Fatigue reliability assessment considering traffic flow variation based on weigh-in-motion data

Abstract

The remaining service life of a bridge is an important criterion for the bridge master to make decisions of maintenance and rehabilitation. In the assessment of fatigue problems of steel and composite bridges, a crucial step is to determine the fatigue stress spectra, which are usually obtained through field measurements. In this study, the standard fatigue vehicle spectrum of Ting Kau Bridge in Hong Kong is obtained from the dynamic weigh-in-motion system that monitors the traffic flow and volume and axle and gross vehicle weights. An advanced probabilistic loading model is proposed to simulate the vehicles running along the bridge based on measurements from the weigh-in-motion system considering not only the diurnal variations of traffic flow within different time intervals but also the rate of change of the annual traffic. The statistical fatigue reliability assessment at critical locations can then be carried out to estimate the remaining service life of the bridge.

Keywords

advanced probabilistic loading model cable-stayed bridge fatigue reliability traffic flow variation weigh-in-motion system

Introduction

The ever-evolving developments in design and construction technologies have made spans of over 1000 m possible for cable-stayed bridges. Steel is normally used in the decks either fully or partly to reduce the self-weight for those with long spans. In the design and construction of cable-stayed bridges, the design life is often 100 years or more. Over the service life, the effects of environmental corrosion, material ageing, long-term load effect, fatigue and so on will inevitably lead to accumulation of damage and possible reduction of strength. In extreme cases, catastrophic failures could happen. Fatigue has been one of the most critical forms of damage for cable-stayed bridges with steel or composite decks, and assessing the remaining fatigue lives of steel bridges has been the primary focus in recent years (Liu et al., 2010). Fatigue of steel girders is caused by cumulative damage under cyclic actions of various kinds of vehicle loads with different weights and physical dimensions.

In the assessment of fatigue problems, a crucial step is to determine the fatigue stress spectra. Usually, the fatigue stress spectra of bridges can be obtained through field measurements such as from structural health monitoring (SHM) systems which are able to record the responses for fatigue assessment and performance prediction (Elkordy et al., 1994). However, SHM systems are always expensive, and there are a lot of structural components on which it is difficult to install strain gauges (Guo and Chen, 2013). As a result, some other effective tools are required to obtain the fatigue stress spectra of the details that are hardly accessible. In the method of Liu and Mahadevan (2007) for predicting stochastic fatigue life under variable amplitude loading, the S-N curve is stochastic and the fatigue damage accumulation rule is nonlinear, making it computationally intensive for long-span cable-stayed bridges.

A statistical load model was proposed by Guo and Chen (2013) recently so that the uncertain information of vehicle types, number of axles, axle weights, axle spacing, transverse position of vehicle and so on could be properly accounted for. As a sequel to this statistical load model, a statistical approach for modelling traffic load is developed based on the probabilistic traffic characteristics extracted from the measured data of weigh-in-motion (WIM) system installed on Ting Kau Bridge in Hong Kong. Lim (1992) noted the diurnal variations in data measured by WIM systems and the correlation between the average steering axle mass for different hours of the day. Vanderstaay et al. (2006) revealed the characteristics that the mass magnitude depended strongly on the axle group type and that the mass during the night was greater than the mass during the day generally. Righiniotis and Chryssanthopoulos (2003) and Righiniotis (2006) pointed out that traffic growth should be taken into account in fatigue life prediction because of the effects of vehicle loads. The variation of annual traffic has been studied for Runyang Bridge in China, giving rise to a traffic growth model to fit the growth trend (Guo et al., 2008). The traffic load model proposed in this study attempts to reflect the diurnal variations of traffic flow within different time intervals and the rate of change of the annual traffic on a bridge (Zhang et al., 2014). In conjunction with this statistical traffic load model, the stress time-histories at critical locations of the bridge are obtained, and the probability density curves of the fatigue life can be constructed for further analyses.

