Longterm condition assessment of suspenders under traffic loads … winn… · Structural health...

STRUCTURAL CONTROL AND HEALTH MONITORING

Long-term condition assessment of suspenders under trafficloads based on structural monitoring system: Application to the

Tsing Ma Bridge

Shunlong Li1,2, Songye Zhu1,�,y, You-Lin Xu1, Zhi-Wei Chen1 and Hui Li2

1Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong2School of Transportation Science and Engineering, Harbin Institute of Technology, Harbin, Peoples’s Republic of China

SUMMARY

Structural health monitoring (SHM) system provides an efficient way to the diagnosis and prognosis ofcritical and large-scale civil infrastructures like long-span bridges. This paper presents a long-termcondition assessment approach of suspenders in a cable-suspension bridge under in-service traffic loadsbased on structural monitoring technique. The traffic loads identified from a monitoring system, includingboth highway and railway traffic loads, and the finite element model of the bridge are employed todetermine the axial force response of the suspender. The stochastic axial force response in the suspender isdescribed by a filtered Poisson process, through which the maximum value distribution of axial forces in itsdesign reference period can be derived using the Poisson Process theory. In this paper, the long-termdeterioration process of steel wires in the suspender considers simultaneously the uniform and pittingcorrosion and the corrosion fatigue induced by both cyclic loading and environmental attack. Such astochastic and coupled corrosion fatigue process of steel wires is simulated using the Monte Carlo method,and the time-variant conditions of the suspender are subsequently assessed in a probabilistic way, such ascrack depth, number of broken wires, ultimate strength, etc. In particular, two load conditions—the trainloads alone and the combination of train load and road traffic load—are examined within this procedure inorder to investigate their respective effects on the deterioration. By employing first-order reliabilitymethod, the reliability indexes of the suspender under the traffic loads are further estimated in terms of thesafety under the extreme traffic load distribution in the design reference period and the serviceabilityspecified in the design specification. The discussions of the life-cycle reliability indexes of the suspenderprovide guidance to the future decision making related to maintenance and replacement of suspenders, andit may also shed light on the long-term condition assessment of other structural members. Copyright r2010 John Wiley & Sons, Ltd.

Received 11 March 2010; Revised 29 October 2010; Accepted 31 October 2010

KEY WORDS: long-term condition assessment; suspender; structural monitoring; reliability analysis

1. INTRODUCTION

Suspenders are always a critical and vulnerable type of structural components in a long-spancable-suspension bridge in normal operation conditions. The importance of their safety andserviceability has been recognized by highway administrations throughout the world in securingproper functions of cable-suspension bridges. In the past decades, considerable efforts have beendevoted to the condition assessment of steel cables. For example, Takena and Miki [1] presented

*Correspondence to: Songye Zhu, Department of Civil and Structural Engineering, The Hong Kong PolytechnicUniversity, Hong Hom, Kowloon, Hong Kong.yE-mail: [email protected]

Copyright r 2010 John Wiley & Sons, Ltd.

Struct. Control Health Monit. 2012; 19:82–101Published online 9 December 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/stc.427

fatigue tests of a parallel-wire large-diameter cable system and conducted a detailed review ofother fatigue results. Matteo and Deodatis [2] presented the methodology to estimate the currentsafety factor of the main suspension cable based on the laboratory testing of wire samplesextracted from real cables for the tensile strength and elongation. Mayrbaurl and Camo [3]studied the cracking and fracture of suspension bridge wires when water enters a suspensionbridge cable. Sih and Tange [4] studied the fatigue crack growth behavior of cables and steelwires for the Runyang bridge. Xu et al. [5] discussed the failure modes and mechanics ofdeteriorated steel wires. Rusk and Hoppe [6] developed the new model for corrosion damagedhigh-strength steel to predict its fatigue life. Toribio and Matos [7] dealt with the fatigue crackgrowth in high-strength wires and discussed the crack front progress by means of aspect ratioevolution with relative crack depth. Most of the previous studies concentrated on thedeterioration of high-strength steel wires based on laboratory testing. Very few studies havebeen reported on the condition assessment of suspenders in operation, which is essentialinformation to optimize their inspection and replacement in bridge maintenance practice.

Although reliability theory has been well established, there are still a few challenges in applyingreliability analysis to performance assessment of real suspenders, one of which is the difficulty ofaccurately modeling complex loading conditions and the corresponding response in suspenders.The epistemic uncertainty due to incomplete knowledge makes them hard to be theoreticallycharacterized, especially for larger-scale long-span bridges with great spatial and temporalcomplexity. The inaccuracy in analytical modeling considerably compromises the credibility of thelong-term condition assessment of suspenders and other structural members in cable-suspensionbridges. In the past two decades, structural monitoring has been gaining increasing attention allover the world. The significant progress in sensing technology and system identification techniqueoffers more convenient and reliable means of obtaining real-time structural information.Structural monitoring techniques can considerably improve the accuracy of long-term conditionassessment and reliability analysis in the sense that: (1) it replaces theoretical models of loadingconditions and structural responses with empirical models derived directly from measurementdata; (2) it offers an efficient way to validate or update the finite element model (FEM) used; and(3) it is able to timely capture any unexpected but potential change of structures, loadings orenvironmental conditions during the relatively long service life of bridges.

This paper presents an investigation of the long-term condition assessment of suspenders in acable-suspension bridge based on the installed monitoring system. Figure 1 summarizes theassessment procedure which comprises four major steps. The first step presents how to acquiresuspender response from the monitoring system. The structural monitoring systems collectmassive amounts of in situ data enabling the identification of traffic loads (highway and railway)and structural parameters (global and local). An updated FEM and traffic load models areemployed together to estimate the probabilistic model of suspender’s response, which is oftendifficult, if not impossible, to be directly measured by sensors. In the second step, the stochasticdeterioration process of the suspender under both highway and railway traffic loads is simulatedusing the Monte Carlo approach, in which a coupled corrosion fatigue process of steel wiresinvolving uniform corrosion, pitting corrosion and cyclic fatigue is taken into account. In thethird step, the time-variant conditions of the suspender due to corrosion fatigue are presented,including the ratio of broken wires, the distribution of crack depth, the remaining cross-sectional area and load capacity of the suspender. Subsequently, the reliability indexes of thesuspender under traffic loads are evaluated in terms of the safety and serviceability criteria.

