工程力学英文版课件13DeflectionofBeamsDuetoBending.ppt

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1、DeflectionofBeamsDuetoBendingEngineeringMechanics1§15–1SlopeandDeflectionofBeams§15–2DeflectionofBeamsbyIntegration§15–3MethodofSuperposition§15–4StaticallyIndeterminateBeamsBendingandShearingStressesinBeamsDeflectionofBeamsDuetoBending2§15-1SlopeandDeflectionofBeamsIntroductionHavingstudiedthes

2、tressessetupinbending,wenowturntotheequallyimportantaspectofbeamstiffness.Inmanystructuralelements,suchasfloorjoistsoraircraftwings,thelimitingconstraintonthedesignisstiffness.Anydesignwhichisstiffenoughwillbestrongenough.DeflectionofBeamsDuetoBending3FxwwTheelasticcurveforabeamThexaxisextendspo

3、sitivetotheright,alongtheinitiallystraightlongitudinalaxisofthebeam.Thewaxisextendspositiveupwardfromthexaxis.Itmeasuresthedeflectionofthecentroidonthecross-sectionalareaoftheelement.DeflectionofBeamsDuetoBending4Theslopeistheturningangleofthecross-sectionalarea.Thepositiveslopeanglewillbemeasur

4、edcounterclockwisefromthexaxiswhenxispositivetotheright.Theslopecanbedeterminedfromdw/dx.DeflectionofBeamsDuetoBendingFxwwTheelasticcurveforabeam5Itisimportantthatweshouldbeabletocalculatethedeflectionofabeamofgivensection,sinceforgivenconditionsofspanandloaditwouldbepossibletoadoptasectionwhich

5、wouldmeetastrengthcriterionbutwouldgiveanunacceptabledeflection.Thetotaldeflectionofabeamisduetoaverylargeextenttothedeflectioncausedbybending,andtoaverymuchsmallerextenttothedeflectioncausedbyshear.DeflectionofBeamsDuetoBending6§15-2DeflectionofBeamsbyIntegrationFromthepreviouschapter,wehave:If

6、thematerialishomogeneousandbehavesinalinear-elasticmanner,thenHooke’slawapplies,.Also,sincetheflexureformulaapplies,.Combiningtheseequationsandsubstitutingintotheaboveequation,wehave:(1)DeflectionofBeamsDuetoBending7Theelasticcurveforabeamcanbeexpressedmathematicallyasw=f(x).Toobtainthisequation

7、,wemustfirstrepresentthecurvatureintermsofwandx.InmostcalculusbooksitisshownthatthisrelationshipisDeflectionofBeamsDuetoBendingSubstitutingintoequation(1),weget(2)8Mostengineeringdesigncodesspecifylimitationsondeflectionsfor