## 5 Numerical Applications
### RBDO of the Three-Bar Truss under the Displacement Constraints  
The proposed RBDO/CS approach is applied to identify the optimal member sizes of a three-bar truss subjected to two forces, \(F_{X}\) and \(F_{Y}\), as illustrated in Figure 4a. All the truss bars have a modulus of elasticity, \(E\). Assume that the forces and modulus of elasticity are random variables, \(\mathbf{x}=(F_{X},F_{Y},E)^{\mathrm{T}}\), following the lognormal distribution. The mean values and coefficients of variance of the random variables are summarized in Table 3. The design variables are the cross-sectional areas of the bars \(\mathbf{d}=(A_{1},A_{2},A_{3})^{\mathrm{T}}\). The objective is to minimize the volume of the truss. The limit-state functions are defined in terms of the displacements at nodes 2 and 3, specifically, \(\mathbf{u}=(u_{2x},u_{2y},u_{3y})^{\mathrm{T}}\) (see Figure 4b), which should not exceed the values of \(\overline{\mathbf{u}}=\left(\overline{u}_{2x},\overline{u}_{2y},\overline{u} _{3y}\right)^{\mathrm{T}}\), as follows:  
\[\begin{array}{l}g_{1}(\mathbf{d},\mathbf{x})=\overline{u}_{2x}-u_{2x}( \mathbf{d},\mathbf{x})\\ g_{2}(\mathbf{d},\mathbf{x})=\overline{u}_{2y}-u_{2y}(\mathbf{d},\mathbf{x}) \\ g_{3}(\mathbf{d},\mathbf{x})=\overline{u}_{3y}-u_{3y}(\mathbf{d},\mathbf{x}) \end{array} \tag{28}\]  
where the nodal displacements, \(\mathbf{u}\), are obtained by solving the global equilibrium equations for the truss as:  
\[\begin{array}{l}u_{2x}(\mathbf{d},\mathbf{x})=\frac{L}{E}\left(\frac{F_{X}} {A_{2}}+\frac{F_{Y}}{A_{2}}\right)\\ u_{2y}(\mathbf{d},\mathbf{x})=\frac{L}{E}\left(\frac{F_{X}}{A_{2}}+\left(\frac{ 1}{A_{1}}+\frac{1}{A_{2}}+\frac{2\sqrt{2}}{A_{3}}\right)F_{Y}\right)\\ u_{3y}(\mathbf{d},\mathbf{x})=\frac{L}{E}\cdot\frac{F_{Y}}{A_{1}}\end{array} \tag{29}\]  
Figure 3: Effect of the step size on the accuracy of the sensitivity analysis performed using the finite difference method (FDM), monte carlo simulation (MCS), and (**a**) CS–SORM and (**b**) CS–SORM-i.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Random Variables** & **Marginal Distribution** & **Mean** & **c.o.v** & **Correlation** \\ \hline \(F_{X}\), kips & lognormal & 100 & 0.2 & \(\rho_{F_{x}F_{y}}=0.3\) \\ \(F_{Y}\), kips & lognormal & 150 & 0.2 & Independent \\ \(E\), ksi & lognormal & 29,000 & 0.2 & Independent \\ \hline \hline \end{tabular}
\end{table}
Table 3: Description of the random variables for the truss problem.  
To investigate the effect of the types of probabilistic distributions on the optimal design and to demonstrate the general applicability of the proposed method, all the random variables in the same problem are assumed to follow the normal and gamma distributions in two cases. All the other parameters such as the mean values and standard deviations of the random variables remain the same. The RBDO results shown in Figure 6 indicate that the minimum and maximum values of the objective function value are obtained when the lognormal and normal distributions are adopted, respectively. This finding highlights the importance of selecting suitable types of random variables to enable the structure design under uncertainties.  
Figure 5: Convergence history of the objective function and design variables pertaining to the deterministic design optimization (DDO) and RBDO/CS–SORM-i, and failure probabilities obtained using the RBDO/CS–SORM-i framework.  
Figure 6: Results of the RBDO/CS–SORM-i framework for normal, lognormal, and gamma distribution cases. (**a**) Optimal volume, and (**b**) optimal areas.