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 defined by the limit-state equation, W(x) = 0. In other words, the failure surface is the boundary that separates the system performance from being unsatisfactory (unsafe) or being satisfactory (safe), that is,
|
{ |
> 0, system performance is satisfactory (or safe region);
= 0, limit-state surface (or failure surface);
< 0, system performance is unsatisfactory (or failure region).
The AFOSM method has been applied to various hydrosystem engineering problems, including storm sewers (Melching and Yen, 1986), dams (Cheng et al., 1982; 1993), sea dikes and barriers (Vrijling, 1987; 1993), freeboard design (Cheng et al., 1986a), bridge scour (Yen and Melching, 1991; Chang, 1994), rainfall-runoff modeling (Melching et al., 1990; Melching, 1992); groundwater pollutant transport (Sitar et al., 1987; Jang et al., 1990), open channel design (Easa, 1992), sediment transport (Bechtler and Maurer, 1992), backwater computations (Cesare, 1991; Singh and Melching, 1993), and water quality modeling (Tung, 1990; Melching and Anmangandla, 1992; Melching and Yoon, 1996; Han et al., 2001; Tolson et al., 2001).
4.1.1 Definitions of stochastic parameter spaces
Before discussing the AFOSM methods, a few notations with regard to the stochastic basic variable space are defined first. In general, the original
stochastic basic variables X could be correlated, non-normal random variables having a vector of mean ix = (pxi, /гХ2,…, pxK)t and covariance matrix Cx as shown in Sec. 2.7.2. The original random variables X can be standardized as
X’ = D-1/2(X — fix) (4.30)
in which X’ = (X^, X2,…, XK)t is a vector of correlated, standardized random variables, and Dx = diag(o2, a|,…, oK) is an K x K diagonal variance matrix. Through the standardization procedure, each standardized variable X’ has the mean zero and unit standard deviation. The covariance matrix of X’ reduces to the correlation matrix of the original random variables X, that is, Cx> = Rx, as shown in Sec. 2.7.2. Note that if the original random variables X are nonnormal, the standardized ones X’ are nonnormal as well. Because it is generally easier to work with the uncorrelated variables in the reliability analysis, the correlated random variables X are often transformed into uncorrelated ones U = T (X), with T ( ) representing transformation, in general. More specifically, orthogonal transforms often are used to obtain uncorrelated random variables from the correlated ones. Two frequently used orthogonal transforms, namely, Cholesky decomposition and spectral decomposition, for dealing with correlated random variables are described in Appendix 4B. In probability evaluation, it is generally convenient to deal with normal random variables. For this reason, orthogonal transformation, normal transformation, and standardization procedures are applied to the original random variables X to obtain independent, standardized normal random variables Z’. Hence this chapter adopts X for stochastic basic variables in the original scale, X’ for the standardized correlated stochastic basic variables, U for the uncorrelated variables, and Z and Zrespectively, for correlated and independent, standardized normal stochastic basic variables.
