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Original Research Paper

Estimating the resistance of aging service-provenbridges with a Gamma process-baseddeterioration model

Cao Wang a,b,*, Kairui Feng b, Long Zhang c, Aming Zou b

a School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australiab Department of Civil Engineering, Tsinghua University, Beijing 100084, Chinac CCCC Highway Consultants CO., Ltd., Beijing 100088, China

h i g h l i g h t s

� A method is proposed to update the resistance of aging bridges using a Gamma process-based deterioration model.

� The difference between the updated distributions of bridge resistance is discussed with different deterioration models.

� A “fully correlated” deterioration process overestimates the bridge resistance compared with a Gamma process-based model.

a r t i c l e i n f o

Article history:

Received 15 July 2018

Received in revised form

5 November 2018

Accepted 10 November 2018

Available online 12 December 2018

Keywords:

Existing aging bridges

Historical load information

Resistance updating

Gamma deterioration process

Bridge safety

* Corresponding author. School of Civil EnE-mail address: cao.wang@sydney.edu.au

Peer review under responsibility of Periodichttps://doi.org/10.1016/j.jtte.2018.11.0012095-7564/© 2018 Periodical Offices of Chanaccess article under the CC BY-NC-ND licen

a b s t r a c t

The environmental or anthropogenic factors, to which the in-service bridges are subjected,

are responsible for the reduction of bridge performance, and finally lead to great service

risk for bridges and increase the probability of substantial economic losses. Probability-

based estimate of bridge resistance is an essential indicator for the bridge condition

evaluation and for optimization of bridge maintenance/repair decisions. It places an

emphasis on the proper probabilistic models of structural properties and assessment

methods. Making full use of historical service load information may improve the accuracy

of bridge performance assessment with reduced epistemic uncertainty for existing aging

bridges. In to-date analyses to update the bridge resistance with past service information,

the models of resistance deterioration have been assumed as either deterministic or fully

correlated, which may differ significantly from the realistic case. With this regard, this

paper proposes a novel method for updating the resistance of service-proven bridges with

a realistic deterioration model. The Gamma stochastic process has been suggested in the

literature to describe the probabilistic behavior of structural time-dependent resistance

and thus is adopted in this paper. An illustrative bridge is presented to demonstrate the

applicability of the proposed method. Parametric examples are conducted to investigate

the role of resistance deterioration model in the updated estimate of bridge resistance with

historical service information.

© 2018 Periodical Offices of Chang'an University. Publishing services by Elsevier B.V. on

behalf of Owner. This is an open access article under the CC BY-NC-ND license (http://

creativecommons.org/licenses/by-nc-nd/4.0/).

gineering, The University(C. Wang).

al Offices of Chang'an Un

g'an University. Publishinse (http://creativecommo

of Sydney, Sydney, NSW 2006, Australia. Tel.: þ61 424003086.

iversity.

g services by Elsevier B.V. on behalf of Owner. This is an openns.org/licenses/by-nc-nd/4.0/).

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e84 77

1. Introduction

The performance of in-service bridges is of great concern to

the asset owners and civil engineers in this era with a rapid

growth in public awareness of infrastructure safety.While the

serviceability goal of new-built bridges has been well accom-

plished in existing codes and standards, the operating or

environmental conditions that the in-service bridges are

exposed to may cause substantial deterioration to structural

strength and/or stiffness below the criteria as assumed for

design, impairing the bridge reliability considerably during a

subsequent service period (Ellingwood, 2005; Li and Elliing-

wood, 2007; Melchers and Beck, 2017; Vu and Stewart, 2000).

To manage the safety of a bridge (or a bridge group within a

traffic network), risk-based performance indicators are

essentially necessary to account for the uncertainties associ-

ated with bridge resistance and applied loads (Kumar and

Gardoni, 2011; Lin et al., 2015; Nowak et al., 2010; Pang and Li,

2016; Saydam et al., 2013). Probability-based assessment of

bridge resistance is a widely-used tool providing quantitative

measure of structural reliability (or probability of failure), both

immediate and long-term, subjected to future extreme events,

which is further the basis for the optimization of mainte-

nance/repair decisions regarding the bridges (Wang et al.,

2016, 2017).

Many existing analytical or predictive approaches lead to

overly conservative estimate of bridge resistance, e.g., Faber

et al. (2000), Saraf and Nowak (1998), Wipf et al. (2003),

implying the necessity of realistic analytical methods for

bridge condition assessment. Although the historical service

load is one of themajor contributors to the reduction of bridge

performance, it is still evident of the bridge resistance (load-

bearing capacity) from the viewpoint of statistics (Ellingwood,

1996; Hall, 1988; Hall and Tsai, 1989; Stewart, 1997). For

instance, for an existing bridge with a service period of T

years, the lower tail of the probabilistic distribution of current

resistance, R(T), is progressively truncated due to the fact that

R(T) is higher than any of the historically imposed loads. In

fact, the successful historical traffic load can be regarded as a

proof load test with uncertainty for an existing service-proven

bridge. In recent years, much effort has been made regarding

the safety issues of existing bridges in many engineering

practices; for instance, the weight-in-motion (WIM) systems

(Vardanega et al., 2016) have been installed on key roads. The

routine condition inspections and evaluations are carried out

to collect traffic flow data and bridge condition information.

Such observed data can be taken full use of in advancing

analysis methods to improve the estimate of bridge

resistance with reduced epistemic uncertainty.

Significant researches have been published in the past

decades regarding the updating of structural performance of

existing aging structures using historical load information.

The work by Hall (1988) was among the first to update the

resistance of existing bridges using historical service

information, which is based on the concept of Bayesian

theorem. The subsequent work by Faber et al. (2000) and

Schweckendiek (2010) also adopted this methodology to

update the reliability of service-proven structures. However,

Hall (1988) ignored the deterioration of bridge resistance

with time, and thus concluded that the updated resistance

after surviving for T years is even higher than the initial

resistance. Such conclusion becomes untenable once the

resistance deterioration is taken into consideration.

Recognizing this, Stewart et al. (Stewart and Val, 1999;

Stewart, 2001) investigated the combined effect of both prior

service loads and resistance deterioration on the reliability

of existing bridges, where the Monte Carlo simulation based

procedure was employed. In order to improve the analysis

efficiency and seek deep insights into the problem, Li et al.

(Li et al., 2015a; Li and Wang, 2015) developed explicit

methods to update the bridge resistance and reliability

estimates with historical service load. However, in the to-

date analysis methods, the models of structural resistance

deterioration have been assumed as either deterministic or

fully-correlated. These models imply that the overall

trajectory of the degradation within a reference period of

interest is determined once the deterioration type and the

deterioration scenario at a specific time are provided. In fact,

the resistance deterioration is unavoidably associated with

uncertainties and it is a non-increasing stochastic process

by nature with the exclusion of rehabilitation or other types

of strengthening. Thus, in the probability-based bridge

performance assessment, a reasonable resistance

deterioration model plays a vital role. It is expected to

ensure that both the temporal autocorrelation and the

variability in the degradation process are properly taken into

account. The Gamma process has been employed in many

researches to model the resistance degradation, e.g., Li et al.

(2015b), Dieulle et al. (2003), Saassouh et al. (2007), Wang

et al. (2015), and thus it is employed in this paper to describe

the probabilistic behavior of time-variant resistance.

The aim of this paper is to investigate the updated resis-

tance of aging service-proven bridges by modeling the resis-

tance deterioration as a Gamma process-based stochastic

process. This paper is organized as follows. After the intro-

duction part in Section 1, the Gamma deteriorationmodel and

the reliability analysis method are briefly introduced in

Section 2, followed by the proposed method for the updating

of bridge resistance in Section 3. Illustrative examples are

presented in Section 4 to demonstrate the applicability of

the proposed method and to investigate the role of

deterioration model in the updated estimate of structural

performance and finally conclusions are formulated in

Section 5.

2. Deterioration model and reliabilityanalysis

The degradation of structural resistance may be posed by

construction defects, severe operating or environmental

conditions or applied loads (Pipinato, 2016). For an aging

bridge, the time-variant resistance changes in time

according to Eq. (1), where R(t) is the resistance at time t; R0

is the initial resistance (a random variable) and D(t) denotes

the deterioration function (a stochastic process). In this

paper, R0 is assumed statistically independent of D(t).

Fig. 1 e Sample degradation curves. (a) Gamma deterioration process. (b) Fully correlated process.

