RELIABILITY ANALYSIS FOR DEFLECTION OF COMPOSITE GIRDER BRIDGES

Transcription

1 RELIABILITY ANALYSIS FOR DEFLECTION OF COMPOSITE GIRDER BRIDGES Pascal Laumt (1) and Andrzj S. Nowak (2) (1) Univrsity of Michigan, Ann Arbor, Michigan, USA (2) Univrsity of Nbraska, Lincoln, Nbraska, USA Abstract Th papr prsnts a probabilistic modl dvlopd for th dflction limit stat of simply supportd composit stl girdr bridgs. Th dflction dpnds on th stiffnss of th stl girdrs and intgrity of th rinforcd concrt slab (composit action). Th structural prformanc can b affctd by dtrioration of componnts, in particular corrosion of stl girdrs, and tim-dpndnt proprtis of concrt. An analytical modl is dvlopd to calculat dflction du to liv loads. Th statistical modls for crp as wll as corrosion of structural paintd stl ar incorporatd in th modl. Th ag-adjustd mthod is slctd. To carry out th tim-dpndnt rliability analysis, an allowabl dflction limit is stablishd basd on th AASHTO LRFD Cod (2004). Mont Carlo simulations ar usd to stimat th dflctions and probabilitis of srvicability failur. Svral diffrnt span lngths, and various nvironmnt xposur conditions ar considrd as wll as a rang of liv load bias factors. This nw and innovativ modl for liv load volution is incorporatd in th probabilistic modl and allows for rliability stimation in cas of liv load volution. 1. INTRODUCTION Composit stl girdr bridgs rprsnt a considrabl prcntag of th bridg population in th Unitd Stats. During th intrstat ntwork dvlopmnt in th 60 s and 70 s, rapid construction and opration was rquird to nsur adquat work progrss. Thrfor, stl bridgs was a natural choic, mad composit with a concrt dck, that allowd for a rapid rction and fast opning to traffic. As a consqunc, ths bridgs rprsnt about 34% of th ntir bridg population. Howvr, today thy also count for as much as 54% of th structurally dficint structurs [1]. Structural dficincis ar du in part to th incras of traffic, but mostly du to corrosion. A composit sction is subjctd corrosion of th stl girdr, but also to th crp and shrinkag of concrt, as wll as th corrosion of th mbddd rinforcmnt. Th rsulting strss rdistribution can b an important factor that has to b accountd for in an analysis. Structural stl corrosion is a major concrn in th Unitd Stats. A study in yar 2002 mandatd by th U.S. Congrss stimatd th total dirct cost of mtal corrosion in 26

2 industrial sctors to b $276 billions dollars pr yar [2]. Of that amount, th annual dirct cost of corrosion for highway bridgs is stimatd to b $8.3 billion, consisting of $3.8 billion to rplac structurally dficint bridgs ovr th nxt tn yars, $2.0 billion for maintnanc and cost of capital for concrt bridg dcks, $2.0 billion for maintnanc and cost of capital for concrt substructurs (without dcks), and $0.5 billion for maintnanc painting of stl bridgs. In th sam rport, th authors also mntiond that th non-stimatd indirct costs to th usrs du to traffic dlays and loss of productivity ar mor than 10 tims th dirct cost of corrosion. Th ffcts of incrasd us of chlorid-basd dicing agnts is dramatically illustratd by th fact that spalls wr quit rar in California until lat 1960s whn th us of chmical dicrs was introducd in this stat. Anothr factor is th incras of salt us on highways during wintr (s Fig. 1), spcially in th snowblt rgions. In fact, ovr 70 % of th nations roads ar locatd in snowy rgions (70% of th U.S. population), which rciv mor than 13 cm (or 5 in.) avrag snowfall annually. Th total amount of salt usd on highways has bn multiplid by 8 sinc 1960 and will most likly continu to grow. Figur 1: US Highway Salts Sals Btwn 1940 and 2004 [3] This papr prsnts a probabilistic modl for rliability analysis of dflctions of composit stl girdr bridgs including paramtrs such as tim-dpndnt matrial proprtis, as wll as liv load volution. Th rliability analysis is basd on th optional dflction srvicability limit stat dfind by AASHTO LRFD (2004). This srvicability failur is dfind to occur whn th dflction inducd by liv load xcds an allowabl dflction limit of lngth/800. Othr limit stats such as long-trm dflctions du to sustaind dad load or cyclic loading, corrosion cracking, ultimat momnt carrying capacity, and shar strngth, ar byond th scop of th prsnt papr. Th dvlopd curvatur-basd dflction modl includs th influnc of instantanous dformation du to liv load as dfind by th cod. Howvr, matrial and nvironmntal paramtrs, modling uncrtaintis, and liv load ar assumd to b random variabls. Thrfor, th modld dflction is a tim-dpndnt random variabl. Mont Carlo

