Elastic arches with large elevation. Singularly perturbed problem of nonlinear elasticity

In this paper we examine the asymptotic expansion of a three-dimensional displacement field defined over a slender, bowlike domain whose shape corresponds to a thin arch. We show that it is possible to derive from the principles of virtual work two families of arch equations whose forms vary with the scale of external forces and the curvature of the arch. The first family of models, for relatively small curvatures, is described by singularly perturbed ordinary differential equations. The second family, when the curvature of the original arch is large, is described by two-dimensional partial differential equations whose form depends on the shape of the cross-section of the solid arch.