## 4 Numerical Examples
### Case in Which the Height \(H\) Is Fixed  
In this scenario, the height \(H\) and the depth \(D_{p}\) are fixed, while the parameters \(B\), \(H^{\prime}\), \(h\), and \(t_{a}\) are tuned within a range (see Table 2). The functional weights used for the calculation are \(p_{m}=0.8\) (N\(\cdot\)m)\({}^{-1}\), \(p_{a}=0.2\) m\({}^{-2}\). Thus, the reduction of the bending moment is more important than increasing the airspace. The results have been obtained for a single run simulation. The best solutions for both cases, linear and nonlinear, are shown in Table 3. The arch shape found by the CRO-SL is basically the same for the linear and the nonlinear cases (see Figure 4). However, the thickness of the structure (\(h\)) for the nonlinear based design is almost the double than for the linear design. This implies that, although the shape of the arch is the same for both scenarios, the increment of the bending stiffness of the arch has lead to a greater thickness of the structure to avoid instabilities issues. The optimal curve shape has been found between the parabola (\(t_{a}=0\)) and the ellipse (\(t_{a}=1\)), for both cases. The parameters used to calculate the functional can be found in Table 3. As expected, the area for both arches is the same. However, the maximum bending moment for the nonlinear case is greater than in the linear case. Then, its functional value reaches a higher \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Parameter** & **Parameter** & **Value [m]** \\ \hline Fixed parameters & \(D_{p}\) & 10 \\ \(H\) & 6 \\ \hline Variable parameters & \(H^{\prime}\) & \(5\leq H^{\prime}\leq 9\) \\ \(B\) & \(15\leq B\leq 30\) \\ \(h\) & \(0.25\leq h\leq 1\) \\ \(t_{a}\) (dimensionless) & \(0\leq t_{a}\leq 1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design parameters for the case in which the height \(H\) is fixed. Figure 4: Arch shapes comparative for linear and nonlinear solutions for the case in which the height \(H\) is fixed. Note that both curve guidelines are coincident and the thickness of the non-linear solution is represented with external dashed lines. value. As it was aforementioned [18], instabilities issues can occur under service loads for many of the arch shapes proposed so far, and this example shows the importance of considering the nonlinear effects in the arch design. The bending moment and the vertical deflections per unit arch length are shown in Figure 5a,b, respectively. The nonlinear analysis implies that instabilities issues may appear for small displacement increments. The CRO-SL has found a stable solution with low vertical deflection. In this first scenario, the coincidence of both shapes (linear and nonlinear solution) allows clarifying the effectiveness of the optimization procedure adopted in this work, as the structural stability is achieved mainly through the arch thickness increase. In this sense, such increase not only augments the bending stiffness as well as bending stresses (as predictable, due to the self-weight increasing), but causes the characteristic redistribution of the bending moment already shown in previous works [18]. However, in this particular case, such redistribution is more relevant for \(B/4\leq x\leq 3B/4\), where the nonlinear bending stresses decreases respect to the linear one, that produces a generalized reduction of the vertical displacements in the arch; this is due to the different stress redistribution that the arch experiments depending on the geometrical behaviour hypotheses: linear or nonlinear. Therefore, under certain design conditions (as the corresponding to this scenario), the second-order optimization of submerged arches is not only feasible (as shown previous works, using other evolutionary strategies, such as the genetic algorithms [18; 20]), but it is necessary in order to obtain a significant reduction in the structural vertical displacements.