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1、DeformationAnalysisofPrestressedContinuousSteel-ConcreteCompositeBeamsJianguoNie1;MuxuanTao2;C.S.Cai3;andShaojingLi4Abstract:Deformationcalculationofprestressedcontinuoussteel-concretecompositebeamsaccountingfortheslipeffectbetweenthesteelandconcreteinterfaceunderserviceloadsisanalyzed.Asimpliedanal

2、yticalmodelispresented.Basedonthismodel,formulasforpredictingthecrackingregionofconcreteslabneartheinteriorsupportsandtheincreaseoftheprestressingtendonforcearederived.Atable for calculating the midspan deection of two-span prestressed continuous composite beams is also proposed. It is found that th

3、einternalforceoftheprestressingtendonunderserviceloadscanbeaccuratelycalculatedusingtheproposedformulas.Byignoringtheincreaseofthetendonforce,thecalculateddeectionareoverestimated,andconsideringtheincreaseofthetendonforcecansignicantlyimprovetheaccuracyofanalyticalpredictions.Asthecalculatedvaluessh

4、owgoodagreementwiththetestresults,theproposedformulascan be reliably applied to the deformation analysis of prestressed continuous composite beams. Finally, based on the formulas forcalculating the deformation of two-span prestressed continuous composite beams, a general method for deformation analy

5、sis of pre-stressedcontinuouscompositebeamsisproposed.DOI:10.1061/ASCEST.1943-541X.0000067CEDatabasesubjectheadings:Prestressedconcrete;Compositebeams;Deformation;Deection;Cracking;Concreteslabs;Con-tinuousbeams.Introduction2 increasing the ultimate loading capacity; 3 decreasing thedeformation unde

6、r service loads; 4 being favorable in crack-widthcontrol;5fullyusingthematerialsandthusreducingthestructural height and overall dead load; and 6 improving thefatigueandfracturebehavior.Continuous steel-concrete composite beams are widely used inbuildingsandbridgesforhigherspan/depthratiosandlessdeec

7、-tionetc.,whichresultsinsuperioreconomicalperformancecom-pared with simply supported composite beams. For continuouscomposite beams, negative bending near interior supports willresultinearlycrackingofconcreteslabandreductionofstiffness.Whenbeamsaredesignedforspanlengthsandloadsgreaterthanusual, the

8、requirement of serviceability limit state due to unac-ceptabledeectionandcrackwidthwouldrequireusingprestress-ingtechnique.Since Szilard 1959 suggested a method for the design andanalysisofprestressedsteel-concretecompositebeamsconsider-ingtheeffectsofconcreteshrinkageandcreep,manyresearchershave de

9、veloped methods for analyzing the behavior of simplysupportedprestressedcompositebeamsHoadley1963;Klaiberetal.1982;Dunkeretal.1986;Saadatmanesh1986;Saadatmaneshetal.1989a,b,c;Albrechtetal.1995,Nieetal.2007 .However,continuous prestressed composite beams have not been re-searched until the late 1980s

10、 Troitsky and Rabbani 1987;Troitsky 1990; DallAsta and Dezi 1998, Ayyub et al. 1990,1992a,b;DallAstaandZona2005.Asaresult,prestressedcon-tinuouscompositebeamshavenotwidelybeenusedpartlyduetothelackofdesigntheory.In fact, the behavior of prestressed continuous compositebeamsdependsontheinteractionbet

11、weenfourmaincomponents:thereinforcedconcreteslab,thesteelproleofbeams,theshearconnections, and the prestressing tendons, which makes pre-stressedcontinuouscompositebeamsmorecomplexthanconven-tional ones. DallAsta and Zona 2005 proposed a nonlinearniteelementmodelsimulatingthebehaviorofprestressedcon

12、-tinuouscompositebeamsaccurately.Thisnumericalapproachisavery powerful research tool for analyzing the externally pre-stressedstructures,butitperhapsistoocomplicatedforaroutinedesignpractice.Comparedwithconventionalsteel-concretecompositebeams,prestressedsteel-concretecompositebeamshaveafewmajorad-v

13、antages: 1 extending the elastic range of structural behavior;1Professor,Dept.ofCivilEngineering,KeyLaboratoryofStructuralEngineeringandVibrationofChinaEducationMinistry,TsinghuaUniv.,Beijing100084,China.2Ph.D. Candidate, Dept. of Civil Engineering, Key Laboratory ofStructural Engineering and Vibrat

14、ion of China Education Ministry,Tsinghua Univ., Beijing 100084, China corresponding author . E-mail:dmh033AssociateProfessor,Dept.ofCivilandEnvironmentalEngineering,LouisianaStateUniv.,BatonRouge,LA,70803;presently,AdjunctPro-fessor,SchoolofCivilEngineeringandArchitecture,ChangshaUniv.ofScienceandTe

