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gradually over time due to concrete creep and shrinkage. Forin prestress losses, which cause stress redistribution and a de-crease of structural safety. Results of the previous time-dependentprestress loss studies are listed in Table 1, in which the structuralage during the measurement ranges from 6 months to 46 years. Itis observed that the prestress losses with time are always presentwhile the specic values of this loss may be inuenced by param-eters such as structural type, ambient temperature, humidity andconcrete material properties, and so on.Considering the importance of accurate modeling of creep,shrinkage, and corrosion to the design and assessment of struc-tures, extensive investigations have been made for decades. Theexisting models regarding concrete creep and shrinkage includelaw for creepBazant and Osman 1976, the ACI 209 modelAmerican Concrete Institute ACI 1982, the double powerlogarithmic law for creepBazant and Chern 1984, the CEB-FIPmodelCEB-FIP 1994, the B3 modelRILEM TC-107-GCS1995, and the GL2000 modelGardner and Lockman 2001.Comparisons of these models were presented by Goel et al.2007, and it was concluded that the recently developed GL2000model, the CEB-FIP model, the B3 model, and the ACI 209model are more reliable than earlier creep models.Corrosion of high-strength reinforcing steel is a complex phe-nomenon that consists of several different but interrelatedmechanisms, such as uniform corrosion, pitting corrosion, andcorrosion-induced cracking. Numerous studies with regard to cor-rosion models have been made and will not be repeated here. Inconjunction with available corrosion studies, time-dependent re-liability analyses have been made. Enright and Frangopol1998investigated the resistance degradation of RC bridge beams underuniform corrosion, and the corrosion initiation time was predictedTime-Dependent Reliability of PSC Box-Girder BridgeConsidering Creep, Shrinkage, and CorrosionTong Guo1; Richard Sause, M.ASCE2; Dan M. Frangopol, Dist.M.ASCE3; and Aiqun Li4Abstract:Bridge performance undergoes time-varying changes when exposed to aggressive environments. While much work has beendone on bridge reliability under corrosion, little is known about the effects of creep and shrinkage on the reliability of concrete bridges.In this paper, the CEB-FIP model for creep and shrinkage is applied by using nite-elementFEanalysis in conjunction with probabilisticconsiderations. Verication is made by comparing the analytical ndings with existing test data. More specically, a time-dependentreliability assessment is made for a composite prestressed concretePSCbox-girder bridge exposed to a chloride environment. Thisrealized via an advanced probabilistic FE method. The postcracking behavior of the thin-walled box girder is described using compositedegenerated shell elements, and importance sampling is used to improve the efciency of the reliability analyses. It is shown that concretecreep and shrinkage dominate during the early stages of bridge structure deterioration. This is accompanied by a decrease in reliabilityowing to the combined action of creep, shrinkage, and corrosion. The reliability indexes for the serviceability and the tendon yielding limitstate fall below the target levels prior to the expected service life. Therefore, early maintenance and/or repair measures are required.DOI:10.1061/ASCEBE.1943-5592.0000135CE Database subject headings:Prestressed concrete; Box girders; Creep; Shrinkage; Corrosion; Probability; Finite element method.Author keywords:Time-dependent reliability; Prestressed concrete box girder; Creep and shrinkage; Corrosion; Probabilistic nite-element method.Introductionprestressed concretePSCstructures, creep and shrinkage resultMaintenance, strengthening, and rehabilitation of the aging civilinfrastructure systems require an accurate assessment of the reli-ability throughout the service life of deteriorating structures. Theprimary factors affecting structural durability should be fullytaken into account. For RC structures, it has long been recognizedthat steel corrosion and concrete creep and shrinkage are themajor causes that produce structural deterioration. Corrosion,cracking, and spalling of the concrete cover, particularly in ag-gressive environments, may lead to a reduction in the cross-sectional area of the reinforcing steel and a decrease in bridgeserviceability. In the worst case, corrosion may even trigger struc-tural collapse. Structural stresses and displacements changethe effective modulus methodMcMillan 1916, the double power1Associate Professor, Key Laboratory of Concrete and PrestressedConcrete Structure, Ministry of Education, Southeast Univ., Nanjing210096, Peoples Republic of China; formerly, Visiting Research Scien-tist, ATLSS Center, Lehigh Univ., 117 ATLSS Dr., Bethlehem, PA 18015corresponding author. E-mail: guotong772Joseph T. Stuart Professor of Structural Engineering and Director,Dept. of Civil and Environmental Engineering, ATLSS Center, LehighUniv., 117 ATLSS Dr., Bethlehem, PA 18015. E-mail: rs0c3Professor and Fazlur R. Khan Endowed Chair of Structural Engineer-ing and Architecture, Dept. of Civil and Environmental Engineering,ATLSS Center, Lehigh Univ., 117 ATLSS Dr., Bethlehem, PA 18015.E-mail: dan.frangopol4Professor, College of Civil Engineering, Southeast Univ., Nanjing210096, Peoples Republic of China. E-mail: aiqunliNote. This manuscript was submitted on October 11, 2009; approvedon May 10, 2010; published online on May 14, 2010. Discussion periodopen until June 1, 2011; separate discussions must be submitted for indi-vidual papers. This paper is part of theJournal of Bridge Engineering,Vol. 16, No. 1, January 1, 2011. ASCE, ISSN 1084-0702/2011/1-2943/$25.00.JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011 /29Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. Visit1 t,t0Ect0 Ec281with a probabilistic