Page 117 - Reliability and Maintainability of In service Pipelines

106 Reliability and Maintainability of In-Service Pipelines

the limit state function of Eq. (4.1), the probability of pipe (structural) failure,
P f , can be determined by:
½
P f tðÞ 5 PG tðÞ # 0Š 5 P½StðÞ $ Rtðފ ð4:3Þ
At a time that P f tðÞ is greater than a maximum acceptable risk in terms of the
probability of pipe failure, P a , it is the time the pipe becomes unsafe or unservice-
able and requires replacement or repairs. This can be determined from the
following:

P f ðT L Þ $ P a ð4:4Þ
where T L is the service life for the pipe for the given assessment criterion and
acceptable risk. In principle, the acceptable risk, P a , can be determined from a
risk cost optimization of the pipeline system during its whole service life. This
can be further studied in Mann and Frey (2011) and Dawotola et al. (2012).
Eq. (4.3) represents a typical upcrossing problem in mathematics and can be
dealt with using time-dependent reliability methods. Time-dependent reliability
problems are those in which either all or some of the basic variables are modeled
as stochastic processes. In this method, the structural failure depends on the time
that is expected to elapse before the first occurrence of the action process S(t)
upcrossing an acceptable limit (the threshold) L(t) sometime during the service life
of the structure [0, T L ]. Equivalently, the probability of the first occurrence of such
an excursion is the probability of failure P f tðÞ during that time period. This is
known as “first passage probability” and can be determined by Melchers (1999):
Ð t
2 νd t
P f tðÞ 5 1 2 1 2 P f 0ðÞ e 0 ð4:5Þ
where P f ð0Þ is the probability of structural failure at time t 5 0 and υ is the mean
rate for the action process S(t) to upcross the threshold R(t).
The upcrossing rate in Eq. (4.5) can be determined from the Rice formula
(Melchers 1999):
N
ð
1
ν 5 ν 5 _ _ _ ð4:6Þ
R S 2 R f _ R; S d _ S
SS
R
1
where ν is the upcrossing rate of the action process S(t) relative to the threshold
R
_
_
R, R is the slope of R with respect to time, SðtÞ is the time-derivative process of
_
S(t), and f _ is the joint probability density function for S and S. An analytical
SS
solution to Eq. (4.6) has been derived for a deterministic threshold R in Li and
Melchers (1993) as follows: