## 5 Numerical Applications
### RBDO of the Truss Cantilever Structure Subjected to Probabilistic Displacement Constraints
\begin{split}\theta&=A_{i}:\quad\
abla_{A_{i}}\mathbf{u }(\mathbf{d},\mathbf{x})=\frac{\partial\mathbf{K}^{-1}(\mathbf{d},\mathbf{x})} {\partial A_{i}}\mathbf{f}(\mathbf{x})=\frac{\Im\left[\mathbf{K}^{-1}\left( \mathbf{d}+i\mathbf{i}\mathbf{e}_{i},\mathbf{x}\right)\mathbf{f}(\mathbf{x}) \right]}{h}\\ \theta&=F_{\text{5y}}:\quad\
abla_{F_{\text{5y}}} \mathbf{u}(\mathbf{d},\mathbf{x})=\mathbf{K}^{-1}(\mathbf{d},\mathbf{x}) \frac{\partial\mathbf{f}(\mathbf{x})}{\partial F_{\text{5y}}}=\frac{\Im\left[ \mathbf{K}^{-1}(\mathbf{d},\mathbf{x})\mathbf{f}\left(\mathbf{x}+i\mathbf{i} \mathbf{e}_{1}\right)\right]}{h}\\ \theta&=F_{\text{21y}}:\quad\
abla_{F_{\text{21y}}} \mathbf{u}(\mathbf{d},\mathbf{x})=\mathbf{K}^{-1}(\mathbf{d},\mathbf{x}) \frac{\partial\mathbf{f}(\mathbf{x})}{\partial F_{\text{21y}}}=\frac{\Im\left[ \mathbf{K}^{-1}\left(\mathbf{d},\mathbf{x}\right)\mathbf{f}\left(\mathbf{x}+i \mathbf{e}_{2}\right)\right]}{h}\\ \theta&=E:\quad\quad\
abla_{E}\mathbf{u}(\mathbf{d},\mathbf{x})=\frac{\partial\mathbf{K}^{-1}(\mathbf{d},\mathbf{x})}{\partial E }\mathbf{f}(\mathbf{x})=\frac{\Im\left[\mathbf{K}^{-1}\left(\mathbf{d}, \mathbf{x}+i\mathbf{e}_{3}\right)\mathbf{f}(\mathbf{x})\right]}{h}\end{split}  \tag{37}\]  
where \(\mathbf{e}_{i}\) is the \(i\)-th column of a 56-th order identity matrix (\(\mathbf{I}_{\text{56}}\)) and \(\mathbf{e}_{k}\) is the \(k\)-th column of a 3-rd order identity matrix (\(\mathbf{I}_{\text{3}}\)). Sensitivities of the limit-state functions with respect to design variables can be computed by using Equation (37). Furthermore, sensitivities of limits-state functions with respect to random variables are used in the FORM for reliability analysis. Notably, the displacement vector \(\mathbf{u}\) and its sensitivity are computed without additional step by taking real and imaginary parts in Equation (37). The initial displacements at nodes 5 and 21 of the truss cantilever structure with mean values of random variables and uniform cross-sectional areas of 5 in\({}^{2}\) for all elements are 0.1339 in and 0.3557 in and corresponding failure probabilities by the FORM are 0.928 and 0.789. Figure 10 illustrates the convergence histories of objective and probabilistic constraint functions, optimized element sizes, and optimal cantilever structure. The proposed RBDO\(-\)CS method quickly identifies the solutions satisfying probabilistic constraints and  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(P_{f}^{\mathbf{J}}\) & \( \begin{array}{c}\mathbf{u}_{5}\\ \mathbf{(in)}\end{array} \) & \( \begin{array}{c}\mathbf{u}_{21}\text{ (in)}\end{array} \) & \(\begin{array}{c}\mathbf{Initial}\  
then reduces the volume to minimize. Optimization results in Figure 10c indicate that elements connected to the roller support in the middle and the bottom chords are primarily strengthened to satisfy probabilistic constraints. Table 12 provides the results from the RBDO\(-\)CS method for the truss cantilever structure. The SL\(-\)KKT method is utilized to verify optimization results. It is observed that the results from the proposed RBDO\(-\)CS and SL\(-\)KKT approaches closely match. Failure probabilities of limit-state functions with optimal design \(\mathbf{d}^{*}\) and random variables are estimated by the FORM, SORM, and MCS with \(10^{7}\) samples (c.o.v = 0.03). This study confirms that the proposed RBDO\(-\)CS method enables the efficient identification of optimal solutions. It is noted that the computational costs of RBDO\(-\)CS and SL\(-\)KKT methods are similar with a larger number of design variables, however, the proposed RBDO\(-\)CS method identifies converged solutions faster than the SL\(-\)KKT method. Next, the force random variables \(F_{5y},\ F_{21y}\) are assumed to follow the normal distributions with the same means and standard deviations. This is to investigate the effect of probabilistic distribution types on the optimal design and to demonstrate the general applicability of the proposed method.