WIM system

Ting Kau Bridge is a cable-stayed bridge in Hong Kong with a total length of 1177 m completed in 1998. As shown in Figure 1, the Ting Kau main span and Tsing Yi main span are 448 and 475 m, respectively, while the two side spans are both 127 m. The bridge connects Ting Kau at the north and Tsing Yi Island at the south over Rambler Channel. The WIM system collects information of vehicle speed, heading direction, lane used, axle weight, axle spacing between neighbouring axles and so on. The system is capable of dynamic measurement of vehicular axle loads and speed with a programmed vehicle classification system. The WIM system is located at the Tsing Yi end of Ting Kau Bridge on both carriageways as shown in Figure 2.

Figure 1.

Schematic plan and elevation of Ting Kau Bridge: (a) plan showing girder labelling and (b) elevation.

Figure 2.

Layout of weigh-in-motion dynamic weigh-bridge system: (a) subsystem for detection of over-weigh vehicles and (b) layout on lanes.

For data cleansing, a record of an individual vehicle moving along Ting Kau Bridge is considered unreliable if

Any one axle spacing or axle weight is zero or negative;

The axle spacing is greater than 16 m;

The vehicle speed is less than 50 km/h;

The gross vehicle weight (GVW) is greater than the maximum legislated weight for a vehicle in Hong Kong, which is 44 tons;

Any one axle weight is less than 0.5 ton with axle spacing larger than 2 m (Enright and O’Brien, 2011).

The data of WIM in 84 days in 24-h records are filtered. After filtering, 5% of vehicles are eliminated from the database.

Variation of traffic flow

Diurnal variation of traffic flow

In order to investigate the features of diurnal variations of WIM data for Ting Kau Bridge, the WIM data collected are divided into eight types according to the number of axles of vehicles moving along the bridge, namely, from 2 to 9 axles as shown in Figure 3. The numbers of the eight vehicle types are counted for each hour, whose results are shown in Figure 4, where n1 to n8 denote the first to the eighth vehicle types, respectively. There are two peaks of the numbers of operating vehicles in a day. The two periods around the peaks are called the morning rush hours and evening rush hours. To take into account the variation of daily transportation properly, the daily traffic in this study is divided into four time intervals for further analysis, including 07:00 to 11:00 (morning rush hours), 11:00 to 17:00, 17:00 to 21:00 (evening rush hours) and 21:00 to 07:00 on the next day. The proportion of vehicles of different types in 1 h is analysed and shown in Table 1 and Figure 5. Figure 5 shows that vehicles are predominantly of type 1 which has two axles, while vehicles of types 3–6 constitute a minor portion. Vehicles of types 7 and 8 are ignored in further analysis because of their extremely small proportion.

Figure 3.

Vehicle type classification for Ting Kau Bridge: (a) vehicle type 1, (b) vehicle type 2, (c) vehicle type 3, (d) vehicle type 4, (e) vehicle type 5, (f) vehicle type 6, (g) vehicle type 7 and (h) vehicle type 8.

Figure 4.

Diurnal variations of numbers of vehicles for Ting Kau Bridge: (a) n₁, (b) n₂–n₅ and (c) n₆–n₈.

Table 1.

Proportion of vehicles of different types in 1 h for the four time intervals.

Time interval	07:00–11:00		11:00–17:00		17:00–21:00		21:00–07:00
Time interval	N	%	N	%	N	%	N	%
Vehicle type 1	17,555	90.24	16,302	80.50	21,961	79.18	16,869	85.79
Vehicle type 2	995	5.11	1412	6.97	1786	6.44	1075	5.47
Vehicle type 3	293	1.51	1020	5.04	1551	5.59	557	2.83
Vehicle type 4	507	2.61	1220	6.02	1934	6.97	936	4.76
Vehicle type 5	103	0.53	295	1.46	499	1.80	222	1.13
Vehicle type 6	1	0.0051	3	0.0148	4	0.0144	3	0.0153
Total number	19,456		20,254		27,736		19,664

Figure 5.

Schematic diagram of percentage of each vehicle type in 1 h for the four time intervals.

Variation of annual traffic

Ting Kau Bridge was opened to traffic in 1999 and the traffic has gained rapid growth in recent years as shown in Table 2. About 5% annual growth of traffic can be observed as shown in Figure 6. In other words, the traffic flow will be doubled in roughly 20 years unless there are other constraints such as traffic flow capacity. As the design life of bridges is 120 years as specified by British Standards (BS) 5400 (1980), this annual growth rate of traffic cannot be ignored; otherwise, any estimates of fatigue life will be grossly erroneous.

