1. Engineering Structures 33 (2011) 3246–3256
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
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, China
b
Department of Civil Engineering, Xiamen University, Xiamen, China
c
Department of Bridge Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China
d
Bridge & Structures Division, Highways Department, Tsing Yi, Hong Kong, China
article info abstract
Article history: Long-span suspension bridges are often subject to multiple types of dynamic loads, especially those
Received 2 March 2011 located in wind-prone regions and carrying both trains and road vehicles. Fatigue assessment shall
Received in revised form be performed to ensure the safety and functionality of the bridges. This paper proposes a framework
9 August 2011
for fatigue analysis of a long-span suspension bridge under multiple loading by integrating computer
Accepted 16 August 2011
Available online 29 September 2011
simulation with structural health monitoring system. By taking the Tsing Ma Bridge in Hong Kong as an
example, a computationally efficient engineering approach is first proposed for dynamic stress analysis of
Keywords:
the bridge under railway, highway and wind loading. The fatigue-critical locations are then determined
Dynamic stress for key bridge components, and databases of the dynamic stress responses at the critical locations are
Fatigue established. The time histories of dynamic stresses induced by individual loading during the design life
Suspension bridges of the bridge are generated based on the databases. The corresponding stress time histories due to the
Structural health monitoring combined action of multiple loading are also compiled. Finally, fatigue analysis is performed to compute
Wind loading the cumulative fatigue damage over the design life of 120 years. The results indicate that it is necessary to
Railway loading consider the combined effect of multiple loading in the fatigue analysis of long-span suspension bridges.
Highway loading © 2011 Elsevier Ltd. All rights reserved.
1. Introduction also not economical to install strain gauges at all critical locations
of a long-span suspension bridge, and not every fatigue-critical
Many long-span suspension bridges have been built around the location is suitable for sensor installation. To overcome these
world, and most of these bridges are steel structures. Research problems, a finite element method (FEM) integrated with field
carried out by the American Society of Civil Engineers (ASCE) in- measurements has been proposed to investigate fatigue damage
dicates that 80%–90% of failures in steel structures are related induced by a particular loading, such as railway loading [8–10],
highway loading [11–13], and wind loading [14,15]. Nevertheless,
to fatigue and fracture [1]. Fatigue analysis is thus essential and
given the long-span period involved in fatigue damage accumu-
imperative in the design of steel bridges [2,3]. There is an ap-
lation in long-span suspension bridges and the complexity of the
proach, which is based on measured strain responses, applied for
dynamic stress responses due to the combined action of multiple
the fatigue assessment of several steel bridges in the past two
loading, a little research has been carried out for fatigue analysis of
decades [4–7]. Although this method is considered to be an accu-
long-span suspension bridges under multiple loading.
rate way to evaluate the fatigue life of steel bridges, it has some This paper proposes a general framework for fatigue analysis
limitations for long-span suspension bridges. For instance, the of a long-span suspension bridge under multiple loading by
evaluation is limited to critical locations installed with strain integrating computer simulation with measurement data from
gauges but the identification of critical locations may not be an easy a Wind and Structural Health Monitoring System (WASHMS).
task for long-span suspension bridges under multiple loading. It is By taking the Tsing Ma suspension bridge in Hong Kong as an
example, a computationally efficient engineering approach is first
proposed for dynamic stress analysis of the bridge under railway,
∗ highway and wind loading. The fatigue-critical locations are then
Corresponding author. Tel.: +852 2766 6050; fax: +852 2365 9291.
E-mail addresses: cezhiwei@xmu.edu.cn (Z.W. Chen), ceylxu@polyu.edu.hk
determined for key bridge components, and databases of the
(Y.L. Xu), ceyxia@polyu.edu.hk (Y. Xia), tongjiliqi@hotmail.com (Q. Li), dynamic stress responses at the critical locations are established.
sebh.bstr@hyd.gov.hk (K.Y. Wong). 120 years of time histories of the dynamic stresses induced
0141-0296/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2011.08.027

2. Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3247
by individual loading are generated based on the databases. 9. Compute the cumulative fatigue damage Dk = Dk−1 + Dk
The corresponding stress time histories due to the combined and the cumulative service time tk = tk−1 + t in the kth time
action of multiple loading are compiled. Fatigue analysis is then step; and
performed to compute the cumulative fatigue damage over the 10. Move to the next time step and go from step (7) to the end of
design life of 120 years. The cumulative fatigue damage induced by the design life.
individual loading and the damage magnification due to multiple
Fatigue damage accumulated in the time step can be calculated
loading are finally investigated.
using the Palmgren–Miner’s rule based on the two-slope S–N
curves in [2]
2. Establishment of framework
D= DH + DL (1)
To establish a framework for fatigue analysis of long-span
where
suspension bridges under combined action of railway, highway,
and wind loading, some key issues need to be considered. First, − ni σrmi
N1
,
given that a great number of stress time histories caused by DH = if σr ,i ≥ σr ,0 and (2)
K2
multiple loading are required for a complete fatigue assessment i=1
of a long-span suspension bridge, it is desirable to develop a − ni σrmi+2
N2
,
computationally efficient engineering approach for dynamic stress DL = if σr ,i < σr ,0 (3)
analysis. Second, as a long-span suspension bridge consists of a i=1
K2 σr2 0
,
large number of bridge components, it is not only impossible but
also unnecessary to carry out fatigue analysis for all the structural where K2 and m are constants relevant to the fatigue detail; K2 is
components. The fatigue-critical locations should be properly determined from constant amplitude experiments corresponding
determined for fatigue analysis. Third, the design life of the bridge to a probability of failure of 2.3%; ni is the applied number of stress
concerned should be designated before the calculation of wind- cycles at the stress range level σr ,i ; N1 and N2 are the number of
induced stress responses for fatigue analysis, because the wind stress range levels σr ,i in the stress time histories above and below
intensity taken into consideration in the fatigue analysis is related σr ,0 , respectively, and σr ,0 is the constant amplitude fatigue limit,
to the design life. Fourth, databases should be established in order which is defined as N = 107 .
