## 4 Numerical Examples
### Case in Which the Span B Is Fixed  
In this scenario, the span of the arch \(B\) and the depth \(D_{p}\) are fixed. The ranges of values for the parameters in which the CRO-SL seeks the optimal solution are defined in Table 4. The functional weights used for this experiment are \(p_{m}=0.7\) (N\(\cdot\)m)\({}^{-1}\) and \(p_{a}=0.3\) m\({}^{-2}\), then the importance of the arch airspace is slightly greater than for the last scenario. The obtained functional values can be found in Table 5. As it can be observed, the area of the arch for the linear analysis based design is lower than for the nonlinear cases. However, the maximum bending moment for the nonlinear case is more than twice as great as the bending moment of the linear case. The parameters of the optimal solution for both cases, linear and nonlinear, are shown in Table 5. The difference in the contour ellipse height \(H^{\prime}\) is very small, 0.50 m. However, the difference between the heights of the arches is more than 1.00 m, see Figure 8. Furthermore, the aim of reducing the vertical deflection in the nonlinear solution to avoid buckling phenomenon has resulted in the optimal solution having a greater thickness than in the linear solution. The shape parameter \(t_{a}\) is very similar in both cases. Both solutions tend to be a parabola curve since \(t_{a}\) are equal to 0.22 and 0.18 for the linear and nonlinear cases, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Parameter** & **Parameter** & **Value [m]** \\ \hline Fixed parameters & \(D_{p}\) & 10 \\ \(B\) & 30 \\ \hline \multirow{4}{*}{Variable parameters} & \(H\) & \(3\leq H\leq 8\) \\  & \(H^{\prime}\) & \(3\leq H^{\prime}\leq 8\) \\ \cline{1-1}  & \(h\) & \(0.25\leq h\leq 1\) \\ \cline{1-1}  & \(t_{a}\) (dimensionless) & \(0\leq t_{a}\leq 1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Design parameters for the case in which the span \(B\) is fixed. Figure 7: Fitness evolution in the CRO-SL for the submerged arch design case in which the height \(H\) is fixed. The weighted functional expression leads to an increment of the airspace of the arch and a reduction of the maximum bending moment. In this scenario, as the spam is fixed, an increment in the height implies an increment in area. Furthermore, this implies a reduction of the water column above the arch structure. However, this does not imply that the maximum bending moment is reduced, especially if the nonlinear behaviour is considered. In fact, the maximum bending moment of the arch when the nonlinear behaviour is considered is greater than in the linear case. This evidences the importance of the self-weight effect in the optimal design of a submerged arch in shallow waters [31], particularly in their second-order behaviour [18]. Moreover, if the optimal linear solution is simulated under nonlinear conditions, the stability is not reached. The bending moment produces an irreversible increment of the vertical deflection. Therefore, it is shown the importance of considering the nonlinear effects in the arch design. As in the previous scenario, the bending moment and the vertical deflections per unit arch length are shown in Figure 9a,b, respectively. The nonlinear analysis also implies that instabilities issues may appear for small displacement increments. Thus, the CRO-SL has found a stable solution, with a low vertical deflection. In addition, the bending moment along the structure is greater for the nonlinear case. Nevertheless, once more, the redistribution effect in the nonlinear bending stresses produces a significant decrease in nonlinear vertical displacements (see Figure 9b); in this last case, the redistribution effect on the vertical displacements is more significance than in the first scenario due to the greater magnitude of the nonlinear bending moments.