the stochastic process components other uncertainties are involved such as stochastic
sequences and non-ergodic random variables.
Consider the example of wave loads on a steel offshore structure. The stresses at any given
time due to the wave loads may be considered to be realisations of a stochastic process.
However due to the characteristics of wave loads on offshore structures, the wave loads may
only be appropriately modelled by stationary stochastic processes for given values of the sea-
states, i.e. the significant wave height and the corresponding zero crossing period. Whereas
the stochastic process describing the wave loads for a given sea-state may be considered a
short-term statistical description of the wave loads the statistical description of sea-states
given e.g. in terms of scatter diagrams providing the frequency of occurrence of different
combinations of significant wave heights and corresponding zero crossing periods is referred
to as a long term statistical wave load modelling. In this context the sea-states may be
considered as random sequences defining the characteristics of the short-term wave load
processes. The variation in time of the random sequence is furthermore much slower than the
wave load process for a given sea-state. Furthermore, the event of failure will only occur if the
stresses within any given sea-state exceed the remaining capacity of the structure. The
capacity may in turn be uncertain itself but does in general not vary or varies very slowly in
time. Such variables are therefore often referred to as being non-ergodic components.
The above example relates directly to offshore engineering and the particulars of the
probabilistic modelling of wave loading. However, other kinds of loads may appropriately be
modelled in the same manner, including traffic loads where the intensity of trucks and the
vehicle loads may exhibit systematic variations as a function of the time of the day or the days
in the week. By introduction of random sequences of truck intensities and truck weights the
stresses in e.g. a bridge in shorter time intervals with given “traffic state” (intensity and
weights) can be assumed to be ergodic stochastic processes. The probability of failure for each
of these traffic states may thus be assessed.
In situations as described above involving in addition to the stochastic process component also
random sequences and non-ergodic components the approximations provided in (8.5) should
be applied in the following way, see also Schall et al. (1991)
1 1
0( ) 1 exp( ( , , ) )t
FtTRQE E 3RQd 38 8
(^89)
99 (8.14)
where EERQ , refer to the expected value operations over the non-ergodic random
variables and the random sequences respectively. Details on how to proceed on the numerical
evaluation of Equation (8.14) may be found in e.g. Strurel (1998).
Situations to Differentiate in Practical Cases
For time variant reliability problems where the resistance is deteriorating and the loading is
time invariant the reliability assessment may be performed by considering the strength
characteristics corresponding to the end point of the service life of the structure.