to generate the stress response time histories of the bridge over its
design life. Databases of railway, highway, and wind loading shall 3. An engineering approach for dynamic stress analysis
be built in different ways because of different properties of loading
type. Wind-induced stress responses are computed in one hour 3.1. Simplifications used in the engineering approach
to build a database for fatigue analysis. As urban passenger trains
often follow a regular timetable that is similar on different days, In the previous work, the authors proposed a coupled dynamic
railway-induced stress time histories are computed in one day, and approach for dynamic stress analysis of long-span suspension
daily time histories are used to compose the database. The database bridges under combined railway, highway, and wind loading [17].
of highway stress time histories is also composed of daily time Though the coupled dynamic approach provides an accurate
histories, as highway traffic conditions among different days are estimation of bridge dynamic stresses, the complexity of the
found to be similar. Fifth, multiple loading-induced fatigue damage framework makes computation very time consuming. It is
should be calculated based on the stress responses induced by impractical to apply the coupled dynamic approach to fatigue
multiple types of loading rather than the summation of damage analysis of a long-span suspension bridge. In this regard, two
induced by individual loading. Fatigue analysis shall therefore major simplifications are adopted here to simplify the coupled
be applied directly to the multiple loading-induced stress time dynamic approach and lead to the engineering approach based
histories, which is the superposition of stress responses induced by on the features and properties of long-span suspension bridges
three individual loadings. Finally, it is recommended that the data under normal operation condition with a trade-off between
measured be adopted in the computation of fatigue damage as far computational accuracy and efficiency.
as possible to represent better the real conditions of the bridge. The first major simplification is to neglect coupled effects of
Taking the above issues into consideration, a framework for the multiple load-induced dynamic stresses. This is because wind
fatigue analysis of a long-span suspension bridge under multiple speed is normally not too high when vehicles are running on the
types of loading within the design life is proposed and summarized bridge. Under extreme wind conditions, such as when a strong
as follows: typhoon is blowing, bridge traffic management systems shall come
1. Develop a computationally efficient engineering approach for into effect and the bridge will be closed to traffic. Therefore, it is
dynamic stress analysis; reasonable to assume that the coupled effects of dynamic stress
2. Designate the design life of the suspension bridge concerned; responses of the bridge induced by railway, highway, and wind
3. Determine the fatigue-critical locations of key structural loading are insignificant under normal operation condition and
components of the bridge; that the bridge motions induced by railway, highway, and wind
4. Establish databases of the dynamic stress responses at the loading are considered to be independent of each other. As a
fatigue-critical locations induced by railway, highway, and result, the bridge stress response at a given point induced by the
wind loading, respectively, using an engineering approach; combined effects of three types of loading can be approximately
5. Generate the multiple loading-induced dynamic stress time obtained by the synchronous superposition of stress responses
histories at the fatigue-critical locations within the design life induced by individual loadings.
based on the databases established in step (4); σb = σrb + σhb + σwb (4)
6. Set the initial damage D0 = 0 and time step t;
7. Count the number of stress cycles at different stress range where σrb , σhb , and σwb are the bridge stress responses induced by
levels from the multiple load-induced stress time history in the railway, highway, and wind loading, respectively.
kth time step using the rainflow counting method [16]; Another major simplification is to neglect the dynamic
8. Compute the increase in the level of fatigue damage Dk in the magnification related to vehicle dynamics. This is because the self-
kth time step for a given fatigue-critical location; weight of a long-span suspension bridge carrying both trains and

3. 3248 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256
road vehicles is much larger than the weight of a train and/or
a series of road vehicles. Furthermore, this study concerns the
dynamic stress response of a long-span suspension bridge rather
than the safety of vehicles. As a result, trains and road vehicles
can be simplified as a series of moving forces on the bridge deck.
Moreover, through the analysis of three resonance conditions, it
is found that the impact factor of a long-span cable-supported
bridge under a series of moving forces is often small [18]. Based
on the above reasons, the bridge stress responses induced by
trains and road vehicles can be calculated based on a series of Fig. 1. Distribution of dynamic strain gauges and anemometers in Tsing Ma Bridge
(unit: m).
static forces and stress influence lines. Therefore, bridge stress
responses induced by railway and highway vehicles are calculated
using the stress influence lines without considering dynamic 5. Calculate the railway load-induced stress σrb by the triple
magnification in this study, but wind-induced stress responses of summation of the product of the stress influence coefficient
a long-span suspension bridge are, however, computed based on Φk,ij and axle load fk,ij ; and
the aerodynamic analysis. 6. Move to the next time instant and go from step (2) to the end of
the given duration of stress responses.
