6 (200

twati

an

East

Univ

ised f

brid

nationwide bridge replacements and rehabilitations are

for structural safety and serviceability for an existing

ing into account the live load increase and material

rating of the bridge. Rating factors for dierent dete-

ces for dierent deteriorating bridgthe same failure m

oad model and deterg author. Tel.: +1-303-4steel are integratess: dan.frangopol@colorad Correspondin7317.

E-mail addre0141-0296/$ - see front matter# 2004 Elsedoi:10.1016/j.engstruct.2004.06.012calculated forpose, a live lconcrete and

92-7165; fax: +1-303-492-

o.edu (D.M. Frangopol).vier Ltd. All rights reserved.stem reliability indi-e types must also beodes. For this pur-ioration models ford into a speciallymembers, or addition failure modesment work, change in strength ofriorating bridge types have to be calculated for various

. Consequently, the sybridge is determined when a maintenance, improve-based on ratings listed in the National Bridge Inven-tory (NBI) database. For distribution of funds, High-way Bridge Repair and Replacement Program(HBRRP) uses the so-called suciency rating, which iscalculated by a formula incorporating structural safety(55%), serviceability and functional obsolescence(30%), and essentiality for public use (15%) [14]. Pos-sessing the highest weight in suciency rating formula,load rating for structural safety is a crucial measure forbridge management and decision making. Load rating

deterioration models, are more commonly used for life-time bridge assessment.For the design of new bridges, the AASHTO LRFD

[1] Specications provides the necessary provisions forobtaining a uniform safety level for bridge compo-nents. Once a bridge is designed and placed in service,the AASHTO Manual for Condition Evaluation ofBridges [2] provides provisions for determination of thesafety and serviceability of existing bridge components.The minimum of the component ratings determines thein the National Bridge Inventory database. Distribution of funds is based on the suciency rating, represented by a formula con-sidering structural safety, functional obsolescence, and essentiality for public use. Possessing the highest weight in suciency rat-ing formula, load rating is a crucial measure for bridge management. While load rating represents the current practice in bridgeevaluation, reliability methods, taking into account live load increase and material deterioration models, are more commonly usedfor lifetime bridge assessment. In this paper, time-dependent relationship between the reliability-based analysis results, represent-ing the future trend in bridge evaluation, and the load ratings is investigated for dierent types of bridges located within an exist-ing bridge network. The comparisons between live load rating factors and reliability indices are made over the lifetime of eachbridge in the network. The ratingreliability prole and ratingreliability interaction envelope concepts are introduced. Further-more, the ratingreliability proles are collectively examined in order to evaluate the time-dependent performance of the overallbridge network.# 2004 Elsevier Ltd. All rights reserved.

Keywords: Bridges; Bridge network; Deterioration; Rating; Reliability; Structures; System reliability; Time-dependent performance

1. Introduction

Prioritization and allocation of federal funds for

of dead load alters the condition or capacity of thestructure. While load rating represents the currentpractice in bridge evaluation, reliability methods, tak-Engineering Structures 2

Time-dependent interaction beof deterior

Ferhat Akgul a, Da Department of Engineering Sciences, Middle

b Department of Civil, Environmental, and Architectural Engineering,

Received 8 December 2003; received in rev

Abstract

Prioritization and allocation of federal funds for nationwide4) 17511765www.elsevier.com/locate/engstruct

een load rating and reliabilityng bridges

M. Frangopol b,

Technical University, 06531 Ankara, Turkey

ersity of Colorado, Campus Box 428, Boulder, CO 80309-0428, USA

orm 12 June 2004; accepted 18 June 2004

ge replacements and rehabilitations are based on ratings listed

portation (CDOT) consists of a letter followed by a

of each bridge in the network. This is followed bydetermination of initial component and system

1752 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765Fig. 1. Bridge network.reliability indices based on system failure models.Detailed descriptions of these procedures are presentedin Akgul and Frangopol [4,6]. Finally, in order to cal-culate the rating factors and reliability indices forbridge components and systems over time, time-depen-dent live load model and corrosion models for concreteand steel are adopted. The theoretical bases for thesemodels are given below. In order to explain how theresults are obtained for a typical bridge, as a represen-tative example, the results of the live load model, cor-rosion deterioration model, and time variation of thereliability index and the rating factor for the compo-nents of bridge E-17-LE are also presented in the fol-lowing sections.

