A hydraulic structure placed in a natural environment over an expected service period is subject to repeated application of loads of varying intensities. The magnitude of load intensity and the number of occurrences of load are, in general, random by nature. Therefore, probabilistic models that properly describe the stochastic mechanisms of load intensity and load occurrence are essential for accurate evaluation of the time-dependent reliability of hydrosystems.
Probability models for load intensity. In the great majority of situations in hydrosystems reliability analysis, the magnitudes of load to be imposed on the system are continuous random variables. Therefore, univariate probability distributions described in Sec. 2.6 potentially can be used to model the intensity of a single random load. In a case in which more than one type of load is considered in the analysis, multivariate distributions should be used. Some commonly used multivariate distribution models are described in Sec. 2.7.
The selection of an appropriate probability model for load intensity depends on the availability of information. In a case for which sample data about the load intensity are available, formal statistical goodness-of-fit tests (see Sec. 3.7) can be applied to identify the best-fit distribution. On the other hand, when data on load intensity are not available, selection of the probability distribution for modeling load intensity has to rely on the analyst’s logical judgment on the basis of the physical processes that produce the load.
Probability models for load occurrence. In time-dependent reliability analysis, the time domain is customarily divided into a number of intervals such as days, months, or years, and the random nature of the load occurrence in each time interval should be considered explicitly. The occurrences of load are discrete by nature, which can be treated as a point random process. In Sec. 2.5, basic features of two types of discrete distributions, namely, binomial and Poisson distributions, for point process were described. This section briefly summarizes two distributions in the context of modeling the load-occurrences. Other load — occurrence models (e. g., renewal process, Polya process) can be found elsewhere (Melchers, 1999; Wen, 1987).
Bernoulli process. A Bernoulli process is characterized by three features:
(1) binary outcomes in each trial, (2) constant probability of occurrence of outcome in each time interval, and (3) the outcomes are independent between trials. In the context of load-occurrence modeling, each time interval represents a trial in which the outcome is either the occurrence or nonoccurrence of the load (with a constant probability) causing failure or nonfailure of the system. Hence the number of occurrences of load follows a binomial distribution, Eq. (2.51), with parameters p (the probability of occurrence of load in each time interval) and n (the number of time intervals). It is interesting to note that the number of intervals until the first occurrence T (the waiting time) in a Bernoulli process follows a geometric distribution with the PMF
g(T = t) = (1 — p)t-1 p (4.97)
The expected value of waiting time T is 1/p, which is the mean occurrence period. It should be noted that the parameter p depends on the time interval used.
Poisson process. In the Bernoulli process, as the time interval shrinks to zero and the number of time intervals increases to infinity, the occurrence of events reduces to a Poisson process. The conditions under which a Poisson process applies are (1) the occurrence of an event is equally likely at any time instant,
(2) the occurrences of events are independent, and (3) only one event occurs at
a given time instant. The PMF describing the number of occurrences of loading in a specified time period (0, t] is given by Eq. (2.55) and is repeated here:
e Xt ( Xt )x
Px (xX, t) =—— for x = 0,1, …
x!
in which X is the average time rate of occurrence of the event of interest. The interarrival time between two successive occurrences is described by an exponential distribution with the PDF
ft(tX) = Xe-t for t > 0 (4.98)
Although condition (1) implies that the Poisson process is stationary, it can be generalized to a nonstationary Poisson process, in which the rate of occurrence is a function of time X(t). Then the Poisson PMF for a nonstationary process can be written as
P(x = x) [/0 x(t) d t]x exp [-/0 x(t) d t]
x!
Equation (4.99) allows one to incorporate the seasonality of many hydrologic events.
