Page 100 - Reliability and Maintainability of In service Pipelines

Methods for Structural Reliability Analysis 89

2
contribution (α ) of each random variable (x) to the variance of the limit state
x
function is introduced as follows (Ahammed and Melchers, 1994):
2

@ G
σ x
2
α 5 @ x ð3:9Þ
x σ 2
G
2
where σ x is standard deviation of the random variable x and σ is the variance of
G
2
the limit state function. Variables with higher values of α contribute more in
x
limit state function than other variables; therefore more focus and study needs to
be carried out to determine the accurate values for such variables.
3.5.2 SENSITIVITY RATIO
A method of sensitivity analysis applied in many different models in science,
engineering, and economics is the SR, also known as the elasticity equation. The
ratio is equal to the percentage change in output (e.g., probability of failure)
divided by the percentage change in input for a specific input variable, as shown
in the following equation (EPA 540, 2001):

3 100%
Y 2 2 Y 1
Y 1
ð3:10Þ
SR 5
3 100%
X 2 2 X 1
X 1
where Y 1 5 the baseline value of the output variable using baseline values of
input variables;
Y 2 5 the value of the output variable after changing the value of one input
variable;
X 1 5 the baseline point estimate for an input variable;
X 2 5 the value of the input variable after changing X 1 :

Risk estimates are considered most sensitive to input variables that yield the
highest absolute value for SR. The basis for this equation can be understood by
examining the fundamental concepts associated with partial derivatives. In fact,
SR is equivalent to the normalized partial derivative. Variables with higher values
of sensitivity ratios are more effective on the limit state function or the probabil-
ity of failure (EPA 540, 2001).

3.5.3 OMISSION SENSITIVITY FACTOR

In computing the reliability index, the basic random variables X can be trans-
formed into standardized normal space U and the limit state function, GðX; tÞ, can
be transformed to gðU; tÞ (Melchers, 1999).