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Time Dependent Reliability of PSC Box Girder Bridge Considering Creep Shrinkage and Corrosion Tong Guo1 Richard Sause M ASCE2 Dan M Frangopol Dist M ASCE3 and Aiqun Li4 Abstract Bridge performance undergoes time varying changes when exposed to aggressive environments While much work has been done 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 fi nite element FE analysis in conjunction with probabilistic considerations Verifi cation is made by comparing the analytical fi ndings with existing test data More specifi cally a time dependent reliability assessment is made for a composite prestressed concrete PSC box girder bridge exposed to a chloride environment This realized via an advanced probabilistic FE method The postcracking behavior of the thin walled box girder is described using composite degenerated shell elements and importance sampling is used to improve the effi ciency of the reliability analyses It is shown that concrete creep and shrinkage dominate during the early stages of bridge structure deterioration This is accompanied by a decrease in reliability owing to the combined action of creep shrinkage and corrosion The reliability indexes for the serviceability and the tendon yielding limit state fall below the target levels prior to the expected service life Therefore early maintenance and or repair measures are required DOI 10 1061 ASCE BE 1943 5592 0000135 CE 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 fi nite element method Introduction Maintenance strengthening and rehabilitation of the aging civil infrastructure systems require an accurate assessment of the reli ability throughout the service life of deteriorating structures The primary factors affecting structural durability should be fully taken into account For RC structures it has long been recognized that steel corrosion and concrete creep and shrinkage are the major 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 bridge serviceability In the worst case corrosion may even trigger struc tural collapse Structural stresses and displacements change gradually over time due to concrete creep and shrinkage For prestressed concrete PSC structures creep and shrinkage result in prestress losses which cause stress redistribution and a de crease of structural safety Results of the previous time dependent prestress loss studies are listed in Table 1 in which the structural age during the measurement ranges from 6 months to 46 years It is observed that the prestress losses with time are always present while the specifi c values of this loss may be infl uenced by param eters such as structural type ambient temperature humidity and concrete 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 The existing models regarding concrete creep and shrinkage include the effective modulus method McMillan 1916 the double power law for creep Bazant and Osman 1976 the ACI 209 model American Concrete Institute ACI 1982 the double power logarithmic law for creep Bazant and Chern 1984 the CEB FIP model CEB FIP 1994 the B3 model RILEM TC 107 GCS 1995 and the GL2000 model Gardner and Lockman 2001 Comparisons of these models were presented by Goel et al 2007 and it was concluded that the recently developed GL2000 model the CEB FIP model the B3 model and the ACI 209 model are more reliable than earlier creep models Corrosion of high strength reinforcing steel is a complex phe nomenon that consists of several different but interrelated mechanisms such as uniform corrosion pitting corrosion and corrosion induced cracking Numerous studies with regard to cor rosion models have been made and will not be repeated here In conjunction with available corrosion studies time dependent re liability analyses have been made Enright and Frangopol 1998 investigated the resistance degradation of RC bridge beams under uniform corrosion and the corrosion initiation time was predicted 1Associate Professor Key Laboratory of Concrete and Prestressed Concrete Structure Ministry of Education Southeast Univ Nanjing 210096 People s Republic of China formerly Visiting Research Scien tist ATLSS Center Lehigh Univ 117 ATLSS Dr Bethlehem PA 18015 corresponding author E mail guotong77 2Joseph T Stuart Professor of Structural Engineering and Director