Analytical formulation and solution of arches defined in global coordinates

•Arches are studied as a particular case of curved beams defined in global coordinates.•The problem is expressed in a unique system of twelve differential equations.•The differential system of linear ordinary equations is lower-triangular.•We obtain the exact analytical solution through successive integrations row by row.•Parabolic arches which have an optimal shape transmitting vertical load are analysed.

This work deals with the arch defined in global coordinates. It shows the procedure followed to obtain its formulation from the differential system of a curved beam. This procedure consists mainly in applying in the differential system, besides the habitual assumptions of the Strength of Materials, the simplifications of geometric conditions of the planar curves. The resulting model includes the analysis of arches of variable section under any action force, moment, rotation or displacement in its plane. The successive integration of the equations of the system permits obtaining the solution that represents the structural behavior of the arch under any type of support. Applying the boundary conditions of each problem it is obtained directly the exact analytical solution. Examples of calculus of parabolic arches are given to show the practical viability of the procedure followed. Results presented in graphs and tables are comparable with those obtained in the literature. The method is suitable for educational purposes.