Page 99 - Reliability and Maintainability of In service Pipelines

88 Reliability and Maintainability of In-Service Pipelines

failure probability for a series system at time t (P f;s ðtÞ) can be estimated by
(Thoft-Christensen and Baker, 1982):
max½P f;i ðtފ # P f;s ðtÞ # 1 2 L n ½1 2 P f;i ðtފ ð3:7Þ
i51
where P f;i ðtÞ is the failure probability of the pipe due to the ith failure mode at
time t and n is the number of failure modes.

3.5 Sensitivity Analysis

Sensitivity analysis is widely accepted as a necessary part of reliability analysis
of structures and infrastructure. The effect of variables on the reliability of a pipe-
line can be analyzed by doing a comprehensive sensitivity analysis. In view of
the large number of variables that affect the limit state function, it is of interest to
identify those variables that affect the failure most so that more research can
focus on those variables.
Sensitivity analysis should be carried out to provide quantitative information
necessary for classifying random variables according to their importance. These
measures are essential for reliability-based service life prediction of deteriorating
pipelines.
Sensitivity analysis provides the degree of variation of limit state functions or
measures at a specific point characterized by a realization of all random variables.
Similarly to the conventional sensitivity measure in the reliability approaches, the
sensitivity measure, S, can be defined as follows (Kong and Frangopol, 2005):

ð
@GðXÞ GX 1 εÞ 2 GðXÞ
S GXðÞ X i 5 5 lim ð3:8Þ
ðÞ
@X i ε-0 ε
where G is a performance function of X; X and ε are vectors; and ε is a small per-
turbation. An element X i of X can be any type of variable or parameter. For
instance, it can be a mean or a standard deviation of a variable, or a deterministic
parameter. For a complex system, the sensitivity measure can be computed by
using the numerical differentiation method rather than by an analytical approach
(Kong and Frangopol, 2005).
Different sensitivity indexes have been introduced. In this section, relative
contribution, sensitivity ratio (SR), and omission factor are discussed.

3.5.1 RELATIVE CONTRIBUTION

A sensitivity index that can be used in a comprehensive reliability analysis is the
relative contribution of each variable in limit state function. The relative