鋼筋混凝土結(jié)構(gòu)

StrengthandDeformationofMemberswithTorsionINTRODUCTIONTorsioninreinforcedconcretestructuresoftenarisesfromcontinuitybetweenmembers.Forthisreasontorsionreceived;relativelyscantattentionduringthefirsthalfofthiscentury,andtheomissionfromdesignconsiderationsapparentlyhadnoseriousconsequences.During;thelast10to15years,agreatincreaseinresearchactivityhasadvancedtheunderstandingoftheproblemsignificantly.Numerousaspectsoftorsioninconcretehavebeen,andcurrentlyarebeing,examinedinvariouspartsoftheworld.ThefirstsignificantorganizedpoolingofknowledgeandresearcheffortinthisfieldwasasymposiumsponsoredbytheAmericanConcreteInstitute.Thesymposiumvolumealsoreviewsmuchofthevaluablepioneeringwork.Mostcodereferencestotorsiontodatehavereliedonideasborrowedfromthebehaviorofhomogeneousisotropicelasticmaterials.ThecurrentACIcode8-2incorporatesforthefirsttimedetaileddesignrecommendationsfortorsion.Theserecommendationsarebasedonaconsiderablevolumeofexperimentalevidence,buttheyarelikelytobefurthermodifiedasadditionalinformationfromcurrentresearcheffortsisconsolidated.Torsionmayariseasaresultofprimaryorsecondaryactions.Thecaseofprimarytorsionoccurswhentheexternalloadhasnoalternativetobeingresistedbutbytorsion.Insuchsituationsthetorsion,requiredtomaintainstaticequilibrium,canbeuniquelydetermined.Thiscasemayalsoberefer-redtoasequilibriumtorsion.Itisprimarilyastrengthproblembecausethestructure,oritscomponent,willcollapse訐thetorsionalresistancecannotbesupplied.Asimplebeam,receivingeccentriclineloadingsalongitsspan,cantileversandeccentricallyloadedboxgirders,asillustratedinFigs.8.1and8.8,areexamplesofprimaryorequilibriumtorsion.Instaticallyindeterminatestructures,torsioncartalsoariseasasecondaryactionfromtherequirementsofcontinuity.Disregardforsuchcontinuityinthedesignmayleadtoexcessivecrackwidthsbutneednothavemoreseriousconsequences.Oftendesignersintuitivelyneglectsuchsecondarytorsionaleffects.Theedgebeamsofframes,supportingslabsorsecondary-beams,aretypicalofthissituation(seeFig.8.2).Inarigidjointedspacestructureitishardlypossibletoavoidtorsionarisingfromthecompatibilityofdeformations.Certainstructures,suchasshellselasticallyrestrainedbyedgebeams,"aremoresensitivetothistypeoftorsionthanareother.Thepresentstateofknowledgeallowsarealisticassessment.ofthetorsionthatmayariseinstaticallyindeterminatereinforcedconcretestructuresatvariousstagesof

theloading.kJFig.8.1.Examplesofprimaryorequihbnumtor&ton.Torsioninconcretestructuresrarelyoccurs.withoutotheractions.Usuallyflexure,shear,andaxialforcesarealsopresent.Agreatmanyofthemorerecentstudieshaveattemptedtoestablishthelawsofinteractionsthatmayexistbetweentorsionandotherstructuralactions.Becauseofthelargenumberofparametersinvolved,someeffortisstillrequiredtoassessreliablyallaspectsofthiscomplexbehavior.

Hgr8+i.Torsioninstaiicatlyindeierminatestrucluresh8.2PLAINCONCRETESUBJECTTOTORSIONThebehaviorofreinforcedconcreteintorsion,beforetheonsetofcracking,canbebasedorsthestudyofplainconcretebecausethecontributionofrein-forcementatthisstageisnegligible.ElasticBehaviorFortheassessmentoftorsionaleffectsinplainconcrete,wecanusethewell-knownapproachpresentedinmosttextsonstructuralmechanics.TheclassicalsolutionofSt.Venantcanbeappliedtothecommonrectangularconcretesection.Accordingly,themaximumtorsionalshearingstressvtisgeneratedatthemiddleofthelongsideandcanbeobtainedfromrx^y

