Engineering Structures 33 (2011) 3246–3256

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Engineering Structures

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Fatigue analysis of long-span suspension bridges under multiple loading:Case study

Z.W. Chen a,b, Y.L. Xu a,∗, Y. Xia a, Q. Li c, K.Y. Wong d

a Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, Chinab Department of Civil Engineering, Xiamen University, Xiamen, Chinac Department of Bridge Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, Chinad Bridge & Structures Division, Highways Department, Tsing Yi, Hong Kong, China

a r t i c l e i n f o

Article history:Received 2 March 2011Received in revised form9 August 2011Accepted 16 August 2011Available online 29 September 2011

Keywords:Dynamic stressFatigueSuspension bridgesStructural health monitoringWind loadingRailway loadingHighway loading

a b s t r a c t

Long-span suspension bridges are often subject to multiple types of dynamic loads, especially thoselocated in wind-prone regions and carrying both trains and road vehicles. Fatigue assessment shallbe performed to ensure the safety and functionality of the bridges. This paper proposes a frameworkfor fatigue analysis of a long-span suspension bridge under multiple loading by integrating computersimulation with structural health monitoring system. By taking the Tsing Ma Bridge in Hong Kong as anexample, a computationally efficient engineering approach is first proposed for dynamic stress analysis ofthe bridge under railway, highway and wind loading. The fatigue-critical locations are then determinedfor key bridge components, and databases of the dynamic stress responses at the critical locations areestablished. The time histories of dynamic stresses induced by individual loading during the design lifeof the bridge are generated based on the databases. The corresponding stress time histories due to thecombined action of multiple loading are also compiled. Finally, fatigue analysis is performed to computethe cumulative fatigue damage over the design life of 120 years. The results indicate that it is necessary toconsider the combined effect of multiple loading in the fatigue analysis of long-span suspension bridges.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Many long-span suspension bridges have been built around theworld, and most of these bridges are steel structures. Researchcarried out by the American Society of Civil Engineers (ASCE) in-dicates that 80%–90% of failures in steel structures are relatedto fatigue and fracture [1]. Fatigue analysis is thus essential andimperative in the design of steel bridges [2,3]. There is an ap-proach, which is based on measured strain responses, applied forthe fatigue assessment of several steel bridges in the past twodecades [4–7]. Although this method is considered to be an accu-rate way to evaluate the fatigue life of steel bridges, it has somelimitations for long-span suspension bridges. For instance, theevaluation is limited to critical locations installed with straingauges but the identification of critical locationsmaynot be an easytask for long-span suspension bridges under multiple loading. It is

∗ Corresponding author. Tel.: +852 2766 6050; fax: +852 2365 9291.E-mail addresses: cezhiwei@xmu.edu.cn (Z.W. Chen), ceylxu@polyu.edu.hk

(Y.L. Xu), ceyxia@polyu.edu.hk (Y. Xia), tongjiliqi@hotmail.com (Q. Li),sebh.bstr@hyd.gov.hk (K.Y. Wong).

0141-0296/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.08.027

also not economical to install strain gauges at all critical locationsof a long-span suspension bridge, and not every fatigue-criticallocation is suitable for sensor installation. To overcome theseproblems, a finite element method (FEM) integrated with fieldmeasurements has been proposed to investigate fatigue damageinduced by a particular loading, such as railway loading [8–10],highway loading [11–13], and wind loading [14,15]. Nevertheless,given the long-span period involved in fatigue damage accumu-lation in long-span suspension bridges and the complexity of thedynamic stress responses due to the combined action of multipleloading, a little research has been carried out for fatigue analysis oflong-span suspension bridges under multiple loading.

This paper proposes a general framework for fatigue analysisof a long-span suspension bridge under multiple loading byintegrating computer simulation with measurement data froma Wind and Structural Health Monitoring System (WASHMS).By taking the Tsing Ma suspension bridge in Hong Kong as anexample, a computationally efficient engineering approach is firstproposed for dynamic stress analysis of the bridge under railway,highway and wind loading. The fatigue-critical locations are thendetermined for key bridge components, and databases of thedynamic stress responses at the critical locations are established.120 years of time histories of the dynamic stresses induced

Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3247

by individual loading are generated based on the databases.The corresponding stress time histories due to the combinedaction of multiple loading are compiled. Fatigue analysis is thenperformed to compute the cumulative fatigue damage over thedesign life of 120 years. The cumulative fatigue damage induced byindividual loading and the damage magnification due to multipleloading are finally investigated.

2. Establishment of framework

To establish a framework for fatigue analysis of long-spansuspension bridges under combined action of railway, highway,and wind loading, some key issues need to be considered. First,given that a great number of stress time histories caused bymultiple loading are required for a complete fatigue assessmentof a long-span suspension bridge, it is desirable to develop acomputationally efficient engineering approach for dynamic stressanalysis. Second, as a long-span suspension bridge consists of alarge number of bridge components, it is not only impossible butalso unnecessary to carry out fatigue analysis for all the structuralcomponents. The fatigue-critical locations should be properlydetermined for fatigue analysis. Third, the design life of the bridgeconcerned should be designated before the calculation of wind-induced stress responses for fatigue analysis, because the windintensity taken into consideration in the fatigue analysis is relatedto the design life. Fourth, databases should be established in orderto generate the stress response time histories of the bridge over itsdesign life. Databases of railway, highway, and wind loading shallbe built in different ways because of different properties of loadingtype. Wind-induced stress responses are computed in one hourto build a database for fatigue analysis. As urban passenger trainsoften follow a regular timetable that is similar on different days,railway-induced stress time histories are computed in one day, anddaily timehistories are used to compose thedatabase. Thedatabaseof highway stress time histories is also composed of daily timehistories, as highway traffic conditions among different days arefound to be similar. Fifth,multiple loading-induced fatigue damageshould be calculated based on the stress responses induced bymultiple types of loading rather than the summation of damageinduced by individual loading. Fatigue analysis shall thereforebe applied directly to the multiple loading-induced stress timehistories, which is the superposition of stress responses induced bythree individual loadings. Finally, it is recommended that the datameasured be adopted in the computation of fatigue damage as faras possible to represent better the real conditions of the bridge.Taking the above issues into consideration, a framework for thefatigue analysis of a long-span suspension bridge under multipletypes of loadingwithin the design life is proposed and summarizedas follows:

