Keywords: thin-walled steel members, 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 novel beam finite element based on a non-linear generalised beam theory (GBT) formulation that extends the scope of an existing one [1] by making it possible to analyse the post-buckling behaviour of thin-walled steel members (i) exhibiting arbitrary open cross-sections (i.e., unbranched or branched sections), (ii) displaying non-standard support conditions (e.g., intermediate supports or in-span localised displacement restraints, such as those stemming from bracing systems) and (iii) subjected to general loading conditions. After a brief review of the main concepts and procedures required to obtain the GBT system of non-linear equilibrium equations, expressed in modal form, the paper describes the steps involved in the numerical implementation of a non-linear beam finite element that (i) adopts Hermite and Lagrange cubic polynomials to approximate the mode amplitude functions, (ii) is capable of handling non-uniform internal forces and moments (e.g., non-uniform bending) and (iii) incorporates the influence of the constraint conditions employed to model localised supports, associated with the full restraint of displacements or rotations taking place at specific points located on the member mid-surface (i.e., exhibiting an intrinsic nodal nature). The ensuing non-linear algebraic equation system is solved by means of an incremental-iterative technique combining Newton-Raphson's method with a load or 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 elastic post-buckling behaviour of unrestrained and restrained simply supported (locally and globally pinned end sections) and uniformly compressed thin-walled lipped channel and lipped I-section steel columns. The GBT-based post-buckling results consist of (i) equilibrium paths describing the variation of displacements characterising the member deformed configuration with the applied load parameter and (ii) figures providing the evolution of longitudinal displacement profiles 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 shell finite element analyses performed using 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 GBT approach developed (i) exhibits a high numerical efficiency and, due to its unique modal nature, and (ii) provides fresh in-depth insight on the mechanics of the column post-buckling behaviour.

1

N. Silvestre, D. Camotim, "Non-linear generalised beam theory for cold-formed steel members", International Journal of Structural Stability and Dynamics, 3(4), 461-490, 2003. doi:10.1142/S0219455403001002

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