A modified scaling line-based SBFEM for buckling analysis of variable thickness beams under axial compression loading

Buckling is a common phenomenon in beam structures, making the study of the buckling behavior of variable thickness beams highly important. This paper develops a modified scaling line-based scaled boundary finite-element method (SBFEM) for analyzing the buckling of variable thickness beams under axial compression loading. By replacing scaling center with a scaling line, the proposed method retains the advantages of traditional SBFEM, which only requires discretization of the bottom-line or axle-line (reference line) of variable thickness beams. Radial solutions can be derived analytically, thereby improving both the efficiency and accuracy of calculations. Furthermore, this method simplifies modeling, as it requires only a single line to continuously and accurately describe the variable thickness beams, while that is difficult to achieve with the traditional SBFEM, thus reducing the complexity of the modeling process. The inherent characteristics of the method also facilitate a reduction in problem dimensionality, transforming the two-dimensional beam analysis into a one-dimensional problem. This paper formulates relevant equations based on elastic theory and derives the governing equation within the framework of the SBFEM coordinate system by integrating fundamental equations with the principle of virtual work. The stiffness matrix and the geometric stiffness matrix of the structure are determined using a precise integration method, which subsequently allows for the determination of the critical buckling load. The effectiveness of the method is demonstrated through a comparative analysis of two numerical examples, showing minimal error in the corresponding critical buckling loads. Subsequently, the critical buckling loads of various types of variable thickness beams are calculated, and the effects of different boundary conditions, shapes, and aspect ratios on the buckling behavior are investigated. Finally, the buckling problem of variable thickness beams in engineering practice was also discussed.