钢筋混凝土结构受扭构件的强度及变形原文.docx

钢筋混凝土结构受扭构件的强度及变形原文.docx

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钢筋混凝土结构受扭构件的强度及变形原文

StrengthandDeformationofMemberswithTorsion

8.1INTRODUCTION

Torsioninreinforcedconcretestructuresoftenarisesfromcontinuitybetweenmembers.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,willcollapseifthetorsionalresistancecannotbesupplied.Asimplebeam,receivingeccentriclineloadingsalongitsspan,cantileversandeccentricallyloadedboxgirders,asillustratedinFigs.8.1and8.8,areexamplesofprimaryorequilibriumtorsion.

Instaticallyindeterminatestructures,torsioncartalsoariseasasecondaryactionfromtherequirementsofcontinuity.Disregardforsuchcontinuityinthedesignmayleadtoexcessivecrackwidthsbutneednothavemoreseriousconsequences.Oftendesignersintuitivelyneglectsuchsecondarytorsionaleffects.Theedgebeamsofframes,supportingslabsorsecondary-beams,aretypicalofthissituation（seeFig.8.2）.Inarigidjointedspacestructureitishardlypossibletoavoidtorsionarisingfromthecompatibilityofdeformations.Certainstructures,suchasshellselasticallyrestrainedbyedgebeams,"aremoresensitivetothistypeoftorsionthanareother.

Thepresentstateofknowledgeallowsarealisticassessment.ofthetorsionthatmayariseinstaticallyindeterminatereinforcedconcretestructuresatvariousstagesoftheloading.

Torsioninconcretestructuresrarelyoccurs.withoutotheractions.

Usuallyflexure,shear,andaxialforcesarealsopresent.Agreatmanyofthemorerecentstudieshaveattemptedtoestablishthelawsofinteractionsthatmayexistbetweentorsionandotherstructuralactions.Becauseofthelargenumberofparametersinvolved,someeffortisstillrequiredtoassessreliablyallaspectsofthiscomplexbehavior.

8.2PLAINCONCRETESUBJECTTOTORSION

Thebehaviorofreinforcedconcreteintorsion,beforetheonsetofcracking,canbebasedorsthestudyofplainconcretebecausethecontributionofrein-forcementatthisstageisnegligible.

8.2.1ElasticBehavior

Fortheassessmentoftorsionaleffectsinplainconcrete,wecanusethewell-knownapproachpresentedinmosttextsonstructuralmechanics.TheclassicalsolutionofSt.Venantcanbeappliedtothecommonrectangularconcretesection.Accordingly,themaximumtorsionalshearingstressvtisgeneratedatthemiddleofthelongsideandcanbeobtainedfrom

whereT=torsionalmomentatthesection

y,x=overalldimensionsoftherectangularsection,x

Ψt=astressfactorbeingafunctiony/x,asgiveninFig.8.3

Itmaybeequallyasimportanttoknowtheload-displacementrelationshipforthemember.Thiscanbederivedfromthefamiliarrelationship.

whereθt,=theangleoftwist

T=theappliedtorque,whichmaybeafunctionofthedistancealongthespan

G=themodulusinshearasdefinedinEq.7.37

C=thetorsionalmomentofinertia,sometimesreferredtoastorsionconstantorequivalentpolarmomentsofinertia

z=distancealongmember

Forrectangularsections,wehave

inwhichβt,acoefficientdependentontheaspectratioy/xofthesection（Fig.8.3）,allowsforthenonlineardistributionofshearstrainsacrossthesection.

Thesetermsenablethetorsionalstiffnessofamemberoflengthsection.ltobedefinedasthemagnitudeofthetorquerequiredtocauseunitangleoftwistoverthislengthas

Inthegeneralelasticanalysisofastaticallyindeterminatestructure,boththetorsionalstiffnessandtheflexuralstiffnessofmembersmayberequired.Equation8.4forthetorsionalstiffnessofamembermaybecomparedwiththeequationfortheflexuralstiffnessofamemberwithfarendrestrained,definedasthemomentrequiredtocauseunitrotation,4EI/1,whereEI=flexuralrigidityofasection.

Thebehaviorofcompoundsections,TandLshapes,ismorecomplex.However,followingBach'ssuggestion,itiscustomarytoassumethatasuitablesubdivisionofthesectionintoitsconstituentrectanglesisanaccept-ableapproximationfordesignpurposes.Accordinglyitisassumedthateach,rectangleresistsaportionoftheexternaltorqueinproportiontoitstorsionalrigidity.AsFig.8.4ashows,theoverhangingpartsoftheflangesshouldbetakenwithoutoverlapping.Inslabsformingtheflangesofbeams,theeffectivelengthofthecontributingrectangleshouldnotbetakenasmorethanthreetimestheslabthickness.Forthecaseofpuretorsion,thisisaconservativeapproximation.