2. STRUCTURAL MONITORING-BASED RESPONSE MODEL OF SUSPENDERS

2.1. FEM of the Tsing Ma bridge

The Tsing Ma Bridge is a cable-suspension bridge located in Hong Kong with an overall lengthof 2160m and a main span of 1377m (as shown in Figure 2). The height of the two towers is206m from the base level to the tower saddle. The two main cables of 36m apart areaccommodated by the four saddles located at the top of the tower legs in the main span, and


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LONG-TERM CONDITION ASSESSMENT OF SUSPENDERS

each cable has a cross section area of 0.759m2 in the main span and 0.801m2 in the side span.The Tsing Ma Bridge carries a dual three-lane highway on the upper level of the bridge deck andtwo railway tracks and two carriageways on the lower level within the bridge deck. The bridge

Monitoring System

Finite Element Model

Model updating

Identified load models

Response model under multi-loads

Uniform

corrosion

Pitting

corrosion

Crack

propagation

Reduction of cross-sectional area and strength in wires

(broken wires, crack depth & deteriorated strength)

Remaining sectional

area of the cable

Remaining resistance

of the cable

Serviceability Safety

Structural

Monitoring

Technique

Corrosion

fatigue

process

Condition

assessment

Reliability

analysis

Structural System

Figure 1. Condition assessment procedure based on structural monitoring.

Total No. of Grids = 10,914

Total No. of CBAR/ CBEAM = 24,234

Total No. of CQUAD4 = 42,906

Total No. of RBAR = 2,854

E21065

Figure 2. Three-dimensional finite element model of the Tsing Ma Bridge.


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deck is a hybrid steel structure 41m wide and 7.63m high. The bridge decks consist ofVierendeel cross frames supported on the two longitudinal trusses acting compositely withstiffened steel plates.

The bridge deck is supported by totally 94 suspenders that are of interest in this study. Thedistance between two suspender units is 18m along the longitudinal axis of the deck. Thesuspender ropes pass over the saddles on the main cable and are anchored by steel sockets to thechords of the bridge deck (as shown in Figure 3(a)). Each suspender unit consists of four standardIWRC (independent wire rope core) wire ropes with an outer diameter of 76mm. The crosssection and construction details of each wire rope are shown in Figures 3(b), (c) [8]. It can be seenfrom Figure 3 that each wire rope is composed of one core strand and six outer strands, denotedby Strand I, II,y,VII. Each outer strand is composed of 41 steel wires of 5 different diameters (i.e.D1, D3, D5, D6 and D8), whereas the core strand is composed of steel wires of 3 different diameters(i.e. D2, D4 and D7) that are distributed in two layers—IWRC layer and central core. There aretotally eight different diameters in the section of a wire rope. In this study, the eight diameters aresorted in an ascending order, denoted by Di, i5 1, 2,y, 8 as marked in Figure 3. Unlike the maincables in the bridge, all the suspenders are exposed to atmosphere directly without sheathprotections, which make the suspenders comparatively vulnerable to corrosion fatigue, acombination result of aggressive environment and cyclic loading.

A comprehensive structural monitoring system has been implemented on the Tsing MaBridge, and it comprises 280 sensors of different types, including anemometers, temperaturesensors, servo-type accelerometers, weigh-in-motion sensors, global positioning systems (GPS),level sensing stations, displacement transducers, strain gauges and CCTV video cameras. Suchstructural monitoring system has been continuously monitoring the loading conditions (e.g.wind, temperature and traffic loads) and bridge response since 1997. A sophisticated three-dimensional FEM of the Tsing Ma Bridge was established and calibrated at local and globallevels (as shown in Figure 2). The model updating was performed based on vibration testingresults and model identification technique. The FEM results are in excellent agreement with thedynamic properties identified from in situ measurements, e.g. first 18 natural frequencies andmode shapes. The details of the modeling, updating and validation can be found in literature[9,10]. It should be noted that although a large number of sensors are installed, it is stillimpractical to directly monitor and assess all critical members in consideration of the scale oflong-span bridges. Therefore, such a delicate FEM of the bridge is an essential tool to providean effective assessment of the bridge performance under service conditions, which is also themajor reason why structural monitoring systems have been increasingly embraced.

2.2. Traffic-induced axial force response in suspenders

The FEM enables the response estimation for the suspenders without sensors under a diversity ofloading. As major gravity load-carrying elements on the Tsing Ma Bridge, the suspenders

I

VVI

VII IV

IIIII

3.607 (D )

413.24

59.59

77.93

16

8+8

8

1

6

1

6

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StrandArea

A(mm )

Number ofWires

Outer

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Core

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King

1

King

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King

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3.302 (D )+2.438 (D )

3.658 (D )

6.350 (D )

3.150 (D )

3.607 (D )

3.962 (D )

WireDiameterD (mm)

StrandLayer

RopeLayer

Main cable

Wire rope

Anchor

Saddle

(a) (b) (c)

Figure 3. Construction details of the standard suspender cable of 76mm diameter: (a) a suspender unit; (b)cross section of a wire rope; and (c) construction details of a wire rope.


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experience a relatively small stress variation under wind load and temperature load. For example,Xu et al. [11] indicated that the stress range in suspenders is less than 8MPa under the mean windspeed of 50m/s at the bridge deck level, which is the design hourly mean wind speed at the site ofthe Tsing Ma Bridge. While in normal operation conditions, the stress variation in suspenders dueto wind load is much smaller and often ignorable. The temperature-induced stress variation in thesuspenders located at mid-span is typically less than 1MPa according to the literature [11].Therefore, only gravity loads, including dead load, superimposed dead load, highway traffic loadand railway traffic load, are taken into account in the suspender response analyses.

