Buffeting-induced fatigue damage assessment of a long suspension bridge

International Journal of Fatigue 31 (2009) 575–586

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Buffeting-induced fatigue damage assessment of a long suspension bridge Y.L. Xu a,*, T.T. Liu b, W.S. Zhang b a b

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China Department of Engineering Mechanics, Dalian University of Technology, Dalian, China

a r t i c l e

i n f o

Article history: Received 21 September 2007 Received in revised form 19 March 2008 Accepted 19 March 2008 Available online 29 March 2008 Keywords: Long suspension bridges Buffeting-induced fatigue Health monitoring system Stress analysis Continuum damage mechanics

a b s t r a c t As modern suspension bridges become longer and longer, buffeting-induced fatigue damage problem for the bridges located in strong wind regions may have to be taken into consideration. Furthermore, there is a trend to install wind and structural health monitoring systems (WASHMS) to long suspension bridges for performance assessment. A systematic framework for assessing long-term buffeting-induced fatigue damage to a long suspension bridge is thus presented in this paper by integrating a few important wind/ structural components with continuum damage mechanics (CDM)-based fatigue damage assessment method. By taking the Tsing Ma Bridge in Hong Kong as an example, a joint probability density function of wind speed and direction is first established based on wind data recorded by the WASHMS installed in the bridge. A structural health monitoring-oriented finite element model of the bridge and a numerical procedure for buffeting-induced stress analysis of the bridge are then used to identify stress characteristics at hot spots of critical steel members under different wind speeds and directions. The accumulative fatigue damage to the critical steel members at hot spots during the bridge design life is finally evaluated using a CDM-based fatigue damage evolution model. The proposed framework is found to be feasible and practical. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction To meet the social and economic needs of the community for efficient and convenient transportation systems, more and more long suspension bridges have been built throughout the world, and the super long suspension bridges with main span length beyond 3000 m are also under consideration. When long suspension bridges are built in wind-prone regions, the bridges will suffer considerable buffeting-induced vibration which appears within a wide range of wind speeds and lasts for almost the whole design life of the bridge. As a result, the frequent occurrence of buffeting response of relatively large amplitude may cause fatigue damage to steel structural members of a long suspension bridge. Although many works have been conducted on traffic-induced fatigue damage of steel bridges [1], there has been very limited research on buffeting-induced fatigue damage of long suspension bridges. Virlogeux [2] analyzed fatigue life of the Normandy cable-stayed bridge in France due to buffeting, in which the background component of buffeting response and the effect of wind direction were not taken into consideration. Gu et al. [3] considered both the background component and wind direction effects when they estimated buffeting-induced fatigue damage of the steel girders of the Yangpu cable-stayed bridge in Shanghai in the frequency–time domain. * Corresponding author. Tel.: +852 2766 6050; fax: +852 2365 9291. E-mail address: [email protected] (Y.L. Xu). 0142-1123/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2008.03.031

They found that the effects of wind direction on the fatigue damage of the Yangpu Bridge are significant but the predicted fatigue life due to buffeting is much longer than the design life of the bridge. The buffeting-induced fatigue analyzes mentioned above were based on the Miner’s law, which is widely used in the fatigue design of steel structures for its simplicity. However, the Miner’s law does not associate fatigue damage with its physical mechanism such as fatigue crack initiation and growth. It does not consider load sequence effects and load cycles below the fatigue limit which can actually propagate micro-cracks if the cracks have been initiated already. On the other hand, the fatigue crack initiation and growth in micro-scale in the vicinity of welds can be well described by the continuum damage mechanics (CDM) [4]. Li et al. [5] recently applied a CDM-based fatigue model to evaluate the effect of one typhoon on fatigue damage to the steel deck of the Tsing Ma Bridge in Hong Kong. They found that the increment of fatigue damage generated by hourly stress spectrum for the typhoon loading could be much greater than that by daily stress spectrum for the normal traffic loading. However, their analysis was based on a single particular typhoon event. The long-term effects of buffeting forces on fatigue damage, associated with the joint probability distribution of wind speed and direction, were not considered. Furthermore, to ensure the normal operation and safety of long span cable-supported bridges, a recent trend is to install a comprehensive wind and structural health monitoring system (WASHMS) in the bridge to monitor its performance and its health status

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Nomenclature Ae An B D D_ N Nb Nj Nr Pu Pu,h Ph PEj Pu T0 U Umax,j ac af ak c

bm f

bm k

bm c

effective cross-section area nominal cross-section area material property damage index rate of fatigue damage number of stress cycles number of blocks total numbers of wind records within the jth wind direction range total number of stress cycles cumulative distribution function of wind speed joint cumulative distribution function of wind speed and wind direction occurrence frequency of wind direction probability of wind speed lower than the maximum wind speed within the jth wind direction range normalized cumulative distribution function of wind speed fatigue damage evolution period wind speed maximum wind speed in the jth wind direction range fitted coefficient in harmonic function for scale parameter fitted coefficient in harmonic function for relative frequency fitted coefficient in harmonic function for shape parameter fitted coefficient in harmonic function for scale parameter fitted coefficient in harmonic function for relative frequency fitted coefficient in harmonic function for shape parameter scale parameter

