International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Time-dependent fatigue reliability assessment of Ting Kau Bridge based on weigh-in-motion data Zhang, Jing; Au, Francis T. K. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Time-Dependent Fatigue Reliability Assessment of Ting Kau Bridge Based on Weigh-In-Motion Data Jing Zhang Associate Professor, Dept. of Civil Engineering, Hefei University of Technology, Hefei, China Francis T.K. Au Professor, Dept. of Civil Engineering, Univ. of Hong Kong, Hong Kong ABSTRACT: An advanced probabilistic loading model is proposed in this paper to simulate the vehicles running along Ting Kau Bridge based on the weigh-in-motion system installed on the bridge. The weigh-in-motion system collects the information of vehicle speed, heading direction, lane used, axle weight, axle spacing between neighboring axles, etc. The data are employed to develop a statistical traffic load model which is able to incorporate the variation of daily traffic flow within different time intervals and the trend of the annual traffic flow as well as the statistical information. The time-dependent fatigue reliability assessment at critical locations can be carried out.  The ever-evolving developments in design and construction technologies have made spans of over 1000m 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. Fatigue has been one of the most critical forms of damage for cable-stayed bridges with steel girders and assessing their remaining fatigue lives has been the primary focus in recent years. 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, etc. will inevitably lead to accumulation of damage and possible reduction of strength. In extreme cases, catastrophic failures could happen. Fatigue of steel bridges 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 systems which are able to record the long-term responses for fatigue assessment and performance prediction (Elkordy et al. 1994). However, structural health monitoring systems are always expensive, and there are a lot of structural components on which it is difficult to install strain gauges. As a result, some other effective tools are required to obtain the fatigue stress spectra of the details that are hardly accessible (Guo and Chen 2013). Recently, a statistical load model has been proposed by Guo and Chen (2013) so that the uncertain information of vehicle types, number of axles, axle weights, axle spacing, transverse position of vehicle, etc. can be accounted for properly. As a sequel to this statistical load model, a statistical approach of modelling the 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. 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). Combined with this statistical traffic load model, the stress time-histories at critical locations of the bridge are 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 obtained, and the probability density curves of the fatigue life can be constructed for further analyses. 1. WEIGH-IN-MOTION SYSTEM Ting Kau Bridge is a cable-stayed bridge in Hong Kong with a total length of 1,177 m completed in 1998. As shown in Figure 1, the Ting Kau main span and Tsing Yi main span are 448 m and 475 m respectively, while the two side spans are both 127m. The bridge connects Ting Kau at the north and Tsing Yi Island at the south over Rambler Channel. A statistical fatigue reliability assessment is performed to estimate the remaining service life of Ting Kau Bridge to support decisions on maintenance and rehabilitation. The dynamic WIM system installed on the bridge monitors the traffic flow and volume, and axle and gross vehicle weights. The system is capable of dynamic measurement of vehicular axle loads and speed with an intelligent vehicle classification system. As shown in Figure 2, the WIM system is located at the Tsing Yi end of Ting Kau Bridge on both carriageways. Data of WIM in 84 days each with 24 hours are filtered. After filtering, 5% of vehicles are eliminated from the database.               Figure 1: Schematic plan and elevation of Ting Kau Bridge.                    Figure 2: Layout of weigh-in-motion dynamic weigh-bridge system. 