Optimized Transfer Matrix Approach for Global Buckling Analysis: Bypassing Zero Matrix Inversion

Abstract

The transfer matrix method has two main disadvantages concerning other numerical methods: numerical instability in extreme cases and the need to calculate the inverse of the zero matrix. This paper attempts to solve the second difficulty of the transfer matrix method, widely used for the global buckling analysis of beams in all engineering fields. In particular, the transfer matrix method necessarily requires the calculation of the inverse of the zero matrix to derive the element transfer matrix, resulting in high computational costs as the number of discretizations and the size of the matrix increases. To mitigate this challenge, this paper presents a transfer matrix method that directly computes the transfer matrix without requiring the inverse of the zero matrix. The method adopts a Laplacian approach, which involves the application of Laplace transforms to the equilibrium equations and subsequent inverse Laplace transforms to express displacements and internal forces relative to the zero point of the coordinate origin significantly reducing computational costs to a minimum. Numerical applications corroborate the effectiveness and superiority of the proposed approach.