PSI - Issue 5

Jing Zhang et al. / Procedia Structural Integrity 5 (2017) 1176–1183 Xia Yang et al./ Structural Integrity Procedia 00 (2017) 000 – 000

1182

7

MRL plot Hill plot

48.6

49.8

53.3

56.6

59.7

47.24

50.59

49.68

51.76

53.62

MSE method

49.5

50.5

52.5

53.0

53.0

As shown in Table 5, thresholds estimated by the three methods increase with the increasing of sample size. The estimated thresholds of the other two parent distributions display in the same way. In addition, a significant drawback of the graphic methods is that the MRL plots and Hill plots of samples generated from the same parent distribution show great instability. Especially for the third parent distribution, more than 80 percent of the MRL plots and Hill plots of their samples are difficult to determine the “ turning point ”, i.e., the threshold. Comparing Table 4 and 5, it can be found that the thresholds estimated by the three methods are smaller than the optimal thresholds generally. Similarly, the threshold estimation results of the other two parent distributions also are not very good, which means that the commonly used methods are not suitable for investigating the vehicle load effect. Based on the estimates at different thresholds of three parent distributions, an empirical threshold estimation technique based on the relationship between the characteristic value and the threshold is proposed in this paper. The steps are as follows: draw the plot of characteristic value versus the threshold and observe the relationship between them. (i) If the sample size is large enough, the characteristic value tends to be stable with the increase of threshold. In this case, the starting point of stable region can be selected as the optimal threshold. (ii) If the sample size is small, the variation magnitude of characteristic value decrease with the increase of threshold, but it does not tend to be stable. In this case, the threshold should be as large as possible under the condition that parameters of GPD can be correctly estimated without large fluctuations or obvious errors. The present paper focuses on the threshold selection in peak over threshold method to estimate the extreme value of vehicle load effect on the bridge. 417 days of strain data of a long cable-stayed bridge located in China are employed to work on this issue. According to the tail distribution of strain data due to vehicle loads, three homothetic distributions are chosen as the parent distributions. On the basis of samples produced by Monte Carlo method from each parent distribution, characteristic values at different thresholds are estimated and compared with the corresponding theoretical value. Then an empirical method for the threshold estimation based on the relationship between the characteristic value and threshold is proposed. The conclusions are drawn as follows: (1) The distribution of strain peaks induced by vehicle loads is a multi-peak distribution, and the tail distribution can be described by a mixed distribution of one Weibull and two Normal distributions. (2) The characteristic value mainly depends on threshold. With the increase of the threshold, the bias of characteristic value decreases firstly and then tends to be stable, while the variance decreases at first and then increases gradually. It means that an optimal threshold involves a trade-off between the bias and the variance. (3) The characteristic value is closely related to the sample size. If the bias does not tend to be stable with the increase of threshold for the shortage of samples, the extreme value cannot be estimated accurately. In that case, the threshold should be as large as possible. (4) The estimated extreme value based on GPD is greater than the theoretical value generally. References ： Beguería, S., Angulo-Martínez, M., Vicente-Serrano, S.M., López-Moreno, J.I., EI-Kenawy, A., 2011. Assessing trends in extreme precipitation events intensity and magnitude using non-stationary peaks-over-threshold analysis: a case study in northeast Spain from 1930 to 2006. International Journal of Climatology 31(14), 2102 – 2114. Brilhante, M.F., Gomes, M.I., Pestana, D., 2013. A simple generalisation of the Hill estimator. Computational Statistics & Data Analysis 57(1), 518-535. Caers, J., Beirlant, J., Maes, M.A., 1999. Statistics for Modeling Heavy Tailed Distributions in Geology: Part I. Methodology. Mathematical geology 31(4), 391 – 410. Davison, A.C., Smith, R.L., 1990. Models for Exceedances over High Thresholds. Journal of the Royal Statistical Society, Series B (Methodological) 52(3), 393 – 442. 4. Conclusions