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e8478

RðtÞ ¼ R0½1� DðtÞ� (1)

In order to reflect the stochastic characteristic of resistance

deterioration, the deterioration function, D(t), is modeled as a

Gamma process, denoted by G(t). Mathematically, if a random

variable,X, followsGammadistributionwith shape parameter

a> 0 and scale parameter b > 0, thenX is written asX ~ Ga (a, b)

and the probability density function (PDF) of X takes the form

of

fXðxÞ ¼ ðx=bÞa�1

bGðaÞ expð�x=bÞ x � 0 (2)

where Gð,Þ denotes the Gamma function, GðuÞ ¼ R∞0 su�1e�sds.

The mean value and variance of X are respectively ab and ab2.

Further, a stochastic process, {X(t), t > 0} is deemed as a

Gamma process if (1) Pr[X (0) ¼ 1] ¼ 1, where Pr($) denotes the

probability of the event in the bracket, and (2) X(t) has inde-

pendent increments and each increment follows Gamma

distribution with an identical scale parameter.

Now we consider the modeling of deterioration process,

G(t). At the end of t years, suppose that the time-variant mean

value and variance of G(t) are respectively m(t) and v(t). Obvi-

ously,m(t) and v(t) are not independent due to the shared scale

parameter of the Gamma process increments. If the shape

parameter and the scale parameter of the deterioration

function increment within time interval (t, t þ dt] are respec-

tively a(t) and b, where dt/0, then

8>>>>>><>>>>>>:

mðtÞ ¼ bZ t

0

aðtÞdt

vðtÞ ¼ b2

Z t

0

aðtÞdt ¼ bmðtÞ(3)

The function m(t) takes the form of

mðtÞ ¼ kta (4)

where k is a constant reflecting the deterioration rate, while a

is a parameter representing the deterioration shape, which

can be determined according to the dominant deterioration

mechanism of interest. For instance, a is 1, 2 and 0.5 corre-

sponding to deteriorations due to corrosion of reinforcement,

sulfate attack and diffusion-controlled aging respectively

(Mori and Ellingwood, 1993). The parameters defining the

stochastic degradation process, b, k and a can be calibrated

by regression analysis of observed data.

Since G(t) has independent increments, the auto-covari-

ance of G(t) at two different time instants, ti and tj (j > i), is

obtained as

Covar�GðtiÞ;G

�tj�� ¼ Var½GðtiÞ� (5)

where Var($) represents the variance of the variable in the

bracket. Further, the linear correlation coefficient of G (ti) and

G (tj), ri,j, is

ri;j ¼Var½GðtiÞ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var½GðtiÞ�Var�G�tj��q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibmðtiÞbm�tj�

s¼

ffiffiffiffitaitaj

s(6)

Eq. (6) indicates that the correlation between G (ti) and G (tj)

increases when the time lag between ti and tj becomes larger

and vice versa. By noting the fact that ri,j is constantly equal

to 1 for the case of fully-correlated deterioration process, it

is seen from Eq. (6) that for the case of Gamma process-

based deterioration, the resistances of an aging bridge at

different times are positively but not fully correlated,

implying that a bridge with slight deterioration at the early

stage of the reference period may still degrade severely at

the latter stage although it has a greater possibility of “good

performance” within the whole service period.

In order to illustrate the behavior of the Gamma deterio-

ration process, Monte Carlo simulations are performed for

b ¼ 0.01, k ¼ 0.0075 and a ¼ 1, and samples of the resulting

degradation curves are presented in Fig. 1. In order to sample a

degradation process within time interval (0, T], we first

discretize this interval into j identical sections with the

maximum duration of the j sections being sufficiently small,

with which the sampling procedure for each simulation run

is as follows:

(1) For i ¼ 1, 2, /, j, sample the resistance deterioration

within the ith interval, di, which is a Gamma random

variable with shape and scale parameters being kata�1Tbj

and b respectively.

(2) Set G

�iTj

�¼Pi

k¼1dk i ¼ 1; 2;/; j.

For the purpose of comparison, the samples of fully

correlated deterioration process are also simulated and pre-

sented in Fig. 1. The fully correlated process means that a

structure with “good” performance at the early stage (i.e.,

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e84 79

minor degradation) is predetermined to behave well within

the whole service period. However, it is not realistic in many

cases, in which the significant deterioration may still occur

at the latter stage. The variability and temporal

autocorrelation can be better reflected by the Gamma

process-based deterioration model.