1. Develop a computationally efficient engineering approach fordynamic stress analysis;

2. Designate the design life of the suspension bridge concerned;3. Determine the fatigue-critical locations of key structural

components of the bridge;4. Establish databases of the dynamic stress responses at the

fatigue-critical locations induced by railway, highway, andwind loading, respectively, using an engineering approach;

5. Generate the multiple loading-induced dynamic stress timehistories at the fatigue-critical locations within the design lifebased on the databases established in step (4);

6. Set the initial damage D0 = 0 and time step 1t;7. Count the number of stress cycles at different stress range

levels from themultiple load-induced stress timehistory in thekth time step using the rainflow counting method [16];

8. Compute the increase in the level of fatigue damage1Dk in thekth time step for a given fatigue-critical location;

9. Compute the cumulative fatigue damage Dk = Dk−1 + 1Dkand the cumulative service time tk = tk−1 +1t in the kth timestep; and

10. Move to the next time step and go from step (7) to the end ofthe design life.

Fatigue damage accumulated in the time step can be calculatedusing the Palmgren–Miner’s rule based on the two-slope S–Ncurves in [2]

1D = 1DH + 1DL (1)

where

1DH =

N1−i=1

ni σmr,i

K2if σr,i ≥ σr,0 and (2)

1DL =

N2−i=1

ni σm+2r,i

K2 σ 2r,0

if σr,i < σr,0 (3)

where K2 and m are constants relevant to the fatigue detail; K2 isdetermined from constant amplitude experiments correspondingto a probability of failure of 2.3%; ni is the applied number of stresscycles at the stress range level σr,i; N1 and N2 are the number ofstress range levels σr,i in the stress time histories above and belowσr,0, respectively, and σr,0 is the constant amplitude fatigue limit,which is defined as N = 107.

3. An engineering approach for dynamic stress analysis

3.1. Simplifications used in the engineering approach

In the previous work, the authors proposed a coupled dynamicapproach for dynamic stress analysis of long-span suspensionbridges under combined railway, highway, and wind loading [17].Though the coupled dynamic approach provides an accurateestimation of bridge dynamic stresses, the complexity of theframework makes computation very time consuming. It isimpractical to apply the coupled dynamic approach to fatigueanalysis of a long-span suspension bridge. In this regard, twomajor simplifications are adopted here to simplify the coupleddynamic approach and lead to the engineering approach basedon the features and properties of long-span suspension bridgesunder normal operation condition with a trade-off betweencomputational accuracy and efficiency.

The first major simplification is to neglect coupled effects ofmultiple load-induced dynamic stresses. This is because windspeed is normally not too high when vehicles are running on thebridge. Under extreme wind conditions, such as when a strongtyphoon is blowing, bridge trafficmanagement systems shall comeinto effect and the bridge will be closed to traffic. Therefore, it isreasonable to assume that the coupled effects of dynamic stressresponses of the bridge induced by railway, highway, and windloading are insignificant under normal operation condition andthat the bridge motions induced by railway, highway, and windloading are considered to be independent of each other. As aresult, the bridge stress response at a given point induced by thecombined effects of three types of loading can be approximatelyobtained by the synchronous superposition of stress responsesinduced by individual loadings.

σb = σrb + σhb + σwb (4)

where σrb, σhb, and σwb are the bridge stress responses induced byrailway, highway, and wind loading, respectively.

Another major simplification is to neglect the dynamicmagnification related to vehicle dynamics. This is because the self-weight of a long-span suspension bridge carrying both trains and

3248 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256

road vehicles is much larger than the weight of a train and/ora series of road vehicles. Furthermore, this study concerns thedynamic stress response of a long-span suspension bridge ratherthan the safety of vehicles. As a result, trains and road vehiclescan be simplified as a series of moving forces on the bridge deck.Moreover, through the analysis of three resonance conditions, itis found that the impact factor of a long-span cable-supportedbridge under a series of moving forces is often small [18]. Basedon the above reasons, the bridge stress responses induced bytrains and road vehicles can be calculated based on a series ofstatic forces and stress influence lines. Therefore, bridge stressresponses induced by railway and highway vehicles are calculatedusing the stress influence lines without considering dynamicmagnification in this study, but wind-induced stress responses ofa long-span suspension bridge are, however, computed based onthe aerodynamic analysis.

3.2. Dynamic stress analysis using the engineering approach

Based on the two major simplifications proposed in thepreceding section, the engineering approach for dynamic stressanalysis of long-span suspension bridges under multiple loadingcan be implemented by the following four steps: (1) analysis ofrailway-induced bridge dynamic stress based on stress influencelines; (2) analysis of highway-induced bridge dynamic stressbased on stress influence lines, (3) analysis of wind-induceddynamic stress using buffeting theory; and (4) combination ofthe stress responses induced by multiple types of loading by thesuperposition method in the time domain. The procedures for thefirst three steps are presented as follows.

