## 6 Concluding Remarks  
An approach to the RBDO of structures is developed using CS approximation and the FORM. The improvement in the accuracy of probabilistic constraints by the CS derivative and its straightforward implementation are presented in this paper. The proposed approach incorporates component reliability targets in RBDO and eliminates the need to analytically derive gradients of functions with respect to the design and random variables for sensitivity analysis. In addition to accounting for the statistical dependence between two-component events, the approach can compute the sensitivities of the failure probability to various parameters, which facilitate the use of gradient-based optimization algorithms. The first numerical example demonstrates the accuracy of the CS approximation without significant reliance on a step size, compared to finite difference methods. Furthermore, the uniform applicability of the proposed approach to RBDO of structures is confirmed through four numerical examples. This paper focuses on the CRBDO with probabilistic constraints of multiple failure modes of a design. The CRBDO problem is formulated such that the optimal structure satisfies each failure mode with target failure probabilities. In some instances, the design constraint needs to be defined as a \"system\" event of multiple constraints, (i.e., a logical (or Boolean) function of multiple failure modes), which is commonly termed a system reliability-based design optimization (SRBDO) problem. In these cases, the sensitivity analysis of the system reliability can be more expensive and complicated than a CRBDO problem due to the statistical dependency among failure events and complexity inherent in the formula. The implementation of -step approximation in SRBDO with diverse reliability analysis methods remains to be further explored. To improve the accuracy of reliability analysis and enhance the use of gradient-based optimization algorithms, areas for further study include sensitivity analysis with CS approximation unified with the SORM or multiple design points approaches, and its incorporation into RBDO.  
Figure 11: Optimal volumes versus correlation between external forces having different marginal distributions.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **RBDO** \\ **Method** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Optimal** \(f^{*}\) \\ **(ft3)** \\ \end{tabular} } & \multicolumn{4}{c}{\(p_{fi}\)} & \multicolumn{2}{c}{**Averaged Computational Time Carried**} & \multicolumn{1}{c}{**Total Number of**} \\ \cline{3-7}  & & \(i\) & **FORM** & **SORM** & **MCS** & **out to Five Iterations, (Seconds)** & **Iterations** \\ \hline RBDO\(-\)CS & 5.828 & 1 & 0.0050 & 0.00516 & 0.00516 & & \\  & 5.828 & 2 & 0.0050 & 0.00516 & 0.00517 & 0.0637 & 37 \\  & 5.827 & 1 & 0.0050 & 0.00517 & 0.00518 & & \\  & 5.827 & 2 & 0.0050 & 0.00518 & 0.00519 & 0.0591 & 175 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Comparison between results from the RBDO\(-\)CS and SL\(-\)KKT for the truss cantilever structure design.  
**Funding:** This research was funded by the Collaboration for Unprecedented Success and Excellence (CUSE) grant program of Syracuse University.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Conflicts of Interest:** The author declares no conflict of interest.