Within the considered time interval (0, T], significant load

events that essentially affect structural safety occur randomly

in time with random intensities. Supposing that the duration

of each load event is sufficiently small so that the loads only

occupy a small fraction of the whole service life, and there is

no dynamic response, we can use a sequence of randomly

occurring pulses with random intensities to model the load

events, say, S1, S2, /, Sn at times t1, t2, /, tn, as illustrated in

Fig. 2. In this paper, the load events are assumed independent

of each other. The reliability over this reference period, L(T), is

defined as

LðTÞ ¼ PrfRðt1Þ>S1∩Rðt2Þ> S2∩/∩RðtnÞ>Sng (7)

Let Gi denote the resistance deterioration within time in-

terval (ti-1, ti], i.e., G(ti) e G(ti-1), and then

LðTÞ ¼ Pr

(∩n

i¼1

"R0

1�

Xi

k¼1

Gk

!>Si

#)(8)

Note that the number of load events, N, is also a random

variable, and thus Eq. (8) is indeed a conditional probability

given N ¼ n. Employing the Poisson process to model the

occurrence of loads, then

PrðN ¼ nÞ ¼ ðlTÞn expð�lTÞn!

n ¼ 0;1; 2;/ (9)

where l is the mean occurrence rate of the loads.

3. Resistance of service-proven structures

If an existing bridge is subjected to a proof load test, then the

PDF of the bridge resistance is simply truncated at this known

load effect (Fujino and Lind, 1977). Treating the service loads

as proof loads with uncertainties, the bridge resistance and

reliability for subsequent service years can be updated with

the historical load information. In the pioneering study by

Hall (1988), assuming that no resistance deterioration occurs

Fig. 2 e Stochastic processes of resistance deterioration

and loads.

during the past time interval (0, T], the updated PDF of

current bridge resistance, f 'RðrÞ, is obtained as

f 'RðrÞ ¼fRðrÞFSðrÞZ ∞

0

fRðrÞFSðrÞdr(10)

where FS($) is cumulative density function (CDF) of the

maximum load effect experienced up to time T, and fR ($) is the

PDF of prior estimate of bridge resistance. It is noticed that the

initial and current resistances are not distinguished in Eq. (10)

since the resistance deterioration is ignored.

Now consider the case where the bridge resistance de-

grades with time during the past service period. If the PDFs of

the initial resistance R0 and the deterioration function G(T) are

known as fR0 ðrÞ and fGðrÞ respectively, prior to the load infor-

mation, the probabilistic distribution of current

resistance, R(T), can be determined according to Eq. (11)

under the assumption of independent R0 and G(T) (Li and

Wang, 2015).

fRðTÞðrÞ ¼Z∞0

fR0ðtÞfG

1� r

t

1tdt (11)

Taking into account the past service information, the

updated PDF of R(T), f 'RðTÞðrÞ, is determined by

f 'RðTÞðrÞ ¼ limdr/0

1dr

Pr½ðr � RðTÞ< rþ drÞjA�

¼ limdr/0

1dr

Pr½ðr � RðTÞ< rþ drÞ∩A�PrðAÞ (12)

where A represents a successful performance (survival) of the

bridge within time period (0, T]. Modeling the resistance

degradation as a Gamma process, the denominator of Eq. (12),

Pr(A), is estimated according to Eq. (8) as

PrðAÞ ¼ Pr

(∩N

i¼1

"R0

1�

Xi

k¼1

Gk

!> Si

#)(13)

while the numerator of Eq. (12) is

Pr½ðr � RðTÞ< rþ drÞ∩A� ¼ Pr

(∩

Nþ1

i¼1

"R0

1�

Xi

k¼1

Gk

!>Sð1Þ

i

#)

� Pr

(∩

Nþ1

i¼1

"R0

1�

Xi

k¼1

Gk

!>Sð2Þ

i

#)(14)

where Sð1Þi ¼ Sð2Þ

i ¼ Si for i ¼ 1, 2, /, N; Sð1ÞNþ1 ¼ r; Sð2Þ

Nþ1 ¼ rþ dr,

and GNþ1 is the resistance deterioration within time interval

(tN, T], i.e., G(T) e G (tN).

