材料力学cai5

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1、Chapter5BendingStressesInthischapter,onlymembershavingsymmetriccrosssectionsandsubjectedtoloadsintheplaneofsymmetryareconsidered.Asaconsequence,thebendingdeflectionsoccurinthisplaneknownastheplaneofbending.Thusthedeflectioncurveisaplanecurvelyingintheplaneofbending.Whenasegmentofabeamisinequilibr

2、iumundertheactionofbendingmomentsalone,suchaconditionisreferredtoaspurebending,orflexure.Studiesinsubsequentchapterswillshowthatusuallythebendingstressesinslenderbeamsaredominant.Therefore,theformulasderivedinthischapterforpurebendingaredirectlyapplicableinnumerousdesignsituations.M5-1Bendingofbe

3、ams1.ThebasicassumptionConsiderahorizontalprismaticbeamhavingacrosssectionwithaverticalaxisofsymmetry;seeFig.5-l(a).Ahorizontallinethroughthecentroidofthecrosssectionwillbereferredtoastheaxisofabeam.Thefundamentalhypothesisoftheflexuretheory:planesectionsthroughabeamtakennormaltoitsaxisremainplan

4、eafterthebeamissubjectedtobending.2.GeometricrelationsofdeflectionInpurebendingofaprismaticbeam,thebeamaxisdeformsintoapartofacircleofradiusρ.Foraelementdefinedbyaninfinitesimalangledθ,thefiberlengthefofthebeamaxisisgivenasds=ρdθ.Hence,dθ1==κdsρwherethereciprocalofρdefinestheaxiscurvatureχ.Inpure

5、bendingofaprismaticbeams,bothρandχareconstant.Thefiberlengthghlocatedonaradiusρ-yis:(ρ-y)dθTherefore,thedifferencebetweenfiberlengthghandefidentifiedhereasducanbeexpressedasfollowsdu=(ρ-y)dθ-ρdθ=-ydθ由于变形很小,∴du≈du(Theaxialfiberdeformation)ds≈dx(弧长近似等于弦长)dudu−ydθ∴ε=≈==−κyxdxds1dθκyε=−κy=−xρ3Stress-

6、strainrelationAbeamisinpurebendingmeansthatitsstressstateisuniaxial.ByusingHooke’slaw:σ=E⋅ε=−Eκyxx4.StaticequilibriumconditionForthepurebendingbeams,∑Fx=0∴σdA=0∫xxA∫−EκydA=−Eκ∫ydA=0AAE≠0,κ≠0∴∫ydA=0AThusyA=∫ydA=0∴y=0Awhere:yisthedistancefromtheorigintothecentroidofanareaA.Thismeansthatalongthezaxi

7、ssochosen,boththenormalstrainεandthenormalstressσarezero.Inbendingtheory,thisaxisxxisreferredtoastheneutralaxisofthecrosssection.Neutralsurface.2M=EκydA=EκIz∫zA24I=ydA--momentofinertiaoftheareaAaroundz-axis.Unit:mz∫AMM