The reliability of a structural component constitutes the basis for performing system reliability of larger structure. In general, a component can fail in one of several failure modes. The treatment of multiple failure modes requires modeling the component behavior as a system. In addition, the system can be defined as a collection or an assemblage of several components that serves some function or purpose (3). A multi-component system can fail in several failure modes. Once the reliability or probability of failures for all of the components that make up the whole systems is evaluated, system reliability can be performed on the overall system. The theory of system reliability is beyond the scope of this chapter. Numerous excellent books and references have been written for the subject.
The reliability of a structural component can be defined as the probability that the component meets some specified demands. For example, the reliability of a structural component such as a beam can be defined as the probability that structural strength of the beam (i.e., ultimate moment capacity) exceeds the applied load (i.e., moment due to the total combined loads). The first step in evaluating the reliability or probability of failure of a structural component is to decide on specific performance function g and the relevant load and resistance variables. The generalized form of the performance function can be expressed as
where g is the performance function, X1, X2, …, Xn are n basic random variables for R and L; and f(.) is a function that gives the relationship between R and L and the basic random variables. The failure in this case is defined in the region where g is less than zero (see Figure 20.6) or R is less than L, that is
whereas the reliability is defined in the region where g is greater than zero (Figure 20.6) or R is greater than L, that is
The limit state is defined when g = 0.
Due to the variability in both strength and loads, there is always a probability of failure that can be defined as
The reliability of a structural component can be defined as the probability that the component meets some specified demands for a specified time frame. Mathematically, it can be given by the following expression:
where Pf = probability of the system or component and Rc = reliability of the component. According to probability theory, since failure and non-failure (or success) constitute two complementary events, therefore,
For the general case, where the basic random variables can be correlated, the probability of failure for the component can be determined by solving the following integral:
Figure 20.6 - Frequency Distribution of Strength R and Load L
where fX = joint probability density function (PDF) of the random vector X = [X1, X2, …, Xn]; and the integration is performed over the region where g = f(.) < 0. The computation of Pf by Equation 13 is called the “full distributional approach” and can be considered the fundamental equation of reliability analysis (22)
In general, the determination of the probability of failure by evaluating the integral of Equation 13 can be a difficult task. In practice, the joint probability density function fX is hard to obtain. Even, if the PDF is obtainable, evaluation of the integral of Equation 13 requires numerical methods. In practice, there are alternative methods for evaluating the above-mentioned integral through the use of analytical approximation procedures such as the First-Order Reliability Method (FORM), which is the focus of our discussion in the next section.
The many advantages and benefits of using reliability-based design methods include the following:
1) They provide the means for the management of uncertainty in loading, strength, and degradation mechanisms.
2) They provide consistency in reliability.
3) They result in efficient and possibly economical use of materials.
4) They provide compatibility and reliability consistency across materials, such as, steel grades, aluminum and composites.
5) They allow for future changes as a result of gained information in prediction models, and material and load characterization.
6) They provide directional cosines and sensitivity factors that can be used for defining future research and development needs.
7) They allow for performing time-dependent reliability analysis that can form the bases for life expectancy assessment, life extension, and development of inspection and maintenance strategies.
8) They are consistent with other industries, AISC, ASHTO, ACI, API, ASME, …, etc.
9) They allow for performing system reliability analysis.