To determine railway-induced dynamic stress responses of along-span suspension bridge, the stress influence lines should beestablished. To derive the stress influence lines for a given fatigue-critical location, the stress response at the designated location dueto a unit vertical force moving along the railway tracks from oneend of the bridge to the other end is computed. The abscissa ofthe stress influence line denotes the position of the unit verticalforce in the longitudinal direction of the bridge, and the ordinate ofthe stress influence line, the so-called stress influence coefficient,Φ , is the stress response induced by the unit vertical force atthe corresponding position. Railway loading is then determinedin terms of a series of moving vertical forces. For example, therailway loading for an eight-car train with 32 wheel-sets can berepresented by 32 vertical forces, with each force coming from onewheel-set. The railway loading information is used to determinethe railway loading for a given train and to simulate the railwayvehicle flow running along the bridge. Such information can beobtained from the train data recorded at the bridge site. Theinformation includes at least the number of trains, the numberand types of railway vehicles in a train, arrival instant, runningspeed, heading direction, railway track in use, number of bogies,bogie weight, and bogie spacing. Underlying assumptions includea constant speed of a typical train running across the bridge ona given railway track. The computational procedure of the stresstime history under railway loading is summarized as follows.

1. Establish the database of railway loading stress influence linesfor a given stress output point;

2. Update the train information at the instant t , which includes thenumber of railway vehicles of a train andwheel-set locations ofthe railway vehicle;

3. Determine the vertical loading fk,ij due to the ith wheel-set inthe jth railway vehicle on the kth railway track using the traininformation obtained;

4. Determine the stress influence coefficient Φk,ij due to the ithwheel-set in the jth railway vehicle on the kth railway trackusing the stress influence line database;

Fig. 1. Distribution of dynamic strain gauges and anemometers in Tsing Ma Bridge(unit: m).

5. Calculate the railway load-induced stress σrb by the triplesummation of the product of the stress influence coefficientΦk,ij and axle load fk,ij; and

6. Move to the next time instant and go from step (2) to the end ofthe given duration of stress responses.

To consider the dynamic stresses induced by road vehiclesrunning along the bridge on different traffic lanes, highway loadingstress influence line for each traffic lane should be established, andhighway loading should be determined based on the measuredroad vehicle data. For instance, the highway loading of a typicalfour-axle road vehicle can be represented by four vertical forceswith each load coming from one axle. To determine not onlythe highway loading due to a given road vehicle but also theroad vehicle flow running along the bridge, the highway loadinginformation should include at least the number and types of roadvehicles, traffic lane in use, arrival time, heading direction, runningspeed, axle number, axle weight, and axle spacing. Underlyingassumptions include a constant speed and no switching of thetraffic lane for a given road vehicle running along the bridge. Thecomputational procedure of the stress time history under highwayloading can be derived in a similar way to that under railwayloading.

Long-span suspension bridges that are built in wind-proneregions suffer considerable buffeting-induced vibration. Therefore,wind-induced dynamic stress responses should also be considered.Wind-induced dynamic stress response time histories can becomputed using a step-by-step procedure. In the first step, windcharacteristics in a given time period, such as one hour or tenminutes, are identified from wind data collected by anemometersinstalled at the bridge site. In the second step, the stochasticwind velocity at the simulation points along the bridge deck andthe normal mean wind speed in the time period of concern aregenerated based on the wind characteristics acquired from themeasured wind data. The buffeting and self-exciting forces on thesurface of the bridge deck are then computed [19]. In the third step,the wind-induced stress responses in the time period of concernare computed at the given stress output points using an integrationmethod. The procedure then moves to the next time period andgoes from the first step to the end of the given duration of stressresponses.

3.3. Verification of the engineering approach

The Tsing Ma Bridge in Hong Kong is a suspension bridge withan overall length of 2132 m (see Fig. 1). The bridge deck is ahybrid steel structure and carries a dual three-lane highway onthe top deck and two railway tracks and two carriageways on alower level within the bridge deck. The dynamic stress responsesof the TsingMaBridge aremainly induced by railway, highway, andwind loading. Both loading conditions and bridge responses aremonitored by theWASHMS installed on the bridge. Therefore, thisbridge is taken as an example to verify the computational accuracyand efficiency of the engineering approach. The information on thetrainwas converted from the strain response time history recorded

Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3249

Fig. 2. Two strain gauges installed at Section L and used in case study.

Fig. 3. 3-D Finite element model of Tsing Ma Bridge [19].

by a special set of strain gauges arranged under the railway beams.The information on heavy road vehicles was recorded by dynamicweigh-in-motion (WIM) stations. Wind data were collected by theanemometers installed at both bridge deck and towers. There are atotal of 110 dynamic strain gauges installed at three deck sections(see Fig. 1). Two of them installed at Section L are selected in thisstudy (see Fig. 2).

Considering the requirement of stress analysis of local bridgecomponents, a complex structural health monitoring orientedfinite elementmodel (FEM) of the TsingMa Bridge was establishedand shown in Fig. 3 [19]. The bridge is modelled using a seriesof beam elements, plate elements, shell elements, and others. Thefinite element model contains 12,898 nodes, 21,946 elements and4788 Multi-Point Connections (MPC). The finite element modelwas also updated using the first 18 measured natural frequenciesand mode shapes of the bridge from the WASHMS. It turned outthat the updated complex finite element model could providecomparable and credible structural dynamicmodal characteristics.

To validate the computational accuracy of the engineeringapproach, the stress responses induced by multiple types ofloading computed using the engineering approach are comparedwith those calculatedusing the coupleddynamic approach. A 140-sdynamic stress timehistorywas computedby the coupled dynamicmethod [13], as shown in Fig. 4. It is used as a reference forcomparison. During this period, there was one train running onthe north track heading towards the Hong Kong Island and 29road vehicles weighing over four tons running along the bridge.

