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When nonnormal random variables are involved, it is advisable to transform them into equivalent normal variables. Rackwitz (1976) and Rackwitz and Fiessler (1978) proposed an approach that transforms a nonnormal distribu­tion into an equivalent normal distribution so that the probability content is preserved. That is, the value of the CDF of the transformed equivalent nor­mal […]

Hasofer-Lind algorithm. In the case that X are independent normal stochastic basic variables, standardization of X according to Eq. (4.30) reduces them to independent standard normal random variables Z’ with mean 0 and covariance matrix I, with I being a K x K identity matrix. Referring to Fig. 4.8, based on the geometric characteristics at […]

In cases for which several stochastic basic variables are involved in a perfor­mance function, the number of possible combinations of such variables satis­fying W (x) = 0 is infinite. From the design viewpoint, one is more concerned with the combination of stochastic basic variables that would yield the lowest reliability or highest failure probability. The […]

The main thrust of the AFOSM method is to improve the deficiencies associated with the MFOSM method, while keeping the simplicity of the first-order approx­imation. Referring to Fig. 4.3, the difference in the AFOSM method is that the expansion point x+ = (xL*, xR+) for the first-order Taylor series is located on the failure surface […]

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 […]

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 […]

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 […]

4.1 Basic Concept The design of a hydrosystem involves analyses of flow processes in hydrology and hydraulics. In a multitude of hydrosystems engineering problems, uncer­tainties in data and in theory, including design and analysis procedures, war­rant a probabilistic treatment of the problems. The risk associated with the potential failure of a hydrosystem is the result […]

Viessman et al. (1977, p. 158) noted that “usually, the length of record as well as the design life for an engineering project are relatively short compared with ge­ologic history and tend to temper, if not justify, the assumption of stationarity.” On the other hand, Klemes (1986) noted that there are many known causes for […]

Most often frequency analysis is applied for the purpose of estimating the mag­nitude of truly rare events, e. g., a 100-year flood, on the basis of short data series. Viessman et al. (1977, pp. 175-176) note that “as a general rule, fre­quency analysis should be avoided… in estimating frequencies of expected hydrologic events greater than […]