3.1. Time-dependent live load model

Live load on a bridge is a function of many para-meters, such as truck weight, axle loads, axle congur-ation, span length, position of vehicles (longitudinal,one to two digit number describing the vertical andhorizontal coordinates of the bridge, respectively, on astatewide grid. The last two letters are the uniqueidentication symbol for the bridge [10].Characteristics of the network bridges are listed in

Table 1 (see also [4]). The bridge network consists ofseven prestressed girder, three steel rolled I-beam, andfour combined welded steel plate and reinforced con-crete girder bridges. Table 1 lists data such as thelength and width of each bridge in addition to the yearin which it was built. The lengths range between 34.1and 82.3 m. The oldest and newest bridges in the net-work were built in 1951 and 1995, respectively, repre-senting a 44-year span between their constructions.

3. Bridge live load, deterioration, and system

reliability

Time-variation of the rating factor and the reliabilityindex are determined based on the fact that while thelive load is increasing due to larger number of truckspassing on the bridge, cross sectional areas of steelreinforcement and structural steel in bridge decks andcomponents will be gradually decreasing starting fromthe onset of corrosion on their surface. A step-by-stepprocess is used to calculate the time-variation of therating factor and the reliability indices. First, initialrating factors are calculated for each bridge componentbridges located within close proximity of each other,preferably along interstate highways, and within thesame transportation and maintenance regions. Bridgedesignation used by Colorado Department of Trans-developed computer program [5]. This program is usedto determine the lifetime reliability proles, while life-time rating calculations are performed separately.Time-dependent relationship between the reliability-

based analysis results, representing the future trend inbridge evaluation, and the load ratings, reecting thecurrent practice in bridge evaluation, is investigatedherein for existing bridges located within a bridge net-work. The comparison between live load rating factorsand reliability indices are made over the lifetime ofeach bridge in the network. The so-called ratingreliability prole and ratingreliability interactionenvelope concepts are introduced. Furthermore, theratingreliability proles are collectively examined inorder to evaluate the time-dependent performance ofthe overall bridge network.

2. Bridge network

The bridge network shown in Fig. 1 [3] is located atnorthwest corner of Denver metropolitan area, andconsists of 14 mixed type highway bridges. This net-work was described in Akgul and Frangopol [4,6]. Theselection criteria for the bridge network was to choose

bridges. Other primary causes for bridge deterioration

in Fig

ength

)

.1

.9

.5

.1

.3

.6

.7

.2

.0

.3

.6

.5

.0

.7

uous

ompo

F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765 1753transverse), number of vehicles, speed of vehicles, sti-

ness of superstructure, and bridge geometry (straight,

curved) [16]. Site specic trac data for such para-

meters is generally collected using weigh-in-motion

(WIM) studies. WIM involves recording weights of

trucks and bridge deections by means of sensors

attached to bridge deck and girders. The results are

used to quantify the load and resistance values such as

actual load eects and girder distribution factors for

the bridge. Subsequently, actual recorded eects are

generally compared to values specied by current

bridge design codes.In this study, the load model developed for the

AASHTO LRFD Bridge Design Specications [1] is

Table 1

Characteristics of 14 Colorado highway bridges in the network shown

Bridge

name

Bridge

type

CDOT

designation

Number

of spans

L

(m

E-16-MU Prestressed CPG 1 34

E-16-LA Prestressed CBGCP 2 77

D-16-DM Prestressed CPGC 2 44

E-16-QI Prestressed CBGCP 2 74

E-16-LY Prestressed CPGC 3 74

E-16-NM Prestressed CPGC 2 64

E-17-MW Prestressed CIC 2 72

E-16-FK Steel I-beam CIC 4 69

E-16-FL Steel I-beam CIC 4 54

E-16-Q Steel I-beam CIC 5 82

E-17-LE Steel plate girder WGCK 4 68

E-17-HS Steel plate girder WGCK 4 64

E-17-HR Steel plate girder WGCK 4 64

E-17-HE Steel plate girder WGCK 4 67

CPG, concrete prestressed girder; CBGCP, concrete box girder contin

crete on rolled I-beam continuous; WGCK, welded girder continuous cused. The extrapolation for the load model used herein

standard AASHTO truck load. A drastic increment inmay be the cracking, dislocation at supports, bearing

damage, excessive vibration, delamination of decks,

and heavy truck use. This study focuses on deterio-the rst few years is followed by a gradual increase

over the lifetime of the bridge.