Dept of Civil and Environmental Engineering ATLSS Center Lehigh Univ 117 ATLSS Dr Bethlehem PA 18015 E mail rs0c lehigh edu 3Professor 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 frangopol lehigh edu 4Professor College of Civil Engineering Southeast Univ Nanjing 210096 People s Republic of China E mail aiqunli Note This manuscript was submitted on October 11 2009 approved on May 10 2010 published online on May 14 2010 Discussion period open until June 1 2011 separate discussions must be submitted for indi vidual papers This paper is part of the Journal of Bridge Engineering Vol 16 No 1 January 1 2011 ASCE ISSN 1084 0702 2011 1 29 43 25 00 JOURNAL OF BRIDGE ENGINEERING ASCE JANUARY FEBRUARY 2011 29 Downloaded 28 Dec 2010 to 61 187 64 11 Redistribution subject to ASCE license or copyright Visithttp www ascelibrary org with a probabilistic model Val and Robert 1997 developed a time dependent corrosion model for reliability analysis of RC slab bridges in which the localized corrosion pitting corrosion was included This model was adopted and improved in work by Stewart 2004 Accelerated pitting corrosion tests were used to obtain spatial and temporal maximum pit depth data for prestress ing wires Darmawan and Stewart 2007 and further investigated by Stewart 2009 were the mechanical behavior of pitting corro sion of fl exural and shear reinforcement and its effect on struc tural reliability However the previous work mainly focused on the infl uence of corrosion while concrete creep and shrinkage were not taken into account The objective of this study is to develop a time dependent reliability evaluation methodology for PSC structures in which the corrosion and concrete creep and shrinkage are all included The methodology is applied to assess the performance of PSC box girders from a bridge exposed to a chloride environment These box girders are widely used in highway bridges and early damage e g cracking is observed in some cases Reasons for the damage include material deterioration construction defects increase in traffi c loads and inaccurate design The unique shear lag effect Luo et al 2002 in thin walled box girders and the complex arrangement of prestressed tendons bring additional dif fi culties into the structural design Although several design codes Deutsches Institut f r Normung DIN 1981 AASHTO 2004 have already provided formulas to account for the shear lag ef fect these formulas are developed for the elastic stage of behavior and are inadequate for reliability analyses in the elastoplastic stage Guo and Li 2009 To make a more rational assessment of the PSC box girders a probabilistic fi nite element FE method is used in which composite degenerated shell elements are used to model the pre and postcracking behaviors of the thin walled box girder and an approximate importance sampling IS method is used to perform the reliability analysis A comparatively wide range of random variables is covered in the analysis such as chloride diffusion rate critical threshold chloride concentration pitting corrosion depth concrete material properties concrete cover thickness and external loads Time Dependent Deterioration Model Concrete Creep and Shrinkage The time dependent change in material properties of concrete is modeled using the CEB FIP model CEB FIP 1994 which takes a number of parameters into consideration such as cement type ambient temperature relative humidity concrete strength and concrete age at loading In the CEB FIP model evolution of concrete creep is de scribed via the creep function J t t0 which is formulated as CEB FIP 1994 J t t0 1 Ec t0 t t0 Ec28 1 where Ec t0 modulus of elasticity at the concrete age of t0and Ec28corresponds to the value at the age of 28 days The so called creep coeffi cient t t0 can be determined from the following hyperbolic power function Table 1 Survey of Time Dependent Prestress Losses Data sourceStructure typeLocationAge after loading Prestress losses Bond information Saiidi et al 1998Posttensioned simply supported box girder bridgeSouthern Nevada U S A 24 months9 16Nonbonded Roller et al 1995Pretensioned high strength concrete bulb tee girdersLouisiana U S A 6 12 18 months9 7 10 8 11 4Bonded Xue et al 2008Prestressed steel concrete composite beamsShanghai P R China1 year15Nonbonded Chouman 