工程力学- 弯曲形变与力学特性

Chapter6Bending16.1ShearandMomentDiagramsBeamBeam–slenderbar,supporttransverseload26.2GraphicalMethodforConstructingShearandMomentDiagramsRegionsofDistributedLoad*Differentialequations:36.3BendingDeformationofaStraightMemberObservations*Horizontallines–curved*Verticallines–remainstraight,butrotated*Upperportion–compressed,lowerportion–stretched*Neutralsurface–nochangeinlongitudinallength4CoordinateSystem*x-axis–longitudinalaxisonneutralplane*y-axis–verticalaxis*z-axis–transverseaxisonneutral*x-zplaneistheneutralplane5Assumptions*Thelongitudinallinedoesnotchangeinlength,butbecomeacurve.*Crosssectionsremainplanar,andperpendiculartolongitudinalaxis.*Deformationofcrosssectionsisneglected.6NormalStrainAtlocationx,cutasmallsegmentoflengthdx7Thestrainreachesitsmaximum,attoporbottom.Oneistension,andanotheriscompression.86.4TheFlexureFormulaNormalStrainandStressBotharelinearlyvaryingwithrespecttoy

(distancetoneutralaxis).Whereistheneutralsurface(axis)?Howtodeterminethecurvatureofthedeformedbeam?Questions!!!9Whereistheneutralsurface(axis)?LocationofNeutralSurfaceForthecrosssection,thereisnolongitudinal(normal)force,10Centroidofcrosssection:*Forsymmetricshapes,theneutralplaneisinmiddle,suchasrectangular,I-beam.Fromthefirstequationofstatics,11FlexureFormulaTheinternalmomentMiscausedbythenormalstress.Note:(i)Aboveneutralaxis,compressivestress(negative

)producespositivemoment.12DefineIasthemomentofinertiaofcrosssectionabouttheneutralaxis(z-axis):-Flexureformula13RelationbetweenthecurvatureandthemomentForcompositeshape(consistingofseveralsimpleshapes)*Centroid:*Momentofinertia:UseParallel-AxisTheorem

IC–momentofinertiaabouttheaxispassingthroughthecentroidI–momentofinertiaaboutneutralaxis

A–totalarea,dy–distancebetweentwoaxes14*Procedure:(i)Dividetheareaintoseveralpartsofsimpleshapes.(ii)Determinetheoverallcentroid(neutralaxis)(iii)Foreachpart,calculatethemomentofinertiaaboutaxispassingthroughitsowncentroid.(iv)Foreachpart,useparallel-axistheoremtocalculatethemomentofinertiaabouttheneutralaxis.(v)Thetotalmomentofinertiaoftheentireareaisthealgebraicsum(+or-)ofthoseofallparts.

1516171819EXAMPLE6.13

2021EXAMPLE6.14

22EXAMPLE6.14(CONTINUED)

23Chapter7TransverseShear247.1ShearinStraightMembers*Abeamsupportsbothshear(V)andmoment(M).*TheshearforceVistheresultoftransverseshearstressdistribution.*Sinceshearmustappearinpairforbalance,thetransverseshearstressandlongitudinalshearstresscoexist.25Thereisrelativedisplacementbetweenadjacentlayers.26Thereareshearforcesbetweenadjacentlayers.Longitudinalshearstressarepresent.277.2TheShearFormulaCutthebeamatlocationx,andtakeasmallsegmentoflengthdx.NormalstressdistributionNormalstressdistribution28ConsiderareaA(abovey),andwriteequilibriumequation2930

-shearstressatpointyofthecrosssectionatxV–inte