3 simulation is usd to prform th rliability analysis basd on th adoptd allowabl dflction limit. Rliability indics (rprsnting th probabilitis of srvicability failur), ar calculatd for typical composit stl bridgs, covring a rang of ralistic bridg span lngths, and th ffct of liv load incras is also invstigatd, as wll as th ffct of th corrosiv nvironmnt. 2. DEFLECTION MODEL Th analysis of tim-dpndnt dformations of statically dtrminat composit bams in bnding was prsntd by Gilbrt [4]. Th dflction limit stat dfind in th AASHTO LRFD cod (2004) only invstigats th instantanous dflctions causd by th dsign truck portion of th liv load (or 25% of th truck and lan loading, whichvr is maximum). Thrfor, no long-trm dflction du to th sustaind dad load is includd in th analysis. Th considrd dflction is th rsult of a short-trm load and th analysis includs timdpndnt variation of th variabls. Providd that th dflctions ar small and that th thory of lasticity is applicabl, th dflction at any point along th bam is obtaind by intgrating th curvatur κ(x) ovr th lngth of th mmbr, as follows ( x) dx dx v = κ (1) whr v = dflction at location x, κ(x) = curvatur at any location x along th mmbr. With an assumption of a parabolic variation of curvatur, th maximum dflction Δ occurring at midspan of a simply supportd bam (s Fig. 2) can b approximatd as 2 L Δ ( t) 96 ( κ ( t) + 10κ ( t) + κ ( t) ) A C B 2 10κ C ( t) L Δ ( t) (2) 96 whr κ A (t), κ B (t) = curvatur at supports at tim t, κ C (t) = curvatur at midspan at tim t, Δ(t) = maximum dflction at midspan at tim t, L = lngth of th mmbr.

4 HS 20 dfind by AASHTO LRFD A C B Δ Momnt Distribution Assumd Parabolic Variation of Curvatur Figur 2: Dflctd Shap of a Simply Supportd Mmbr undr Liv Load Th analysis rquird for dflction du to th short-trm liv load dos not account for th tim-dpndnt rlaxation and strss rdistribution du to crp and shrinkag of concrt. Th strains inducd by concrt rlaxation undr sustaind loading ar ignord in th prsnt analysis sinc only instantanous dflctions ar considrd. Thrfor, Eq. 2 can b rducd to its final form sinc th strains at supports, shrinkag-causd, ar ngligibl. Th curvatur κ C at any tim t causd by liv load can b dtrmind as A ( t) M κ C ( t) = (3) E ( t, τ ) 2 [ A ( t) I ( t) B ( t) ] whr M = momnt inducd by liv load, A (t) = quivalnt ara of th transformd sction at tim t, B (t) and I (t) = first and scond momnt ara of th transformd sction at tim t, E (t,τ) = ag-adjustd ffctiv modulus at tim t, τ = ag at first loading. b D c Dss D s Cntroid of Stl Girdr Sction A ss, I ss Transformd stl Sction n ss A ss, n ss I ss (a) Actual Cross-Sction (b) Transformd Cross-Sction Figur 3: Actual and Transformd Cross-Sction Bazant [5] formulatd a mthod that accountd for aging of concrt. H introducd th ag-adjustd ffctiv modulus mthod, somtims calld th Trost-Bazant Mthod. To

