Keywords: web-tapered I-beam, stability analysis, lateral-torsional buckling, finite element, non-linear strain, energetic principles, equilibrium equations.

Since steel members have been extensively used in structural engineering, designers have always been trying to achieve the greatest material economy. Combined with the general tendency to increase structure and member slenderness and with the development of high strength steels, instability problems become more and more important. As a consequence, the optimisation of steel structures gets with increasing complexity.

In that way, tapered beams and columns present not only a great economical interest, because they allow a non negligible material saving, but also make the conception rather consistent. Because geometrical characteristics are still not constant along the beam, modelling their behaviour appears to be much more difficult than for prismatic ones, especially for stability aspects. Computer assistance becomes then essential. However, a few number of analytical solutions, related to some particular cases, are available in the literature, as well as simplified methods and abacus. But these methods cannot cover all cases, and may not be accurate enough to justify the increased complex design compared to material economy. Then, resorting to numerical methods (i.e. the finite element method) should also provide more accurate and accessible solutions. Indeed, it is a common use to divide the member into several prismatic finite elements, in order to approximate the variation of the cross-sections. And of course, the best the meshing is, the higher the precision. But this technique requires a lot of elements for being accurate, and might lead to erroneous results, as it is shown here.

The objective of this paper is first to propose a complete analytic expression for the non linear axial strain of generally tapered thin-walled beams, based on an adapted beam theory. It contains also the presentation of a new beam finite tapered element, able to account with more accuracy for the particularities of these members. The model has been built using Vlassov's transversally rigid section assumptions, and energetic principles. Its originality lies in the fact that it contains several terms that are not considered in the classical prismatic models, in the definition of the non linear axial strain, what results in qualitatively different equilibrium equations. All of them are given and briefly commented, for the particular case of web-tapered I-beams. The additional terms, only due to cross-section variations, concern all cases where torsional aspects play a role, like lateral-torsional buckling for example. It is then obvious that the segmentation technique with prismatic elements cannot lead to exact results in such cases, even with a large number of elements.

A special beam finite element have then been carried out in order to obtain more correct results, and is presented here. It has been derived accordingly with the analytic model, and developed to account for a complete set of cases: spatial behaviour, large displacements, material non linearity, linear or non linear stability, monosymmetric cross-sections, influence of initial imperfections and of beam deformations during loading. The paper explains how it was implemented in a finite element software, and special attention is devoted to particular aspects, like the way locking problems are solved for instance.

For the particular case of linear elasticity, several results are presented. First, the case of plane strong axis flexural buckling is investigated, in order to show the convergence properties of the element. Then, the case of linear torsion is analysed, because first order linear analysis must also be adapted. Comparison with classical prismatic elements solution is also given. In the case of torsional buckling, it is shown that the segmentation technique may also lead to about 30% inexact results in a critical load calculation. Finally, the paper presents the more practical case of lateral-torsional buckling, through a tapered cantilever and a simply supported beam. These examples enable to highlight the differences compared to the segmentation technique and to other authors solutions.

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