2. STATISTICAL LOAD MODEL The WIM data collected are divided into 6 vehicle types according to the number of vehicle axles, the first two types as shown in Figure 3 for example. To take into account the variation of daily traffic properly, the diurnal variations of highway loading are described by 4 unequal time intervals, i.e. 7:00-11:00, 11:00-17:00, 17:00-21:00 and 21:00-7:00 on the following day, to cover the morning and evening rush hours separately as well as the off-peak hours based on the hourly statistics (Zhang et al. 2014). Considering the random variables of each vehicle type, the probability density functions 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 4 to 6. Furthermore, the parameters for all distributions are listed in Tables 1 and 2. μ is the location parameter (mean value), σ is the scale parameter (standard deviation) and ν is a shape parameter for t location-scale distribution. The symbol “(1)” in the first column is used to denote those for the first time interval. It is found that most of the axle spacing of the 6 vehicle types follows the t location-scale distribution (‘③’), while most of the axle weights follow the normal (‘①’) and lognormal distributions (‘②’).  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3             Figure 3: Vehicle type classification for Ting Kau Bridge.   Figure 4: Distribution of axle weight for the first axle of vehicle type 2 in the 3rd time interval (17:00-21:00).    Figure 5: Distribution of axle weight for the third axle of vehicle type 3 in the 1st time interval (7:00-11:00).    Figure 6: Distribution of axle spacing for the first axle of vehicle type 1 in the 4th time interval (21:00-7:00 next day).   Table 1: Parameters of distribution for random axle spacing of 6 vehicle types in the first time interval (‘Dis’ means ‘Distribution’). Variable ‘Dis’ μ σ ν AS11(1) ③ 2.5676 0.13052 1.2024 AS21(1) ③ 5.2354 0.30080 1.4579 AS22(1) ③ 1.4355 0.093464 1.7610 AS31(1) ① 2.7697 0.88170 - AS32(1) ① 5.2634 2.1288 - AS33(1) ③ 1.2316 0.068948 1.0879 AS41(1) ③ 3.1878 0.20994 4.0925 AS42(1) ③ 6.8215 0.32511 1.6377 AS43(1) ③ 1.2512 0.070962 1.5072 AS44(1) ③ 1.2568 0.079235 3.1590 AS51(1) ③ 3.0459 0.20093 2.9606 AS52(1) ③ 1.2747 0.071586 1.2049 AS53(1) ③ 6.3169 0.29711 1.3438 AS54(1) ③ 1.2527 0.075765 1.5450 AS55(1) ③ 1.2513 0.074239 1.5141 AS61(1) ① 2.4479 0.97129 - AS62(1) ③ 1.2743 0.15662 0.83262 AS63(1) ① 0.98317 0.91200 - AS64(1) ② 0.74483 0.75062 - AS65(1) ② 0.21142 0.37379 - AS66(1) ③ 1.2164 0.13626 0.98418  0 50 100 150 200 250 300 350 40000.0050.010.0150.020.025Axle weight (kN)Probability density  AW21_3Normal0 20 40 60 80 100 120 140 160 18000.0050.010.0150.02Axle weight (kN)Probability density  AW33_1Lognormal0 2 4 6 8 10 12 14 1600.20.40.60.811.21.41.61.82Axle spacing (m)Probability density  AS11_4t location-scale(a) Vehicle type 1 AW11 AW12 AS11 (b) Vehicle type 2 AW21 AW22 AS21 AW23 AS22 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 Table 2: Parameters of distribution for random axle weight of 6 vehicle types in the first time interval. Variable ‘Dis’ μ σ ν AW11(1) ② 2.0982 0.65419 - AW12(1) ② 2.0467 0.76570 - AW21(1) ① 55.021 15.734 - AW22(1) ② 2.0467 0.7657 - AW23(1) ① 70.085 22.936  AW31(1) ① 48.084 15.737 - AW32(1) ② 3.7438 0.59541 - AW33(1) ② 3.4977 0.72807 - AW34(1) ② 3.6509 0.71893 - AW41(1) ① 54.140 11.187 - AW42(1) ② 3.9697 0.49263 - AW43(1) ② 3.4002 0.60631 - AW44(1) ② 3.4839 0.49764 - AW45(1) ② 3.4304 0.57072 - AW51(1) ① 51.546 13.477 - AW52(1) ① 45.800 21.067 - AW53(1) ① 47.053 22.043 - AW54(1) ② 3.3481 0.62303 - AW55(1) ② 3.4752 0.59020 - AW56(1) ② 3.5000 0.68282 - AW61(1) ③ 44.965 6.3000 1.4798 AW62(1) ① 36.147 25.495 - AW63(1) ① 38.853 25.311 - AW64(1) ① 32.018 20.567 - AW65(1) ① 31.179 17.650 - AW66(1) ① 34.918 20.390 - AW67(1) ② 2.8771 1.0300 - 3. STATISTICAL FATIGUE LIFE ESTIMATION 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 3 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 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 is therefore also uncertain. Therefore Monte Carlo simulation can be used to simulate the large number of occurrences so that the variations can be revealed.   