Geometric stiffness of membranes using symbolic algebra

Abstract

This paper is concerned with derivation of the geometric stiffness matrix for membrane shells which are represented by constant stress triangular finite elements. Symbolic algebra is used to calculate the gradient of the member nodal force vector of each element when the stresses are kept fixed. This gradient defines the geometric stiffness matrix of the element in global coordinates. The present approach follows the earlier works associated with trusses, plane frames and space frames. It has the advantage of explicitness in derivation, while showing clear physical insight. For the case of small rotations, all the mathematical manipulations can be handled by hand. However, for the case of finite rotations, one must have recourse to symbolic algebra programs. The geometric stiffness matrices derived were implanted into an existing nonlinear membrane analysis program that was used to study two examples from the available literature.