Figure 7.3: Example of the variance reduction factor according to Madsen et al. [25].
Figure 7.3 clearly shows that the variance of the stochastic field mostly depends on the
influence area A, whereas the type of the covariance function and (A) have minor
importance. Considering realistic influence areas, it is evident that the influence of the
variance of the stochastic field is negligible. Therefore, the equivalent uniformly distributed
load can be approximated estimated by:
E Qequ mq and
2
Var Q equ V (7.10)It has been found that the sustained live load is best represented by a Gamma distribution. In
particular, in the important upper tail, the Gamma distribution describes the observed data
better than the Normal and the Lognormal distribution. In Melchers (1987), the type I extreme
value distribution is also suggested for the representation of the maximum sustained live
loading corresponding to a given reference period. For reasons of numerical convenience the
type I extreme value distribution is often used instead of the Gamma distribution.
If it can be assumed that load changes occur as events of a Poisson process (see section 2.10)
with rate ( the probability distribution function of the maximum load within a given
reference period T is given by the exponential distribution:
FxQ,max() exp((TFx(1 Q()) (7.11)where is the so-called random point in time probability distribution function of the load
(the probability distribution function of the maximum load in a reference period equal to
FxQ()1/().
Although, transient live load events normally occur in the form of concentrated loads,
transient loads are usually represented in the probabilistic modelling in the form of a
stochastic field (JCSS (2001)). Therefore the following moments for an equivalent uniformly
distributed load Pequ due to transient loads may be derived as:
E Pequ mp and
2
Var P equ V (7.12)