## 5 Numerical Applications
### RBDO of the Truss Cantilever Structure Subjected to Probabilistic Displacement Constraints  
Consider a truss cantilever structure consisting of 56 bar elements and 21 nodes (Figure 9c,d). The truss cantilever structure subjected to two external forces at nodes 5 and 21 contains three-roller supports as shown in Figure 9a. All of the truss elements have the same modulus of elasticity \(E\). The objective of RBDO is to minimize the weight of the truss cantilever structure and constraints are vertical displacements \((u_{5y},\ u_{21y})\) at nodes 5 and 21. Limit-state functions are defined as follows:  
\[ \begin{array}{l}g_{1}(\mathbf{d},\mathbf{x})=\overline{u}_{5}-u_{5y}( \mathbf{d},\mathbf{x})\\ g_{2}(\mathbf{d},\mathbf{x})=\overline{u}_{21}-u_{21y}(\mathbf{d},\mathbf{,x} )\end{array}  \tag{34}\]  
where \(\overline{u}_{5},\ \overline{u}_{21}\) are the maximum allowable displacements in the vertical direction, respectively. Two external forces and modulus of elasticity are assumed to be random variables \(\mathbf{x}=\left(F_{5y},\ F_{21y},E\right)\), and cross-sectional areas are design variables \(\mathbf{d}=\left(A_{i},\ i=1,\ldots,56\right)\). Table 10 provides marginal distributions, mean values, and second moments of the random variables. Parameters for reliability analysis and optimization are listed in Table 11. The RBDO problem can be written in a nested formulation as follows:  
\[ \begin{array}{l}\underset{\mathbf{d}}{\text{min}}f(\mathbf{d})=\sum_{i=1}^{56 }A_{i}L_{i}\\ \text{s.t.}\ P_{fi}=P[g_{i}(\mathbf{d},\mathbf{x})\leq 0]\leq{P_{fi}}^{\text{t}},\ i=1,2 \ \mathbf{d}^{\text{t}}\leq\mathbf{d}\leq\mathbf{d}^{\text{u}}\\ \text{with}\ \mathbf{K}(\mathbf{d},\mathbf{x})\mathbf{u}(\mathbf{d},\mathbf{x})= \mathbf{f}(\mathbf{x})\end{array}  \tag{35}\]  
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 \(\mathbf{K}(\mathbf{d},\mathbf{x})\mathbf{u}(\mathbf{d},\mathbf{x})=\mathbf{f }(\mathbf{x})\). The derivative of the displacement vector with respect to a parameter \(\theta(=\mathbf{d}\ \mathbf{or}\ \mathbf{x})\) is obtained analytically as  
\[\
abla_{\theta}u(\mathbf{d},\mathbf{x})=\frac{\partial\mathbf{K}^{-1}( \mathbf{d},\mathbf{x})}{\partial\theta}f(\mathbf{x})+\mathbf{K}^{-1}(\mathbf{ d},\mathbf{x})\frac{\partial\mathbf{f}(\mathbf{x})}{\partial\theta} \tag{36}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random** & **Marginal** & **Mean** & **c.o.v** & **Correlation** \\
**Variables** & **Distribution** & & & \\ \hline \(F_{5y}\), kips & Joint lognormal & 120 & 0.1 & \(\rho_{F_{5y},F_{21y}}=0.3\) \\ \(F_{21y}\), kips & Joint lognormal & 140 & 0.1 & \(\rho_{F_{5y},F_{21y}}=0.3\) \\ \(E\), kisi & Normal & 29000 & 0.2 & Independent \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Description of random variables for the truss structure optimization problem. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Optimal \(f\times\)L, in\({}^{3}\)** & \multicolumn{3}{c}{**Optimal \(A_{i}^{*}\), in\({}^{2}\)**} & \multicolumn{3}{c}{\(P_{fi}\)} \\ \cline{2-6}
**RBDO\(-\)CS/SL\(-\)KKT** & **Element** & **RBDO\(-\)CS/SL\(-\)KKT** & **FORM** & **SORM** & **MCS** \\ \hline \multirow{6}{*}{\(177.46\)} & \(1\) & \(31.37\) & \(1.667\times 10^{-4}\) & \(1.667\times 10^{-4}\) & \(1.655\times 10^{-4}\) \\  & \(2\) & \(31.37\) & \(1.667\times 10^{-4}\) & \(1.667\times 10^{-4}\) & \(1.672\times 10^{-4}\) \\ \cline{1-1}  & \(3\) & \(22.18\) & \(1. The sensitivity analysis in Equation (36) requires gradients of inversed stiffness matrix and force vector, which are computationally expensive and implicitly defined. Rather than directly solving Equation (36) for sensitivity analysis, applying the CS derivative approximation to Equation (36) yields  
\[