The normal mean wind speed is 11.91 m/s. The standard deviationof turbulent wind is 1.310 m/s in the horizontal direction and0.679 m/s in the vertical direction. The integral length scales are256.7 m in the horizontal direction and 40.8 m in the verticaldirection.

To use the engineering approach to compute 140-s dynamicstress time history, the railway and highway loading stressinfluence lines of the Tsing Ma Bridge should be established.For each of the two strain gauges specified, two stress influencelines corresponding to the two railway tracks are established. Sixstress influence lines corresponding to the six highway trafficlanes are also established. All stress influence lines are generatedby structural analyses based on the finite element model of thebridge. The details on how to generate these influence lines are notgiven because of the limitation of paper length. In the computationof stress response, the train and road vehicle information isupdated at each time step, and the length of the time step 1t is0.02 s. The acquired wind characteristics are adopted to generatethe stochastic wind velocity field of the entire bridge deck, andthen the buffeting and self-excited forces on the bridge deck areestimated. The stress time histories under wind loading at thelocations concerned are computed. Based on the stress responsesinduced by railway, highway, and wind loading, respectively, themultiple load-induced stress responses are computed using thesuperposition method. The result computed by the engineeringmethod is also plotted in Fig. 4 for comparison. The figure showsthat the stress time histories computed using the engineeringapproach match well with those from the coupled dynamicapproach. The relative differences in the peak-to-peak stressresponses (the response obtained by the coupled dynamic methodminus that by the engineering approach, divided by one predictedby the coupled dynamic method) at the location of strain gaugesSP-TLS-02 and SS-TLS-12 are 16.1% and 5.4%, respectively. The16.1% error is the worst case and this error will not exaggeratethe final fatigue damage because fatigue damage depends ona large number of stress ranges rather than peak stresses. Theresults demonstrate that the level of computational accuracy of theengineering approach is acceptable.

In addition to the computational accuracy, the computationalefficiency is also an important factor for the engineering approach.Most of the trains running across the bridge follow a timetable on adaily basis; thus the cycle of railway loading is close to one day. Ashundreds of trains and thousands of road vehicles run across thebridge every day, the computational efficiency of the engineeringapproach is tested for one day only. The measured train, roadvehicle, wind, and strain data collected on 19 November 2005 areused for dynamic stress computation and comparison. This daywaschosen as the windwas relatively strong and the traffic was heavy.

8

6

4

2

0

–2

–4

–6

–80 20 40 60 80 100 120 140

Time(s)

Stre

ss (

MPa

)

Time (s)

Stre

ss (

MPa

)

a b

0 20 40 60 80 100 120 140

0

–10

–5

–15

10

5

Coupled dynamic methodEngineering method

Coupled dynamic methodEngineering method

Fig. 4. Stress time histories under railway, highway, and wind loading: (a) SP-TLS-02; (b) SS-TLS-12.

3250 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256

Time (hour)

Stre

ss (

MPa

)St

ress

(M

Pa)

a

b

0 5 10 15 20

0 5 10 15 20

0

–10

–20

10

20

0

–10

–20

10

20

Fig. 5. Daily stress time histories under multiple types of loading at SP-TLS-02:(a) Calculated; (b) measured.

The gross train weight (GTW) ranges from 282.7 to 402.2 tons.Their running speed ranges from 25 to 38 m/s. On the same day,8225 and 8623 heavy road vehicles weighing over 30 kN ran acrossthe bridge using the north and south three-lane carriageway,respectively. The gross vehicle weight (GVW) ranges from 4 to 54tons. The mean wind speed and direction are obtained from winddata recorded by the anemometers installed at the middle of thebridge deck. The hourly mean wind speed perpendicular to thebridge axis ranges from 2 to 13 m/s on that day. The turbulenceintensities are taken as 24% and 17% in the horizontal and verticaldirections by considering the most turbulent cases in the field. Theintegral length scales are taken as 251 and 56 m in the horizontaland vertical directions, respectively. As wind-induced dynamicstress responses are dominated by vibration modes of a relativelylow frequency, only the first 153 modes of vibration up to 2 Hzare included in this case for the stress response computation.24-h time periods of the railway-, highway-, and wind-inducedstress responses are calculated using the engineering approach.Based on the daily stress responses induced by the three individualloadings, the multiple load-induced stress response is obtainedby superposition. The daily multiple load-induced stress timehistories computed using the engineering approach at the locationof strain gauges SS-TLS-12 and SP-TLS-02 are shown in Figs. 5(a)and 6(a), respectively. The measured ones are shown in Figs. 5(b)and 6(b) for comparison. It can be seen that the computed stresstime histories agree well with the measured ones. The relativedifferences in the root mean square (RMS) of the stress responsesare calculated to determine the relative differences (the measuredvalue minus the computed one, divided by the measured one) atthe two typical locations. The relative differences in the RMSs ofthe stress responses are 12.9% and 8.4% for strain gauges SS-TLS-12 and SP-TLS-02, respectively. The coupled dynamic approach isactually not applicable for the computation of the daily dynamicstress responses as it takes an intolerably long time, whereas onlyseveral minutes are required for the engineering approach if thestress influence lines are available. The small relative differencesbetween the computed and measured time histories and a shortcomputation time for the engineering approach demonstrate thehigh level of computational efficiency and acceptable level ofcomputational accuracy.

