Page 121 - Reliability and Maintainability of In service Pipelines

110 Reliability and Maintainability of In-Service Pipelines

TABLE 4.1 Typical Values for Exponential Parameter, b; in Different Deterioration
Types
Deterioration type Exponential References
parameter, b
Degradation of concrete due 1 Ellingwood & Mori (1993)
to reinforcement corrosion
Sulfate attack 2 Ellingwood & Mori (1993)
Diffusion-controlled aging 0.5 Ellingwood & Mori (1993)
Creep 1/8 Cinlar et al. (1977)
Expected scour-hole depth 0.4 Hoffmans & Pilarczyk (1995) and van
Noortwijk & Klatter (1999)

4.3.2 DEVELOPING GAMMA DISTRIBUTED DEGRADATION MODEL WITH AVAILABLE
CORROSION DEPTH DATA

In this section using the gamma distributed degradation (GDD) model for reliabil-
ity analysis of corrosion affected pipes in case of availability of corrosion depth
data is discussed. The data of corrosion depth can be achieved by periodical
inspections.
b
To model the corrosion as a gamma process with shape function α tðÞ 5 ct and
scale parameter λ, the parameters c and λ should be estimated. For this purpose,
statistical methods are suggested. The two most common methods that can be used
for parameter estimation are the maximum likelihood and method of moments.
Both methods for deriving the estimators of c and λ were initially presented by
Cinlar et al. (1977) and were developed by van Noortwijk and Pandey (2003).

4.3.2.1 Maximum Likelihood Estimation

In statistics, maximum-likelihood estimation (MLE) is a method of estimating the
parameters of a statistical model. When applied to a data set and given a statisti-
cal model, MLE provides estimates for the model’s parameters.
In general, for a fixed set of data and underlying statistical model, the method
of maximum likelihood selects values of the model parameters that produce a dis-
tribution that gives the observed data the greatest probability (i.e., parameters that
maximize the likelihood function). Given that n observations are denoted by
x 1 ; x 2 ; ... ; x n ; the principle of maximum likelihood assumes that the sample data
set is representative of the population. This has a probability density function of
f x ðx 1 ; x 2 ; ... ; x n ; θÞ, and chooses that value for θ (unknown parameter) that most
likely caused the observed data to occur, i.e., once observations x 1 ; x 2 ; ... ; x n are