through the analysis and synthesis of real-time measurement data [6]. These real-time data provide the information on wind, global bridge response, local strain response and others. However, many key issues remain unsolved as how to take full advantage of the real-time data for effective and reliable health assessment of the bridges. Li et al. [5] used the strain data recorded by the WASHMS installed in the Tsing Ma Bridge during Typhoon York to evaluate a single typhoon-induced fatigue damage of the bridge at the strain gauge points. It should be noted that the number of sensors is always limited for a long suspension bridge and the locations of structural defects or degradation may not be at the same positions as the sensors. There is the possibility of that the worst structural condition may not be directly monitored by sensors. In this regard, a numerical procedure for buffeting-induced stress analysis of a long suspension bridge using the structural health monitoring-oriented finite element model has been recently proposed by the authors [7]. A significant improvement of the proposed procedure is that the local structural behaviour associated with strain/stress, which is prone to cause local damage, can be analyzed directly. The field measurement data including strain data recorded by the WASHMS during Typhoon York have been analyzed and compared with the numerical results. The results showed that the proposed procedure has advantages over the traditional equivalent beam finite element models and could be used to find out most critical locations of stress of the bridge. This paper aims at presenting a systematic framework for assessing long-term buffeting-induced fatigue damage to a long

ccm cfm ckm fu fh fu,h k ka mb no nf nc nk zd zt Xj a a0 aj b h ra rc rm rr ra,j rr,j rr,max  r DU Dh

fitted coefficient in harmonic function for scale parameter fitted coefficient in harmonic function for relative frequency fitted coefficient in harmonic function for shape parameter probability density function of wind speed relative frequency of wind direction joint density function of wind speed and wind direction shape parameter parameter related to material property a total number of stress cycles in one block number of wind records per year for all wind directions order of harmonic function for relative frequency order of harmonic function for scale parameter order of harmonic function for shape parameter height of the average deck level height of the top of tower jth wind direction range material property; exponent of wind profile parameter related to material property a material property in jth stress range material property wind direction stress amplitude stress limit to fatigue mean value of each stress cycle stress range of each stress cycle jth stress amplitude jth stress range maximum stress range mean value of stress cycle interval of wind speed interval of wind direction

suspension bridge by integrating a few important wind/structural components with the CDM-based fatigue damage assessment method. By taking the Tsing Ma Bridge as an example, a joint probability density function of wind speed and direction will be first established based on wind data recorded by the WASHMS within the period between 1 January 2000 and 31 December 2005. The numerical procedure for buffeting-induced stress analysis of the bridge using the structural health monitoring-oriented finite element model was established and verified by the authors [7]. It will then be used in this study to identify stress characteristics at hot spots of critical steel members at different levels of wind speed and direction. The accumulative fatigue damage to the critical members at their hot spots during the bridge design life will be finally evaluated by using a CDM-based fatigue damage model, taking into consideration of long-term effects of buffeting forces.

2. Joint probability density function of wind speed and direction 2.1. Tsing Ma Bridge and WASHMS The Tsing Ma Bridge in Hong Kong is a suspension bridge with a main span of 1377 m between the Tsing Yi tower in the east and the Ma Wan tower in the west (see Fig. 1a). The height of the two reinforced concrete towers is 206 m. The two main cables of 1.1 m diameter and 36 m apart in the north and south are accommodated

Y.L. Xu et al. / International Journal of Fatigue 31 (2009) 575–586

Fig. 1. Tsing Ma Bridge in Hong Kong.

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by the four saddles located at the top of the tower legs in the main span. The bridge deck is a hybrid steel structure consisting of a series cross-frames supported on two outer-longitudinal trusses and two inter-longitudinal trusses acting compositely with stiffened steel plates (see Fig. 1b). The bridge deck is suspended by suspenders in the main span and in the Ma Wan side span and by three piers in the Tsing Yi side span. The bridge deck carries a dual three-lane highway on the upper level of the deck and two railway tracks and two carriageways on the lower level within the bridge deck. The alignment of bridge deck deviates for 17° in counterclockwise from the east–west axis (see Fig. 1c). A WASHMS was installed in the bridge by the Highways Department of Hong Kong in 1997 [8]. There were nine types of sensors in the WASHMS, including 6 anemometers and 110 strain gauges. Two digital Gill Wind Master Ultrasonic Anemometers (AneU) were installed on the north side and south side, respectively, of the bridge deck at the mid-main span (Section J in Fig. 1). Two analogue mechanical anemometers (AneM) were installed on two sides of the bridge deck near the middle of the Ma Wan side span (Section B in Fig. 1a). Each analogue mechanical anemometer consisted of a horizontal component (RM YOUNG 05106) and a vertical component (RM YOUNG 27106). Another two analogue mechanical anemometers (AneM) of horizontal component only were arranged at the level of 11 m above the top of each bridge tower on the south side. The sampling frequencies of all anemometers were set as 2.56 Hz. A total of 110 strain gauges were installed on the three typical bridge deck sections to measure the change in strain of structural members under different loading conditions. There were 49, 32, and 29 strain gauges installed in Sections L, E and D, respectively (see Fig. 1a). The instrumented locations included chord members of the longitudinal trusses, cross-frame chord members, and plan bracing members, deck through and rocker bearing at the Ma Wan tower. The sampling frequencies of all strain gauges were 51.2 Hz. 2.2. Joint probability density function of wind speed and direction For the estimation of wind-induced fatigue damage, it is necessary to have information on the distribution of the complete population of wind speed at a bridge site [9]. Various probability density functions (PDF) have been proposed to model a complete population of wind speed [10] but the convenience of the twoparameter Weibull distribution has encouraged its greater use than the other distributions [11]. With the lower bound being zero, the cumulative distribution function (CDF) and PDF of the Weibull form are, respectively "   # k U ð1Þ Pu ðUÞ ¼ 1  exp  c " #  k1  k k U U ð2Þ exp  fu ðUÞ ¼ c c c where U is the wind speed; c (>0) is the scale parameter with the same unit as wind speed; and k (>0) is the shape parameter without dimension. One possible weakness in the two-parameter Weibull distribution for wind speed is that it neglects the effect of wind direction [12]. Since wind-induced fatigue damage to a bridge is closely related to wind direction, a joint probability distribution of wind speed and wind direction has to be used for this study. To this end, a practical joint probability distribution function is adopted in this study for a complete population of wind speed and wind direction based on two assumptions: (1) the distribution of the component of wind speed for any given wind direction follows the Weibull distribution; and (2) the interdependence of wind dis-