4. Determination of fatigue-critical locations

The above section proposes an engineering approach fordynamic stress analysis. In the next step, the fatigue-critical

Time (hour)

Stre

ss (

MPa

)St

ress

(M

Pa)

a

b

–10

–30

–20

0

10

0 5 10 15 20

0

–10

–30

–20

10

0 5 10 15 20

Fig. 6. Daily stress time histories under multiple types of loading at SS-TLS-12:(a) Calculated; (b) measured.

locations shall be determined for the key structural componentsof a long-span suspension bridge. Given that the main structuralcomponents of and loadings on one long-span suspension bridgecan be quite different from another, the determination of fatigue-critical locations is case-dependent. The Tsing Ma suspensionbridge is taken as an example for illustration. The key structuralcomponents of the Tsing Ma Bridge can be classified into 55components in 15 groups. The details of the classification of thecomponents in each group are given in Table 1 [10]. The fatigue-critical locations are determined through the stress analysis of eachcomponent. To make sure that the size of the FEM is not too largeto be used for dynamic analysis, some types of bridge componentscannot be modelled exactly. For instance, if the orthotropic deckof the bridge were modelled using shell elements, the size of theFEMwould be too large to be used. Therefore, the orthotropic deckbetween the two adjacent cross frames at an interval of 4.5 m wassimplymodelled by a plate element that was fixed to the two crossframes at its two ends in the longitudinal direction and free on theother two sides in the lateral direction. Such a modelling makesit impossible to obtain actual stresses of the orthotropic deck.Apart from these components, some other types are neglectedbecause they are not critical to fatigue in practice. The bridgecomponents taken into consideration for fatigue analysis in thisstudy are highlighted in grey in Table 1.

As the fatigue damage of the Tsing Ma Bridge is induced bythe combined effect of railway, highway, and wind loading, thefatigue-critical locations should be determined based on the mul-tiple types of loading. However, it is very difficult because somanystress analyses are required for a great number of structural com-ponents under a large number of loading combinations in whichdifferent intensities of the three loadings shall be considered. Somesimplifications are therefore necessary. The fatigue damage in-duced by railway and highway loading was separately investi-gated, and it was found that for most bridge components exceptfor the upper deck, the fatigue damage is mainly caused by mov-ing trains, and that the contribution of moving road vehicles issmall. In addition, wind-induced fatigue damage to the bridge isnot significant [11]. Therefore, railway loading is a dominant fac-tor for the fatigue damage of the bridge. Given that almost all of thetrains running across the Tsing Ma Bridge since November 2005are 16-bogie trains, a standard train is defined to represent all 16-bogie trains by taking the weight of each bogie as themeanweightof the relevant bogies of all 16-bogie trains in November 2005.

Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3251

Table 1Classification of the structural components of Tsing Ma Bridge [10].

Name of group Name of component Group no. Component no. Serial no.

Suspension cables

Main cables

1

(a) 1Strand shoes (b) 2Shoe anchor rods (c) 3Anchor bolts (d) 4Cable clamps & bands (e) 5

SuspendersHangers

2(a) 6

Hanger connections: Stiffeners (b) 7Hanger connections: Bearing plates (c) 8

TowersLegs

3(a) 9

Portals (b) 10Saddles (c) 11

AnchoragesChambers

4(a) 12

Prestressing anchors (b) 13Saddles (c) 14

Piers: M1,M2, T1, T2, T3 Legs 5 (a) 15Cross beams (b) 16

Outer-longitudinal trusses

Top chord

6

(a) 17

Diagonal (b) 18

Vertical post (c) 19

Bottom chord (d) 20

Inner-longitudinal trusses

Top chord

7

(a) 21

Diagonal (b) 22

Vertical post (c) 23

Bottom chord (d) 24

Main cross frames

Top web

8

(a) 25Sloping web (b) 26Bottom web (c) 27

Bottom chord (d) 28

Intermediate cross frames

Top web

9

(a) 29Sloping web (b) 30Bottom web (c) 31

Bottom chord (d) 32

Plan bracingsUpper deck

10(a) 33

Lower deck (b) 34

Deck Troughs 11 (a) 35Plates (b) 36

Railway beamsT -sections

12(a) 37

Top flanges (b) 38

Connections (c) 39

Bearings

Rocker bearings at Ma Wan tower

13

(a) 40PTFE bearings at Tsing Yi tower (b) 41PTFE bearings at Pier T1 (c) 42PTFE bearings at Pier T2 (d) 43PTFE bearings at Pier T3 (e) 44PTFE bearings at Tsing Yi Anchorage (f) 45Rocker bearings atM2 (g) 46PTFE bearings atM1 (h) 47Hinge bearing at Lantau Anchorage (i) 48

Movement joints Highway movement joint 14 (a) 49Railway movement joint (b) 50

Tsing Yi approach deck

Top chord

15

(a) 51

Diagonal (b) 52

Vertical post (c) 53

Bottom chord (d) 54

Diagonals (K -bracings) (e) 55

The standard train has a fixed configuration, and the railway load-ing of the train is represented by 32 vertical forces. The standardtrain is then adopted to compute the railway-induced dynamicstress responses of members in a given type of bridge component,and then the responses are compared to each other to determinethe fatigue-critical members and locations.

Eqs. (2) and (3) indicate that fatigue damage is the function ofthe stress range level σr and number of stress cycles n. To simplify

the criteria for determining fatigue-critical location, an assumptionis adopted that the number of stress cycles induced by a standardtrain is almost the same for all components of the same type, anddifference only exists in the stress range level. This assumption isacceptable because the stress fluctuations at all components of thesame type induced by the standard train running across are foundto have a similar pattern. In addition, the equations demonstratethat the damage is most sensitive to the maximum stress range

3252 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256

Fig. 7. Major structural components of bridge deck.

