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Statistical analysis of stress spectra for fatigue life assessment of steel bridges

with structural health monitoring data

X.W. Ye a, Y.Q. Ni a,⇑, K.Y. Wong b, J.M. Ko a

a Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong b Highways Department, The Hong Kong SAR Government, Hong Kong

a r t i c l e i n f o

Article history:

Received 2 January 2011

Revised 9 June 2012

Accepted 14 June 2012

Available online 24 July 2012

Keywords:

Steel bridge

Structural health monitoring

Fatigue life assessment

Stress spectrum

Typhoon effect

a b s t r a c t

This paper aims at developing a monitoring-based method for fatigue life assessment of steel bridges

with use of long-term monitoring data of dynamic strain. A standard daily stress spectrum is derived

by statistically analyzing the stress spectra accounting for highway traffic, railway traffic, and typhoon

effects. The optimal number of daily strain data for derivation of the standard daily stress spectrum is

determined by examining the predominant factors which affect the prediction of fatigue life. With the

continuously measured dynamic strain responses from the instrumented Tsing Ma Bridge carrying both

highway and railway traffic, the proposed method is exemplified to evaluate the fatigue life of fatigue-

critical welded details on the bridge.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

Steel bridges are subjected to a large number of repetitive load-

ings of different magnitudes caused mainly by the passage of vehi-

cles. They are expected to be vulnerable to fatigue-related damage

and failure [1–3]. Therefore, it is essential to assess the fatigue

behavior of structural components, especially welded details on

steel bridges. Among the current fatigue analysis methodologies,

the traditional stress–life (S –N ) approach has been widely used

for fatigue-related design and evaluation of aircraft, offshore struc-

tures, and steel bridges [4–9]. When the stress–life approach stip-

ulated in specifications is adopted for bridge fatigue damage and

life evaluation, it is requisite to know the stress spectra of welded

details in critical locations and the S –N curves of the details

[10,11].

The stress spectra can be acquired from a theoretical stress

analysis by assuming a fatigue loading model and a structural

model. The fatigue loading models are usually assumed according

to the specifications or obtained based on the measurement data

for a relatively short time, and seldom constructed according to

the statistical treatment of long-term monitoring data under actual

traffic conditions. As to the structural model, the uncertainties in

material properties, geometric configurations, and boundary con-

ditions cannot be entirely considered while the finite element

technique provides a powerful tool in structural modeling and

analysis. Therefore, the accuracy of the simulated stress may be

very limited and a considerable discrepancy between the theoret-

ical analysis results and actual stress status exists in many cases

[12]. The consequence of inaccurate estimation of the stress spec-

tra would be significant in view of the fact that the predicted fati-

gue life is inversely proportional to at least the cube of the stress

range [13].

It has been shown that fatigue analysis based on the design

specifications usually underestimates the remaining fatigue life of

existing bridges by overestimating the live load stress ranges [14].

Fatigue evaluation by using field monitoring data of strain or stress

under actual traffic loads proves to be a more accurate and efficient

method for existing bridges [15,16]. This can be achieved by using

non-destructive evaluation (NDE) technologies such as the

load-controlled diagnostic load testing and short-term in-service

monitoring. Research efforts have been devoted to fatigue life

assessment of bridges through field strain measurement [17–22].

However, most of these strategies were encountered withthe prob-

lems either in disturbing the normal operation of the tested bridge

or only procuring limited data under normal traffic conditions with

a designated time period. Particularly, the NDE technologies are

impossible to be performed and thus incapable of recording the

field strain time history data under extreme conditions, such as

typhoon, earthquake, and ship collision.

On-line structural health monitoring (SHM), enabled by recent

advances in sensing and data acquisition technologies, communica-

tion algorithms, computational techniques, and data management

systems, has gained worldwide applications in civil engineering

0141-0296/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.06.016

⇑ Corresponding author. Tel.: +852 2766 6004; fax: +852 2334 6389.

E-mail address: [email protected] (Y.Q. Ni).

Engineering Structures 45 (2012) 166–176

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field over the past decade [23–31]. It has increasingly become an

important tool for diagnosing and prognosing the structural health

conditionand safety status of bridges. Withan instrumented bridge

health monitoring system, the continuously field-measured

dynamic strain data can be acquired and transferred into the stress

spectra resulting from actual traffic and environmental conditions.