3.2. Deterioration models for concrete and steel

Two essential materials used for bridge construction,

concrete and steel, are both aected by environmental

stressors. They deteriorate progressively when exposed

to atmosphere and chlorides. Deterioration of these

materials is one of the major causes of deterioration of

. 1 (see also [4])

Width Year built ADTT

(trucks/day)(ft) (m) (ft)

112.0 11.6 38.0 1994 810

255.5 39.2 128.5 1983 450

146.0 14.2 46.5 1990 390

243.2 30.7 100.7 1995 1335

243.7 34.1 112.0 1985 1610

212.0 28.0 92.0 1991 2955

238.6 30.5 100.0 1987 230

227.0 10.4 34.0 1951 1370

177.0 10.4 34.0 1951 765

270.0 12.2 40.0 1953 890

225.0 19.7 64.5 1972 992

211.7 10.4 34.0 1963 5

209.8 10.4 34.0 1962 306

222.2 10.4 34.0 1962 1290

prestressed; CPGC, concrete prestressed girder continuous; CIC, con-

site.ration due to corrosion only, since time variant models

d relative humidity),

sure nitia

orientation, angle of exposure, time of wetness, atmos-the multiplier (uncertainty factor) for exural moment

in slab is shown for the 75-year time period for the pheric pollutants, deicing salt, and debris) [7,12]. ForFig. 2 demo

slab of bridgenstrates how this

E-17-LE. In thismodel is used

gure, the varfor the

iation of and expoconcrete deteconditions (irioration model, Fl climate, sicks secoheltering,corrosion include temperature an

initial time t 0, i.e. due to a single truck. atmosphere (environmental conditions aecting steel

standard deviation of maximum moment or shear at on the metal (composition of alloys in metal), local

ber (0.577216), and lX and rX are the mean value and action. Severity of steel corrosion, in general, depends

of Type I largest value distribution, c is the Euler num-

where un and an are the location and scale parametersalkali-silica reaction, sulphate attack and freezethawfrequently observed, followed by deterioration due to6an by carbonation and chloride contamination are most

rYn p rX 2 Refs. [8,9,13,15]. For steel corrosion, the types causedpYn X n X anXmodels for steel corrosion in concrete can be found in

l l u r c r 1 this eld. For instance, treatment of various types ofvarious models have been developed by researchers intime, corresponding to a sample size of n trucks:moments and shears in bridge components at a future For deterioration of concrete and steel in bridges,is based on the mean value lYn and standard deviationrYn of Type I largest value asymptotic distribution for

for other deterioration types have not yet been fully

developed in this eld.nd law of

Fig. 2. Time-variation of the mean of the maximum moment in the slab of bridge E-17-LE.

1754 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765diusion due to chloride is used:

@C

@t Dc @

2C

@x23

where C is the concentration of chloride ions, Dc is theeective diusion coecient, t is time, and x is the dis-

tance from outer surface of the solid.

Fig. 3. Time-variation of the slab reinModels developed to predict time-dependent cor-

rosion penetration in steel are usually empirical for-

mulas intending to capture the actual corrosion

process. They generally include the time variable

accompanied by several regression coecients in the

form of a power formula. In this study, a power func-forcing area for bridge E-17-LE.

tion is em

p botb1where boyears, resmic transThe d

second lacrete deckconcretesteel (4) ithe applicthe slabdom vartime foryears. Withe corrorcorr indic

ated forstructureders) areple, theridge E-for the

and (b),

e modele bridgean pre-is con-

s, exureons arebridges,ed bothre of thethe slab

tem f

F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765 1755and employs the First Order System ReliabilityMethod to determine the reliability proles for bridgemembers and systems. System reliability analysis of thebridges required the description of a failure model for

Fig. 4. (a) Girder layout, (b) systributions of the slab reinforcement area are plotted at10-year intervals using the Monte Carlo simulationmethod. The prole shown represents the mean valueof these distributions over time.