3.2. Dynamic stress analysis using the engineering approach To consider the dynamic stresses induced by road vehicles
running along the bridge on different traffic lanes, highway loading
Based on the two major simplifications proposed in the stress influence line for each traffic lane should be established, and
preceding section, the engineering approach for dynamic stress highway loading should be determined based on the measured
analysis of long-span suspension bridges under multiple loading road vehicle data. For instance, the highway loading of a typical
can be implemented by the following four steps: (1) analysis of four-axle road vehicle can be represented by four vertical forces
railway-induced bridge dynamic stress based on stress influence with each load coming from one axle. To determine not only
lines; (2) analysis of highway-induced bridge dynamic stress the highway loading due to a given road vehicle but also the
based on stress influence lines, (3) analysis of wind-induced road vehicle flow running along the bridge, the highway loading
dynamic stress using buffeting theory; and (4) combination of information should include at least the number and types of road
the stress responses induced by multiple types of loading by the vehicles, traffic lane in use, arrival time, heading direction, running
superposition method in the time domain. The procedures for the speed, axle number, axle weight, and axle spacing. Underlying
first three steps are presented as follows. assumptions include a constant speed and no switching of the
To determine railway-induced dynamic stress responses of a traffic lane for a given road vehicle running along the bridge. The
long-span suspension bridge, the stress influence lines should be computational procedure of the stress time history under highway
established. To derive the stress influence lines for a given fatigue- loading can be derived in a similar way to that under railway
critical location, the stress response at the designated location due loading.
to a unit vertical force moving along the railway tracks from one Long-span suspension bridges that are built in wind-prone
end of the bridge to the other end is computed. The abscissa of regions suffer considerable buffeting-induced vibration. Therefore,
the stress influence line denotes the position of the unit vertical wind-induced dynamic stress responses should also be considered.
force in the longitudinal direction of the bridge, and the ordinate of Wind-induced dynamic stress response time histories can be
the stress influence line, the so-called stress influence coefficient, computed using a step-by-step procedure. In the first step, wind
Φ , is the stress response induced by the unit vertical force at characteristics in a given time period, such as one hour or ten
the corresponding position. Railway loading is then determined minutes, are identified from wind data collected by anemometers
in terms of a series of moving vertical forces. For example, the installed at the bridge site. In the second step, the stochastic
railway loading for an eight-car train with 32 wheel-sets can be wind velocity at the simulation points along the bridge deck and
represented by 32 vertical forces, with each force coming from one the normal mean wind speed in the time period of concern are
wheel-set. The railway loading information is used to determine generated based on the wind characteristics acquired from the
the railway loading for a given train and to simulate the railway measured wind data. The buffeting and self-exciting forces on the
vehicle flow running along the bridge. Such information can be surface of the bridge deck are then computed [19]. In the third step,
obtained from the train data recorded at the bridge site. The the wind-induced stress responses in the time period of concern
information includes at least the number of trains, the number are computed at the given stress output points using an integration
and types of railway vehicles in a train, arrival instant, running method. The procedure then moves to the next time period and
speed, heading direction, railway track in use, number of bogies, goes from the first step to the end of the given duration of stress
bogie weight, and bogie spacing. Underlying assumptions include responses.
a constant speed of a typical train running across the bridge on
a given railway track. The computational procedure of the stress 3.3. Verification of the engineering approach
time history under railway loading is summarized as follows.
1. Establish the database of railway loading stress influence lines The Tsing Ma Bridge in Hong Kong is a suspension bridge with
for a given stress output point; an overall length of 2132 m (see Fig. 1). The bridge deck is a
2. Update the train information at the instant t, which includes the hybrid steel structure and carries a dual three-lane highway on
number of railway vehicles of a train and wheel-set locations of the top deck and two railway tracks and two carriageways on a
the railway vehicle; lower level within the bridge deck. The dynamic stress responses
3. Determine the vertical loading fk,ij due to the ith wheel-set in of the Tsing Ma Bridge are mainly induced by railway, highway, and
the jth railway vehicle on the kth railway track using the train wind loading. Both loading conditions and bridge responses are
information obtained; monitored by the WASHMS installed on the bridge. Therefore, this
4. Determine the stress influence coefficient Φk,ij due to the ith bridge is taken as an example to verify the computational accuracy
wheel-set in the jth railway vehicle on the kth railway track and efficiency of the engineering approach. The information on the
using the stress influence line database; train was converted from the strain response time history recorded

4. Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3249
The normal mean wind speed is 11.91 m/s. The standard deviation
of turbulent wind is 1.310 m/s in the horizontal direction and
0.679 m/s in the vertical direction. The integral length scales are
256.7 m in the horizontal direction and 40.8 m in the vertical
direction.
To use the engineering approach to compute 140-s dynamic
stress time history, the railway and highway loading stress
influence lines of the Tsing Ma Bridge should be established.
