Page 23 - Dynamic Loading and Design of Structures

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ascertain whether, for this set, failure (i.e. g(x)≤0) has occurred. The experiment is repeated
many times and the probability of failure, P f, is estimated from the fraction of trials leading to
failure divided by the total number of trials. This socalled Direct or Crude Monte Carlo
method is not likely to be of use in practical problems because of the large number of trials
required in order to estimate with a certain degree of confidence the failure probability. Note
that the number of trials increases as the failure probability decreases. Simple rules may be
found, of the form N>C/Pf, where N is the required sample size and C is a constant related to
the confidence level and the type of function being evaluated.
Thus, the objective of more advanced simulation methods, currently used for reliability
evaluation, is to reduce the variance of the estimate of P. Such methods can be divided into
f
two categories, namely indicator function methods (such as Importance Sampling) and
conditional expectation methods (such as Directional Simulation). Simulation methods are
also described in a number of textbooks (e.g. Ang and Tang, 1984; Augusti et al., 1984;
Melchers, 1999).

1.3 FRAMEWORK FOR RELIABILITY ANALYSIS

The main steps in a reliability analysis of a structural component are the following:

(1) define limit state function for the particular design situation considered;
(2) specify appropriate time reference period;
(3) identify basic variables and develop appropriate probabilistic models;
(4) compute reliability index and failure probability;
(5) perform sensitivity studies.

Step (1) is essentially the same as for deterministic analysis. Step (2) should be considered
carefully, since it affects the probabilistic modelling of many variables, particularly live and
accidental loading. Step (3) is perhaps the most important because the considerations made in
developing the probabilistic models have a major effect on the results obtained. Step (4)
should be undertaken with one of the methods summarized above, depending on the
application. Step (5) is necessary insofar as the sensitivity of any results (deterministic or
probabilistic) should be assessed.

1.3.1 Probabilistic modelling

For the particular limit state under consideration, uncertainty modelling must be undertaken
with respect to those variables in the corresponding limit state function whose variability is
judged to be important (basic random variables). Most engineering structures are affected by
the following types of uncertainty:

●Intrinsic physical or mechanical uncertainty; when considered at a fundamental level, this
uncertainty source is often best described by stochastic processes in time and space,
although it is often modelled more simply in engineering applications through random
variables.