## 3 Applications and Results
### Reliability of Reinforced Concrete Beam under Flexure Moment
As an example of the simulated bending moments, histograms of \(D\) and \(V\) are shown in Appendix A (Figure A1) for \(1\times 10^{6}\) MCS and \(r_{V/D}=1.0\); it can be observed that the values are comparable in average (because \(r_{V/D}=1.0\) is considered), and that histograms for \(D\) and \(V\) clearly resemble normal and Gumbel distributions, respectively, which is expected given that these variables were sampled from such PDFs. Figure 4: Deviations from the exact reliability index as percentage error for RCB; (**a**) \(r_{V/D}=0.4\) and (**b**) \(r_{V/D}=2.0\). Figure 3: Reliability index as a function of the number of simulations for RCB. (**a**), (**b**), (**c**) and (**d**) are different runs for the MCS. From Figures 3 and 4 similar conclusions to those drawn from Figures 1 and 2 can be extracted. Some additional observations worth to mention are that \(n_{sim}=1\times 10^{4}\) seems a reasonable number of simulations for the normality polynomial approach, if a compromise between \(n_{sim}\) and error (in terms of mean and standard deviation) is envisaged, that the fitted polynomials lead to better results than the FORM for increasing \(n_{sim}\) and that larger errors are obtained for smaller \(r_{V/D}\); this latter aspect could be attributed to a better approximation of the failure surface for larger \(r_{V/D}\), since the first order approximation to the failure surface by the other method shown in Figure 3 (i.e., the FORM), also deviates more from the exact value for decreasing \(r_{V/D}\). By observing Figure 4, it is pointed out once more that error and its uncertainty is less for normality polynomial when decreasing \(n_{sim}\) than for MCS for this case too and, as previously mentioned, is not always possible to estimate the error for MCS for decreasing number of simulations. The error for \(\beta\) obtained with the normality polynomial approach exhibits an asymptotic behavior towards approximately \(\mu_{\varepsilon}=1\%\) for large \(n_{sim}\). As before, power laws fit adequately the error and its uncertainty and are defined as  
\[\mu_{\varepsilon}=\delta\times n_{sim}^{-\gamma}+1 \tag{23}\]  
\[\sigma_{\varepsilon}=\kappa\times n_{sim}^{-\tau} \tag{24}\]  
where \(\delta=656\), \(1103\), and \(1349\), \(\gamma=0.7401\), \(0.8493\), and \(0.9062\), \(\kappa=129.8\), \(201\), and \(179.1\), \(\tau=0.4937\), \(0.5609\), and \(0.5703\) for \(r_{V/D}=0.4\), \(1.0\), and \(2.0\), respectively. In Equation (23) the constant unity is included to shift the curve upwards to reproduce the asymptotic behavior mentioned; nonetheless, it could be skipped, and the equations will still fairly adequately describe the mean error. The fitting for \(\mu_{\varepsilon}\) was performed for the whole range of \(n_{sim}\), while for \(\sigma_{\varepsilon}\), \(n_{sim}\) from \(250\) and over was employed. Note that although the range for the fitting could be established based on practical grounds and fitting improvement, in any case the errors and their uncertainties follow a power law; this is the case for the three case studies carried out in this study. It is noteworthy that the normality polynomial approach leads to comparable \(\mu_{\varepsilon}\) and \(\sigma_{\varepsilon}\) for Figures 2 and 4, considering that the LSF for the RCB is a more complex (non-linear) function, and that it has much more random variables and several PDFs. The fitted coefficients of the polynomials for Figure 3 (corresponding to \(r_{V/D}=0.4\)) and selected \(n_{sim}\) are listed in Table 1 (middle set of values). If normality polynomials of order higher than \(3\) are used, no further accuracy is gained (or even higher inaccuracies could be obtained; [11]). This is confirmed by carrying out a single case for \(r_{V/D}=0.4\) using a 4 th-order polynomial, since the results are comparable to those of the 3 rd-order polynomial case (Table 1, lower set of values).