tribution in different wind directions can be reflected by the relative frequency of occurrence of wind: "  kðhÞ #! U Pu;h ðU; hÞ ¼ P h ðhÞ 1  exp  cðhÞ ZZ ð3Þ ¼ fh ðhÞfu;h ðU; kðhÞ; cðhÞÞdu dh "   kðhÞ1 kðhÞ # kðhÞ U U fu;h ðU; kðhÞ; cðhÞÞ ¼ exp  cðhÞ cðhÞ cðhÞ

Ph ðhÞ ¼

Z

ð4Þ

h

fh ðhÞdh

ð5Þ

0

in which 0 6 h < 2p; Ph(h) is the relative frequency of occurrence of wind in wind direction h. The occurrence frequency Ph(h) as well as the distribution parameters k(h) and c(h) can be estimated using wind data recorded at the bridge site. 2.3. Statistical analysis of wind data from WASHMS Wind records of hourly mean wind speed and direction within the period between 1 January 2000 and 31 December 2005 from the anemometer installed on the top of the Ma Wan tower are used in this study to find the joint probability density function of hourly mean wind speed and direction. The height of the anemometer is 214 m above the sea level. The wind records are carefully checked, by which abnormal records are eliminated first. Wind records having an hourly mean wind speed lower than 1 m/s are then removed in order to avoid any adverse effect on the statistics. The qualified data are further divided into two categories: monsoon and typhoon according to the records of typhoon warning signal hoisted by the Hong Kong Observatory (HKO) during the period concerned. As a result, 19,775 hourly monsoon records are available for calculation of the joint probability density function of wind speed and direction. The number of hourly typhoon wind records during the concerned period is so small that the corresponding joint probability density function could not be obtained appropriately. All the monsoon records are classified into 16 sectors of the compass with an interval of Dh = 22.5° according to the hourly mean wind direction (see Fig. 1c). In each sector, mean wind speed is further divided into 16 ranges from zero to 32 m/s with an interval of DU = 2 m/s. This leads to a total of 256 cells, and the relative frequency of hourly mean wind speed and wind direction in each cell is listed in Table 1. The last row in Table 1 gives the relative frequency of mean wind speed without considering wind direction, and the last column in Table 1 shows the relative frequency of mean wind direction without considering wind speed. Based on the relative frequencies of wind speed and wind direction calculated above, the theoretical expression of joint probability density function is deduced based on Eq. (3). The Weibull function is used to fit the histogram of hourly mean wind speed for each wind direction, and the typical results in the east, south and west directions are depicted in Fig. 2a–c, respectively. The Weibull parameters identified for each wind direction are listed in Table 2 together with the relative frequency of mean wind direction. The coefficient of determination to measure the quality of fitting is also given in Table 2 for each wind direction. It can be seen that the Weibull function fits wind data satisfactorily with the coefficient of determination greater than 90% for a majority of wind directions. Only for the north direction, the coefficient of determination is about 70%. The Weibull function is also applied to the complete wind records without considering wind direction, as shown in Fig. 2d. The Weibull parameters identified for this case are given in the last row of Table 2. The results show that the Weibull function also fits the complete wind data adequately.

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Y.L. Xu et al. / International Journal of Fatigue 31 (2009) 575–586 Table 1 Relative frequency of hourly mean wind speed and direction Direction

0–2

2–4

4–6

N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW

0.081 0.101 0.086 0.142 0.243 0.253 0.293 0.172 0.283 0.187 0.071 0.101 0.101 0.091 0.051 0.308

0.465 0.753 0.870 3.075 3.798 3.004 2.908 1.760 1.350 0.910 0.531 0.278 0.602 0.475 0.379 1.102

0.126 0.647 0.728 4.420 6.144 4.733 3.965 2.528 2.336 1.163 0.855 0.455 0.637 0.435 0.142 0.622

Sum

2.564

22.260

29.937

6–8

8–10

10–12

12–14

14–16

16–18

18–20

20–22

22–24

24–26

26–28

28–30

30–32

0.152 0.647 0.389 3.363 6.528 4.243 2.655 2.215 2.599 1.077 0.753 0.329 0.607 0.303 0.147 0.389

0.081 0.440 0.288 1.290 2.190 1.446 0.860 0.753 0.925 0.531 0.314 0.066 0.182 0.061 0.051 0.223

0.071 0.324 0.197 0.435 0.501 0.470 0.430 0.354 0.465 0.142 0.051 0.030 0.071 0.015 0.035 0.131

0.076 0.314 0.167 0.177 0.329 0.217 0.142 0.121 0.228 0.167 0.015 0.010 0.015 0.010 0.000 0.025

0.106 0.197 0.025 0.278 0.253 0.137 0.051 0.091 0.086 0.137 0.005 0.005 0.005 0.000 0.000 0.035

0.071 0.142 0.010 0.263 0.152 0.056 0.040 0.051 0.025 0.106 0.005 0.005 0.000 0.000 0.000 0.030