2003a bPosttensioned simply supported beamsLeeds U K 1 year4 04 20 32Bonded and nonbonded Halew and Russell 2006Pretensioned simply supported I beamsOklahoma U S A 1 year27 37 6Bonded Barr et al 2008Prestressed high performance concrete girdersWashington D C U S A 3 years16 2 27 5Bonded Kowalsky et al 2001Prestressed high performance concrete bridge girdersNorth Carolina U S A 12 9 19 1 Naito and Sause 2008Three span precast prestressed spread box beam bridge Pennsylvania U S A 12 years32 7Bonded Azizinamini et al 1996Pretensioned simply supported bridge I beamsNebraska U S A 25 years20 7Bonded Pessiki et al 1996Pretensioned simply supported bridge I beamsPennsylvania U S A 28 years17 2 18 1Bonded Anderson 2005Reactor containmentSweden30 years5 10Nonbonded Naito et al 2006Prestressed box beam bridgePennsylvania U S A 46 years32 9 38 4 41 7Bonded 30 JOURNAL OF BRIDGE ENGINEERING ASCE JANUARY FEBRUARY 2011 Downloaded 28 Dec 2010 to 61 187 64 11 Redistribution subject to ASCE license or copyright Visithttp www ascelibrary org t t0 1 1 RH RH0 0 46 h 100 1 3 5 3 0 1fcm28 1 0 1 t0 1 5 t t0 H t t0 0 3 2 where RH relative environmental humidity RH0equals to 100 and h nominal size of the concrete member mm defi ned as 2Ac u Acis the cross sectional area and u is the perimeter in contact with the atmosphere fcm28stands for the mean compres sive strength at the age of 28 days and H min 1 500 150 1 1 2RH 18 h 100 250 The evolution of concrete strength with time is described by Eq 3a fcm t cc t fcm28 3a where cc t exp s 1 28 teq 3b is a time dependent coeffi cient and s takes the values of 0 20 0 25 and 0 38 for rapid hardening high strength cement normal and rapid hardening cement and slowly hardening cement re spectively The equivalent age of concrete teq is defi ned as teq 0 t 4 000 1 273 1 T d 3c where T temperature of concrete at days The modulus of elasticity of concrete at t days can be esti mated as Ec t cc t Ec28 4 The shrinkage strains s t ts at an age of t days is s t ts 160 10 9 0 1fcm28 10 6 RH t ts 350 h 100 2 t ts 5a where shrinkage coeffi cient dependent on cement type ts age of concrete at the beginning of shrinkage day and RHis related to the environmental humidity RH as follows RH 1 55 1 RH 100 3 40 RH 99 0 5RH 99 5b The above equations concerning creep and shrinkage are inte grated into the nonlinear FE code In this way the infl uences of structural details such as the reinforcement arrangement and the time dependent prestress level can be considered automatically by the FE program Meanwhile the change in concrete strength and elastic modulus due to creep can also be included CEB FIP 1994 Stress Relaxation Similar to the modeling of creep stress relaxation is modeled via the relaxation function by using a generalized Maxwell model which can be physically interpreted as a set of parallel springs and dampers i e a set of Maxwell elements as shown in Fig 1 where Eiand idenote the stiffness of each spring and damper respectively The stress strain relationship for the generalized Maxwell model is expressed as t t E t d 6a where E t denotes the relaxation function which can be ex panded into a Dirichlet series E t i 0 n Ei e t i 6b where Ei time dependent stiffness of a Maxwell element see Fig 1 The relaxation time for a Maxwell element is i i Ei 6c Substituting Eq 6b into Eq 6a for time t and time t t respectively and after summation integration and subtraction the stress increment in t is i 0 n 1 e t i E t i t i t 7 where t sampling point at the midpoint of the time increment namely t t 2 More details regarding the generalized Maxwell model and the relaxation function can be found in Bazant and Wu 1974 Time Dependent Corrosion Model It is well recognized that corrosion is a primary cause of structural deterioration of RC structures One direct consequence of corro sion is the cross sectional area loss of reinforcement and in the most common case uniform corrosion the diameter of a corrod ing reinforcing bar D at time t can be estimated as Val and Robert 1997 D t D0 0 0232 t ti icorr 8 where icorr corrosion current density A cm2 D0 initial di ameter of the reinforcing bar cm and ti time of corrosion ini tiation year In addition to the uniform corrosion which is generally a mac rocell corrosion process pitting corrosion in a chloride