XFEM based method for buckling analysis of thin-walled beams

Buckling of thin-walled beams involves complex deformation modes, making their analyses challenging. Most of the methods available for the computation of buckling loads of these elements rely on beam theories, which provide high computational efficiency, yet are limited in accuracy and versatility. On the other hand, the finite element method provides reliable solutions for a variety of structures, albeit with higher computational costs. This paper presents a new approach for the buckling analysis of thin-walled beams. The proposed model is developed by utilizing the eXtended Finite Element Method (XFEM). Global enrichment functions, based on the solutions of beam theory, are added to a standard 3D Finite Element (FE) coarse mesh in the longitudinal direction to enhance its performance and reduce the number of degrees of freedom. We consider the enrichment functions to be a set of modes derived from dynamic free vibrations of a beam or those derived from buckling analysis. The enriched nodes can be viewed as connected lines along the sections of the beam, enabling the computation of lateral-torsional buckling. While our XFEM method can be viewed as a semi-analytical approach, it does overcome well-known limitations of similar methods, e.g., the Finite Prism Method (FPM). For example, our method can in principle be extended to beams with non-prismatic cross-sections (e.g., tapered beams), open cross-sections, or those with perforations. However, these extensions are not discussed in the current paper. Furthermore, the XFEM method can also be applied to beams with different boundary conditions along the beam, such as lateral restraints. Additionally, one can improve the solution by sectional and/or longitudinal mesh refinements.The performance of the proposed method is investigated, and the results are compared with solutions obtained from FE models with a very fine mesh. It is shown that the proposed approach enables modeling thin-walled beams for buckling analysis using coarse FE mesh. In some cases, the number of degrees of freedom in XFEM is less than 5% of those required in the 3D FE model to achieve the same error in computing the first buckling load.