(1.9.6)-6.5 Bending of Unsymmetric Beams材料力学材料力学

BendingofUnsymmetricBeamsChapter6StressesinBeams(AdvancedTopics)UnsymmetriccompositebeammadeupfromchannelsectionandoldwoodbeamUnsymmetricBeamsUnsymmetriccrosssectionofananti-collisionbeamInengineeringstructures,beamsmayhaveunsymmetriccrosssections,forexample,*thecrosssectionofananti-collisionbeam,and*anunsymmetriccompositebeammadeupfromchannelsectionandoldwoodbeam.Inthissection,onlyunsymmetricbeamsinpurebendingareconsidered,andthestressesandthepositionoftheneutralaxisinbeamsareofinterest.Inlatersections,theeffectsoflateralloadsareinvestigated.1.NeutralAxisAssumethatthezaxisistheneutralaxis.①TheresultantforceinxaxisEandκyareconstantsThezaxis(theneutralaxis)mustpassthroughthecentroid.TheoriginoftheyandzaxesforanunsymmetricbeammustbeplacedatthecentroidC.Tofindthestressesandnetrualaxis,atthisstageoftheanalysis,thereisnodirectwayofdeterminingthesequantities.Therefore,anindirectapproachmustbeused.Takethis*unsymmetricbeamforexamplewithabendingmomentMactingattheend.First*constructtwoperpendicularaxes,yandzaxesatanarbitrarilyselectedpointC.*Assumethatthez

axisistheneutralaxis.Consequently,thebeamdeflectsinthex-yplane.Sincethebeamisinpurebending,theresultantforceinthexaxisovertheentirecrosssectionmustbezero*.where*σxrepresentsthenormalstressactingonanelementofareadAlocatedatdistanceyfromtheneutralaxis,Eisthemodulusofelasticity

andkyisthecurvature.Atanygivencrosssection,*Eandkyareconstants,therefore*.Thisequationshowsthat*thezaxispassesthroughthecentroidCofthecrosssection.Whenassumethattheyaxisisthe

neutralaxis,sameconclusioncanbeobtaiend.Thatisneutralaxismustpassthroughthecentroid.Itfollowsthat*theoriginoftheyandzaxesforanunsymmetricbeammustbeplacedatthecentroidCofthecrosssection.Whenanunsymmetricbeamisinpurebending,theplaneinwhichthebendingmomentactsisperpendiculartotheneutralsurface,onlyiftheyandzaxesareprincipalcentroidalaxes

ofthecrosssectionandthebendingmomentactsinoneofthetwoprincipalplanes(x-yplaneorx-zplane).②Theresultantmomentofthestressesσx

Assumethatthezaxisistheneutralaxis.Andwhendiscussingthemomentresultantofthestressesσx,itleadstothefollowingimportantconclusion*:Whenanunsymmetricbeamisinpurebending,theplaneinwhichthebendingmomentactsisperpendiculartotheneutralsurfaceonlyiftheyandzaxesareprincipalcentroidalaxesofthecrosssectionandthebendingmomentactsinoneofthetwoprincipalplanes,thex-yplaneorthex-zplane.Thisconclusionleadstoadirectmethodforfindingthestressesinanunsymmetricbeamsubjectedtoabendingmomentactinginanarbitrarydirection.2.Procedureforanalyzinganunsymmetricbeam②ResolvethebendingmomentMintocomponents③Thesuperpositionofthebendingstresses①LocatethecentroidCandconstructasetofprincipalaxesTheanglebetweentheneutralaxis

andthezaxis:Nowlet’slookatageneralprocedureforanalyzinganunsymmetricbeamsubjectedtoanybendingmomentM.*FirstlocatethecentroidCofthecrosssectionandconstructingasetofprincipalaxesatthecentroidC,theyandzaxesinthefigure.Next,*resolvethebendingmomentMintocomponentsMyandMz.Astheusualformulasforpurebendingcanbeappliedhere,thestressesarecomputedusingthemomentsMyandMzactingseparately.*Superposingthebendingstresses,theresultantstressatanypointisgiven.Also,*theequationoftheneutralaxisnnisobtainedbysettingσxequaltozeroandsimplifying.Samehere,thisequationshowsingeneraltheanglesβandθarenotequal;hencetheneutralaxisisgenerallynotperpendiculartotheplaneinwhichtheappliedcoupleMacts.Theonlyexceptionsarethethreespecialcasesdescribedintheprecedingsection.ThenormalstressatpointA:3.Alternateprocedureforanalyzinganunsymmetricbeam-ageneralizedbendingtheory②ResolvethebendingmomentMintocomponents③Thegeneralizedflexureformula①LocatethecentroidCandconstructasetofnonprincipalaxesEquilibriumrelations:Forsuchunsymmetricbeams,likeZ-section,whentheorientationoftheprincipalaxescannotbeobtainedbyinspectionorfromtables,itmaybeeasiertoworkwiththenonprincipalcentroidalaxesthatarealignedwiththesidesofthecrosssection.Thatisanalternateprocedureforanalyzinganunsymmetricbeam,whichwillgiveageneralizedbendingtheory.*Consideranunsymmetriccrosssection*withyandzaxeshavingtheiroriginatthecentroid,buttheyarenotprincipalaxes.*ThebendingmomentMresolvedintocomponentsMyandMz,andbendingofthebeamoccursinboththex-yandx-zplanes,neitherofwhichisaprincipalplane.*Togetthegeneralizedflexuralformulaforcalculatingthenormalstresssσxatanypointinanunsymmetricbeam,*thenormalstressatanypointAis.thecurvatureskyandkzareunknown,whicharefoundfrom*equilibriumrelationships,theresultant(axial)forceequalstozero,*Myequalstothemomentstressresultantabouttheyaxis,and*Mzequalstothemomentstressresultantaboutthezaxis.Thefirstequilibriumequationissatisfiedautomaticallybecausetheoriginoftheaxesisatthecentroidofthecrosssection.Solvethelasttwoequationssimultaneouslytoobtaintheexpressionsforcurvaturesintermsofthebendingmoments.Theorientationoftheneutralaxisnn：③ThegeneralizedflexureformulaThensubstitutecurvatureskyandkzessionsintotheexpressionofthenormalstressσx*.Thisisthegeneralizedflexureformulaforanunsymmetricbeamactedonbymomentsaboutperpendicularcentroidalaxesthatarenotnecessarilyprincipalaxes.Theorientationoftheneutralaxisnnisobtainedbyequatingσxtozerotofindanexpressionfor