## 4 Numerical Examples
### Example 3--MBB beam  
Consider the MBB beam problem (Figure 8); the maximum deflection is given by  
\[\delta_{max}=\frac{1}{48}\frac{PL^{3}}{EI} \tag{21}\]  
where \(E\) is the material's elasticity modulus, \(I\) is the inertia moment, \(P\) is the applied load, and \(L\) is the span length. The average values and standard deviations of the random variables are given in Table 2. Thus, the failure criterion can be assumed as  
\[\frac{\delta_{max}}{L}>\frac{1}{300} \tag{22}\]  
The limit state in the function of L can be obtained with a substitution of Equations (21) into (22), resulting in  
\[G(E,I,P)=48 EI-300 PL^{2} \tag{23}\]  
Table 3 shows the mean and standard deviation values of the normally distributed random variables used in the RBTO analysis. The SESO method is used to perform the MBB beam's reliability analysis and to assess the influence of the target reliability index and the new uncertain variables on the optimal configuration. In particular, the DTO SESO's optimal configuration is compared with RBTO SESO, and the compliance and displacement values, \(C_{DTO}\) and \(\delta^{*}\), respectively, resulting \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Vol \%** & **Objective Function** & **Displacement** & **Computational** \\  & **(mm\({}^{3}\))** & **(N\(\cdot\)mm)** & **(mm)** & **Cost (Sec)** \\ \hline
**RBTO-ESO** & 0.202 & 0.5578 & 0.00558 & 1288.04 \\ \hline
**RBTO-SESO** & 0.202 & 0.6556 & 0.00655 & 1427.02 \\ \hline
**RBTO-SERA** & 0.202 & 0.6560 & 0.00656 & 859.23 \\ \hline
**RBTO-SIMP** & 0.202 & 0.7089 & 0.00709 & 2274.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reliability-based topology optimization using evolutionary methods. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Variable**} & **Average** & **Standard Deviation** & **Distribution** \\ \hline
**P (kN)** & \(100\) & \(10\) & Normal \\ \hline
**E (kN/m\({}^{2}\))** & \(10^{5}\) & \(10^{4}\) & Normal \\ \hline
**I (m\({}^{4}\))** & \(10^{-5}\) & \(10^{-6}\) & Normal \\ \hline
**nelx (length) (mm)** & 120 & 12 & Normal \\ \hline
**nely (height) (mm)** & 20 & 2.0 & Normal \\ \hline
**nelz (width) (mm)** & 20 & 2.0 & Normal \\ \hline
**volume (mm\({}^{3}\))** & 0.20 & 0.02 & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Random variables for the MBB beam. from this analysis, are defined as boundary constraints in the RBTO analysis. Thus, the problem can be defined as  
\[ \begin{split}&\text{\emph{Minimize}}\quad V\\ &\text{Subject to :}\quad C-C_{\text{DTO}}\leq 0\\ &\delta-\delta^{*}\leq 0\\ & P[G<0]\leq P_{f}=\beta_{u}\end{split}  \tag{24}\]  
Figure 12 displays the optimal RBTO SESO settings. It is possible to see clearly that the optimal settings are different when compared to the DTO, see Figure 9, as they present fewer bars. This shows that the proposed approach can produce a more reliable structure compared to DTO when uncertainties are considered. Note also that in the MBB beam's design, when the \(\beta_{t}=3.5\), that is, a failure probability \(P_{f}=2.3262\times 10^{-4}\) it takes only seven iterations to reach convergence in the reliability calculation. Thus, the proposed approach only needs to evaluate the finite element analysis (FEA) 68 times, while the traditional RBTO method needs to process FEA 476 times. Therefore, the proposed approach is shown to have high efficiency. Figure 13 shows the graphic of the \(C_{DTO}\) and compliance RBTO (\(C_{RBTO}\)) variation as a function of the target reliability index. Based on the graphic, it can state that the design produced by the RBTO analysis is more rigid than that of the DTO, that is, as the reliability index increases, \(C_{RBTO}<C_{DTO}\). Therefore, the lower the compliance, the higher the stiffness. In deterministic structural optimization, the designer aims to reduce the construction cost without worrying about the effects of uncertainties in materials, geometry, and loading.