## 5 Numerical Applications
### RBDO of the Truss Cantilever Structure under the Displacement Constraints  
Consider a truss cantilever structure consisting of 56 bar members and 21 nodes (see Figure 12c,d). The truss cantilever structure subjected to two external forces at nodes 5 and 21 contains three-roller supports, as illustrated in Figure 12a. All of the truss members have the same modulus of elasticity, \(E\). The objective of RBDO is to minimize the weight of the  
Figure 11: Influence of the changes in the (**a**) correlation coefficients of the forces random variables, and (**b**) coefficients of variation of the moment capacity random variables on the FORM/SORM-i-based RBDO results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(i\) & \(P_{\hat{H}}\) & \(\hat{\beta_{i}}\) & **Optimal d\({}^{\star}\), m** & **Optimal/\({}^{\star}\), m** \\ \hline RBDO–FORM & 1 & 0.0005 & 3.310 & & \\  & 2 & 0.0030 & 2.748 & (3.362, 5.148) & 13.658 \\  & 3 & 0.0030 & 2.748 & & \\ \hline RBDO/CS–SORM & 1 & 0.0003 & 3.417 & & \\  & 2 & 0.0030 & 2.748 & (3.277, 5.196) & 13.669 \\  & 3 & 0.0030 & 2.748 & & \\ \hline RBDO/CS–SORM-i & 1 & 0.0004 & 3.349 & & \\  & 2 & 0.0030 & 2.748 & (3.264, 5.201) & 13.666 \\  & 3 & 0.0030 & 2.748 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Results of the RBDO/CS-SORM/SORM-i and RBDO-FORM for the ductile frame structure. truss cantilever structure under the constraints on horizontal and vertical displacements (\(u_{5x}\), \(u_{21y}\)) at nodes 5 and 21. Limit-state functions are defined as follows:  
\[ \begin{array}{l}g_{1}(\mathbf{d},\mathbf{x})=\overline{u}_{5x}-u_{5x}( \mathbf{d},\mathbf{x})\\ g_{2}(\mathbf{d},\mathbf{x})=\overline{u}_{21y}-u_{21y}(\mathbf{d},\mathbf{x}) \end{array}  \tag{35}\]  
where \(\overline{u}_{5x}\), \(\overline{u}_{21y}\) are the maximum allowable displacements in the horizontal and vertical direction at nodes 5 and 21 respectively, and \(\mathbf{x}=(F_{5x,F21y,E})^{\mathrm{T}}\) and \(\mathbf{d}=(A_{1},A_{2},\ldots,A_{56})^{\mathrm{T}}\) represent the vectors of random variables and design variables. Table 13 provides marginal distributions, mean values, and second moments of the random variables. Parameters for reliability analysis and optimization are listed in Table 14. The RBDO problem can be written in a nested formulation as follows:  
\[ \begin{array}{ll}\underset{\mathbf{d}}{\text{min}}&f(\mathbf{d})=\sum \limits_{i=1}^{56}A_{i}L_{i}\\ \text{s.t.}&P[g_{i}(\mathbf{d},\mathbf{x})\leq 0]\leq P_{fi^{\prime}}^{ \mathrm{t}},\,i=1,2\\ \text{with}&\mathbf{K}(\mathbf{d},\mathbf{x})\mathbf{u}(\mathbf{d},\mathbf{x })=\mathbf{f}(\mathbf{d},\mathbf{x})\\ &\mathbf{d}^{\mathrm{l}}\leq\mathbf{d}\leq\mathbf{d}^{\mathrm{u}}\end{array}  \tag{36}\]  
where \(\mathbf{K}\) is the global stiffness matrix of the truss cantilever structure, \(\mathbf{u}\) is the global displacement vector, and \(\mathbf{f}\) is the global external force vector. Note that \(\mathbf{u}\) is an implicit function defined through the equilibrium equations \(\mathbf{K}(\mathbf{d},\mathbf{x})\mathbf{u}(\mathbf{d},\mathbf{x})=\mathbf{f} (\mathbf{d},\mathbf{x})\). The derivative of the displacement vector with respect to a parameter and sensitivity analysis using the CS approach is discussed in [42]. The initial displacements at nodes 5 and 21, \(u_{5x}(\mathbf{d}^{0},\mathbf{\mu}_{\mathbf{x}}),\,\,u_{21y}(\mathbf{d}^{0}, \mathbf{\mu}_{\mathbf{x}})\), with mean values of random variables as well as the uniform cross-sectional areas of 5 in 2 for all members, are 0.306 in and 0.726 in, and their corresponding failure probabilities assessed using the SORM-i are 0.525 and 0.773, respectively. Figure 12: Truss cantilever structure: (**a**) dimension, loading, and boundary conditions, (**b**) deformed shape and displacements of interest, (**c**) truss member numbers, and (**d**) node numbers. Figure 13 illustrates the convergence histories of objective and probabilistic constraint functions, optimized member sizes using RBDO/CS-SORM-i, and optimized cantilever structure.