A purely analytical solution would have existed if the es-

timates of the probabilities in Eqs. (13) and (14) can be solved

numerically. However, multiple-fold integral is involved in

Eqs. (13) and (14) so that it is difficult to find the numerical

solution in practice. Therefore, we adopt a simulation-based

approach herein to estimate the posterior PDF of current

resistance. The procedure for each simulation run is

illustrated in Fig. 3. Note that the service load information can

also be used to update the estimates of initial

resistance, thus the updated PDF of R0 is also obtained in the

simulation.

Fig. 4 e Girder section of Qingfang Bridge (unit: mm) (Li and

Wang, 2015).

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e8480

4. Illustrative examples

4.1. Structural configuration

The Qingfang Bridge as illustrated in Li and Wang (2015) is

adopted in this paper to demonstrate the application of the

proposed updating method and to investigate the role of

deterioration model in the updated estimate of bridge

resistance. This bridge, with a service age of 22 years, is

located in a coastal city in China; as a result, its dominant

deterioration mechanism is chloride-induced corrosion of

the steel bar, which is regarded independent of the load

process. Since the bridge is a simply supported RC beam

bridge, we focus on the bending moment capacity

(resistance) of the girder, whose section is given in Fig. 4.

The probabilistic models of material properties and

geometric variables are listed in Table 1, from which the

mean value and standard deviation of the initial resistance

are determined respectively as 4022 and 479 kN$m based on

Monte Carlo simulation. A condition inspection at the end of

22 years revealed that the mean value of G(22) is 0.35. As

there is no evidence on the variability of the deterioration

function, we assume that the coefficient of variation (COV,

defined as the ratio of standard deviation to the mean value)

of G(22) equals 0.25. The total bridge load consists of the

dead load and the truck live load. The mean occurrence rate

of the live load is set equal to 1/year, and the load process is

assumed to be stationary for the purpose of simplicity.

4.2. Application of the proposed method

The updated distributions of initial resistance (R0) and current

resistance (R(22)) are obtained using 100,000 simulation runs

Fig. 3 e Flow chart to simulate the updated PDFs of R(T) and

R0.

and are plotted in Fig. 5, where the deterioration type is

assumed linear (i.e., a ¼ 1 in Eq. (4)). The influence of

historical service load on the estimates of bridge resistances

is obvious. For instance, the mean values of R0 and R(22)

increase from 4022 kN$m and 2614 kN$m to 4061 kN$m and

2676 kN$m respectively, while the COV of R0 and R(22)

decrease from 0.1191 and 0.1801 to 0.1149 and 0.1615

accordingly.

4.3. Updated bridge resistances with differentdeterioration models

In order to investigate the effect of deteriorationmodel choice

(i.e., the fully correlated and Gamma process-based models)

on the updated distribution of initial and current resistances

for bridges subjected to different service conditions, three

deterioration mechanisms (i.e., linear, parabolic and square-

root, c.f. Eq. (4)) and two load models as presented in Table 2

are considered in the parametric examples.

The updated PDFs of the bridge initial and current re-

sistances using the Gamma process-based deterioration

model are presented in Figs. 6 and 7, where the resistance

deterioration type is assumed linear, and the service load

models in Table 2 are considered. For all cases, the lower tail of

the resistance distribution is truncated as a result of the

Table 1 e Statistical descriptions of bridge resistance andloads.

Items Variable Mean Standard deviation Distribution

Model x 1.05 0.11 Normal

Geometry b (mm) 2200 82 Normal

d (mm) 1350 44 Normal

AS (mm2) 8042 / Deterministic

Material fy (MPa) 375 38 Normal

f 'c (MPa) 28.5 6 Lognormal

Load* D (kN$m) 1.05Dn / Deterministic

L (kN$m) 0.46Ln 0.18Ln Extreme type I

Note: * Dn and Ln are the nominal dead load and live load respec-

tively, Dn ¼ 1113 kN$m, and Ln ¼ 1215 kN$m.

Fig. 5 e Prior and updated PDFs of bridge resistances. (a) R0. (b) R(22).

Table 2 e Summarization of live load models.

Load model Mean Standard deviation Distribution

Service load 1 (SL1) 0.46Ln 0.18Ln Extreme type I

Service load 2 (SL2) 0.60Ln 0.18Ln Extreme type I

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e84 81

failure of some deficient bridges caused by the service loads,

and this truncation is enhanced by greater load intensity.