For each of the two strain gauges specified, two stress influence
lines corresponding to the two railway tracks are established. Six
Fig. 2. Two strain gauges installed at Section L and used in case study. stress influence lines corresponding to the six highway traffic
lanes are also established. All stress influence lines are generated
by structural analyses based on the finite element model of the
bridge. The details on how to generate these influence lines are not
given because of the limitation of paper length. In the computation
of stress response, the train and road vehicle information is
updated at each time step, and the length of the time step t is
0.02 s. The acquired wind characteristics are adopted to generate
the stochastic wind velocity field of the entire bridge deck, and
Fig. 3. 3-D Finite element model of Tsing Ma Bridge [19]. then the buffeting and self-excited forces on the bridge deck are
estimated. The stress time histories under wind loading at the
by a special set of strain gauges arranged under the railway beams. locations concerned are computed. Based on the stress responses
The information on heavy road vehicles was recorded by dynamic induced by railway, highway, and wind loading, respectively, the
weigh-in-motion (WIM) stations. Wind data were collected by the multiple load-induced stress responses are computed using the
anemometers installed at both bridge deck and towers. There are a superposition method. The result computed by the engineering
total of 110 dynamic strain gauges installed at three deck sections method is also plotted in Fig. 4 for comparison. The figure shows
(see Fig. 1). Two of them installed at Section L are selected in this that the stress time histories computed using the engineering
study (see Fig. 2). approach match well with those from the coupled dynamic
Considering the requirement of stress analysis of local bridge approach. The relative differences in the peak-to-peak stress
components, a complex structural health monitoring oriented responses (the response obtained by the coupled dynamic method
finite element model (FEM) of the Tsing Ma Bridge was established minus that by the engineering approach, divided by one predicted
and shown in Fig. 3 [19]. The bridge is modelled using a series by the coupled dynamic method) at the location of strain gauges
of beam elements, plate elements, shell elements, and others. The SP-TLS-02 and SS-TLS-12 are 16.1% and 5.4%, respectively. The
finite element model contains 12,898 nodes, 21,946 elements and 16.1% error is the worst case and this error will not exaggerate
4788 Multi-Point Connections (MPC). The finite element model the final fatigue damage because fatigue damage depends on
was also updated using the first 18 measured natural frequencies a large number of stress ranges rather than peak stresses. The
and mode shapes of the bridge from the WASHMS. It turned out results demonstrate that the level of computational accuracy of the
that the updated complex finite element model could provide engineering approach is acceptable.
comparable and credible structural dynamic modal characteristics. In addition to the computational accuracy, the computational
To validate the computational accuracy of the engineering efficiency is also an important factor for the engineering approach.
approach, the stress responses induced by multiple types of Most of the trains running across the bridge follow a timetable on a
loading computed using the engineering approach are compared daily basis; thus the cycle of railway loading is close to one day. As
with those calculated using the coupled dynamic approach. A 140-s hundreds of trains and thousands of road vehicles run across the
dynamic stress time history was computed by the coupled dynamic bridge every day, the computational efficiency of the engineering
method [13], as shown in Fig. 4. It is used as a reference for approach is tested for one day only. The measured train, road
comparison. During this period, there was one train running on vehicle, wind, and strain data collected on 19 November 2005 are
the north track heading towards the Hong Kong Island and 29 used for dynamic stress computation and comparison. This day was
road vehicles weighing over four tons running along the bridge. chosen as the wind was relatively strong and the traffic was heavy.
a 8 b 10
Coupled dynamic method Coupled dynamic method
6 Engineering method Engineering method
5
4
2 0
Stress (MPa)
Stress (MPa)
0
–5
–2
–10
–4
–6 –15
–8
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
Time(s) Time (s)
Fig. 4. Stress time histories under railway, highway, and wind loading: (a) SP-TLS-02; (b) SS-TLS-12.

5. 3250 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256
a 20 a 10
10
0
Stress (MPa)
Stress (MPa)
0
–10
–10
–20
–20
–30
0 5 10 15 20 0 5 10 15 20
b 20 b 10
10
Stress (MPa)
0
Stress (MPa)
0
–10
–10
–20
–20
0 5 10 15 20 –30
0 5 10 15 20
Time (hour) Time (hour)
Fig. 5. Daily stress time histories under multiple types of loading at SP-TLS-02: Fig. 6. Daily stress time histories under multiple types of loading at SS-TLS-12:
(a) Calculated; (b) measured. (a) Calculated; (b) measured.
The gross train weight (GTW) ranges from 282.7 to 402.2 tons. locations shall be determined for the key structural components
Their running speed ranges from 25 to 38 m/s. On the same day, of a long-span suspension bridge. Given that the main structural
8225 and 8623 heavy road vehicles weighing over 30 kN ran across components of and loadings on one long-span suspension bridge
the bridge using the north and south three-lane carriageway, can be quite different from another, the determination of fatigue-
respectively. The gross vehicle weight (GVW) ranges from 4 to 54 critical locations is case-dependent. The Tsing Ma suspension
tons. The mean wind speed and direction are obtained from wind
bridge is taken as an example for illustration. The key structural
data recorded by the anemometers installed at the middle of the
components of the Tsing Ma Bridge can be classified into 55
bridge deck. The hourly mean wind speed perpendicular to the
components in 15 groups. The details of the classification of the
bridge axis ranges from 2 to 13 m/s on that day. The turbulence
components in each group are given in Table 1 [10]. The fatigue-
intensities are taken as 24% and 17% in the horizontal and vertical
directions by considering the most turbulent cases in the field. The critical locations are determined through the stress analysis of each
integral length scales are taken as 251 and 56 m in the horizontal component. To make sure that the size of the FEM is not too large
and vertical directions, respectively. As wind-induced dynamic to be used for dynamic analysis, some types of bridge components
stress responses are dominated by vibration modes of a relatively cannot be modelled exactly. For instance, if the orthotropic deck
low frequency, only the first 153 modes of vibration up to 2 Hz of the bridge were modelled using shell elements, the size of the
are included in this case for the stress response computation. FEM would be too large to be used. Therefore, the orthotropic deck
24-h time periods of the railway-, highway-, and wind-induced between the two adjacent cross frames at an interval of 4.5 m was
stress responses are calculated using the engineering approach. simply modelled by a plate element that was fixed to the two cross
Based on the daily stress responses induced by the three individual frames at its two ends in the longitudinal direction and free on the
loadings, the multiple load-induced stress response is obtained other two sides in the lateral direction. Such a modelling makes
by superposition. The daily multiple load-induced stress time it impossible to obtain actual stresses of the orthotropic deck.