The highway load on the Tsing Ma Bridge is monitored by seven dynamic weigh-in-motionstations, and the measured information includes axle number, axle spacing, vehicle speed, vehiclecategory, lane number, arrival time, etc. The train load is identified through a set of strain gaugesinstalled on the inner way beam under two rail tracks. The bogie load data include the traindirection, train speed, the number of train bogies, bogie load and arrival time of each train. Moredetails about the traffic load monitoring on the Tsing Ma Bridge are described in the literature [12].

Axial force time histories in various members under highway and railway traffic loads can becomputed by means of the influence lines corresponding to railway tracks and highway laneswhich are derived from the FEM. In order to differentiate the effects of traffic loads, the axialforces in the suspenders are computed respectively under the railway load alone, and under thecombination of highway and railway loads. The results of the suspender E21065 (shown inFigure 2), which has been identified as the most critical suspender after axial force time historyanalyses, are presented hereinafter as an example. However, the presented method can beapplied to the conditions assessment of any other suspenders in the bridge.

2.2.1. Under railway load. Figure 4(a) illustrates a typical axial force time history in thesuspender E21065 under train load, as well as dead load and superimposed dead load. Figures4(b) shows the distribution of the time intervals between trains that can be modeled by anexponential distribution, while Figure 4(c) presents the distribution of the magnitudes of thepeak axial forces in axial force cycles, where we can observe two distinct local maxima. Thesecond maximum corresponds to the special moment when two trains pass over the bridgesimultaneously. Therefore, the peak axial forces, xM, can be modeled by a bimodaldistribution—a superposition of two weighted normal distributions:

FxM ðxMÞ ¼ p1FxM � mM1

sM1

� �1p2F

xM � mM2

sM2

� �ð1Þ

where p11p2 5 1, p140, p240, F( � ) represents the cumulative probability function of standardnormal distribution. The parameters of the model can be evaluated by the maximum likelihoodestimation as p1 5 0.962, p2 5 0.038, mM1 5 621.92, mM2 5 644.31, sM1 5 2.98 and sM2 5 5.70.Both empirical and theoretical distributions are shown in Figures 4(c), and it is seen that thelatter match the former quite well.

According to the literature [13,14], such a stochastic process of trainload-induced responsecan be described by a Filtered Poisson Process, which is used in Section 3 to simulate thestochastic railway load. In order to conduct safety assessment in Section 4, it is also necessary toestimate the extreme value distribution of loading effects in a certain period. For this purpose,the second peak in Figure 4(c) would apparently govern the extreme value distribution. Thestochastic process corresponding to the second peak can be treated as an independent FilteredPoisson Process. Following the recommendation by Miao [13], the extreme value distribution oftrainload-induced axial force in a certain period can be estimated by

FMðxÞ ¼ exp �lp2Ts 1� Fx� mM2

sM2

� �� ð2Þ

where FM(x) is the cumulative distribution function of the extreme value, the Poisson parameter lrepresents the average occurrence rate per second, which is determined from the time intervaldistribution (l5 0.0066 in this section); Ts is the period in unit of second. Figure 4(d) shows theprobability density functions of extreme axial force distributions in different service periods, namely


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1 week, 1 year and 120 years, where 120 years represent the design reference period of the Tsing MaBridge. It should be pointed out that the accuracy of the extreme value distribution is often sensitiveto the probabilistic model of traffic loads which is determined from the observed data. Theapproach recommended by Xu and Chen [12] may not be accurate enough when the monitoringperiod is limited. However, the accuracy of the extreme value distribution could be considerablyimproved by long-term structural monitoring data. With the continuous accumulation of trafficmonitoring data in future, more accurate extreme value distribution would be directly inferred fromthe annual maximum data in future. The condition assessment approach for suspenders presented inthis study could be easily adapted to the new probability model.

2.2.2. Under railway load1highway load. This subsection characterizes the axial force responsein the suspender under the railway and highway traffic loads simultaneously in order to examinethe contribution of highway load to the deterioration process of the suspender. Figure 5(a)illustrates one-week axial force time history caused by the railway and highway loads collectedby the structural health monitoring (SHM) system. Similar to the response under the train loadalone, the distribution of the time intervals can be modeled by an exponential distribution, andthe stochastic process of the axial force response can be simulated by a Filtered Poisson Process(Poisson parameter l5 0.1067 in this section). Through the fitting analysis, the peak magnitudesof the axial force, xM, caused by dual traffic loads, can be approximately modeled by asuperposition of two weighted Generalized Extreme Value distributions and one weightednormal distribution as follows:

FxM ðxMÞ ¼p1 1� exp � 11xM1

x� mM1

sM1

� �� 1xM1

" #( )

1p2 1� exp � 11xM2

x� mM2

sM2

� �� 1xM2

" #( )1ð1� p1 � p2ÞF

xM � mM3

sM3

� �ð3Þ

0 24 48 72 96 120 144 168590

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Axi

al f

orce

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ende

r (k

N)

(a)0

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abili

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ensi

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Empirical probability density Theoretical probability density Empirical cumulative distribution Theoretical cumulative distribution

(b)

6150.00

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ensi

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(c) Axial force of suspender (kN)

Cum

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ive

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6300.00

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(d)

Origin1 Year120 Year

Axial force of suspender (kN)

Prob

abili

ty D

ensi

ty

100 200 300 400 500 600

620 625 630 635 640 645 650 655 635 640 645 650 655 660 665 670 675 680

Figure 4. Axial force response under railway load: (a) time history in one week; (b) distribution of timeintervals; (c) distribution of maximum axial forces; and (d) extreme value distribution in different return periods.