From the comparison of Figs. 6 and 7, it is seen that the

impact of load history on the updated distribution is more

significant for the current resistance. This is because the

service load is used to update both the initial resistance and

the deterioration process, and the combined effect is

reflected in the updated distribution of R(22). For the purpose

of comparison, the updated distributions of initial and

current resistances using the fully-correlated deterioration

Fig. 6 e Updated PDFs of R0 with different

Fig. 7 e Updated PDFs of R(22) with differen

model are also calculated and are presented in Figs. 6 and 7,

and the updated mean value and COV of R0 and R(22) are

listed in Tables 3 and 4. It is seen that the difference

between the updated distributions of R0 associated with

both deterioration models is negligible. This is because the

Gamma process and fully correlated deterioration processes

result in close estimates of reliability in the past service

period (Li et al., 2015b), and the truncation of the lower tail is

caused by the “failure” of the bridges. The influence of

deterioration model choice on the updated distribution is

more significant for R(22). The updated mean value of R(22)

associated with the fully correlated process is greater than

that associated with the Gamma deterioration model,

indicating that the estimate of current resistance may be

overestimated with the fully correlated deterioration model.

This observation can be explained by the fact of the

approximately identical probability of structural failure

deterioration models. (a) SL1. (b) SL2.

t deterioration models. (a) SL1. (b) SL2.

Table 3 e Updatedmean value and COV of R0 for differentservice loads.

Servicemodel

Statistics Prior Updated

Gammaprocess model

Fully-correlatedmodel

SL1 Mean (kN$m) 4022 4061 4050

COV 0.1191 0.1149 0.1160

SL2 Mean (kN$m) 4022 4085 4077

COV 0.1191 0.1124 0.1132

Table 4 e Updated mean value and COV of R(22) fordifferent service loads.

Servicemodel

Statistics Prior Updated

Gammaprocess model

Fully-correlatedmodel

SL1 Mean (kN$m) 2614 2676 2727

COV 0.1801 0.1615 0.1614

SL2 Mean (kN$m) 2614 2715 2763

COV 0.1801 0.1523 0.1536

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e8482

probability (i.e., the probability that the sampled resistance is

not recorded). Compared with the Gamma process-based

deterioration, with which the bridges with low current

resistance may have a relatively high resistance at the early

stage during the service period, the fully correlated

deterioration model enables that the sampled bridges with

Fig. 8 e Updated PDFs of R0 with different deterioration model

Fig. 9 e Updated PDFs of R(22) with different deterioration mode

low current resistance have greater probability of failure

(smaller probability of record) because the performance of

these bridges during the overall service period is always

“weak” (i.e., with severe deterioration).

In order to investigate the effect of deterioration type on

the difference between the updated resistance distributions

associated with different deterioration models, Figs. 8 and 9

show the updated PDFs of initial and current resistances

assuming square root and parabolic deterioration models,

where the live load intensity is 0.6Ln for all cases. The updated

mean value and COV of R0 and R(22) are presented in Tables 5

and 6. It is seen that the difference between the updated dis-

tributions of R0 associated with both deterioration models is

as before slight, indicating that one may employ either dete-

rioration model to update the estimate of initial resistance

with negligible error. Comparing Figs. 7(b) and 9, it is seen that

the parabolic deterioration model leads to the greatest dif-

ference between the updated distribution of R(22) using the

two deterioration models, followed by those associated with

the linear and square root deterioration types. This is because

with the parabolic model, the resistance deterioration occurs

mainly at the latter stage of the service life, and as a result the

truncation of the lower tail caused by the structural failure

also occurs at the latter stage. Thus, the fully-correlated

deterioration model unavoidably overestimates the estimate

of current resistance, and this overestimation is enhanced

with the parabolic deterioration mechanism. As a result, it is

more important to choose a proper resistance deterioration

model to update the bridge resistance for the cases where the

dominant deterioration mechanism is parabolic, such as the

RC bridges subjected to sulfate attack.

s. (a) Square root deterioration. (b) Parabolic deterioration.

ls. (a) Square root deterioration. (b) Parabolic deterioration.

Table 5 e Updatedmean value and COV of R0 for differentdeterioration mechanisms.

