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

1180

5

Note: P 1 ， P 2 ， P 3 denote the weighting coefficients; μ N , σ N denote the mean value and standard deviation of Normal distribution; μ L , σ L denote the mean value and standard deviation of Lognormal distribution; σ w , ξ w , μ w denote the scale parameter, shape parameter and position parameter of Weibull distribution. With reference to the measured data of Taiping Lake Bridge, the 100 strain peaks, which are in the top 10 percent, are supposed to be produced every day. From each parent distribution, 100 times respectively for sampling time of a day, a month, six months, a year, ten years and fifty years are randomly sampled by Monte Carlo method, which means 100 groups respectively with the sample size of 100, 3000, 18000, 36000, 360000, 1800000 from each parent distribution. Only the first parent distribution is described in detail due to length limitation while conclusions of other two parent distributions are introduced here.

3.2. The effect of threshold chosen on EV estimation

A series of values chosen as thresholds are listed in Table 2. The mean number of exceedances over a chosen threshold is calculated and also shown in the table. For samples of each group, the exceedances over each chosen threshold are fitted by GPD models, and then 100-yearly EV at each chosen threshold is estimated by POT method. The parameters of GPD are estimated by the maximum likelihood. The fitting residual can be obtained as follows

N

^

1



( F x

( F x

)

)



Fitting residual =

(3)

, i N

, i N

i N 

1

which is used as a goodness-of-fit function for GPD. F ( x i,N ) denotes the empirical cumulative probability at value of x i,N , and ^ , ( ) i N F x denotes the cumulative probability calculated from GPD at x i,N . The mean values of fitting residual of 100 group samples with the same sampling length are listed in Table 3. Characteristic value of the expectation of each 100-yealy EV distributions is calculated. The mean value, the standard deviation and 95% confidence interval (CI) of the characteristic values are shown in Table 4. Since the first parent distribution is known, the theoretical characteristic value can be obtained which is 68.44.

Table 2. The number of exceedances over different thresholds

Threshold ( u )

Sampling time/ d

48

50

52

54

56

58

1

1.25

0.59

0.39

0.18 4.39

0.04 1.85

0.01 0.55 3.46 6.92 72.4

30

42.30

21.87

10.24 60.84

180 360

243.93 496.51 4919.9 24621

129.07 157.78 2574.7 12889

26.98 53.19

10.42 20.52

124.34

3600

1234 6156

531.82 2648.8

205.96 1035.5

18000

363.49

Table 3. Fitting residual of GPD model

Threshold ( u ) 48

Sampling time/ d

50

52

54

56

58

30

0.0364 0.0240

0.0491 0.0254

0.0755 0.0323

0.1751 0.0426

-

- -

180

0.0803