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Analysis and calculation of factors on curvature ductility of unbonded prestressed concrete beams

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Analysis and calculation of factors on curvature ductility of unbonded prestressed concrete beams

    Analysis and calculation of factors on

    curvature ductility of unbonded

    prestressed concrete beams

    JournalofHarbInstituteofTechnolo(NewSeries),Vo1.14,No.1,2007

    Analysisandcalculationoffactorsoncurvatureductilityof

    ZHENGWenzhong,XIEHengyah,YANGChun-feng

    郑文忠,解恒燕,杨春峰

    (SchoolofCivilEngineering,HarbinInstituteofTechnology,Harbin150090,China) Abstract:Inconsiderationthatbehaviorofcurvatureductilityofinteriorsupportdirectlyinfluencesthedegree

    ofmomentmodificationofunbondedprestressedconcrete(UPC)continuousstructures.constitutiverelation

    shipsofconcrete,non

    prestressedreinforcementandprestressedreinforcementusedfornonlinearanalysisare ven.Tbroughsimulationanalysisonsimplebeamssubjectedtosingleloadingatthemiddleofthespan,the

    lawoffactorsinfluencingcurvatureductility,suchasglobalreinforcingindex,prestressingdegree,effective

    prestress,strengthofconcreteandgradeofnon-prestressedreinforcementareexplored.Basedontheseresear-

    ches,calculatingformulaofcurvatureductilitycoefficientofUPCbeamsisestablished.whichprovidesbasic

    dataforfurtherresearchonplasticdesignofUPCjndeterminatestructures. Keywords:unbondedprestressedconcrete;continuousbeam;curvatureductilitycoefficient;globalreinfor-

    cingindex

CLCnumber:TU378,5Documentcode:AArticleID:l005-9ll3(2007)0l-00l8-05

    Whethertheprospectiveplasticintemalforceredis

    tributioninunbondedprestressedconcrete(UPC)con

    tinuousstructurescanoccurornotismainlydetermined bytherotatingcapacityofplastichingeininteriorsup

    portregions.Whiletherotatingcapacityofplastichinge jscloselyrelatedtothecurvatureductilityofthejnterior supportregaons,so,itisessentialtodoresearcheson thelawofcurvatureductilityofUPCcontinuousbeams aswellasthequantitativemethod.Atpresent.curva

    tureductilityisexpressedbythecurvatureductilityco

    efficient.Itcanbecalculatedasfollows:

    '

    (1)

    where西..istheultimatecurvatureofthecalculatedsee

    tion;.istheyieldcurvatureofthecalculatedsection. Inordertoresearchintothelawofcurvatureduc

    tilityofinteriorsupportofUPCcontinuousbeams. meanwhile,consideringthemechanicalpropertyofin

    teriorsupportofUPCcontinuousbeamsandregionsbe

    tweenthetwoflexuraIpointofthebeamaresimilarto thatofsimplebeam.wedoresearchonUPCsimple beamsubjectedtosingleloadingatthemiddleofthe span.Byanalyzingfactorsinfluencingcurvatureductil

    ity,globalreinforcingindex,prestressingdegreeAp, effectiveprestressorPe,strengthofconcreteandgradeof non.prestressedreinforcement.theformulaforcalculat

    ingcurvatureductilitycoefficientiSestablished.which willprovidebasicdataforfurtherresearchesonplastic

Received2oo5062O

    designofUPCindeterminatestructures. 1ConstituteRelationshipandSelectionofMateri?

    alandDimensionoftheSimulateMember 1.1ConstituteRelationship

    Stresstrainrelationshipofconcreteincompression

    isgivenbyEq.(2):

    orc=

    ?1)0<c<.,(2)

    L=.<<,

    =2一南(50),

    o=0.002+0.5(.

    

    50)x10一?0.002,

    =

    0.0033(.50)x10'?0.0033.

    AlJIhesymbolsandtheirmeaningintheabovee

    quationsarethesamewithRef.[1]. Stress--strainrelationshipofnon--prestressedrein--

    forcementisgivenbyEq.(3):

    E?y,

    i<.(3)

    Stressstrainrelationshipotprestressedreinforce

    mentisshowninFig.1.Itisathreestraightlinerood

    elgivenbyEq.(4)L2J:

    rEplpp<pl

    orp={J+Ep2(PPJ)Pl?<,(4)

    +E(p)p2?p?.

