Beam elements in instability problems Original Research Article

This paper deals with the formulation of beam elements for the numerical analysis of instability phenomena in frame-type structures. Total versus co-rotational approaches are discussed comparatively, for both two-dimensional and three-dimensional problems, and the similarities between the two types are outlined. In the context of 3D beam elements, special attention is given to the parameterisation of the orthogonal transformation used to define the rotational field of the beam. The technique advocated in the paper is based on the so-called rotational vector. This leads to symmetric stiffness matrices and avoids the need for special updating procedures for the rotational variables. A set of test problems, for which the critical behaviour is governed by fold, cusp and butterfly catastrophes, is used to assess the performances of the considered element types. It is shown that analytically verified identities in element formulation, also hold in numerical application. The examples also show how complex instability behaviour can be reproduced by all elements, where sufficient accuracy is introduced into the kinematic expressions. The analytical derivation of element expressions, with symbolic manipulations from stated basic assumptions, is consistently used in the paper.