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where p1 5 0.910, p2 5 0.088, xM1 5�0.087, xM2 5�0.059, mM1 5 600.20, mM2 5 624.72,mM3 5 651.96, sM1 5 3.38, sM2 5 4.78, sM3 5 5.57. Similar to Equation (2), the extreme valueof the axial force responses in a certain period can be calculated, and the results are illustrated inFigure 5(d). In Figure 5(c), the first peak is mainly caused by highway traffic, whereas theother two peaks are mainly due to train load. Although the highway traffic-induced stressrange is smaller than that due to train load, its occurrence frequency is much higherthan train load. Furthermore, the maximum value distributions shown in Figure 5(d) areslightly larger than that in Figure 4(d), because of the superposition effect of the train load andhighway load.

3. DETERIORATION PROCESS OF SUSPENDER UNDER TRAFFIC LOADS

3.1. Theory for corrosion fatigue process

Corrosion is a common form of steel deterioration as a result of chemical and electrochemicalactions. For the cable corrosion process, two common types of corrosion, namely uniformcorrosion that is presumed to be uniform on a wide zone and pitting corrosion that may belocally concentrated. On the other hand, fatigue is a type of material damage resulting fromrepeated stress applications such as cyclic loading. In aging steel cables, local pits due tocorrosion may provide sites for fatigue crack initiation. In addition, fatigue crack propagationrate can be accelerated by some corrosive agents (e.g. seawater). Such a coupled corrosion andfatigue process is considered as one of the most severe deterioration mechanisms. Corrosionfatigue deterioration process can be basically divided into three stages: (1) corrosion initiationand pit nucleation; (2) pit to crack transition; (3) corrosion fatigue crack growth. The first stagecovers the time when the zinc layer is depleted and the inner bare steel suffers from uniform

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0.35 Origin 1 Year 120 Year

(d) Axial force of suspender (kN)

Prob

abil

ity

Den

sity

Figure 5. Traffic-induced axial force analysis: (a) time history; (b) distribution fitting of time interval;(c) distribution of axial force; (d) effect of return period on probability density.


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corrosion and pit corrosion. Pit nucleation and growth involve electrochemical process affectedpredominantly by environmental factors. The transition from pit growth to corrosion fatiguecrack is governed by the competition between the processes of pit growth and corrosion fatiguecrack growth. Two optional criteria may be taken to describe the transition process [15,16]: (1)the stress intensity factor of the maximum pit depth reaches the threshold for corrosion fatiguecrack nucleation; (2) the corrosion fatigue crack rate exceeds the pit growth rate. In this study,the models for pit growth and corrosion fatigue crack are investigated and the rate competitioncriterion is employed. In the last stage, corrosion fatigue crack growth depends on the cyclicloading, and it can be evaluated by the linear elastic fracture mechanics (LEFM). It should beemphasized that localized pit corrosion causes significant stress concentration, and thusaccelerates the initiation and prorogation of cracks at critical locations. For example, water ormoisture tends to accumulate at the bottom socket connections of the suspenders whosecompact details make proper maintenance difficult. Therefore, the suspender bottom end in acollar where the environmental conditions are often favorable to corrosion is identifiedas the most critical location in a suspender [1,17]. According to the literature [1], in realcable bridges, the fatigue phenomena of cables only occur at the location of anchors. In thisstudy, the suspender E21065 was identified as the most critical suspender subjected to the largeststress amplitude under traffic loads. Hence, the deterioration process at the section close to thebottom anchor in the suspender E21065 is simulated in the following section. Its entiredeterioration process would be the results of coupled actions of cyclic loading in a corrosiveenvironment.

3.1.1. Uniform corrosion model. Previous investigations have shown that the uniform corro-sion depths of high-strength steel wires in a cable are mainly influenced by the corrosiontime, the corrosion velocity of galvanized layer and inner steel. The exponential model[18,19] is one of widely used empirical models for predicting the corrosion depth in steelcomponents

uðtÞ ¼ Cðt � t0Þg ð4Þ

where u(t) is the corrosion depth at time t(in years); t0 represents the time when zinc layeris consumed and steel starts to corrode; C and n are model parameters depending on metaltype and environment. References [20,21] indicate the initiation time t0 is dependent ondifferent local environment conditions (e.g. humidity and temperature), which may beaffected by sunshine and locations of steel wires. According to experimental results bySuzumura and Nakamura [20] and Furuyu et al. [21] the initiation time t0 can be categorizedinto four categories dependent on local environment conditions of steel wires: 1.7 yearif constantly wet, 1.7 year if cyclically wet and dry, 6.9 year if soaked in the water and 34 yearif highly humid day and night. Different parts of a cable section are often correspondingto different categories or local environment conditions [20]. In this study, the local environmentis assumed to be cyclically wet and dry for the outer strands, and highly humid day and nightfor the core strand. The corrosion process is a spatial and temporal random process.When bridges are situated in the marine environment (e.g. bay bridges), NaCl is the mostaggressive medium in bridge operational environment. Lan [22] studied the uniform corrosionof a cable-stayed bridge near the Gulf of Chihli, and their data distribution are analyzed andadopted in this study, as no in situ corrosion data are available for the suspender of interest. Thevalue of g is taken as 0.5, and the corrosion rate C is proved to follow lognormal distributionwhose expected value and coefficient of variation are equal to 7.91 mm/year and 1.14,respectively.

3.1.2. Pit corrosion model. The studies [23,24] indicated that the penetration of pitting can betreated as a random variable described by the extreme value theory. The ratio of maximumpenetration of pitting a(t) to uniform corrosion depth u(t), i.e. k5 a(t)/u(t), follows the Extreme


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Value Type I distribution, and its cumulative distribution function can be expressed as

F ðkÞ ¼ exp � exp �ðk� b0Þ

a0

� �� ð5Þ

where a0 and b0 are the distribution parameters. Hawn [23] estimated these two parameters forwires with 125mm length and 8mm diameter based on the experimental study, a0 5 1.02,b05 5.08. Then the distribution parameters for wires with any given length can be expressed as [23]

bk ¼ b011

a0ln

Ak

A0

� �; ak ¼ a0 ð6Þ

where Ak represents the surface area of a given high-strength wire; A0 is the surface area of a wirewith 125mm length and 8mm diameter. The finite difference format of the pit growth rate can beexpressed as

da ¼ aðt11Þ � aðtÞ ¼ k uðt11Þ � uðtÞ½ � ð7Þ

where the pit growth rate remains a random variable during the service life of steel wires.

