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 of the combined effects of inherent randomness of external loads and various uncertainties involved in the analysis, design, construction, and operational procedures. Hence, to evaluate the probability that a hydrosystem will function as designed requires uncertainty and reliability analyses.
As discussed in Sec. 1.5, failure of an engineering system can be defined as the load L (external forces or demands) on the system exceeding the resistance R (strength, capacity, or supply) of the system. The reliability ps is defined as the probability of safe (or nonfailure) operation, in which the resistance of the structure exceeds or equals to the load, that is,
Ps = P (L < R) (4.1)
in which P(■) denotes the probability. Conversely, failure probability pf can be computed as
Pf = P (L > R) = 1 — ps (4.2)
The definitions of reliability and failure probability, Eqs. (4.1) and (4.2), are equally applicable to component reliability, as well as total system
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reliability. In hydrosystems engineering analyses, the resistance and load frequently are functions of several stochastic basic variables, that is, L = g( Xl) = g( X ь X 2,…, Xm) and R = h( Xr ) = h( Xm+1, Xm+2,…, Xk ), where Xі, X2,…, XK are stochastic basic variables defining the load function g(Xl) and the resistance function h(Xr ). Accordingly, the failure probability and reliability are functions of stochastic basic variables, that is,
ps = P [g(Xl) < h(Xr)] (4.3)
Note that the foregoing presentation of load and resistance in reliability analysis should be interpreted in a very general context. For example, in the design and analysis hydrosystems infrastructures, such as urban drainage systems, the load could be the inflow to the sewer system, whereas the resistance is the sewer conveyance capacity; in water quality assessment, the load may be the concentration or mass of pollutant entering the environmental system, whereas the resistance is the permissible pollutant concentration set by water quality regulations; in the economic analysis of a hydrosystem, the load could be the total cost, whereas the resistance is the total benefit.
Evaluation of reliability or failure probability by Eqs. (4.1) through (4.3) does not consider the time-dependent nature of the load and resistance if statistical properties of the elements in Xl and Xr do not change with time. This procedure generally is applied when the performance of the system subject to a single worst-load event is considered. From the reliability computation viewpoint, this is referred to as static reliability analysis.
In general, a hydrosystem infrastructure is expected to serve its designated function over an expected period of time. Engineers frequently are interested in knowing the reliability of the structure over its intended service life. In such circumstances, elements of service period, randomness of load occurrences, and possible change in resistance characteristics over time must be considered. Reliability models incorporating these elements are called time-dependent reliability models (Kapur and Lamberson, 1977; Tung and Mays, 1980; Wen, 1987). Computations of the time-dependent reliability of a hydrosystem infrastructure initially require the evaluation of static reliability. Sections 4.3 through
4.6 describe methods for static reliability analysis, and Sec. 4.7 briefly describes some basic methods for dealing with the time-dependent nature of reliability analysis.
As discussed in the preceding chapters, the natural randomness of hydrologic and geophysical variables, such as flood and precipitation, is an important part of the uncertainty in the design of hydrosystems infrastructures. However, other uncertainties also may be significant and should not be ignored. Failure to account for the other uncertainties in the reliability analysis in the past (as discussed in Sec. 1.3) hindered progress in evaluation of failure probability associated with hydrosystems infrastructures. As noted by Cornell (1969) with respect to traditional frequency-based analyses of system safety:
It is important in engineering applications that we avoid the tendency to model only those probabilistic aspects that we think we know how to analyze. It is far better to have an approximated model of the whole problem than an exact model of only a portion of it.