whereT=torsionalmomentatthesectiony,x=overalldimensionsoftherectangularsection,x<y屮t=astressfactorbeingafunctiony/x,asgiveninFig.8.3Itmaybeequallyasimportanttoknowtheload-displacementrelationshipforthemember.Thiscanbederivedfromthefamiliarrelationship.d0tTd0tTGCwhere0t,=theangleoftwistT=theappliedtorque,whichmaybeafunctionofthedistancealongthespanG=themodulusinshearasdefinedinEq.7.37C=thetorsionalmomentofinertia,sometimesreferredtoastorsionconstantorequivalentpolarmomentsofinertiaz=distancealongmemberForrectangularsections,wehavec=fitx3y (8.3)inwhich0t,acoefficientdependentontheaspectratioy/xofthesection(Fig.8.3),allowsforthenonlineardistributionofshearstrainsacrossthesection.Thesetermsenablethetorsionalstiffnessofamemberoflengthsection.ltobedefinedasthemagnitudeofthetorquerequiredtocauseunitangleoftwistoverthislengthasInthegeneralelasticanalysisofastaticallyindeterminatestructure,boththetorsionalstiffnessandtheflexuralstiffnessofmembersmayberequired.Equation8.4forthetorsionalstiffnessofamembermaybecomparedwiththeequationfortheflexuralstiffnessofamemberwithfarendrestrained,definedasthemomentrequiredtocauseunitrotation,4EI/1,whereEI=flexuralrigidityofasection.

0141710 14 18 2,2 2.6 30 3.5 4.0 4屆5.0678910豐83.Stiffnessandstressfacersforrectangularsecuonssubjectedtotorsion.Thebehaviorofcompoundsections,TandLshapes,ismorecomplex.However,followingBach'ssuggestion,itiscustomarytoassumethatasuitablesubdivisionofthesectionintoitsconstituentrectanglesisanaccept-ableapproximationfordesignpurposes.Accordinglyitisassumedthateach,rectangleresistsaportionoftheexternaltorqueinproportiontoitstorsionalrigidity.AsFig.8.4ashows,theoverhangingpartsoftheflangesshouldbetakenwithoutoverlapping.Inslabsformingtheflangesofbeams,theeffectivelengthofthecontributingrectangleshouldnotbetakenasmorethanthreetimestheslabthickness.Forthecaseofpuretorsion,thisisaconservativeapproximation.F^.8-4.ThesubdiveionofcompoundsectionsforF^.8-4.Thesubdiveionofcompoundsectionsfortorsionalanalysis.UsingBach'sapproximation,8-5theportionofthetotaltorqueTresistedbyelement2inFig.8.4ais(85)andtheresultingmaximumtorsionalshearstressisfromEq.8.1Theapproximationisconservativebecausethe"junctioneffect"hasbeenneglected.Compoundsectionsinwhichshearmustbesubdividedinadifferentway.Theelastictorsionalshearstressflowcanoccur,asinboxsections,Figure8.4cillustratestheprocedure.distributionovercompoundcrosssectionsmaybebestvisualizedbyPrandtl'smembraneanalogy,theprinciplesofwhichmaybefoundinstandardworksconcretestructures,weseldomencountertheonelasticity."Inreinforcedforegoingassumptionsassociatedwithlinearconditionsunderwhichtheelasticbehavioraresatisfied.PlasticBehaviorInductilematerialsitispossibletoattainastateatwhichyieldinshearoccuroverthewholeareaofaparticularcrosssection.Ifyieldingoccursoverthewholesection,theplastictorquecanbecomputedwithrelativeease.ConsiderthesquaresectionappearinginFig.8.5,whereyieldinshearVtyhassetinConsiderthesquaresectionappearinginFig.8.5,whereyieldinshearVtyhassetinthequadrants.ThetotalshearforceVactingoveronequadrantis,11b<5.,11b<5.Torsionalyieldingofasquaresec俎on.TTThesameresultsmaybeobtainedusingNadai's‘sandheapanalogy.'Accordingtothisanalogythevolumeofsandplacedoverthegivencrosssectionisproportionaltotheplastictorquesustainedbythissection.theheap(orroof)overtherectangularsection(seeFig.8.6)hasaheightxv.Fi誓乩亂Fi誓乩亂Nadai?ssandheapanalogywherex=smalldimensionofthecrosssection.midoverthesquaresection(Fig.8.5)is（呦Thevolumeoftheheapovertheoblongsection(Fig.8.6)is