3.1.3. Corrosion fatigue crack growth. The fatigue crack grows under cyclic stress variationinduced by traffic loads. In this study, the axial force response in suspenders is simulated usingthe Monte Carlo simulation based on the Filtered Poisson Process model presented in the lastsection. The corrosion fatigue crack growth of corrode wires was estimated at the central(deepest) point of the crack front. For the environmental-assisted fatigue situations, the cracksgrow only when the stress intensity factor range DK is larger than the corrosion fatigue threshold[25], and in this study the threshold for high-strength steel wires is taken as 2:8 MPa �

ffiffiffiffim

paccording to Xu et al. [5]. The test results of high-strength steel wires by Huneau and Mendez [26]indicated that the Paris–Erdogan law [27] can be used for the corrosion fatigue crack growthevaluation

dadN¼ CDKm ð8Þ

where da/dN indicates the crack growth rate; m and C are the exponent and coefficientof the Paris–Erdogan law, respectively. DK5Kmax�Kmin demonstrates the stress intensityfactor range. The stress intensity factor range DK at center point of crack front can becomputed by

DK ¼ YDsffiffiffiffiffiffipa

pð9Þ

where Ds indicate the axial stress range in the wire; a represents the depth of crack at time t. Thedimensionless stress intensity factor Y for the geometry provided by Astiz [28] was employed,which was generated by the polynomial fitting of the finite element analysis results of cylindricalcracked bars in tension. The stress intensity factors are expressed as a function of the relativecrack depth (a/D) and the crack aspect ratio (a/b)

Y ða; b;DÞ ¼X4

i¼0i6¼1

X3j¼0

Yijða=DÞiða=bÞj ð10Þ

where the value of coefficients Yij can be found in Reference [28]. Reference [7] investigatedthe propagation of fatigue crack front in steel wires and bars through fatigue crackingtests at room temperature, and Figure 6(a) shows the optical-microscopy picture of onefracture surface obtained by Toribio and Matos [7]. Thus, the crack front could bemodeled as an ellipse with its center located at the wire surface. As shown in Figure 6(b), a(i.e. crack depth) and b are semi axes of the ellipse. The diameter of corroded steel wires D isdemonstrated as

D ¼ D0 � 2uðtÞ ð11Þ

where D0 is the initial diameter of the steel wires.


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The crack growth rate due to corrosion fatigue often consists of three regions dependingon the amplitude of stress intensity factor DK, and each region is generally modeled by a powerlaw defined by Equation (8) [26]. In the first region, the crack growth rate due to corrosionfatigue is higher than that in air; the second region is associated with a frequency-dependentplateau that implies a nearly constant crack growth rate; while in the third region thecrack growth rate is similar to that in air. However, Huneau and Mondez [26] conductedcorrosion fatigue tests of high-strength steel wires with a diameter of 5mm in 3.5% NaClsolution, and measured crack growth rates of small cracks. Different power-law regions werenot observed in their testing results, because the crack growth rates remain below the valuecorresponding to the plateau observed in long crack da/dN–DK curves [26]. It should be notedthat 3.5wt% NaCl solution is a common and rational selection in various corrosion test tosimulate real corrosion environments of offshore bridges (e.g. [29,30]). In this study, the wirediameters are from 2.44 to 6.35mm, and the estimated ranges of Ds and sK are even smallerthan the values used in [26]. Hence, it is assumed that only small cracks will be developed inthese high-strength steel wires, and the crack growth rates can be appropriately modeled by asingle power law. The parameters C5 5.50� 10�12 and m5 3.9 obtained in [26] are adoptedin this study. Following [31], the parameter m is treated as a deterministic value, whereasthe parameter C is taken as a lognormally distributed variable with coefficient of variationequal to 0.1.

When the corroded wire is subjected to the ith stress cycle Dsi, the crack depth ai thatpropagates from ai�1can be computed by

ai ¼ ½0:5ð2� mÞCðYDsffiffiffip

pÞm1a0:5ð2�mÞ

i�1 �2

2�m ðm 6¼ 2Þ ð12Þ

As mentioned before, the pit growth and fatigue crack growth can be treated as twocompeting processes [7]. In each time step, the possible crack growths da due to pit growth andfatigue crack growth are calculated, respectively, and the actual crack front propagation isgoverned by the larger one. This actual crack front propagation will be used to compute the pitgrowth and the fatigue crack growth in the next step.

3.1.4. Ultimate force of the corrode wire. Because of excessive crack propagation, the steel wiresmay fracture finally when their stress intensity factor reaches the fracture toughness KC for cablewires. In this study, KC is assumed to be a normally distributed random variable. Determinedfrom the experiment results in [3], the mean value of �KC ¼ 62:2MPa �

ffiffiffiffim

pand the coefficient of

variation 0.159 were employed in this study.To evaluate the ultimate load-carrying capacity of the cable later on, the time-variant

ultimate load capacity of deteriorated high-strength steel wires is calculated by

rðl;iÞðtÞ ¼ sðl;iÞu ðtÞAðl;iÞr ðtÞ ð13Þ

(a) (b)

Figure 6. Pit configuration and fatigue growth of surface cracks [7]: (a) crack fronts in a hot rolled steel barand (b) crack modeling.


