Time-dependent interaction between load rating and reliability of deteriorating bridges

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7317.

E-mail address: dan.frangopol@colorad

0141-0296/$ - see front matter # 2004 Else

doi:10.1016/j.engstruct.2004.06.012

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vier Ltd. All rights reserved.

Engineering Structures 26 (2004) 1751–1765
www.elsevier.com/locate/engstruct
Time-dependent interaction between load rating and reliabilityof deteriorating bridges

Ferhat Akgul a, Dan M. Frangopol b,�

a Department of Engineering Sciences, Middle East Technical University, 06531 Ankara, Turkeyb Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Campus Box 428, Boulder, CO 80309-0428, USA

Received 8 December 2003; received in revised form 12 June 2004; accepted 18 June 2004

Abstract

Prioritization and allocation of federal funds for nationwide bridge replacements and rehabilitations are based on ratings listedin the National Bridge Inventory database. Distribution of funds is based on the sufficiency rating, represented by a formula con-sidering structural safety, functional obsolescence, and essentiality for public use. Possessing the highest weight in sufficiency 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 different 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 rating–reliability profile and rating–reliability interaction envelope concepts are introduced. Further-more, the rating–reliability profiles 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 fornationwide bridge replacements and rehabilitations arebased 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 sufficiency 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 sufficiency rating formula,load rating for structural safety is a crucial measure forbridge management and decision making. Load ratingfor structural safety and serviceability for an existingbridge is determined when a maintenance, improve-ment work, change in strength of members, or addition

of dead load alters the condition or capacity of thestructure. While load rating represents the currentpractice in bridge evaluation, reliability methods, tak-ing into account the live load increase and materialdeterioration models, are more commonly used for life-time bridge assessment.For the design of new bridges, the AASHTO LRFD

[1] Specifications 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 therating of the bridge. Rating factors for different dete-riorating bridge types have to be calculated for variousfailure modes. Consequently, the system reliability indi-ces for different deteriorating bridge types must also becalculated for the same failure modes. For this pur-pose, a live load model and deterioration models forconcrete and steel are integrated into a specially

1752 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 1751–1765

developed computer program [5]. This program is usedto determine the lifetime reliability profiles, 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, reflecting 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 rating–reliability profile and rating–reliability interactionenvelope concepts are introduced. Furthermore, therating–reliability profiles 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 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-portation (CDOT) consists of a letter followed by aone to two digit number describing the vertical andhorizontal coordinates of the bridge, respectively, on astatewide grid. The last two letters are the uniqueidentification 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 componentof each bridge in the network. This is followed bydetermination of initial component and systemreliability 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 configur-ation, span length, position of vehicles (longitudinal,
Fig. 1. Bridge network.

F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004) 1751–1765 1753

transverse), number of vehicles, speed of vehicles, stiff-

ness of superstructure, and bridge geometry (straight,

curved) [16]. Site specific traffic data for such para-

meters is generally collected using weigh-in-motion

(WIM) studies. WIM involves recording weights of

trucks and bridge deflections 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 effects and girder distribution factors for

the bridge. Subsequently, actual recorded effects are

generally compared to values specified by current

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

AASHTO LRFD Bridge Design Specifications [1] is

used. The extrapolation for the load model used herein

is based on the mean value lYnand standard deviation

rYnof Type I largest value asymptotic distribution for

moments and shears in bridge components at a future

time, corresponding to a sample size of n trucks:

lYn¼ lX þ unrX þ c

anrX ð1Þ

rYn¼ p

ffiffiffi

6p

anrX ð2Þ

where un and an are the location and scale parameters

of Type I largest value distribution, c is the Euler num-ber (0.577216), and lX and rX are the mean value and

standard deviation of maximum moment or shear at

initial time t ¼ 0, i.e. due to a single truck.Fig. 2 demonstrates how this model is used for the

slab of bridge E-17-LE. In this figure, the variation of

the multiplier (uncertainty factor) for flexural moment

in slab is shown for the 75-year time period for the

standard AASHTO truck load. A drastic increment in

the first 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 affected 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

