## 5 Numerical Applications
### RBDO of the Truss Structure under Probabilistic Strength Constraints  
The proposed RBDO\(-\)CS method is applied to identify the optimal element sizes of an indeterminate truss structure by [PERSON] and [PERSON] [51] (see Figure 7). The deterministic design variables \(\mathbf{d}=\left(A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}\right)^{\mathrm{T}}\) are the cross-sectional areas of elements. The applied force \(F_{A}\) and yield strengths \(\sigma_{i},\ i=1,\ldots,6\) of the elements are the random variables \(\mathbf{x}=\left(\sigma_{1},\ \sigma_{2},\ldots,\sigma_{y},F_{A}\right)^{\mathrm{T}}\), which are assumed to follow the normal distribution. All random variables are statistically independent of each other, and the means and standard deviations of them are summarized in Table 7. In this example, RBDO aims to minimize the volume of the truss structure while satisfying the probabilistic constraints in terms of yield strengths. In this example, the truss element forces are derived in terms of the applied load with the assumption of the two diagonal elements carrying equal forces. Limit-state functions are defined as:  
\[g_{i}(\mathbf{d},\mathbf{x}) =A_{i}\sigma_{i}-\frac{\sqrt{2}}{2}F_{A},\quad i=1,2\] \[=A_{i}\sigma_{i}-\frac{1}{2}F_{A},\quad i=3,4,5,6\] \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’(m)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Optimal** \\ **d’(m)** \\ \end{tabular} } & \multicolumn{2}{c}{\(P_{H}\)} & \multirow{2}{*}{\
\begin{tabular}{c} **Averaged Computational Time Carried** \\ **out to Five Iterations, (Seconds)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Total Number of** \\ **Iterations** \\ \end{tabular} } \\ \cline{3-3} \cline{5-6}  & & & \(i=1\) & 0.0004 & 0.0004 & & \\ RBDO\(-\)CS & 13.658 & (3.362,5.148) & \(i=2\) & 0.0030 & 0.0028 & 0.105 & 12 \\  & & & \(i=3\) & 0.0030 & 0.0032 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison between results from the RBDO\(-\)CS and SL\(-\)KKT methods for the frame structure problem. Figure 7: Six-element truss structure. The RBDO problem can be defined as follows:  
\[ \begin{array}{ll}\underset{\mathbf{d}}{\text{min}}f(\mathbf{d})=\sqrt{2}\sum_{i= 1}^{2}A_{i}+\sum_{j=3}^{6}A_{j}\\ \text{s.t.}&P_{fi}=P[g_{i}(\mathbf{d},\mathbf{x})\leq 0]\leq P_{fi}^{t},\ i=1, \ldots,6\\ \mathbf{d}^{1}\leq\mathbf{d}\leq\mathbf{d}^{\mathbf{u}}\\ \end{array}  \tag{33}\]  
The initial design variables \(\mathbf{d}^{0}\) are set to 12 in\({}^{2}\), and the corresponding failure probabilities of the constraints by the FORM are \(P_{fi}=(0.999,0.999,0.865,0.865,0.865,0.865)\). The proposed RBDO\(-\)CS method with the parameters listed in Table 8 is able to find an optimal solution quickly, whose convergence history is plotted in Figure 8. Furthermore, the optimal designs are confirmed with the SL\(-\)KKT approach. Table 9 represents the optimal design and failure probabilities of constraints. The results of the SORM and MCS with \(10^{8}\) samples (c.o.v = 0.02) confirm the failure probability \(P_{fi}\) of the optimal design by the proposed method. It should be noted that the FORM \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variables** & **Marginal Distribution** & **Mean** & **Standard Deviation** & **Correlation** \\ \hline \(F_{A}\), kips & Normal & 1000 & 100 & Independent \\ \hline \(\sigma_{1}\), ksi & & & & \\ \(\sigma_{2}\), ksi & & & & \\ \(\sigma_{3}\), ksi & & & & \\ \(\sigma_{4}\), ksi & Normal & 36 & 3 & Independent \\ \(\sigma_{5}\), ksi & & & & \\ \(\sigma_{6}\), ksi & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Description of random variables for the truss optimization problem. Figure 8: Convergence histories of the proposed RBDO\(-\)CS method: (**a**) optimal volume; (**b**) design variables; and (**c**) failure probabilities. and SORM approximations of the probability of failure are sufficiently accurate for this problem.