Pf = Jfx(x)dx (11.3) gmo where fx (x) is the joint probability density function of the random variables X . This integral is, however, non-trivial to solve and numerical approximations are expedient. Various methods for the solution of the integral in Equation (11.3) have been proposed including numerical integration techniques, Monte Carlo simulation and asymptotic Laplace expansions. Numerical integration techniques very rapidly become inefficient for increasing dimension of the vector X and are in general irrelevant. In the following we will direct the focus on the widely applied and quite efficient FORM methods, which furthermore can be shown to be consistent with the solutions obtained by asymptotic Laplace integral expansions.
11.3 Linear Limit State Functions and Normal Distributed Variables For illustrative purposes we will first consider the case where the limit state function g(x) is a linear function of the basic random variables x . Then we may write the limit state function as:
g( x )= ao +Ea ix, (11 -I) If the basic random variables are normally distributed we furthermore have that the !meat safety margin M defined through: M = ao+ZaiX, ( 1 1 5)
is also normally distributed with mean value and variance
Pm = au +Zahux, QM =Ea;oi +E Epoi,aiaorri
where p, are the correlation coefficients between the variables X, and X i . Defining the failure event by Equation (11.1) we can write the probability of failure as: PF= P(g(X) 5 0) = P(M 0) (11.7) which in this simple case reduces to the evaluation of the standard normal distribution function:
P, =4)(4) (11.8) where fi the so-called reliability index (due to Cornell (1969) and Basler (1961)) is given as: