## 5 Numerical Applications
### RBDO of the Ductile Frame Structure Subjected to Probabilistic Moment Strength Constraints  
The frame structure under uncertainty in the magnitudes of external forces and moment capacities is illustrated in Figure 5a. Assume that the frame structure is constructed with ductile members that have plastic moment capacities \(m_{i},\ i=1,\ldots,5\), at joints. Under the externally applied forces \(h,\ v\), this frame may fail due to any of the sway, beam, and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Target Failure Probability \(P_{f}^{t}\)** & **Initial Design \(d^{0}\), m** & **Lower Bound \(d^{l}\), m** & **Upper Bound \(d^{u}\), m** & **Convergence Criterion** \\ \hline
0.01 & 0.25 & 0.1 & 2.0 & \(10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for the target failure probability and optimization. Figure 4: Convergence history of the proposed RBDO\(-\)CS and SL\(-\)KKT approaches: (**a**) optimal area; (**b**) design variable; and (**c**) failure probability. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**RBDO** & **Optimal** & **Optimal** & \multicolumn{2}{c}{\(P_{f}\)} & **Averaged Computational Time Carried** & **Total Number** \\
**Method** & \(f(m^{2})\) & \(d^{*}(m)\) & **FORM** & **SORM** & **out to Five Iterations, (Seconds)** & **of Iterations** \\ \hline RBDO\(-\)CS & 0.2293 & 0.3909 & 0.0100 & 0.0126 & 0.0130 & 11 \\ SL\(-\)KKT & 0.2293 & 0.3908 & 0.0101 & 0.0127 & 0.0108 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between results from the RBDO\(-\)CS and SL\(-\)KKT approaches for the column design problem. combined mechanisms illustrated in Figure 5b. Using the principle of virtual work, the limit-state functions regarding these mechanisms are formulated [13] as  
\[ \begin{array}{l}g_{1}(\mathbf{d},\mathbf{x})=m_{1}+m_{2}+m_{4}+m_{5}-hd_{1}\\ g_{2}(\mathbf{d},\mathbf{x})=m_{2}+2m_{3}+m_{4}-vd_{2}\\ g_{3}(\mathbf{d},\mathbf{x})=m_{1}+2m_{3}+2m_{4}+m_{5}-hd_{1}-vd_{2}\end{array}  \tag{30}\]  
where \(\mathbf{x}=(m_{1},\ m_{2},\ldots,m_{5},h,v)^{\mathrm{T}}\) is the vector of random variables, and \(\mathbf{d}=(d_{1},d_{2})^{\mathrm{T}}\) is the vector of design variables (\(d_{1}\): height, \(d_{2}\): width). Assumed distributions and second moments of the random variables are tabulated in Table 4. The maximization of the height and width of the ductile frame structure and the three limit-state functions in terms of the moment capacities are considered the objective and probabilistic constraint functions, respectively. The RBDO problem of the ductile structure can be stated as  
\[ \begin{array}{ll}\underset{\mathbf{d}}{\text{max}}&f(\mathbf{d})=d_{1}+2d_ {2}\\ \text{s.t.}&P_{f1}=P[g_{1}(\mathbf{d},\mathbf{x})\leq 0]\leq P_{f1}{}^{\mathrm{t}}\\ &P_{f2}=P[g_{2}(\mathbf{d},\mathbf{x})\leq 0]\leq P_{f2}{}^{\mathrm{t}}\\ &P_{f3}=P[g_{3}(\mathbf{d},\mathbf{x})\leq 0]\leq P_{f3}{}^{\mathrm{t}}\\ &\mathbf{d}^{\mathrm{l}}\leq\mathbf{d}\leq\mathbf{d}^{\mathrm{u}}\end{array}  \tag{31}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variables** &
\begin{tabular}{c} **Marginal** \\ **Distribution** \\ \end{tabular} & **Mean** & **c.o.v** & **Correlation** \\ \hline \(m_{1}\), k\(\mathrm{N}\mathrm{m}\) & & & & \\ \(m_{2}\), k\(\mathrm{N}\mathrm{m}\) & & & & \\ \(m_{3}\), k\(\mathrm{N}\mathrm{m}\) & Joint lognormal & 150 & 0.2 & \(\rho_{m_{i}m_{j}}=0.3\) \\ \(m_{4}\), k\(\mathrm{N}\mathrm{m}\) & & & &, \(i\
eq j\) \\ \(m_{5}\), k\(\mathrm{N}\mathrm{m}\) & & & & \\ \hline \(h\), k\(\mathrm{N}\) & Gumbel & 50 & 0.4 & Independent \\ \(v\), k\(\mathrm{N}\) & Gamma & 60 & 0.2 & Independent \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Description of random variables for the ductile frame optimization problem. Figure 5: (a) Ductile frame structure, and (b) failure mechanisms. A target failure probability of 0.003 is assigned to all probabilistic constraints.