3.3. System reliability model

Once the models for live load and material deterio-ration (for dierent bridge member types) were ident-ied, they were implemented as program modules intothe network-level lifetime system reliability programRELNET (reliability of system networks) [5].The program uses the Monte Carlo simulation tech-

nique to calculate the resistance degradation proles

the network. As representative examples, time variationof the reliability index and the operating rating factorfor the slab and girder of bridge E-17-LE are presentedin Figs. 5 and 6. Once these proles are obtained foreach bridge in the network, the results of time-dependent rating calculations and reliability proles are

ailure model for bridge E-17-LE.or exural or shear failure of two adjacent girders.When calculating the system reliability index for thebridge superstructure, a correlation coecient of 0.5 isassumed between girder resistances, representing themidpoint between the ideal cases of independence andperfect correlation.

4. Time-dependent rating factor vs. reliability index

for individual bridges

The proles for resistance degradation and reliabilityindex are systematically generated for each bridge inployed for corrosion of structural steel [11,17]

4and p are the corrosion losses after one and tpectively, and b1 is the slope of the logarith-formation of (4).eterioration model associated with Ficksw of diusion (3) is used for reinforced con-s, reinforced concrete girders, and prestressedgirders. Similarly, the corrosion model fors applied to steel girders. Fig. 3 demonstratesation of the concrete deterioration model forreinforcement for bridge E-17-LE. The ran-iable Tisr, indicating the corrosion initiationslab reinforcement has a mean value of 3.61th the lognormal (LN) density distributions ofsion initiation time Tisr and corrosion rateated in the gure, the probability density dis-

each bridge type. Since bridges are generally rsuperstructure components, only the supermembers of the bridges (i.e., slab and the girconsidered in this investigation. As an examcross sectional view of the superstructure of b17-LE and the corresponding failure modelsuperstructure are displayed in Fig. 4(a)respectively.The topology of series-parallel system failur

for the superstructure changed depending on thtype. For instance, for girders of simple spstressed bridges, exure of girder at midspansidered while for continuous prestressed girderat both midspan and pier support locatiincluded in the system failure model. For steelsuch as E-17-LE, girder failure modes includexure and shear. As shown in Fig. 4(b), failusteel bridge is dened as the exural failure of

Fig. 5. Time-variation of the reliability index for the slab and girder of bridge E-17-LE.

1756 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765combined, emphasizing the interaction between rating

and reliability proles.Time variation of the operating rating factor RF(t) vs.

system reliability index b(t) for several representative

bridges of dierent types in the network are presented inFig. 6. Time-variation of the operating rating factFigs. 713. Network bridges include a variety of girder

types such as the prestressed concrete, reinforced con-

crete, steel rolled I-beam, and steel welded plate girder

bridges. Girder type is indicated in each graph. Bridge

operating rating factor is plotted against the systemor for the slab and girder of bridge E-17-LE.

F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765 1757reliability index since the operating rating and the

reliability index are representative of the overload con-

dition and the safety of a bridge, respectively. Operating

rating factor for a bridge indicates the maximum per-

missible live load value that the bridge can carry. HavingFig. 8. Time-dependent operating rating vsa large number of vehicles at operating rating weight tra-

veling over the bridge may shorten the life of the bridge.Time-dependent rating factor vs. reliability index

graphs have inverted horizontal axis for the reliability

index values. Thus, the upper-left hand region in theseFig. 7. Time-dependent operating rating vs. reliability index for bridge E-16-MU.. reliability index for bridge D-16-DM.

Fig. 9. Time-dependent operating rating vs. reliability index for bridge E-17-MW.

1758 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765graphs represents the highest rating and reliability,while lower-right hand region represents the lowest rat-ing and reliability for a bridge component or system.Therefore, over time, the point representing bridge

rating factorreliability index pair is expected to move

Fig. 10. Time-dependent operating rating vfrom upper left to lower right hand corner in all graphsdue to member deterioration and increase in loadeects due to trac.Since the theoretical basis used in these graphs is thesame, only the graph for bridge E-16-MU, shown ins. reliability index for bridge E-16-FK.