The stress spectra thus obtained provide the most authentic and

practical information for fatigue assessment of steel bridges as well

as for decision-making on bridge inspection and maintenance

actions.

In this paper, a monitoring-based stress–life fatigue evaluation

method is proposed and verified using long-term measurement

data acquired by an on-line SHM system permanently installed

on the suspension Tsing Ma Bridge. The daily stress spectra at a

specific location are procured using field-measured strain time

history data and rainflow counting algorithm. In recognizing the

resemblance of the daily stress spectra obtained under normal

traffic and wind conditions, a standard traffic-stress spectrum is

derived by averaging an appropriate number of daily stress spectra.

Likewise, a standard typhoon-stress spectrum is derived using the

field-measured data during typhoon periods. Then a standard daily

stress spectrum accounting for highway traffic, railway traffic, and

typhoon effects is formulated by synthesizing the two kinds of

stress spectra in accordance with the proportion of typhoon days

in a year. The optimal number of daily strain data for derivation

of a standard daily stress spectrum is determined by examining

the effect of different influencing factors including the number of

daily strain data, weekend and workday traffic variations, and

seasonal changes in traffic patterns. With such obtained standard

daily stress spectrum, the fatigue life of a critical welded detail

on the bridge is evaluated.

2. Strain measurement data from SHM system

2.1. Tsing Ma Bridge and its SHM system

The Tsing Ma Bridge in Hong Kong, as shown in Fig. 1, is a steel

suspension bridge with a main span of 1377 m and an overall

length of 2160 m. It carries a dual three-lane highway on its upper

level of the bridge deck while two railway tracks and twin single-

lane sheltered carriageways are located on the lower level of the

bridge deck. As a combined highway and railway transport connec-

tion between the Tsing Yi Island and the Lantau Island, it forms a

key part of the most essential transportation network linking the

Hong Kong International Airport to the urban areas.

With the purpose of tracing the structural health conditions of

the bridge, a long-term SHM system, named WASHMS (Wind

And Structural Health Monitoring System) has been implemented

and operated on the Tsing Ma Bridge by the Highways Department

of the Hong Kong Special Administrative Region since its opening

to public traffic in 1997 [32,33]. Based on a modular design concept

[34], the WASHMS was devised to be composed of six integrated

modules, namely Module 1 – sensory system (SS); Module 2 – data

acquisition and transmission system (DATS); Module 3 – data pro-

cessing and control system (DPCS); Module 4 – structural health

evaluation system (SHES); Module 5 – structural health data man-

agement system (SHDMS); and Module 6 – inspection and mainte-

nance system (IMS).

The SS consists of 283 sensors in eight types, including ane-

mometers, servo-type accelerometers, temperature sensors, dy-

namic strain gauges, global positioning systems, displacement

transducers, level sensors, and dynamic weigh-in-motion sensors.

It is purposed to monitor structural and environmental conditions

under operation and to evaluate structural degradation as its oc-

curs through the long-term monitoring of four categories of

parameters, i.e., environmental loads and status, traffic loads,

bridge features, and bridge responses. The DATS comprises on-

structure data acquisition units and optical fiber cabling networks

for the acquisition, conversion, temporary storage, and transmis-

sion of signals. On-structure data acquisition units are indispens-

able to SHM systems deployed on long-span bridges and

essential to assuring quality and fidelity of the acquired data

[26]. The DPCS refers to the hardware and software of a high-per-

formance computer for system control, operation display, and pro-

cessing and analysis of data. The DPCS and SHES, which are key

modules for executing offline structural health evaluation, are

composed of servers, workstations, and software facilities for data

interpretation, health evaluation, and data management [35]. The

IMS is a laptop-computer-aided portable system for the inspection

and maintenance of the SHM system itself. The SHM system de-

ployed on the Tsing Ma Bridge has operated continuously and stea-

dily for 15 years.