15、chnology,Changsha,China.4Formerly,GraduateStudent,Dept.ofCivilEngineering,KeyLabo-ratoryofStructuralEngineeringandVibrationofChinaEducationMin-istry,TsinghuaUniv.,Beijing100084,China.Note.ThismanuscriptwassubmittedonAugust10,2008;approvedon April 20, 2009; published online on October 15, 2009. Discu

16、ssionperiodopenuntilApril1,2010;separatediscussionsmustbesubmittedfor individual papers. This paper is part of the Journal of StructuralEngineering,Vol.135,No.11,November1,2009.©ASCE,ISSN0733-9445/2009/11-13771389/$25.00.Asprestressingtechniqueisaneffectivewaytoreducedefor-mation and crack widt

17、h under service loads, particular attentionhastobepaidtothedeformationcalculationofprestressingcon-tinuouscompositebeams.Themainobjectiveofthisresearchistodevelopcalculationmethodsforthedeformationofprestress-ing continuous composite beams based on the reduced stiffnessJOURNALOFSTRUCTURALENGINEERING

18、©ASCE/NOVEMBER2009/1377Downloaded 19 Feb 2012 to 30. Redistribution subject to ASCE license or copyright. Visit Thedownwardconcentratedforceappliedbytendonsattheinte-rior support is not shown in the gure as the force is applieddirectlyonthesupport.Therigidi

19、tyalongthebeamcanbecon-sideredasunchangedinthisstagesincethecrackingofconcreteusually does not occur.The section properties can be calculatedby the transformed section method ignoring the slip effect be-tweensteelandconcreteinterfaceatthisstage.Itisassumedthatthedistributionofmomentalongthebeamdueto

20、theprestressingforcekeepsunchanged.Oncealltheparametershavebeendeter-mined,deformationintherststage f1canbedirectlycalculatedbymethodsofstructuremechanics.Fig.1.Sketchoftwo-spanprestressedcontinuouscompositebeammethodthatwasdevelopedforconventionalcontinuouscompos-itebeamsNieandCai2003.Theproposedme

21、thod,veriedbytestresults,issuitablefordesignpractice.In the second stage shown in Fig. 2b, application of theexternal force P results in the increase of downward deectionf andachangeofprestressingtendonforceT.Intheregionof2TheoreticalStudysaggingmoment,thereducedexuralstiffnessB=E1I1/1+ isusedduetot

22、heslipeffects,whereisstiffnessreductioncoef-cient according to the reduced stiffness method Nie and Cai2003,andtheaxialstiffnessEAiscalculatedbythetransformedsectionmethod.IntheregionofhoggingmomentintherangeofnL neareachsideoftheinteriorsupports,concreteisconsideredno longer in service due to crack

23、ing. In this case the bendingrigidityE2I2 andaxialrigidityE2A2 canonlyincludethecontri-butionofthereinforcementandsteelmaterials,andparameterandaredenedas=B/E2I2,and=EA/E2A2.Actually,inthesecondstage,concreteinthehoggingmomentmaystillcontributetostiffnessbecauseoftheprestressingforce.Therefore, the

24、partial interaction between the steel and concreteshould be considered for a rational analysis. For simplicity, thiskind of interaction effect is considered in the present study byadjustingthevalueofnL insteadofactuallymodifyingthestiff-nessofcompositebeamsnearthesupports,whichresultsinonlysmallerro

25、rsaswillbeveriedbytheexperimentsanddiscussedlater.AnalyticalModelPrestressed continuous composite beams discussed in this paperareshowninFig.1wheretheprestressingtendonsarelaidoutasfold lines or straight lines for the convenience of construction.Thestraightlinescanbeconsideredasaspecialcaseofthefold

26、-linetypewith=0incalculation.Thepositionoftendonscanbeeitherinternalorexternal,whichwillnotinuencethemethodofanalysis.Thus,theresearchinterestinthispaperisconcentratedonatwo-spanprestressedcontinuouscompositebeamwithfold-line tendons as shown in Fig. 1, and the methodology can beappliedtootherkindso

27、fprestressedcontinuouscompositebeams.Thecalculationmodelofprestressedsteel-concretecompositebeamsisshowninFig.2.Theprocessofloadingcanbedividedinto two stages. In the rst stage shown in Fig. 2a, beams areinitially prestressed by tendons and the equivalent loads appliedto the continuous beams by tend

28、ons are composed of two parts.TherstpartincludesaxialcompressionforceT andmoment0T0e0atthebeamends,wheree0=distancefromthebeamanchortotheneutralaxisofthetransformedsection,positivebelowneu-tral axis. The second part includes vertical concentrated loadsappliedbytendons.ForceequilibriumshowninFig.3giv

