Page 30 - Dynamic Loading and Design of Structures

Page 17

interest is

(1.20)

If the loads are independent, replacing X by max {p (t)}+max {p (t)} leads to very
T
1
T
2
conservative results. However, the distribution of X can be derived in few cases only. One
possible way of dealing with this problem, which also leads to a relatively simple
deterministic code format, is to replace X with the following

(1.21)

This rule (Turkstra’s rule) suggests that the maximum value of the sum of two independent
load processes occurs when one of the processes attains its maximum value. This result may
be generalized for several independent time varying loads. The conditions which render this
rule adequate for failure probability estimation are discussed in standard texts. From a
theoretical point, the rule leads to an underestimation of the probability of failure, since it is
assumed that failure must be associated with the maximum of at least one load process,
whereas in reality failure can also occur in other instances.
The failure probability associated with the sum of a special type of independent identically
distributed processes (so-called Ferry Borges-Castanheta (FBC) process) can be calculated in
a more accurate way, as will be outlined below. Other results have been obtained for
combinations of a number of other processes, starting from Rice’s barrier crossing formula.
The FBC process is generated by a sequence of independent identically distributed random
variables, each acting over a given (deterministic) time interval. This is shown in Figure 1 .7
where the total reference period T is made up of ni repetitions where ni=T/Ti. Hence, the FBC
process is a rectangular pulse process with changes in amplitude occurring at equal intervals.
Because of independence, the maximum value in the reference period T is given by

(1.22)

When a number of FBC processes act in combination and the ratios of their repetition
numbers within a given reference period are given by positive integers it is, in principle,
possible to obtain the extreme value distribution of the combination through a recursive
formula. More importantly, it is possible to deal with the sum of FBC processes by
implementing the Rackwitz-Fiessler algorithm in a FORM/ SORM analysis.
A deterministic code format, compatible with the above rules, leads to the introduction of
combination factors for each time varying load. In principle, these factors express ratios
between fractiles in the extreme value and point in time distributions so that the probability of
exceeding the design value arising from a combination of loads is of the same order as the
probability of exceeding the design value caused by one load. For time varying loads, they
depend on distribution parameters, target reliability, FORM/SORM sensitivity factors and on
the