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

1178

3

u N

1 ( ) 1 ( u i N   

n e u

), X u X u  

(1)

i

i

where N u denotes the number of exceedances over u . The set data of { u , e n ( u )} represents the MRL plot. Generally, the value of u above which the plot is an approximately straight line can be selected as the optimal threshold. 2.2. Hill estimator Let X (1,n) > X (2,n) > … > X ( n,n ) be associated descending order statistics of ( X 1 , X 2 , …, X n ) which are independent and identically distributed random variables (Brilhante, 2013). Assuming that the distribution of these random variables is heavy-tailed, the Hill estimator, a well-known estimator of ξ , is defined as

X

1 log( k



(2)

( , ) i n

H

k n 

) ,



, k n

k

X

i

1



( , ) k n

Obviously, the Hill estimator is function of these extreme random variables { X (1, n ) , X (2, n ) , …, X ( k , n ) } which depends on the chosen threshold. A Hill plot is constructed by the Hill estimator of a range of k value versus the value of k or the threshold. The value of X k , n above which the Hill estimator tends to be stable can be chosen as the optimal threshold.

2.3. Mean square error

The mean square error (MSE) can be evaluated on any tail characteristics θ such as the shape parameter of GPD, an extreme quantile or a return period. The optimal threshold can be obtained by evaluating the MSE for a range of thresholds. The bootstrap is chosen as the technique to analyze estimation bias and variance of the samples at each threshold. Since threshold selection greatly influences the parameters estimation of GPD, the shape parameter ξ is chosen as the tail characteristic θ in this paper .

3. Threshold estimation based on the extreme value extrapolation of strain data

3.1. Simulation of strain peaks distribution induced by vehicle loads

It is demonstrated that the EV estimation corresponds to the tail data. To ascertain the probability distribution model of tail data of vehicle load effect, the strain data of 417 days of a long cable-stayed bridge in China is taken as the real example. The strain due to vehicle loads can be obtained by decomposing the raw strain data through analytical model decomposition method. Fig. 1 shows the strain time history due to vehicle loads of 27 th December, 2015. Each “ jump ” in the figure represents the strain time history induced by a vehicle passing through the bridge. The maximum value of each “ jump ” is defined as strain peak which is assumed to be independent (Ruan, 2012; Ou and Li, 2011). According to the suggestion of DuMouchel (1983), the upper 10 percent of these stain peaks, which is larger than 18.1 με , are collected. Since the histogram of collected data shows three peaks, three mixed distributions (a mixed distribution of three Normal distributions, a mixed distribution of one Lognormal and two Normal distributions, a mixed distribution of one Weibull and two Normal distributions) are chosen to fit these collected data shown in Fig. 2. The Kolmogorov Smirnov test for goodness-of-fit is used. Results show that all the three models coincide with the tail data well, and the third model, i.e., a mixed distribution of one Weibull and two Normal distributions, is the best. In order to investigate the influence of threshold on EV estimation based on GPD, three homothetic parent distributions are given to simulate the tail distribution of vehicle load effect as follows (i) A mixed distribution of three Normal distributions; (ii) A mixed distribution of one Lognormal and two Normal distributions; (iii) A mixed distribution of one Weibull and two Normal distributions with different parameters. Parameters of the three parent distributions are listed in Table 1.