A finite element fundamental to thin shell theory

Abstract

This work is intended to contribute to the search for an explanation of the way a really good finite element should behave during analysis of a general shell problem. The finite element analysis of thin shells is currently receiving much attention in the international literature. Indeed it seems that in almost every new issue of the engineering journals there is a proposal for a new and more efficient shell element. The reason for this is of course the underlying complexity of the shell problem and, more specifically the difficulty of taking bending into correct account. In order to elicit an understanding of the use of the finite element method in shell problems an in-depth study is presented of the behaviour of a vehicle shell finite element. This element is the very simple combined constant membrane stress and constant bending moment flat triangle. The examination of its behaviour reveals that the characteristics of an assembly of these elements are such as to enable recovery, in a remarkable way, of each of the types of deformation identified by the classical first approximation theory. Recovery of rigid body movement, inextensional. bending, membrane action and edge effect- is achieved to an accuracy consistent with the order of magnitude of inherent errors of the classical theory. Thus, the element is seen to hold a position of fundamental importance with regard to the numerical analysis of thin shells. Special attention is given to the sensitive low energy bending response. This reveals that there are two quite different roles for the element bending freedoms. One role concerns inextensional bending movements which extend over the whole model. The other role concerns local rotational movements which accompany the curvature changes of inextensional bending and edge effect. Extensive numerical comparisons are made against solutions obtained from the classical theory for shells which are very deep with strongly negative Gaussian curvature and which are considered to provide very severe tests. Investigation of edge effect concerns a cantilevered circular cylinder under edge moment. To complete this examination of bending details are given of a matrix procedure which is intended to assess thin shell finite element models in their response to inextensional bending. To conclude this work the results of a preliminary study of the mathematical details of convergence of the vehicle element are presented. This investigation is specific to the geometry of a circular cylinder and clamped boundary conditions. It is shown that, despite the highly nonconforming nature of the element, O(h) asymptotic convergence in the energy norm is achieved and in this respect is similar in behaviour to the Clough-Johnson flat plate shell finite element.