Table 2.

Annual traffic flow of Ting Kau Bridge.

Year	Total traffic flow
2000	13852104
2001	13877867
2002	15030028
2003	14799159
2004	17553296
2005	18793501
2006	19783091
2007	20784378
2008	20651031
2009	20063672
2010	22672072
2011	23923671
2012	25139219

Figure 6.

Annual traffic growth of Ting Kau Bridge.

Statistical fatigue life estimation

Statistical load model

Considering the random variables of each vehicle type, the probability density functions (PDFs) of axle weights and axle spacing in four different time intervals of each day can be obtained. Some plots of the distribution considering the diurnal variation for different time intervals are shown in Figures 7 to 9. Furthermore, the parameters for all distributions are listed in Tables 3 and 4. The symbol ‘(1)’ in the first column is used to denote those for the first time interval and so on. It is found that most of the axle spacing of the six vehicle types follows the t location-scale distribution, while most of the axle weights follow the normal and lognormal distributions.

Figure 7.

Distribution of axle weight for the first axle of vehicle type 2 in four time intervals: (a) 7:00–11:00, (b) 11:00–17:00, (c) 17:00–21:00 and (d) 21:00–7:00 next day.

Figure 8.

Distribution of axle weight for the third axle of vehicle type 3 in four time intervals: (a) 7:00–11:00, (b) 11:00–17:00, (c) 17:00–21:00 and (d) 21:00–7:00 next day.

Figure 9.

Distribution of axle spacing for the first axle of vehicle type 1 in four time intervals: (a) 7:00–11:00, (b) 11:00–17:00, (c) 17:00–21:00 and (d) 21:00–7:00 next day.

Table 3.

Parameters of distribution for random axle spacing of six vehicle types in the first time interval.

Variable	Distribution	µ	σ	ν
AS ₁₁ (1)	t location-scale	2.56757	0.130518	1.2024
AS ₂₁ (1)	t location-scale	5.2354	0.3008	1.45787
AS ₂₂ (1)	t location-scale	1.43549	0.0934636	1.76101
AS ₃₁ (1)	Normal	2.76969	0.881701	–
AS ₃₂(1)	Normal	5.26343	2.12879	–
AS ₃₃(1)	t location-scale	1.2316	0.0689478	1.08788
AS ₄₁(1)	t location-scale	3.1878	0.209944	4.0925
AS ₄₂(1)	t location-scale	6.82146	0.325105	1.63771
AS ₄₃(1)	t location-scale	1.25117	0.0709615	1.50719
AS ₄₄(1)	t location-scale	1.25682	0.0792352	3.159
AS ₅₁(1)	t location-scale	3.04588	0.200932	2.96062
AS ₅₂(1)	t location-scale	1.27471	0.0715863	1.20488
AS ₅₃(1)	t location-scale	6.31688	0.297105	1.34377
AS ₅₄(1)	t location-scale	1.25269	0.0757651	1.54495
AS ₅₅(1)	t location-scale	1.25126	0.0742391	1.51411
AS ₆₁(1)	Normal	2.44792	0.971289	–
AS ₆₂(1)	t location-scale	1.27432	0.156619	0.832623
AS ₆₃(1)	Normal	0.983169	0.912002	–
AS ₆₄(1)	Lognormal	0.744827	0.750618	–
AS ₆₅(1)	Lognormal	0.211422	0.373785	–
AS ₆₆(1)	t location-scale	1.21641	0.136261	0.984184

Table 4.

Parameters of distribution for random axle weight of six vehicle types in the first time interval.

