtooperatealsoo nthenaturalboundaryconditions,if thesearenotsatisfiedbythetrialfunctions.

9 ]RITZANALYSISMETHODL etnbethefunctionaloftheem-1variationalpr oblemthatisequivalenttothedifferentialfo rmulationgivenin( )and( ).In theRitzmethodwesubstitutethetrialfunctio ns<pgivenin( )intonandgeneratensimul taneousequationsforthepara metersaiusingthestationaryconditiononn ,2 ;=1,2, ,n( )Analysisofcontinuoussystems;differentia landvariationalformulationsProperties SincetheapplicationofoIl=0generatesthepr incipleofvirtualdisplacements,weineffect usethisprincipleintheRitzanalysis. Byinvoking0II= Asymmetriccoefficientmatrixisgenerated,o fformK U=RExampleR=100N2 Area=1em(.._-x,u---- -- ----~---r;;;-== ;;B;",.,. ~~--- 11 AnalysisofCOitiDlOISsystems;differeatial ad,ariali"fOllDaialiODSH erewehave1180IT=1EA(~)2dx2axo-100uIx =180andtheessentialboundaryconditionisuI x=O=0 LetusassumethedisplacementsCase1u=a1x+a2 iCase2~u=I1JO0<x <100100< x <180 WenotethatinvokingoIT=0weobtain1180oIT=( EA~~)o(~~)dx-100 OUIx=180o=0ortheprincipleofvirtualdispla cements 180(~~u)(EA~~)dx=100 OUIx=180oJETTdV= 12 Analysisofcontinuoussystems;differential andvariationalformulationsExactSolutionU singintegrationbypartsweobtain~(EA~)=0ax axEA~=100axx=180 Thesolutionisu = 1~Ox; 0<x<100100<x<180 Thestressesinthebararea= 100;0<x<100a=100; 100<x<180(l+x-l00)2402 13 Analysisofcontinuoussystems;differential andvariationalformulationsPerformingnowt heRitzanalysis:Case1f180dx+I(1+x-l00)224 0100 Invokingthatorr=0weobtainE[ ,wehavetheapproximatesolutionu= -E2x2 14a=.)

10 DifferentialandvariationalformulationsCa se2 Herewehave100EJ12n=2(100uB)af180dx+I(1+x -l00)2240100 Invokingagainon=0weobtainE[ ][~:]=[~oo]240-1313 Hence, <x < >100o=802-15 AulysisofCOilinDmassystems;diUerenliaian dvarialiOlla1I01'1 BDlaIiGlSuEXACT~------::.:--~~~. " ~..,-__--r-__--.,r---~X15000E10000E5000- E-100180 CALCULATEDDISPLACEMENTS(J50100-I=:::==-= =_==_:=os:=_=_=,==_=_==""EXACT"I~~SOLUTI ON1I-<, ~. +---,~--------r-------~X100180 CALCULATEDSTRESSES2 18balysisofcoatiDloassystms;diBerenlialu dvariationalfonnllatioasWenotethatinthis lastanalysise ,butthederiva (m-1)stderivativesofthefunc tions;inthisproblemm=1 .edomainsA -BandB 17 FORMULATIONOFTHEDISPLACEMENT-BASEDFINITE ELEMENTMETHODLECTURE358 MINUTES3 1 Formulationofthedisplacement-basedfinite elementmethodLECTURE3 Generaleffectiveformulationofthedisplace ment-basedfiniteelementmethodPrincipleof virtualdisplacementsDiscussionofvariousi nterpolationandelementmatricesPhysicalex planationofderivationsandequa.)