Fig. 11. Time-dependent operating rating vs. reliability index for bridge E-17-LE.

F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765 1759Fig. 7, will be explained herein. This will also serve as a

description of the format used for the other graphs.

For the prestressed concrete bridge E-16-MU, Fig. 7Fig. 12. Time-dependent operating rating vs. reliability inshows three plots: slab exure, girder exure and the

system ratingreliability interaction curve denoted by

the label (RF(t) , b(t) ). This designation indicatesbridge sysdex for slab and concrete girders of bridge E-17-HS.

bility

1760 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765that the time-dependent rating factor RF(t)bridge is forthe bridge, representing the minimum of all superstruc-ture component ratings, and the time-dependent

Fig. 13. Time-dependent operating rating vs. reliareliability index b(t)sys is for the system representing theseries-parallel system failure model for the bridge

ingreliability pand system ratigure are overlvalues for the bt 0, are not splayed considerasingle average tringreliability cuto immediate apthe graphs startrst month, reexposure for theSince the sud

values due to inded, the remainles shown in tdeterioration, alto trac is stillmonth, the pointhe bridge E-16nent ratingreliatime domain hidseparate points in time are displayed consistently for allbridges: 1, 6 month, 1, 10, 20, 30, 40, 50, 60, and 75years. For bridge E-16-MU, these points are clearlyvisible in both the slab and the girder proles. How-

index for slab and steel girders of bridge E-17-HS.ever, for several other bridges, it was necessary to dis-play a few representative points instead of all 10 points

liability graphsrvations can beehavior, there-nt bridge typesm a broad per-liability curvesroup type. Forthe prestressedE-16-MU and

y, are generallyability domain.irders over pierity values areE-17-MW (seetressed concretegreliability fory the extensionexure toward

-17-LE, shownery low ratingich is indicatedince the slab exure controls the rat-role for the bridge, the slab exurengreliability interaction curves in thisapped. The rating vs. reliability indexridge components at initial time, i.e.hown in these graphs since they dis-bly higher reliability indices due to auck and revealed sudden drops in rat-rves at initial stage of service life dueplication of the trac load [4]. Instead,from a time, chosen as the end of thepresenting a reasonable trac loadbridge.den initial drop in ratingreliabilityitial application of trac load is exclu-ing reduction in ratingreliability pro-hese gures is mainly due to materialthough increase in live load eect duea contributing factor. Starting from 1ts in time until 75 year life period for-MU are indicated along each compo-bility prole. In order to reveal theden in these graphs, the following 10

mentioned above.By comparing the lifetime ratingre

for dierent bridges, the following obsemade: each bridge displayed a unique bfore, distinct generalizations for dierewould not be appropriate. However, frospective, general location of ratingreshow similarities within a given bridge ginstance, ratingreliability curves forconcrete bridges, as shown for bridgesD-16-DM in Figs. 7 and 8, respectivellocated near the middle of ratingreliFor exure of continuous prestressed glocation, higher rating and reliabilobserved as shown in Fig. 8. BridgeFig. 9), however, although being a presbridge, displays a relatively low ratinthe exure of slab which is indicated bof the ratingreliability curve for slabthe lower-right corner of the graph.Steel bridges such as E-16-FK and E

in Figs. 10 and 11, respectively, have vreliability values for exure of slabs whsuperstructure. Sby the extension of the ratingreliability prole toward

the lower-right c

period, slab rat

reach the lowe

bridges. This is d

the oldest ones

quently, they

strength reinforcBridges with s

E-17-HS shown

vior similar to th

arate graphs are

concrete and steIn general,

reliability prole

nent having th

reliability. For r

since the bridge

minimum com

reliability point

ingreliability p

reliability analy

used.Bridge rating

determined considering both the slab and girder rat-

ings. Consequently, following the same process, the

nteraction

he network arependent bridgestem reliability17, The time-he interaction

ting factors forly, is shown ins are listed inperating bridge, and for 1, 20,gure enablesctor values fortheir lifetimes.stressed girdertween 1 monththe other hand,ges is reectedg factors for a

liability indicesfor all network bridges, based on girders only, is shownin Fig. 15, and the corresponding values are listed in

erati

F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765 1761system reliability indices in these graphs were calcu-

lated using the system failure model including both

slab and girders.