Variable	Distribution	µ	σ	ν
AW ₁₁ (1)	Lognormal	2.09822	0.654189	–
AW ₁₂ (1)	Lognormal	2.0467	0.765704	–
AW ₂₁ (1)	Normal	55.0218	15.7337	–
AW ₂₂ (1)	Lognormal	2.0467	0.765704	–
AW ₂₃ (1)	Normal	70.0851	22.9357	–
AW ₃₁ (1)	Normal	48.0842	15.7365	–
AW ₃₂ (1)	Lognormal	3.74376	0.595411	–
AW ₃₃ (1)	Lognormal	3.49771	0.728073	–
AW ₃₄ (1)	Lognormal	3.65088	0.718928	–
AW ₄₁ (1)	Normal	54.14	11.1867	–
AW ₄₂ (1)	Lognormal	3.96965	0.492631	–
AW ₄₃ (1)	Lognormal	3.40015	0.606311	–
AW ₄₄ (1)	Lognormal	3.48393	0.497636	–
AW ₄₅ (1)	Lognormal	3.43039	0.570716	–
AW ₅₁ (1)	Normal	51.5456	13.4769	–
AW ₅₂ (1)	Normal	45.7997	21.0665	–
AW ₅₃ (1)	Normal	47.0527	22.0427	–
AW ₅₄ (1)	Lognormal	3.3481	0.623026	–
AW ₅₅ (1)	Lognormal	3.47518	0.590202	–
AW ₅₆ (1)	Lognormal	3.49999	0.682821	–
AW ₆₁ (1)	t location-scale	44.9652	6.300000	1.47975
AW ₆₂ (1)	Normal	36.1465	25.4953	–
AW ₆₃ (1)	Normal	38.8528	25.3107	–
AW ₆₄ (1)	Normal	32.0181	20.5672	–
AW ₆₅ (1)	Normal	31.1792	17.6498	–
AW ₆₆ (1)	Normal	34.9181	20.3899	–
AW ₆₇ (1)	Lognormal	2.87714	1.03004	–

When the statistical loading is applied for fatigue life estimation, the parameters of each individual vehicle and the arrangement of vehicles in each lane follow certain statistical distributions derived from the traffic data measured by the WIM system. Table 5 shows the proportion of traffic volume for different vehicle types and different lanes at different time intervals. The statistical traffic load model is employed to obtain the stress time-histories at critical locations of the bridge. Based on the simulated statistical stress time-histories, rain-flow counting (Downing, 1982) is used to calculate the number of statistical cycles. The probabilistic distributions of fatigue lives can be calculated by using the statistical distribution of number of cycles. The results of fatigue life obtained using the proposed method are therefore also uncertain. Therefore, Monte Carlo simulation can be used to simulate the large number of occurrences so that the variations can be revealed. Based on the principle of the mathematical statistics, the Monte Carlo method as a kind of numerical method adopting statistical sampling theory solves mathematical and physical problems approximately. Hence, it can also be considered as a statistical experiment or stochastic simulation. The mean stress effect may cause significant difference in fatigue life assessment because, with the revised stress ranges, the cycles to fatigue vary greatly. When considering the mean stress effect, the fatigue life can also be predicted (Zhang and Au, 2013).

Table 5.

Proportion of traffic in each lane.

Vehicle type	Proportion of traffic volume (%)	Proportion in each lane (%)
Vehicle type	Proportion of traffic volume (%)	Lane 1	Lane 2	Lane 3	Lane 4	Lane 5	Lane 6
		Time interval 1 (07:00–11:00)
1	90.23	15.50	28.57	15.74	12.96	17.90	9.33
2	5.11	46.21	11.37	0.36	37.72	3.82	0.52
3	1.51	39.05	12.31	1.44	30.04	11.86	5.30
4	2.61	40.50	11.45	0.27	32.37	13.53	1.88
5	0.53	30.26	20.06	1.06	29.11	14.79	4.73
6	0.00881	24.31	45.14	0.69	6.25	17.36	6.25
		Time interval 2 (11:00–17:00)
1	80.49	9.26	17.07	10.92	15.02	20.99	26.75
2	6.97	30.35	15.74	0.67	40.79	11.41	1.04
3	50.37	31.82	17.33	0.68	29.11	17.53	3.53
4	6.03	32.70	18.04	0.44	27.37	18.85	2.61
5	1.46	23.80	25.87	0.45	22.54	23.27	4.06
6	0.019	10.80	54.94	0.31	4.01	22.53	7.41
		Time interval 3 (17:00–21:00)
1	79.18	11.68	21.95	15.21	13.25	20.08	17.82
2	6.44	33.93	20.04	0.93	35.18	9.04	0.88
3	5.59	32.43	20.04	0.85	27.96	15.20	3.52
4	6.97	33.08	18.99	0.52	27.09	17.39	2.93
5	1.80	29.14	28.63	0.54	19.49	17.73	4.48
6	0.017	16.75	34.54	2.32	7.47	29.38	9.54
		Time interval 4 (21:00–07:00 next day)
1	85.79	13.27	24.00	20.92	10.82	16.34	14.64
2	54.67	36.10	22.95	0.84	34.37	5.10	0.65
3	28.36	29.98	20.21	1.71	29.33	14.27	4.51
4	47.60	30.08	16.46	0.46	32.30	17.72	2.98
5	1.13	26.61	28.79	0.85	21.77	17.11	4.86
6	0.017	8.42	54.04	3.16	5.61	20.00	8.77

