8.1 Introduction
Both resistances and loads may in principle be functions of both time and or space. This is e.g.
the case when considering the concrete compressive strength in a larger structure where both
the mean value and the standard variation of the compressive strength may vary from area to
area due to variations in the concrete production but also due to variations in the execution
and curing. Another example is e.g. the wind loading acting on high rise buildings or bridge
pylons where mean values and standard deviations of the wind pressure not only vary as a
function of the location on the structure but also as a function of time. In both mentioned
examples the additional aspect of stochastic dependency in both time and space also plays an
important role as this is determining for the scale of variation.
As a consequence of the time/spatial variability resistances and loads may not always be
appropriately modelled by time invariant basic random variables and time variant reliability
analysis is thus an important issue in structural reliability analysis.
Previously the task of estimating the probability of failure has been considered in cases where
the uncertain resistances and load characteristics:
can be assumed to be time invariant
are indeed time variant but exhibit sufficient ergodicity such that e.g. by use of extreme
value considerations time invariant probabilistic idealisations may be established.
In the present chapter the cases are considered where time is a direct parameter in the
probabilistic modelling of the resistance and load characteristics and where it is not
immediately possible to idealise the probabilistic modelling into a time invariant formulation.
The following presentation of time variant reliability analysis is by no means complete but
aims to provide a basis for understanding when it is appropriate to represent time variant
problems by equivalent time invariant problems as well as how time variant problems may be
approached in a practical way by approximate methods.
8.2 General Formulation
For structural reliability problems where the uncertainties are modelled by stochastic
processes the event of failure can normally be related to the event that the structural response
process (stresses, displacements, etc.) has an excursion out of a safe domain bounded by some
critical level or sets of levels, i.e. failure domain surfaces.
In order to introduce the basic concepts relating to the problem of assessing the probability of
such events the cases of scalar valued stochastic processes are considered. To this end first a
special class of discrete processes, namely filtered Poisson processes are introduced and
thereafter a special class of continuous processes, namely the Normal processes are
considered.