Keywords: Generalized Beam Theory (GBT), thin-walled members, non-linear elastic-plastic members, plastic buckling, local-plate modes, distortional modes, global modes.

Most cold-formed metal members display very slender thin-walled cross-sections, a feature making them highly susceptible to both local (local-plate and distortional) and global buckling phenomena. In fact, their structural efficiency is greatly affected by these phenomena. Moreover, unlike carbon (mild) steels, aluminium and stainless steel alloys display non-linear elastic-plastic stress-strain relationships, often exhibiting a considerable amount of strain hardening. This material difference implies that buckling (bifurcation) frequently occurs in the non-linear (elastic-plastic) range and, therefore, it is necessary to account for the material non-linear behaviour along the fundamental equilibrium path. Up until now, the structural analysis of such thin-walled members was only possible by resorting to computationally intensive numerical techniques, essentially adopting fine meshes of either shell elements or finite strips.

The Generalized Beam Theory (GBT), originally developed by Schardt (1989) and subsequently employed by Davies et al. (1996, 1998) and Silvestre & Camotim (2002, 2003, 2004), has proven to be a rather powerful, elegant and clarifying analytical tool to study the local and global (elastic) bifurcation, dynamic and post-buckling behaviour of cold-formed steel members. However, up until last year, the available GBT formulations could only be applied to linear elastic members and, for instance, it was not possible to use them to analyse members made of non-linear materials. This limitation was (partially) overcome by the authors (2003, 2004), who derived and numerically implemented a materially non-linear GBT formulation, specifically aimed at studying the buckling behaviour of thin-walled members acted by simple loadings (e.g., columns or doubly symmetric beams under uniform compression or bending). In this paper, an extension the above materially non-linear GBT formulation is derived, numerically implemented and illustrated. This extended formulation makes it possible to analyse the buckling (bifurcation) behaviour of open and closed (single-cell hollow) thin-walled elastic-plastic members subjected to more general loading conditions than before (e.g., members subjected to non-uniform internal force and moment diagrams). Indeed, the loading conditions that can be considered are now only restricted by the basic hypothesis of the available plastic buckling theory for elastic-plastic solids, which states that no unloading may take place along the fundamental equilibrium path (Hill 1958). When describing the main steps involved in the derivation of this novel materially non-linear GBT formulation, particular attention is paid to the implications of the non-linear constitutive modelling in the GBT procedure, namely in the cross-section and member analyses.

Concerning the GBT member analysis, the paper also presents the development of a 2-node bar (beam) finite element, which is based on the use of Hermite cubic polynomials to approximate each mode rate amplitude function and has 4 degrees of freedom (n is the number of deformation modes included in the analysis). Although bearing some resemblance with the finite element formulated by Silvestre & Camotim (2003), this bar element exhibits a few novel aspects: (i) is valid for non-linear materials, (ii) allows for the variation of internal forces and moments along the member and (iii) is obtained directly from the virtual work equation.

After performing the GBT cross-section analysis and defining the member tangent stiffness and geometric matrices ( and , respectively), both dependent on the load parameter (through the instantaneous elastic moduli and internal stress distribution), it is necessary to solve the non-linear eigenvalue problem defined by the "adjacent equilibrium condition" of the hypoelastic "comparison member" (Hutchinson 1974), to evaluate the critical load parameter. In the general case, this requires a "trial-and-error" procedure involving the performance of pairs of (i) pre-buckling materially non-linear (but geometrically linear) analyses, to obtain the active elastic moduli and stress values at the fundamental path configuration associated with given values, and (ii) hypoelastic linear stability analyses, to determine critical load parameter values . The procedure ends when the and values are "sufficiently close", for a given tolerance. Then, provides an accurate estimate of the true critical load value.

Finally, in order to illustrate the application and capabilities of the proposed non-linear GBT formulation and bar FEM numerical implementation, several numerical results (critical stresses/loads and buckling mode shapes) are presented and discussed. They concern the buckling behaviour of (i) elastic-plastic uniformly compressed simply supported rectangular plates, (ii) elastic and elastic-plastic simply supported RHS beams under uniform major axis bending and (iii) elastic lipped channel cantilevers subjected to point loads applied at the shear centre of the free end section. While in the first two (simply supported) cases sinusoidal shape functions are used to approximate the modal rate amplitude functions (they provide exact solutions), the analysis of the cantilevers employs the bar FEM implementation formulated in this paper. Moreover, both -deformation and -flow small-strain plasticity theories are considered in the analysis of the elastic-plastic plates and RHS beams. Several elastic GBT results are compared with values yielded by shell FEM analyses performed in the codes ABAQUS or ADINA. However, due to software limitations only the plastic bifurcation behaviour of the uniformly compressed rectangular plates is adequately validated - through the comparison with an exact analytical solution (Handelman & Prager 1948).

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