As part of the SHM system, 110 weldable foil-type strain gauges

were installed to measure the strain of structural members under

different loading conditions. As shown in Fig. 1, the locations of

strain gauges include the rail track section at Chainage (CH)24662.5, bridge-deck trough section at CH24664.75, deck at tower

and rocker bearing links at CH23623, and bridge-deck section at

CH23488. Most of the strain gauges were attached to the fatigue-

prone portions which were identified during the design of the

monitoring system.

In order to assess the fatigue condition of the bridge, 1-year (the

year of 1999) monitoring data of dynamic strain from all 110 strain

gauges have been acquired for strain-based fatigue and condition

assessment. In this paper, the fatigue life evaluation for the most

fatigue-prone detail monitored by the strain gauge SSTLS13 on

the rail track section at CH24662.5 is exemplified. Fig. 2 illustrates

the location of the strain gauge SSTLS13 which was installed under

the bottom flange of the railway beam composed of two inverted

1377m355.5m76.5m23m 300m

72m 72m 72m 72m

206.4m 206.4m

Anchorage

TsingYi Island

Anchorage

Ma Wan Island78.58m

Tsing Yi TowerMa Wan Tower

Str-L(29) Str-L(30)Str-R(2)

Str-L(47)Str-R(2)

C H 2

3 4 8 8

C H 2

3 6 2 3

C H 2

4 6 6 2 . 5

Str-L: linear strain gauge (106)Str-R: rosette strain gauge (4)

C H 2

4 6 6 4 . 7 5

Fig. 1. Tsing Ma Bridge and layout of strain gauges.

X.W. Ye et al. / Engineering Structures 45 (2012) 166–176 167



T-beams welded to the top flange plate, denoted as Detail H inFig. 2a. Fig. 2b shows the sectional view of Detail H.

2.2. Strain measurement data under normal traffic and wind

conditions

The strain data were recorded continuously at a sampling fre-

quency of 51.2 Hz. Fig. 3 shows three typical daily strain time his-

tories obtained from the strain gauge SSTLS13 under normal traffic

and wind conditions (the strain time history measured on a typical

day in 2011 is specifically provided for comparison with the daily

strain time histories obtained in 1999). By examining the measure-

ment data, the following observations are made: (i) the daily

strain–time curves exhibit some common characteristics in the

shape of curves and magnitude of cycles, and there are almost nostrain pulses from 1:30 am to 5:30 am since the airport railway

ceases its service during this time period; (ii) strain pulses with

large amplitudes appear in each daily strain time history, which

are the strain responses caused by train traffic; and (iii) the overall

drift of the strain–time curves is significant, which is attributed to

the daily cycle effect of temperature variation after confirming its

synchronism with the longitudinal displacement time history con-

currently measured at the expansion joint. This drift does not affect

the calculation of stress range since the stress range depends only

on the difference between the peak and the valley of each stress

cycle. The high coincidence between the strain response data ac-

quired in 1999 (Fig. 3a and b) and those acquired in 2011

(Fig. 3c) demonstrates high durability and excellent performance

of the strain gauges deployed on the Tsing Ma Bridge during a longworking period exceeding 10 years.

The measured strain responses stem mainly from four effects:highway traffic, railway traffic, wind, and temperature (the static

strain due to initial dead loads is unable to obtain as the strain

gauges were installed after the completion of construction). The

overall drift (slow-varying ingredient) caused by temperature ef-

fect, although quite large, contributes little to the stress because

the majority of temperature-induced strain is released by move-

ment and rotation of the bridge deck at the expansion joint and

bearings. When fatigue assessment methods take into account

stress range only (such as the method used in this study), it is

unnecessary to separate the temperature-induced strain because

it does not affect the calculation of stress range. When mean stress

is required in addition to stress range for fatigue life assessment

[36], stress-irrelevant components have to be eliminated from

the strain measurement data before inferring the correspondingstress. A wavelet-based multi-component decomposition method

[37] has been developed to eliminate the temperature-induced

ingredient from raw measurement data of strain. It uses the de-

correlation and perfect reconstruction properties of discrete wave-

let transform in multi-resolution analysis and embeds a selection

criterion for physical source extraction. In addition to separating

the temperature-induced strain components, this method is also

competent to remove the spikes and noises corrupted in the mea-

surement data.