0

10

20

30

40

50

0 500 1000 1500 2000

Fig. 8. Maximum stress ranges of diagonal members in north outer-longitudinaltrusses.

1σmax because fatigue damage is a function of σmr or σm+2

r . 1σmaxis therefore selected as the index for determining the fatigue-critical locations of bridge components of the same type. To makethe problemmanageable,1σmax is approximately decided by usingthe difference of the maximum and minimum stress in the stresstime history based on the principle of the level crossingmethod. Asfatigue is critical to the tension and reversal members, additionalstructural analysis should be performed to check the net stress inthe member due to the dead and super-imposed dead loads plusan extreme live load. If it is positive, then the member is finallydefined as a fatigue-sensitive member.

Let us take the diagonal members of outer-longitudinal trussesas an example to illustrate the determination of fatigue-criticallocations (see Fig. 7). Given the symmetry of the cross sectionsof the bridge, the standard train is supposed to run on the northrailway track and accordingly only the outer-longitudinal trusson the north needs to be considered. The stress time histories atthe stress output points of all of the diagonal members of thenorth outer-longitudinal truss are computed based on the standardtrain running across the bridge on the north railway track. Themaximum stress ranges are subsequently estimated. Fig. 8 showsthe maximum stress ranges of the diagonal members of the northouter-longitudinal truss due to the standard train running onthe north side of the bridge deck. The potential fatigue-criticallocations in the diagonal members of the north outer-longitudinaltruss can be determined from the figure: the diagonal memberE32123 (T ) close to the Ma Wan tower in the main span, and thediagonal member E32403 (T ) close to the Tsing Yi tower in themain span. ‘‘T ’’ or ‘‘B’’ in brackets denotes that the potential fatigue-critical location is at the top or bottom flange of the cross section

of the member at the two ends. Similar procedures are applied tothe other bridge components to determine the potential fatigue-critical locations. Furthermore, the net stresses at the potentialfatigue-critical locations are also checked for determining the finalfatigue-critical locations. The results demonstrate that the fatigue-critical sections of the bridge deck are around the bridge towers.Within the fatigue-critical sections, six of the strain points arechosen for fatigue analysis, that is, the elements E32123 (T ) atthe top flange of the outer-longitudinal diagonal member close tothe Ma Wan Tower, E34415 (B) at the bottom flange of the outer-longitudinal bottom chord of the Tsing Yi Tower, E40056 (T ) at thetop flange of the inner-longitudinal top chord of the Tsing Yi Tower,E40906 (B) at the bottom flange of the inner-longitudinal bottomchord of the Tsing Yi Tower, E39417 (B) at the bottom flange of theT -section of the railway beam of the Tsing Yi Tower, and E55406(T ) at the top flange of the bottom web of the cross frame close tothe Tsing Yi Tower.

5. Databases of dynamic stress responses to different loadings

In this section, the databases of dynamic stress responsesinduced by railway, highway, and wind loading at the criticallocations of the Tsing Ma Bridge are established based on theloading information recorded by the WASHMS.

5.1. Database of wind-induced dynamic stress response

Long-span suspension bridges built in wind-prone regionssuffer from considerable wind-induced vibration, which appearswithin a wide range of wind speeds and lasts for almost the wholedesign life of the bridge. A joint probability distribution functionof the mean wind speed and direction is utilized to describe windintensity at the bridge site [11]. The distribution of wind speedfor any given wind direction is assumed to follow the Weibulldistribution. The parameters in the distribution are determinedfrom monsoon wind records of hourly mean wind speed anddirection during the period from 1 January 2000 to 31 December2005, which were collected by an anemometer installed on thetop of the Ma Wan tower. Given that the measured typhoon windrecords are not enough to establish a reliable joint probabilitydistribution, only monsoon wind is of concern in this study. Themaximum wind speed at the top of the tower in each winddirection within the 120-year design life is then obtained from thejoint probability distribution. The maximum wind speed obtainedat the top of the tower is converted to the average deck level of thebridge. The maximum hourly mean wind speed at the deck levelis 25.89 m/s in the north direction for winds over the over-landfetch, and 15.47m/s in the south direction forwinds over the open-sea fetch [11]. Finally, a database of hourly wind-induced dynamicstress responses at the fatigue-critical locations is established:

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from 5 to 26 m/s at an interval of 1 m/s for winds over the over-land fetch, and from 5 to 16 m/s at an interval of 1 m/s for windsover the open-sea fetch. The stress fluctuations induced by windof a normal hourly mean wind speed less than 5 m/s are neglectedas they contribute little to fatigue. The database includes a totalof 34 one-hour time histories for each fatigue-critical location.The nominal stress of each fatigue-critical member is computedbased on the stresses at five points of the two ends of the member.The hot-spot stresses, which reflect the stress concentration atwelded joints, should be considered in fatigue analysis [16]. Thehot-spot stresses at the fatigue-critical locations are determinedby multiplying the nominal stresses by the stress concentrationfactor (SCF). It is noted that the value of SCF depends largely onthe local geometry of the connection details. Nevertheless, thereare a number of fatigue-critical locations in this bridge, and thelocal geometry of the connection details at these locations is quitedifferent. Considering that the SCF of 1.4 was used in the design ofthe bridge concerned for almost all connections, this number is alsoused for the six identified fatigue-critical locations in this study.The fatigue damage at fatigue-critical locations refers hereafter tothe fatigue damage at these hot spots.