model. Val and Robert1997developed atime-dependent corrosion model for reliability analysis of RCslab bridges, in which the localized corrosionpitting corrosionwas included. This model was adopted and improved in work byStewart2004. Accelerated pitting corrosion tests were used toobtain spatial and temporal maximum pit-depth data for prestress-ing wiresDarmawan and Stewart 2007, and further investigatedby Stewart2009were the mechanical behavior of pitting corro-sion of exural and shear reinforcement and its effect on struc-tural reliability. However, the previous work mainly focused onthe inuence of corrosion while concrete creep and shrinkagewere not taken into account.The objective of this study is to develop a time-dependentreliability evaluation methodology for PSC structures, in whichthe corrosion and concrete creep and shrinkage are all included.The methodology is applied to assess the performance of PSCbox girders from a bridge exposed to a chloride environment.These box girders are widely used in highway bridges, and earlydamagee.g., crackingis observed in some cases. Reasons forthe damage include material deterioration, construction defects,increase in trafc loads, and inaccurate design. The unique shearlag effectLuo et al. 2002in thin-walled box girders and thecomplex arrangement of prestressed tendons bring additional dif-culties into the structural design. Although several design codesDeutsches Institut fr NormungDIN1981; AASHTO 2004have already provided formulas to account for the shear lag ef-fect, these formulas are developed for the elastic stage of behaviorand are inadequate for reliability analyses in the elastoplasticstageGuo and Li 2009. To make a more rational assessment ofthe PSC box girders, a probabilistic nite-elementFEmethod isused, in which composite degenerated shell elements are used tomodel the pre- and postcracking behaviors of the thin-walled boxgirder, and an approximate importance samplingISmethod isused to perform the reliability analysis. A comparatively widerange of random variables is covered in the analysis, such aschloride diffusion rate, critical threshold chloride concentration,pitting corrosion depth, concrete material properties, concretecover thickness, and external loads.Time-Dependent Deterioration ModelConcrete Creep and ShrinkageThe time-dependent change in material properties of concrete ismodeled using the CEB-FIP modelCEB-FIP 1994, which takesa number of parameters into consideration, such as cement type,ambient temperature, relative humidity, concrete strength, andconcrete age at loading.In the CEB-FIP model, evolution of concrete creep is de-scribed via the creep functionJt,t0, which is formulated asCEB-FIP 1994Jt,t0= +whereEct0=modulus of elasticity at the concrete age oft0andEc28corresponds to the value at the age of 28 days. The so-calledcreep coefcientt,t0can be determined from the followinghyperbolic power function:30/ JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. Visit1RH/RH0 5.3 10.46h/1001/3 0.1 +t1/500.32E0Maxwell elementFig. 1.Generalized Maxwell modelh100+ 250denote the stiffness of each spring and damper,respectively.3at3bdenotes the relaxation function, which can be ex-panded into a Dirichlet seriesnEt,=Eiet/ii=0whereEi=time-dependent stiffness of a Maxwell elementseeFig. 1. The relaxation time for a Maxwell element isi=i/Ei 6cSubstituting Eq.6binto Eq.6afor timetand timet+t,respectively, and after summation, integration, and subtraction,41ecm28tts350h/1002+tts1974.Time-Dependent Corrosion Model 1.551RH/100% 40%RH99%0.5 RH99%most common caseuniform corrosion, the diameter of a corrod-Dat timetcan be estimated asVal andRobert 1997=D0 0.0232ttiicorrtiationyear.rocell corrosion process, pitting corrosion in a chloride environ-ment, being a microcell corrosion process, may lead to a faster0.1fcm28E1dampersi.e., a set of Maxwell elements, as shown in Fig. 1,andimodel is expressed aswhereEt,6bthe stress increment intisnt/iEti=0wheret=sampling point at the midpoint of the time increment,5amodel and the relaxation function can be found in Bazant and Wu3deterioration of RC structures. One direct consequence of corro-sion is the cross-sectional area loss of reinforcement, and in the8E2itnamely,t+t/2. More details regarding the generalized MaxwellIt is well recognized that corrosion is a primary cause of structuralEnt,t0= 1+tt0H+tt0where RH=relative environmental humidity; RH0 equals to100%; andh=nominal size of the concrete membermmdenedas 2Ac/u.Acis the cross-sectional area anduis the perimeter incontact with the atmosphere.fcm28stands for the mean compres-sive strength at the age of 28 days, andH= min 1,500;1501+1.2RH18The evolution of concrete strength with time is described bywhereEiEq.3aThe stress-strain relationship for the generalized Maxwellfcmt=cctfcm28wheret= Et,d 6acct= exps128/teqis a time-dependent coefcient andstakes the values of 0.20,0.25, and 0.38 for rapid hardening high-strength cement, normaland rapid hardening cement, and slowly hardening cement, re-spectively. The equivalent age of concreteteqis dened astteq= 4,0001/273 1/Td 3c0whereT=temperature of concrete at days.The modulus of elasticity of concrete attdays can be esti-mated asEct=cctEc28The shrinkage strainsst,tsat an age oftdays is= it 7st,ts=160 + 109 0.1f106RHwhere=shrinkage coefcient dependent on cement type;ts=age of concrete at the beginning of shrinkageday; andRHisrelated to the environmental humidityRHas follows:RH=5bThe above equations concerning creep and shrinkage are inte-ing reinforcing bargrated into the nonlinear FE code. In this way, the inuences ofstructural details such as the reinforcement arrangement and theDttime-dependent prestress level can be considered automatically bythe FE program. Meanwhile, the change in concrete strength andwhereicorr=corrosion current densityA/cm2;D0=initial di-elastic modulus due to creep can also be includedCEB-FIP ameter of the reinforcing barcm; andti=time of