histories computed using the engineering approach at the location Apart from these components, some other types are neglected
of strain gauges SS-TLS-12 and SP-TLS-02 are shown in Figs. 5(a) because they are not critical to fatigue in practice. The bridge
and 6(a), respectively. The measured ones are shown in Figs. 5(b) components taken into consideration for fatigue analysis in this
and 6(b) for comparison. It can be seen that the computed stress study are highlighted in grey in Table 1.
time histories agree well with the measured ones. The relative As the fatigue damage of the Tsing Ma Bridge is induced by
differences in the root mean square (RMS) of the stress responses the combined effect of railway, highway, and wind loading, the
are calculated to determine the relative differences (the measured fatigue-critical locations should be determined based on the mul-
value minus the computed one, divided by the measured one) at tiple types of loading. However, it is very difficult because so many
the two typical locations. The relative differences in the RMSs of
stress analyses are required for a great number of structural com-
the stress responses are 12.9% and 8.4% for strain gauges SS-TLS-
ponents under a large number of loading combinations in which
12 and SP-TLS-02, respectively. The coupled dynamic approach is
different intensities of the three loadings shall be considered. Some
actually not applicable for the computation of the daily dynamic
simplifications are therefore necessary. The fatigue damage in-
stress responses as it takes an intolerably long time, whereas only
duced by railway and highway loading was separately investi-
several minutes are required for the engineering approach if the
stress influence lines are available. The small relative differences gated, and it was found that for most bridge components except
between the computed and measured time histories and a short for the upper deck, the fatigue damage is mainly caused by mov-
computation time for the engineering approach demonstrate the ing trains, and that the contribution of moving road vehicles is
high level of computational efficiency and acceptable level of small. In addition, wind-induced fatigue damage to the bridge is
computational accuracy. not significant [11]. Therefore, railway loading is a dominant fac-
tor for the fatigue damage of the bridge. Given that almost all of the
4. Determination of fatigue-critical locations trains running across the Tsing Ma Bridge since November 2005
are 16-bogie trains, a standard train is defined to represent all 16-
The above section proposes an engineering approach for bogie trains by taking the weight of each bogie as the mean weight
dynamic stress analysis. In the next step, the fatigue-critical of the relevant bogies of all 16-bogie trains in November 2005.

6. Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3251
Table 1
Classification of the structural components of Tsing Ma Bridge [10].
Name of group Name of component Group no. Component no. Serial no.
Main cables (a) 1
Strand shoes (b) 2
Suspension cables Shoe anchor rods 1 (c) 3
Anchor bolts (d) 4
Cable clamps & bands (e) 5
Hangers (a) 6
Suspenders Hanger connections: Stiffeners 2 (b) 7
Hanger connections: Bearing plates (c) 8
Legs (a) 9
Towers Portals 3 (b) 10
Saddles (c) 11
Chambers (a) 12
Anchorages Prestressing anchors 4 (b) 13
Saddles (c) 14
Legs (a) 15
Piers: M1, M2, T 1, T 2, T 3 5
Cross beams (b) 16
Top chord (a) 17
Outer-longitudinal trusses Diagonal 6 (b) 18
Vertical post (c) 19
Bottom chord (d) 20
Top chord (a) 21
Inner-longitudinal trusses Diagonal 7 (b) 22
Vertical post (c) 23
Bottom chord (d) 24
Top web (a) 25
Main cross frames Sloping web 8 (b) 26
Bottom web (c) 27
Bottom chord (d) 28
Top web (a) 29
Intermediate cross frames Sloping web 9 (b) 30
Bottom web (c) 31
Bottom chord (d) 32
Upper deck (a) 33
Plan bracings 10
Lower deck (b) 34
Troughs (a) 35
Deck 11
Plates (b) 36
T -sections (a) 37
Railway beams Top flanges 12 (b) 38
Connections (c) 39
Rocker bearings at Ma Wan tower (a) 40
PTFE bearings at Tsing Yi tower (b) 41
PTFE bearings at Pier T 1 (c) 42
PTFE bearings at Pier T 2 (d) 43
Bearings PTFE bearings at Pier T 3 13 (e) 44
PTFE bearings at Tsing Yi Anchorage (f) 45
Rocker bearings at M2 (g) 46
PTFE bearings at M1 (h) 47
Hinge bearing at Lantau Anchorage (i) 48
Highway movement joint (a) 49
Movement joints 14
Railway movement joint (b) 50
Top chord (a) 51
Diagonal (b) 52
Tsing Yi approach deck 15
Vertical post (c) 53
Bottom chord (d) 54
Diagonals (K -bracings) (e) 55
The standard train has a fixed configuration, and the railway load- the criteria for determining fatigue-critical location, an assumption
ing of the train is represented by 32 vertical forces. The standard is adopted that the number of stress cycles induced by a standard
train is then adopted to compute the railway-induced dynamic train is almost the same for all components of the same type, and
stress responses of members in a given type of bridge component, difference only exists in the stress range level. This assumption is
and then the responses are compared to each other to determine acceptable because the stress fluctuations at all components of the
the fatigue-critical members and locations. same type induced by the standard train running across are found
Eqs. (2) and (3) indicate that fatigue damage is the function of to have a similar pattern. In addition, the equations demonstrate
the stress range level σr and number of stress cycles n. To simplify that the damage is most sensitive to the maximum stress range

7. 3252 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256
Fig. 7. Major structural components of bridge deck.
of the member at the two ends. Similar procedures are applied to
50
the other bridge components to determine the potential fatigue-
critical locations. Furthermore, the net stresses at the potential
40 fatigue-critical locations are also checked for determining the final
fatigue-critical locations. The results demonstrate that the fatigue-
critical sections of the bridge deck are around the bridge towers.
