## 5 Numerical Applications
### Comparative Study on the Accuracy in the Reliability Assessment and Sensitivity Analysis  
The accuracy in the assessment of the failure probability and sensitivity analysis by the FORM/SORM is studied and compared to that pertaining to the results obtained using the finite difference method and MCS. Consider a nonlinear function of random variables \(\mathbf{x}=\left(x_{1},x_{2},x_{3}\right)^{\mathrm{T}}\) and deterministic variables \(\mathbf{d}=\left(d_{1},d_{2}\right)^{\mathrm{T}}\):  
\[f(\mathbf{d},\mathbf{x})=7-x_{1}x_{2}x_{3}\Bigg{(}\frac{d_{1}}{2d_{2}^{2}} \Bigg{)} \tag{27}\]  
Assume that the random variables follow the normal distribution with the means and standard deviations listed in Table 1. The correlation coefficients are assumed to be uniform, \(\rho_{x_{1}x_{j}}=0.3\left(i,j=1,2,3,\ i\
eq j\right)\). The failure probability of the nonlinear function \(P_{f}=P[f(\mathbf{d}=(0.7,0.8),\mathbf{x})\leq 0]\) is computed using the FORM, SORM, and SORM-i. The sensitivity analysis of the function at \(\mathbf{d}=(0.7,0.8)\) is performed using the proposed method with the SORM and SORM-i techniques, as described in Equations (24) and (25), and the FDM with a step size of \(h=10^{-5}\). The results are verified through the MCS involving \(10^{7}\) samples (coefficient of variance, c.o.v = 0.001). The results of the failure probability and sensitivity calculations are summarized in Table 2. The SORM estimate of the failure probability according to [PERSON]'s formula is \(P_{f}=0.1332\), whereas the result based on [PERSON]'s formula is \(P_{f}=0.1291\). These values closely match the \"exact\" result obtained through the MCS. However, the failure probability obtained using the FORM exhibits significant (more than 10%) differences. Since the failure surface has a high nonlinearity, the failure probability estimated using the FORM is less accurate than the quadratic approximation obtained through the SORM. The results of the sensitivity analysis implemented using the proposed method exhibit a satisfactory agreement with the results of the FDM, as validated through the MCS. Furthermore, the impact of the step size on the sensitivity of the second-order failure probability is investigated by varying \(h\) from \(10^{-14}\) to \(10^{-1}\). The sensitivity analysis is performed using the proposed method integrated with [PERSON]'s approximation and [PERSON]'s approach, FDM, and MCS, with \(10^{7}\) samples. The numerical results, presented in Figure 3, show that the sensitivities obtained using the proposed methods are consistent for different step sizes. This finding indicates that a section of the step size does not affect the calculation accuracy. In contrast, the FDM breaks down as the step size becomes smaller than \(10^{-6}\) or larger than \(10^{-3}\). Therefore, the proposed method using the CS derivative approximation is more stable and accurate for small step sizes at which the finite difference approach cannot ensure reasonable accuracy. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Random** \\ **Variables** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Marginal** \\ **Distribution** \\ \end{tabular} } & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**c.o.v**} & \multicolumn{2}{c}{**Correlation Coefficient**} \\ \cline{3-5}  & & & & \(\boldsymbol{x_{1}}\) & \(\boldsymbol{x_{2}}\) & \(\boldsymbol{x_{3}}\) \\ \hline \(x_{1}\) & Normal & 2 & 0.25 & 1 & 0.3 & 0.3 \\ \hline \(x_{2}\) & Normal & 2.5 & 0.25 & 0.3 & 1 & 0.3 \\ \hline \(x_{3}\) & Normal & 1.5 & 0.25 & 0.3 & 0.3 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the random variables for the comparative study.