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where su and Ar represent, respectively, the ultimate stress and remaining cross-sectional area ofsteel wires with cracks, and the superscripts illustrate that the deterioration process is dependenton the locations (l5 Strand I,II,y,VII) and the diameters (i5 1,2,y,8) of steel wires. Theultimate strength su of deteriorated high-strength wires is affected by the existence of unilateralcrack geometry, and it can be estimated by the toughness criterion [31]. The ultimate strength ofcracked wire su is calculated according to the following relationship [32]:

su ¼KC

Y ðac; bc;DÞ �ffiffiffiffiffiffiffipac

p ð14Þ

where Y(ac, bc, D) is the normalized stress intensity factor determined by Equation (10); ac, bcare the critical crack depth and crack width when the axial stress equals to the ultimate strengthsu; KC is fracture toughness for cracked bridge wires, which should be different from theASTM-based KIC value due to the wire size considerations [33]. Therefore, the KC valueobtained in fracture tests of degraded suspension bridge wires [3] is used in this study.

As mentioned before, the crack front is modeled as an ellipse, and thus the remaining crosssection area Ar of the corroded steel wires can be estimated through the geometry shown inFigure 6(b). The intersecting point of the crack front and the circumference of uncorroded corehas the coordinate,

xA ¼�a2D1a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2D214b4 � 4a2b2

p2ðb2 � a2Þ

; yA ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDxA � x2A

q; a 6¼ b

xA ¼a2

D; yA ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � x2A

q; a ¼ b

ð15Þ

The remaining sectional area Ar of the deteriorated steel wires due to corrosion fatigue can beexpressed as

Ar ¼pD2=4� ðA1 � A2Þ; xApD=2A3 � A1; D=2oxA and apD0; a4D

8<: ð16Þ

where

A1¼ 2

Z yA

0

ab

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2�y2

pdy; A2¼ 2

Z yA

0

D�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2�4y2

p2

dy; A3 ¼ 2

Z yA

0

D1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2�4y2

p2

dy ð17Þ

3.2. Results and discussions

The stochastic corrosion fatigue processes under two load cases—railway load alone and thecombination of railway and highway loads—are simulated using the Monte Carlo approach, forsteel wires with different diameters (i5 1,2,y,8) and different locations (l5 Strand I,II,y,VII).Some representative results are summarized and discussed in this section.

3.2.1. Deterioration under railway load. Figures 7 and 8 show the statistical results underrepeated railway load actions for steel wires with a diameter of 3.607mm in Strand II (i.e.l5 Strand II and i5 4). Figure 7(a) shows the distribution of corrosion fatigue crack depth after120 year operation time, and Kolmogorov–Smirnov (K-S) Hypothesis Testing at a significancelevel of 5% indicates that the distribution does not reject the Weibull Distribution. Thecumulative probability function of the Weibull distribution is

FaðxÞ ¼ 1� exp �xZ

� �m� �ð18Þ

The theoretical distribution functions are also shown in Figure 7(a). Figure 7(b) illustrates thecrack evolution of the steel wires with time, including the mean value and standard deviation ofthe crack depth, and the mean value of the ratio (or percentage) of broken wires. It can be seenthat all three variables increase nonlinearly over time.


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Figure 8(a) illustrates the distribution of ultimate load capacity of the corroded wires at 120-year,while Figure 8(b) shows its evolution during bridge’s service life. It can be seen that the corrosionfatigue process not only leads to load capacity reduction but also increases the dispersion in wires’load capacity. K-S test at a significance level of 5% indicates that the distribution of the ultimatestrength does not reject the normal distribution, and the theoretical distributions are shown inFigure 8(a).

3.2.2. Deterioration under railway and highway loads. Figures 9 and 10 show the simulatedcorrosion fatigue process of steel wires under both railway and highway load actions. Again, thesteel wires with a diameter of 3.607mm in Strain II (i.e. l5 Strand II and i5 4) are shown as arepresentative case. In comparison with Figures 7 and 8, similar qualitative observations can bemade from Figures 9 and 10. However, slightly higher ratio of broken wires and standarddeviation of the remaining wire load capacity can be observed in Figures 8(b) and 10(b) becauseof the superposition of highway traffic action.

Figure 11 summarizes the time-variant coefficients of variation of remaining wire loadcapacity for steel wires with different diameters. The trend that the variability of remaining loadcapacity increases over time can be observed in all diameters. In particular, the wires of smallerdiameter are associated with larger coefficients of variation.

Figure 12 compares the effect of two load cases—the railway load, and the combination ofthe railway and highway loads. Figure 12(a), (b) show the distributions of the 120-year crackdepth and ultimate load capacity for the aforementioned steel wires. It can be seen that thesuperposition of highway load effect slightly accelerates the crack propagation, and adversely

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20.00

0.03

0.06

0.09

0.12

0.15

0.18

Cum

ulat

ive

Prob

abili

ty

Prob

abili

ty D

ensi

ty

(a) Wire Crack Depth (mm)

0.0

0.2

0.4

0.6

0.8

1.0


20 40 60 80 100 120 1400.0

0.3

0.6

0.9

1.2

1.5

Rat

io o

f B

roke

n W

ire

(%)

Operation Time (Year)

Wir

e C

rack

Dep

th (

mm

)

(b)

Mean STD Ratio

0

1

2

3

4

5

Figure 7. Evolution of wire crack depth under railway load (D5 3.607mm, l5 Strand II): (a) distributionof crack depth at 120 year and (b) evolution of crack depth and broken wire ratio over time.


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affects the crack depth and ultimate load capacity. However, the difference is nearly ignorable inFigure 12, which implies that the corrosion fatigue process of the suspenders on the Tsing MaBridge is dominated by the stress cycles induced by train loads. Although the occurrencefrequency of highway loads is considerably higher than railway loads, the stress range inducedby the former one is much smaller (as shown in Figure 4 and 5), which imposes minimalinfluence on the corrosion fatigue process in this study. Similar observations can also be made tothe wires of other diameters.

