铁木辛柯梁

AnIntroductiontoTimoshenkoBeamFormulationanditsFEMimplementationChanYumJiCOME,TechnischeUniversitätMünchenContentofpresentation„Introduction„FormulationofTimoshenkoBeamElements„FEMimplementation…Example„ProblemwithFEMimplementation…Reason„p-versionFEMimplementation…Example„QuestionsandAnswersReferences„Bathe,K.-J.:FiniteElementProcedures(PrenticeHall,EnglewoodCliffs,1996)„Bischoff,M.:LectureNotesoncourseAdvancedFiniteMethods,TUM0.1Introduction:ReviewofEuler-BernoulliBeamTheory„Beamiscondensedtoan1-Dcontinuum„Assumptions„Mid-surfaceplaneremainsinmid-surfaceafterbending„Crosssectionsremainstraightandperpendiculartomid-surface„Onevariable(displacement)ateachpoint„Applicabletothinbeams0.2Howaboutthickbeams?„Shearingforceexistsinsidebeam„Assumption“Crosssectionsremainperpendiculartocentroidalplane”nolongervalids„Timoshenkotheory0.3Timoshenkobeamtheory„Beamiscondensedtoan1-Dcontinuum„Assumption„Mid-surfaceplaneremainsinmid-surfaceafterbending„Crosssectionsremainstraightandperpendiculartomid-surface„Twoindependentvariables(displacementandrotation)ateachpoint„Distributivemomentstakenintoaccount1.1Governingequations„Kinematicequations„Equilibrium„Constitutiveequations(MaterialLaws)DisplacementsStrainsStressesForcesKinematicequationsMaterialLawsEquilibrium1.2Kinematicequations„RemembertheequationsforEuler-Bernoullibeams……dxdw=β22dxwddxd−=−=βκ1.2Kinematicequations„…andherecomestheequationsforTimoshenkobeams!…Westillassumecrosssectionremainsstraightatthemomentγβ−=dxdwdxdβκ−=1.3Equilibrium„ConsiderapartofthebeamQMQdxdMmQdxdQq+−=+−=−=−=''1.4Constitutiveequations(MaterialLaws)„Bendingpart„Shearingpart…αtakesintoaccountofnon-straightcrosssectionsκEIM=γαGAQ=1.5Summaryofallequations„Kinematicrelations„Equilibrium„MaterialLawsγβ−=dxdwdxdβκ−=γακGAQEIM==QMQdxdMmQdxdQq+−=+−=−=−=''1.6Boundaryconditions„Displacement/Essential/Dirichlet„Force/Neumann00)0()0(MMQQ==llMlMQlQ−=−=)()(00ˆ)0(ˆ)0(ββ==wwlllwlwββˆ)(ˆ)(==2.1FiniteElementMethod–Weakformulation„FEMisanumericalmethodoffindingapproximatesolutions„“Weak”formulation…Thethreeequationsarenotsatisfiedateachpoint,butonlyingeneralsense„Virtualworkprinciple:0int=+extWWδδ2.2Virtualworkprinciple„Externalvirtualwork„Internalvirtualwork„As,()lllllextMMwQwQdxmwqWδβδβδδδβδδ+++++=∫00000()∫+=−ldxMQW0intδκδγδ0int=+extWWδδ()lllllMMwQwQdxMQmwqδβδβδδδκδγδβδ++++−−+=∫0000002.3Virtualworkprinciple–inMatrices=βwu∂∂−∂∂=xx01*L=EIGA00αC=MQσ=κγε∂∂∂∂=xx01L()0d000=⋅−⋅−⋅−⋅∫lTlTlTxδuPδuPδupδεσ=mqp2.4Discretisation„FEMcannotdealwithcontinuousfunctions„Unknowncoefficients(d)withpre-assignedshapefunctions(N)…nodalvaluesasunknowns…twonodesmakesupanelement…twolinearshapefunctionsforanelement„Matrixform:u=N·d2.4DiscretisationBecauseandandsupposedNuu⋅=≈h()0000=⋅−⋅−⋅−⋅∫lTlTlTdxδuPδuPδupδεσ()[]0d00=⋅−⋅−⋅∫bllxδuPPδupδuCLLuTTεCσ⋅=uLε⋅=()[]0d00=⋅−⋅⋅−⋅∫δdPPδdNpδdCBBdTTllx+⋅=⋅∫∫l0TPPNpdCBBxxlldd00StiffnessMatrixLoadVectorUnknown2.5Implementation„Mapleexample„Comparison:WithEuler-BernoulliBeamLP=t33.1ProblemwithFEMimplementation„Displacementmuchsmallerthanexpected„Extremelyslowconvergingrate„Addingelementsdoesnothelp„Resultdependsononecriticalparameter…Displacement=0whenparameterreachesinfinity„Locking3.1LockingbehaviourexhibitsslowconvergingrateConvergingbehaviourofFEsolution00.20.40.60.811.2051015202530NumberofelementsRelativedisplacemenEulerBernoulli(Analytical)Timoshenko(FEapproximation)3.1LockingbehaviourdependsonslendernessChangeofestimateddisplacementagainstslenderness0.010.111002468101214161820SlendernessRelativedisplacementEulerBernoulli(Analytical)Timoshenko(FEapproximation)3.2ReasonsoflockingFirstReason„Equilibrium:„Whentissmall,sheardominatesifw’andβdonotbalance()()+′+′′=+′+′′−=+′−=βαββαβwGbtEbtwGAEIQMm2123.2ReasonsoflockingSecondreason„Kinematicequation:„Here,wislinear(setbyN1andN2)„Thenw’becomesconstant„Theonlysolutionforβ=constant„Zeroshearifslendernessistowardsinfinityγβ−=dxdw4.1Solvingproblem„Theprocess…Formulation…FEMImplementation…Discretisation„Methodsonimplementation„Methodsondiscretisation4.2HighOrderfunctions„Changethediscretisationscheme…Allowhigherordertermsinshapefunctions…βneedsnottobeconstant„Hierarchicshapefunctions…Nodalmodes…Bubblemodes…Advantages4.3Example„Maplesheet4.4Graphshowingconvergenceofp-methodShapesofdeflectionwithdifferentordersconsidered00.511.522.53012345LengthDeflection1storder2ndorder3rdorderExact5Conclusion„Timoshenkobeamtheoryisapplicableforboththickandthinbeams„Itsuffersfromseverelockingbehaviourwhenlinearshapefunctionsareapplieddirectly„Employinghighorderfunctionscansolvetheproblem6QuestionsandAnswersYourcommentsarealsowelcomed