Page 20 - Dynamic Loading and Design of Structures

Page 7

Table 1.1 Relationship between βand P .
f

P f 10– 1 10– 2 10– 3 10– 4 10– 5 10– 6 10– 7
ß 1.3 2.3 3.1 3.7 4.3 4.7 5.2

mathematical model which idealizes the limit state. In this respect, the probability of failure
evaluated from eqn (1.5a) or (1.6) is a point estimate given a particular set of assumptions
regarding probabilistic modelling and a particular mathematical model for g( · ).
The uncertainties associated with these models can be represented in terms of a vector of
random parameters , and hence the limit state function may be rewritten as g(X, ). It is
important to note that the nature of uncertainties represented by the basic random variables X
and the parameters is different. Whereas uncertainties in X cannot be influenced without
changing the physical characteristics of the problem, uncertainties in can be influenced by
the use of alternative methods and collection of additional data.
In this context, eqn (1.6) may be recast as follows

(1.9)

where Pf( ) is the conditional probability of failure for a given set of values of the parameters
and f X| (x| ) is the conditional probability density function of X for given .
In order to account for the influence of parameter uncertainty on failure probability, one
may evaluate the expected value of the conditional probability of failure, i.e.

(1.10a)

where f ( ) is the joint probability density function of . The corresponding reliability index
is given by

(1.10b)

The main objective of reliability analysis is to estimate the failure probability (or, the
reliability index). Hence, it replaces the deterministic safety checking format (e.g. eqn (1.4)),
with a probabilistic assessment of the safety of the structure, typically eqn (1.6) but also in a
few cases eqn (1.9). Depending on the nature of the limit state considered, the uncertainty
sources and their implications for probabilistic modelling, the characteristics of the
calculation model and the degree of accuracy required, an appropriate methodology has to be
developed. In many respects, this is similar to the considerations made in formulating a
methodology for deterministic structural analysis but the problem is now set in a probabilistic
framework.