Page 118 - Reliability and Maintainability of In service Pipelines

Time-Dependent Reliability Analysis 107

( ! !)
R 2 μ R2μ _ SjS R 2 μ _ SjS R2μ _ SjS
_ _ _
σ _ SjS
1
υ 5 [ S [2 2 Φ 2 ð4:7Þ
R
σ S σ S σ _ SjS σ _ SjS σ _ SjS
where [ and Φ are standard normal density and distribution functions, respec-
_
tively, μ and σ denote the mean and standard deviation of S and S, represented
by subscripts and “|” denotes the condition. For a given Gaussian stochastic pro-
cess with mean function μ ðtÞ, and autocovariance function C SS ðt i ; t j Þ, all terms in
S
Eq. (4.7) can be determined, based on the theory of stochastic processes as
detailed in Papoulis and Pillai (2002) as follows.

1 ρ
σ _ S
S
μ _ SjS 5 E SS 5 R 5 μ _ S ðR 2 μ Þ ð4:8aÞ
_

σ S
2 2 1=2
σ _ SjS 5 ½σ _ S ð12ρ ފ ð4:8bÞ
where
dμ ðtÞ
S
5
μ _ S ð4:8cÞ
dt

2
# 1=2
2
5 @ C SS ðt i ; t j Þ
σ _ S 4 : i5j ð4:8dÞ
@t i @t j

C _ðt i ; t j Þ
SS
ð4:8eÞ
1=2
ρ 5
ð
C SS t i ; t i Þ:C _ S _ S t j ; t j
And the cross-covariance function is:
@C SS ðt i ; t j Þ
C _ t i ; t j 5 ð4:8fÞ

SS
@t j
Because it is unlikely that the corrosion depth in a given pipe exceeds the wall
thickness at the beginning of structural service, the probability of failure due to
corrosion at t 5 0 is zero, i.e., P f 0ðÞ 5 0. The solution to Eq. (4.5) can be
expressed, after substituting Eq. (4.7) into Eq. (4.5), and considering that R is
_
constant (R 5 0) therefore:
( ! !)
ð t
σ _ SjS ðtÞ R 2 μ ðtÞ μ _ SjS ðtÞ μ _ SjS ðtÞ μ _ SjS ðtÞ
S
P f tðÞ 5 [ [2 1 Φ d τ
0 σ S ðtÞ σ S ðtÞ σ _ SjS ðtÞ σ _ SjS ðtÞ σ _ SjS ðtÞ
ð4:9Þ
The application of Eq. (4.9) for calculation of the failure probability for differ-
ent case studies will be presented in sections 5.1.1 and 5.4.1.1 in Chapter 5.