Servicemodel

Statistics Prior Updated

Gammaprocess model

Fully-correlatedmodel

Square

root

Mean (kN$m) 4022 4108 4099

COV 0.1191 0.1112 0.1113

Parabolic Mean (kN$m) 4022 4068 4054

COV 0.1191 0.1144 0.1151

Table 6 e Updated mean value and COV of R(22) fordifferent deterioration mechanisms.

Servicemodel

Statistics Prior Updated

Gammaprocess model

Fully-correlatedmodel

Square

root

Mean (kN$m) 2614 2747 2766

COV 0.1801 0.1458 0.1465

Parabolic Mean (kN$m) 2614 2685 2789

COV 0.1801 0.1597 0.1604

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e84 83

Finally, it is noted that the proposed updating method in

this paper is based on the assumption of independent resis-

tance deterioration and loading processes, implying that the

effect of shock deterioration caused by extreme loads on the

updated resistance remains unaddressed, which deserves

more effort before the updating method becomes a practical

tool for bridge performance assessment.

5. Conclusions

A method has been developed in this paper to update the

resistance of existing aging bridges with prior service loads

employing a realistic deterioration model. For the purpose of

illustration, the proposed method is applied to an actual RC

beam bridge as suggested by Li and Wang (2015), based

on which parametrically representative examples are

conducted to show the difference between the updated

estimates of bridge resistances using different deterioration

models (i.e., the fully correlated and Gamma process-based

models). It is found that the deterioration model choice has

negligible impact on the updated estimate of bridge initial

resistance regardless of the dominant deterioration

mechanism. However, the difference between the updated

distributions of current resistance is significant posed by the

choice of deterioration model; such difference associated

with the parabolic deterioration type is largest, followed by

the linear and square root deterioration models. Moreover,

the fully correlated deterioration process may considerably

overestimate the updated mean value of current resistance

compared with the Gamma process-based model, indicating

that the unexpected risk will be induced in the probability-

based maintenance decisions if one employs the fully

correlated deterioration process.

Conflict of interest

The authors do not have any conflict of interest with other

entities or researchers.

Acknowledgments

This research has been supported by the National Natural

Science Foundation of China (Grant Nos. 51578315, 51778337),

the National Key Research and Development Program of

China (Grant No. 2016YFC0701404), and the Faculty of Engi-

neering and IT PhD Research Scholarship (SC1911) from the

University of Sydney. These supports are gratefully acknowl-

edged. The authors would like to acknowledge the thoughtful

suggestions of three anonymous reviewers, which substan-

tially improved the present paper.

r e f e r e n c e s

Dieulle, L., B�erenguer, C., Grall, A., et al., 2003. Sequentialcondition-based maintenance scheduling for a deterioratingsystem. European Journal of Operational Research 150 (2),451e461.

Ellingwood, B.R., 1996. Reliability-based condition assessmentand LRFD for existing structures. Structural Safety 18 (2e3),67e80.

Ellingwood, B.R., 2005. Risk-informed condition assessment ofcivil infrastructure: state of practice and research issues.Structure and Infrastructure Engineering 1 (1), 7e18.

Faber, M.H., Val, D.V., Stewart, M.G., 2000. Proof load testing forbridge assessment and upgrading. Engineering Structures 22(12), 1677e1689.

Fujino, Y., Lind, N.C., 1977. Proof-load factors and reliability.Journal of Structural Division 103 (4), 853e870.

Hall, W.B., 1988. Reliability of service-proven structures. Journalof Structural Engineering 114 (3), 608e624.

Hall, W.B., Tsai, M., 1989. Load testing, structural reliability andtest evaluation. Structural Safety 6 (2e4), 285e302.

Kumar, R., Gardoni, P., 2011. Modeling structural degradation of RCbridge columns subjected to earthquakes and their fragilityestimates. Journal of Structural Engineering 138 (1), 42e51.

Li, Q., Ellingwood, B.R., 2007. Performance evaluation and damageassessment of steel frame buildings under mainshockeaftershock earthquake sequences. EarthquakeEngineering and Structural Dynamics 36 (3), 405e427.

Li, Q., Wang, C., 2015. Updating the assessment of resistance andreliability of existing aging bridges with prior service loads.Journal of Structural Engineering 141 (12), 04015072.