0.035 0.066 0.015 0.147 0.116 0.020 0.025 0.010 0.025 0.025 0.000 0.000 0.000 0.000 0.005 0.005

0.020 0.046 0.005 0.056 0.116 0.005 0.005 0.010 0.005 0.000 0.005 0.000 0.000 0.000 0.000 0.010

0.005 0.015 0.000 0.046 0.046 0.010 0.010 0.005 0.015 0.005 0.000 0.000 0.000 0.000 0.000 0.000

0.010 0.000 0.000 0.010 0.020 0.010 0.000 0.005 0.010 0.000 0.000 0.000 0.000 0.000 0.000 0.005

0.000 0.000 0.000 0.010 0.005 0.000 0.000 0.000 0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.005 0.000 0.000 0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

26.397

9.699

3.722

2.013

1.411

0.956

0.496

0.283

0.157

0.071

0.020

0.005

0.010

Sum 1.305 3.692 2.781 13.714 20.440 14.604 11.388 8.076 8.359 4.450 2.604 1.279 2.220 1.391 0.809 2.887 100.0

Fig. 2. Weibull distribution and relative frequency of mean wind speed.

The relative frequency of wind direction and the scale and shape parameters of the Weibull function calculated above are given in polar plot in Fig. 3a–c, respectively. It can be seen that the dominant monsoon direction is the east, and the scale and shape parameters do not vary significantly with wind direction. For the convenience of subsequent calculation, the data given in Table 2 and Fig. 3, regarding the relative frequency of wind direction fh(h), the scale parameter c(h) and the shape parameterk(h), are fitted by the following harmonic functions [13]:

fh ðhÞ ¼ af þ

nf X

f

ð6Þ

c

ð7Þ

k

ð8Þ

bm cosðmh  cfm Þ

m¼1

cðhÞ ¼ ac þ

nc X

bm cosðmh  ccm Þ

m¼1

kðhÞ ¼ ak þ

nk X m¼1

bm cosðmh  ckm Þ

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Y.L. Xu et al. / International Journal of Fatigue 31 (2009) 575–586

harmonic functions for the relative frequency of wind direction, the scale and shape parameters, respectively. Five terms are used to fit the shape parameter and the relative frequency. For the scale parameter, only four terms are needed.

Table 2 Identified parameters in different wind direction sectors Direction N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Total

Record number

fh(h)

c(h)

k(h)

Coefficient of determination

258 730 550 2712 4042 2888 2252 1597 1653 880 515 253 439 275 160 571

0.013 0.037 0.028 0.137 0.204 0.146 0.114 0.081 0.084 0.045 0.026 0.013 0.022 0.014 0.008 0.029

8.501 9.100 6.931 7.270 7.222 6.764 6.384 6.819 7.266 7.460 6.465 5.904 6.080 5.392 5.454 5.745

1.394 1.823 1.940 1.913 2.146 2.344 2.215 2.242 2.221 1.949 2.521 2.295 2.462 2.438 1.992 1.649

0.6943 0.9165 0.8914 0.9826 0.9840 0.9929 0.9916 0.9887 0.9753 0.9723 0.9987 0.9823 0.9771 0.9840 0.8268 0.9248

19,775

1.000

6.995

2.042

0.9869

2.4. Maximum wind speed When the above joint probability density function is used to estimate fatigue damage to the bridge, the maximum wind speed Umax for a given wind direction should be determined for a designated fatigue damage evolution period. The probability of the wind speed lower than the maximum wind speed within the jth wind direction range Xj, i.e. P Ej can be determined by the probability distribution function of wind speed:  Z Z Umax;j PEj ¼ fu;h ðU; kðhÞ; cðhÞÞdU dh Xj

¼

Z Xj

"  kðhÞ #! U max;j dh 1  exp  cðhÞ

ð9Þ

The exceedance probability of the maximum wind speed within the jth wind direction range can be expressed as

where a, bm and cm are the coefficients to be determined, whose superscripts f, c and k denote the relative frequency, the scale and shape parameters, respectively; and nf, nc and nk are the order of harmonic functions. Fig. 4a–c displays the histograms and the fitted

a

0

1  PEj ¼

1 Nj þ 1

ð10Þ

0.0 337.5

22.5

0.25 0.20

315.0

45.0

0.15 292.5

67.5

0.10 0.05 0.00 0.00

270.0

90.0

0.05 0.10

247.5

112.5

0.15

225.0

135.0

0.20 0.25

202.5

157.5

180.0

Relative frequency of wind direction

b

c

0.0 337.5

10

22.5

8

315.0

45.0

0.0 337.5

22.5

315.0

45.0 2

6 292.5

3

67.5

4

292.5

67.5 1

2 00

270.0

90.0

270.0

90.0

0

2 1

4

247.5

112.5

247.5

112.5

6

225.0

2

135.0

8

202.5

10

180.0

157.5

Weibull scale parameter

225.0

135.0 202.5

3

180.0

157.5

Weibull shape parameter

Fig. 3. Relative frequency of wind direction and Weibull scale and shape parameters.