Basic procedures of fatigue analysis integrating WIM data

The probabilistic analysis of the fatigue lives of Ting Kau Bridge can be performed by combining with the statistical traffic load model according to the following procedures as shown in Figure 10.

1. Establishment of statistical load model

The axle weights and axle spacing of each vehicle type for different time intervals are treated as random variables, whose probability density distributions can be obtained by curve fitting.

2. Calculation of influence line

A moving unit load is applied on different lanes (i.e. slow, middle and fast lanes) of the baseline finite element model. As a result, influence lines of stresses at various critical locations can be obtained for different structural components. A number of load steps corresponding to a static analysis are defined for the bridge.

3. Fatigue analysis

(a) Determination of vehicle type

Table 5 shows the proportion of traffic volume for different vehicle types at different time intervals. Based on the information, vehicle samples with the amount equal to the number of vehicles crossing the bridge in each time interval can be simulated. The vehicle type for each sample can be determined. A random number T_vehicle uniformly distributed in the interval [0, 1] is generated first and then the vehicle type is determined by comparing the value of T_vehicle with the cumulative density of different kinds of vehicle. For example, for T_vehicle ≤ 0.9023 in the first time interval, loads of vehicle type 1 will be applied, while loads of vehicle type 2 are used for 0.9023 < T_vehicle ≤ 0.9534.

(b) Selection of lane information

Table 5 also shows the proportion of the traffic volume for different lanes at different time intervals. Vehicle samples travelling along different lanes during each time interval can be obtained. A random number T_lane uniformly distributed in the interval [0, 1] is generated first and then the lane is determined by comparing the value of T_lane with the cumulative density of different lanes. For example, if loads of vehicle type 1 are to be applied and T_lane ≤ 0.155, the simulated vehicle is applied on the first lane, while the vehicle is applied on the second lane for 0.155 < T_lane ≤ 0.4407.

Once the vehicle type and the lane information are known, the next target is to generate the samples of axle weights and spacing for each vehicle. The random number following normal PDF, lognormal PDF and t location distribution can be generated by the truncated Latin Hypercube sampling method (Helton and Davis, 2003), and the correlations between the axle weights and axle spacing can be neglected to simplify the analysis.

(d) Calculation of stress time-histories for each vehicle sample

According to the vehicle type, the lane, axle loads and the axle spacing of the sample determined in the above steps, loads of a vehicle are generated. The influence lines of the details concerned for the specified lane are used for determining the stress time-histories under the moving axles obtained. The results associated with the influence lines obtained from Step (2) are multiplied by the axle weights, and the results of all axles are superimposed according to the axle spacing, so as to obtain the stress time-histories.

(e) Assessment of fatigue life considering mean stress effect

According to the simulated stress time-histories, rain-flow counting is carried out in the MATLAB environment to calculate the number of cycles. The steps from (a) to (e) should be repeated for each sample of vehicle in different time intervals. The number of repetitions should be equal to the number of vehicles crossing the bridge. The daily accumulated number of cycles can be used to estimate the fatigue life for different critical locations considering the mean stress effect.

4. Statistical distribution of fatigue life

The Monte Carlo method is used here to repeat the procedures of fatigue analysis for a significant number of times so that a significant number of random fatigue lives can be obtained, from which the statistical distribution can be obtained using the fitting approach. A total of 100 repetitions are adopted here.

