Page 38 - Dynamic Loading and Design of Structures

Page 25

Thus, for the above example, the return period of the characteristic value of the load Qk
which represents the average time between events Q>Qk is given by

or

Return periods of 50 to 100 years are reasonable for characteristic values of variable actions
used in the design of ordinary permanent buildings. For accidental actions, a longer return
period might be appropriate, especially if ultimate or collapse limit states are considered.
)
Bearing in mind the notation introduced above for the reference (tr) and unit observation (τ
periods, the return period may be written as

(1.30)

where p is defined for a reference period tr and n>1. The last expression is asymptotically
p
correct as (1−) tends to unity, which is compatible with the notion of specifying
characteristic values on the basis of fairly rare events; note that the return period becomes
.
independent of the unit observation period τ
The probability distribution of extreme values is often closely approximated by one of the
asymptotic extreme value distributions (Types I, II and III). The characteristics of extreme
distributions depend on the initial, or parent, distribution and on the number of repetitions, n.
In general, distributions shift to the right with increasing n. Which of the three types is
relevant depends on the shape of the upper tail of the parent distribution. Of particular
importance in the context of timevarying loads is the Gumbel or Type I extreme distribution
for maxima, which is obtained if the initial distribution has an exponentially decreasing upper
tail. It has the following probability distribution function

(1.31)

where un and an are the distribution parameters. The mean and variance of Qn are related to
the distribution parameters through the following expressions

An interesting property of this distribution is that the variance is independent of the number of
repetitions (i.e. it remains constant). On the other hand, the mean value increases with the
number of repetitions. Ang and Tang (1984) present an exposition of extreme value theory as
applied to a variety of civil engineering problems.

Accidental actions