Fig. 14. Time-variation of the bridge opTable 3. Using this gure, similar observations andcomparisons can be made for the network bridgesbased on system reliability indices. The gure displays

ng rating factor for the network bridges.orner of the graph. Within 1040-year

ingreliability proles of these bridges

st observed value for the network

ue to the fact that the steel bridges are

among the network bridges. Conse-

were built with slabs having lower

ing steel.teel and concrete girders such as bridge

in Figs. 12 and 13, displayed a beha-

at of the steel bridges. For clarity, sep-

provided in these gures for reinforced

el girders of the bridge E-17-HS.the position of the system rating

was close to the prole of the compo-

e minimum rating and the lowest

ating, this is a reasonable performance

rating in practice is controlled by the

ponent rating. However, from the

of view, the position of the system rat-

role reected the result of the system

sis based on the system failure model

factors in ratingreliability graphs were

5. Time-dependent ratingreliability i

at network-level

In Figs. 14 and 15, all bridges in tplotted together, showing the time-deoperating rating factors and the syindices, respectively. In Figs. 16 anddependent network results show tbetween rating and reliability proles.The variation of operating bridge ra

all network bridges, based on girders onFig. 14, and the corresponding valueTable 2. In Fig. 14, the variation of orating factors are indicated for 1 month50, and 75 years for each bridge. Thiseasy comparison of the bridge rating faall bridges in the network throughoutLarge lifetime deteriorations for prebridges are characterized by big gaps beand 75 year marks for each bridge. Onsmall deterioration for steel girder bridby very small vertical gaps among ratingiven bridge.Similarly, the variation of system re

Fig. 15. Time-variation of the system reliability index for the network bridges.

1762 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765the snapshots of system reliability indices for allbridges at various discrete points in time.Figs. 16 and 17, on the other hand, provide a sum-mary of the ndings reported in Figs. 14 and 15 in

Fig. 16. Time-variation of the bridge operating rating factor vs. systemterms of bridge operating rating factor vs. systemreliability index for all bridges in the network consider-ing slab and girders, and girders only, respectively. In

Figs. 16 and 17, rating reliability envelopes are shownreliability index for the network bridges based on slab and girders.

Fig. 17. Time-variation of the bridge operating rating factor vs. system reliability index for the network bridges based on girders only.

F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765 1763for the network bridges both for time t 1 month andfor the lifetime interaction proles. The envelope for

time t 1 month, represented by the square region, isa snapshot of ratingreliability values for all bridges inthe network at that initial time. The remaining data

Table 3

Time-variation of th

Time b(t)sys

E-16-

MU

1 month 3.73

1 year 3.48

20 year 3.22

50 year 2.67

75 year 1.81

Table 2

Time-variation of th

Time RF(t)OPE,

E-16-

MU

1 month 1.565

1 year 1.442

20 year 1.328

50 year 1.202

75 year 0.983points represent the ratingreliability values of the

bridges at subsequent discrete points in time through-

out the lifetime of each bridge. Therefore, the network-

level time-dependent ratingreliability interactionenvelope, represented by upper and lower bounds,