UsingBach'sapproximation,8.5theportionofthetotaltorqueTresistedbyelement2inFig.8.4ais

andtheresultingmaximumtorsionalshearstressisfromEq.8.1

Theapproximationisconservativebecausethe"junctioneffect"hasbeenneglected.

Compoundsectionsinwhichshearmustbesubdividedinadifferentway.Theelastictorsionalshearstressflowcanoccur,asinboxsections,Figure8.4cillustratestheprocedure.distributionovercompoundcrosssectionsmaybebestvisualizedbyPrandtl'smembraneanalogy,theprinciplesofwhichmaybefoundinstandardworksconcretestructures,weseldomencountertheonelasticity."Inreinforcedforegoingassumptionsassociatedwithlinearconditionsunderwhichtheelasticbehavioraresatisfied.

8.2.2PlasticBehavior

Inductilematerialsitispossibletoattainastateatwhichyieldinshearoccuroverthewholeareaofaparticularcrosssection.Ifyieldingoccursoverthewholesection,theplastictorquecanbecomputedwithrelativeease.

ConsiderthesquaresectionappearinginFig.8.5,whereyieldinshearVtyhassetinthequadrants.ThetotalshearforceVactingoveronequadrantis

ThesameresultsmaybeobtainedusingNadai's‘sandheapanalogy.’Accordingtothisanalogythevolumeofsandplacedoverthegivencrosssectionisproportionaltotheplastictorquesustainedbythissection.theheap（orroof）overtherectangularsection（seeFig.8.6）hasaheightxv.

wherex=smalldimensionofthecrosssection.midoverthesquaresection（Fig.8.5）is

Thevolumeoftheheapovertheoblongsection（Fig.8.6）is

ItisevidentthatΨty=3whenx/y=IandO,y=2whenx/y=0

ItmaybeseenthatEq.8.7issimilartotheexpressionobtainedforelasticbehavior,Eq.8.1.

Concreteisnotductileenough,particularlyintension,topermitaperfectplasticdistributionofshearstresses.Thereforetheultimatetorsionalstrengthofaplainconcretesectionwillbebetweenthevaluespredictedbythemembrane（fullyelastic）andsandheap（fullyplastic）analogies.Shearstressescausediagonal（principal）tensilestresses,whichinitiate,thefailure.Inthelightoftheforegoingapproximationsandthevariabilityofthetensilestrengthofconcrete,thesimplifieddesignequationforthedeterminationofthenominalultimatesections,proposedbyshearstressinducedbytorsioninplainconcreteACI318-71,isacceptable:

wherex≤y.

Thevalueof3fortisorty,3,isaminimumfortheelastictheoryandamaxi-mumfortheplastictheory（seeFig.8.3andEq.8.7a）.

TheultimatetorsionalresistanceofcompoundsectionscanbematedbythesummationofthecontributionoftheconstituentsectionssuchasthoseinFig.8.4,theapproximationis

wherex≤yforeachrectangle.

Theprincipalstress（tensilestrength）conceptwouldsuggestthatfailurecracksshoulddevelopateachfaceofthebeamalongaspiralrunningat450tothebeamaxis.However,thisisnotpossiblebecausetheboundaryofthefailuresurfacemustformaclosedloop.Hsuhassuggestedthatbendingoccursaboutanaxisparalleltotheplanesthatisatapproximately450tothebeamaxisandofthelongfacesofarectangularbeam.Thisbendingcausescompressionbeam.Thelattertensioncrackingeventuallyandtensilestressesinthe450planeacrosstheinitiatesasurfacecrack.Assoonasflexuraloccurstheflexuralstrengthofthesectionisreduced,thecrackrapidlypropagates,andsuddenfailurefollows.Hsuobservedthissequenceoffailurewiththeaidofhigh-speedmotionpictures.Formoststructureslittleusecanbemadeofthetorsional（tensile）strengthofunreinforcedconcretemembers.

8.2.3TubularSections

Becauseoftheadvantageousefficientinresistingdistributionofshearstresses,tubularsectionsaremostresistingtorsion.Theyarewidelyusedinbridgeconstruction.Figure8.7illustratesthebasicformsusedforb