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Relative Frequency (%)

a

Table 3 Maximum hourly mean wind speeds of 120 year return period

0.25

Direction

nf=5

0.20

0.15

0.10

0.05

0.00

0

45

90

135

180

225

270

315

Wind Direction (Degree)

Relative frequency of wind direction

b

12

Weibull Scale Parameter

At deck level

Perpendicular to the deck

39.63 31.45 21.89 25.34 22.36 18.79 18.63 19.36 20.86 24.03 15.62 15.02 14.90 13.04 15.56 22.28

27.06 21.47 14.94 17.31 15.27 12.83 14.44 15.01 16.18 18.63 10.66 10.26 10.17 8.91 10.62 15.21

25.89 16.57 7.02 1.66 4.46 8.16 12.75 14.94 15.47 14.38 5.01 0.98 2.97 5.66 9.38 15.14

Bridge for buffeting-induced stress analysis using the following equation:  a zd Uðzd Þ ¼ Uðzt Þ ð12Þ zt

8 6 4 2 0

0

45

90

135

180

225

270

315

Wind Direction (Degree)

Weibull scale parameter 4.0 3.5

Weibull Shape Parameter

At tower top

nc=4

10

c

N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW

Umax (m/s)

where U(zd) is the mean wind speed at the average deck level zd; U(zt) is the mean wind speed at the top of the tower zt; and a is the exponent of wind profile. In this study, the average deck level is taken as 60 m. The exponent a is taken as 0.30 for winds over the over-land fetch (see Fig. 1c, clockwise from the SW to SE) and 0.2 for winds over the open-sea fetch (clockwise from the SE to SW). The converted maximum wind speeds at the average deck level and their components perpendicular to the bridge deck are also listed in Table 3, in which the positive sign indicates wind from the south and the negative sign denotes wind from the north.

nk=5

3.0

3. Buffeting-induced stress analysis and stress characteristics

2.5

3.1. Structural health monitoring-oriented finite element model

2.0 1.5 1.0 0.5 0.0

0

45

90

135

180

225

270

315

Wind Direction (Degree)

Weibull shape parameter Fig. 4. Histograms and fitted harmonic functions.

in which Nj is the total numbers of wind records within the jth wind direction range, which can be given by Z N j ¼ T 0 n0 fh ðhÞdh ð11Þ Xj

where T0 is the fatigue damage evolution period and n0 is the number of wind records per year for all directions. By using Eqs. (9)– (11), the maximum wind speed Umax,j for the jth wind direction range can be obtained, and the results are listed in Table 3 for a wind return period of 120 years which is actually the design life of the bridge. The maximum wind speed obtained at the top of the tower is converted to the average deck level of the Tsing Ma

The sensors for strain measurement in the WASHMS for a long suspension bridge are always limited: not all the stress responses of all the local components can be directly monitored. To facilitate an effective assessment of stress-related bridge safety, a structural health monitoring-oriented finite element model (FEM) is needed for a long suspension bridge so that stresses/strains in all important bridge components can be directly computed and some of them can be compared with the measured ones for verification. However, the currently conducted buffeting analyzes of long span bridges are often based on a simplified spine beam FEM of equivalent sectional properties [14,15]. Such simplified model is effective to capture the dynamic characteristics and global structural behaviour of the bridge under strong winds without heavy computational effort. However, local structural behaviour linked to stress and strain, which is prone to cause local damage, could not be estimated directly. On the other hand, with the rapid development of information technology, the improvement of speed and memory capacity of personal computer (PC) has made it possible to establish a structural health monitoring-oriented finite element model for a long suspension bridge. In this regards, a complex structural health monitoring (SHM)-oriented finite element (FE) model has recently been established by the authors for the Tsing Ma Bridge with significant modelling features of the bridge deck included for the good replication of geometric details of the as-built complicated deck. The proposed SHM-oriented FE model has also been

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updated using the first 18 natural frequencies and mode shapes of the bridge measured on the site with the updated parameters being material properties only because the geometric features and supports of bridge deck have been modelled in a great detail in the proposed SHM-oriented FE model. It turns out that the updated complex FE model could provide comparable structural dynamic modal characteristics. More details can be found in the literature [16]. 3.2. Buffeting-induced stress analysis Based on the established SHM-oriented FE model, a numerical procedure for buffeting-induced stress analysis of long suspension bridges has been proposed by the authors [7]. Significant improvements of the proposed procedure are that the effects of the spatial distribution of both buffeting forces and self-excited forces on a bridge deck structure are taken into account, as opposed to lumping all buffeting forces and self-excited forces at the centre of elasticity as in the case of an equivalent beam finite element model. Local strains and stresses in structural members of the bridge deck, which are prone to cause local damage, are predicted directly using the mode superposition technique in the time domain. The field measurement data including wind, acceleration and stress recorded by the anemometers, accelerometers, and strain gauges in the WASHMS installed on the bridge during Typhoon York have been analyzed. The buffeting-induced acceleration responses at the locations of 12 accelerometers installed in the bridge has been computed using the mode superposition method and compared with the measured results. The buffeting-induced stress responses at the locations of nine strain gauges installed in the bridge have been computed through the modal stress analysis and compared with the measured ones. The comparative results show that the computed acceleration and stress time histories are similar in both pattern and magnitude with the measured ones. In this study, this procedure is used to find out the time histories of critical stress responses due to buffeting and to obtain hot spot stress characteristics for buffeting-induced fatigue damage assessment. 3.3. Critical stresses and hot spot stresses In the SHM-oriented FE model of the bridge, there are a total of 15,904 beam elements used to model the bridge deck [16]. To find the most critical beam elements and the corresponding most critical stresses in the bridge deck, a buffeting-induced stress analysis is carried out now by considering a 15 m/s mean wind perpendicular to the bridge axis from the south for an hour. The von Karman spectra together with other wind characteristics are first used for the numerical simulation of the stochastic wind velocity field for the entire bridge deck. The stochastic wind velocity field comprises a series of time histories of fluctuating wind velocity in the horizontal and vertical directions at various points along the bridge deck. In the simulation, the turbulence intensity is taken 24% in the horizontal direction and 17% in the vertical direction by considering the most turbulent cases in the field. The integral length scale is taken as 251 m in the horizontal direction and 56 m in the vertical direction. The wind incidence is assumed to be zero. The exponential form of coherence function is adopted to reflect turbulent wind correlation along the bridge deck in both horizontal and vertical directions. The exponential decay coefficient is selected as 16 in the simulation [17]. A fast spectral representation approach proposed by Cao et al. [18] is adopted for the digital simulation of stochastic wind velocity field. A total of 120 points along the bridge with an interval 18.0 m are considered in the simulation of wind field. The average elevation of the bridge deck is taken as 60 m. Since this study concerns buffeting-induced stresses other than traffic-

