机械设计05弯曲剪切和扭转

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1、05Bending,ShearandTorsionShigley,Fig.4-15Theneutralplanepassesthroughthecentroidofthecrosssectionalarea.Stress(strain)iszeroattheneutralplane.BendingofStraightBeams:DeformedBeamsDeformedBeamsThematerialisisotropic,homogeneousandlinearelastic.Thebeamisinitiallystraight.Planecrosssec

2、tionsremainplaneduringbending(i.e.,thedeformationduetotransverseshearisignored).Thebeamaxesaretheprincipalaxesofthesection,i.e.,Iyz=0.Thebeamwouldfailbybendingratherthanbycrushing,wrinkling(localbuckling)orglobalbuckling.Assump-tions2Thereisatleastonesetofcoordinatesystemaboutwhich

3、Iyz=0.TheaxisaboutwhichIyz=0iscalledtheprincipalaxis.Ifthesectionissymmetricaboutanaxis,thisaxisisaprincipalaxis.Formostshapes,thereareonlytwomutuallyperpendicularprincipalaxes.StraightBeams:MomentsofInertiaDefinitionPrincipalAxis3UnderAllLoadsUnderN:UnderMyUnderMzIfIyz≠0,StraightB

4、eams:NormalStressesNormalStressNotes4UnderMzStraightBeams:NormalStressDistributionEquation5dxbShearStressinBeams:ShearStressForequilibriumofbeamelement6ShearStressinBeams:FirstMomentofAreaShearStressinaBeamForbuiltupsectionShearStressFirstMomentofAreaNoteShearFlowbisthewidthofthecr

5、osssectionattheparticulardistancey1.IisthesecondmomentofareaoftheentiresectionaboutN.A.y1=the1stmomentofareaoftheisolatedareaisthedistancefromtheneutralaxis(N.A.)tothecentroid(C.G.)oftheisolatedarea(Forceperunitlength)7ShearStressinaNarrowRectangularBeamFreeBodyDiagramShearStresspa

6、rabolicA:Areaoftheshadedarea.A0:Areaoftheentirecrosssection8IBeam:ShearStresstxyShearStressForWideBeams9IBeam:ShearStresstxzintheFlangeShearStresstxzqinWeb10ExampleAlongbeamismadeofthreeplanks,nailedtogether.Knowingthatthespacingbetweennailsis25mmandthattheverticalshearinthebeamisV

7、=500N,determinetheshearforceineachnail.Problem11Example:SolutionStrategyIfthebeamwereonepiece,therewouldbeshearstress(shearflow)betweenthecapandtheweb.SolutionStrategyThenailswillcarrytheforceresultedfromtheshearstress.12Example:SolutionShearflowqonthelowersurfaceoftheupperplank.IS

8、hearFlowShearForceinEachNa