4. CONDITION ASSESSMENT AND RELIABILITY ANALYSIS

4.1. Ultimate capacity prediction of suspender

Daniels [34] and Phoenix [33] showed that the strength of fiber/wire bundle would follow thenormal distribution. And the cables or suspenders would be approximately modeled byparalleled steel wires bundles. In this study, an alternative methodology is employed and thetime-variant ultimate load-carrying capacity of the suspender cable is calculated by the synthesisof the remaining load capacity of the unbroken steel wires of different diameters Di

(i5 1,2,y,8) at different locations (l5 Strand I, II,y, VII). With the number of broken wireskðl;iÞðtÞ at time t (obtained in the Monte Carlo simulations) among totally N ðl;iÞ in each category,where the superscripts represent the corresponding location and diameter, the remaining area

0 2 4 6 8 10 12 14 160.00

0.04

0.08

0.12

0.16

0.20

Ultimate Capacity of Steel Wire (kN)

Cum

ulat

ive

Prob

abil

ity

Prob

abil

ity D

ensi

ty

(a)

0.0

0.2

0.4

0.6

0.8

1.0


20 40 60 80 100 120 1400

5

10

15

20

(b)

Stan

dard

Dev

iati

on (

kN)


Ulti

mat

e C

apac

ity

of S

teel

Wir

e (k

N)

2.30

2.35

2.40

2.45

Mean STD Mean STD

Figure 8. Evolution of ultimate load-carrying capacity of corroded steel wires under railway load(D5 3.607mm, l5 Strand II): (a) distribution of ultimate strength at 120 year and (b) evolution of ultimate

strength over time.


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and load capacity of unbroken wires can be estimated by

AðtÞ ¼X

l

Xi

Xj

Aðl;iÞr ðtÞ; RðtÞ ¼X

l

Xi

Xj

rðl;iÞðtÞ

tXt0 j ¼ 1; 2; . . . ;N ðl;iÞ � kðl;iÞð19Þ

where A(t) and R(t) are the remaining cross-sectional area and the resultant load-carryingcapacity of the deteriorated suspender, Aðl;iÞr and rðl;iÞ represent the cross-sectional areas andultimate strength of individual wires which follow the distributions described before. As steelwires in one strand are supposed to be in a similar local environment, it is assumed that thecorrosion fatigue processes of steel wires are fully correlated in one strand but are independentamong different strands. Figure 13 shows the evolution of the suspender’s conditions (includingthe loss of the net area and the reduction of load-carrying capacity) in the design referenceperiod of the bridge due to the corrosion fatigue-induced deterioration. The results are alsoobtained from the Monte Carlo simulations.

4.2. Reliability assessment under traffic loads

4.2.1. Serviceability assessment. According to China’s Technical Code of Maintenance for CityBridge (J281-2003), the cable in cable-supported bridges should be replaced when the loss ofcable’s cross-sectional area caused by corrosion exceeds 10%. This criterion is taken as the

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20.00

0.03

0.06

0.09

0.12

0.15

Cum

ulat

ive

Prob

abili

ty

Prob

abili

ty D

ensi

ty

(a) Wire Crack Depth (mm)

0.0

0.2

0.4

0.6

0.8

1.0


20 40 60 80 100 120 1400.0

0.3

0.6

0.9

1.2

1.5

Rat

io o

f B

roke

n W

ire

(%)


Wir

e C

rack

Dep

th (

mm

)

(b)

Mean STD Ratio

0

1

2

3

4

5

Figure 9. Evolution of wire crack depth under railway and highway loads (D5 3.607mm, l5 Strand II):(a) distribution of crack depth at 120 year and (b) evolution of crack depth and broken wire

ratio over time.


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serviceability limit state of the suspender examined in this paper. Based on the time-variantdistributions of cross-sectional area loss of the suspender due to corrosion fatigue, the long-termreliability corresponding to the serviceability is assessed in the service life of the bridge, and the

0 2 4 6 8 10 12 14 160.00

0.03

0.06

0.09

0.12

0.15

0.18

0.21

Ultimate Capacity of Steel Wire (kN)

Cum

ulat

ive

Pro

babi

lity

Pro

babi

lity

Den

sity

(a)

0.0

0.2

0.4

0.6

0.8

1.0


20 40 60 80 100 120 1400

5

10

15

20

(b)

Stan

dard

Dev

iatio

n (k

N)


Ult

imat

e C

apac

ity

of S

teel

Wir

e (k

N)

Mean STD

2.25

2.30

2.35

2.40

2.45

2.50

Figure 10. Evolution of ultimate load-carrying capacity of corroded steel wires under railway and highwayloads (D5 3.607mm, l5 Strand II): (a) distribution of ultimate strength at 120 year and (b) evolution of

ultimate strength over time.

100.16

0.20

0.24

0.28

0.32 2.438mm 3.150mm 3.302mm 3.607mm 3.658mm 3.785mm 3.962mm 6.350mm

Coe

ffic

ient

of V

aria

nce

Operation time (Year)

20 30 40 50 60 70 80 90 100 110 120

Figure 11. Coefficient of variation of wire load-carrying capacity for different diameters under railwayload (l5 Strand II).


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deterioration time history is shown in Figure 14(a). The reliability index of the serviceability ofthe suspender is less than zero after 50-year service.

4.2.2. Safety assessment. The limit state function of safety assessment for existing or newly builtbridges can be expressed as

ZðtÞ ¼ RðtÞ � SðtÞ ð20Þ

in which R(t) and S(t) represent the stochastic processes of resistance and load effects,respectively. The load effects S can be described as the linear or nonlinear functions of single orcombination of several load effects. For the aforementioned reasons, only highway and railwaytraffic loads are taken into account in the safety assessment of the suspender of interest. In thisstudy, the extreme value distribution of traffic load effects in the service life of the Tsing MaBridge, i.e. 120 years, is employed, and the safety of the suspender under such extreme load caseis assessed. The resistance R is a time-variant random variable due to the above-describedcorrosion fatigue process. Consequently, the reliability of the suspender’s safety under theextreme traffic loads is a function of time t, and it can be computed by

PsðtÞ ¼ PfZðtÞ40g ¼ PfRðtÞ4SðtÞ; t 2 ½0;t�g ð21Þ

In this study, the corresponding reliability index b can be computed iteratively using theHasofer–Lind–Rackwitz–Fiessler algorithm [35,36]. Figure 14(b) illustrates the evolution of the

0.00.00

0.03

0.06

0.09

0.12

0.15

(a)

Prob

abili

ty D

ensi

ty

Wire Crack Depth (mm)

Traffic LoadingTrain Loading

0.9

0.000

0.005

0.010

0.015

0.020

0.025

0.030

00.00

0.03

0.06

0.09

0.12

0.15

0.18

(b)

Prob

abili

ty D

ensi

ty

Ultimate axial force of wire (kN)

Traffic Loading Train Loading

0.3 0.6 0.9 1.2 1.5 1.8 2.1

2 4 6 8 10 12 14 16

1.2 1.5 1.8 2.1

Figure 12. Probability density comparison between train loading and traffic load after 120 years(D5 3.607mm, l5 Strand II): (a) wire crack depth and (b) ultimate axial force of wire.