induced stresses, the sampling frequency and duration used in the simulation of wind speeds are, respectively, 16 Hz and 3600 s. The method proposed by the authors [7] is then used to estimate the buffeting forces and self-excited forces at the nodes of the FEM of the bridge. In the determination of the buffeting forces at the nodes, the drag, lift, and moment coefficients of the bridge deck are taken as 0.135, 0.090, and 0.063, respectively, at the zero wind angle of attack with respect to the deck width of 41 m based on the wind tunnel test results [15]. The first derivatives of the drag, lift, and moment coefficients with respect to the same wind angle of attack are 0.253, 1.324, and 0.278, respectively. The aerodynamic transfer functions between the simulated fluctuating wind velocities and the buffeting forces are assumed to be one. Due to the lack of wind tunnel test results on lateral flutter derivatives, only the vertical and rotational motions of the bridge deck are taken into account in the simulation of self-excited forces. A total of 12 frequency independent coefficients are determined by using the measured flutter derivatives and the least squares fitting method. They are used to determine the aeroelastic stiffness matrix and the aeroelastic damping matrix, by which the self-excited forces at the nodes of the FEM of the bridge can be determined. The buffeting-induced stress responses of the bridge are finally computed using the mode superposition technique. The first 80 modes of vibration of the bridge are considered in the computation. The highest frequency involved in the computation is 1.1 Hz. The damping ratios for all the modes of vibration are taken as 0.5%. The modal stresses multiplied by the generalized displacement vector yield the stress time histories at five points of the end section of each element. Among five points, four points are located at each corner of the end section and one point is situated at the centroid of the end section. By comparing the maximum values and the standard deviations of all stress time histories, the crosssection of the bridge deck at the Ma Wan tower is identified as the most critical section (CH23623 in Fig. 1a), in which the six elements of no. 34111 and 38111, 40881 and 48611, 58111 and 59111 are identified as the most critical elements. The elements 34111 and 38111 are the bottom chords of the outer north and south longitudinal trusses, respectively, on the main span side (see Fig. 5). The elements 40881 and 48611 are the bottom chords of the inner north and south longitudinal trusses, respectively, on the main span side. The elements 58111 and 59111 are the bottom chords in the middle of the cross-frame close to the north and south inner longitudinal trusses, respectively. The hot spot stress approach, which considers the stress concentration at welded joints, has been widely used in hollow steel tubular structure fatigue design and analysis [19]. To apply the hot spot stress approach to the fatigue damage assessment of the bridge under traffic loading, Chan et al. [20] computed the stress concentration factors (SCF) of typical welded joints of the bridge deck based on both global and local finite element models. The hot spot stress block cycles were then determined by multiplying the nominal stress block cycles by the SCF. This approach and the SCF obtained by Chen et al. [20] are used in this study. In this regard, the maximum stress at the middle section of each critical element is computed based on the stresses at the two ends of the element and is taken as the nominal stress of the element. The hot spot stress is then determined by multiplying the nominal stress by the corresponding SCF. The type of welded connection for the elements 34111, 38111, 40881 and 48611 in this study is classified as F2 according to BS5400 [21] and the SCF is taken as 1.95 [20]. The type of welded connection for the elements 58111 and 59111 is classified as F and the SCF is taken as 1.44. Depicted in Fig. 6 is the 1-h time history of the hot spot stress for the element 40881.

Y.L. Xu et al. / International Journal of Fatigue 31 (2009) 575–586

583

Fig. 5. Critical deck section and critical elements identified.

Fig. 6. Time history of hot spot stress of element 40881.

3.4. Hot spot stress characteristics For the subsequent buffeting-induced fatigue damage assessment of the bridge deck at the six hot spot stress locations for a wind return period of 120 years, the preceding exercise has to be repeated for the mean wind speeds from 5 to 30 m/s with an interval of 5 m/s for winds over the over-land fetch and from 5 to 20 m/ s at an interval of 5 m/s for winds over the open-sea fetch, respectively. This yields a total of 60 1-h time histories of the hot spot stresses for the bridge deck. For each of 60 1-h time histories, the Rainflow counting method [22] is applied to obtain the hot spot stress characteristics within 1 h for different wind speeds and directions. The hot spot stress characteristics include the total number of stress cycles (Nr), the stress range (rr), the mean value (rm) of each stress cycle and the maximum stress range (rr,max). The mean value of each stress cycle includes the mean stress caused by the mean wind speed. As a result, a total of 60 data sets of hot spot stress characteristics are produced. Fig. 7 displays the total number of stress cycles and the maximum stress range against mean wind speed and wind terrain for the hot spot stress

at element 40881. It can be seen that the variations of the total number of stress cycles with mean wind speed and wind terrain are not considerable within the range from 2200 to 2300 cycles. The variation of the maximum stress range with wind terrain is very small but the effect of mean wind speeds on the maximum stress range is significant. 4. Fatigue damage assessment with continuum damage mechanics (CDM) 4.1. Damage evolution model The damage growth of the material is considered as a progressive internal deterioration, which induces a loss of effective cross-section area that carries loads. The damage index D for the isotropic damage in the continuum damage mechanics (CDM) is often defined as D¼

