Page 170 - Reliability and Maintainability of In service Pipelines

Case Studies on the Application of Structural Reliability 155

coefficient for the wall thickness reduction between two points in time t i and t j .
Therefore all variables in Eq. (5.31) can be determined.
To estimate the probability of failure due to corrosion, a critical limit for the
wall thickness reduction should be established. ASCE manual No.69 (1989) con-
siders exposure of reinforcement as a criterion for failure. Therefore the maxi-
mum acceptable limit for wall thickness reduction (i:e:; d max ) can be considered
equal to the thickness of concrete cover (d max 5 a 0 ; concrete cover ).
Using Gamma Distributed Degradation (GDD) Model The average rate of wall
thickness reduction within time in a concrete sewer is calculated through
Equation 1.19 as explained in Section 1.6.1.2. Defining failure as the time when
all concrete cover is corroded, the developed algorithm (GDD model) in
Section 4.3.3 is used for formulation of the probability of failure based on the
gamma process concept.
Given that a random deterioration (i.e., corrosion depth, d) has a gamma dis-
tribution with shape parameter α . 0 and scale parameter λ . 0, formulation for
calculation of the probability of failure for a corrosion affected pipe is developed
as it was mentioned in Section 4.3.
The failure was defined as the time that all concrete cover on the reinforce-
ment is corroded (ASCE manual No.69 (1989)), therefore the concrete sewer is
said to fail when its corrosion depth, denoted by dtðÞ, is more than a special value
(for instance, concrete cover, a 0 Þ. Assuming that the time at which failure occurs
is denoted by the lifetime T, due to the gamma distributed deterioration,
Eq. (4.14) can be used for calculation of the probability of failure.

ð
a 0 Γðα tðÞ; a o λÞ
FtðÞ 5 Pr T # tÞ 5 Pr dtðÞ $ a 0 Þ 5 f dtðÞ dðÞd d 5 ð4:14Þ
ð
ð
0 Γðα tðÞÞ
b
Where α tðÞ 5 ct is the shape parameter with physical constants c . 0 and
b . 0 and λ is the scale parameter. The parameters α tðÞ and λ can be estimated
by using the estimation method explained in Section 4.3.3. For the exponential
parameter b, a value of one is assumed (b 5 1) based on some examples of
expected deterioration that have been presented in the Table 4.1.
5.4.1.2 Results and Analysis

The probability of failure due to wall thickness reduction is computed using first
passage probability (Eq. 5.31) and the results are shown in Fig. 5.23. As can be
concluded from this figure, the effect of the autocorrelation coefficient (ρ )on
d
the probability of failure is negligible, specially for the area of interest (i.e., lower
probability of failure).
Fig. 5.24 shows the results obtained for the probability of pipe failure by using
the GDD method.