Convergent thin-shell models using cartesian components of the displacements

Starting from the establishment of a deep-shell theory using cartesian components of the displacements, two simplifications of it are considered in the frame of moderate deflections, in view of their finite element implementation. The first one, consisting simply of the use of plane elements, is proved to converge to the deep-shell solution when the mesh is refined, and the rate of convergence is evaluated. A more refined approach is presented, which gains one order of convergence from the former one and reduces to the Marguerre shallow-shell theory when this one is applicable, a condition which is made precise in the text.