An  Ae An

ð13Þ

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Fig. 7. Variations of total number of cycles and maximum stress range of hot spot stress at element 40881.

where An is the nominal cross-section area; and Ae is the effective cross-section area considering area loss due to damage. Based on thermodynamics and potential of dissipation, the rate of damage for high-cycle fatigue can be expressed as a function of the accumulated micro-plastic strain, the strain energy density release rate and the current state of damage in CDM [22]. The micro-plastic strain, often neglected in a low-cycle fatigue problem, and its accumulation must be considered for high-cycle fatigue damage, even if macro-plastic strain does not exist [24]. In one-dimensional situa_ can be written tion, the equation for the rate of fatigue damage, D, as jb hri _ r2 jr  r D_ ¼ ð1  DÞa B

ð14Þ

 ¼ rm is the mean stress; the symbol h i denotes the McCauley where r brackets where hxi = x for x > 0 and hxi = 0 for x < 0; a, b and B are the material properties. These properties can be determined using the Woehler curves obtained through uniaxial periodic fatigue tests under strain-controlled condition and direct damage measurements of the material [23,24]. Eq. (14) is a general constitutive model for highcycle fatigue and it can be integrated over time for the cycles with different mean stresses and stress ranges. For instance, when considering rm = 0 and neglecting the variation of (1  D)a in one stress cycle, integrating Eq. (14) over the cycle yields dD rbþ3 a ¼ dN Bðb þ 3Þð1  DÞa

mb X rbþ3 dD a;j ¼ Bð1  DÞaj ðb þ 3Þ dN b j¼1

ð16Þ

where mb is the total number of stress cycles in the block; and Nb is the number of blocks. If mean stress is not equal to zero, the fatigue damage rate generated by one block of stress cycles can be determined by the following expression [24]:  bþ3 mb  X rr;j þ 2rm;j rr;j 2 dD ¼ aj Bð1  DÞ ðb þ 3Þ dN b j¼1

ð17Þ

where rr,j = 2ra,j is the jth stress range; and aj depends on the jth stress range, that is, aj = f(rr,j). Within 1 h block, the maximum stress range affects the damage increment most. Eq. (17) can be rewritten as bþ3

ð1  DÞae dD ¼

mb X ½ðrr;j þ 2rm;j Þrr;j  2 dN b Bð1  DÞaj ae ðb þ 3Þ j¼1

ð18Þ

where ae is determined by the maximum stress range rr,max through the function aj = f(rr,j). Let us consider the damage accumulation in the kth block. Integrating Eq. (18) over the kth block yields ! bþ3 Z Dk Z Nb;k X mb;k ½ðrr;jk þ 2rm;jk Þrr;jk  2 ð1  DÞae;k aj;k dN b ð1  DÞae;k dD ¼ Bðb þ 3Þ Dk1 N b;k1 j¼1

ð15Þ

where ra is the stress amplitude of the cycle and N is the number of cycles. In this study, the Rainflow counting method has been applied to obtain the hot spot stress characteristics within 1 h for different wind speeds and directions. The hot spot stress characteristics include the total number of stress cycles (Nr), the stress range (rr), the mean value (rm) of each stress cycle and the maximum stress range (rr,max) within 1 h. Since a normal fatigue life of a bridge may be over 100 years, the effects of load sequence on damage accumulation within 1 h can be neglected. Nevertheless, the effects of load sequence are taken into consideration in this study at 1 h interval. As a result, by considering 1 h as one block the fatigue damage rate generated by one block of stress cycles with zero mean stress can be expressed as

ð19Þ ae;k aj;k

ae;k aj;k

¼ ð1  Dk1 Þ By making the approximation ð1  DÞ [23], the damage evolution model for fatigue damage assessment of a long suspension bridge in this study is given as  ðae;k þ 1Þ Dk ¼ 1  ð1  Dk1 Þae;k þ1  Bðb þ 3Þ !)1þa1 mb;k e;k X bþ3 ½ðrr;jk þ 2rm;jk Þrr;jk  2 ð1  Dk1 Þae;k aj;k ð20Þ  j¼1

Buffeting-induced fatigue in welded joints of a bridge is a cumulative process over many years. Buffeting-induced fatigue damage accumulation including both fatigue crack initiation and growth in micro-scale can be well estimated by using Eq. (19) evolutionally as shown in the subsequent section. For the bridge

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concerned in this study, aj = karr,j + a0 is used. The parameters ka and ao are taken as 0.135 and 101.4 MPa, respectively, for both the connection types F2 and F. (b + 3) = 3.01 is used for both the connection types. [B(b + 3)] = 3.41  1013 is used for the connection type F2 and [B(b + 3)] = 5.11  1013 is used for the connection type F. According to BS5400 [21], if the stress range rr,j is less than 2 r rc, it should be reduced in the proportion rr;jc , where rc is the stress limit to fatigue. The stress limit to fatigue used in this study is 35 MPa for the connection type F2 and 40 MPa for the connection type F. 4.2. Buffeting-induced fatigue damage assessment

no;j ¼ no

Z

fh ðhÞdh

ð21Þ

Xj

where no = 8760 is the total number of hourly wind records in 1 year; and fh(h) is the relative frequency of wind direction (see Eq. (6)). After the number of wind records in the jth wind direction range is determined, the number of hourly wind records in the ith wind speed range within the jth wind direction range shall then be determined. This can be done by considering a normalized cumulative distribution function as follows:  kðhj Þ U 1  exp  cðhi;jj Þ  ð22Þ Pu ðU i ; hj Þ ¼

