Page 169 - Reliability and Maintainability of In service Pipelines

154 Reliability and Maintainability of In-Service Pipelines

Using First Passage Probability Method Eq. (5.30) is a typical upcrossing problem
that can be solved by using first passage probability theory. In a time-dependent reli-
ability problem all or some of the basic random variables are modeled as stochastic
processes. For the above problem, the sewer failure depends on the time that is
expected to elapse before the first occurrence of the stochastic process, d(t), upcrosses
a critical limit (the threshold, d max ) sometime during the service life of the sewer.
As it was described in Section 4.3, the probability of failure of a pipe can be
determined by using first passage probability theory from Eq. (4.9). Considering
Eq. (5.30) as the failure definition, reduction of wall thickness (d) is the action
(load) and d max is its effect or the acceptable limit (resistance). Therefore
Eq. (4.9) can be reproduced with d replacing S and d max replacing R.
( ! !)
ð t
σ _ djd ðtÞ d max 2 μ ðtÞ μ _ djd ðtÞ μ _ djd ðtÞ μ _ djd ðtÞ
d
P f tðÞ 5 [ [2 1 Φ d τ
0 σ d ðtÞ σ d ðtÞ σ _ djd ðtÞ σ _ djd ðtÞ σ _ djd ðtÞ
ð5:31Þ
For a given Gaussian stochastic process with mean function μ tðÞ, and autoco-
d

variance function C dd t i ; t j , all terms in Eq. 5.31 can be determined as outlined
in Section 4.3 by using the following formulations:
_ 1 ρ
σ _ d
μ _ djd 5 E djd 5 d max 5 μ _ d ðd max 2 μ Þ ð5:32aÞ
d d
σ d
2 2 1=2
σ _ djd 5 ½σ _ d ð12ρ ފ ð5:32bÞ
d
where
dμ ðtÞ
d
5
μ _ d ð5:32cÞ
dt
" # 1=2
2
5 @ C dd ðt i ; t j Þ
σ _ d ð5:32dÞ
@t i @t j
i5j
C _ðt i ; t j Þ
ρ 5 dd ð5:32eÞ
d 1=2
ðt j ; t j Š
½C dd ðt i ; t i ÞUC _ d _ d
and the cross-covariance function is
@C dd ðt i ; t j Þ
C _ðt i ; t j Þ 5 ð5:32fÞ
dd
@t j
2
C dd ðt i ; t j Þ 5 λ ρ μ ðt i Þμ ðt j Þ ð5:32gÞ
d d d d
where λ d is the coefficient of variation of the wall thickness reduction which is
determined based on Monte Carlo simulations and ρ d is (auto-) correlation