HenceT_

x2yHenceT_

x2y?7}where(8,7a)Itisevidentthat屮ty=3whenx/y=IandO,y=2whenx/y=OItmaybeseenthatEq.8.7issimilartotheexpressionobtainedforelasticbehavior,Eq.8.1.Concreteisnotductileenough,particularlyintension,topermitaperfectplasticdistributionofshearstresses.Thereforetheultimatetorsionalstrengthofaplainconcretesectionwillbebetweenthevaluespredictedbythemembrane(fullyelastic)andsandheap(fullyplastic)analogies.Shearstressescausediagonal(principal)tensilestresses,whichinitiate,thefailure.Inthelightoftheforegoingapproximationsandthevariabilityofthetensilestrengthofconcrete,thesimplifieddesignequationforthedeterminationofthenominalultimatesections,proposedbyshearstressinducedbytorsioninplainconcreteACI318-71,isacceptable:wherexWy.Thevalueof3fortisorty,3,isaminimumfortheelastictheoryandamaxi-mumfortheplastictheory(seeFig.8.3andEq.8.7a).TheultimatetorsionalresistanceofcompoundsectionscanbematedbythesummationofthecontributionoftheconstituentsectionssuchasthoseinFig.8.4,theapproximationis37；37；wherexWyforeachrectangle.(8.8a)

Theprincipalstress(tensilestrength)conceptwouldsuggestthatfailurecracksshoulddevelopateachfaceofthebeamalongaspiralrunningat450tothebeamaxis.However,thisisnotpossiblebecausetheboundaryofthefailuresurfacemustformaclosedloop.Hsuhassuggestedthatbendingoccursaboutanaxisparalleltotheplanesthatisatapproximately450tothebeamaxisandofthelongfacesofarectangularbeam.Thisbendingcausescompressionbeam.Thelattertensioncrackingeventuallyandtensilestressesinthe450planeacrosstheinitiatesasurfacecrack.Assoonasflexuraloccurstheflexuralstrengthofthesectionisreduced,thecrackrapidlypropagates,andsuddenfailurefollows.Hsuobservedthissequenceoffailurewiththeaidofhigh-speedmotionpictures.Formoststructureslittleusecanbemadeofthetorsional(tensile)strengthofunreinforcedconcretemembers.TubularSectionsBecauseoftheadvantageousefficientinresistingdistributionofshearstresses,tubularsectionsaremostresistingtorsion.Theyarewidelyusedinbridgeconstruction.Figure8.7illustratesthebasicformsusedforbridgegirders.ThetorsionalpropertiesofthegirdersimproveinprogressingfromFigs.8.7ato8.7g.Fig.&7.E師icformsusedforbridgecrcs?sections』'Whenthewallthicknesshissmallrelativetotheoveralldimensionsofthesection,uniformshearstressacrossthethicknesscanbeassumed.Byconsideringthemomentsexertedaboutasuitablepointbytheshearstresses,actingoverinfinitesimalelementsofthetubesection,asinFig.8.8a,thetorqueofresistancecanbeexpressed.as(8+9時(shí)T=Jhvt(8+9時(shí)（時(shí)fo（時(shí)foTheproducthvt=v。istermedtheshearflow,.andthisisconstant;thusT T衛(wèi) orp.— r押囪 2AQhwhereAo=theareaenclosedbythecenterJineofthetubewall(shadedareainFig.8.8).Theconceptofshearflowaroundthethinwalltubeisusefulwhentheroleofreinforcementintorsionisconsidered.TheACIcode&2suggeststhattheequationrelevanttosolidsections.8.&beusedalsoforhollowsections,withthefollowingmodificationwhenthewallthicknessisnotlessthanx/lO(seeFig.8.8c):(8,10)wherex<y.Equation8.9bfollowsfromfirstprinciplesandhastheadvantageofbeingapplicabletoboththeelasticandfullyplasticstateofstress.Thetorque-twistrelationshipforhollowsectionsmaybereadilyderivedfromstrainenergyconsiderations.Byequatingtheworkdonebytheappliedtorque(externalwork)tothatoftheshearstresses(internalwork),thetorsionconstantCOfortubularsectionscanbefoundthus:internalwork=internalwork=(sumofshearstressshearstrainactingonthetubeelements)fvt ((unitlength)=-xO叭-^hdsx1J Gexterna!work=(appliedtorque)(angleoftwistperunitJengthofmember)Tx?thetwoexpressionsandusingEq.8.9b,therelationshipHencebyequatingthetwoexpressionsandusingEq.8.9b,therelationshipbetweentorqueandangleoftwistisfoundtobe

