结构力学英文课件 Chapter7.ppt

Chapter7Slope-DeflectionMethod,Section1Introduction,Slope-deflectionmethod(orsimplythedisplacementmethod)isanothermethodtoanalyzethestaticallyindeterminatestructure.Inthelastchapter(forcemethod),theunknownsofprimarystructureareforces.Afterobtainingunknownforce,displacementofthestructurecanbesolved.Infact,thedisplacement(ordeflection)canbeproducedundertheactionofexternalloads,sothedisplacementsofthestructurehaveinherentrelationtotheexternalload.Thatistosay,thedisplacementsofthestructurecanbealsousedasprimaryunknowns.Thismethodisreferredtoasdisplacementmethod.Rotationdisplacementandtranslationdisplacementaregeneralizeddisplacement.Sothedisplacementmethodisalsocalledslope-deflectionmethod.,EX.,Notice:(1)Theflexuraldeformationoftheflexuralmembersintheframeistakenintoaccount,buttheshearingandaxialdeformationsofwhichareneglected.(2)Straightmembersofconstantsectionbetweenthetwoendsareconsidered.,Section2Determineunknownsandprimarystructure,1、Primaryunknown(1)、Rotationdisplacementsunknown(2)、Translationunknown,A,B,C,D,B,C,B,C,2、Basicassumptions(1)、Smalldisplacementissupposed.(2)、Axialforceandshearingforcearedisregarded.,3、Howtodecidethenumberofprimaryunknowns,(1)、Thenumberofrotationdisplacements()isequaltothenumberofrigidjoints.(2)、Thenumberoftranslationunknowns()Themethodofdeterminingthetranslationunknownsisasfollowing.Allrigidjoints(includingfixedsupports)arechangedintohinges;calculatedegreeoffreedomofthenewstructure.Thedegreeoffreedomofthenewstructureisequaltothenumberoftranslationunknownoftheoriginalstructure.,Howtodecidethenumberofprimaryunknowns,Thereare4rigidjointsinFig.Sotherotationdisplacementunknownsare4.,Fig(a)ischangedintoFig(b),degreeoffreedomofFig(b):sotranslationunknownsoforiginalstructureare2.,4,Degreeofindeterminacy,Thedegreeofindeterminacy(n)is:,5,Primarystructure,Whenindeterminatestructureisanalyzedbyusingdisplacementmethod,everymemberisconsideredasastaticallyindeterminatebeamwithsinglespan.Sotheprimarystructureisthateverymemberischangedintoanindeterminatebeamwithsinglespan.Arigidarmisaddedateveryrigidjointtopreventrotationofthejoint(butcannotpreventtranslation)atthesametime,alinkisaddedatjointwheretranslationispossible.Thelinkpreventstranslationofthejoint.Afterthese,thestructureisprimarystructure.,EX：,Finddegreeofindeterminacy,setupprimarystructure：,Section3Slope-Deflectionequation,Allmembersinstructureareconsideredasabeamwithasinglespan,thebeamisfixedattwoends.Letsdiscusstheinternalforce-bendingmomentattwoendsundertheactionsofrotationandtranslationdisplacementsandexternalloads.Fig.showsabeam,itisfixedattwoends.EIisconstant.ThebeamissubjectedtoaforceP,rotationatendAis,androtationatendBis,thetranslationatendBrelativeendAis,bendingmomentsatendA,Baredesired.,Solution:letsolveitusingforcemethod.(1)Theprimarystructure(2)Canonicalequations,Because,Solution:(3)Determinecoefficients(GraphMultiplication),linearrigidity,let,Wehave,Wehave,isreferredtoas“slope-deflectionequation”,iforiginalstructureisthatoneendisfixedandtheotherendishinged,itsslope-deflectioncanbededuced.SupposeendBishinged.,Thefollowingsignisestablished:Bendingmomentclockwiseispositiverelativetoendofmember;counterclockwiseisnegativerelativetorigidjointorsupport.,Shearingforceclockwiseispositiveaboutisolatingbody(freebody).,Rotationangleclockwiseispositive,Thefollowingsignisestablished:Translationclockwiserotationofthewholememberispositive.,Theendmoment,endshearingforceandendreactionarelistedintableforconvenience.,Section4AnalysisofindeterminateStructureusingdisplacementmethod,Solution:(1)Itisindeterminatetotheseconddegree(2)PrimaryStructure:,(3)Canonicalequations,Inordertosolvethisproblem,wemustfindoutthedifferencesbetweenoriginalstructureandprimarystructureatfirst.,(a)Standingonthesideofdeformation,joint1cantrotate(Z1)duetorigidarm;translation(Z2)atjoint2cantexitduetolinkintheprimarystructure.But,Z1,Z2areexistencesinoriginalstructure.Inordertoeliminatethedifference,arotationZ1andtranslationZ2canbeenforcedatjoint1andjoint2respectivelyinprimarystructure.,(b)Standingonthesideofinternalforces.Z1,Z2areenforcedinprimarystructure.Thereactionmoment(R1)andreactionforce(R2)mustbeproducedinrigidjoint1andthelinkrespectively.But,R1,R2arenotexistencesinoriginalstructure.So,theproblemisthatthereactionsR1,R2are0undertheactionsofZ1,Z2andexternalloadsintheprimarystructure.Sothecanonicalequationscanbeexpressedas:,denotereactionsdueto,respectivelyin.Itiscalledcanonicalequations.Theequationsmean:Allattachedreactionsinaddedarmsandlinksare0undertheactionsofrotations,translationsandexternalloadsinprimarystructure.Iftherearenunknownsinstructure,thecanonicalequationscanbewrittenas:,arecalledmaincoefficients.arecalledsecondarycoefficients.(3)Calculate,Therearetwotypescoefficientintheequations,onetypeisreactionmoments,andtheotherisreactionforce.Inordertoworkoutthecoefficients,thetableofendmomentsandreactionsisused.DrawMdiagramsduetoexternalforcesandrespectively.Thecoefficientscanbeobtainedfromequilibriumconditionsoffreebody,whichisfromthebendingmomentdiagrams.,(4)DrawtotalM,StepstosoluteProblem:P270,11-3(d)（1）todeterminethedegreeofindeterminacy；(2)tosetupprimarystructure;（3）towritecanonicalequations；（4）todrawMp,Mi；（5）to