Li, Q., Wang, C., Zhang, L., 2015a. Updating for bearing capacity ofexisting bridges considering structural deterioration andloading history. Journal of Tsinghua University Science andTechnology 55 (1), 8e13.

Li, Q., Wang, C., Ellingwood, B.R., 2015b. Time-dependentreliability of aging structures in the presence of non-stationary loads and degradation. Structural Safety 52,132e141.

Lin, X., Zong, Z., Niu, J., 2015. Finite element model validation ofbridge based on structural health monitoring. Part II:uncertainty propagation and model validation. Journal ofTraffic and Transportation Engineering (English Edition) 2 (4),279e289.

J. Traffic Transp. Eng. (Engl. Ed.) 2019; 6 (1): 76e8484

Melchers, R.E., Beck, A.T., 2017. Structural Reliability Analysis andPrediction. John Wiley & Sons, New York.

Mori, Y., Ellingwood, B.R., 1993. Reliability-based service-lifeassessment of aging concrete structures. Journal ofStructural Engineering 119 (5), 1600e1621.

Nowak, A.S., Lutomirska, M., Sheikh Ibrahim, F., 2010. Thedevelopment of live load for long span bridges. BridgeStructures 6 (12), 73e79.

Pang, L., Li, Q., 2016. Service life prediction of RC structures inmarine environment using long term chloride ingress data:comparison between exposure trials and real structuresurveys. Construction and Building Materials 113, 979e987.

Pipinato, A., 2016. Bridge assessment, retrofit, and management.In: Pipinato, A. (Ed.), Innovative Bridge Design Handbook.Butterworth-Heinemann, Oxford, pp. 721e757.

Saassouh, B., Dieulle, L., Grall, A., 2007. Online maintenancepolicy for a deteriorating system with random change ofmode. Reliability Engineering and System Safety 92 (12),1677e1685.

Saraf, V., Nowak, A.S., 1998. Proof load testing of deterioratedsteel girder bridges. Journal of Bridge Engineering 3 (2), 82e89.

Saydam, D., Bocchini, P., Frangopol, D.M., 2013. Time-dependentrisk associated with deterioration of highway bridgenetworks. Engineering Structures 54, 221e233.

Schweckendiek, T., 2010. Reassessing reliability based onsurvived loads. In: 32nd Conference on Coastal Engineering,Shanghai, 2010.

Stewart, M.G., 1997. Time-dependent reliability of existing RCstructures. Journal of Structural Engineering 123 (7), 896e902.

Stewart, M.G., 2001. Effect of construction and service loads onreliability of existing RC buildings. Journal of StructuralEngineering 127 (10), 1232e1235.

Stewart, M.G., Val, D.V., 1999. Role of load history in reliability-based decision analysis of aging bridges. Journal ofStructural Engineering 125 (7), 776e783.

Vardanega, P.J., Webb, G.T., Fidler, P.R.A., et al., 2016. Bridgemonitoring. In: Pipinato, A. (Ed.), Innovative Bridge DesignHandbook. Butterworth-Heinemann, Oxford, pp. 759e775.

Vu, K.A.T., Stewart, M.G., 2000. Structural reliability of concretebridges including improved chloride-induced corrosionmodels. Structural Safety 22 (4), 313e333.

Wang, C., Li, Q., Zou, A., et al., 2015. A realistic resistancedeterioration model for time-dependent reliability analysisof aging bridges. Journal of Zhejiang University Science A 16(7), 513e524.

Wang, C., Li, Q., Ellingwood, B.R., 2016. Time-dependent reliabilityof ageing structures: an approximate approach. Structure andInfrastructure Engineering 12 (12), 1566e1572.

Wang, C., Zhang, H., Li, Q., 2017. Time-dependent reliabilityassessment of aging series systems subjected to non-stationary loads. Structure and Infrastructure Engineering 13(12), 1513e1522.

Wipf, T.J., Phares, B., Klaiber, F., et al., 2003. Development ofBridge Load Testing Process for Load Evaluation. TR-445.Center for Transportation Research and Education, IowaState University, Arnes.

Cao Wang received his M.E. degree in July2015 from the Department of Civil Engi-neering, Tsinghua University. Currently, heis pursuing a PhD degree in the School ofCivil Engineering, The University of Sydney,Australia. His research interests includestructural reliability analysis and naturalhazards assessment.