Table 3: Proportion of traffic in each lane.  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   12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 Once the vehicle type and the lane information are found, the next target is to generate the samples of axle weights and spacing for each vehicle. Random numbers following the normal probability density function, lognormal probability density function and t location distribution can be generated by the truncated Latin Hypercube sampling method, and the correlations between the axle weights and axle spacing are neglected to simplify the analysis. According to the vehicle type, the lane, axle loads and the axle spacing of the sample determined, the 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 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. 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. 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, i.e. the 1st and the 4th main girders, as shown in Figure 1. Combining with statistical distribution of number of cycles, the statistical distributions of fatigue lives for different critical locations can be obtained. Taking into account diurnal variations, the statistical fatigue lives at locations ‘SS-GLE-04’ and ‘SS-GLW-04’ follow the normal distribution as shown in Figures 7 and 8. Using the rain-flow counting method, the fatigue lives obtained from the data on 25 Nov 2007 of ‘SSGLE04’ and ‘SSGLW04’ are 490 years and 286 years, respectively. The fatigue lives variations in 5 years at locations of ‘SS-GLE-04’ and ‘SS-GLW-04’ considering an annual growth rate of traffic of 5% are estimated as shown in Figures 9 and 10 respectively.   Figure 7: Probability density curve of fatigue life at location ‘SS-GLE-04’ considering diurnal variation.    Figure 8: Probability density curve of fatigue life at location ‘SS-GLW-04’ considering diurnal variations.   The mean values of fatigue lives calculated using the probabilistic approach here are very close to those obtained from strain gauge measurements, which verifies the applicability of this method. Results also show that the effect of annual growth rate of traffic on fatigue life prediction cannot be ignored. 160 180 200 220 240 260 280 300 320 34000.0020.0040.0060.0080.010.012Fatigue life (year)Probability densityConsidering mean stress effect  SS-GLE-04Normal (237,34)250 300 350 400 450 500 550012345678x 10-3Fatigue life (year)Probability densityConsidering mean stress effect  SS-GLW-04Normal (394,56)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6   Figure 9: Fatigue life variation at location of ‘SS-GLE-04’ considering annual growth of traffic of 5%.     Figure 10: Fatigue life variation at location of ‘SS-GLW-04’ considering annual growth of traffic of 5%.  4. ACKNOWLEDGMENTS The authors gratefully acknowledge the Highways Department of the Hong Kong Government for assistance received in producing this paper as well as permission of its publication. Any opinions expressed or conclusions reached in the paper are entirely of the authors.  5. REFERENCES Elkordy M.F., Chang K.C. and Lee G.C. (1994). A structural damage neural network monitoring system, Computer-Aided Civil and Infrastructure Engineering, 9(2): 83-96. Guo T. and Chen Y.W. (2013). Fatigue reliability analysis of steel bridge details based on field-monitored data and linear elastic fracture mechanics. Structure and Infrastructure Engineering, 9(5): 496-505. Zhang J., Au F.T.K. and Ren W.X. (2014). Variation of traffic flow of Ting-Kau Bridge based on weigh-in-motion data. Proceedings of the 13th International Symposium on Structural Engineering, 1912-1921. 0 1 2 3 4180190200210220230240Increased number of yearFatigue life (year)Considering annual growth of traffic  SS-GLE-04(2,206)(1,221)(3,191)(4,184)(0,237)0 1 2 3 43003103203303435036037038039400Increased number of yearFatigue life (year)Considering annual growth of traffic  SS-GLW-04(0,394)(1,367)(2,340)(3,319)(4,305)Number of years with growth of traffic Number of years with growth of traffic 

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