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Figure 1.7 Realization of an FBC process.

frequency characteristics (i.e. the base period assumed for stationary events) of loads
considered within any particular combination. This is further discussed in Section 1.5.

1.4.3 Introduction to crossing theory

In considering a time dependent safety margin (i.e. M(t)=g(X(t)), the problem is to establish
the probability that M(t) becomes zero or less in a reference time period, tL. As mentioned
previously, this constitutes a so-called ‘crossing’ problem. The time at which M(t) becomes
less than zero for the first time is called the ‘time to failure’ and is a random variable, see
Figure 1.8. The probability that M(t)≤0 occurs during t is called the ‘first-passage’
L
probability. Clearly, it is identical to the probability of failure during time tL.
The determination of the first passage probability requires an understanding of the theory of
random processes. Herein, only some basic concepts are briefly introduced in order to see
how the methods described above have to be modified in dealing with crossing problems.
Melchers (1999) provides a detailed treatment of time-dependent reliability aspects.
The first-passage probability Pf (t) during a period [0, tL] is

(1.23)

where signifies that the process X(t) starts in the safe domain and N(tL) is the
number of outcrossings in the interval [0, t . The second probability term is equivalent to 1—
L
P (0), where P (0) is the probability of failure at t=0. Equation (1.23) can be rewritten as
f
f
(1.24)

from which different approximations may be derived depending on the relative magnitude of
the terms. A useful bound is

(1.25)