U max;j kðhj Þ 1  exp  cðh Þ j

To assess the buffeting-induced fatigue damage to the bridge deck at the identified six hot spot stress locations for a wind return period of 120 years, the occurrence sequences of 1,051,200 (=24  365  120) blocks in 1 h duration should be determined in consideration of different wind speeds and directions before Eq. (19) can be applied. The number of hourly wind records in the jth wind direction range Xj in 1 year can be determined by Table 4 Distribution of the number of hourly wind records in 1 year Direction N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Sum

0–5

10–15

15–20

20–25

25–

Sum

98 66 165 455 602 526 404 320 235 152 100 74 73 78 68 83

5–10 91 84 206 527 802 720 476 399 334 212 121 79 75 59 44 64

46 41 73 139 189 118 62 66 90 54 7 2 1 0 1 18

17 11 1 11 4 0 0 1 4 4 0 0 0 0 0 1

5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

258 203 445 1132 1597 1364 942 786 663 422 228 155 149 137 113 166

3499

4293

907

54

6

1

8760

If the hourly mean wind speeds in the jth wind direction range are divided into several wind speed ranges from zero to Umax,j at an interval of 5 m/s, the number of hourly wind records in the ith wind speed range within the jth wind direction range is then given as no ði; jÞ ¼ no;j ½P u ð5ðiÞ; hj Þ  P u ð5ði  1Þ; hj Þ

ði ¼ 1; 2; . . .Þ

ð23Þ

By Eqs. (21) and (23), the distribution of the number of hourly wind records in 1 year within the designated wind direction and wind speed ranges can be obtained and the results from this study are listed in Table 4. Because only monsoon wind effect on the bridge is considered in this study and the monsoon wind in Hong Kong normally is southerly (from 90° to 270° in Fig. 3) in summer and northerly (from 270° to 90° in Fig. 3) in winter, it is assumed that monsoon wind blows over the open-sea fetch in summer and over the over-land fetch in winter. Two random permutation sequences of uniform distribution are then generated according to the total number of wind records in summer and in winter, respectively, in a particular year. The first random permutation sequence actually brings out the occurrence sequence of wind records over the open-sea fetch while the second random permutation sequence leads to the occurrence sequence of wind records over the over-land fetch for that particular year. Each wind record is then converted to each wind block according to its wind direction and wind speed. The hot spot stress characteristics corresponding to each wind block can be best found from one of the

Fig. 8. Damage evolution during the 120 year return period.

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Table 5 Fatigue damage at the end of 120 years Locations Damage index

38111 0.0121

34111 0.0096

40881 0.0169

48611 0.0183

58111 0.0110

59111 0.0105

term typhoon wind-induced fatigue damage assessment when long-term field measurement data on typhoons are available to the authors. Acknowledgements

60 data sets of the hot spot stress characteristics obtained in advance. The fatigue damage accumulation of the bridge deck at each hot spot stress location can finally be processed using the damage evolution model (Eq. (19)) 1 year after another until 120 years by assuming the zero damage at the beginning of fatigue damage accumulation. Fig. 8 shows the damage evolution of the bridge deck at the six hot spot stress locations during 120 years period. It can be seen that damage index increases with time. Slight nonlinear relationship between the damage index and time can be observed for the hot spot stress of the element 48611, which indicates the nonlinear nature of fatigue initiation and growth and the capability of the damage evolution model used in this study. Listed in Table 5 are the fatigue damage indices of the bridge deck at the six hot spot stress locations at the end of 120 years. It can be seen that monsoon wind-induced fatigue damage to the bridge deck is not significant. It should be noticed that this study does not take into account typhoon effects and traffic effects. Nevertheless, the procedure proposed in this study can also be applied to typhoon wind-induced fatigue damage. It will be done by the authors when long-term field measurement data on typhoons are available. 5. Concluding remarks The framework for assessing long-term wind-induced fatigue damage to a long suspension bridge has been proposed in this study by integration of the joint probability density distribution of wind speed and direction, the structural health monitoring-oriented finite element model of the bridge, the numerical procedure for buffeting-induced stress analysis, and the continuum damage mechanics (CDM)-based fatigue damage assessment method. By taking the Tsing Ma Bridge as an example, the joint probability density function of wind speed and direction were first established based on wind data recorded by the wind and structural health monitoring system within the period between 1 January 2000 and 31 December 2005. The numerical procedure for buffeting-induced stress analysis of the bridge based on its structural health monitoring-oriented finite element model was then used to identify hot spot stress characteristics for different wind speeds and directions. The accumulative fatigue damage to the critical members at their hot spots during the bridge design life of 120 years was finally evaluated using a CDM-based fatigue damage evolution model taking into consideration of long-term effects of buffeting forces. The results show that the proposed procedure can well assess the longterm buffeting-induced fatigue damage to a long suspension bridge, in which the fatigue damage accumulation including both fatigue crack initiation and growth can be demonstrated. The results also show that monsoon wind-induced fatigue damage to the bridge is not significant. The proposed procedure will be applied to long-

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