## 3 Applications and Results
### A Classical Limit State Funtion
Figure 2 shows that is not always possible to compute the statistics for the MCS; this happens when not a single failure is reported in one or more of the 1000 runs for a given \(n_{sim}\) (at least \(5\times 10^{4}\) are necessary). This is not a problem when using the fitted polynomials. Additionally, the MCS approach always tend to larger mean errors and standard deviations for a decreasing number of simulations. This makes the normality polynomial approach more adequate for estimating the reliability indices; however, for few simulations the errors are too large. Nevertheless, the designer could decide which precision level (quantitatively) is willing to accept using information like the one in Figure 2 as an aid (and reduce the number of required simulations as a function of such an accepted error). If desired, the curves in Figure 2 could be casted as a mathematical expression. For instance, the following power equations fit very well the mean (\(\mu_{\varepsilon}\)) and standard deviation (\(\sigma_{\varepsilon}\)) of the error in Figure 2 (by fitting from \(2.5\times 10^{2}\) simulations and over)  
\[\mu_{\varepsilon} =257.2\times n_{sim}^{-0.5244} \tag{19}\] \[\sigma_{\varepsilon} =157.3\times n_{sim}^{-0.5068} \tag{20}\]  
If the power in Equations (19) and (20) is assumed as \(-0.5\) in both expressions, the coefficient of variation of the error is \(\
u_{\varepsilon}\approx 150/250\approx 0.6\), i.e., it is constant and roughly independent of the number of simulations; the actual \(\
u_{\varepsilon}\) obtained from the 1000 runs does exhibit such a roughly constant behavior for this case, except that it is \(\approx\)0.7 (difference related to the actual different powers in Equations (19) and (20)). Power equations like Equations (19) and (20) can be linearized by taking logarithms on both sides. Therefore, if the involved variables are transformed into the logarithmic space, a linear fitting can be performed. In this study, we simply used a built-in function in the commercial software for the fitting. Coefficients for the fitted normality polynomial in Figure 1a are shown in Table 1 (upper set of values) for selected values of \(n_{sim}\). The computing of these coefficients is based on minimizing the error in Equation (4) and was implemented in the coded program by using a built-in function of the programming language employed (MATLAB). As mentioned before, the coefficients \(a_{0}\) can be linked to the generalized reliability index. Coefficients for results in Figure 1b-d (or those associated to Figure 2) were computed but not shown for brevity. Figure 2: Deviations from the exact reliability index as percentage error. As regards the reliability index, design point and sensitivity factors derived from the simulations (i.e., those obtained with Equations (11), (12), and (14)), they are compared with those obtained by applying the FORM to Equation (15). They are summarized in Table 2 for selected values of \(n_{sim}\). Results listed in Table 2 indicate that when the points obtained by applying the criterion in Equation (8) were enough to successfully perform a multi-linear regression (also with a built-in function, as in the case of the normality polynomial fitting), \(\beta\) did not deviate from the exact value, but marginally, just like the reliability index obtained with FORM. However, a relatively large \(n_{sim}\) was required, usually at least \(1\times 10^{5}\) simulations (by inspecting all the \(1000\times n_{sim}\) cases used to derived Figure 2); this depends on each run (implicitly the generated random numbers in each simulation), and sometimes less than \(1\times 10^{5}\) simulations are required.