corrosion ini-1994.In addition to the uniform corrosion, which is generally a mac-Stress RelaxationSimilar to the modeling of creep, stress relaxation is modeled viadeterioration. According to Gonzlez et al.1995, the maximumthe relaxation function by using a generalized Maxwell model,penetration of pitting is about four to eight times that associatedwhich can be physically interpreted as a set of parallel springs andwith uniform corrosion. Under attack from pitting corrosion, theJOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011 /31Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitAr(T)D0Fig. 4.Pit congurationadapted from Val and Robert1997a=2pt1 11bA1= a ; A2= 2pt2aptD011c1= 2 arcsin 2= 2 arcsin 11dX C0C4D Cfyt=1Pcorr100fy0wherefyt=deteriorated yield strength at timet;fy0correspondsto its original value;=empirical coefcient; and according tothe regression analyses of Du et al.2005and Vu et al.2009,avalue of 0.0054 is taken for reinforcing bars and 0.0075 is takenfor stranded wires, respectively.Pcorris the percentage of corro-sion loss, which can be obtained from Eq.8or Eq.11in termsof area loss.A1A2, D0A1A2, pt D0Composite Degenerated Shell ElementOne challenge in the probabilistic FE analysis of thin-walledstructures is analysis efciency because both nonlinear FE analy-For this reason, shell elements rather than more common solidelements are used.respectively. To avoid membrane and shear locking, which resultsin an excessively stiff behavior, a reduced 22 integrationscheme is used over the element areain the and directions,ptD01 D0 pt 12 2 2 D0 2a aD 2pt2net cross-sectional area is also reduced by corrosion as follows:122D222 2 2Many laboratory results have indicated that the yield stress on theProbabilistic FE Model0ap(t)Fig. 2.Rupture of stranded wires due to stress and corrosionreinforcement is prone to failure due to stress concentration. Forstranded wires with high prestress, the pitting corrosion processmay be acceleratedVu et al. 2009, and brittle rupture of thestranded wires may occur earlier than expected. Research carriedout by Naito et al.2006investigated corrosion of prestressingstrands in PSC box girders. PSC box beams were taken from a46-year-old bridge and assessed. Rupture of the prestressedstranded wires in the bottom plate was observed under the com-bined action of stress and corrosionFig. 2. Heavy pitting, de-ned as a pit greater than 20% of the wire section area, wasobserved, most likely from chloride attackFig. 3.According to Vals work, the radius of the pit at timetcan beestimated aspt= 0.0116ttiicorrR 9whereR=penetration ratio between the maximum and averagepenetration. The time of corrosion initiationtiis predicted by thefollowing equationEnright and Frangopol 1998:ti=whereX=concrete covercm;Dcrepresents the chloride diffu-sion coefcientcm2/year;C0denotes the chloride concentrationat the concrete surface% weight of concrete; and Ccr=threshold chloride concentration% weight of concrete.The net cross-sectional area of a corroded rebarArtat timetis estimated by Eq.11Stewart 2009. Fig. 4 illustrates therelationship betweenArtandptD20 24 2Art= 11a0, pt D0withsis and probabilistic analysis involve a large number of iterations.Pitting corrosionIn this study, an eight-node composite degenerated shell ele-ment in the FE code DIANA DIANA2008is used Fig. 5a.This shell element is capable of modeling pre- and postcrackingbehaviors of thin-walled RC structures and enables easy modelingof distributed reinforcement as well as prestressed tendons. Forthis shell element, each node has ve degrees of freedom: threedisplacements uX,uY, anduZin the global XYZdirections and tworotations and, respectively, around the local and axes,Fig. 3.Pit observation with optical microscopeprole taken by therst writer at the ATLSS Center32/ JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitReinforcing barteqz(c)XX(e)element boundaries. These intersections serve as location points,which dene the relationship of the tendons to their surroundingshell elements. The tendons are divided into segments by theselocation points, and numerical integration of each tendon segmentis performed separately. The embedded tendons are constrained tothe FE mesh. They add stiffness to the FE model and their strainsare calculated from the displacement eld of the surrounding shellelements. That is, the tendons are bonded to surrounding shellelements. Using the NOBOND input option in DIANA, tendonscan be specied as unbonded to the surrounding shell elements. Inthis case the stiffness of tendons does not contribute to the stiff-, dene the positions of the segments in the shellchange directly with the deformation of the surrounding shellelement. More details regarding automatic generation of tendonscan be found in Oh and Jeon2004.Crack Model and Tension Softening RelationThe composite degenerated shell element uses a smeared crackingmodelDe Borst 1987, and a tension softening model for con-crete developed by Hordijk1991, see Fig. 6, in which the rela-tionship between crack stresscrand crack straincris dened bytness of the surrounding elements, nor do the tendon strainsReinforcement gridY1(b)Reinforcement gridX(a)YYsIntegration pointLocation pointElement node(d)Fig. 5.Composite degenerated shell elementadapted fromDIANA2008:acomposite shell element;bmodeling of reinforcing bars viareinforcement grid;clocation of reinforcement grid in shell element;dsegment of reinforcement; andeautogeneration of tendonwhile the integration in the thickness direction is three-pointSimpson integration.Nonprestressed reinforcement is modeled using a reinforce-ment grid, which is embedded in the shell element, see Fig. 5a.The reinforcement ratios in two perpendicular directions are rep-resented by two equivalent thicknessesteqthe area of cross sec-tion per unit lengthof the grid, see Fig. 5b. Each reinforcementgrid is distinguished by the position parameterz, which denesthe distance from