Non-linear analysis of structures by the finite element method as a supplement to a linear analysis

The discrete elements of finite dimensions which replace the structural continuum in the finite element method can always be chosen sufficiently small that the linear relations between element deformations and element stresses remain valid to the same degree of approximation as is considered acceptable in the linear theory of elasticity. This observation formed the basis for the treatment of geometrical nonlinearities by Argyris and his co-workers in their natural mode technique [1]and [2].Here we give an alternative development of the theory. The element deformations, linearly related to nodal displacements and rotations in a local coordinate system, are expressed as analytic functions of the nodal coordinates in the global system. Then, for structures with an initially linear behaviour, the stability and postbuckling analysis is developed on the basis of the general theory founded by Koiter [3].The theory is illustrated by the example of frame-structures. The location of the nodal points is defined in terms of the displacement vector, while the orientation of an orthogonal triad attached to each nodal point is described by means of modified angular coordinates of Euler. The accuracy of the analysis is demonstrated for a problem solved analytically by Koiter [5]and verified experimentally by Roorda [4].