A systematic deduction of linear natural approach equations

Abstract

Natural Approach equations of a finite element are the discrete counterpart of the continuous field elastic equations (equilibrium, compatibility and constitutive), and their use has clear advantages, in different contexts, over classical stiffness procedures. This paper presents a systematic, algorithmic approach for the building of the Natural Approach equations for displacement finite elements in the linear case. The proposed method starts from the global static matrix of the element and all the discrete kinematic and static relations are obtained from the kernel and generalized inverse of this matrix, which allows equilibrium and compatibility discrete matrices to be obtained by explicit analytical formulas. Alternatively, kernel and generalized inverse matrix can be built numerically, using an algorithm included in the paper. A procedure to achieve a diagonal discrete constitutive matrix is developed. The proposed method is applicable to any kind of curvilinear isoparametric element, and explicit results for plane and solid elements are given.