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1、ShearingStressinBeams06IntroductionFormostbeams,boththebendingmomentandshearforcewillpresent.Whenshearingstressesareexertedontheverticalfacesofanelement,equalstressesmustbeexertedonthehorizontalfaces.Thepossiblestresscomponentsatanypointonthecrosssectionwillbesx,txyandtxz.xyzsxtxztxyIfV=0(purebend
2、ing),txy=txz=0NowV0,bothtxyandtxzmaynotbezero.Distributionofshearingstressesisunknownbutsatisfies(txy0)txzmaynotbezero,buttheaverageiszero2CalculationofShearingStress:PreliminaryThedistributionoftxycannotbedeterminedfromstaticsalone.Analysisofdeformationmaynothelptoomuch.However,weknowthatshear
3、ingstresscomesinpair,arrowtoarrowandtailtotail,asshown.Ifwecandeterminetyx,thenfromtxy=tyx,wecanobtaintxy.xyzsxtxztxyWeknowthattxzmaynotbezero,buttheaverageiszero.However,foranarrowbeam(widthmuchsmallerthanheight),wecanassumethattxz=0zero,sincetzx0ontheleftandrightsurfaces.Therefore,thistopicwill
4、befocusingontxyonly.txztzx=03ShearontheHorizontalFaceofaBeamElementMVV+dVM+dMtxytyxss+dsdH(Resultantoftyx)F1F2D′C"D"dxA*yForthebeamshown,consideranelementCDD′C′tocalculateshearstressaty1.C′CDA*4ShearontheHorizontalFace:FirstMomentofAreaShearFlowFirstMomentofAreaIisthesecondmomentofareaoftheentires
5、ectionaboutthecentroidalaxisoftheentiresection.A*isareaoftheisolatedarea,fromwheretheshearflowiscalculatedtoafreesurface.Horizontalshearflow,thehorizontalshearforceperunitlength,inabeamisgivenbyqss+ds=the1stmomentofareaoftheisolatedareaA*.yA*C*y*y1Forbuiltupsectionsy*isthedistancefromthecentroidal
6、axis(neutralaxis,N.A.)tothecentroid(C*)oftheisolatedarea.5Example1ProblemAbeamismadeofthreeplanks,nailedtogether.Knowingthatthespacingbetweennailsis25mmandthattheverticalshearinthebeamisV=500N,determinetheshearforceineachnail.6Example1:SolutionStrategyIfthebeamwereonepiece,therewouldbeshearstress(
7、shearflow)betweenthecapandtheweb.SolutionStrategyThenailswillcarrytheforceresultedfromtheshearflow.7FirstMomentofArea.Fortheisolatedarea(theflange),Qcanbecalculatedas**Example1:SolutionShearFlow.Thehorizontalshea