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In the first-order methods, the performance function W (X), defined on the basis of the loading and resistance functions g(XL) and h(XR), are expanded in a Taylor series at a reference point. The second – and higher-order terms in the series expansion are truncated, resulting in an approximation involving only the first two statistical moments of the variables. This simplification greatly en­hances the practicality of the first-order methods because, in many problems, it is rather difficult, if not impossible, to find the PDF of the variables, whereas it is relatively simple to estimate the first two statistical moments. The pro­cedure is based on the first-order variance estimation (FOVE) method, which is summarized below...

From Eqs. (4.1) and (4.4) one realizes that the computation of reliability requires knowledge of the probability distributions of the load and resistance or of the performance function W. In terms of the joint PDF of the load and resistance, Eq. (4.1) can be expressed as

in which f R L(r, t) is the joint PDF of random load L and resistance R, r and t are dummy arguments for the resistance and load, respectively, and (r1, r2) and (t1, t2) are the lower and upper bounds for the resistance and load, respectively. The failure probability can be computed as

This computation of reliability is commonly referred to as load-resistance interference.

TABLE 4.1 Reliability Formulas for Selected Distributions

Distribution of W

Coefficient of v...

In reliability analysis, Eq. (4.3) alternatively can be written in terms of a per­formance function W (X) = W (XL, XR) as

ps = P [W(Xl, Xr) > 0] = P [W(X) > 0] (4.4)

in which X is the vector of basic stochastic variables in the load and resistance functions. In reliability analysis, the system state is divided into the safe (sat­isfactory) set defined by W (X) > 0 and the failure (unsatisfactory) set defined by W (X) < 0 (Fig. 4.1). The boundary that separates the safe set and failure set is a surface, called the failure surface, defined by the function W(X) = 0, called the limit-state function. Since the performance function W(X ) defines the condition of the system, it is sometimes called system-state function.