Keywords: thin-walled steel frames, post-buckling analysis, non-linear generalised beam theory, generalised beam theory-based beam finite element.

This paper presents the development and illustrates the application of a beam finite element based on generalised beam theory (GBT) and is intended to analyse the local, distortional and global post-buckling behaviour of thin-walled steel frames. After briefly reviewing the main concepts and procedures required to (i) obtain the GBT cross-section (global, distortional, local, warping shear and transverse extension) deformation modes and (ii) establish the corresponding system of non-linear equilibrium equations, the paper describes in detail the steps involved in the numerical implementation of a non-linear beam finite element that incorporates the influence of the frame joint behaviour. In particular, one addresses the constraint conditions developed to model the compatibility between the end section wall warping and transverse bending displacements of two non-aligned lipped/plain channel members connected at a frame joint. The ensuing non-linear algebraic equation system, which governs the frame elastic post-buckling behaviour, is solved by means of an incremental-iterative technique combining Newton-Raphson's method with either a load or a displacement control strategy.

The application and capabilities of the proposed GBT based beam finite element approach are illustrated by presenting and discussing numerical results concerning the local, distortional and/or global post-buckling behaviour of (i) "L-shaped" frames with short and long members and (ii) a symmetric portal frame, all exhibiting flange-continuity joints and containing critical-mode initial imperfections. The GBT based post-buckling results consist of (i) equilibrium paths describing the variation of displacements characterising the frame deformed configuration with the applied load parameter, (ii) GBT modal participation evolution diagrams and (iii) figures providing the evolution of frame deformed configurations as post-buckling progresses - some equilibrium paths are determined with different deformation mode combinations, thus making it possible to assess and quantify the relative importance of the various cross-section deformation modes. For validation purposes, most of the above post-buckling results are also compared with values yielded by beam and shell finite element analyses performed in the code ANSYS - an excellent agreement is found in all cases, despite the huge difference between the numbers of degrees of freedom involved in the two analyses. Therefore, it seems fair to say that the developed GBT beam finite element approach (i) exhibits a high numerical efficiency and, due to its unique modal nature, (ii) provides the means to obtain fresh in-depth insight on the mechanics of the frame post-buckling behaviour.

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