this grid to the shell midsurface, see Fig. 5c,andzshould be no greater than the shell thicknesst. The totalarea of the grid is divided into several segments, see Fig. 5d,which contribute to the stiffness of the shell element. The locationpoints, markedelement.For PSC girders, the prestressing tendonsPTsoften have acomplex curved shape, which requires one to specify the inter-secting points of a tendon with the FE mesh. However, such de-tailed input of the tendon geometry is difcult and often leads toerrors in locating tendons with respect to the elements. In thisstudy, an automatic tendon generation scheme included inDIANA2008is used, as shown in Fig. 5e. For this generation scheme,tendons are dened with a few location points and shape func-tionsstraight, quadratic, or cubic curve. DIANA searches theshell elements for the intersections of the tendons with the shellJOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011 /33Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitGeneration of random values/hfromRandraw(Xi, ,M)FE analysis forf 1+c expccr crultcrcrX*13bXiiYesXi,k for k=L,N1,from Randraw(Xi, ,N)fEhfor kf(Xi=1ki=1Pf=EndFig. 7.Approximate IS owchartPf=whereXk=vector of the discrete sample values of the randomIGxfx IGxfxx xIS density function can be established at a so-called “designpoint”Au and Beck 1999, which is a point on the failure surfacehaving the highest probability density. The design point can befound by solving a constrained optimization problem when thelimit-state function is explicit. Unfortunately, the design point isnot as easy to nd using a probabilistic FE method where thelimit-state function is implicit. A search process for the approxi-mate design point, based on a number of MCSs, is as follows, seeFig. 7.Msets of values for the random variablesXkk=1,.,Maregenerated according to their distributions. A deterministic FEforXi,k for k=1,L,MiX for kcr crt 1ult 2ultfor k=1,L,M=XkminGXk X*=Xkmax()f()XkG(X*)0*FE analysis for=1,L,N G(X )k)= , ,i1N1 IGXkfXkN Xk14affecting the accuracy and efciency of the method. Usually, theXi=X*k3 c2 0crultcr()()NoXdetermine)= fi()Xi,k,i,ik=1N15k=1variables and the subscriptkdenotes the sample number.The IS density functionxin Eq.14is a crucial factori=L,MG(X )crult1+ckkni ik iN1,k31expcrn()XIG(XNok k kG(X )0)f(X )/ )YeskStartftXi=i i=1,L,nGIfcrFig. 6.Nonlinear tension softeningadapted from Hordijk1991determinecr=013awhere the parametersc1=3 andc2=6.93, the ultimate crack strainultcrcan be achieved byIultcr= 5.136Gfhft&GIfin Eq.13brepresents the Mode-I fracture energy andhis theequivalent crack length.ftin Eq.13bis the reduced tensilestrength, which equals toIft= 0.739G 13cConstant shear retention is used to describe the reduction in theshear stiffness due to crack.Sampling MethodologyFor a large structure with a small probability of failure, a largenumber of samplingse.g., 10,000may be required for MonteCarlo samplingMCS. A more efcient form of MCS, Latinhypercube samplingLHS, avoids clustering samples and forcesthe tails of a distribution to participate in the sampling processSheskin 1997. Thus, the LHS may require 2040% fewer sam-plings than MCS to deliver the same results with the same accu-racythe number is problem dependent.To minimize the required number of samplings, an IS method-ology is used in this study. The IS methodology uses MCS withsamples which are selected to be more frequently in the failureregion because only such samples contribute to the evaluation ofPf.Givennrandom variablesxii=1,.,n, represented in thevectorx, the failure probabilityPfis written in the following formAu and Beck 1999:+Pf= xdx=EwhereGx=limit-state function andfx=joint probability den-sity function, which is expressed asfx=ni=1fixi, assuming thatthe random variables are independent.fixiis the probabilitydensity function for the variablexi. For each random variable,fixiand the associated distribution function are dened by adistribution type, a mean valuei, and a standard deviationi.xis an IS density function assumed to have the form ofx=ni=1ixi, whereixi=IS density function of the variablexi.IGxis the indicator function, whereIGx=1 ifGx0 andIGx=0 otherwise. Applying MCSwithNsamplesof therandom variables, the probability of failurePfis estimated by34/ JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitDetermination of sampling centerGeneration of random numberskINPUT OUTPUTStress & displacementsAccording to the available test descriptions, eight rectangularbeams were chosen for analysis and comparison. These beams aresubjected to their self-weight loads only and are simply sup-i=1iXi,k, where i,k=10StirrupNR1,0/C10010StirrupNR2,0/CPrestressed tendon, diameter=7mm, duct diameter=20mmMild steel bar, diameter=6mmCold-work deformed bar, diameter=10mmCold-work deformed bar, diameter=16mmFig. 9.Cross section and reinforcement detailsadapted from Chou-man2003a,b, dimension in millimetersXPfMaterial nonlinearityGeometric nonlinearitynk=NR1,1/C NR1,2/C NR1,3/CNR2,1/C NR2,2/C NR2,3/CiFILE READAnalysisresult files100Probabilistic analysis codeFILE WRITECalculation of corrosionCalculation of , andG()XRun FE codeFinite element codeParametricallydefined modelFig. 8.Probabilistic analysis data owanalysis is performed for each set of valuesXk. Among theXkwithGXk0, the set of valuesXwhich has the largestfXkisa candidate design point; otherwise, the set of valuesXcorre-sponding to the minimumGXkshould be taken as the candidate ported. Fig. 9 illustrates the beam dimensions and the reinforce-design point. Sets of valuesXkk=1,.,Mare then generated ments. The prestressed tendons used were made of 7-mm-from distributions centered atX. The distribution for each ran-diameter, plain cold-drawn high tensile wires with a strength ofdom variableXiis modied by replacing the mean valueiwith 65.0 kN. The jacking stress was 1,395 MPa, and tendons in theseXi. The search for the design point is repeated.Msets of values eight beams were not bonded to the concrete. Different reinforce-for the random variables are generated; deterministic FE analysisment ratios were used to study their inuence on creep andis performed and newXiis collected. The process is then repeatedshrinkage, and stirrups with a diameter of 10 mm were used foruntilGX0, and the change inXi for each random variable all specimens except NR 1.0/c and NR 2.0/c, as shown in Fig. 9.between two repetitions is within the predened tolerancei. The The beams were in a constant environment of 453% RH andnalXis the approximate design point. 