A simple ‘mixed’ method for plate and shell problems

Abstract

Plates and shells are usually analyzed by the finite element method, and particularly by the deformation method. In this method the most interesting interior forces and moments are obtained as differences of the unknown deformations, which often leads to a considerable loss in accuracy. To avoid this loss in accuracy, the interior forces and strains are introduced as unknowns directly. These forces and strains as well as the geometrical curvatures of the shell and loads will be approximated linearly from point to point. The structures may be divided into triangular and rectangular elements. The interior forces and strains do not satisfy the equilibrium and compatibility conditions, and therefore equilibrium and compatibility equations have to be formulated as in a mixed method. This is done by means of the principles of virtual work, but in a very simple manner, for example as virtual displacements rigid body motions of the elements are chosen. The interior forces act only along the grid lines, thus it is quite simple to establish the equations. By using the duality between bending and stretching the amount of computer programming is reduced. Both problems can be treated with the same program, which must be slightly extended for application to shallow shells. In the case of the deformation method the stiffness matrices for stretching and bending unfortunately have a quite different structure and it is not easy to develop simple stiffness matrices for double curved shells. Furthermore, the stretching and bending stresses have different approximations and they are discontinuous at the boundaries of the elements. In the method described in this paper each stress and strain has the same approximation with no discontinuities at the boundaries of the elements. Results will be shown for stress problems in plates and shells.