Time-dependent reliability models

Reliability computations for time-dependent models can be made for determin­istic and random cycle times. The development of a model for deterministic cycles is given first, which naturally leads to the model for random cycle times.

Number of occurrences is deterministic. Consider a hydrosystem with a fixed resistance (or capacity) R = r subject to n repeated loads L1, L2,…, Ln. When the number of loads n and system capacity r are fixed, the reliability of the system after n loadings ps(n, r) can be expressed as

Ps(n, r ) = P [(L1 < r ) n (L2 < r ) П—П (Ln < r )] = P (Lmax < Г ) (4.100)

where Lmax = max{L1, L2,…, Ln}, which also is a random variable. If all ran­dom loadings L are independent with their own distributions, Eq. (4.100) can be written as

Ps(n, r) = [FLi(r)] (4.101)

i=1

where FLi (r) is the CDF of the ith load. In the case that all loadings are gener­ated by the same statistical process, that is, all Ls are identically distributed with FLi(r) = FL(r), for i = 1, 2,…, n, Eq. (4.101) can further be reduced to

Подпись: (4.102)Ps(n, r) = [FL(r )]

Подпись: Ps(n) Time-dependent reliability models Подпись: (4.103)

If the resistance of the system also is a random variable, the system reliability under the fixed number of loads n can be expressed as

Number of occurrences is random. Since the loadings to hydrosystems are re­lated to hydrologic events, the occurrence of the number of loads, in general, is uncertain. The reliability of the system under random loading in the specified time interval [0, t] can be expressed as

TO

Подпись: (4.104)Ps(t) = n (t |n) Ps(n)

n=0

in which n(t |n) is the probability of n loadings occurring in the time interval [0, t]. A Poisson distribution can be used to describe the probability of the num­ber of events occurring in a given time interval. In fact, the Poisson distribution has been found to be an appropriate model for the number of occurrences of hy­drologic events (Clark, 1998; Todorovic and Yevjevich, 1969; Zelenhasic, 1970). Referring to Eq. (2.55), n(t |n) can be expressed as

e Xt ( X t )n

n (t |n) =————- (— (4.105)

n!

where X is the mean rate of occurrence of the loading in [0, t], which can be estimated from historical data.

Substituting Eq. (4.105) in Eq. (4.104), the time-dependent reliability for the random independent load and random-fixed resistance can be expressed as

Подпись: e Xt (Xt)n" n! Подпись: Ps(t) = n= 0 Подпись: Ps (n, r) fR (r) drПодпись: 0(4.106)

Under the condition that random loads are independently and identically dis­tributed, Eq. (4.106) can be simplified as

n TO

Ps (t) = e- t [1-FL(r)] fR (r) dr (4.107)

0

Updated: 18 ноября, 2015 — 1:00 пп