202C.Pfcan be estimated using Eq.15with the approximate de- The 1-year tendon stresses after prestressing, obtained fromsign point. N sets of values for the random variables Xk,k FE analyses, are shown in Fig. 10a, where it is observed that the=1,.,Nare generated from distributions centered on the nalprestress losses are strongly affected by the presence of nonpre-X. Deterministic FE analysis is performed andGkXkis calcu-lated and used in Eq.15.InEq.15,Xkis calculated asX iX fiXi,k,Xi,i, and fXk=ni=1fiXi,k, wherefiXi,k=fiXi,k,i,i. This process is sum-marized in Fig. 7, in which “Randraw”Guy and Podgaetsky2005denotes a MATLAB code for generatingMvalues of arandom variable from a truncated normal distribution. In thisstudy,Mequals 10 andNequals 200. It usually takes about 20iterations to nd the design point.A probabilistic analysis code was developed using the aboveapproximate IS method, and to call the FE code. The data ow inthe probabilistic analysis is shown in Fig. 8.Comparison of Creep and Shrinkage Analysis withExperimental ResultsThe numerical method to account for concrete creep and shrink-age is veried by comparing the calculated results with availablelaboratory tests, which were carried out at the University ofLeeds, U.K.Chouman 2003a,b. Twenty partially PSC beamswith an effective span of 2.7 m and an overall depth of 180 mmwere tested for a period of over 1 year to study the time-dependent prestress losses and deection. These single-spanspecimens have rectangular- or I-shaped cross sections with dif-ferent percentages and distributions of nonprestressed and pre-stressed steels.JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011 /35Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitNR1,0/CNR2,1/C13501300125012001150NR1,1/C(FEA)1100NR1,3/C(Test)0 50 100 150 200 250 300 350Time since prestressing (day)0.010NR1,0/CNR2,0/C0.0060.0040.0020.000-0.002-0.004Time since prestressing (day)Generally speaking, the accuracy of the FE analysis to account forcreep and shrinkage is acceptable.Time-Dependent Reliability of PSC Box-GirderBridgeBridge DescriptionThe time-dependent reliability evaluation methodology was ap-plied to an existing composite multibox-girder bridge constructedin Jiangsu Province, China. The bridge consists of ve spans witha span length of 25 m and a constant depth of 1.4 m. The bridgehas an overall width of 34.5 m and contains eight lanesfourlanes in each direction. The precast box girders were initiallyerected as simply supported spans. Then nonprestressed reinforc-ing bars and PTs in the top ange were connected in situ at thesupport to form a continuous-span bridge, see Fig. 11a. The vegirders are connected by cross beams at the midspan and sup-ports. Figs. 11b and cshow the reinforcing bar and the tendonarrangement, respectively.FE 16.70 12.62 9.68 6.81 23.51 17.7116.57 8.58FE 4.24 2.550.48NR1,1/CNR2,2/CNR1,2/C(FEA)NR1,3/C(FEA)1050NR1,1/CNR2,1/C0 50 100 150 200 250 300 35012.26 7.810.493.01 7.46 5.06 1.77 1.65NR1,2/CNR2,3/CNR1,0/C(Test)0 50 100 150 200 250 300 350NR1,2/CNR2,2/C2.81 7.98 5.32 1.77 1.82NR1,3/CNR1,1/C(Test)NR1,3/CNR2,3/C1400NR1,0/C(FEA)0.008NR1,1/C(FEA)NR1,3/C(FEA)NR1,0/C(Test)NR1,2/C(FEA)NR1,0/C(Test)NR1,0/C(Test)NR1,0/C(Test)1400NR2,0/C13501300125012001150NR1,0/C(FEA)1100NR1,2/C(Test)1050Time since prestressing (day)(a) Time-variant tendon stresses (from FE analysis) (b) Comparison of tendon stresses from FE analysis and tests0.0100.0080.0060.0040.0020.000-0.002-0.0040 50 100 150 200 250 300 350Time since prestressing (day)(c) Time-variant deflections (from FE analysis) (d) Comparison of deflections from FE analysis and testsFig. 10.Time-dependent tendon stress and deectionstressed reinforcing steel, and larger decrease in tendon stressis seen in beams with lower ratios of nonprestressed reinforce-ment. Moreover, the presence of larger compressive stress in-creases the level of creep and shrinkage, resulting in fasterreduction in tendon stress. Results of four specimens, NR1,0/C,NR1,1/C, NR1,2/C, and NR1,3/C, are compared with the testresults, see Fig. 10b. The time-dependent test results of theother four specimens, however, are not provided by Chouman2003a. Fig. 10cshows the calculated time-dependent deec-tion, and it is found that the reinforcement ratio and prestresslevel also have a signicant inuence on it. Comparison of time-dependent deections between FE and test results is illustrated inFig. 10d.Table 2 summarizes the comparison of the FE analysis resultswith the test results at the time of 1 year, and a satisfactory agree-ment is observed except in the case of the underlined group. Themeasured prestress losses in NR2,2/C appear to be inconsistentlylarge; however, no explanation was given by Chouman2003a,b.Table 2.One Year Prestress Losses and DeectionNR1,0/C NR1,1/C NR1,2/C NR1,3/C NR2,0/C NR2,1/C NR2,2/C NR2,3/CPrestress loss%Test 15.80 12.38 10.47 7.30 20.32 16.53DeectionmmTest 3.93 2.52Note: The measured deections in Table 2 have accounted for the initial cambers.36/ JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. Visitsis results and is not considered.Random VariablesThe random variables in the study are summarized in Table 3. Thedead weight model, suggested by Ghosn et al.2009, is used,which follows a normal distribution with a mean value of 1.05and a coefcient of variationCOVof 0.1. The live load is de-scribed by an extreme