Differential formulation and numerical solution for elastic arches with variable curvature and tapered cross-sections

In this paper, an alternative analytical and numerical formulation is presented for the solution of a system of static and kinematic ordinary differential equations for curved beams. The formulation is represented here as being useful in the structural evaluation of arch structures and it is propaedeutic to be used in optimization frameworks. Using a finite-difference method, this approach enables the evaluation of the best solution sets accounting for (1) various arch shapes (i.e. circular, quadratic and quartic polynomial forms); (2) different loading combinations; (3) cross-sections varying along the arch span; and (4) different global radius of arch curvature. The presented original approach based on finite-difference, produces solutions with a good precision in a reasonable computational time. This method is applied to 4 different arch case studies with varying rise, cross-section, loading and boundary conditions. Results show good agreement with those obtained using a numerical finite element approach. The presented approach is useful for the (preliminary) design of arches, a common and efficient structural typology for road and railway bridges and large span roofs. As far as buckling verifications are concerned, the comparisons in terms of maximum acting axial force and critical axial force computed is reported in order to consider the effect of instability phenomena coming to trace for each arch configuration the feasible domain.