30 Within the fatigue-critical sections, six of the strain points are
chosen for fatigue analysis, that is, the elements E32123 (T ) at
the top flange of the outer-longitudinal diagonal member close to
20
the Ma Wan Tower, E34415 (B) at the bottom flange of the outer-
longitudinal bottom chord of the Tsing Yi Tower, E40056 (T ) at the
10 top flange of the inner-longitudinal top chord of the Tsing Yi Tower,
E40906 (B) at the bottom flange of the inner-longitudinal bottom
chord of the Tsing Yi Tower, E39417 (B) at the bottom flange of the
0 T -section of the railway beam of the Tsing Yi Tower, and E55406
0 500 1000 1500 2000 (T ) at the top flange of the bottom web of the cross frame close to
the Tsing Yi Tower.
Fig. 8. Maximum stress ranges of diagonal members in north outer-longitudinal
trusses.
5. Databases of dynamic stress responses to different loadings
σmax because fatigue damage is a function of σrm or σrm+2 . σmax In this section, the databases of dynamic stress responses
is therefore selected as the index for determining the fatigue- induced by railway, highway, and wind loading at the critical
critical locations of bridge components of the same type. To make locations of the Tsing Ma Bridge are established based on the
the problem manageable, σmax is approximately decided by using loading information recorded by the WASHMS.
the difference of the maximum and minimum stress in the stress
time history based on the principle of the level crossing method. As 5.1. Database of wind-induced dynamic stress response
fatigue is critical to the tension and reversal members, additional
structural analysis should be performed to check the net stress in Long-span suspension bridges built in wind-prone regions
the member due to the dead and super-imposed dead loads plus suffer from considerable wind-induced vibration, which appears
an extreme live load. If it is positive, then the member is finally within a wide range of wind speeds and lasts for almost the whole
defined as a fatigue-sensitive member. design life of the bridge. A joint probability distribution function
Let us take the diagonal members of outer-longitudinal trusses of the mean wind speed and direction is utilized to describe wind
as an example to illustrate the determination of fatigue-critical intensity at the bridge site [11]. The distribution of wind speed
locations (see Fig. 7). Given the symmetry of the cross sections for any given wind direction is assumed to follow the Weibull
of the bridge, the standard train is supposed to run on the north distribution. The parameters in the distribution are determined
railway track and accordingly only the outer-longitudinal truss from monsoon wind records of hourly mean wind speed and
on the north needs to be considered. The stress time histories at direction during the period from 1 January 2000 to 31 December
the stress output points of all of the diagonal members of the 2005, which were collected by an anemometer installed on the
north outer-longitudinal truss are computed based on the standard top of the Ma Wan tower. Given that the measured typhoon wind
train running across the bridge on the north railway track. The records are not enough to establish a reliable joint probability
maximum stress ranges are subsequently estimated. Fig. 8 shows distribution, only monsoon wind is of concern in this study. The
the maximum stress ranges of the diagonal members of the north maximum wind speed at the top of the tower in each wind
outer-longitudinal truss due to the standard train running on direction within the 120-year design life is then obtained from the
the north side of the bridge deck. The potential fatigue-critical joint probability distribution. The maximum wind speed obtained
locations in the diagonal members of the north outer-longitudinal at the top of the tower is converted to the average deck level of the
truss can be determined from the figure: the diagonal member bridge. The maximum hourly mean wind speed at the deck level
E32123 (T ) close to the Ma Wan tower in the main span, and the is 25.89 m/s in the north direction for winds over the over-land
diagonal member E32403 (T ) close to the Tsing Yi tower in the fetch, and 15.47 m/s in the south direction for winds over the open-
main span. ‘‘T ’’ or ‘‘B’’ in brackets denotes that the potential fatigue- sea fetch [11]. Finally, a database of hourly wind-induced dynamic
critical location is at the top or bottom flange of the cross section stress responses at the fatigue-critical locations is established:

8. Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3253
from 5 to 26 m/s at an interval of 1 m/s for winds over the over- 60
Stress time history (MPa)
land fetch, and from 5 to 16 m/s at an interval of 1 m/s for winds
40
over the open-sea fetch. The stress fluctuations induced by wind
of a normal hourly mean wind speed less than 5 m/s are neglected 20
as they contribute little to fatigue. The database includes a total
of 34 one-hour time histories for each fatigue-critical location. 0
The nominal stress of each fatigue-critical member is computed
–20
based on the stresses at five points of the two ends of the member. 0 5 10 15 20
The hot-spot stresses, which reflect the stress concentration at Time (hour)
welded joints, should be considered in fatigue analysis [16]. The
hot-spot stresses at the fatigue-critical locations are determined Fig. 9. A sample stress time history due to multiple types of loading.
