## 3 Applications and Results
### A Classical Limit State Funtion
For the considered runs, \(2\times 10^{5}\) simulations seem to guarantee the obtaining of the values reported in Table 2. In any case, when the muti-linear regression is successfully performed, the results are quite adequate and invariant for increasing number of simulations; this is also the case for the coefficients of the regression (i.e., they remain independent of \(n_{sim}\)), which are \(c=0.5837\), \(b_{1}=0.0998\), and \(b_{2}=-0.1333\). If the tolerance \(e_{l}\) in Equation (8) is increased, the minimum number of required simulations can be decreased. For instance, if \(e=0.25\) were used (instead of the actual used \(e=0.05\)), \(5\times 10^{4}\) simulations would be enough for a successful multi-linear regression. Moreover, the design point, sensitivity factors and reliability index would be the same as those reported in Table 2. The opposite would occur if \(e=0.005\) were used (instead of the actual used \(e=0.05\)), i.e., a much larger number of simulations would be required to determine the reliability parameters from the regression. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & **Classic LSE**, **Equation (15)** & \\ \hline \(n_{sim}\) & \(a_{0}\) & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) \\ \hline \(1\times 10^{3}\) & \(-3.1193\) & \(3.9078\) & \(3.6571\) & \(-1.9371\) & — \\ \(1\times 10^{4}\) & \(-3.3195\) & \(4.9672\) & \(1.8106\) & \(-0.9905\) & — \\ \(1\times 10^{5}\) & \(-3.4813\) & \(5.8709\) & \(0.2415\) & \(-0.1324\) & — \\ \hline \multicolumn{5}{c}{**Structural LSE**, **Equation (21)**} & \(r_{\text{V/D}}=0.4\) \\ \hline \(n_{sim}\) & \(a_{0}\) & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) \\ \hline \(1\times 10^{3}\) & \(-3.0950\) & \(6.3170\times 10^{-07}\) & \(3.8257\times 10^{-14}\) & \(-3.2619\times 10^{-21}\) & — \\ \(1\times 10^{4}\) & \(-3.2923\) & \(7.3372\times 10^{-07}\) & \(2.1457\times 10^{-14}\) & \(-2.3097\times 10^{-21}\) & — \\ \(1\times 10^{5}\) & \(-3.3341\) & \(7.8972\times 10^{-07}\) & \(5.2658\times 10^{-15}\) & \(-1.0534\times 10^{-21}\) & — \\ \hline \multicolumn{5}{c}{**Structural LSE**, **Equation (21)**} & \(r_{\text{V/D}}=0.4\) **4 th-order** \\ \hline \(n_{sim}\) & \(a_{0}\) & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) \\ \hline \(1\times 10^{3}\) & \(-2.8726\) & \(1.7464\times 10^{-07}\) & \(2.7997\times 10^{-13}\) & \(-5.0461\times 10^{-20}\) & \(3.1134\times 10^{-27}\) \\ \(1\times 10^{4}\) & \(-3.2802\) & \(6.7401\times 10^{-07}\) & \(6.0193\times 10^{-14}\) & \(-1.0616\times 10^{-20}\) & \(5.7321\times 10^{-28}\) \\ \(1\times 10^{5}\) & \(-3.2799\) & \(7.1276\times 10^{-07}\) & \(3.6215\times 10^{-14}\) & \(-5.7785\times 10^{-21}\) & \(2.4322\times 10^{-28}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coefficients of the fitted normality polynomials. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **Design Point** & **Sensitivity Factors** \\ \hline \(n_{sim}\) & \(\beta\) & \(x_{R}\) & \(x_{L}\) & \(\alpha_{R}\) & \(\alpha_{L}\) \\ \hline \(1\times 10^{4}\) & — & — & — & — & — \\ \(5\times 10^{4}\) & — & — & — & — & — \\ \(1\times 10^{5}\) & 3.5055 & 8.0699 & 8.0699 & \(-0.5990\) & 0.8007 \\ \(Therefore, this approach based on multi-linear regression can be quite an adequate alternative by itself to compute \(\beta\), conditioned on the feasibility of performing enough simulations, the number of which can be decreased by using a large \(e\); with the additional advantage that the design point and sensitivity factors can also be determined. In the following section a more realistic LSF for a structural application is used to further investigate the revisited methods.