Complete Study Guide - Finite Element Procedures …

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.

2 , ,listedbelowtheshortdescriptionofeachlec ture, ; ;examplesSAp, , ; valueproblemsAnalysisofdiscretesystems:e xampleanalysisofaspringsystemBasicsoluti onrequirementsUseandexplanationofthemode rndirectstiff nessmethodVariationalformulationTEXTBOOK : Thefiniteelementmethodisnowwidelyusedfor analysisofstructuralengineeringproblems. 'ncivil,aeronautical,mechanical,ocean,mi ning,nuclear,biomechani cal,..engineering Sincethefirstapplicationstwodecadesago,- wenowseeapplicationsinlinear,nonlinear, variouscomputerprogramsareavailableandin significantuseMyobjectiveinthissetoflect uresis: tointroducetoyoufiniteelementmethodsfort helinearanalysisofsolidsandstructures.

3 ["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.]

4 ( ). -~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.

5 - 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:steady-stateanalysisofsystemofrigi dcartsinterconnectedbyspringsPhysicallay outELEMENTSU1U3I~:~\l).

6 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.

7 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 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!

8 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;differential andvariationallonnDlationsLECTURE2 Basicconceptsintheanalysisofcontinuoussy stemsDifferentialandvariationalformulati onsEssentialandnaturalboundaryconditions Definitionofem-IvariationalproblemPrinci pleofvirtualdisplacementsRelationbetween stationarityoftotalpotential,theprincipl eofvirtualdisplacements,andthediffer entialformulationWeightedresidualmethods ,Galerkin,leastsquaresmethodsRitzanalysi smethodPropertiesoftheweightedresidualan dRitzmethodsExampleanalysisofanonuniform bar,solutionaccuracy,introductiontothefi niteelementmethodTEXTBOOK: , , , , , , , , , , , , 2 Analysisofcontinuoussystems.

9 DifferentialandvariationalformulationsBA SICCONCEPTSOFFINITEELEMENTANALYSIS 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~.-;+~~dxI~AreaA,massdens ityp2=pAau~Theconstitutiverelationisaua= E axCombiningthetwoequationsaboveweobtain2 4baIysis01 COitiDlOUsystems;differatialaDdvariation aliOl'lDDlatiODST heboundaryconditionsareu(O,t}= EA~~(L,t)=ROwithinitialconditionsu(x,O}= ~(xO)= at'9essential(displ.))

10 (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!]