## 5 Numerical Applications
### RBDO of the Ductile Frame Structure under the Moment Strength Constraints  
The frame structure subjected to external forces is illustrated in Figure 9a. Assume that the frame structure is constructed using ductile members having plastic moment capacities \(m_{i}\)\(i=1,\,\ldots\,5\), at the joints. The three failure mechanisms under the externally applied forces, \(h\) and \(v\), are illustrated in Figure 9b. Assume that the external forces and moment capacities are random variables, and the height and width of the structure are the design variables. The limit-state functions related to these failure mechanisms can be defined using the principle of virtual work [6], as:  
\[ \begin{array}{l}g_{1}(\mathbf{d},\mathbf{x})=m_{1}+m_{2}+m_{4}+m_{5}-hd_{1} \\ g_{2}(\mathbf{d},\mathbf{x})=m_{2}+2m_{3}+m_{4}-vd_{2}\\ g_{3}(\mathbf{d},\mathbf{x})=m_{1}+2m_{3}+2m_{4}+m_{5}-hd_{1}-vd_{2}\end{array}  \tag{33}\]  
where \(\mathbf{x}=\left(m_{1},m_{2},\ldots,m_{5},h,v\right)^{\mathrm{T}}\) and \(\mathbf{d}=\left(d_{1},d_{2}\right)^{\mathrm{T}}\) represent the vectors of random variables and design variables (\(d_{1}\): height, \(d_{2}\): width), respectively. The considered distributions and second moments of the random variables are presented in Table 9. The objective is  
Figure 8: Convergence history of the objective function, design variables, and probabilistic constraints (by RBDO/CS–SORM-i). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**RBDO Method** & **Optimal d\({}^{\star}\), in** & \(f\) (d\({}^{\star}\), \(\mu_{\lambda}\)), in\({}^{\mathrm{3}}\) & \(P_{f}\) \\ \hline RBDO/CS–SORM & \((34.5,56.2,72.1)\) & 16,128 & 0.005 \\ RBDO/CS–SORM-i & \((34.6,56.4,72.3)\) & 16,185 & 0.005 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results obtained using the RBDO/CS–SORM/SORM-i for the cantilever beam. Figure 10 shows the convergence histories obtained using the proposed method RBDO/CS-SORM/SORM-i and the FORM-based RBDO approach. The three methods can promptly identify the feasible solutions satisfying the probabilistic constraints. Table 12 compares the results obtained using the proposed approaches and the RBDO using the FORM. The RBDO/CS-SORM/SORM-i approaches result in a less conservative design in terms of the width but more conservative designs in terms of the height compared to results pertaining to the FORM-based RBDO, while satisfying the same requirements pertaining to the component level reliability. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Limit-State Function**} & \multicolumn{2}{c}{**FORM**} & \multicolumn{2}{c}{**SORM**} & \multicolumn{2}{c}{**SORM-i**} \\ \cline{2-7}  & \(\beta\) & \(P_{f}\) & \(\beta\) & \(P_{f}\) & \(\beta\) & \(P_{f}\) \\ \hline \(g_{1}(\mathbf{d}^{0},\mathbf{x})\) & 1.452 & 0.0732 & 1.468 & 0.0711 & 1.472 & 0.0705 \\ \(g_{2}(\mathbf{d}^{0},\mathbf{x})\) & 1.435 & 0.0757 & 1.468 & 0.0710 & 1.479 & 0.0696 \\ \(g_{3}(\mathbf{d}^{0},\mathbf{x})\) & 0.701 & 0.2415 & 0.688 & 0.2457 & 0.674 & 0.2503 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Failure probabilities and reliability indices of initial limit-state functions estimated using the FORM, SORM, and SORM-i. Figure 10: Convergence history of the objective function, design variables, and failure probability pertaining to the (a) RBDO–FORM, (b) RBDO/CS–SORM, and (c) RBDO/CS–SORM-i. The effects of the coefficient of variations of \(m_{i}\)\(i=1,\ldots,5\) and the correlations between the random variables \(v\) and \(h\) on the optimal result are investigated. The RBDO problem is solved again considering the same target system probability of 0.003, while varying the coefficient of variations and correlation coefficients. For simplicity, the same types of distributions and mean values, coefficient of variations, and correlation coefficients listed in Table 9 are used, except for the varying parameters of interest. Figure 11a shows the influence of the increase in the correlation coefficients (from 0.1 to 0.5) on the optimal \(f\).