Geometry of Submerged Funicular Arches in Cartesian Coordinates

The derivation of funicular shapes for submerged arches has been the subject of a number of studies. In a recent paper, the solution governing the shape of the arch was derived in a coordinate system involving the arc length s and tangent angle θ of the arch. The solution in $s-\theta$ coordinates, however, is not convenient for defining the actual shape of the funicular arch. In this paper, the solution for submerged funicular arches is extended to Cartesian coordinates since the resulting geometric expressions can be more readily used in engineering construction. It will be shown that the submerged funicular arch in Cartesian coordinates involves a combination of elliptic integrals of the first and second kind. An important finding that results from the present formulation is the existence of a minimum compressive force, below which the shape of the funicular arch cannot be determined. Other parameters that are of interest in design are also discussed. The proposed solution for the submerged funicular arch is applied to a design example to illustrate the procedure and effectiveness of the approach.