The development of hybrid axisymmetric elements based on the Hellinger-Reissner variational principle

We present a general procedure for the development of hybrid axisymmetric elements based on the Hellinger–Reissner principle within the context of linear elasticity. Similar to planar elements, the stress interpolation is obtained by an identification of the zero-energy modes. We illustrate our procedure by designing a lower-order (four-node) and a higher-order (nine-node) element. Both elements are of correct rank, and moreover use the minimum number of stress parameters, namely seven and 17. Several examples are presented to show the excellent performance of both elements under various demanding situations such as when the material is almost incompressible, when the thickness to radius ratio is very small, etc. When the variation of the field variables is along the radial direction alone, when the mesh is uniform, and the loading is of pressure type, the developed elements are superconvergent, i.e. they yield the exact nodal displacement values