Page 35 - Dynamic Loading and Design of Structures

Page 22

According to ISO 2394 (ISO, 1998) and Eurocode 1 (European Standard, 2000), the
characteristic value(s) of a permanent action G may be obtained as:

●one single value Gk typically the mean value, if the variability of G is small (CoV≤0.05);
●two values G , and G k,sup typically representing the 5 per cent and 95 per cent fractiles, if
k inf
the CoV cannot be considered small.

In both cases it may be assumed that the distribution of G is Gaussian.

Variable actions
For single variable loads, the form of the point in time distribution is seldom of immediate use
in design; often the important variable is the magnitude of the largest extreme load that occurs
during a specified reference period for which the probability of failure is calculated (e.g.
annual, lifetime). In some cases, the probability distribution of the lowest extreme might also
be of interest (water level in rivers/lakes).
Consider a random variable X with distribution function Fx(x). If samples of size n are
taken from the population of X : (x1, x2,…, xn), each observation may itself be considered as a
random variable (since it is unpredictable prior to observation). Hence, the extreme values of
a sample of size n are random variables, which may be written as

The probability distributions of Y and Y may be derived from the probability of the initial
1
n
variate X. Assuming random sampling, the variables X ,X ,…,X are statistically independent
n
2
1
and identically distributed as X, hence

The distribution of F Yn (y) is thus given by

(1.26)

which can be written as

(1.27)

Similar principles may be used to derive the distribution of the lowest extreme.
For a time varying load Q the distribution on the left-hand side of equation (1.27) can be
interpreted as the maximum load in a specified reference period t whereas the distribution on
r
the right-hand side represents the maximum load occurring during a much shorter period,
.
sometimes called the unit observation time τIn this case, the exponent is equal to the ratio
between the two (i.e. n=t /τand n>1). Equation (1.27) may thus be written as
r