## R

nn where oR and oS are the standard deviations of the resistance and the ¿th load, respectively. Equations 12.3 and 12.4 indicate that several parameters are needed to evaluate the reliability-based resistance and load factors and the nature of mathematical sophistication required to evaluate them, even when all the variables are assumed to be normal.

When R and S are independent lognormal random variables, expressions similar to Equations 12.3 and 12.4 can be derived [14]. In general, R and S are not independent normal or lognormal random variables, and the probability of failure using Equation 12.2 is expected to be very challenging. They are functions of many different random variables, and their exact probabilistic characteristics are very difficult to evaluate. Historically, several methods with various degrees of sophistication were proposed to evaluate the probability of failure or reliability [14]. Reliability is calculated as (1.0 — probability of failure). It is not possible to discuss the details of all the methods here. They are simply identified here for the completeness of discussion.

Initially, in the late 1960s, the first-order second-moment (FOSM) method, also known as the mean value first-order second-moment (MVFOSM), was proposed to calculate the probability of failure neglecting the distributional information on the random variables present in the problem. This deficiency was overcome by the advanced first-order second-moment method where all the variables are assumed to be normal and independent as proposed by Hasofer and Lind [15]. Rackwitz [16] proposed a more general formulation applicable to different types of distributions. Currently, it is the most widely used reliability evaluation technique. Using this concept, the probability of failure has been estimated using two types of approximations to the limit state at the design point (defined in the following section): first order (leading to the name first-order reliability method or FORM) and second order (leading to the name second-order reliability method or SORM). Since FORM is a commonly used reliability evaluation technique, it is discussed in more detail below. A person without a sophisticated background in probability and statistics can use simulation to evaluate the underlying risk or reliability, as discussed in Section 12.7.

### 12.4.1 First-Order Reliability Method

The basic idea behind reliability-based structural design is to design a structural member satisfying several performance criteria and considering the uncertainties in the relevant load- and resistance-related random variables, called the basic variables X;. Since the R and S random variables in Equation 12.2 are functions of many other load- and resistance-related random variables, they are generally treated as basic random variables. The relationship between the basic random variables and the performance criterion, known as the performance or limit state function, can be mathematically represented as

The failure surface or the limit state of interest can then be defined as Z = 0. The limit state equation plays an important role in evaluating reliability using FORM. It represents the boundary between the safe and unsafe regions and a state beyond which a structure can no longer fulfill the function for which it was designed. Assuming R and S are the two basic random variables, the limit state equation, and the safe and unsafe regions are shown in Figure 12.2. A limit state equation can be an explicit or implicit function of the basic random variables and can be linear or nonlinear. Reliability estimation using explicit limit state functions is discussed here. Haldar and Mahadevan [13] discussed reliability evaluation techniques for implicit limit state functions.

Two types of performance functions are generally used in structural engineering: strength and serviceability. Strength performance functions relate to the safety of the structures and serviceability performance functions are related to the serviceability (deflection, vibration, etc.) of the structure. The reliabilities underlying the strength and serviceability performance functions are expected to be different.

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