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,xΨt=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