Derivation of element stiffness matrices by assumed strain distributions

Abstract At the present time, the stiffness matrices of most elements are derived either from an assumed displacement distribution, as in the case of isoparametric elements, or from an assumed stress distribution, as in the case of hybrid elements. In the former case, the displacement distribution is differentiated to find strains, and energy principles are used to complete the derivation of the stiffness matrix. In the latter case the assumed stress distribution is integrated along boundaries to find nodal point forces, and energy principles are used to complete the derivation. The author and his colleagues have, in the past, used the assumed displacement approach almost exclusively in the development of finite elements for the MSC/NASTRAN program and have wholeheartedly embraced the isoparametric method, using ad-hoc fixes (such as reduced order and selective integration) to overcome its numerous deficiencies. Recently, we encountered a case of severe locking in the design of a six-noded, curved shell element, where no amount of tinkering with the isoparametric method would work. The difficulty was finally overcome by applying a radically different approach to the evaluation of the membrane strains. With this approach, a strain field is assumed and then related to nodal displacements by line integration of the strain field along straight line segments between pairs of nodal points. 1 The resulting TRIA6 element compares favorably with our eight-noded QUAD8 element for doubly-curved shell applications. Success led us to speculate about other applications of the new approach, and we soon discovered that we had previously used it for the evaluation of transverse shear strains in our three-noded TRIA3 plate element. This knowledge led us to substitute the new approach for the existing calculation of transverse shear strains in our four-noded QUAD 4 plate element, thereby enabling an arbitrarily shaped element to pass the patch test for all combinations of bending and twisting moments. Most recently we have attempted to design a four-noded membrane element with the new approach. The result was not an unqualified success in that the element does not pass the patch test for arbitrary shape, even though it eliminates the locking phenomenon observed in some applications of the corresponding reduced shear isoparametric element.