5. Statistical fatigue life considering annual growth of traffic

According to the stress time-histories simulated from the influence lines at various critical locations and the statistical traffic information, the traditional Monte Carlo method is used to repeat the procedures of fatigue analysis for a significant number of times so that the fatigue lives can be estimated taking into account the annual growth of traffic. A flat rate of increase is assumed for different types of vehicles. Combining with statistical distribution of number of cycles, the statistical distributions of fatigue lives for different critical locations can be obtained.

Figure 10.

Flowchart of probabilistic approach of fatigue life assessment.

Results and discussions

For comparison, the measurements at the linear strain gauges of ‘SS-GLE-04’ and ‘SS-GLW-04’ are studied, which are deployed to record the strains at the lower flange of the two outer girders, that is, the first and the fourth main girders. In interpreting the labels, ‘E’ means the east side where the strain gauge is deployed on the first girder, ‘W’ means the west side where the strain gauge is deployed on the fourth girder and ‘L’ is the location as shown in Figure 1. The strain time-histories on 22 November 2007 are as shown in Figure 11. Using the rain-flow counting method, the fatigue lives obtained from data of strain gauges are 490 and 286 years, respectively.

Figure 11.

Data segments of strain time-histories of (a) ‘SS-GLE-04’ and (b) ‘SS-GLW-04’ on 22 November 2007.

According to the process described above, the probability density curves of fatigue life without considering diurnal variations at locations ‘SS-GLE-04’ and ‘SS-GLW-04’ are presented in Figures 12 and 13 which roughly follow the normal distribution. With the diurnal variations considered, the fatigue lives at the same locations still follow the normal distribution as shown in Figures 14 and 15, but the values may be different. As shown in Figures 16 and 17, the fatigue lives considering the mean stress effect are almost one-third of those without considering the mean stress effect. The mean values of fatigue lives calculated using the probabilistic approach here are very close to those obtained from strain gauge measurements, which indicates reasonable accuracy of this method. Finally, the fatigue life variations in 5 years at locations of ‘SS-GLE-04’ and ‘SS-GLW-04’ considering mean stress effect and the hypothetical annual growth rate of traffic of 5% are shown in Figures 18 and 19, respectively. However, the results should be interpreted cautiously as the growth rate may be limited by the traffic flow capacity.

Figure 12.

Probability density curve of fatigue life at location ‘SS-GLE-04’ without considering diurnal variations.

Figure 13.

Probability density curve of fatigue life at location ‘SS-GLW-04’ without considering diurnal variations.

Figure 14.

Probability density curve of fatigue life at location ‘SS-GLE-04’ considering diurnal variations but without considering mean stress effect.

Figure 15.

Probability density curve of fatigue life at location ‘SS-GLW-04’ considering diurnal variations but without considering mean stress effect.

Figure 16.

Probability density curve of fatigue life at location ‘SS-GLE-04’ considering diurnal variations and mean stress effect.

Figure 17.

Probability density curve of fatigue life at location ‘SS-GLW-04’ considering diurnal variations and mean stress effect.

Figure 18.

Fatigue life variation at location of ‘SS-GLE-04’ considering annual growth of traffic of 5%.

Figure 19.

Fatigue life variation at location of ‘SS-GLW-04’ considering annual growth of traffic of 5%.

Conclusion

A statistical traffic load model is developed based on WIM data. The proposed advanced traffic load model is able to incorporate the variation of daily traffic flow within different time intervals and the trend of the annual traffic flow. Moreover, the uncertain information of vehicle type, number of axles, axle weight, axle spacing, transverse position of vehicle, vehicle speed and heading direction can be accounted for properly. Combining with the statistical traffic load model, a probabilistic approach is proposed to evaluate the variation of fatigue lives of Ting Kau Bridge with time of service. The fatigue lives at the monitored sections of the bridge are all within their designed limit of 120 years.

Footnotes

Acknowledgements

The authors gratefully acknowledge the Highways Department of the Hong Kong Government for assistance received in producing this article as well as permission of its publication.

Declaration of conflicting interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship and/or publication of this article: Any opinions expressed or conclusions reached in this article are entirely of the authors.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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