by multiplying the nominal stresses by the stress concentration
factor (SCF). It is noted that the value of SCF depends largely on railway, highway, and wind loading need to be generated for
the local geometry of the connection details. Nevertheless, there fatigue analysis. To generate them, the hourly mean wind speeds
are a number of fatigue-critical locations in this bridge, and the and directions for 120 years should be first obtained. A two-step
local geometry of the connection details at these locations is quite Monte Carlo simulation (MCS) method is adopted to draw out
different. Considering that the SCF of 1.4 was used in the design of 1051,200 (120 × 365 × 24) pairs of hourly mean wind speed and
the bridge concerned for almost all connections, this number is also direction for 120 years. In the first step, the mean wind direction
used for the six identified fatigue-critical locations in this study. is extracted through MCS according to the relative frequency of
The fatigue damage at fatigue-critical locations refers hereafter to the mean wind direction without considering wind speed. In the
the fatigue damage at these hot spots. second step, the mean wind speed at the top of the tower is
drawn out through MCS according to the probability distribution
5.2. Database of railway-induced dynamic stress response of the mean wind speed at the given mean wind direction under
the condition that it is not larger than the maximum wind speed
Since it is almost impossible to predict railway traffic volume in this direction. Finally, the mean wind speed and direction are
in the distant future for the Tsing Ma Bridge, one month of paired after two steps of MCS, and then converted into the hourly
railway traffic that is close to the current traffic conditions is normal mean wind speed to generate a sequence of 120 years.
adopted here to establish the database of railway-induced stress As the monsoon wind in Hong Kong normally is southerly (from
90° to 270°) in summer and northerly (from 270° to 90°) in
responses at the fatigue-critical locations for fatigue analysis.
winter, the sequence of hourly normal mean wind speeds in each
Monthly railway traffic in November 2005 is selected to establish
year is adjusted to consider this. For each hourly normal mean
the database, and more than 90% of trains are of the 16-bogie
wind speed in the sequence, the corresponding wind-induced
type. In addition, the daily average number of trains in this month
dynamic stress response can be found in the database established
reaches the maximum in the time period of concern. The daily
in the previous section. As wind blowing in two directions is of
time histories of railway-induced stress responses at the fatigue-
concern, the mean wind direction in each hour of the sequence
critical locations are computed using the stress influence lines for
is adopted to determine whether the wind is blowing in the
railway loading and the railway loading information measured in
direction of the over-land fetch or open-sea fetch. As the database
each day of November 2005. No large difference can be found in
is established for different levels of mean wind speed at an interval
the stress time histories among these days. The railway loading
of 1 m/s, rounding towards infinity is adopted to handle the hourly
information in each day of November 2005 is adopted to compute
normal mean wind speeds in the sequence. Finally, 1,051,200 h of
30 daily railway-induced stress time histories at the fatigue-critical
wind-induced dynamic stress time histories are generated at each
locations, to compile the database of railway-induced dynamic
fatigue-critical location to compose a time history of 120 years.
stress responses.
In addition, 120 years of railway-induced stress time histories
are generated at the fatigue-critical locations based on the
5.3. Database of highway-induced dynamic stress response database of 30 daily time histories. An integer between one and
thirty is randomly drawn out for each day to generate a random
The highway traffic on the Tsing Ma Bridge has been monitored number sequence of 120 years. For each one in the sequence,
through seven dynamic weigh-in-motion (WIM) stations installed the corresponding daily railway-induced dynamic stress responses
near the Lantau Administration Building since August 1998. The can be found in the database established in the aforementioned
road vehicle data in November 2005 are adopted to build a section. Finally, 43,800 daily time histories at each fatigue-critical
database of highway-induced stress responses because this month location are used to compose 120 years of railway-induced
reached a maximum number of monthly vehicles and other dynamic stresses. As the database of highway-induced stress
months had slightly less vehicle numbers. Highway-induced stress responses is also based on 30 daily time histories, a similar
time histories are also computed in one-day units. The daily processing method is applied to obtain 120 years of highway-
time histories of highway-induced stress responses at the fatigue- induced dynamic stresses at the fatigue-critical locations.
critical locations are computed using the stress influence lines for Based on the engineering approach proposed in the previous
highway loading and the highway loading information measured section, the stress responses at the critical locations induced by
in each day of November 2005. No large differences can be found the combined effects of railway, highway, and wind loading can
in the stress time histories among these days. Finally, 30 daily be approximately obtained from the three responses induced by
stress response time histories at the fatigue-critical locations are individual loadings by superposition. Therefore, a 120-year time
computed to create the database of highway-induced dynamic history of the stress induced by multiple types of loading is
stress response. determined from those induced by railway, highway, and wind
loading individually. It should be noted that the bridge is closed to
6. Multiple load-induced dynamic stress time histories in traffic when the mean wind speed recorded on site is very high;
design life therefore, the bridge stress responses under this condition are
induced by wind loading only. Fig. 9 shows a sample daily multiple
The design life of the Tsing Ma Bridge is 120 years; therefore, load-induced hot-spot stress time history at the critical location
120 years of time histories of the dynamic stresses induced by E32123.

