Page 126 - Reliability and Maintainability of In service Pipelines

114 Reliability and Maintainability of In-Service Pipelines

in a Taylor series about the point defined by the vector of the means
ðμ ; μ ; ... ; μ Þ. By truncating the series, the mean and variance are (Papoulis
X 1 X 2 X n
and Pillai, 2002):
n
2
n
1 X X @ Y
EYðÞ Y μ ; μ ; ...; μ 1 covðX i ; X j Þ ð4:22Þ

X 1 X 2 X n
2 @X i @X j
i51 j51
n n
X X
varðYÞ c i c j covðX i ; X j Þ ð4:23Þ
i j
The flowchart in Fig. 4.4 illustrates the gamma distributed degradation model in
the case that corrosion measurements are not available. The procedure will be used
for reliability analysis of case study pipelines in Chapter 5.
4.4 Monte Carlo Simulation Method

Monte Carlo simulation has been successfully used for reliability analysis of dif-
ferent structures and infrastructure (e.g., Camarinopoulos et al., 1999; Melchers,
1999; Sadiq et al., 2004; Yamini, 2009). Hence, the method can be used as a veri-
fication method to check the results which are obtained from the application of
the two time-dependent analytical method (i.e., first passage probability method
and gamma distributed degradation model).
Monte Carlo simulation techniques involve sampling at random to artificially
simulate a large number of experiments and to observe the results. To use this
method in structural reliability analysis, a value for each random variable is
selected randomly ( ^ x i ) and the limit state function (Gð^ xÞ) is checked. If the limit
state function is violated (i.e., Gð^ xÞ # 0), the structure or the system has failed.
The experiment is repeated many times, each time with randomly chosen vari-
ables. If N trials are conducted, the probability of failure then can be estimated by
dividing the number of failures to the total number of iterations:

nðGð^ xÞ # 0Þ
P f ð4:24Þ
N
The accuracy of the Monte Carlo simulation result depends on the sample size
generated and, in the case when the probability of failure is estimated, on value of the
probability (the smaller the probability of failure, the larger the sample size needed to
ensure the same accuracy). The accuracy of the failure probability estimates can be
checked by calculating their coefficient of variation (e.g., Melchers, 1999).
In order to improve the accuracy of estimating the probability of ultimate
strength failure, while keeping the computation time within reasonable limits,