(1.9.10)-6.9 Shear Centers of Thin-Walled材料力学材料力学

MechanicsofMaterialsShearCentersofThin-WalledOpenSectionsChapter6StressesinBeams(AdvancedTopics)Welcometomechanicsofmaterials.Inthissection,howtolocateshearcentersinbeamsofThin-WalledOpenCrossSectionsisexamined.-beamswithsinglysymmetricorunsymmetriccrosssections.ShearCentersofThin-WalledOpenSectionsBecausetheshearcenterofadoublysymmetriccrosssectionisknowntobelocatedatthecentroid,*onlybeamswithsinglysymmetricorunsymmetriccrosssectionsaretobeconsideredhere,suchas*crosssectionsgivenbelow.Twoprincipalsteps:1.Evaluatetheshearstressesactingonthecrosssectionwhenbendingoccursaboutoneoftheprincipalaxes.2.Determinetheresultantofthosestresses.Theshearcenterislocatedonthelineofactionoftheresultant.ProcedureforlocatingshearcentersTheprocedureforlocatingtheshearcenterconsistsof*twoprincipalsteps:*first,evaluatetheshearstressesactingonthecrosssectionwhenbendingoccursaboutoneoftheprincipalaxes,and*second,determinetheresultantofthosestresses.Theshearcenterislocatedonthelineofactionoftheresultant.Considerbendingaboutbothprincipalaxestodeterminethepositionoftheshearcenter.Asintheprivoussections,useonlycenterlinedimensionswhenderivingformulasandmakingcalculations.Thisprocedureissatisfactoryifthebeamisthin-walled,thatis,ifthethicknessofthebeamissmallcomparedtotheotherdimensionsofthecrosssection.1.ChannelsectionSinglySymmetric.Theshearcentermust

belocatedontheaxisofsymmetry,thezaxis.Shearstressformula①

τ1

intheflange:②

τ2

intheweb:③

τmax

intheweb:Thefirstbeamtobeanalyzedisa*channelsection,*singlysymmetricaboutthezaxis.Basedonthepreviousdiscussioninsection6.6,theshearcentermustbelocatedonthezaxis.AndtheoriginCisthecentroid,sothatboththeyandzaxesareprincipalcentroidalaxes.Tofindthepositionoftheshearcenter,assumethatthebeamisbentaboutthezaxisastheneutralaxis,andthendeterminethelineofactionoftheresultantshearforceVy,whichisparalleltotheyaxis.Then,TheshearcenterislocatedwherethelineofactionofVyintersectsthezaxis.*BaseduponthediscussionsinSection6.8,theshearstressesinachannelvarylinearlyintheflangesandparabolicallyintheweb.Theresultantofthosestressescanbefoundifthemaximumstressτ1intheflange,thestressτ2atthetopoftheweb,andthemaximumstressτmaxinthewebare

known.Tofindthestressτ1,τ2andτmax,*theshearstressformulaisused.

*Forthestressτ1intheflange,*Qzequaltothefirstmomentoftheflangeareaaboutthezaxis,euqaltobtftimesh/2.SubstitueQzintotheaboveformula,*τ1isfound.Inasimilarmanner,*τ2and*τmaxcanfound.④ThehorizontalshearforceF1intheflanges:⑤TheverticalforceF2intheweb:Stessdiagrammadeoftwoparts:-arectangleofarea:τ2h-aparabolicsegmentofarea:

*ThehorizontalresultantshearforceF1intheflangescanbefoundfromthetriangularstressdiagrams.*F1isequaltotheareaofthestresstrianglemultipliedbythethicknessoftheflange.*Inasimilarmanner,thevertical

resultantshearforceF2inthe

webcanbefound.

*Theareaofthestressdiagramismadeupoftwoparts—*arectangleofareaτ2hand*aparabolicsegmentofareathen

*F2inthe

webcanbefound.Ifsubstituteτ2,andτmaxintotheexpressionforF2,itisobtainedthat*theverticalforceF2mustbeequaltotheshearforceVy,sincethe

forcesintheflangeshavenoverticalcomponents.⑥Thepositionoftheshearcenter:Momentrelationship:Themomentofthethreeforcesaboutanypointinthecrosssectionmustbe

equaltothemomentoftheforceVyaboutthatsamepoint.Therefore,thethreeforcesactingonthecrosssection

havearesultant*VythatintersectsthezaxisattheshearcenterS.*Hence,themomentofthethreeforcesaboutanypointinthecrosssectionmustbeequaltothemomentoftheforceVyaboutthatsamepoint.Thismomentrelationshipprovidesanequationfromwhichthepositionoftheshearcentermaybefound.