distribution with a mean value of 1.0 and aCOV of 0.19Ghosn et al. 2009. A dynamic load amplicationDLAfactor is applied to the static truck load with a mean valueof 1.10 and a standard deviation of 0.08National CooperativeHighway Research ProgramNCHRP2007.Concrete strength is assumed to follow a normal distributionwith a mean value of 56.40 MPa and a standard deviation of 6.35MPa, see Fig. 14a. The thickness of the concrete cover is mod-eled by a normal distribution with a mean value of 9.7 mm greaterthan the design value of 40 mm and a standard deviation of 7.6mm, see Fig. 14b. The plate thickness of the box girder, whichaffects the stresses and cracking behavior, is modeled accordingto the statistical results of Li and Bao1997.The annual average temperature and ambient humidity nearbridge site are provided by the China Meteorological Administra-tion, which are recorded from the year of 1951 to 2008, see Figs.15a and c. Normal distributions are used to model the ambienttemperature and humidity, as shown in Figs. 15b and d.Surface chloride contentC0, threshold chloride concentrationCcr, and diffusion coefcientDcare adopted from the model pro-posed by Val and Pavel2008. According to Gonzlez et al.1995, the mean current densityicorris estimated by a normalrandom variable with a mean value of 1.0A/cm2and a COV of0.2. The penetration ratioRis dened as a normal random vari-able with a mean value of 3.0 and a COV of 0.33 according toFE Modeling of PSC Box Girder145kNbridge is modeled. Though a more accurate model would includeall ve spans, the computation cost was considered to be too high.In the ve-span bridge, the adjacent spans provide longitudinaland rotational constraints to the single interior span considered inthis study. Therefore, the longitudinal and rotational displace-ments of the interior span at its supports are relatively small, so145kNIn this study, only one of the ve interior spans of the continuous(a) Bridge cross-section in the mid-span (dimension in cm)Fig. 13.Transverse truck load positionsdimensions in centimeters(b) Rebar arrangement (mid-span section, dimension in cm)with one truck in each laneproduces the largest load effect.Therefore, this transverse load case is used in the following analy-sis. The transverse position of tire loads within a laneCzarneckiand Nowak 2008seems to have a smaller inuence on the analy-(c)Tendon arrangement(side view, dimensions in cm)Fig. 11.Bridge congurationLoad CasesFor short or medium span bridges, the dominant loads includedead weight, trafc loads, and prestressing loads. The deadweight of the box girder is taken into account by the FE programautomatically and the dead weight of the pavement is treated asuniform pressure on the top ange of the box girders.The prestressing load is applied by bundles of tendons with adiameter of 15.24 mm, which have a cross-sectional area of140 mm2. The jacking stressconis 1,395 MPa, which is 75% ofthe yield stress1,860 MPaof the tendons. Jacking forces areapplied symmetrically at two ends when the concrete strength ofbox girders reaches 90% of its value, and the jacking sequence oftendons is T1, T2, T3, T4, and T5. The instantaneous prestresslosses, caused by friction, anchorage set, and elastic shorteningAASHTO 2004, are estimated to be 90 MPa.The live load model is dened as the truck shown in Fig. 12superimposed with a uniformly distributed lane load of 9.3 kN/mAASHTO 2004. The longitudinal position of the truck load isdetermined from an inuence line analysis to determine theworst-case scenarioe.g., the maximum deection or the largeststrain. For the maximum deection of the midspan section, thefront wheel is placed 6.6 m away from the left support, as shownin Fig. 12. Transversely, four specic truck load positions areinvestigated, as shown in Fig. 13, with the multiple presence fac-tors of 1.00, 0.85, and 0.65 that are used for two trucks, threetrucks, and four trucks, respectivelyAASHTO 2004, to accountfor the probabilities of simultaneous lane occupation. Position Iprevious studies Stewart and Rosowsky 1998. As shown inTable 3, the distributions of the random variables are truncated atcertain values to avoid erroneous sampling.Box-girder6.6 9.84.3 4.335kNPier25.0Fig. 12.Longitudinal truck load positiondimensions in metersJOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011 /37Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitActual dead weightActual live loadDLA factor 1.10 0.073 Normaltruncated at 1Actual plate thicknessConcrete strength 56.4 0.113 Normaltruncated at 40Thickness of concrete cover 4.97 0.153 Normaltruncated at 0.17Annual relative humidity 77.1 0.039 Normaltruncated at 100Annual average temperature 15.4 0.044 Normal China Meteorological AdministrationSurface chloride content 15 0.20 Normaltruncated at 0Threshold chloride concentration 2.0 0.20 Normaltruncated at 0Diffusion coefcient 0.631 0.2 Lognormaltruncated at 0Corrosion current density 1 0.2 Normaltruncated at 0Penetration ratio 3 0.33 Normaltruncated at 1In the FE analysis, the inuence of corrosion, concrete creep,and corrosion is considered separately. For corrosion, the level ofcorrosion at a given service year is calculatedconsidering thegenerated values of the random variables such as thickness ofconcrete cover, surface chloride content, threshold chloride con-centration, and diffusion coefcientbefore the FE analysis isperformed, and the cross-sectional areas of tendons and reinforc-ing bars are modied to have the remaining cross-sectional areacalculated from Eq.11.The inuence of creep and shrinkage is considered through atwo-step analysis. First, the loads live loads, dead weight, pre-stressing loads, etc. are applied on the FE model considering thegenerated values of the random variables such as concretestrength, plate thickness, environment temperature, relative hu-midity, etc., and then FE analysis provides the strain information.Second, the service time is used and the