材料力学06_梁的剪应力

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1、ShearingStressinBeams06IntroductionFormostbeams,boththebendingmomentandshearforcewillpresent.Whenshearingstressesareexertedontheverticalfacesofanelement,equalstressesmustbeexertedonthehorizontalfaces.Thepossiblestresscomponentsatanypointonthecrosssectionwillbesx,txya

2、ndtxz.xyzsxtxztxyIfV=0(purebending),txy=txz=0NowV0,bothtxyandtxzmaynotbezero.Distributionofshearingstressesisunknownbutsatisfies(txy0)txzmaynotbezero,buttheaverageiszero2CalculationofShearingStress:PreliminaryThedistributionoftxycannotbedeterminedfromstaticsalone.

3、Analysisofdeformationmaynothelptoomuch.However,weknowthatshearingstresscomesinpair,arrowtoarrowandtailtotail,asshown.Ifwecandeterminetyx,thenfromtxy=tyx,wecanobtaintxy.xyzsxtxztxyWeknowthattxzmaynotbezero,buttheaverageiszero.However,foranarrowbeam(widthmuchsmallertha

4、nheight),wecanassumethattxz=0zero,sincetzx0ontheleftandrightsurfaces.Therefore,thistopicwillbefocusingontxyonly.txztzx=03ShearontheHorizontalFaceofaBeamElementMVV+dVM+dMtxytyxss+dsdH(Resultantoftyx)F1F2D′C"D"dxA*yForthebeamshown,consideranelementCDD′C′tocalculateshe

5、arstressaty1.C′CDA*4ShearontheHorizontalFace:FirstMomentofAreaShearFlowFirstMomentofAreaIisthesecondmomentofareaoftheentiresectionaboutthecentroidalaxisoftheentiresection.A*isareaoftheisolatedarea,fromwheretheshearflowiscalculatedtoafreesurface.Horizontalshearflow,th

6、ehorizontalshearforceperunitlength,inabeamisgivenbyqss+ds=the1stmomentofareaoftheisolatedareaA*.yA*C*y*y1Forbuiltupsectionsy*isthedistancefromthecentroidalaxis(neutralaxis,N.A.)tothecentroid(C*)oftheisolatedarea.5Example1ProblemAbeamismadeofthreeplanks,nailedtogether

7、.Knowingthatthespacingbetweennailsis25mmandthattheverticalshearinthebeamisV=500N,determinetheshearforceineachnail.6Example1:SolutionStrategyIfthebeamwereonepiece,therewouldbeshearstress(shearflow)betweenthecapandtheweb.SolutionStrategyThenailswillcarrytheforceresulte

8、dfromtheshearflow.7FirstMomentofArea.Fortheisolatedarea(theflange),Qcanbecalculatedas**Example1:SolutionShearFlow.Thehorizontalshea