5.2. Database of railway-induced dynamic stress response

Since it is almost impossible to predict railway traffic volumein the distant future for the Tsing Ma Bridge, one month ofrailway traffic that is close to the current traffic conditions isadopted here to establish the database of railway-induced stressresponses at the fatigue-critical locations for fatigue analysis.Monthly railway traffic in November 2005 is selected to establishthe database, and more than 90% of trains are of the 16-bogietype. In addition, the daily average number of trains in this monthreaches the maximum in the time period of concern. The dailytime histories of railway-induced stress responses at the fatigue-critical locations are computed using the stress influence lines forrailway loading and the railway loading information measured ineach day of November 2005. No large difference can be found inthe stress time histories among these days. The railway loadinginformation in each day of November 2005 is adopted to compute30 daily railway-induced stress timehistories at the fatigue-criticallocations, to compile the database of railway-induced dynamicstress responses.

5.3. Database of highway-induced dynamic stress response

The highway traffic on the Tsing Ma Bridge has beenmonitoredthrough seven dynamic weigh-in-motion (WIM) stations installednear the Lantau Administration Building since August 1998. Theroad vehicle data in November 2005 are adopted to build adatabase of highway-induced stress responses because this monthreached a maximum number of monthly vehicles and othermonths had slightly less vehicle numbers. Highway-induced stresstime histories are also computed in one-day units. The dailytime histories of highway-induced stress responses at the fatigue-critical locations are computed using the stress influence lines forhighway loading and the highway loading information measuredin each day of November 2005. No large differences can be foundin the stress time histories among these days. Finally, 30 dailystress response time histories at the fatigue-critical locations arecomputed to create the database of highway-induced dynamicstress response.

6. Multiple load-induced dynamic stress time histories indesign life

The design life of the Tsing Ma Bridge is 120 years; therefore,120 years of time histories of the dynamic stresses induced by

Time (hour)

Stre

ss ti

me

hist

ory

(MPa

)

40

20

0

–20

60

0 5 10 15 20

Fig. 9. A sample stress time history due to multiple types of loading.

railway, highway, and wind loading need to be generated forfatigue analysis. To generate them, the hourly mean wind speedsand directions for 120 years should be first obtained. A two-stepMonte Carlo simulation (MCS) method is adopted to draw out1051,200 (120 × 365 × 24) pairs of hourly mean wind speed anddirection for 120 years. In the first step, the mean wind directionis extracted through MCS according to the relative frequency ofthe mean wind direction without considering wind speed. In thesecond step, the mean wind speed at the top of the tower isdrawn out through MCS according to the probability distributionof the mean wind speed at the given mean wind direction underthe condition that it is not larger than the maximum wind speedin this direction. Finally, the mean wind speed and direction arepaired after two steps of MCS, and then converted into the hourlynormal mean wind speed to generate a sequence of 120 years.As the monsoon wind in Hong Kong normally is southerly (from90° to 270°) in summer and northerly (from 270° to 90°) inwinter, the sequence of hourly normal mean wind speeds in eachyear is adjusted to consider this. For each hourly normal meanwind speed in the sequence, the corresponding wind-induceddynamic stress response can be found in the database establishedin the previous section. As wind blowing in two directions is ofconcern, the mean wind direction in each hour of the sequenceis adopted to determine whether the wind is blowing in thedirection of the over-land fetch or open-sea fetch. As the databaseis established for different levels of meanwind speed at an intervalof 1m/s, rounding towards infinity is adopted to handle the hourlynormal mean wind speeds in the sequence. Finally, 1,051,200 h ofwind-induced dynamic stress time histories are generated at eachfatigue-critical location to compose a time history of 120 years.

In addition, 120 years of railway-induced stress time historiesare generated at the fatigue-critical locations based on thedatabase of 30 daily time histories. An integer between one andthirty is randomly drawn out for each day to generate a randomnumber sequence of 120 years. For each one in the sequence,the corresponding daily railway-induced dynamic stress responsescan be found in the database established in the aforementionedsection. Finally, 43,800 daily time histories at each fatigue-criticallocation are used to compose 120 years of railway-induceddynamic stresses. As the database of highway-induced stressresponses is also based on 30 daily time histories, a similarprocessing method is applied to obtain 120 years of highway-induced dynamic stresses at the fatigue-critical locations.

Based on the engineering approach proposed in the previoussection, the stress responses at the critical locations induced bythe combined effects of railway, highway, and wind loading canbe approximately obtained from the three responses induced byindividual loadings by superposition. Therefore, a 120-year timehistory of the stress induced by multiple types of loading isdetermined from those induced by railway, highway, and windloading individually. It should be noted that the bridge is closed totraffic when the mean wind speed recorded on site is very high;therefore, the bridge stress responses under this condition areinduced bywind loading only. Fig. 9 shows a sample daily multipleload-induced hot-spot stress time history at the critical locationE32123.

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Fig. 10. Multiple load-related spectra: (a) Stress range spectrum; (b) fatigue damage spectrum.

7. Fatigue analysis at fatigue-critical locations

The rainflow counting method is applied to the 120-yearmultiple load-induced stress timehistory, and the number of stresscycles in different stress range levels can be obtained. A stressrange spectrum is defined as the percentage of the number ofstress ranges in each stress range set to the total number in allsets. Fig. 10(a) displays the stress range spectrum at E32123. Itdemonstrates that most of the stress ranges are at the low levels,as 92.0% are less than 8MPa. Because fatigue damage ismuchmoresensitive to high level stress range rather than low level one, astress range of large amplitude may make a great contribution tofatigue damage although it occurs less frequently. Fatigue damagein each stress range bin is computed using Eq. (2) or Eq. (3). Thetype of welded connection at the six fatigue-critical locations inthis study is classified as F according to British Standard [20] withσr,0 = 40 MPa, K2 = 6.3 × 1011, and m = 3. A fatigue damagespectrum is defined as the percentage of fatigue damage in eachstress range set to the total damage in all sets. Fig. 10(b) displaysthe fatigue damage spectrum at E32123. The figure shows that thecontribution of stress ranges in the low levels (less than 8 MPa) tofatigue damage is small, and that the greatest fatigue damage is inthe stress range of 36–44 MPa.

