Page 21 - Dynamic Loading and Design of Structures

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1.2.4 Computation of structural reliability

An important class of limit states are those for which all the variables are treated as time
independent, either by neglecting time variations in cases where this is considered acceptable
or by transforming time dependent processes into time invariant variables (e.g. by using
extreme value distributions). For these problems so-called asymptotic or simulation methods
may be used, described in a number of reliability textbooks (e.g. Ang and Tang, 1984;
Ditlevsen and Madsen, 1996; Madsen et al., 1986; Melchers, 1999; Thoft-Christensen and
Baker, 1982).

Asymptotic approximate methods
Although these methods first emerged with basic random variables described through
‘second-moment’ information (i.e. with their mean value and standard deviation, but without
assigning any probability distributions), it is nowadays possible in many cases to have a full
description of the random vector X (as a result of data collection and probabilistic modelling
studies). In such cases, the probability of failure could be calculated via first or second order
reliability methods (FORM and SORM respectively). Their implementation relies on:

(1) Transformation techniques

(1.11)

where U , U ,…, U are independent standard normal variables (i.e. with zero mean value and
1
2
n
unit standard deviation). Hence, the basic variable space (including the limit state function) is
transformed into a standard normal space, see Figures 1.2(a) and 1.2(b). The special
properties of the standard normal space lead to several important results, as discussed below.

(2) Search techniques
In standard normal space, see Figure 1.2(b), the objective is to determine a suitable checking
point: this is shown to be the point on the limit—state surface which is closest to the origin,
the so-called ‘design point’. In this rotationally symmetric space, it is the most likely failure
point, in other words its co-ordinates define the combination of variables that are most likely
to cause failure. This is because the joint standard normal density function, whose bell-shaped
peak lies directly above the origin, decreases exponentially as the distance from the origin
increases. To determine this point, a search procedure is generally required.
Denoting the co-ordinates of this point by