2010节段平衡悬臂桥的时间依赖性分析

EngineeringStructures32(2010)1038–1045ContentslistsavailableatScienceDirectEngineeringStructuresjournalhomepage:www.elsevier.com/locate/engstructTime-dependentanalysesofsegmentallyconstructedbalancedcantileverbridgesRichardMalm∗,HåkanSundquistDepartmentofCivilandArchitecturalEngineering,RoyalInstituteofTechnology(KTH),SE-10044Stockholm,SwedenarticleinfoArticlehistory:Received21January2009Receivedinrevisedform24July2009Accepted22December2009Availableonline15January2010Keywords:BalancedcantileverSegmentalconstructionCast-in-placeCreepShrinkageDeflectionabstractSegmentallyconstructedconcretecantileverbridgesoftenexhibitlargerdeflectionsthanthosepredictedbythedesigncalculations.Theslenderandlongspansincombinationwiththefactthatpermanentloadsareonlypartiallycompensatedforbyprestressingarereasonsforthelargedeflectionsthatincreaseduringthelifetimeofthebridge,althoughatadecreasingrate.Therateofdryingshrinkagemaybeonereasonfortheacceleratingdisplacementofcast-in-placebridges.Theconstructionofcontinuousspansinsteadofintroducingjointshasbothcomfortanddurabilityadvantages.Thecontinuousspanishowevermorecomplicatedtodesign,andsecondaryrestraintmomentsduetocreep,shrinkageandthermaleffectsdevelopattheconnection.Theresultsofanalysesofthestepwisecast-in-placeconstructionofabalancedcantileverbridgewithtime-dependentmaterialpropertiesshowbothhigherdeflectionthanthoseoriginallyassumedinthedesigncalculationsandhighstressesinthewebsduetostressingofthetendonsinthebottomflange.Theanalysesshowsignificanteffectsofcreepduringcantileveringandofanon-uniformdryingshrinkagerateonthecontinuousbridge.©2010ElsevierLtd.Allrightsreserved.1.IntroductionPrestressedsegmentallyconstructedconcretebridgesaresensitivetoalong-termincreaseindeflectionandareoftensub-jectedtoanincreasinglong-termdeflection.Thetotalverticaldisplacementofsuchbridgesisaresultofalargedownwarddis-placementduetothedeadload,liveloadsandalargeupwarddisplacementduetoprestress.Thelong-termincreaseindisplace-mentsisofgreatimportancefortheserviceability,durabilityandreliability.Duetothis,itisimportanttobeabletoobtainaccuratepredictionsofthedeformationofthesebridgesduringconstruc-tionandtheirservicelife.Severalbridgeshavebeenclosedorre-pairedduetoexcessivedeflectionbeforetheendoftheirinitiallyassumedservicelife.Thecostofareducedservicelifeistremen-dousforsociety,theownersandusers.Box-girderbridgesaretraditionallyanalysedaccordingtotheoryofbendingwherethecross-sectionsareassumedtoremainplane.Thistheoryis,however,toosimplifiedtocapturethedeformationofbox–girderbridgesaccurately.Themaindeficiencyofthistheoryisthatitcannotcapturetheshearlageffectintheslabsduetothedeadweightandtheprestress.Theshearlagcausesanonlineardistributionofnormalstressesoverthetopandbottomflangesinthecross-section.Neglectingtheshearlageffectmayleadtoaconsiderableunderestimationofthelong-term∗Correspondingauthor.Tel.:+4687908585;fax:+468216949.E-mailaddress:Richard.Malm@byv.kth.se(R.Malm).0141-0296/$–seefrontmatter©2010ElsevierLtd.Allrightsreserved.doi:10.1016/j.engstruct.2009.12.030deflectionsofbox–girderbridges.A3Dfiniteelement(FE)modelconsistingofshellorsolidelementscanautomaticallycapturetheeffectofshearlagandcanalsocapturetheeffectsofdifferentialshrinkageanddryingcreepifsuitablematerialdescriptionsareused.Fewexampleswheretime-dependenteffectshaveresultedincrackingincast-in-placebalancedcantileverbridgesarefoundintheliterature.Theliteratureregardingtime-dependenteffectsinthistypeofbridgemainlyfocusesonlargelong-termdeflec-tions[1–5].InthestudyofKristekandVrablik[6],aprogramtooptimizethetendonlayouttocounteracttheincreasinglong-termdeflectionsispresented.Previousstepwiseanalysesofbalancedcantileverbridgesincludevisco-elasticcreep,butdonotincludeanon-uniformshrinkagerate,instudiesofprecastconcrete[7]andwiththeprestressingimplementedasequivalentnodalforces[8].Afteronlyafewyearsofservice,twosimilarbridgesinSwe-den,bothsegmentallyconstructedwiththebalancedcantilevertechnique,hadtobeclosedtotrafficduetoextensivecrackinginthewebs.Thehypothesisisthatthecracksinthesebridgesareduetothestressingofthetendonsinthebottomflangeincom-binationwiththefactthattherearenotendonsintheweb.Thepurposeofthispaperistoreportastudyoftheinfluenceoftime-dependenteffectsintheconstructionstage.Itisparticularlyim-portanttostudywhicheffectsarelikelytocausecrackingandmustthereforebeincludedinordertocreateanaccuratemodelthatcandescribethecracking.Thisstudyisbasedonafiniteelementanalysisofasegmentallyconstructedbalancedcantileverbridgethatdescribesthestepwiseconstructionwiththenonlineartime-dependentdevelopmentofthematerialproperties.R.Malm,H.Sundquist/EngineeringStructures32(2010)1038–104510391120H11323456789100V13434333332741207054Fig.1.ElevationoftheGröndalbridgeintheconstructionstage.784444444442222444444444222222222224Fig.2.ElevationoftheGröndalbridgewiththetendonarrangementandtheextentofcrackingonthewebfacingsouth.1.1.DescriptionofthebridgesTheGröndalbridgeandtheAlvikbridgehadtobeclosedtotraf-ficduetoextensivecrackingintheweboftheirbox–girdersec-tions.Thesecrackswerefirstfoundonlyafewyearsafterservice.Thebridgesarepartsofthelight-railcommuterlineinStockholm,Sweden.Theinclinedwebcrackswerefirstobservedinaninspec-tiononly2yearsaftercompletion.Subsequentbridgeinspectionsshowedthatthecrackswereincreasingbothinnumberandinsize.Thelargestcrackswereobservednearthequarter-pointofthewebsattheinsideofthebox–girdersectionandtheywereupto0.6mmwide.Sincethewebsweremorecrackedontheinsideofthebox–girdersectionitwasconsideredprobablethatthermaleffects,undersummerconditions,mightbeoneofthefactorscaus-ingthecracks.Thedesignersfearedthatashearfailuremightbeimminentunlessthebridgeswereclosedtotraffic.Theinclinedwebcrackswereinitiallyassumedtojeopardisetheultimatelimitsafety.Duringatemporaryclosure,thebridgeswerestrengthened.Informationregardingthestrengtheningusingacombinationofcarbon-fibrelaminatesandverticalDywidagtendonscanbefoundintheliterature[9–11].Apreviousinvestigation[12]suggestedthatthecrackingwasduetoinadequateshearreinforcementinthewebsintheserviceabilitylimitstate.BoththeGröndalbridgeandtheAlvikbridgeareprestressedcontinuoushollowbox–girderbridges.TheGröndalbridgeconsistsof11spanswithatotallengthof430m.Fig.1showstheelevationofthisbridge.Themainandthetwoadjacentspanswereconstructedwiththebalancedcantileverconstructiontechniquewhilethesidespanswereerectedspanbyspanonasupportingscaffold.TenofthetwelvepiersoftheGröndalbridgehavearockfoundationwhiletheremainingtwopiersarebuiltonpiles.ThehighestpierontheGröndalbridgeis34m.Thecross-sectionalheightofthesuperstructureisapproxi-mately7.50mabovethepiersandabout2.75minthemid-span.Thewebsarerelativelyslenderwithathicknessof0.35mandhavearatherlowamountofreinforcement:horizontalreinforcementwithadiameterof12mmand200mmspacingandverticalwebreinforcementwithadiameterof16mmand200mmspacing.Theamountofreinforcementisincreasedinthemid-spantoadiam-eterof20mminthehorizontalbars.Prestressingcablesarepro-videdintheupperflangeastheyarenecessaryintheconstructionstage,andthecablesinthebottomflangearepost-tensionedafterthecompletionofthesuperstructurewhenthecentresegmentiscast.Thetendonarrangementforthemain-spanisshowninFig.2togetherwithasketchoftheextentofcrackingintheweb.1.2.BalancedcantileverconstructionTheprincipleofthefreecantileverconstructionmethodisthatapreviouslycastsegmentservesastheworkbasisfortheexecutionofthenextsegment.Aformtravellerisattachedtothepreviouslycastsegmentandcarriestheformworkforthenewsegmentthatistobecast.AnillustrationofaformtravellerisshowninFig.4.AccordingtoHewson[13],theweightofthetravellerusedforin-situconstructionwiththebalancedcantilevertechniqueisusually40–120tonnesforspansbetween50and200m.ThisintervalintheweightoftravellerisslightlysmalleraccordingtoTakács[3]whereittypicallyweights500–900kN.Afterasegmentispouredthetravellerremainsasasupportforthenewlycastsegmentuntilithasreachedsufficientstrengthandcanbestressedtotheexistingcantileverarmwithpost-tensionedtendonsanchoredinthenewsegment[14].Tocompensateforthelong-termdeflections,anupwarddisplacementduringcantileveringoccursduetotensioningofthetendons.Theseplanneddisplacementsarecommonlyreferredtoascamber.Themain-spanoftheGröndalbridgewassymmetricallycastfrompierssevenandeight,seeFig.1.Thecantileverarmsconsistof13segments,each4mlong,fromthepiers,andwheretwoadjacentcantileversmeettheyarejoinedwithone1.4mlongcentresegmenttoclosethestructure.Thesegmentswerecastwithatravellingformatintervalsof1week.2.FiniteelementanalysisTheanalysespresentedinthispaperhavebeenperformedwiththefiniteelement(FE)softwareAbaqus/Standard6.7[15].Themodellingapproachusedisathree-dimensionalmodelusingshell1040R.Malm,H.Sundquist/EngineeringStructures32(2010)1038–1045Fig.3.FEmodelofthebridge.AASECTIONA-AFig.4.Illustrationofaformtraveller.elements.Thereby,theFEmodelautomaticallycapturestheeffectofshearlag.Anumericalanalysiswiththeprogramisdividedintosteps,eachcorrespondingtoaloadchangefromonemagnitudetoanother.Inthiscase,asteprepresentsthecastingofonesegment.Thesegmentalcastinghasbeenmodelledinseparatestepswherenewelementshavebeenintroducedintothemodelineachstep.Thenewlyintroducedelementsaregivenmaterialpropertiesthatdevelopovertimetodescribethattheconcretecures.TheFEmodelusedfortheanalysisconsistsofshellelementsinthemainandthetwoadjacentspans.Theremainingspansandallpiershavebeendefinedasbeamelements,asshowninFig.3.2.1.EvolutionofmaterialpropertiesThereareseveralmaterialpropertiesandphenomenathathavesomeeffectontheresponseofthestructure.Theevolutionofmaterialparameterssuchastheelasticmodulus,creep,relaxationandshrinkagehasbeendescribedaccordingtothemethodsinthedesigncodesCEB-FIPModelCode1990[16]andEurocode2[17].Onemajorproblemwithmostconcretematerialmodelsusedtodescribecreepisthattheycannotbecombinedwiththematerialmodelsusedtodescribecracking.Thisisthecasewiththevisco-elasticmaterialmodel,viscoinAbaqus,whichcannotbecombinedwiththematerialmodelsuitablefordescribingconcretecracking,concretedamagedplasticityinAbaqus.Thismeansthattoanalysetheeffectsofcreepandcracking,somestrategytocompensateforthishastobeadopted.Thestudypresentedinthispaperfocusesonidentifyingthetime-dependenteffectsthathavetobeincludedtoaccuratelydescribethecrackingthatoccurredintheGröndalbridge.2.1.1.ElasticmodulusInthisconstructionprocess,theconcreteisloadedatanearlyage,wheretheconcretehastocarryaloadatalowdegreeofmaturity.Concreteincreasesinstrengthandstiffnessasaresultofcuring.Atanearlyage,thestrengthandstiffnessincreasequicklyandtheincreasethengraduallystagnatesbutdoesnotstopcompletely.Inthedevelopmentofthematerialinthefiniteelementanalyses,theevolutionoftheelasticmoduluswasimplementedaccordingtotheCEB-FIPModelCode1990,[16].Thematerialpropertiescorrespondingtoaconcreteageof28dayscanbecalculatedbasedonthecompressivestrengthaccordingtoEc=αE(fcmfcm0)1/3(1)whereEcisthemodulusofelasticityofconcreteatanageof28days(MPa),αE=2.15·104(MPa),fcmisthecompressivestrengthofconcreteatanageof28days(MPa)andfcm0=10(MPa).Whenanelasticanalysisisperformed,alowervalueofthemodulusofelasticityshouldbeusedtotakeintoaccounttheinitialplasticcrackingduetotheplasticshrinkage.ItissuggestedthatthisisdonebydecreasingtheelasticmodulusaccordingtoEcs=0.85Ec,whereEcsisthesecantmodulusofelasticityintheelasticrangeforconcrete.Totakeintoaccountconcreteofanarbitraryage,thetime-dependentfunctionmaybeused:Ec(t)=√exps(1−√28t)·Ecs(2)wheretistheageoftheconcreteindaysandsisacoefficientdependingonthecementtypeandisequalto0.20forrapidlyhardeningcementforhighstrengthconcrete,0.25fornormalandrapidlyhardeningcementand0.38forslowlyhardeningcement.Theimplementeddevelopmentoftheelasticmodulusisillus-tratedinFig.5(a).Thecalculationisbasedonaveragevaluesofthecompressivestrengthof10specimensmadefromtheconcretemixusedintheGröndalbridge.Toincludetheincreaseinelasticmod-ulusinthefiniteelementanalysis,afieldvariablewasintroducedthatdescribedtheevolution.Thefieldvariablewasdefinedtocor-respondtothetimeaftercastingforeachsegmentintheFEanal-ysis.Afteranewsegmenthadbeenintroduced,thematerialwasgivenaninitialelasticmoduluswhichincreasedasthetotaltimeintheanalysisincreased,asshowninFig.5(a).R.Malm,H.Sundquist/EngineeringStructures32(2010)1038–10451041abcdFig.5.Developmentof(a)elasticmodulus,(b)creepcoefficient,(c)shrinkagestrainand(d)non-uniformshrinkagerate.2.1.2.ShrinkageShrinkagehasbeenimplementedaccordingtoEurocode2[17]wherethetotalshrinkageisthesumoftheautogenousandthedryingshrinkage.Thedryingshrinkageisdefinedasεcd=βds(t,t0)khεcd,0(3)withβds(t,t0)=t−ts(t−ts)+0.04√h3(4)andthebasicdryingshrinkagestrainiscalculatedasεcd,0=0.85·10−6βRH(200+110αds1)exp−αds2fcmfc0(5)withβRH=−1.55(1−(RH100)3)(6)whereRHistherelativehumidityoftheambientenvironment(%),h=2Ac/uisthenotionalsizeofthestructuralmember(mm),Acistheareaofthecross-section(mm2),uistheperimeterofthecross-sectionincontactwiththeatmosphere(mm),fcmistheaveragecompressivestrengthofconcreteatanageof28days(MPa),fcm0=10(MPa),αds1andαds2arecoefficientsdependingonthecementtypeandareequalto6and0.11respectivelyforrapidlyhardeninghighstrengthcement,4and0.12respectivelyfornormalandrapidlyhardeningcementand3and0.13respectivelyforslowlyhardeningcement.Theautogenousshrinkagestraindevelopsduetochemicalreactionsduringhardeningintheearlyageconcrete.Autogenousshrinkageisofspecialimportancewhenyoungerconcreteiscastagainstolderalreadyhardenedconcrete[3,17].Itcan,accordingto[17],beexpressedasεca(t)=βas(t)εca(∞)(7)whereεca(∞)=2.5(fck−10)·10−6(8)andβas=1−exp−0.2√t.(9)TheshrinkagecanbedescribedbyapplyinganexternalthermalloadinaFEanalysis.WhenshrinkageisintroducedintheFEanalyses,differentialshrinkagebetweenthesegmentsisalwaysconsidered.Themostcommonoptionwhenincludingshrinkageistoassumethattheshrinkageisconstantoverthecross-section.Inthiscase,thebottomflangeisatmostfivetimesthickerthanthetopflange.Thiswillhaveaconsiderableeffectonthetimethedryingshrinkageoccurs,butthedryingcreepwillnotbeaffectedasmuch,accordingtoBazantandBajewa[18].Kristeketal.[4]describeacasewherethebottomflangeisalmosttwiceashighasthatintheGröndalbridgeandthetopflangeisofcomparablethickness.Thedifferenceindryingcreepforthesetwoflangethicknessesislessthan10%.Inthefollowinganalyses,twoapproacheshavebeenmadetostudytheinfluenceofshrinkage.Inthefirstapproach,eachsegmentisassignedasingleshrinkagecurveforthewholecross-section,i.e.onenotationalsizeforthewholecross-sectionofeachsegment,asshowninFig.5(c).Inthesecondapproach,referredtoasanon-uniformshrinkagerate,thewebs,thetopandbottom1042R.Malm,H.Sundquist/EngineeringStructures32(2010)1038–1045flangesofeachsegmentareassigneddifferentshrinkagecurves,asshowninFig.5(d).Thenotationalsizeiscalculatedseparatelyforthesepartsinthecross-sectiontoincludetheeffectofthicknessdependenceonthedryingshrinkage.Asimilarapproachwasmadein[3],withsatisfactoryaccuracycomparedtothemoreadvancedcreepandshrinkagemodelB3developedin[18].Theshrinkagestrainisrepresentedbyacorrespondingtemperatureandintroducedintothefiniteelementmodel,wherenegativetemperaturesareintroducedtodescribeshrinkage.2.1.3.CreepCreepisaccountedforbyusingthedescriptioninCEB-FIPModelCode1990,[16].ModelCodewasalsousedbythecompanydesigningthebridges.Thecreepcoefficientiscalculatedasϕ(t,t0)=ϕ0βc(t−t0)(10)where,ϕ0isthenotionalcreepcoefficientandβc(t−t0)isthetimefunctiondescribingthedevelopmentofcreepwithtime.Thenotionalcreepcoefficientisestimatedasϕ0=ϕRHβ(fcm)β(t0)(11)withϕRH=1+1−RH/1000.46(h/100)1/3(12)β(fcm)=5.3√fcm/10(13)β(t0)=10.1+√t0.(14)Thetime-developmentfunctionisdescribedbyβc(t−t0)=(t−t0βH+t−t0)0.3(15)withβH=150(1+(1.2RH100)18)h100+250≤1500.(16)Creep-relatedproblemscanbedescribedinAbaquswithtwodifferentmaterialmodels,eitherthematerialmodelcreeporbyusingavisco-elasticdescription,visco.Accordingto[19],thematerialmodelcreepinAbaqusisnotsuitablefortheanalysisofconcreteifthestressesvaryandespeciallynotifitinvolvesunloading.Becauseofthis,creephasinthisstudybeenincludedinthemodelwiththevisco-elasticmaterialdefinitionviscoandaquasi-staticnumericalintegration.Thevisco-elasticmaterialhasbeendefinedassumingaconstantbulkmodulus,i.e.atime-independentdilatationalresponseaccordingtotheexpression:K0=E03(1−2ν).(17)TheinstantaneousshearmodulusisdefinedasG0=E02(1+ν).