Overview

Note

Remember, the concept of component reliability analysis is introduced in the textbook here. The pages in this workbook provide additional information to help you through the exercises and workshop assignments related to this material.

There are several key ingredients needed for a component reliability analysis:

1. Random variables, \(X\), each of which has a marginal distribution

2. A multivariate probability distribution, \(f_X(x)\)

3. A function that takes the random variables as inputs and describes the performance of the component of interest, \(q_X(x)\) (i.e., a function of random variables)

4. A criteria that describes a “region of interest,” a subset of the random variables, referred to as \(\Omega\)

5. An algorithm that solves for the probability of observing the conditions described by \(\Omega\), defined by the following integral:

\[ P[\Omega] = \int_{\Omega} f_X(x) \; \mathrm{d} X \]

In many civil engineering applications, we wish to evaluate failure probability (the inverse of reliability) of a particular object (i.e., the component). If we can formulate a function of random variables such that the output defines failure of the component, then \(P[\Omega]\) is the failure probability, \(p_f\).

By convention, we prefer to write our function of random variables \(q_X(x)\) such that values less than zero denote a failed condition; we also use the notation \(g_X(x)\). The failure probability integral thus becomes:

\[ p_f = P[g_X(x)<0] = \int_{\Omega} f_X(x) \; \mathrm{d}X \]

The equation \(g_X(x)\) is called the limit state function.

Note that if we can evaluate the distribution of \(g_X(x)\) (denoted \(f_g(x)\)), then the integral is simply:

\[ p_f = P[g<0] = F_g(g=0) = \int_{-\infty}^{0} f_g(x) \; \mathrm{d}X \]

In some cases this can be done analytically, but for most practical cases it is not possible.