## 2 Problem Statement: Nonlinear Submerged Arches Design
### Alternative Approach: Shape Parameterization  
In the case of a nonuniform arch of arbitrary shape, it becomes impossible to present the buckling solution in analytical form [32]. For these cases, the solution may be obtained using only approximate procedures, such as the variational methods. In the case of arches in shallow waters, the solution of differential equation of stability may be obtained in closed form, as previously indicated in the first epigraph of this Section. Many important solutions related to stability of arches have been obtained by [PERSON] [35] or [PERSON] [1]. Likewise, buckling of funicular arches under gravity loading and overburden (the so-called _Inglis problem_[36; 37]) has been solved numerically by several authors [1; 38], under different support conditions, reaching in all the cases a good adjustment. The buckling loads were calculated for an arch with a relatively high slenderness greater than 100. Moreover, an approximated solution to the buckling load has been also developed for the hinged support problem [39].  
On the other hand, the optimal design of the submerged arches in terms of their second-order response may be approached using a parameterization of the arch's centerline. In this sense, note that, in practice, the submerged arches are not strictly funicular due to aspects like geometric imperfections, or design load variations (e.g., tidal variation), among other reasons [31]. For these cases, bending moments cannot be completely eliminated, but they can be reduced resulting in a well-dimensioned arch with a negligible bending stress. In this sense, alternative design procedures, based on the arch shape optimization using parametric functions [40], are feasible. This is the strategy adopted in this work.  
The solution of Equation (1) without considering self-weight load lays halfway between parabolic and elliptical types for intermediate values of ratios \(D_{p}/H\) and/or \(H/B\)[41]. A geometrical parameterization over these two conical forms is proposed in [20]. Thus, the equations of the parabola and the ellipse intersecting at the apex and the support of the arch (see Figure 2), and having the same pole, may be expressed, respectively, as  
\[r_{p}(\theta,B,H,H^{\prime})=\frac{B^{2}sec^{2}(\theta)\Bigg{[}\sqrt{\frac{4 cos^{2}(\theta)\left[1+\frac{H-H^{\prime}}{H^{\prime}}\right]}{B^{2}}+\frac{ sin^{2}(\theta)}{H^{\prime 2}}-\frac{sin(\theta)}{H^{\prime}}}\Bigg{]}}{2 \left[1+\frac{H-H^{\prime}}{H^{\prime}}\right]}, \tag{3}\]  
\[r_{e}(\theta,B^{\prime},H^{\prime})=\frac{1}{\sqrt{\frac{cos^{2}(\theta)}{B^{ \prime 2}}+\frac{sin^{2}(\theta)}{H^{2}}}}, \tag{4}\]  
where \(r\) and \(\theta\) are the radial coordinate and the angular coordinate, respectively, and \(H^{\prime}\) and \(B^{\prime}\) are the lengths of the semi-minor axis and the semi-major axis of the ellipse, respectively. From the above considerations, the funicular solution is easily approximated using a linear combination of the two conic types:  
\[r_{pe}(\theta,B,H,H^{\prime},t_{a})=(1-t_{a})\cdot r_{p}\big{(}\theta,B,H,H^{ \prime}\big{)}+t_{a}\cdot r_{e}\big{(}\theta,B^{\prime},H^{\prime}\big{)} \tag{5}\]  
where \(t_{a}\in[0,1]\) and \(r_{pe}\) is the radial coordinate of the intermediate curve corresponding to the parameter \(t_{a}\) and the angular coordinate \(\theta\). Equation (5) represents a parametric family of curves between parabolic (\(t_{a}=0\)) and elliptical (\(t_{a}=1\)) forms, where the shape parameterization is posed as a function of four parameters: \(t_{a}\), \(H^{\prime}\), \(B\) and \(B^{\prime}\). However, only three of them are independent, as the arch support is assumed to be contained in the ellipse defined in Equation (4), and in this case, \(H^{\prime}\) and \(B^{\prime}\) must verify  
\[B^{\prime}=\frac{B}{\sqrt{1-\left[\frac{H-H^{\prime}}{H^{\prime}}\right]^{2}}} \tag{6}\]