E-17-

HR

E-17-

HE

3.00 3.46

2.70 3.20

2.07 2.89

0.89 2.03

0.08 0.88

E-17-

HR

E-17-

HE

0.802 0.835

0.736 0.773

0.673 0.711

0.507 0.675

0.377 0.479e system reliability index for the network bridges

E-16-

LA

D-16-

DM

E-16-

QI

E-16-

LY

E-16-

NM

E-17-

MW

E-16-

FK

E-16-

FL

E-16-

Q

E-17-

LE

E-17-

HS

3.28 4.04 4.07 2.59 3.92 3.40 3.33 2.67 2.66 3.47 3.94

3.06 3.75 3.84 2.34 3.67 3.13 3.04 2.36 2.35 3.15 3.54

2.82 3.18 3.59 2.08 3.40 2.85 2.74 2.03 2.02 2.79 2.76

2.24 2.06 2.87 1.95 2.43 2.74 2.65 1.93 1.92 2.68 1.54

1.32 1.35 1.81 1.48 1.34 2.45 2.60 1.89 1.87 2.62 0.56

e bridge operating rating factor for the network bridges

bridge

E-16-

LA

D-16-

DM

E-16-

QI

E-16-

LY

E-16-

NM

E-17-

MW

E-16-

FK

E-16-

FL

E-16-

Q

E-17-

LE

E-17-

HS

1.495 1.595 1.793 1.143 1.616 1.469 0.870 0.763 0.762 0.939 0.949

1.370 1.467 1.656 1.060 1.500 1.339 0.807 0.706 0.705 0.865 0.883

1.256 1.276 1.529 0.982 1.389 1.225 0.746 0.651 0.649 0.790 0.754

1.097 0.953 1.355 0.952 1.184 1.190 0.729 0.635 0.633 0.767 0.563

0.828 0.786 1.063 0.877 0.895 1.133 0.721 0.627 0.626 0.756 0.433

(1) Lifetime rating and reliability analyses for dier-

Consequently, based on the time-dependent rating

1 to 0.75, and 93% for reliability, i.e. from 4.1 to 0.25).

1764 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765ent bridge types in an existing network have to incor-porate time-dependent models for both live loadincrease and resistance deterioration. A live load modeland separate deterioration models for concrete and steeldue to environmental stressors, such as chloride ingress,for dierent member types were investigated and theintegration of these models into a network-level, time-dependent, system reliability analysis program isaccomplished. When updated with eld data, such pro-grams can become highly crucial tools for lifetimebridge evaluations in future practice, and can be furtherimproved to aid maintenance and repair decisions.(2) Currently, bridge ratings are calculated at dis-

crete and irregular time periods determined by changesin loading and/or capacity of the structure. Althoughrecent load and resistance factor rating procedures aredeveloped with the aim of incorporating time-depen-dent changes in structural loads and resistance in theform of load and resistance factors, live load increaseand structural deterioration are not directly taken intoconsideration during the rating process (i.e., usingsound and proven analytical and experimental models).This study demonstrated that it is possible to predictthe load rating and reliability index of a bridge usinglive load and resistance deterioration models integratedinto a single computational platform.shows the safety of the network bridges throughouttheir lifetime.Fig. 16 revealed an extremely close behavior for all

bridges considering the lifetime system ratingreliabilityproles. Although having unique initial ratingreliability values slightly dispersed at the beginning,ratingreliability values for the network bridges, basedon both the slab and the girders, followed almost thesame path when member degradation and live loadincrease were taken into account. A lifetime ratingreliability interaction envelope is also dened for thenetwork bridges as indicated in Fig. 16. Based on thisgraph, it is possible to conclude that the networkbridges show variability in system ratingreliabilityvalues initially, i.e. t 1 month, however, after the rstmonth of trac, the values for all bridges on the aver-age gradually converge.A similar behavior is also observed in Fig. 17, where

the ratingreliability values are based on girders only.In this case, however, there is a visible distinction,characterized by a wide gap between lifetime ratingreliability proles of prestressed and steel bridges. Pre-stressed bridges converged toward rating and reliabilityindex values signicantly larger than those associatedwith steel bridges.

6. ConclusionsSince system reliability index evaluates the actual safety

of the structure, while rating factor reects the live load

capacity only, such a drastic reduction in actual safety

of the structure deserves more attention as compared to

the reduction in live load capacity. Therefore, it may be

more appropriate to base lifetime bridge evaluation on

reliability index rather than the load rating.

Acknowledgments

The partial nancial support of the US National

Science Foundation through grants CMS-9912525 and

CMS-0217290 is gratefully acknowledged. The support

provided by the Colorado Department of Transpor-

tation is also gratefully acknowledged. The opinions

and conclusions presented in this paper are those of the

writers and do not necessarily reect the views of the

sponsoring agencies.

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F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 17511765 1765

Time-dependent interaction between load rating and reliability of deteriorating bridgesIntroductionBridge networkBridge live load, deterioration, and system reliabilityTime-dependent live load modelDeterioration models for concrete and steelSystem reliability model

Time-dependent rating factor vs. reliability index for individual bridgesTime-dependent rating-reliability interaction at network-levelConclusionsAcknowledgmentsReferences