## 4 Numerical Examples
### Example 4--An Optimal Bridge  
The optimization of a bridge-type structure with the deck at the top was studied by [51]. The dimensions of the design domain, the loading conditions, and the boundary conditions are given in Figure 14. The material properties are considered to be \(=210\) GPa and \(v=0.30\). To ensure that pressure is applied over the entire top surface, a solid section of 1.5 m thick is defined as a non-projectable layer. The design aims to maximize the stiffness subject to a prescribed volume of \(V_{f}=0.25\). Figure 12: Optimal topologies for different reliability indexes. From the DTO results, for each method proposed in this study, the OF's maximum value is obtained, which is a reciprocal measure of the stiffness and quantifies the structure's energy. The next step is to include the uncertainties in the geometry, loading, and volume (structure dimensions' function) and define the RBTO problem, whose target is to obtain a minimum volume design as rigid as the DTO design when the uncertainties are considered. This problem requires the assessment of probabilistic constraints, which are expressed in terms of the boundary state function G, defined as  
\[G=1-\frac{1}{C_{DTO}V_{f}} \tag{25}\]  
where \(C_{DTO}\) is the DTO's compliance in the optimal setting, and \(V_{f}\) is the volume fraction. Generally, \(G>0\) denotes a safe region where \(G<0\) corresponds to a failure region. The probabilistic constraints induce the RBTO to obtain a structural framework at least as rigid as the one provided by the DTO, assuming a target \(P_{f}\). Consequently, the RBTO problem is formulated as indicated by Equation (26):  
\[ \begin{split}\text{\emph{Minimize}}&\quad V\\ \text{Subject to:}&\quad C-C_{DTO}\leq 0\\ \delta-\delta^{*}\leq 0\\ P[G<0]\leq P_{f}=\beta_{u}\end{split}  \tag{26}\]  
where P [\(\cdot\)] is the probability operator. The initial design parameters are given in Table 4, where nelx (length), nely (height), and nelz (width) represent the structure's geometry, P, which represents the external load  
Figure 14: Design domain, loading and boundary conditions for a bridge, where the solid section denotes the non-designable material. Figure 13: Variation of DTO and RBTO compliance as a reliability index function. distributed on the bridge deck, and they are considered random variables with normal distribution, while the material properties have constant distribution. Table 5 shows the proposed approach's efficiency by comparing the variation in the number of iterations to achieve convergence in the reliability calculation procedure and the number of iterations involved in the TO procedure. In addition, it shows that finite element analysis (FEA), with this formulation, has a lower computational cost than that used in traditional RBTO. Figure 15 displays the optimal topologies achieved by the proposed algorithms using \(\beta_{u}=3.8\). Note that the volume found in the RBTO analysis is \(V_{f}=0.225\), which means a 9% reduction compared to the DTO analysis. However, the optimal topologies found by ESO, SESO and SERA are similar but different when compared to the optimal topology presented by the SIMP technique. For the 3D results' comparison, the volumes, displacements, OF and computation time due to each method, for the bridge, are listed in Table 6.  
The RBTO formulation of [16] is adapted to the three-dimensional RBTO analysis. In this example, two studies are performed: (1) random variables--geometry, volume and applied force (the same as considered by the authors); and (2) random variables--geometry, volume, force, with the inclusion of the elasticity modulus, and compliance. The first analysis compared to the DTO design shows a decrease in structural mass. However, it can notice an increase in the structure's compliance. In this sense, the design obtained by DTO is more rigid than the one proposed by RBTO analysis. Figure 16a shows the graphic of force by the structure's failure probability and Figure 16b illustrates the volume and force random variables by reliability index \(\beta_{i}\).