Titel_SS06

In case of non-stationary Normal processes it can be shown, see e.g. Madsen et al. (1986), that

the MOR may be determined as:

(()) X ()(( ) ( ))

XX X

t

 

E --

 

(^) ,





 (8.13)

where

() () ()

()

X

X

t tt

t

 E 



 

.

In the foregoing only the simplest cases have been considered, i.e. the case of scalar valued

stochastic processes where the failure domain is given as a simple one-sided boundary to the

sample space of the process. However, the presented theory may easily be generalised to

vector valued stochastic processes and also to random fields. In the specialist literature, see

e.g. Bryla et al. (1991), Faber et al. (1989) and Faber (1989) more general situations are

considered and efficient approximate solutions are given.

8.3 Approximations to the Time Variant Reliability Problem

The exact assessment of the time variant reliability problem, i.e. the first passage problem, is

hardly possible for the types of stochastic processes which are relevant for civil engineering

purposes and it is therefore necessary to approach the problem by means of approximations

and or simplifications of the considered problem.

One of the most commonly adapted approximations is to assess the probability of failure

conditional on events which may be assumed to follow a Poisson process. This could e.g. be

relevant when assessing the probability of failure in regard to earthquakes, floods, ship

impacts, explosions, fires and other rare events. In this case the reliability problem may be

assessed using the Poisson spike process model in which case the first passage problem is

readily solved using Equation (8.5).

For phenomena occurring continuously in time such as stresses in a steel structure subject to

wave loading the Poisson spike model is no longer appropriate as no particular event can be

said to be more critical than others. Failure could e.g. occur as a result of a combination of

fatigue crack growth and extreme stresses and whether the structure will fail due to instable

crack growth or plastic rupture depends on the crack geometry and the stresses at any given

time. Also in this case help may be found from the results derived from Poisson processes. It

can be shown that under certain regularity conditions, which are usually fulfilled then the

failure events will occur as realisations of a simple Poisson process in which case the result of

Equation (8.5) may readily be applied.

Non-ergodic Components and Random Sequences

Without going into details it is, however, emphasised that care should be taken when applying

the results in Equation (8.5) as the characteristics of the considered problem could lead to a

gross misuse of these. This is for engineering purposes typically the case when in addition to