2.3. Strain measurement data under typhoon conditions

The Tsing Ma Bridge is located at a region with typhoon wind

climate. A total of five typhoons, named ‘‘Leo’’, ‘‘Maggie’’, ‘‘Sam’’,‘‘York’’, and ‘‘Dan’’ buffeted Hong Kong in 1999. The wind informa-

CL OF BRIDGE

STIFFENER

RAILWAY BEAM WEB

RAILWAY BEAM FLANGE

CROSS FRAME FLANGE

50mm755mm 755mm

1050mm

2050mm

1050mm

2050mm

1 4 0 0

m m

SSTLS13

CL

CROSS FRAME FLANGE

TRACK PLATE

(a)

(b)

Fig. 2. Strain gauge SSTLS13 on rail track section at CH24662.5: (a) deck cross-section at CH24662.5; (b) sectional view of Detail H.

168 X.W. Ye et al. / Engineering Structures 45 (2012) 166–176



tion during the typhoon attacks was measured by the digital ultra-

sonic anemometers which were deployed on the north and south

sides of the bridge deck at the middle of the main span of the Tsing

Ma Bridge. Each anemometer is able to measure three components

of wind velocity simultaneously. Fig. 4 shows the 10-min mean

wind speed and wind direction, respectively, obtained from the

north anemometer denoted as WITJN01 on August 22, 1999 during

typhoon ‘‘Sam’’. The wind direction is measured from east anti-

clockwise. It is seen from Fig. 4 that the maximum 10-min mean

wind speed was 24.03 m/s at about 15:40, August 22, 1999 during

typhoon ‘‘Sam’’.

The strain time history data recorded by the monitoring system

during the typhoon attacks are highly valuable for assessing

typhoon-induced fatigue of the bridge. It is usually impossible to

carry out field testing on a bridge when a typhoon passes over

the bridge because all actions on the bridge have to be stopped

during that period. For the Tsing Ma Bridge, trains across the bridge

to the Hong Kong Airport were still in service during the typhoon

attacks. Fig. 5 gives the measured strain time history acquired by

the strain gauge SSTLS13 on August 22, 1999 during typhoon

‘‘Sam’’. It is observed that the strain time history due to typhoon,

when the highway road was closed, has a pattern different from

00:00 04:00 08:00 12:00 16:00 20:00 24:00

-250

-200

-150

-100

-50

0

Time (hour)

S t r a i n ( 1 0 - 6 )

(a)

00:00 04:00 08:00 12:00 16:00 20:00 24:00

-250

-200

-150

-100

-50

0

Time (hour)

S t r a i n ( 1 0 - 6 )

(b)

00:00 04:00 08:00 12:00 16:00 20:00 24:00

-150

-100

-50

0

50

100

Time (hour)

S t r a i n ( 1 0 - 6 )

(c)

Fig. 3. Measured daily strain time histories under normal traffic and wind conditions: (a) on August 8, 1999; (b) on September 24, 1999; (c) on July 3, 2011.




that under normal traffic and wind conditions as shown in Fig. 3. A

further observation into Fig. 5 reveals that the typhoon-induced

strain–time curve has much more cycles of strain range in large

amplitudes, and the strain values obtained during 12:30–16:00

are larger than others, which are in coincidence with the track

and time of typhoon ‘‘Sam’’ arriving around the Tsing Ma Bridge.

3. Statistical analysis of stress spectra

3.1. Rainflow cycle counting technique

When the strain time histories are available, the stress time his-

tories are obtained through simply multiplying the measured

strain data by the elasticity modulus of steel in assumption of elas-

tic strain. The material of steel components of the Tsing Ma Bridgeis Grade Fe 510C as specified in BS EN 10 025 (1990) and the elas-

ticity modulus is 205 GPa. As the strain time histories are com-

posed of many pulses as shown in Fig. 3, it is difficult to define a

stress cycle directly. Instead, a cycle counting method should be

pursued to transfer the complex irregular stress time histories into

a set of constant stress range frequency data (stress spectrum). In

this study, the rainflow cycle counting technique [38–40] is

adopted to seek the peaks and valleys in the stress time history

and then extract stress ranges and the number of cycles for each

specific stress range.