creep and shrinkageanalysis is made using the previous strain information to obtaintendon stresses and deections.For the nonlinear analyses, Newton-Raphson iteration is used.The convergence criterion is a combination of force and displace-ment, and the convergence tolerance is 0.01. For each FE simu-=56.40=6.35105036 40 44 48 52 56 60 64 68 72Measured thickness minus design value (mm)1.05 0.10 Normaltruncated at 01.0 0.19 Extreme Itruncated at 0National Cooperative Highway1.052 0.114 Normaltruncated at 0.85Field testField testChina Meteorological AdministrationVal and Pavel2008Val and Pavel2008Val and Pavel2008Gonzlez et al.1995Stewart and Rosowsky199825=7.6-8 -4 0 4 8 12 16 20 24 28Ghosn et al.2009Ghosn et al.2009Li2004=9.7Table 3.Description of Random VariablesVariable Properties Mean COV Distribution SourceDWNominal dead weightLLAASHTO truck load plus lane loadDLAResearch ProgramNCHRP2007PTDesign plate thicknessCSMPaXcmRH%TCC0kg/m3Ccrkg/m3Dccm2/yearicorrA/cm2Rthe two supports are xed in the FE model. As a result of thissimplied model, the stress at midspan of the FE model may besmaller and the stress at the supports of the FE model may belarger than in the actual ve-span bridge under live load.Fig. 16ashows the FE model mesh. The plates at the ange-slab connection adopt a varying thicknessfromt1tot2to bettermodel the behavior at the connection and decrease the stress con-centration. Nonprestressed reinforcement is simulated by a seriesof rebar grids shown for only one of the ve box girders in Fig.16bfor clarity. Fig. 16cillustrates the prestressed tendons forone box girder. Initial material properties of concrete and rein-forcing steel are dened by using their standard properties. Forconcrete, a density of 2,400 kg/m3, a modulus of elasticity of34,500 MPa, a Poissons ratio of 0.17, and the tension softeningcurve shown in Fig. 5 are used. For steel reinforcing bars, a den-sity of 7,850 kg/m3, a modulus of elasticity of 200,000 MPa, aPoissons ratio of 0.3, and a bilinear kinematic hardening modelwith a yield strength of 335 MPa are used. For prestressed ten-dons, a density of 7,850 kg/m3, a modulus of elasticity of195,000 MPa, a Poissons ratio of 0.3, and a bilinear kinematichardening model with a yield strength of 1,860 MPa are used.3025202015151050Concrete Strength (MPa)(a) Concrete strength (b) Concrete cover thicknessFig. 14.Statistical results of eld data38/ JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. Visit121086214.5 15.0 15.5 16.0 16.5 17.0 17.5Annual temperature (0C)18=77.1=2.99930Annual ambient humidity(%)(d) Statistics of ambient humidity(a) Concrete substrate(b) Rebar grids(c)Prestressed tendonsFig. 16.FE model for PSC box girder=15.4=0.6770 75 80 8517.517.016.516.015.5415.014.51950 1960 1970 1980 1990 2000 2010Year(a) Annual average temperature (b) Statistics of temperature84158012766721950 1960 1970 1980 1990 2000 2010Year(c) Annual average ambient humidityFig. 15.Annual average temperature and ambient humiditylation, a step size of 0.1 of the load increment is used and for eachload step, the maximum number of iterations is 50.Denition of Limit StateTwo limit states are considered in the reliability analyses: theserviceability limit stateSLSand the tendon yielding limit stateTYLS. The SLS is usually related to gradual deterioration, usercomfort, or maintenance costs. Cracking, excessive deection,and vibration are typical SLS variables. In this work, the SLS isreached when the deection exceeds the threshold value, and adeection limit equivalent to span length/800 is adoptedAASHTO 2004. The TYLS is assumed to be reached when themaximum tendon stress exceeds the yield stress. Similar deni-tions of the TYLS can be found in the work of Rajashekhar andEllingwood1995. Thus the well-known performance functionGxcan be expressed asGxSLS=RsSs=Deflection limitMaximum deflection16aandGxTYLS=RTYSTY=Yielding stressMaximum tendon stress16bwhereRandSrepresent resistance and load effect, respectively.For the TYLS, the applied loadsor load effectsinclude deadload, live load, and time-dependent effects such as creep and cor-rosion. For the SLS, creep analysis is made rst under dead load,and then live load is applied to the bridge to obtain the deectionand compare with the deection limiti.e.,L/800, whereL=span length.JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011 /39Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitYielding stressTendon stress40101700 1750 1800 1850 19000Yielding stress & tendon stress (MPa)fall into the failure region, even though the sampling number issmaller than that of the direct MCS. This comparison shows theadvantage of using the IS.Additional information that can be obtained from the MCS arethe sensitivities of the response variables to the random variables.The sensitivity analysis is based on regressions of the responsedata versus the normalized random variables in the vicinity of theapproximate design pointFigiel and Kamiski 2009. Figs. 18aand bshow the normalized sensitivities of deection and tendonstress, respectivelyin the 50th service year, to the random vari-ables. It is shown that the concrete strength, the dead weight, and0.012345678910111213-0.1-0.2-0.3Variable number0.012345678910111213-0.1-0.2-0.3Variable number1LL12RHYielding stress50Variable number2DW13TTendon stress3PT 4CS 5X 6Dc 7C0 8Ccr(See Table 3 for the meaning of the variables. )60800600304002020001650 1700 1750 1800 1850Yielding stress & tendon stress (MPa)(a)Direct MC sampling(with 1,000 samples)(b)Importance sampling(with 200 samples)Fig. 17.Comparison between direct MCS and IS resultsResults and DiscussionFirst, a comparison was made between results from direct MCSand from the IS method. Taking the reliability analysis for theTYLS as an example, the direct MCS with 1,000 samples wasmade under the assumption that the bridge had been in service for50 years. Both the tendon stress and the tendon yielding stress arerandom variables. It is observed from Fig. 17athat most oftendon stresses are far less than the yielding stress, which meansthat many more samples are required in order to obtain the failureprobability. Instead, the IS method with 200 samples was per-formed as discussed earlier. As shown in Fig. 17b, more samples0.01 2 3 4 5 6 7 8 9 10 11 12 13-0.1-0.2-0.3Variable number(a) Deflection sensitivities(in the 50th year) (b) Tendon stress (in the 50th year)0.012345678910111213-0.1-0.2-0.3(c) Deflection sensitivities (in the 100th year) (d) Tendon stress (in the 100th year)Key to variable numbers:9icorr 10R 11DLAFig. 