5 account for th aging of concrt, th ag-adjustd ffctiv modulus of lasticity is calculatd as E ( t, τ ) Ec () t ( t, τ ) φ( t, τ ) = (4) 1+ χ whr E (t,τ) = ag-adjustd ffctiv modulus of lasticity; τ = ag of concrt at th tim of loading, E c = initial modulus of lasticity of concrt, χ(t,τ) = aging cofficint, φ(t,τ) = crp cofficint. A valu of χ(t,τ) = 0.8 can b assumd for most practical cass. Aftr rviw of svral modls and thir associatd uncrtaintis, th crp cofficint is modld using th B3 modl by Bazant [6]. Its complt dscription can b found in th litratur. Crp is a complx phnomnon to modl and it offrs a grat variability in th outcom. Howvr, Bazant s formulation, through advancd analytical modling, smd to hav ovrcom most challngs. Th B3 modl lads to th most accurat rsults [6-7], spcially for long-trm prdiction (>1000 days). Th rror function Ψ 1 is dfind as a lognormal variabl with man valu of 1 and a cofficint of variation 14.6%. Th transformd sction proprtis (s Fig. 3), such as th ara, and th first and scond momnts of ara, ar dfind as follows A ( t) = bd n ( t) A ( t) (5) c + ss 2 c + ss 2 B ( t) = 1 bd n ( t) B ( t ) (6) ss ss 2 [ I ( t) A ( t D ] I = + (7) 1 3 ) ( t) 3 bdc + nss ( t) ss ss ss whr b = width of concrt slab, D c = thicknss of th slab, A ss (t) = ara of th stl girdr at tim t, B ss (t) and I ss (t) = first and scond momnts of ara at tim t, n ss (t) = ag-adjustd modular ratio at tim t dfind as Es nss ( t) = (8) E ( t, τ ) whr E s = lastic modulus of th stl girdr (supposd to b a constant). Th girdr sction proprtis ar tim-dpndnt, and thy ar rlatd to th corrosion dvlopmnt, as xplaind in th nxt sction. 2.1 Corrosion of structural stl of bridg girdrs Corrosion of th suprstructur may caus a considrabl rduction of rsistanc. Corrosion may not only caus fractur, but also yilding or buckling of mmbrs. Also of primary importanc, corrosion may induc an incras in strss, a chang in gomtric proprtis (by a dcras of sction modulus), or a buildup of corrosion products. Ths changs ar mostly associatd with a loss of matrial. Th loss can b on a local or microscopic lvl, such as pitting; or in a gnral ara, such as surfac corrosion. A consqunc of surfac corrosion is th rduction in mmbr cross sction proprtis, such as th sction modulus or th slndrnss ratio. Such proprtis ar critical in a mmbr s ability to rsist bnding momnts or axial forcs.