JottrnafofHarbinInstituteofTechno(NewSeries),Vol.14,No.1,2007

JI

    3/

    

    opIP2,.

    Fig.1Stressstrainrelationshipofprestressedreinforce- ment

    1.2SelectionofMaterial

    lateMembers

    dex.Atthesametime,prestressingdegreeA,effee

    tiveprestressOrpe,strengthofconcreteandgradeof nonprestressedreinforcementarealsobeenstudied. 2.1influenceofGlobalReinforcingIndexand PrestressingDegree

    Globalreinforcingindexisgradedinto0.1, 0.15,0.2,0.25and0.3.PrestressingdegreeAis gradedinto0.4,0.5,0.6and0.7.Alldataandin

    crementsofultimatestressAtrPofthesimulatebeams areshowninTab.1.

    andDimensionofSimuTab?1Scheduleofsimulatebeamsforanalysisofcnrva?

    Designedgradeofconcreteforsimulatemembers isC40.C45andC50.andtheircubicaveragecorn

    pressionstrengthare49.84N/mm.56.24N/ram and62.49N/mmrespectively.Gradeofnon.pres tressedreinforcementisHRB335andHRB400.and theirelasticmodulusis=2.0x10'N/mm.The averageyieldstrengthis=420N/mmand483N/ mmrespectively.西5milldimeterhightensilesteel

    wireisusedasprestressedreinforcementanditsBOrn inalyieldstrengthisf02:1540N/ram,anditsulti

    matestrengthis1738N/mm.Theelasticmodulusis

E=2.0x10N/mm.

    Dimensionandreinforeementsofsimulatemem. bersareshowninFig.2.

    Fig.2Dimensionandreinforcementsofsimulatemembers 2AnalysisofFactorsInfluencingCurvatureDuc

    tilityCoefficient

    GlobalreinforcingindexcanbecalculatedbyEq. (5):

    /3o:Ap+AZAtsfs(5)

    wherehisthedistancefromresuhanlforcepointof prestressedreinforcementtotheedgeofconcretecorn

    pressionregion.

    Theessenceofglobalreinforcingindexissimilar torelativedepthofconcretecompressionregion.They areallkeyfactorsthatindicatetherotationcapacityof section.Lotsofresearchesshow:thoughthereare manyfactorsinfluencingsectionductilityofbeam, whenprestressingdegreeandeffectiveprestressarein normalrange,globalreinforcingindexisthedecisive factor.S0analysisoffactorsinfluencingcurvature ductilityshouldcentersaroundglobalreinforcingin

    tureductilityinfluencedby0and

    Notes:Compressionstrength0fconcreteis=33.4N/mm;Effective prestressofunbondedtend.nsistr~=950N/ram;Strengthofnonpres

    tressedreinforcementis:420N/mm.

    Yieldcurvature,ultimatecurvatureandsee. tioncurvatureductilitycoefficientglobalreinforcing

    index130andprestressingdegreeApareshowninFigs.3 and4.

    0

Fig.3and巾一8oand^p

    (NewSeries),Vo1.14,No.1,2007JoMr,tZofHarbinInstituteofTechnolo l4

    12

    10

    8

    6

    4

    2

    0.40.50.60.7

    Fig.4Bhandp

    FromFigs.3and4,wecanseethatwhenglobal

    reinforcingindexisconstant,withprestressingdegree A.increasing,yieldcurvaturevandultimatecurvature decreasegradually.Itismoreobviouswhenglobal reinforcingindexissmal1.WithprestressingdegreeA increasing,curvatureductility/xincreases,butthein

    crementisverysmal1.Thecauseliesin:whenglobal reinforcingindexisconstant,ifprestressingdegreeis greater,thequantityofnonprestressedreinforcement

    decreases.Thenthesectionwillyieldearlierandthe yieldcurvaturewillbesmaller.Whenglobalreinforcing indexisconstant,ultimateincrementofstressofan

    bondedtendonvarieslittlewiththechangeofprestress- ingdegree.Resultantforceofprestressedreinforcement andnonprestressedreinforcementwillbegreaterwith higherprestressingdegreeandmoreprestressedrein

    forcements.Accordingly,therelativedepthofcompres

sionregionisgreater.sotheultimatecurvatureissmal

    ler.Butsinceultimatecurvaturechangeslittlethan yieldcurvature,curvatureductilityincreasesalongwith theincreaseofprestressingdegree.