bridges. Other primary causes for bridge deterioration

may be the cracking, dislocation at supports, bearing

damage, excessive vibration, delamination of decks,

and heavy truck use. This study focuses on deterio-

ration due to corrosion only, since time variant models

for other deterioration types have not yet been fully

developed in this field.For deterioration of concrete and steel in bridges,

various models have been developed by researchers in

this field. For instance, treatment of various types of

models for steel corrosion in concrete can be found in

Refs. [8,9,13,15]. For steel corrosion, the types caused

by carbonation and chloride contamination are most

frequently observed, followed by deterioration due to

alkali-silica reaction, sulphate attack and freeze–thaw

action. Severity of steel corrosion, in general, depends

on the metal (composition of alloys in metal), local

atmosphere (environmental conditions affecting steel

corrosion include temperature and relative humidity),

and exposure conditions (initial climate, sheltering,

orientation, angle of exposure, time of wetness, atmos-

pheric pollutants, deicing salt, and debris) [7,12]. For

concrete deterioration model, Fick’s second law of

Table 1

Characteristics of 14 Colorado highway bridges in the network shown in Fig. 1 (see also [4])

Bridge

name

B

t

ridge

ype

CDOT

designation

N

o

umber

f spans

L
ength Width Year built A DTT
(trucks/day)
(m) (f t) (m) ( ft)
E-16-MU P
restressed CPG 1 3 4.1 1 12.0 11.6 38.0 1 994 810
E-16-LA P
restressed CBGCP 2 7 7.9 2 55.5 39.2 1 28.5 1 983 450
D-16-DM P
restressed CPGC 2 4 4.5 1 46.0 14.2 46.5 1 990 390
E-16-QI P
restressed CBGCP 2 7 4.1 2 43.2 30.7 1 00.7 1 995 1 335
E-16-LY P
restressed CPGC 3 7 4.3 2 43.7 34.1 1 12.0 1 985 1 610
E-16-NM P
restressed CPGC 2 6 4.6 2 12.0 28.0 92.0 1 991 2 955
E-17-MW P
restressed CIC 2 7 2.7 2 38.6 30.5 1 00.0 1 987 230
E-16-FK S
teel I-beam CIC 4 6 9.2 2 27.0 10.4 34.0 1 951 1 370
E-16-FL S
teel I-beam CIC 4 5 4.0 1 77.0 10.4 34.0 1 951 765
E-16-Q S
teel I-beam CIC 5 8 2.3 2 70.0 12.2 40.0 1 953 890
E-17-LE S
teel plate girder WGCK 4 6 8.6 2 25.0 19.7 64.5 1 972 992
E-17-HS S
E-17-HR S
E-17-HE S
teel plate girder WGCK 4 6 7.7 2 22.2 10.4 34.0 1 962 1 290
CPG, concrete prestressed girder; CBGCP, concrete box girder continuous prestressed; CPGC, concrete prestressed girder continuous; CIC, con-

crete on rolled I-beam continuous; WGCK, welded girder continuous composite.


diffusion due to chloride is used:

@C

@t¼ Dc

@2C

@x2ð3Þ

where C is the concentration of chloride ions, Dc is theeffective diffusion coefficient, t is time, and x is the dis-tance from outer surface of the solid.

Models 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 coefficients in the

form of a power formula. In this study, a power func-

Fig. 3. Time-variation of the slab reinforcing area for bridge E-17-LE.

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


tion is employed for corrosion of structural steel [11,17]

p ¼ botb1 ð4Þ

where bo and p are the corrosion losses after one and tyears, respectively, and b1 is the slope of the logarith-mic transformation of (4).The deterioration model associated with Fick’s

second law of diffusion (3) is used for reinforced con-crete decks, reinforced concrete girders, and prestressedconcrete girders. Similarly, the corrosion model forsteel (4) is applied to steel girders. Fig. 3 demonstratesthe application of the concrete deterioration model forthe slab reinforcement for bridge E-17-LE. The ran-dom variable Tisr, indicating the corrosion initiationtime for slab reinforcement has a mean value of 3.61years. With the lognormal (LN) density distributions ofthe corrosion initiation time Tisr and corrosion ratercorr indicated in the figure, the probability density dis-tributions of the slab reinforcement area are plotted at10-year intervals using the Monte Carlo simulationmethod. The profile 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 different bridge member types) were ident-ified, 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 profilesand employs the First Order System ReliabilityMethod to determine the reliability profiles for bridgemembers and systems. System reliability analysis of thebridges required the description of a failure model for