29、esthevalueoftheequivalentconcentrateforceFappliedbythetendonsasT0sin,whichequalstoT0approximatelyasisverysmall.Inordertoobtainthedeectionofthecompositebeamsinthisstage, the length of cracking region of concrete slab at interiorsupports, dened by n, should be determined rst. For conven-tionalcontinuo

30、uscompositebeams,itisfoundinpreviousstudiesandexperimentsthattaking0.15forthenvaluewillbeaccurateenough for design Nie et al. 2004. However, for prestressedcontinuous composite beams, the length of cracking region ofconcreteslabissmallerthantheconventionones.Furthermore,nisrelatedtotheprestressingde

31、greedirectly,whichhasbeenveri-ed by tests. The other parameter T is also very essential forcalculatingthedeection.Since the materials are generally linear elastic under serviceload conditions, the principle of superposition can be used toobtainthetotaldeectionas f1+f2,where f1canbecalculateddirectly

32、bymethodsofstructuralmechanics.Inthisstudy,wearemore concerned about the increase of deection under serviceloads, i.e., f2. Therefore, this paper will only investigate theincrease of deection in the second stage, and for convenience,f2 will be rewritten as f hereafter.According to the discussionmade

33、above,thecoreofdeformationcalculationistodeterminethe values of n and T, which will be discussed further in thefollowingparts.Fig. 2. Calculation model of prestressed continuous steel-concretecompositebeam:arstloadingstage;bsecondloadingstageThecableslipatthesaddlepointsisacomplexbehavioroftheextern

34、ally prestressed composite beams. The slip friction at thesaddle points can inuence the behavior of beams under serviceloads. Negligible friction occurs by using individually coatedsingle-strand tendons Conti et al. 1993 and the assumption ofnegligiblefrictioncanbefoundinthepreviousmodelDallAstaand

35、Zona 2005. This assumption is also used in the followinganalyticalstudies.Fig.3.Equivalentloadappliedtothebeambytendons1378/JOURNALOFSTRUCTURALENGINEERING©ASCE/NOVEMBER2009Downloaded 19 Feb 2012 to 30. Redistribution subject to ASCE license or copyright. Visit http:/www.ascelibrary

36、.org51Mk=0.85Mek= m1mPkL640where Mek=moment due to Pk ignoring the moment redistribu-tion.The relationship between the service load and the initial pre-stressingforcecanbederivedusingEqs.5and6as 40T051m1mL 2eW20T017 0Pk=+7AUndertheapplicationofexternalforceandprestressingforce,thedistributionofmomen

37、talongthebeamisshownasFigs.4 bandc, respectively. The tension stress at the top of concrete at theboundaryofthecrackingregionequalstozero,whichleadstoMTx=nL+MPx=nLT=08WAFig.4.Theoreticalanalysisofthelengthofcrackingregionofcon-creteslab:acalculationmodeloftwo-spanprestressedcontinuouscompositebeams;

38、bmomentdistributionduetoprestressingtendonforce;andcmomentdistributionduetoexternalloadswhere T=tendon force under service load conditions. Comparedwiththeinitialprestressingforce,theincreaseoftendonforceisrelatively small, and T can be taken proximately as T0; MTx=moment distribution along the beam

39、 due to the prestressingforce,andMPx=momentdistributionalongthebeamduetotheserviceload.TheyarecalculatedasPredictionofCrackingRegionofConcreteSlabx=3Te02 Lx21Te0+ 23m2+32m+1 TxIn this part, the length of cracking region of concrete slab overinteriorsupportswillbetheoreticallyanalyzedbasedonthecal-cu

40、lation model shown in Fig. 4a. After the initial force T0 isprestressed,astructuralanalysisgivesthesaggingmomentattheinteriorsupportasMT23m1mTL0xnL951405151=T0e0+32m1mT0LMPx=m240m1 Pkx+ m1mPkL0xnLMT0140210Accordingly,theinitialcompressivestressatthetopofconcreteslabattheinteriorsupportiscalculatedas

41、Introducing Eqs. 7, 9, and 10 into Eq. 8 leads to the ex-pressionofnasafunctionofpc=MT0+T0=AT0e0+2W3m1mT0L2W+TA2A10n=BCA11Wwhere W=section modulus of transformed composite section atthe top of concrete ange and A=cross-sectional area of trans-formedsection.Themomentneededtoeliminatethecompressivestr

42、essattheinteriorsupportisobtainedaswhereA,B,andCcanbecalculatedas1W+321mme0LA=2+Ae0 3331 mL+mB=2+ m+22e0M0=pcW=12T0e0+32m1mT0L+TA0W3C=51m251m4051m251mTheprestressingdegreeisdenedas=MM40From Eq. 11 we can see that the main factors inuencing therangeofconcretecrackingregionincludetheprestressingdegree