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reliability index in terms of the suspender’s safety during the bridge design life. The reliabilityindex drops considerably in the early service period, while the deterioration speed would becomeslow in the subsequent service period. At the end of the service period, the smallest reliabilityindex is estimated to be higher than 4, implying a good safety margin of the suspender under theextreme traffic loads during the entire design life.

Through the comparison of Figures 14(a), (b), it can be observed that the reliability in termsof the serviceability limit state is much lower than that of the safety limit state. As a result,the maintenance and replacement of the suspender will be likely governed by the service-ability criterion if applied. It should be emphasized that such a serviceability criterionin the design code is typically specified for stay cables that often experience much higherstress level in comparison to the suspender in this study. The results of this study imply thatthe direct adoption of this criterion may result in a very conservative condition for thesuspenders.

It should be pointed out that the condition assessment and reliability analysis for suspendersin this study are based on the past monitoring data of traffic loads. The possible changes oftraffic loads in the long service life of the bridge are not taken into account. With long-term real-time structural monitoring system, however, such unexpected changes would be timely identifiedand characterized. As a result, more accurate assessment would be obtained using the updatedstochastic models in future.

0 100 200 300 400 500 600 700 800 9000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

(a)

Pro

babi

lity

Den

sity

Area Loss of Rope (mm2)

20 Year 40 Year 80 Year 120 Year

1000 1500 2000 2500 3000 3500 4000 45000.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

(b) Ultimate Axial Force of Rope (kN)

Pro

babi

lity

Den

sity

20 Year 40 Year 80 Year 120 Year

Figure 13. Long-term condition assessment of the suspender subjected to corrosion fatigue: (a) the loss ofcross sectional area over time and (b) the reduction of load-carrying capacity over time.


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5. CONCLUSION

This study presents a long-term condition assessment approach of suspenders in a cable-suspension bridge, the Tsing Ma Bridge, based on structural monitoring techniques. With theSHM-identified traffic loads and the FEM of the bridge, the response of the suspender withoutdirect measurement is estimated under in-service traffic loads, including both highway andrailway traffic loads. The results indicate that the axial force distribution under both highwayand railway traffic loads can be well fitted by the superposition of three weighted distribution.By modeling the axial force response of the suspender with a filtered Poisson process, themaximum value distribution of axial forces in different periods can be derived.

Subsequently, the long-term stochastic deterioration process of a steel suspender cable undertraffic loads is simulated using the Monte Carlo method, in which a coupled corrosion fatigueprocess involving uniform corrosion, pitting corrosion and cyclic stress-induced fatigue is takeninto account. The time-variant conditions of steel wires (such as crack depth, number of brokenwires, ultimate strength) are evaluated in a probabilistic way within this procedure, and thedeteriorating conditions of the suspender cable, including the loss of the cross-sectional area andthe remaining load-carrying capacity are obtained by synthesizing the results of steel wires. Thecomparative study indicates that the corrosion fatigue process of the studied suspender isdominated by the railway traffic load, upon which the highway traffic load imposes nearlyignorable effect despite its much higher occurrence frequency than the railway load.

-2

0

2

4

6

8

(a)

Rel

iabi

lity

Inde

x β

104

5

6

7

8

9

10

11

12

(b)

Rel

iabi

lity

Inde

x β


20 30 40 50 60 70 80 90 100 110 120

10


20 30 40 50 60 70 80 90 100 110 120

Figure 14. Long-term reliability analysis of the suspender: (a) serviceability assessment and (b) safetyassessment under extreme traffic loads in 120 years.


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The life-cycle reliability of the suspender in terms of the serviceability and safety is computedand discussed based on the time-variant conditions of the suspender (the loss of cross-sectionalarea and the remaining resistance). The serviceability limit state is defined with regard to theratio of broken wires while the safety reliability is assessed under the extreme value distributionof the axial force in the design reference period of the bridge. The life-cycle safety reliabilityindex shows a good safety margin of the suspender under extreme traffic loads in the wholeservice life, while the serviceability reliability index is much lower, which implies that theperiodic maintenance and replacement are necessary in the service life. It is pointed out that thisserviceability criterion in the design code may be too conservative for the suspender of interest inthis study, as it is typically intended for stay cables that are often associated with higher stresslevel. Although only the results of the most critical suspender E21065 are presented in this study,the same method can be directly applied to any other suspenders in the bridge. If the spatialcorrelation of corrosion fatigue process is available in future through in situ measurement, thisstudy could be further extended to the system reliability of the whole bridge based on thereliability of each suspender element.

In practice, the stochastic structural response under multiple external loads and the possiblechanges of external loads in the long service life of bridges are often difficult to be characterizedand modeled without in situ measurement. The presented approach to incorporate real-timestructural monitoring data into condition assessment and reliability analysis would lead to abetter description of deterioration trend of various structural members in their service life.

ACKNOWLEDGEMENTS

The authors are grateful for the financial support from The Hong Kong Polytechnic University through aNiche Areas Funding Scheme (PolyU-1-BB6X) and from NSFC (Grant Nos. 51008096 and 50830203).Findings and opinions expressed here, however, are those of the authors alone, not necessarily the views ofthe sponsors.

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