(18)Therearefourwaysofdefiningthevisco-elasticrelaxationparametersinAbaqus:bydirectspecificationofthepronyseries,byinclusionofcreepdata,byinclusionofrelaxationtestdataorbyinclusionoffrequency-dependentdataobtainedfromsinusoidaloscillationexperiments.Inthepresentcase,therelaxationtestdatawereusedtospecifythevisco-elasticbehaviour.Thenormalisedshearandbulkmoduli,gR(t)andkR(t),weredefinedasfunctionsofthecreepcoefficient:gR(t)=GR(t)G0=E02(1+ν)(1+ϕ(t,t0))E02(1+ν)=1(1+ϕ(t,t0))(19)kR(t)=KR(t)K0=E03(1−2ν)(1+ϕ(t,t0))E03(1−2ν)=1(1+ϕ(t,t0)).(20)Thecreepcoefficientusedtodescribethevisco-elasticmaterialbehaviourintheFEanalysisisshownforonesegmentinFig.5(b).Thecreepiscalculatedbasedonanotionalsizeforeachsegment.Thecreepbehaviourisincludedinthematerialdefinitionofthefiniteelementmodel.Thiswillleadtoeachelementhavingdifferentcreepstrainsdependingonwhenintroducedintothemodelandtheirstresslevel.Thiswillresultindifferentamountofcreepstrainsoverthecross-sectionofthesegments.2.1.4.RelaxationTherelaxationoftheprestressingtendonscanbedefinedwiththesamevisco-elasticmaterialmodelthatisusedtodefinecreepinconcrete.TherelaxationhasbeenimplementedaccordingtoEurocode2[17]where,forlowrelaxationwireandstrands,therelaxationlosscanbecalculatedaccordingtoχ=1σprσpi=0.66·10−5ρ1000exp9.1µ(t1000)0.75(1−µ)(21)where1σpristheabsolutevalueoftherelaxationlossesoftheprestress,σpiistheabsolutevalueoftheinitialprestressforpost-tensioning,tisthetimeaftertensioning(inh),µ=σpi/fpkwherefpk=1770(MPa)isthecharacteristicvalueofthetensilestrengthoftheprestressingsteel,andρ1000=2.5%isthevalueofrelaxationlossat1000haftertensioningandatanaveragetemperatureof+20◦C.Relaxationisimplementedasrelaxationtestdata,withthenormalizedshearandbulkmoduli,gR(t)andkR(t),definedasfunctionsoftherelaxationlossgR(t)=GR(t)G0=1−χ(22)kR(t)=KR(t)K0=1−χ.(23)Therelaxationisincludedinthematerialpropertiesfortheten-dons.Theprestresslossmaybehigherthanpredictedbytextbookformulasthatiswhythetotallossofprestressiscalculatedintheanalysiswhereitisdependentonthecreepandshrinkageoftheconcrete.2.2.SegmentalconstructionphaseThesegmentalconstructionhasbeenperformedinthefiniteelementanalyseswherethewholestructurewasinitiallymodelledwiththegeometryoftheplannedfinalstructureaccordingtotheconstructiondrawings.Asafirststep,allelementsinthecantileverarmsweredeactivated,i.e.removedfromthecalculation.Insubsequentsteps,thecastingsequenceinthecantileveringprocesswassimulated,thecorrespondingsegmentsbeingactivatedinthefiniteelementmodel.TheelementsareactivatedwiththeAbaquscommandwithstrainsothattheyareaddedatazerostateinasmoothconjunctionwithpreviouslycastsegments.Eachcalculationstepcorrespondsto1weektosimulatethecastingsequenceandthedevelopmentofthematerialproperties.Theelementsareintroducedinthebeginningofeachstepwiththeirdeadweightandalowvalueoftheelasticmodulus.Asthecalculationofthestepprogresses,theconcretecuresandasaresulttheelasticmodulus,creepandshrinkageincrease.Inthebeginningofthenextstep,thissegmentispost-tensionedandanewsegmentR.Malm,H.Sundquist/EngineeringStructures32(2010)1038–10451043isactivatedasstressfree.Inreality,thepost-tensioningofthesegmentisperformedafterapproximately5daysofcuring,butinthemodelithasbeenintroducedinstantaneouslyinthebeginningofthesubsequentstep.Thisreducesthenumberofste