结构力学英文课件Chapter-9

Chapter9principleofvirtualworkanddisplacementsofstructuresAbstractofthechapterTheattentionofthechapterismainlyfocusedonthecalculationofthedisplacementofstaticallydeterminatestructuresbyusingtheprincipleofvirtualwork.Intheanalysisofstaticallydeterminateandindeterminatestructures,thechapterplaysaroleofthelinkbetweentheprecedingandthefollowingofthetext;itisnotonlytheendoftheanalysisofstaticallydeterminatestructuresbutalsothebeginningoftheanalysisofstaticallyindeterminatestructures.First,wewilldiscusstheprincipleofvirtualworkforrigidbodiesanditstwokindsofapplications:determiningtherestraintforcesanddeterminingthedisplacementofstructures.Then,wemainlydiscusstheprincipleofvirtualworkfordeformablebodies,thegeneralequationsofstructuresduetoloadings,temperaturechangesandsupportsettlements,andthemethodofgraphmultiplicationandsoon.Finally,introducethereciprocallawsforlinearly-elasticdeformablesystemsTherearethreefactorsthatcauseastructuretoyielddisplacements,loadings,temperaturechangesandmaterialexpansion,supportsettlements.9.1introductionforcalculationofstructuraldisplacementThecalculationofthedisplacementofstaticallydeterminatestructuresisaveryimportantportionofthecontentofstructuralmechanics,anditisalsothefoundationoftheanalysisforstaticallyindeterminatestructures.Theevaluationofthestiffnessrequirementforastructurehavetwosortsofpurpose.Thefirstpurposeisforcheckingthestiffnessrequirementofthestructure.Indesignstageofastructure,itisnecessarythatastructurehastosatisfynotonlytherequirementofstrengthbuttherequirementofstiffnessaswell.Thatis,thedeflectionofthestructuremustnotexceedtheallowablevaluespecifiedbydesigncode.Thesecondpurposeisforthepreparationofanalysisforstaticallyindeterminatestructures.becauseintheanalysisofstaticallyindeterminatestructures,wehavetoconsidernotonlystaticequilibriumconditionsbutthecompatibleconditionsofthedisplacementsofastructure.9.2Virtualworkthetermvirtualisusedheretodistinguishthetraditionalconceptofrealworkandthenewconceptofvirtualwork.Theso-calledvirtualworksimplymeansimaginarywork,ofwhichtheforcehasnorelationwiththedisplacementundergonebytheforceshasnorelationwiththedisplacementundergonebytheforceduringperformingthework.Thatis,thedisplacementisnotcausedbytheforcedoingthework;orwhenperformingtheworkthroughthedisplacement,theforceisaconstantvalue.9.2.2TwokindsofapplicationsofprincipleofvirtualworkforrigidbodysystemsIfthestrainofprincipleofthesystemisnottakenintoconsideration,onlythemovementsofthemembersofthesystemareconcernedduringitsmovingprocess,thesystemisrecognizedasarigidbodysystem,nowtwokindsofitsapplicationswillbediscussed.Thereexisttwostates,oneisforcestateandanotherisdisplacementstate.Sincetheforcestateandthedisplacementstateareindependenttoeachotherforthevirtualworkperformedbytheforcestateactingonastructurewillbedeterminedbyimaginingalikelydisplacementstateforthestructure;whereasanunknowndisplacementofagivendisplacementstateofastructurewillbeobtainedbyimaginingalikelybalancedforcestateforthestructure.(1)Determinationofunknownrestraintforcesforstaticallydeterminatestructures(2)Determinationofdisplacementsforstaticallydeterminatestructures9.2.3PrincipleofvirtualworkfordeformablebodiesWhenasystemundergoesadeformingprocessnotonlyeachmembersofthesystemdeveloprigid-bodydeformation,thematerialsmakingofthesystemyielddeformationaswell.Thesystembelongstothecategoryofdeformablesystems.Oneofapplicationsofprincipleofvirtualworktodeformablebodiesistheprincipleofvirtualforcesanditcanbestatedasfollows:Ifadeformablestructureisinequilibriumundertheactionofavirtualsystemofforcesandifitissubjectedtoanysmallrealdeformationconsistentwiththesupportandcontinuityconditionsofthestructure,thenthevirtualexternalworkdonebythevirtualexternalforcesactingthroughtherealexternaldisplacementsisequaltothevirtualinternalworkdonebythevirtualinternalforcesactingthroughtherealinternaldisplacements.Inthisstatement,thetermvirtualisassociatedwiththeforcestoindicatethattheforcesystemisarbitraryandhasnorelationwiththeactioncausingtherealdeformation.ThestatementmaybeexpressedmathematicallyasWe=Wi

Werepresentsthevirtualworkdonebyexternalforcesystem;Wirepresentsthevirtualworkdonebyinternalforcesystem.Considerasimplebeamisinequilibriumundertheactionofasystemofforcesasshown.fig9.8(a).Adisplacementanddeformedstateofthesimplebeamcausedbyotherreasonotherthanthesystemoftheforces.Takeadifferentialsegmentdsofthebeamasadiscussingobjectasshowninfig.theinternalforcesactingondscausedbytheforcesysteminfig(a).Isshowninfig(c).andtherelativedeformationofdsduetothedeformationofthestateshowninfig(b)isshowninfig(d)Thevirtualworkdonebytheinternalforcesshowninfig9.8(c)actingthroughthedeformationshowninfig9.8(d)canbedefinedasConsequently,theexpressionforthevirtualworkdonebytheinternalforcesofthesimplebeamwillbeexpressedasFortheframedstructure,theexpressionwillbecomeIntroducingtheexpressionforexternalvirtualwork,theequationWe=Wicanberewrittenas:SinceTheequationwillbecomeThisequationisthemathematicalexpressionofprincipleofvirtualwork,whichiscommonlytermedvirtualworkequationfordeformablebodies.Thereisavarietyofapplicationofprincipleofvirtualwork.However,thechapterwillonlydiscussoneofitsapplicationstothecalculationofdisplacementsforvariousdeformablestructures.9.3GeneralequationandunitloadmethodforcomputingdisplacementsTheunitloadmethodfordeformablebodiescanbeveryeasilyextendedbythatforrigidbodies.Nowassumearealdeformedstateofastructurehasbeengiven.Ourpurposeistodetermineoneofitsdisplacements△ofthestructurebyequation.Bysubstitutingtheunitloadp=1anditscorrespondingreactionsRKandinternalforcesintoequation,thegeneralexpressionforcomputingadesireddisplacement△byusingloadmethodcanbeexpressedas:Theproceduretofindadesireddisplacement△ofthestructuremaybestatedasfollows:(1)Imposeaunitloadp=1atthelocationandinthedirectionofthedesireddisplacement△(2)Determinethereactionsandinternalforcesofthestructurecausedbytheunitloadp=1accordingtothestaticequilibriumconditions.(3)Calculatethedesireddisplacement△byusingtheequationsIfadesireddisplacement△determinedbyequationhaspositivesign,itmeansthatthedirectionof△isthesameasthatofp=1andviceversa.9.4CalculationofdisplacementscausedbyloadsComputingformulaandprocedurefordeterminationofdisplacementsForelasticframedstructures,thestrainsinequationscanbedeterminedbyusingHook’slawlearnedinStrengthofMaterials.ThestraincanbeexpressedasSubstitutingthestrainsintheequation,thegeneralformulaforcalculatingelasticdisplacementsduetotheactionofloadswillbewrittenasThefollowingstep-by-stepprocedurecanbeusedtodeterminethedisplacementsofastructureduetoactionofexternalloadsbyunitloadmethod.(1)RealsystemWriteouttheexpressionsexpressingtheinternalforcesofthestructureduetoitsrealexternalloadsbystaticequilibriumequations,andthenconstructtheirdiagrams.(2)VirtualsystemRemoveallthegivenloadsfromthestructure;thenapplyageneralizedunitloadatthelocationandinthedirectionofthedesireddisplacement△.Findouttheexpressionsexpressingthevirtualinternalforcesofthestructureduetothegeneralizedunitloadbystaticequilibriumequations,andthenconstructtheirdiagrams.Thenthedesireddisplacement△ofthestructurenowcanbedetermined.FormulateforcalculatingdisplacementsofvarioustypesofstructuresEquationisageneralofequationforcomputingelasticdeflectionsofastaticallydeterminatestructureduetoexternalloads.Thedifferentsimplifiedequationsofequationswillthuslybeobtainedbyconsideration,onwhichthemaineffectofdeformationistakenandnegligibleeffectofdeformationisdiscarded,ofcharacteristicsofdeformationofastructure.BeamandrigidframesForbeamsandrigidframes,theirdeformationismainlycausedbybendingmoments.Therefore,theequationcanbesimplifiedas(2)TrussesFortrusses,eachofthemembersissubjectedtoonlyaxialforce.generally.Theequationisthuslysimplifiedas(3)CompositestructuresForcompositestructures,beam-typedmemberaremainlysubjectedtobendingmoments,andtwo-forcemembersaremainlyundergoneaxialforces.Thereforeequationmaybemodifiedas(4)ArchesThedeformationsduetobendingmomentsandaxialforceshavetobetakenintoconsiderationforarchesExamplesforcalculatingdeflectionsduetoexternalloads9.5graph-multiplicationmethodRecallfromprevioussectionthatwhencalculatingthedisplacementsofbeamsandframes,thefollowingintegralhavetobecarriedout:Theexpressioninvolvestheintegralofaproductoftwointernalforcefunctions.Directlyperformingtheintegralisoftentediousandtimeconsuming.However,forastraightmemberorasegmentofastraightmemberwithaconstantflexuralrigidityEI,theintegralcanbeconvenientlyperformedintermsofso-calledgraph-multiplicationmethodthatcantakeadvantageofthefactthatthediagramofoneofthetwofunctionsiscomposedbystraightlinestofacilitatetheevaluationoftheintegral.Calculatingequationsofgraph-multiplicationmethodConsidertwobendingmomentdiagramsofastraightsegmentABofamember,oneofwhich(Mi)isastraight-linediagramasshowninfig.iftheflexuralrigidityEIisconstantoverthelengthofthesegment,EIcanbeputoutsidetheintegral,thenwewriteSincethediagramofMiisastraightlinewithaslopeofɑ,iforientingtheintersectingpointbetweenthelinerepresentingMiandthebaselinexastheorigin,theordinateofMiatanarbitraryxpositionwillbeTherefore,wewriteisthemomentofthedifferentialareawithrespecttocoordinateaxisy;theintegralmeansthesummationofthemomentsofalldifferentialareasoverthesegmentABwithrespecttoaxisy,recallingfromthetheoremofmomentofanarea,theintegralequalstothemomentoftheareaofthebendingmomentMkdiagramwithrespecttoaxisy.ifletx0representthedistancebetweenthecentroidofthediagramMkandtheaxisy,wewriteSubstituting,weobtainThisexpressionistheequationofcalculatingdisplacementsintermsofgraph-multiplicationmethod.Ittransformsthecalculationofanintegralwiththeproductoftwofunctions,oneofwhichisalinearfunction,intothecalculationoftheirgraphicareas,loca