## 5 Numerical Applications
### RBDO of the Short Column under a Probabilistic Strength Constraint  
The proposed RBDO-CS approach is applied to identify the optimal member size of a short column subjected to an axial force \(P\) and bending moments \(M_{1}\) and \(M_{2}\), as shown in Figure 3. Assume an elastic perfectly plastic material with yield stress \(\sigma_{y}\). The limit-state function regarding the failure of the column is defined by  
\[g(d,\mathbf{x})=1-\frac{M_{1}}{S_{1}(d)\sigma_{y}}-\frac{M_{2}}{S_{2}(d) \sigma_{y}}-\left(\frac{p}{A(d)\sigma_{y}}\right)^{q} \tag{28}\]  
Figure 2: (**a**) Comparison of first derivatives by finite difference and complex-step (CS) methods, (**b**) absolute errors from computing the first derivatives with varying step sizes. where \(\mathbf{x}=\left(M_{1},M_{2},P,\sigma_{y}\right)^{\mathrm{T}}\) denotes the vector of random variables, \(q=2\) is the limit-state function parameter, \(d\) is the design variable, \(A(d)=1.5d^{2}\) is the cross-sectional area, and \(S_{1}(d)=0.375d^{3}\) and \(S_{2}(d)=0.25d^{3}\) are the flexural moduli of the plastic column section. Assume that \(M_{1},M_{2},P,\) and \(\sigma_{y}\) have the Nataf distribution with the marginal distributions and coefficient of variance (c.o.v) listed in Table 1. The objective function for this problem is taken as the minimization of the cross-sectional area, and the probabilistic constraint is defined in terms of yield strength. The RBDO problem is formulated as  
\[ \begin{array}{ll}\underset{d}{\text{min}}&A(d)\\ \text{s.t.}&P_{f}=P[g(d,\mathbf{x})\leq 0]\leq P_{f}\text{t}\\ &d^{\mathrm{I}}\leq d\leq d^{\mathrm{u}}\end{array}  \tag{29}\]  
Table 2 summarizes the target failure probability and optimization parameters. A sensitivity analysis is performed by the proposed method to utilize a gradient-based optimization algorithm--the method of moving asymptotes (MMA; [44]). The optimization process is repeated until the convergence criteria on the maximum number of interactions or the relative change in the design variable is satisfied. The initial reliability analysis of the limit-state function \(g\left(d^{0},\mathbf{x}\right)\) by the FORM results in \(\mathbf{u}^{*}=\left(-2.0084,-0.9605,-0.3399,0.4619\right)^{\mathrm{T}}\), \(\beta=-2.2989\), and \(P_{f}=0.9892\). In this study, the single loop approach using the Karush-Kuhn-Tucker conditions (SL-KKT) [18] is utilized to compare the proposed RBDO-CS method. The CS approximation is also implemented to compute sensitivities in the SL-KKT. The same optimization parameters and convergence criteria are considered for the comparison. Table 3 represents the optimal area and design variables obtained by the proposed RBDO-CS and SL-KKT methods, as well as the averaged computational time carried out to five optimization iterations (a modern (2020) eight-core 3.8 GHz Intel process (16 GB RAM) is used for the analyzes). Because the SL-KKT approach does not perform reliability analysis during optimization procedures, the failure probability of limit-state function is computed by the FORM with optimal column design. The optimal results from both approaches are in agreement with each other. Figure 4 shows the convergence histories of the RBDO-CS and the SL-KKT methods regarding the objective function, design variable, and failure probability. The proposed RBDO-CS method enables the identification of a feasible solution while satisfying the probabilistic constraint. Although the proposed RBDO-CS \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Random** & **Marginal** & **Mean** & **c.o.v** & \multicolumn{4}{c}{**Correlation Coefficient**} \\
**Variables** & **Distribution** & **Mean** & **c.o.v** & \multicolumn{4}{c}{**\(M_{1}\)**} & \(M_{2}\) & \(P\) & \(\sigma_{y}\) \\ \hline \(M_{1}\), kNm & Normal & 250 & 0.3 & 1.0 & & & & \\ \(M_{2}\), kNm & Normal & 125 & 0.3 & 0.5 & 1.0 & & & \\ \(P\), kN & Gumbel & 2500 & 0.2 & 0.3 & 0.3 & 1.0 & & \\ \(\sigma_{y}\), MPa & Weibull & 40 & 0.1 & 0 & 0 & 0 & 1.0 \\ \hline \hline \end{tabular}
\end{table}