Gaussian approximation failure

The use of Stirling s approximation assumes a sufficiently large value of Njj. This assumption fails at high degrees of crosslinking and high extensions. The failure leads to non-Gaussian behavior. Note the similarity of the last term of this sum and that evaluated in the... 

An approximation of the parameter distributions by a Gaussian density enables an efficient estimation of the failure probability variation which agreed very well with the results of the sample analysis in the investigated example. 

Equation 5 provides an exact estimate of the failure probability provided that the limit state function is linear with respect to the Gaussian distributed vector of uncertain parameters. Under more general conditions, Equation 5 yields only approximate results. Moreover, it should be noted that EORM does not produce any measure of the error introduced by the linearization assumption. 

This section discusses a class of methods known as the first-order reliability methods to compute the probability of failure of structural systems. These methods are based on the first-order Taylor s series expansion of the performance function G(X). The first-method, known as the first-order second-moment (FOSM) method, focuses on approximating the mean and standard deviation of G and uses this information to compute Pf. Then, the FOSM method is extended to the advanced FOSM method in two steps first, the methodology is developed for the case where all the variables in X are Gaussian (normal) and, second, the methodology is extended to the general case of non-normal variables. 

Once the mean and standard deviation of G is calculated, then G is approximated as a Gaussian distribution, with mean and standard deviation (T(3. Then, the failure probability can be calculated as... 

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