3.2. Derivation of standard daily stress spectrum

By executing the rainflow cycle counting technique to the stresstime history data covering 1 day, a daily stress spectrum is ob-

tained. Fig. 6 shows the histograms of three typical daily stress

spectra attained under normal traffic and wind conditions by spec-

ifying a resolution of 1 MPa for the stress range interval. The stress

cycles with amplitudes less than 2 MPa are discarded because the

lower valid limit of the strain gauge is 10 micro-strains. A compar-

ison of Fig. 6c with Fig. 6a and b evidences that the traffic amount

on the Tsing Ma Bridge does not have noticeable growth in the past

decade.

An insight into Fig. 6 reveals that the daily stress spectra are

alike for different days under normal traffic and wind conditions.

It is therefore reasonable to average an appropriate number of dai-

ly stress spectra resulting from different days to form a standard

traffic-stress spectrum, as shown in Fig. 7. Similarly, a standard ty-phoon-stress spectrum can be derived using the strain measure-

ment data obtained under typhoon conditions, as shown in

Fig. 8. It is observed from Fig. 8 that the stress spectrum under ty-

phoon conditions has a pattern different from that under normal

traffic and wind conditions.

After obtaining the standard traffic-stress spectrum and the

standard typhoon-stress spectrum, a more general standard daily

stress spectrum accounting for both traffic (highway and railway)

and typhoon effects can be formulated by proportionally combin-

ing the two standard stress spectra. With such a standard daily

stress spectrum, the fatigue life of the structural detail under

00:00 04:00 08:00 12:00 16:00 20:00 24:000

5

10

15

20

25

Time (hour)

M e a n w i n d s

p e e d ( m / s )

(a)

00:00 04:00 08:00 12:00 16:00 20:00 24:000

50

100

150

200

Time (hour)

M e a n w i n d d i r e c t i o n ( d e g r e e )

(b)

Fig. 4. Mean wind speed and wind direction acquired by anemometer WITJN01: (a)

10-min mean wind speed; (b) 10-min mean wind direction.

00:00 04:00 08:00 12:00 16:00 20:00 24:00

-160

-120

-80

-40

0

20

Time (hour)

S t r a i n ( 1 0 - 6 )

Fig. 5. Measured strain time history on August 22, 1999 during typhoon ‘‘Sam’’.




monitoring can be evaluated by regarding that the monitored de-

tail suffers from a block of daily repeated cycles in accordance with

the standard daily stress spectrum. The determination of optimal

number of daily strain data for derivation of a standard daily stress

spectrum will be described in the next section.

4. Monitoring-based fatigue life assessment

4.1. Presentation of procedure

BS5400 Part 10 [10] is a commonly adopted standard for bridgedesign, which specifies methods for fatigue damage and life assess-

ment. The Tsing Ma Bridge was designed according to this stan-

dard. Therefore, the monitoring-based fatigue life evaluation will

also be carried out in accordance with this standard. The methods

for fatigue design and assessment stipulated in BS5400 Part 10 are

based on the Miner’s rule [41] for fatigue damage accumulation,

which is expressed by

D ¼Xi

ni

N ið1Þ

where D is the fatigue damage accumulation index; ni is the speci-

fied number of cycles for the ith stress range, S i; and N i is the corre-sponding number of cycles to failure for the ith stress range.

0 10 20 30 40 50 600

500

1000

1500

2000

2500

Stress range (MPa)

C y

c l e s / d a y

(a)

0 10 20 30 40 50 600

500

1000

1500

2000

2500

Stress range (MPa)

C y c l e s / d a y

(b)

0 10 20 30 40 50 600

500

1000

1500

2000

2500

Stress range (MPa)

C y c l e s / d a y

(c)

Fig. 6. Histograms of daily stress spectra under normal traffic and wind conditions: (a) on August 8, 1999; (b) on September 24, 1999; (c) on July 3, 2011.




As stated before, the stress ranges and the corresponding num-

ber of cycles suffered by the structural components under monitor-

ing have been obtained by rainflow cycle counting of the strain

measurement data. It is observed in Fig. 6 that the obtained stress

spectra contain a large number of small stress ranges. This is a

common situation for steel bridges [11]. The procedure to treat

the low stress ranges recommended by BS5400 is to reduce the

number of repetitions of each stress range less than the fatigue

limit, S 0, by multiplying a reducing factor, k i, which is defined as

ki ¼ ðS i=S 0Þ

2if S i < S 0

1 if S i > S 0

( ð2Þ

In order to calculate the cumulative fatigue damage on a struc-

tural component by the Miner’s rule, the number of repetitions tofailure of the specified stress range is needed. It is obtained from

the S –N curves or S –N relationships, which are established from

the experimental results for different materials and different cate-

gories of welded details. The S –N curves provided in BS5400 were

formulated through statistical analyses of available experimental

data (linear regression analysis of logS and logN ) with minor

empirical adjustments to ensure the compatibility of results be-

tween various categories, in which the stress–life relation is repre-

sented by

N S m ¼ K 0 D

d ð3Þ

where N is the predicted number of cycles to failure for a stress

range, S ; m is the inverse slope of the mean-line logS –logN curve;K 0 is a constant relating to the mean-line log S –logN curve; D is

the reciprocal of the anti-log of the standard deviation of logN ;

and d is the number of standard deviations below the mean-line

logS –logN curve, which is also called the probability factor of which

different values correspond to different probabilities of failure.

Each connection detail subject to fluctuating stress should,

where possible, have a particular class designated in BS5400. The

detail of the weld joint near to the strain gauge SSTLS13 is shown

in Fig. 2. It is categorized as class F 2 according to BS5400.

According to the Miner’s rule, the total cumulative fatigue dam-

age shouldnot be greater than unity; otherwise the structural com-

ponent will be considered to have fatigue failure. The Miner’s

summation in accordance with the standard daily stress spectrum

represents the cumulative fatigue damage generated per day on

the structural component monitored. Consequently, when the fati-

gue life F is represented in terms of number of years, it is obtained

from the standard daily stress spectrum andthe S –N relationshipas

F ¼ 1

365 Xi

kini

N i

ð4Þ

A flowchart illustrating the monitoring-based fatigue life

assessment method is given in Fig. 9. Firstly, the original measure-

ment data of structural strain are retrieved from the SHM system

and preprocessed to remove the spikes and noises [37]. Then, the

strain time history is converted into the stress time history by sim-

ply multiplying the elasticity modulus of material. After obtaining

the stress time history, the rainflow counting algorithm is used to

determine the stress range and number of cycles. Subsequently,the standard stress spectrum is derived by statistically analyzing

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

Stress range (MPa)

C y

c l e s / d a y

Fig. 7. Standard traffic-stress spectrum.

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

Stress range (MPa)

C y c l e s / d a y

Fig. 8. Standard typhoon-stress spectrum.




the stress spectra accounting for the traffic characteristics and

external loads. Finally, the fatigue life of the structural component

is calculated by using the S –N curve method together with the

Miner’s rule.

4.2. Determination of optimal number of daily strain data

A major issue in converting the field strain data to construct

stress range frequency is the duration of the data acquisition pro-

cess. In general, a 2- or 3-day period for highway bridges and a 5-

to 10-day period for railway bridges are sufficient since during

such a period nearly all stress ranges experienced by critical com-

ponents in a bridge can be captured [19]. For the Tsing Ma Bridge

which carries not only highway but also railway traffic, the strain

data have been acquired continuously by the on-line SHM system.

It provides a good chance to establish a realistic representation of

entire stress ranges and their frequencies. On the other hand, it is

impractical and time-consuming to take all daily strain data into

account in the fatigue life evaluation. In the following, we investi-

gate the influence of number of daily strain data on the predictedfatigue life with the purpose of seeking an optimal number of daily

strain data to formulate the standard daily stress spectrum.

Eighty-day measurement data obtained from the strain gauge

SSTLS13 under normal traffic and wind conditions are selected

for this purpose. Principally, each daily stress spectrumcan be used

to predict a fatigue life at this detail. Fig. 10 shows the predicted

fatigue life from the individual daily stress spectra sorted in an

ascending order. It is obvious that the predicted fatigue life is dif-

ferent from different daily stress spectra, ranging from 620 to

1001 years. The reason which can be easily understood is because

the live load really exerted on the bridge is different in each spe-

cific day.

