Page 95 - Reliability and Maintainability of In service Pipelines

84 Reliability and Maintainability of In-Service Pipelines

samples are generated. Monte Carlo method is the major simulation method
for structural reliability analysis of pipelines.
FORM techniques: In First-Order Reliability Methods (FORM) the limit state
function (failure function) is linearized and the reliability is estimated using
level II or III methods.
SORM techniques: In Second-Order Reliability Methods (SORM) a quadratic
approximation to the failure function is determined and the probability of fail-
ure for the quadratic failure surface is estimated.
Time-dependent reliability techniques: When a pipeline is subjected to a time-
dependent degradation process, probabilistic time-dependent methods can be
used. First passage probability theory has been introduced for time-dependent
reliability analysis (Melchers 1999). Gamma process concept also has the
potential of usage as a model for reliability analysis of pipelines subject to
monotonic degradation processes (Mahmoodian and Alani 2014). These meth-
ods are discussed and developed in Chapter 4, Time-Dependent Reliability
Analysis, for reliability analysis of pipelines.

For a detailed introduction to structural reliability theory references are made
to the following textbooks: Melchers (1999), Thoft-Christensen and Baker (1982),
and Ditlevsen and Madsen (1996).

3.3 Generalization of a Basic Reliability Problem

In a basic reliability problem only one load effect, S, can be resisted by one resis-
tance, R. The load and the resistance are expressed by a known probability den-
sity function, f S and f R , respectively.
Considering the definition of safety, the pipeline will be marked as failed if its
resistance, R, is less than the stress resultant, S, action on it. Therefore the proba-
bility of failure can be stated as follows:
P f 5 PR 2 S # 0Š 5 PGðR; SÞ # 0Š ð3:2Þ
½
½
where GðR; SÞ is termed the limit state function, and the probability of failure is
identical to the probability of limit state violation. In Fig. 3.1, the above equation
is represented by the hatched failure domain D, so that the failure probability
becomes (Melchers 1999):

ð ð
P f 5 PðR 2 S # 0Þ 5 f RS ðr; sÞdrds ð3:3Þ
D
where f RS ðr; sÞ is the joint (bivariate) density function.