弹性力学课件34

xyh/2h/2MMh1Forconvenience,weconsideronlyaunitwidthofthebeam.LetthebendingmomentontheunitwidthbeM,whichhasadimensionof[force][length]/[length],orsimply[force].SubstitutionoftheexpressionforxinEqs.(2)intoEqs.(3)yieldsOneitheroftheverticaledges,thesurfaceforcesmusthavearesultantforceequaltozeroandaresultantmomentequaltoM.Thisrequires(3)Thefirstequationisalwayssatisfiedwhilethesecondequationrequires6a=12M/h3.Since：ThemomentofinertiaofthebeamsectionThus,Eqs.(2)become：(4)Eqs.(4)mayberewrittenas(5)Eqs(5)coincidecompletelywiththeelementarysolutiongiveninmechanicsofmaterials.Note：thesolution(5)isexactonlywhenthesurfaceforcesontheendsofthebeamconsistofnormaltractionsproportionaltoy.Ifthecouplesontheendsareappliedinanyothermanner,thissolutionisnolongerexact,becauseitwillnotpreciselysatisfytheboundaryconditionsontheends.AccordingtoSaint-Venant’sprinciple,however,itdoesrepresentthestressdistributiononanycrosssectionatalargedistancefromtheends.3.2DETERMINATIONOFDISPLACEMENTSWhenthestresscomponentshavebeenfoundbysolutionintermsofstresses,wecanfindthedisplacementcomponentsbyphysicalandgeometricequations.Thiswillbeillustratedwiththeproblemofpurebendingasanexample.Inthecaseofplanestress,thestraincomponentsare:ThenwehaveSubstitutingu,vinto：Itcanberewrittenas：Wenotethattheleftsideisafunctionofyalonewhiletherightsideisafunctionofx

alone.

Afunctionofycanbeequaltoafunctionofxonlywhentheyarebothequaltoaconstant,say.Byintegration,wehave：（1）（2）Thus,thedisplacementcomponentscanbeexpressedas：(1)(2)From(1),wecanseethat,nomatterhowthebeamissupported,theangleofrotationofanyverticallineelementinthebeamisOnagivencrosssectionofthebeam,isconstant,sincexisconstant.Thisshowsthatalltheverticallineelementsinthecrosssectionhavethesameangleofrotationduetobendingofthebeam,andconsequentlyprovesthatthecrosssectionremainsplaneafterbending.From(2),weseethatallthelongitudinallines(fibres)ofthebeamwillhavethesamecurvatureafterbending.Thisisthebasicformulaforcomputationofbeamdeflectionsinmechanicsofmaterials.Ifthebeamissimplysupported：xyh/2h/2MMTherewillbeneitherhorizontalnorverticaldisplacementsatthehingedsupport,andtherewillbenoverticaldisplacementeither,atthebarsupport.TheconditionsofconstraintareFromwhichwehaveInsertionofthesevaluesinto(1)and(2)yieldsThedeflectionofthebeamaxisis：Whichisthesameastheresultobtainedinmechanicsofmaterials.Inthecaseofacantileverbeamxyh/2h/2MMTheconditionsofconstraint：Fromwhichwehave：Thuswehave：Thedeflectionofthebeamaxisis：Whic