*Here,selecttheshearcenteritselfastheasthecenterofmoments,equatingthemomentsofF1,F2andVy,gives*,whereeisthedistancefromthecenterlineofthewebtotheshearcenter.Thensolvefor*e.Thus,thepositionoftheshearcenterofachannelsectionhasbeendetermined.SinglySymmetric.Theshearcentermust

belocatedontheaxisofsymmetry,thezaxis.2.Anglesection①TheshearstressesatdistancesfromtheedgeinthelegsThenextshapetobeconsideredisan*equal-leganglesection,inwhicheachlegoftheanglehaslengthbandthicknesst.*ThezaxisisanaxisofsymmetryandtheoriginofcoordinatesisatthecentroidC;therefore,boththeyandzaxesareprincipalcentroidalaxes.Tolocatetheshearcenter,followthesamegeneralprocedureasthatdescribedforachannelsection.*Forthispurpose,selectasectionbblocatedatdistancesfrompointa,*thenusetheshearstressformulatofindthecorrespondingshearstressesinthelegsoftheangle.*thefirstmomentoftheareabetweenpointaandsectionbb

*isequalto*itsareastmultipliedby*itscentroidaldistancefromtheneutralaxis.Then,*shearstresscanbeobtained.*Shearstresscanbeexpressedinanotherform,bysubstitutingIzintotheaboveexpression.*ThemomentofinertiaIzcanbeobtainedfromCase24ofAppendixE.Ats=b③TheshearforceFineachleg：②Themaximumshearstresses:Thisequationgivestheshearstressatanypointalongthelegoftheangle.*Thestressvariesquadraticallywiths,asshowninthefigure.*Themaximumvalueoftheshearstressoccursattheintersectionofthelegsoftheangle,*wheres=b,τmax=3Vy/2bt√2.*TheshearforceFineachlegis*equaltotheareaoftheparabolicstressdiagramtimesthethicknesstofthelegs.SincethehorizontalcomponentsoftheforcesFcanceleachother,onlytheverticalcomponentsremain.*EachverticalcomponentisequaltoF/√2,or*Vy/2,sotheresultantverticalforceisequaltotheshearforceVy,asexpected.SinceVypassesthroughtheintersectionpointofthelinesofactionofthetwoforcesF,*theshearcenterSislocatedatthejunctionofthetwolegsoftheangle.Asimplelineofreasoning:Thepointofintersectionofthetwoforcesinthelegsistheshearcenter.3.SectionsConsistingofTwoIntersectingNarrowRectanglesIntheprecedingdiscussionofananglesection,theshearstressesandtheforcesinthelegswereevaluatedtolocatetheshearcenter.However,ifthesoleobjectiveistolocatetheshearcenter,itisnotnecessarytoevaluatethestressesandforces.*Sincetheshearstressesandtheirresultantsareparalleltothecenterlinesofthelegs,theresultantofthetwoforcesFis

asingleforcethatpassesthroughtheirpointofintersection.Consequently,thispointmustbetheshearcenter.

Thus,theshearcenterofanequal-leganglesectioncanbefoundbyasimplelineofreasoning,withoutmakinganycalculations.Thisisvalid*forallcrosssectionsconsistingoftwothin,intersectingrectangles.Ineachcase,theresultantsoftheshearstressesareforcesthatintersectatthejunctionoftherectangles.Therefore,theshearcenterSislocatedatthatpoint,asshowninthefigure.NoaxesofsymmetrybutsymmetricaboutthecentroidC.TheshearcenteroftheZ-sectioncoincideswiththecentroid.4.ZSectionNowdeterminethelocationoftheshearcenterofa*Z-sectionhavingthinwalls.Thesectionhas*noaxesofsymmetrybutissymmetricaboutthecentroidC.PleaserefertheSectionD.1ofAppendixDforadiscussionofsymmetryabout

apoint.Theyandzaxesareprincipalaxesthroughthecentroid.AssumingthatashearforceVyactsparalleltotheyaxisandcausesbendingaboutthezaxisastheneutralaxis.ThentheshearstressesintheflangesandwebwillbedirectedasshowninFigure.*Fromsymmetry,theforcesF1inthetwoflangesmustbeequ