6 Kaysr and Nowak [8] stablishd a rlationship btwn th annual corrosion loss, th stl typ and th nvironmnt xposur, basd on th study by Albrcht [9]. This rlationship is dfind as C Ψ2 B = At (9) whr C = avrag corrosion pntration or actual corrosion loss (μm), t = numbr of yars, A and B = paramtrs dtrmind from th analysis of xprimntal data, Ψ 1 = modl rror. Tabl 1: Avrag Valus for Corrosion Paramtrs A and B, for Carbon Stl [9]. Environmnt Carbon Stl A B Rural Urban Marin Valus of A and B dpnd on th nvironmntal conditions at th bridg location, as wll as th typ of stl that is usd (carbon mostly). In this study, th typs of nvironmnt ar classifid as rural, urban, or marin. Avrag A and B valus ar summarizd in Tabl 1 for unprotctd carbon stl. Ths valus wr xtrapolatd from th rsults obtaind from tsts on small mtals spcimns placd on sit with diffrnt xposurs. Park [10] xplaind that th xtrapolation to full-siz bridgs may lad to rrors du to variation in componnt siz, orintation, and local nvironmnt. Thrfor, ths cofficints ar tratd as a random variabl by adding th rror function Ψ 1. Carbon stl was th matrial of choic in th 1960 s and 1970 s, but it rquirs painting for corrosion protction. Evn though painting provids a dcnt protction for 15 to 20 yars, th bridgs rquir rpainting aftr yars (dpnding on th Stat DOT critical thrsholds). Th dlaying ffct is shown in Figur 2. Indd, Park [10] dfind thr corrosion curvs for paintd carbon stl for thr typs of nvironmnt. Th corrosion losss ar dlayd but ultimatly follow th original curvs (dfind by Eq. 9 and Tabl 1). Indd, onc th paint cracks and starts to pl off, an acclratd corrosion can bgin. Thrfor, th rlationship btwn th rat of corrosion and tim can follow an S-shap curv. Th low corrosion rat on Fig. 2 corrsponds to a rural nvironmnt, mdium to urban, and high to marin, rspctivly. Corrosion of stl girdr occurs only on th wb and on th lowr flang. This rduction of th stl sction is takn into account in th prsntd analysis. Also, aftr rviw of th rlvant litratur [10, 11, 12], th rror function 2 is dfind as a lognormal variabl with th man valu of 1.0 and cofficint of variation of 20%.

7 Figur 4: Effct of Painting on th Thicknss Loss Du to Corrosion for Carbon Stl 2.2 Random paramtrs In th dflction analysis considrd, th random paramtrs can b groupd into thr catgoris such as (1) matrial proprty paramtrs, (2) modling rrors, (3) applid load paramtrs. Th dimnsional proprtis ar tratd as dtrministic valus sinc thir associatd cofficints of variation ar usually small and ngligibl. Matrials paramtrs such as concrt strngth f c, and concrt modulus of lasticity E c ar considrd. Th modling rrors hav bn introducd prviously for th crp cofficint and corrosion prdictions. Th applid load paramtrs considr th liv load only, sinc it govrns in th short-trm load analysis. Th concrt modulus of lasticity E c is dscribd as a random variabl, but it is ssntially rlatd to th concrt comprssiv strngth by th following formula Ec = 4734 f c (10) whr f c = concrt comprssiv strngth at 28 days (MPa). Thrfor, th variation of th concrt modulus of lasticity is closly rlatd to th concrt comprssiv strngth. Ths random variabls accounts for th variability inhrntly associatd with practical applications. Th rlvant statistical paramtrs such as th bias factor, man valu, and standard dviation ar summarizd in Tabl 2. All th prsntd random variabls hav lognormal distributions. In Tabl 2, th man valu of th truck loading of th liv load is dfind as (0.78GDF*1.79(M HS20 )). In his study, Eom [14] found that th girdr distribution factor dfind by AASHTO can b vry consrvativ. Basd in xtnsiv xprimnts, a closr valu of th GDF is 0.78*GDF givn by AASHTO (2004). Also, a study mandatd by th Transportation Rsarch Board in 1999 [15] has shown that th adquat bias factor for momnt causd by HS20 truck loading is It is includd in th man valu for variational purpos (s Fig. 7).