each bridge type. Since bridges are generally rated forsuperstructure components, only the superstructuremembers of the bridges (i.e., slab and the girders) areconsidered in this investigation. As an example, thecross sectional view of the superstructure of bridge E-17-LE and the corresponding failure model for thesuperstructure are displayed in Fig. 4(a) and (b),respectively.The topology of series-parallel system failure model

for the superstructure changed depending on the bridgetype. For instance, for girders of simple span pre-stressed bridges, flexure of girder at midspan is con-sidered while for continuous prestressed girders, flexureat both midspan and pier support locations areincluded in the system failure model. For steel bridges,such as E-17-LE, girder failure modes included bothflexure and shear. As shown in Fig. 4(b), failure of thesteel bridge is defined as the flexural failure of the slabor flexural or shear failure of two adjacent girders.When calculating the system reliability index for thebridge superstructure, a correlation coefficient 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 profiles for resistance degradation and reliabilityindex are systematically generated for each bridge inthe 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 profiles are obtained foreach bridge in the network, the results of time-dependent rating calculations and reliability profiles are

Fig. 4. (a) Girder layout, (b) system failure model for bridge E-17-LE.


combined, emphasizing the interaction between rating

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

system reliability index b(t) for several representative

bridges of different types in the network are presented in

Figs. 7–13. 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 system

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

Fig. 6. Time-variation of the operating rating factor for the slab and girder of bridge E-17-LE.


reliability 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. Having

a 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 these

Fig. 7. Time-dependent operating rating vs. reliability index for bridge E-16-MU.

Fig. 8. Time-dependent operating rating vs. reliability index for bridge D-16-DM.


graphs 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 bridgerating factor—reliability index pair is expected to move

from upper left to lower right hand corner in all graphsdue to member deterioration and increase in loadeffects due to traffic.Since the theoretical basis used in these graphs is the

same, only the graph for bridge E-16-MU, shown in

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

Fig. 10. Time-dependent operating rating vs. reliability index for bridge E-16-FK.


Fig. 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. 7

shows three plots: slab flexure, girder flexure and the

system rating–reliability interaction curve denoted by

the label (RF(t)bridge, b(t)sys). This designation indicates

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

. 12. Time-dependent operating rating vs. reliability index for slab and concrete girders of bridge E-17-H
Fig S.


that the time-dependent rating factor RF(t)bridge is forthe bridge, representing the minimum of all superstruc-ture component ratings, and the time-dependentreliability index b(t)sys is for the system representing theseries-parallel system failure model for the bridgesuperstructure. Since the slab flexure controls the rat-ing–reliability profile for the bridge, the slab flexureand system rating–reliability interaction curves in thisfigure are overlapped. The rating vs. reliability indexvalues for the bridge components at initial time, i.e.t ¼ 0, are not shown in these graphs since they dis-played considerably higher reliability indices due to asingle average truck and revealed sudden drops in rat-ing–reliability curves at initial stage of service life dueto immediate application of the traffic load [4]. Instead,the graphs start from a time, chosen as the end of thefirst month, representing a reasonable traffic loadexposure for the bridge.Since the sudden initial drop in rating–reliability

values due to initial application of traffic load is exclu-ded, the remaining reduction in rating–reliability pro-files shown in these figures is mainly due to materialdeterioration, although increase in live load effect dueto traffic is still a contributing factor. Starting from 1month, the points in time until 75 year life period forthe bridge E-16-MU are indicated along each compo-nent rating–reliability profile. In order to reveal thetime domain hidden in these graphs, the following 10separate points in time are displayed consistently for all

bridges: 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 profiles. How-ever, for several other bridges, it was necessary to dis-play a few representative points instead of all 10 pointsmentioned above.By comparing the lifetime rating–reliability graphs

for different bridges, the following observations can bemade: each bridge displayed a unique behavior, there-fore, distinct generalizations for different bridge typeswould not be appropriate. However, from a broad per-spective, general location of rating–reliability curvesshow similarities within a given bridge group type. Forinstance, rating–reliability curves for the prestressedconcrete bridges, as shown for bridges E-16-MU andD-16-DM in Figs. 7 and 8, respectively, are generallylocated near the middle of rating–reliability domain.For flexure of continuous prestressed girders over pierlocation, higher rating and reliability values areobserved as shown in Fig. 8. Bridge E-17-MW (seeFig. 9), however, although being a prestressed concretebridge, displays a relatively low rating–reliability forthe flexure of slab which is indicated by the extensionof the rating–reliability curve for slab flexure towardthe lower-right corner of the graph.Steel bridges such as E-16-FK and E-17-LE, shown