Table 1 lists the fatigue life predicted using different number of

daily strain data randomly extracted from the 80 days. Firstly, 5daily strain data are used and drawn out randomly from 80 daily

strain data, and the involved dates will be included in the next cal-culation; the rest is deduced by analogy. It is seen that the pre-

Collection of data from SHM

Preparation of strain time history

Convert the strain time history

into stress time history

Elasticity modulus

of material

Determine the stress range and

number of cycles

Rainflow counting

algorithm

Obtain the standard stress spectrumIdentify the joint detail Establish the S - N curve

Calculate the fatigue life of the

structural component

Miner’s rule

Statistical analysis

Fig. 9. Flowchart of monitoring-based fatigue life assessment method.

0 10 20 30 40 50 60 70 80600

650

700

750

800

850

900

950

1000

1050

Sample number

F a t i g u e l i f e ( Y e a

r )

Fig. 10. Fatigue life predicted using different daily stress spectra.

Table 1

Predicted fatigue lives with different number

of daily strain data.

Number of daily

strain data

Fatigue life

(year)

5 780

10 746

15 754

20 768

25 768

30 767

80 765




dicted fatigue life is varying slightly when using more than 10 daily

strain data and keeps almost the same when using more than 20

daily strain data. Therefore, it is reliable to assess the fatigue life

by using a standard daily stress spectrum obtained from adequate

daily strain data. For the specific case of the Tsing Ma Bridge, it is

appropriate to use 20 daily strain data in formulating a standard

daily stress spectrum.

It is preferable to further observe the effects of weekend and

workday traffic variations as well as seasonal changes in traffic

patterns on the fatigue life evaluation. Table 2 lists the fatigue life

prediction results when considering different traffic patterns: (i) 20

daily strain data on weekend, (ii) 20 daily strain data on workday,

(iii) 10 daily strain data on weekend and 10 daily strain data on

workday, (iv) 20 daily strain data in summer, (v) 20 daily strain

data in winter, and (vi) 10 daily strain data in summer and 10 daily

strain data in winter. It is observed from Table 2 that these two fac-

tors only have little influence on the fatigue life prediction results.

Because the Tsing Ma Bridge plays a very important role in the

transportation between the Hong Kong International Airport and

downtown, the traffic volume on the bridge is substantive and sta-

tionary over the whole year except for some special periods such as

the duration of typhoons. According to the Annual Traffic Census

performed by the Traffic and Transport Survey Division of the

Transport Department of Hong Kong SAR Government [42], the

highway traffic volume has become steady since the opening of

the new Hong Kong International Airport in July 1998, and the rail-way traffic is regular and would not grow noticeably.

4.3. Fatigue life assessment of Tsing Ma Bridge

When using the standard daily stress spectrum method, the

strain/stress data obtained under typhoon conditions should be

included for fatigue life evaluation. A total of 22 days out of

365 days were under typhoon conditions in 1999. According to this

proportion, strain data acquired from 20 days including 1 day

under typhoon conditions are chosen to construct the standard

daily stress spectrum which is illustrated in Fig. 11. With this

standard daily stress spectrum, the fatigue life of the welded detail

monitored by the strain gauge SSTLS13 will be evaluated.

Since the weld joint near the strain gauge SSTLS13 is catego-

rized as F 2, the parameters K 0, D, and m are determined from

BS5400 as K 0 = 1.23 1012, D = 0.592, and m = 3.0. When the prob-

ability factor is assumed as d = 2 which corresponds to a failureprobability of 2.3% and the standard S –N design curve [10], the fa-

tigue life of the welded detail monitored by the strain gauge

SSTLS13 is calculated as 718 years. It is only slightly different from

the predicted fatigue life by using the standard traffic-stress spec-

trum. This is because the strain data under typhoon conditions ac-

count for a small portion in 1 year in comparison with the strain

data under normal traffic and wind conditions. For the Tsing Ma

Bridge, the fatigue damage of structural components is mainly

caused by the highway and railway traffic.

Table 3 lists the predicted fatigue lives of the welded detail

monitored by the strain gauge SSTLS13 with different probabilities

of failure by using the formulated standard daily stress spectrum,

Table 2

Predicted fatigue lives with different traffic

patterns.