    2.2InfluenceofGlobalReinforcingIndexand EffectivePrestress

    Globalreinforcingindexisgradedinto0.1,

    0.15,0.2,0.25and0.3.Effectiveprestressisgra

    dedintoor=800N/mm,950N/mmand1100N/

    mm.AlldataandincrementsofultimatestressAo-p.of thesimulatebeamsareshowninTab.2.

    Yieldcurvature,ultimatecurvatureandsection curvatureductilitycoefficientglobalreinforcingindex

    andeffectiveprestressIareshowninFigs.5and6.

    FromFigs.5and6,wecanseethatwhenglobal

    reinforcingindexJB0isconstant,witheffectiveprestress increasing,yieldcurvatureofsection,,isalmost invariable,whileultimatecurvatureincreasegradu

    ally.Itgrowsobviouswhenglobalreinforcingindex issmaller.Sowhenglobalreinforcingindex3oissinai

    ler,curvatureductilitycoefficientdbecomesgreater withtheincreaseofeffectiveprestressor.While,is invariablewhenglobalreinforcingindexisgreater. Thecauseliesin:Atultimatestate,wheneffective ?

    20?

    prestressisgreater.ultimatestressofunbondedtendon ispossibletoexceedproportionallimit,andplasticity ofprestressedtendondevelopsfast.sostressincrement ofunbondedtendonslowsdownandultimatecurvature

    increases.Asaresult.curvatureductilityincreases. Especially,whenglobalreinforcingindexissmaller, theultimatestressincrementisgreater.Thenitismore possiblethatstressofunbondedtendonexceedspropor- tionallimit.sothisphenomenabecomesmoreobvious. Tab.2Scheduleofsimulatebeamsforanalysisofcurva- tureductilityinfluencedbyand

    Notes:Compressionstrengthofconcreteis~:33.4N/mm;Strengthof

    nonprestressedreinforcementisfy420N/ram;Prestressingdegreeis

    A=0.7.

    14

    l2

    lO

    e8

    6

    4

    2

    O

    l2

    11

    lO

    9

    8

    7

    6

    5

    4

    2

    1

    0

×lOY(mill')

    O.1O.2O.3

    0

    Fig?5and一卢0and

    

    Fig.6一卢0and6

JournalofHarbinInstituteofTechnolo(NewSeries),Vo1.14,No.1,2007

    2.3InfluenceofGlobaIReinforcingIndexand StrengthofConcrete

    Globalreinforcingindexisgradedinto0.1, 0.2and0.3.Strengthofconcreteisgradedintof= 33.4N/mm(C40),37.68N/mm(C45)and41.87

    N/mm(C50).A1ldataandincrementsofultimate stress?ofthesimulatebeamsareshowninTab.3. Curvatureductilitycoefficient/x,t,-globalreinforcing indexBandgradeofconcrete{isshowninFig.7 FromFig.7,wecanseethatwhenglobalreinfor. cingindex0isconstant,thehigherofcompression strengthofconcreteis,thesmallercurvatureductility coefficient/.Lbecomes.Thecauseliesin:ultimate stressincrementbecomesgreaterifstrengthofconcrete ishigher.Asaresult,atultimatestate,

    relativedepth

    ofconcretecompressionregionisgreater, socurvature

    ductilitybecomessmaller.

    Tab.3Scheduleofsimulatebeamsforanalys~ofcurva. tureductilityinfluencedbyandstrengthofcon? crete

Notes:Strengthofn0n-prestressedreinforcementisf,=420N/mm;Ef-

    feetiveprestressofunbondedtendonsis=950N/mm;Prestressing

    degreeisA=0.7.

    16

    l4

    l2

    10

    e

    8

    6

    4

    2

    C40C45C50

    gradeofconcrete

    Fig-7恤一nand|l

    2.4InnuenceofGlobal

    StrengthGradeof

    forcement

    ReinforcingIndexand

    Non-prestressedRein-

    Globalreinforcingindex30isgradedinto0.1, 0.2and0.3.Srengthofnonprestressedreinforcement isgradedintof,.=420N/mm(HRB335)and= 483N/mm(HRB400).Alldataandincrements0u1. timatestress?ofthesimulatebeamsareshownin Tab.4.

    Curvatureductility

    cingindex30andgrade

    mentisshowninFig.8.

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