environ ment being a microcell corrosion process may lead to a faster deterioration According to Gonz lez et al 1995 the maximum penetration of pitting is about four to eight times that associated with uniform corrosion Under attack from pitting corrosion the 0 E 1 E 2 E n E 1 2 n 0 Maxwell element Fig 1 Generalized Maxwell model JOURNAL OF BRIDGE ENGINEERING ASCE JANUARY FEBRUARY 2011 31 Downloaded 28 Dec 2010 to 61 187 64 11 Redistribution subject to ASCE license or copyright Visithttp www ascelibrary org reinforcement is prone to failure due to stress concentration For stranded wires with high prestress the pitting corrosion process may be accelerated Vu et al 2009 and brittle rupture of the stranded wires may occur earlier than expected Research carried out by Naito et al 2006 investigated corrosion of prestressing strands in PSC box girders PSC box beams were taken from a 46 year old bridge and assessed Rupture of the prestressed stranded wires in the bottom plate was observed under the com bined action of stress and corrosion Fig 2 Heavy pitting de fi ned as a pit greater than 20 of the wire section area was observed most likely from chloride attack Fig 3 According to Val s work the radius of the pit at time t can be estimated as p t 0 0116 t ti icorrR 9 where R penetration ratio between the maximum and average penetration The time of corrosion initiation tiis predicted by the following equation Enright and Frangopol 1998 ti X2 4Dc erf 1 C0 Ccr C0 2 10 where X concrete cover cm Dcrepresents the chloride diffu sion coeffi cient cm2 year C0denotes the chloride concentration at the concrete surface weight of concrete and Ccr threshold chloride concentration weight of concrete The net cross sectional area of a corroded rebar Ar t at time t is estimated by Eq 11 Stewart 2009 Fig 4 illustrates the relationship between Ar t and p t Ar t D0 2 4 A1 A2 p t 2 2 D0 A1 A2 2 2 D0 p t D0 0 p t D0 11a with a 2p t 1 p t D0 2 11b A1 1 2 1 D0 2 2 a D0 2 p t 2 D0 A2 1 2 2p t 2 ap t 2 D0 11c 1 2 arcsin a D0 2 2 arcsin a 2p t 11d Many laboratory results have indicated that the yield stress on the net cross sectional area is also reduced by corrosion as follows fy t 1 Pcorr 100 fy0 12 where fy t deteriorated yield strength at time t fy0corresponds to its original value empirical coeffi cient and according to the regression analyses of Du et al 2005 and Vu et al 2009 a value of 0 0054 is taken for reinforcing bars and 0 0075 is taken for stranded wires respectively Pcorris the percentage of corro sion loss which can be obtained from Eq 8 or Eq 11 in terms of area loss Probabilistic FE Model Composite Degenerated Shell Element One challenge in the probabilistic FE analysis of thin walled structures is analysis effi ciency because both nonlinear FE analy sis and probabilistic analysis involve a large number of iterations For this reason shell elements rather than more common solid elements are used In this study an eight node composite degenerated shell ele ment in the FE code DIANA DIANA 2008 is used Fig 5 a This shell element is capable of modeling pre and postcracking behaviors of thin walled RC structures and enables easy modeling of distributed reinforcement as well as prestressed tendons For this shell element each node has fi ve degrees of freedom three displacements uX uY and uZin the global XYZ directions and two rotations and respectively around the local and axes respectively To avoid membrane and shear locking which results in an excessively stiff behavior a reduced 2 2 integration scheme is used over the element area in the and directions Fig 2 Rupture of stranded wires due to stress and corrosion Pitting corrosion Fig 3 Pit observation with optical microscope profi le taken by the fi rst writer at the ATLSS Center 0 D TAr 1 2 a tp Fig 4 Pit confi guration adapted from Val and Robert 1997 32 JOURNAL OF BRIDGE ENGINEERING ASCE JANUARY FEBRUARY 2011 Downloaded 28 Dec 2010 to 61 187 64 11 Redistribution subject to ASCE license or copyright Visithttp www ascelibrary org while the integration in the thickness direction is three point Simpson integration Nonprestressed reinforcement is modeled using a reinforce ment grid which is embedded in the shell element see Fig 5 a The reinforcement ratios in two perpendicular directions are rep resent