andthetorsionalstiffnessofsuchmemberistherefore(8.4a)whereC0istheequivalentpolarmomentofinertiaofthetubularsectionandisgivenby(&⑴wheresismeasuredaroundthewallcenterline.Thesameexpressionforthemorecommonformofboxsection(Fig.8.8b)becomes(8.Ila)ForuniformwallthicknessEq.8.11reducesfurtherto(8.11b)wherepistheperimetermeasuredalongthetubecenterline.Itisemphasizedthattheprecedingdiscussiononelasticandplasticbehaviorrelatestoplainconcert.andthepropositionsareapplicableonlyatlowloadintensitiesbeforecracking.Theymaybeusedforpredictingtheoneofdiagonalcracking.BEAMSWITHOUTWEBREINFORCEMENTSUBJECTTOFLEXUREANDTORSIONThefailuremechanismofbeamssubjectedtotorsionandbendingdependsonthepredominanceofoneortheother.Theratioofultimatetorquetomoment,TJ/MUisasuitableparametertomeasuretherelativemagnitudeoftheseactions.Theflexuralresistancedependsprimarilyontheamountofflexuralreinforcement.The-torsionalbehaviorofaconcretebeamwithoutwebreinforcementismoredifficulttoassessinthepresenceofflexure.

Flexuralstressesinitiatediagonalcracksinthecaseoftorsion,muchastheydointhecaseofshear.Inthepresenceofflexurethesecracksarearrestedinthecompressionzone.Forthisreasonadiagonallycrackedbeamiscapableofcarryingacertainamountortorsion.Themannerinwhichthistorsionisresistedis,atpresent,amatter,ofspeculation.Clearlythecompressionzoneofthebeamiscapableofresistingalimitedamountoftorsion,.andhorizontalreinforcementcanalsocontributetotorsionalresistancebymeansofdowelaction.Ithasbeenfound(e.g.,byMattock")thatthetorsionalresistanceofacrackedsectionisapproximatelyone-halftheultimatetorsionalstrengthoftheuncrackedsection,providedacertain-amountofbendingispresent.Thusonehalfthetorquecausingcrackingcanbesustainedaftertheformationofcracks.Thetorquethuscarriedissosmallthatitsinfluenceonflexureoncanbeignored.Thenominaltorsionalshearstress,correspondingtothislimitedtorsionisconservativelyassumedbyACI318-71.tobe40%ofacrackingstressof6^7；W丿刁N/mm2).二％斗0.4(67?；)“你psiWA/7；N/mm2)⑻⑵andthetorquesuppliedbythe.concretesectiononly,aftertheonsetofcracking,isrevealedbyEq.8.8tobe(8J3)Similarly,forcompoundsections,Eq.8.8agives(8J3a)withthelimitationsonoverhangingpartsasindicatedinFig.8.4.WhenT./M>0.5(i.e.,whentorsionissignificant),brittlefailurehasbeenobserved.Whenthebendingmomentismorepronounced,(i.e.,whenT/Mu<0.5),amoreductilefailurecanbeexpected.Thetorsionalstrengthofabeamcanbeincreasedonlywiththeadditionofwebreinforcement.Theamountofflexuralreinforcementappearstohavenoinfluenceonthetorsionalcapacityoftheconcretesection,T.InTorLbeamstheoverhangingpartoftheflangescontributetotorsional.strength.Thishasbeenverifiedonisolatedbeams.Theeffectivewidthofflanges,whenthesearepartofafloorslab,isdifficulttoassess.Whenayieldlinecandevelopalonganedgebeambecauseofnegativebendingmomentintheslab,asillustratedinFig.8.9itisunlikelythatmuchoftheflangecancontributetowardtorsionalstrength.Fig.乩dYieldlinealonganedgebeam.TORSIONANDSHEARINBEAMSWITHOUTWEBREINFORCEMENTItisevidentthatinsuperposition;theshearstressesgeneratedbytorsionandshearingforceareadditivealongonesideandsubtractivealongtheoppositesideofarectangularbeamsection.Thecriticaldiagonaltensilestressesthatensuearefurtheraffectedbyflexuraltensilestressesintheconcrete,becauseitisimpossibletoapplyshearingforceswithoutsimultaneouslyinducingflexure.Afullyr