in Figs. 10 and 11, respectively, have very low rating–reliability values for flexure of slabs which is indicatedby the extension of the rating–reliability profile toward

ig. 13. Time-dependent operating rating vs. reliability index for slab and steel girders of bridge E-17-HS.
F


the lower-right corner of the graph. Within 10–40-year

period, slab rating–reliability profiles of these bridges

reach the lowest observed value for the network

bridges. This is due to the fact that the steel bridges are

the oldest ones among the network bridges. Conse-

quently, they were built with slabs having lower

strength reinforcing steel.Bridges with steel and concrete girders such as bridge

E-17-HS shown in Figs. 12 and 13, displayed a beha-

vior similar to that of the steel bridges. For clarity, sep-

arate graphs are provided in these figures for reinforced

concrete and steel girders of the bridge E-17-HS.In general, the position of the system rating–

reliability profile was close to the profile of the compo-

nent having the minimum rating and the lowest

reliability. For rating, this is a reasonable performance

since the bridge rating in practice is controlled by the

minimum component rating. However, from the

reliability point of view, the position of the system rat-

ing–reliability profile reflected the result of the system

reliability analysis based on the system failure model

used.Bridge rating factors in rating–reliability graphs were

determined considering both the slab and girder rat-

ings. Consequently, following the same process, the

system reliability indices in these graphs were calcu-

lated using the system failure model including both

slab and girders.

5. Time-dependent rating–reliability interaction

at network-level

In Figs. 14 and 15, all bridges in the network areplotted together, showing the time-dependent bridgeoperating rating factors and the system reliabilityindices, respectively. In Figs. 16 and 17, The time-dependent network results show the interactionbetween rating and reliability profiles.The variation of operating bridge rating factors for

all network bridges, based on girders only, is shown inFig. 14, and the corresponding values are listed inTable 2. In Fig. 14, the variation of operating bridgerating factors are indicated for 1 month, and for 1, 20,50, and 75 years for each bridge. This figure enableseasy comparison of the bridge rating factor values forall bridges in the network throughout their lifetimes.Large lifetime deteriorations for prestressed girderbridges are characterized by big gaps between 1 monthand 75 year marks for each bridge. On the other hand,small deterioration for steel girder bridges is reflectedby very small vertical gaps among rating factors for agiven bridge.Similarly, the variation of system reliability indices

for all network bridges, based on girders only, is shownin Fig. 15, and the corresponding values are listed inTable 3. Using this figure, similar observations andcomparisons can be made for the network bridgesbased on system reliability indices. The figure displays

Fig. 14. Time-variation of the bridge operating rating factor for the network bridges.


the 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 findings reported in Figs. 14 and 15 in

terms of bridge operating rating factor vs. systemreliability index for all bridges in the network consider-ing slab and girders, and girders only, respectively. InFigs. 16 and 17, rating reliability envelopes are shown

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

ariation of the bridge operating rating factor vs. system reliability index for the network bridges based on
Fig. 16. Time-v slab and girders.


for the network bridges both for time t ¼ 1 month and

for the lifetime interaction profiles. The envelope for

time t ¼ 1 month, represented by the square region, is

a snapshot of rating–reliability values for all bridges in

the network at that initial time. The remaining data

points represent the rating–reliability values of the

bridges at subsequent discrete points in time through-

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

level time-dependent rating–reliability interaction

envelope, represented by upper and lower bounds,

-variation of the bridge operating rating factor vs. system reliability index for the network bridges based o
Fig. 17. Time n girders only.
Table 3

