Page 137 - Reliability and Maintainability of In service Pipelines

124 Reliability and Maintainability of In-Service Pipelines

suitable for the failures assessment of corroded pipes. According to the theory of
systems reliability the failure probability for a series system at time t; (P f;s ðtÞ),
can be estimated by an equation the same as Eq. (3.7):

n
max P f;i ðtÞ # P f;s ðtÞ # 1 2 L 1 2 P f;i ðtÞ ð5:5Þ
i51
th
where P f;i ðtÞ is the failure probability of the pipe due to the failure of i corrosion
pit on the pipe wall (determined by Eq. (5.3)) at time t and n is the number of
corrosion pits existing in the pipeline.
At a time that P f;s ðtÞ is greater than a maximum allowable risk in terms of the
failure probability, P a , (assuming the same consequences) it is the time the pipe-
line system is broken. This can be presented as follows:

P f;s ðT f Þ $ P a ð5:6Þ
where T f denotes the time the pipeline fails due to corrosion induced strength
loss. In principle, P a can be estimated from a risk cost optimization analysis of
the pipeline during its whole service life.
For Eq. (5.3) to be of practical use, i.e., determining the failure probability
due to strength loss over time, the key is to develop a stochastic model for the
residual strength. This is dealt with in the next section.

5.1.1.1 Model for Residual Strength and Corrosion
To determine the remaining strength of a corroded pipe, equations are mainly
based on the ratio of the cross-section of the corrosion pit to the original cross-
section. An analytical model developed by Kiefner and Vieth (1990), calculates
the residual strength of a corroded pipe through the following equation:

2dσ f 1 2 A=A o
Q 5 ð5:7Þ
ð
D o 1 2 A= MA o Þ
where d is pipe wall thickness, D o is pipe diameter, σ f is the flow stress, A is the
cross-section area of the corrosion pit projected onto the longitudinal axis of pipe,
A o is the original cross-section area before corrosion, and M is the Folias factor
that accounts for bulging of the pipe before failure. The corroded cross-section
area, A can be approximated as A 5 a:l, where a and l denote the average corro-
sion depth and longitudinal length of the corrosion pit, respectively.
The cross-section area before corrosion is A o 5 d:l. Flow stress, σ f , is gener-
ally defined as a function of the yield stress σ y . Assuming σ f 5 1:15σ y and