Repeated loadings on a hydrosystem are characterized by the time each load is applied and the behavior of time intervals between load applications. From a reliability theory viewpoint, the uncertainty about the loading and resistance variables may be classified into three categories: deterministic, random fixed, and random independent (Kapur and Lamberson, 1977). For the deterministic category, the loadings assume values that are exactly known a priori. For the random-fixed case, the randomness of loadings varies in time in a known manner. For the random-independent case, the loading is not only random, but the successive values assumed by the loading are statistically independent.
Deterministic. A variable that is deterministic can be quantified as a constant without uncertainty. A system with deterministic resistance and load implies that the behavior of the system is completely controllable, which is an idealized case. However, in some situations, a random variable can be treated as deterministic if its uncertainty is small and can be ignored.
Random fixed. A random-fixed variable is one whose initial condition is random in nature, and after its realization, the variable value is a known function of time. This can be expressed as
Xt = X0 g(T) for t > 0 (4.95)
where X0 and Xt are, respectively, the random variable X at times t = 0 and t = t, and g(T) is a known function involving time. Although Xt is a random variable, its PDF, however, is completely dependent on that of X0. Therefore, once the value of the random initial condition X0 is realized or observed, the value of subsequent time can be uniquely determined. For this case, given the PDF of X0, the PDF and statistical moments of Xt can be obtained easily. For instance, the mean and variance of Xt can be obtained, in terms of those of X0, as
E(Xt) = E(X0)g(t) for t > 0 (4.96a)
Var(Xt) = Var(X0)g2(t) for t > 0 (4.96b)
in which E(X0) and E(Xt) are the means of X0 and Xt, respectively, andVar(X0) and Var(Xt) are the variances of X0 and Xt, respectively.
Random independent. A random-independent variable, unlike the random-fixed variable, whose values occurred at different times are not only random but also independent each other. There is no known relationship between the values of X0 and Xt.