Time-variation of the system reliability index for the network bridges

Time b
(t)sys
E-16-

MU

E-16-

LA

D

D

-16-

M

E-16-

QI

E

L

-16-

Y

E-16-

NM

E

M

-17-

W

E-16-

FK

E

F

-16-

L

E-16-

Q

E

L

-17-

E

E

H

-17-

S

E

H

-17-

R

E-17-

HE

1 month 3
.73 3.28 4 .04 4.07 2 .59 3.92 3 .40 3.33 2 .67 2.66 3 .47 3 .94 3.00 3.46
1 year 3
.48 3.06 3 .75 3.84 2 .34 3.67 3 .13 3.04 2 .36 2.35 3 .15 3 .54 2.70 3.20
20 year 3
.22 2.82 3 .18 3.59 2 .08 3.40 2 .85 2.74 2 .03 2.02 2 .79 2 .76 2.07 2.89
50 year 2
.67 2.24 2 .06 2.87 1 .95 2.43 2 .74 2.65 1 .93 1.92 2 .68 1 .54 0.89 2.03
75 year 1
.81 1.32 1 .35 1.81 1 .48 1.34 2 .45 2.60 1 .89 1.87 2 .62 0 .56 � 0.08 0.88
Table 2

Time-variation of the bridge operating rating factor for the network bridges

Time R
F(t)OPE,bridge
E-16-

MU

E-16-

LA

D

D

-16-

M

E-16-

QI

E

L

-16-

Y

E

N

-16-

M

E

M

-17-

W

E

F

-16-

K

E

F

-16-

L

E

Q

-16- E

L

-17-

E

E

H

-17-

S

E

H

-17-

R

E-17-

HE

1 month 1
.565 1.495 1 .595 1.793 1 .143 1 .616 1 .469 0 .870 0 .763 0 .762 0 .939 0 .949 0 .802 0.835
1 year 1
.442 1.370 1 .467 1.656 1 .060 1 .500 1 .339 0 .807 0 .706 0 .705 0 .865 0 .883 0 .736 0.773
20 year 1
.328 1.256 1 .276 1.529 0 .982 1 .389 1 .225 0 .746 0 .651 0 .649 0 .790 0 .754 0 .673 0.711
50 year 1
.202 1.097 0 .953 1.355 0 .952 1 .184 1 .190 0 .729 0 .635 0 .633 0 .767 0 .563 0 .507 0.675
75 year 0
.983 0.828 0 .786 1.063 0 .877 0 .895 1 .133 0 .721 0 .627 0 .626 0 .756 0 .433 0 .377 0.479


shows the safety of the network bridges throughouttheir lifetime.Fig. 16 revealed an extremely close behavior for all

bridges considering the lifetime system rating–reliabilityprofiles. Although having unique initial rating–reliability values slightly dispersed at the beginning,rating–reliability 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 rating–reliability interaction envelope is also defined for thenetwork bridges as indicated in Fig. 16. Based on thisgraph, it is possible to conclude that the networkbridges show variability in system rating–reliabilityvalues initially, i.e. t ¼ 1 month, however, after the firstmonth of traffic, the values for all bridges on the aver-age gradually converge.A similar behavior is also observed in Fig. 17, where

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

6. Conclusions

(1) Lifetime rating and reliability analyses for differ-ent 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 different member types were investigated and theintegration of these models into a network-level, time-dependent, system reliability analysis program isaccomplished. When updated with field 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.

(3) A time-dependent rating–reliability envelope can

serve as a useful tool for quantifying the interaction

between rating and reliability of individual bridges

within a bridge network. Such an envelope can be con-

sidered as a characteristic set of measures representing

the structural health of an overall bridge network.

Using the developed envelopes, it is possible to esti-

mate both the live load capacity and the corresponding

system reliability of any bridge in the network at any

point in time within a selected time horizon.Consequently, based on the time-dependent rating–

reliability interaction, the following general observation

can be made. Rating factor may not reflect the actual

safety of a bridge over its lifetime. For instance, the

operating rating factor for girder flexure over lifetime

of bridge E-16-MU (see Fig. 7) decreased by 37% (from

1.565 to 0.983), whereas system reliability index

decreased by 51% (from 3.73 to 1.81). This case repre-

sents a smaller relative reduction in rating vs.

reliability. For the slabs, such as the slab of bridge

E-16-MU, a substantially larger relative reduction in

rating vs. reliability is visible (25% for rating, i.e. from

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

Since system reliability index evaluates the actual safety

of the structure, while rating factor reflects 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 financial 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 reflect the views of the

sponsoring agencies.

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Time-dependent interaction between load rating and reliability of deteriorating bridges

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