## 3 Applications and Results  
### A Classical Limit State Funtion  
The two approaches described in the previous section are applied here to the simplest classical LSF  
\[g=R-L \tag{15}\]  
where \(R\) and \(L\) can be considered, in a broad sense, as random variables for the capacity and demand of a system element. For the sake of simplicity and illustration purposes, units are skipped and both random variables are assumed independent and characterized with lognormal distributions, with mean values and standard deviations \(m_{R}=10\), \(m_{L}=5.6\), \(\sigma_{R}=1\), and \(\sigma_{L}=0.75\) for the capacity and demand, respectively. These values are arbitrary, except by the fact that they lead to a reliability index equal to practically 3.5, which is a common reference for code calibration and that can be computed with the following expression [21; 22]  
\[\beta_{RE}=\frac{\left(ln\;m_{R}-\frac{1}{2}\ln(1+v_{R}^{2})\right)-\left(ln\; m_{L}-\frac{1}{2}\ln(1+v_{L}^{2})\right)}{\sqrt{\ln(1+v_{R}^{2})+\ln(1+v_{L}^{2})}} \approx\frac{\ln m_{R}-\ln m_{L}}{\sqrt{v_{R}^{2}+v_{L}^{2}}} \tag{16}\]  
where \(v_{R}\) and \(v_{L}\) are the coefficients of variation of \(R\) and \(L\), respectively. This reliability index is shown in Figure 1a (dashed line), as a reference to inspect how close are the \(\beta\)s obtained by crude Monte Carlo simulations as a function of the number of simulations (shown in logarithmic scale in the horizontal axis from \(2\times 10^{1}\) to \(2\times 10^{7}\)) to the exact value. The reliability index for Equation (15) using Monte Carlo simulations (dashed-dotted line in Figure 1a) is obtained by plugging into Equation (7) the following probability of failure  
\[p_{f}=\
icefrac{{n_{fail}}}{{n_{sim}}} \tag{17}\]  
which is simply the ratio of number of failures, \(n_{fill}\), to the total number of simulations. The latter (i.e., \(n_{sim}\)) is also the number of fractile constraints when the reliability index is computed with the normality polynomial approach, also depicted in Figure 1a (solid line). Additional runs are shown in Figure 1b-d, which indicate that the results are different and dependent on the generated random numbers of each run, but they stabilize if enough simulations are performed or enough fractile constraints are used. Other observations from Figure 1 include that \(\beta\) cannot always be computed with the Monte Carlo simulations (MCS) (not a single failure is obtained), while the opposite occurs when using normality polynomials, although significant deviations are observed for a limited number of simulations, that the fitted normality polynomials tends to deviate less from the exact reliability index for fewer \(n_{sim}\), and that such error in the precision may not be large for a relative small \(n_{sim}\), (e.g., \(1\times 10^{3}\)). To quantitatively inspect these deviations, \(\beta_{RE}\) is to be used as benchmark to assess the exactness of the used methods (i.e., normality polynomials and MCS) in terms of the relative percentual error given by  
\[\varepsilon=\left|\frac{\beta_{bench}-\beta_{oth}}{\beta_{bench}}\right|\times 100 \tag{18}\]  
where \(\beta_{bench}\) denotes the reliability index considered as benchmark and \(\beta_{oth}\) is the reliability index computed with any other method. A total of 1000 runs are performed for each \(n_{sim}\), and \(\varepsilon\) is computed for each of the runs. Then, the mean values of the error and their uncertainties (the unbiased standard deviation) are computed and plotted for the whole range of \(n_{sim}\) in Figure 2 (solid lines and dashed lines for normality polynomial and MCS mean errors, respectively), where the mean errors \(\pm\) one standard deviation are also depicted in grey lines. Figure 1: Reliability index as a function of the number of simulations. (a), (b), (c) and (d) are different runs for the MCS.