Keywords: generalized beam theory, thin-walled members, cold-formed steel, post-buckling, bifurcation analysis, natural discretization.

The paper contains a study of the post-buckling behaviour of thin-walled channel columns in the context of the generalized beam theory (GBT) [1]. The present analysis models the member by means of a GBT Total Potential Energy (TPE) formulation [2] and the equilibrium system is solved through the Rayleigh-Ritz method, whose coordinate functions are orthonormal polynomials directly derived from the relevant boundary conditions [3]. The scheme relaxes some assumptions of the classical GBT theory [1], thus enabling a more precise stability analysis of thin-walled prismatic members, the full stress-state definition at any point of the member and the account of all relevant energy terms of the TPE. The relaxed assumptions [4] are: i) the null membrane transversal extensions, ii-a) the null shear distortion in open sections or ii-b) the constant shear flow in closed cells, and iii) the linear warping displacements between two consecutive main nodes of the cross section (nodes at the cross section edges or at the folding lines). The analysis of the critical state and the search of post-critical equilibrium paths are made by the adequate numerical techniques, respecting the bifurcational behaviour of the member's equilibrium system, and thus enabling all relevant information about modal interaction at the bifurcational state and beyond bifurcation [5].

To illustrate the concepts just presented, a thin-walled channel section is analysed in the buckling and post-buckling domain. It is noted that four buckling zones occur, depending of the member's length, although only three were presented here - see Figure 38.1. For a member whose length falls in the distortional length range, the post-buckling behaviour was analysed by computing the equilibrium paths and the evolution of stresses and displacements beyond bifurcation. The post-critical behaviour is shown to be stable, although little additional load capacity occurs after buckling. Figure 38.2 illustrates the member general configuration for a load level P equal to 1.078 times the critical load.

1

R. Schardt, "Verallgemeinerte Technische Biegetheorie", Springer-Verlag, Berlin-Heidelberg, Germany, 1989.

2

P. Simão, L. Simões da Silva, "A unified energy formulation for the stability analysis of open and closed thin-walled members in the framework of the Generalized Beam Theory", Thin-Walled Structures, 42(10), 1495-1517, 2004. doi:10.1016/j.tws.2004.03.021

3

P. Simão, "Post-buckling analysis of thin-walled prismatic members in the context of the Generalized Beam Theory", PhD Thesis, University of Coimbra, Coimbra, Portugal, 2005 (soon available at http://www.dec.uc.pt/~pedro).

4

P. Simão, L. Simões da Silva, "Post-buckling behaviour of open cross-section thin-walled columns in the context of the Generalized Beam Theory", V.P. Iu, L.N. Lamas, Y.-P. Pi, K.M. Mok (Eds.), Computational Methods in Engineering and Science - Proceedings of the 9th International Conference on Enhancement and Promotion of Computational Methods in Engineering and Science - EPMESC IX, A.A. Balkema Publishers, 891-898, 2003.

5

P. Simão, L. Simões da Silva, "A numerical scheme for post-buckling analysis of thin-walled members in the context of GBT", Proceedings of the International Conference on Computational and Experimental Engineering and Sciences, ICCES 2004, Funchal, Madeira, 2079-2086, 2004.

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