Traffic pattern Fatigue life (year)

i 749

ii 739

iii 754

iv 772

v 747vi 763

0 10 20 30 40 50 600

500

1000

1500

2000

2500

Stress range (MPa)

C y c l e s / d a y

Fig. 11. Histogram of standard daily stress spectrum.

Table 3

Predicted fatigue lives with different probabilities of failure.

Probability of failure (%) Probability factor, d Fatigue life (year)

50 0 3836

31 0.5 2528

16 1.0 1664

2.3 2.0 718

0.14 3.0 323

010203040500

500

1000

1500

2000

2500

3000

3500

4000

Percentage of failure probability

F a t i g u e l i f e ( y e a r )

Calculated value

Fitted curve

Fig. 12. Predicted fatigue lives with different probabilities of failure.




which are also illustrated in Fig. 12. It is seen from Fig. 12 that the

predicted fatigue life decreases with the decrease of the failure

probability. The predicted fatigue life for the welded detail near

the strain gauge SSTLS13 is longer than the design fatigue life of

500 years adopted in designing the Tsing Ma Bridge. This is par-

tially because the existent volume of traffic, especially the highway

traffic, is quite low in comparison with the maximum allowable

traffic volume designated in the bridge design.

5. Conclusions

SHM has become an increasingly accepted technology for diag-

nosing and prognosing bridge condition and safety. The continu-

ously measured strain data from an on-line SHM system can be

used to assess the status of fatigue which is among the most crit-

ical forms of damage potentially occurring in steel bridges. In this

paper, a standard daily stress spectrum method for fatigue life

assessment of steel bridges using SHM data has been proposed

and applied to assess the fatigue status of critical welded details

on the instrumented Tsing Ma Bridge carrying both highway and

railway traffic. While the proposed method was demonstrated

using the monitoring data from a specific bridge, it is viable to con-struct a standard stress spectrum from long-term monitoring data

for the purpose of facilitating the fatigue life assessment of instru-

mented bridges. The proposed method makes it convenient to

simultaneously consider the effects of different loads (highway

traffic, railway traffic, monsoon, typhoon) with the use of a single

standard stress spectrum. When noticeable annual growth in traf-

fic volume is observed, the proposed method is still applicable by

deriving a growth factor for each dominant stress range when

the monitoring data acquired in different years are available. In

applying the proposed method, it is unnecessary to separate the

temperature-induced ingredient and slow-varying drift from the

raw measurement data, thereby enabling the usability of the strain

measurement data containing stress-irrelevant components and

static drift and exempting the use of a multi-component decompo-sition method.

Based on the statistical analysis of the stress spectra and the fa-

tigue life assessment results obtained for the Tsing Ma Bridge, the

following specific conclusions can be drawn: (i) a standard daily

stress spectrum accounting for highway traffic, railway traffic,

and typhoon effects can be formulated from the long-term moni-

toring data by combining the standard traffic-stress spectrum

and standard typhoon-stress spectrum proportionally. Due to hea-

vy highway and railway traffic on the bridge, the predicted fatigue

life using the general standard daily stress spectrum is not much

different from that predicted by the standard traffic-stress spec-

trum obtained under normal traffic and wind conditions; (ii) for

the Tsing Ma Bridge, the predicted fatigue life is varying slightly

when using more than 10 daily strain data and keeps almost the

same when using more than 20 daily strain data. As a result, it is

appropriate to use about 20 daily strain data in formulating the

standard daily stress spectrum. It is also found that the influence

of different traffic patterns and seasonal changes is insignificant

for fatigue life prediction; and (iii) the proposed method provides

a feasible approach for fatigue life assessment of welded details

on steel bridges by using field monitoring data from an SHM sys-

tem, which in turn helps bridge management authority and engi-

neers to make decisions in prioritizing inspection maintenance

activities.

Acknowledgments

The work described in this paper was supported by The HongKong Polytechnic University under the Grant G-U845 and through

the Development of Niche Areas Programme (Project No. 1-BB68).

The writers also wish to thank the engineers at the Highways

Department of the Hong Kong SAR Government for their support

throughout the work.

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