18.Probabilistic sensitivities40/ JOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitPhase II=3.5Phase I0 1020304050607080901006Phase II3210Service time (year)(b) Serviceabilitylimit stateFig. 20.Reliability index prolesby corrosion. Area losses in tendons as well as nonprestressedreinforcing bars result in a higher stress level under external loadsand larger deections.ConclusionsThis paper presents a reliability assessment of a PSC box-girderbridge with consideration given to creep, shrinkage, and corro-sion. The pre- and postcracking behaviors of the thin-walled PSCbox girders are modeled using a degenerated composite shell el-ement with tension softening behavior. An IS method is used toobtain reliability indexes from implicit performance functions. Abalance between efciency and accuracy is provided by the ISmethod. From this study, the following conclusions are drawn.Concrete creep and shrinkage have a signicant inuence onthe decrease in reliability indexes with regard to tendon yieldingand deection, particularly in the early age of the bridge. Underthe combined action of creep, shrinkage, and corrosion, reliabilityindexes undergo a three-phase reduction and fall below the targetreliability level before the expected life of 100 years.The sensitivities of the response variables to the random inputvariables also vary with time. Before corrosion becomes serious,the dominant variables inuencing the structural reliability arelive loads, plate thickness, and DLA. In this study, corrosion be-comes signicant around 50 years after the bridge opens to trafc.Thereafter, the corrosion-related variables gradually become thecontrolling factors for tendon yielding. The proposed methodol-ogy can be used for a better understanding of the deteriorationPhase IIIPhase III=1.5Phase I10 20 30 40 50 60 70 80 90 10065432Cracks1(a) 3D perspective viewService time (year)(a) Tendon yielding limit state(b) Side ViewFig. 19.Typical crack patternfor one girder54the live loads have the greatest inuence on the girder deection,while the variables related to corrosion have little inuence. Forthe tendon stress, the external loads and the corrosion-related fac-tors have the greatest inuence. Note that these sensitivities mayvary with time, as shown in Figs. 18c and d, where the inuenceof corrosion-related variables becomes more signicant in the100th year.A typical crack pattern of the box girder is illustrated in Fig.19, in which the bending cracks in the top plate and the bottomplate are observed. Although the bridge is partially prestressedand cracks are allowedbut the crack width should be withinlimits, the cracks in the top plate facilitate chloride and waterpenetration, and therefore, it is from this point of view that cracksin the top plate are more detrimental.Performing the reliability analysis at certain service years, theevolution of the reliability index is obtained, which can be gen-erally divided into three phases. In Phase Iabout 1012 yearsafter loading, a decrease in the reliability of tendon yielding isobserved, see Fig. 20a. Meanwhile, a faster increase in deec-tion occurs due to creep and shrinkage, which results in a signi-cant decrease in the reliability index for SLS, as shown in Fig.20b. In Phase II, a mild decrease in the reliability index is ob-served. A signicant decrease is found again in Phase IIIabout5060 years after loading. The reliability index in the YLS fallsbelow the target value of 3.5National Cooperative Highway Re-search ProgramNCHRP2007after 54 years of service. Ac-cording to ISO 2394ISO 1996, the target value is chosen as 1.5;therefore the reliability index in the SLS falls below the targetvalue after about 65 years of service. According to past experi-ence, the self-equilibrating prestress force generally does notchange the reliabilities for the ultimate limit state; however, creepand shrinkage may produce a stress redistribution, making certaintendons sustain higher stresses under live loads and more prone toyielding. Meanwhile, creep and shrinkage cause the time-dependent reduction in the elastic modulus of concrete, whichmakes tendons bear additional external forces. This may explainthe decrease in reliability index of YLS in Phase I. The decreasein reliability index becomes less signicant in Phase II, and this isbecause concrete creep and shrinkage are most signicant in therst 1015 years. Meanwhile, in Phase II corrosion is not yetsevere. The decrease in reliability index for the TYLS in PhaseIII, however, becomes faster again and this is mainly controlledJOURNAL OF BRIDGE ENGINEERING ASCE / JANUARY/FEBRUARY 2011 /41Downloaded 28 Dec 2010 to 1. Redistribution subject to ASCE license or copyright. VisitEnright, M. P., and Frangopol, D. M.1998. “Probabilistic analysis ofresistance degradation of reinforced concrete bridge beams under cor-rosion.”Eng. Struct.,2011, 960971.Figiel, ., and Kami ski, M.2009. “Numerical probabilistic approachto sensitivity analysis in a fatigue delamination problem of a two layercomposite.” Appl. Math. Comput., 2091, 7590.Gardner, N. J., and Lockman, M. J. 2001. “Design provisions for dryingshrinkage and creep and normal-strength concrete.” ACI Mater. J.,982, 159167.Ghosn, M., Moses, F., and Frangopol, D. M. 2009. “Redundancy andGoel, R., Kumar, R., and Paul, D. 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