8 Tabl 2: Summary of Statistical Paramtrs. Random variabl Bias Factor Man Valu Cofficint of Variation Rfrnc Matrial Proprtis Concrt comprssiv strngth f c (MPa) % Ellingwood [13] Concrt modulus of lasticity E c (MPa) - Eq Modling rrors Crp modling rror Ψ % Bazant [7] Corrosion modling rror Ψ % [ ] Applid Loads Liv Load (HS 20 Truck) 1.0 (0.78GDF) *1.79*M HS % [14 15] 3. DEFLECTION SERVICEABILITY FAILURE Excssiv dflction may caus damag to th concrt dck by tnsion cracking, buckling of th stl girdr, or discomfort to th usrs. AASHTO LRFD (2004) rcommnds an allowabl dflction limit. This limit stat is optional and up to th ownr of th structur during th dsign. Th allowabl dflction limit is dfind as L/800, whr L = span lngth. Th dflction causd by th dsign truck is to b chckd. Howvr, as statd prviously, th ral momnt obsrvd during bridg opration (undr ral traffic) can b in fact 1.78 tims largr than that corrsponding to th dsign momnt calculatd at th dsign stag. Thrfor, an allowabl dflction limit stat of L/400 is adoptd in this study. Th srvic limit stat is givn by L G( t) = Δ( t) (11) 400 whr L = span lngth of th bridg, Δ(t) = maximum midspan dflction at tim t. Mont Carlo simulation is usd to gnrat sts of random variabls (1,000,000 for ach random variabl, for ach tim intrval of 1 yar) that hav bn dfind in Tabl 2. A midspan dflction is calculatd for ach run. Th probability of srvicability failur is th probability of xcding th adoptd allowabl dflction limit, and it is dscribd as follows p f [ ( t) 0] n G ( t) = (12) n t whr n[g(t) 0] = numbr of runs whn th dflction xcds th allowabl limit, n t = total numbr of simulations for ach tim incrmnt. Th associatd rliability indx β(t) is dfind as [ p ( t) ] 1 β ( t) = Φ (13) f whr β(t) = rliability indx at tim t, p f (t) = probability of failur at tim t, Φ -1 = invrs standard normal distribution function.

9 4. EXAMPLES OF APPLICATION Th structurs considrd ar simply supportd composit stl girdr bridg bam lmnts. A varity of lngth and corrosion nvironmnts ar considrd. Thy hav bn dsignd following th AASHTO LRFD (2004) dsign cod (U.S. Units). For convninc, Tabl 3 summarizs th bam slction for all cass considrd. Tabl 3: Summary of Invstigatd Sctions. Span (m - ft) Girdr Spacing Stl Sction W24x55 W30x90 W33x130 W40x167 W40x215 W44x262 Th dtrministic matrial paramtrs includd in th analysis ar: thicknss of th concrt slab h s = cm (9 in.) no asphalt covr; Typ I cmnt; ag at loading τ = 28 days; ag whn drying bgins τ = 28 days; rlativ humidity H = 70% (typical in Michigan, Unitd Stats); cmnt contnt = 360 kg/m 3 ; watr/cmnt ratio = 0.45; aggrgat/cmnt ratio = 5.0. Th influnc of th corrosion nvironmnt is shown in Fig. 5. Th rliability indx drops dramatically whn th nvironmnt bcoms harshr. Typical snow blt stats such as Michigan hav an nvironmnt clos to th marin xposur (bcaus of xtnsiv us of salt as a dicing agnt in wintr months), which is vry damaging for th structurs. Th ffcts of painting can also includd. Fig. 6 illustrats th rliability indx volution as a function of th bridg lngth. Th AASHTO LRFD (2004), associatd with th slctd allowabl dflction limit, provids a consistnt targt valu btwn 3.5 and 4.2, dpnding of th dgr of ovrdsign in th sction slction. A bias factor snsitivity analysis is prformd, as shown in Fig. 7, to dmonstrat its ffct on th tim-dpndnt rliability indx. In cas of th growth of liv load, or incras of th bias factor (ratio of actual ovr dsign load), thr is a drastic drop in th rliability indx (as sn in Fig. 7). For xampl, aftr 100 yars, if th bias factor is incrasd to 1.3, thn th rliability indx β = 0.85, corrsponding to th probability of failur of 20%.