Based on the multiple load-induced stress time histories overthe period of 120 years and the time step 1t = 1/365 year,the curves of cumulative fatigue damage within 120 years at thefatigue-critical locations can be computed. The cumulative fatiguedamage 1Dk in the kth day is calculated based on the daily stresstime history using Eqs. (1)–(3), and the cumulative fatigue damageDk is updated by adding the new damage on this day. Fig. 11shows the cumulative fatigue damage curves at the fatigue-criticallocations within a design life of 120 years. It is noted that thestructure is in danger when the cumulative fatigue damage isgreater than one. The maximum of the 120 years of cumulativefatigue damage at the fatigue-critical locations of the Tsing MaBridge is very close to one, which implies that the health conditionof the bridge is satisfactory. In addition, the cumulative damagecurves seem to be very linear. That is because Miner’s model isa linear damage model, and traffic loading is assumed to remainstable over the design life

In addition to the fatigue damage induced by multiple types ofloading, the fatigue damage induced by each individual loadingtype is also investigated. The 120 years of cumulative fatiguedamage induced by railway, highway, and wind loading arerespectively computed based on the three stress responses underthe different loadings. The results of the damage at differentfatigue-critical locations are listed in Table 2. It is found thatrailway loading plays a dominant role in the fatigue damage ofthe Tsing Ma Bridge, and that the damage induced by highway

Fig. 11. Cumulative fatigue damage curves at fatigue-critical locations.

loading is greater than that due to wind loading at some locationswhereas other locations are in reverse. It is also found that fatiguedamage due to combined effects of railway, highway, and windloading is larger than the sum of fatigue damage due to each ofindividual loadings, for fatigue damage is the function ofm-powerstress range (nonlinear relationship), and stress ranges induced bymultiple loading are larger than those caused by individual loading.In addition, the fatigue damage spectra of railway, highway, andwind loading are investigated based on the 120-year timehistories,and the results are shown in Fig. 12(a–c). The figure shows thatthe spectra are quite different. For example, the greatest fatiguedamage induced by railway loading is in the stress range of32–40 MPa, that induced by highway loading is in the range of0–4 and 8–24 MPa, and that induced by wind loading in therange of 0–12 MPa. To study the combined effect of multipletypes of loading on fatigue damage, a multiple load magnificationfactor is defined as the ratio of the fatigue damage due to thecombined effect of the three loadings to the sum of the damagedue to each individual loading. The factors at the six fatigue-criticallocations are computed and range from1.06 to 1.35. Themaximumfactor is at critical locations E32123 and E34415, at which thefatigue damage induced by highway and wind loading is muchcloser to that induced by railway loading than at the other criticallocations. The results indicate that the combined effect of multipleloads must be considered in a bridge subject to multiple typesof loading, especially in the case in which the contributions ofdifferent loadings to fatigue damage are close.

8. Conclusions

A general framework has been proposed for fatigue analysisof a long-span suspension bridge under multiple loading over its

Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3255

Table 2120 years of cumulative fatigue damage under different loading types.

Fatigue-critical locations Loading typesRailway (R) Highway (H) Wind (W ) R+H +W

E32123 (T ) 0.70 0.048 0.011 1.02E34415 (B) 0.66 0.044 0.0092 0.96E40056 (T ) 0.52 0.0022 0.0057 0.68E40906 (B) 0.42 0.0025 0.0052 0.54E55406 (T ) 0.34 0.0037 0.0016 0.41E39417 (B) 0.48 0.0020 0.0074 0.52

Fig. 12. Fatigue damage spectra: (a) Railway; (b) highway; (c) wind.

design life in this paper. The framework was applied to the TsingMa suspension bridge in Hong Kong. An engineering approachfor dynamic stress analysis of a long-span suspension bridgeunder multiple types of loading was first proposed. The TsingMa Bridge and the measurement data recorded by the WASHMSwere employed to verify the feasibility of the proposed approach.The fatigue-critical locations of the bridge were determined forthe key structural components. Based on the measurement datarecorded by the WASHMS installed on the bridge, the databasesof wind, railway, and highway loadings as well as stress responsesat the fatigue-critical locations were established to generate 120-year time histories of multiple loading-induced stress responses.Finally, the fatigue analysis based on the 120-year stress timehistories was performed to compute the cumulative fatiguedamage over the bridge’s design life. The results indicate thatthe health condition of the bridge is satisfactory. The cumulativefatigue damage induced by individual loading and the damagemagnification due to the combined action of three types of loadingswere also investigated. The results show that railway loadingplays the dominant role in the fatigue damage of the bridge. Thedamage induced by highway loading is greater than that due towind loading at some locations, whereas the reverse is the case

in other locations. Furthermore, it is necessary to consider thecombined effect of multiple types of loading in the fatigue analysisof long-span suspension bridges. In reality, uncertainties exist inexternal loadings, structural modelling, and structural parameterin fatigue assessment. The fatigue reliability analysis of multi-loading bridges deserves further study.

Acknowledgements

The authorswish to acknowledge the financial support from theResearch Grants Council of Hong Kong (PolyU 5327/08E), the HongKong Polytechnic University (PolyU-1-BB68), and the NationalNatural Science Foundation of China (NSFC-50830203 and NSFC-51108395). Sincere thanks go to theHighwaysDepartment ofHongKong for providing the authors with the field measurement data.Any opinions and concluding remarks presented in this paper areentirely those of the authors.

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