Technical notes

Abstract

A decade has elapsed since triangular elements under constant strain and stress were independently introduced within the matrix Displacement Method by the Turner group at Boeing and the Aeronautical Structures Department at Imperial College. They proved a considerable success in the practical analysis of complex membrane shapes as occur in modern aircraft wings. However, the original formulation based on a local cartesian system of axes and the classical concept of stress and strain inevitably leads to an unsymmetrical and aesthetically (as well as computationally) unfavourable expression for the stiffness matrix k. Moreover, the method was restricted to isotropic media. The necessity of extending the theory to finite displacements ultimately showed the inadequacy of the past approach and guided the writer to the creation of new concepts, e.g. the natural stiffness of an element. This natural stiffness, which ignores the rigid body motions and is hence of dimensions (3 X 3) for a triangle with three nodal points, relates in a concise manner the nodal force components parallel to the sides and the elongations of the sides themselves. A powerful tool in its derivation and further development was a new definition of stress and strain vectors which excludes shearing contributions and only works with direct components parallel to the sides. The idea has also been extended to tetrahedra and many standard elements which are incorporated as a basic feature in the structural language ASKA (Automatic System for Kinematic Analysis) developed at ISD by the Applied Programming Group under Dr. H. Kamel. These fundamental ideas were emphasised in the author's recent lecture to the Society.