10 Figur 5: Rliability Indx Evolution of W90x30 for Various Environmnt Exposurs, (Span = 28.3 m, 60 ft; Girdr Spacing = 2.43 m, 8 ft) Figur 6: Rliability Indx Evolution in Function of th Bridg Lngth for Urban Environmnt (Girdr Spacing 2.43 m, 8 ft)

11 Figur 7: Effct of th Bias Factor on th Rliability Indx for Urban Environmnt (Span = 28.3 m, 60 ft; Girdr Spacing = 2.43 m, 8 ft) 5. CONCLUSIONS A probabilistic tim-dpndnt dflction modl for simply-supportd composit stl girdr bridgs is prsntd. Diffrnt corrosion nvironmnts ar considrd and corrosion modling is includd in th dtaild analytical solution. An allowabl dflction limit of L/400 is adoptd and provids an accptabl rang of initial rliability indics for th considrd structurs. Th tim-dpndnt rsults suggst that th dflction volution ovr th yars can b of major concrn, vn ignoring th long-trm ffcts du to sustaind loading (dad load). Th influnc of th bias factor of th liv load has bn invstigatd and suggsts that th incras in traffic loads should b monitord closly in ordr to account for th volution of liv load. REFERENCES [1]. FHWA, FHWA bridg programs matrial typ of structur by stat, [2]. Koch G. H., Brongrs M. P., Thompson N. G., Virmani Y. P, Payr J. H., Corrosion cost and prvntiv stratgis in th Unitd Stats, Rport No. FHWA-RD , Offic of Infrastructur Rsarch and Dvlopmnt, Fdral Highway Administration, McLan, Va, [3]. Salt Institut, U.S. Salt Sals, [4]. Gilbrt R. I., Tim ffcts in concrt structurs, Elsvir, Amstrdam, [5]. Bazant Z. P., Prdiction of concrt crp ffcts using th ag-adjustd ffctiv modulus mthod, ACI Journal, Vol. 69, April 1972, pp [6]. Bazant Z. P., Bawja S., Crp and shrinkag prdiction modl for analysis and dsign of concrt structurs: Modl B3, RILEM Rcommndations, Matrials and Structurs, Vol. 28, 1995, pp

12 [7]. Fabourakis G. C., Ballim Y., Prdiciting crp dformation of concrt: a comparison of rsults from diffrnt invstigations, Procdings of th 11 th FIG Symposium on Dformation Masurmnts, Santorini, Grc, [8]. Kaysr J. R., Nowak A. S., Rliability of corrodd stl girdr bridgs, Structural Safty, Vol. 6, 1989, pp [9]. Albrcht P., Nami A. H., Prformanc of wathring stl in bridgs, National Cooprativ Highway Rsarch Program, Rport 272, July [10]. Park C. H., Nowak A. S., Das, P. C.,Flint A. R.,1998, Tim-Varying Rliability Modl of Stl Girdr Bridgs, Procdings Institution of Civil Enginrs, Structurs and Buildings, Thomas Tlford Publishing, Vol. 128, Issu 4, Nov. 1998, pp [11]. McCun R. H., Albrcht P., Composit modling of atmosphric corrosion pntration data, Application of Acclratd Corrosion Tsts to Srvic Lif Prdiction of Matrials, ASTM STP 1194, Gustavo Cragnolino and Narasi Sridhar, Eds. Amrican Socity for Tsting and Matrials, Philadlphia, 1994, pp [12]. Mlchrs R. E., Corrosion uncrtainty modling for stl structurs, Journal of Constructional Rsarch, Vol. 52, 1999, pp [13]. Ellingwood B., Galambos T. V., McGrgor J. G., Cornll C. A., Dvlopmnt of a probability basd load critrion for Amrican national standard A58, National Burau of Standards, NBS Spcial Publication 577, Washington D.C., [14]. Eom J., Vrification of th bridg rliability basd on fild tsting, PhD Dissrtation, Univrsity of Michigan, [15]. Nowak A. S., Calibration of LRFD Bridg Dsign Cod, National Cooprativ Highway Rsarch Program, NCHRP Rport 368, TRB, 1999.