r - The Hong Kong Polytechnic University

Case Study
With time segment of 8s
0.05
With time segment of 50s
Probability Density Function
0.2
0.15
0.1
0.05
Probabilily Density Function
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0
0 20 40 60 80 100 120 140 160
Strain (1.0E-6)
0.005
0
20 40 60 80 100 120 140 160
Strain (1.0E-6)
0.025
With time segment of 90s
0.012
With time segment of 300s
Probability Density Function
0.02
0.015
0.01
0.005
Probability D ensity Function
0.01
0.008
0.006
0.004
0.002
0
0 20 40 60 80 100 120 140 160
Strain (1.0E-6)
0
0 20 40 60 80 100 120 140 160
Strain (1.0E-6)
Figure 1 Initial distribution fitting using Gumbel distribution: time segment of (a) 8s; (b) 50s; (c) 90s; (d) 300s
Time history record of strains at the mid-span of a steel girder in the CORIBM Bridge on route LA 70 in District
61, Assumption Parish, Louisiana, were recorded. Since only three hours monitoring data is available, it is not
advisable to estimate extreme strain distribution for a very long mean recurrence interval. The extreme strain
distribution (Gumbel distribution) for mean recurrence intervals of 1 day, 10 days, 30 days, 180 days and one
year were estimated using maximum likelihood estimation method as shown in Fig . 1. The yearly extreme
strain distribution can be derived from the initial distributions with time segments of 90s or 300s as
(22)
where the initial distributions and have been obtained from distribution fitting previously. It is
difficult to prove directly that Eq. (22) is tenable and it is difficult to derive the parameters of yearly extreme
response distribution through an analytical method. An alternative method is to generate samples using Monte
Carlo simulation following the distribution functions on the right side of Eq. (22), and then, fit the generated
samples with the selected distribution function, Gumbel distribution (maximum cases). The factors derived from
distribution fitting were shown in Fig. 2. From Fig. 2, it is seen that both the location factor μ and the scale factor
σ show converging property as the length of time segment increases. For different mean recurrence intervals, μ
converges to different values, but σ converges to a fixed value. The location factor μ determines the mode value of
the distribution while the shape factor σ determines the variance or the standard deviation of the distribution. Its
location shifts to the right direction as the mean recurrence interval increases. The distributions have different
mode values but same variance for different mean recurrence intervals. The PDF of extreme strain distribution for
mean recurrence intervals of 1 day, 10 days, 30 days and one year are shown in Fig. 3.
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