## 4 Numerical Examples
### Example 6--Reliability Analysis--Effects of Boundary Conditions and Self-Weight on Bridge Topology  
Figure 11 displays the design domain and boundary conditions of a bridge with three different support locations. The objective is to analyze the influence of the location of the supports on the optimal configuration for different reliability indices. Furthermore, we consider geometry, volume, loading, compliance, and elastic modulus as random variables with a normal distribution and a standard deviation equal to 0.1. In this study, we consider the self-weight of the structure during the optimization process. The material density was defined for a steel structure: \(\rho=7800\,\mathrm{kg/m^{3}}\). The volume fraction is set to \(V=0.25\) and the filter radius is set to \(r=1.5\) m. In each case, the mesh is defined as 120 \(\times\) 30 \(\times\) 10, totaling 36,000 hexahedral finite elements with eight nodes according to [PERSON] et al. (2014). The deck with a thickness \(\mathrm{t}=1.0\) m is defined as a non-design domain area shown as the  
Figure 10: RBTO with own weight: (**a**) BESO and (**b**) SESO. Figure 9: RBTO: (**a**) BESO and (**b**) SESO. darker area in Figure 11, and \(H=14.5\) m, \(L=90\) m, \(c=15\) m, and \(B=10.0\) m. Therefore, the region in the highlighted region will not be allowed to remove solid elements. Figure 12 displays the deterministic optimal topologies obtained by the two methods covered in this article. Although the topologies are identical, the SESO method required a computational cost of \(1.5\) h, resulting in a compliance of \(\mathrm{C}=6.479\times 10^{5}\) N.m, while the BESO method required time \(=1.6\) h of computational cost and resulted in a compliance of \(\mathrm{C}=6.495\times 10^{5}\) N.m. In the optimal configurations shown in the Figure 13a BESO and Figure 13b SESO, reliability analysis was considered with an equal reliability index \(\beta_{t}=3\), but the structure's weight was not considered. It is noted that the topologies are similar, with a final volume \(15\%\) smaller than in deterministic topologies and computational costs \(8\%\) lower for SESO. It is worth mentioning that the compliance presented by BESO is \(0.2\%\) lower than in SESO. Additionally, SESO introduced more tensile cables on tensile arch bridges to help support the bridge load and distribute it effectively. In a tensile arch bridge, the arch supports most of the load through compression, but the tensile cables help distribute some of this load laterally to the supporting piers and ends of the bridge. These tensile cables help to stabilize the bridge structure and prevent excessive deformation under load, thus ensuring the safety and stability of the bridge. Figure 11: Design domain and boundary conditions. Figure 12: Deterministic procedure topology optimization: (**a**) BESO and (**b**) SESO. Figure 13: RBTO considering self-weight: (**a**) BESO and (**b**) SESO. The example represented in Figure 11 was analyzed for different reliability indices, as shown in Table 2, for the two methods proposed in this article. The random variables follow the data in Table 1. It is noteworthy that the RBTO models provided a volume reduction in relation to the DTO models of approximately 9%. It can be seen that the optimal topologies presented while considering the structure's weight are different from the deterministic optimal topologies without considering the self-weight. This is because the tensile arch is mainly responsible for supporting compression loads, while the bridge deck, which is under compression, is responsible for transmitting these loads to the supports. In this case, cables are not needed because the loads are being transferred directly through the arch and deck without the need for an additional tensile cable structure. When considering the structure's self-weight, a fully distributed load and the influence of gravity throughout the analyzed domain are taken into account. Therefore, it results in lighter and more efficient structures, as expected, according to [32].