43、,theparameterW/Ae0,theparametermL/e0,andtheloadingpositionm.TheireffectsonnareplottedinFigs.57.FromFigs.57 we can see that the length of concrete cracking region fallsmoreandmorequicklyastheprestressingdegreerises.Whentheprestressingdegreeistakenas1,thelengthofconcretecrackingregion is zero, referre

44、d to as fully prestressed composite beams.Similarly,azerooftheprestressingdegreeresultsinthelengthofconcrete cracking region being as 1/C, which depends only ontheloadingpositionmandcorrespondstoconventionalcompositebeams. Fig. 5 indicates how n varies within the usual range ofparameter W/Ae0 when t

45、he other parameters are xed. It iskwhereMk=momentattheinteriorsupportduetoserviceload Pkexcludingprestressingeffect.IntroducingEq.3intoEq.4givesMk=T20e0+3m1mT0L+T0W52AIt is found in experiments that the moment redistribution coef-cientaattheinteriorsupportcanreachabout15%underserviceloadconditions.T

46、herefore,15%isusedtocalculatethemomentattheinteriorsupportunderserviceloadsapproximatelyasJOURNALOFSTRUCTURALENGINEERING©ASCE/NOVEMBER2009/1379Downloaded 19 Feb 2012 to 30. Redistribution subject to ASCE license or copyright. Visit Fig.5.InuenceofparameterW

47、/Ae0onnFig. 8. Comparison among test results, theoretical results and sim-pliedtheoreticalresultsfoundthattheinuenceofparameterW/Ae0onnisveryslightandcanbeignored.In most cases, the neutral axis in the region of positive mo-mentisadjacenttothesteeltopange,andtheprestressingten-dons are adjacent to t

48、he steel bottom ange. According to thesketchshowninFig.1,mLrepresentstheverticaldistancefromthebeamanchortothecenteroftendonstakenproximatelyasthepositionofthesteelbottomange,leadingtothefollowing:gion,andinlowprestressingdegreeregionitvariesfrom0.15to0.20approximatelywhenmvarieswithintheusualrange.

49、Sincetheactuallengthofconcretecrackingregionisslightlyshorter than the theoretical result due to the assumption that thetensile strength of concrete and the increase of tendon force arenegligible, Eq. 11 should be modied to a certain extent. Fur-thermore, except for , the other three parameters all

50、slightlyinuencethen value.Thus,Eq.11canbesimpliedconsider-ingthefollowingfactors:mL+e0hsmLhs112e0 e01.RelationshipformatbetweennandasdenedbyEq.11is maintained by adjusting only the coefcient in the equa-tion.Sincetheheightofthesteelbeamh isabout4to8timesofthesanchoreccentricitye0,theparametermL/e0va

51、riesfrom3to7.WithinthisrangewecanconcludefromFig.6thatthevariationofparametermL/e0willnotsignicantlyinuencethevalueofn.2.3.Thenewequationcanreducetoconventionalnonprestressedcase,i.e.,when=0,n =0.15.The parameter W/Ae0 and mL/e0 can be taken as theaveragevalueswithintheusualrange.Consequently,Eq.11i

52、ssimpliedasFig. 7 shows how the loading position m inuences the nvalue.The n approaches to unity in high prestressing degree re-n=14320113Thecomparisonbetweentestresultsdiscussedlater ,theoreticalresults, and simplied modied theoretical results is shown inFig.8,whichprovesthatEq.13isreasonableandacc

53、urateforthecalculation.PredictionofTendonForceTheincrementoftendonforceduetoexternalloadscanbepre-dicted by developing the equilibrium equation, the deformationcompatibilityequation,andthephysicalequationforthestructuresystem.Theexternalloadsmainlyresultinbeammomentswhosedistribution depends on the

54、loading conditions. The change ofprestressing tendon forces mainly result in axial forces and mo-ments, which can be solved by a simple structural analysis asshowninFig.9.Fig.6.InuenceofparameterL/e0onnTheeffectofprestressingforceincrementTisresolvedintotwo parts in Fig. 9a, namely the equivalent ve

55、rtical forces andhorizontalaxialforcesatbeamends.Thepositionchangeofneu-tralaxisintheregionofhoggingmomentneartheinteriorsupportinuencesthemomentdistributionduetoprestressingforce.Asaresult,acoefcient=e02/e0isdenedheretodescribeit,wheree02representstheverticaldistancefromtheprestressingtendontotheelasticneutralaxisintheregionofhoggingmomentasshowninFig.9awithpositiveforbeingbelowtheneutralaxis.In order to solve the expression of R1 and R2 in Fig. 9a,Fig.