9. 3254 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256
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-year
multiple load-induced stress time history, and the number of stress
cycles in different stress range levels can be obtained. A stress
range spectrum is defined as the percentage of the number of
stress ranges in each stress range set to the total number in all
sets. Fig. 10(a) displays the stress range spectrum at E32123. It
demonstrates that most of the stress ranges are at the low levels,
as 92.0% are less than 8 MPa. Because fatigue damage is much more
sensitive to high level stress range rather than low level one, a
stress range of large amplitude may make a great contribution to
fatigue damage although it occurs less frequently. Fatigue damage
in each stress range bin is computed using Eq. (2) or Eq. (3). The
type of welded connection at the six fatigue-critical locations in
this study is classified as F according to British Standard [20] with
σr ,0 = 40 MPa, K2 = 6.3 × 1011 , and m = 3. A fatigue damage Fig. 11. Cumulative fatigue damage curves at fatigue-critical locations.
spectrum is defined as the percentage of fatigue damage in each
stress range set to the total damage in all sets. Fig. 10(b) displays loading is greater than that due to wind loading at some locations
the fatigue damage spectrum at E32123. The figure shows that the whereas other locations are in reverse. It is also found that fatigue
contribution of stress ranges in the low levels (less than 8 MPa) to damage due to combined effects of railway, highway, and wind
fatigue damage is small, and that the greatest fatigue damage is in loading is larger than the sum of fatigue damage due to each of
the stress range of 36–44 MPa. individual loadings, for fatigue damage is the function of m-power
Based on the multiple load-induced stress time histories over stress range (nonlinear relationship), and stress ranges induced by
the period of 120 years and the time step t = 1/365 year, multiple loading are larger than those caused by individual loading.
the curves of cumulative fatigue damage within 120 years at the In addition, the fatigue damage spectra of railway, highway, and
fatigue-critical locations can be computed. The cumulative fatigue wind loading are investigated based on the 120-year time histories,
damage Dk in the kth day is calculated based on the daily stress and the results are shown in Fig. 12(a–c). The figure shows that
time history using Eqs. (1)–(3), and the cumulative fatigue damage the spectra are quite different. For example, the greatest fatigue
Dk is updated by adding the new damage on this day. Fig. 11 damage induced by railway loading is in the stress range of
shows the cumulative fatigue damage curves at the fatigue-critical 32–40 MPa, that induced by highway loading is in the range of
locations within a design life of 120 years. It is noted that the 0–4 and 8–24 MPa, and that induced by wind loading in the
structure is in danger when the cumulative fatigue damage is range of 0–12 MPa. To study the combined effect of multiple
types of loading on fatigue damage, a multiple load magnification
greater than one. The maximum of the 120 years of cumulative
factor is defined as the ratio of the fatigue damage due to the
fatigue damage at the fatigue-critical locations of the Tsing Ma
combined effect of the three loadings to the sum of the damage
Bridge is very close to one, which implies that the health condition
due to each individual loading. The factors at the six fatigue-critical
of the bridge is satisfactory. In addition, the cumulative damage
locations are computed and range from 1.06 to 1.35. The maximum
curves seem to be very linear. That is because Miner’s model is factor is at critical locations E32123 and E34415, at which the
a linear damage model, and traffic loading is assumed to remain fatigue damage induced by highway and wind loading is much
stable over the design life closer to that induced by railway loading than at the other critical
In addition to the fatigue damage induced by multiple types of locations. The results indicate that the combined effect of multiple
loading, the fatigue damage induced by each individual loading loads must be considered in a bridge subject to multiple types
type is also investigated. The 120 years of cumulative fatigue of loading, especially in the case in which the contributions of
damage induced by railway, highway, and wind loading are different loadings to fatigue damage are close.
respectively computed based on the three stress responses under
the different loadings. The results of the damage at different 8. Conclusions
fatigue-critical locations are listed in Table 2. It is found that
railway loading plays a dominant role in the fatigue damage of A general framework has been proposed for fatigue analysis
the Tsing Ma Bridge, and that the damage induced by highway of a long-span suspension bridge under multiple loading over its

10. Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3255
Table 2
120 years of cumulative fatigue damage under different loading types.
Fatigue-critical locations Loading types
Railway (R) Highway (H) Wind (W ) R+H +W
E32123 (T ) 0.70 0.048 0.011 1.02
E34415 (B) 0.66 0.044 0.0092 0.96
E40056 (T ) 0.52 0.0022 0.0057 0.68
E40906 (B) 0.42 0.0025 0.0052 0.54
E55406 (T ) 0.34 0.0037 0.0016 0.41
E39417 (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 Tsing in other locations. Furthermore, it is necessary to consider the
Ma suspension bridge in Hong Kong. An engineering approach combined effect of multiple types of loading in the fatigue analysis
for dynamic stress analysis of a long-span suspension bridge of long-span suspension bridges. In reality, uncertainties exist in
under multiple types of loading was first proposed. The Tsing external loadings, structural modelling, and structural parameter
Ma Bridge and the measurement data recorded by the WASHMS in fatigue assessment. The fatigue reliability analysis of multi-
were employed to verify the feasibility of the proposed approach. loading bridges deserves further study.
The fatigue-critical locations of the bridge were determined for
the key structural components. Based on the measurement data Acknowledgements
recorded by the WASHMS installed on the bridge, the databases
of wind, railway, and highway loadings as well as stress responses The authors wish to acknowledge the financial support from the
at the fatigue-critical locations were established to generate 120- Research Grants Council of Hong Kong (PolyU 5327/08E), the Hong
year time histories of multiple loading-induced stress responses. Kong Polytechnic University (PolyU-1-BB68), and the National
Finally, the fatigue analysis based on the 120-year stress time Natural Science Foundation of China (NSFC-50830203 and NSFC-
histories was performed to compute the cumulative fatigue 51108395). Sincere thanks go to the Highways Department of Hong
damage over the bridge’s design life. The results indicate that Kong for providing the authors with the field measurement data.
the health condition of the bridge is satisfactory. The cumulative Any opinions and concluding remarks presented in this paper are
fatigue damage induced by individual loading and the damage entirely those of the authors.
magnification due to the combined action of three types of loadings
were also investigated. The results show that railway loading References
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