Stability of a Non-uniform Column resting on a Foundation, calculated with the Finite Element Method

This paper presents the application of the finite element method to columns with non-uniform cross-sections resting on elastic foundations, solving the eigenvalue problem of finding the critical load. The formula for calculating the stiffness matrix resulting from column bending, elastic foundation, and geometric stiffness is established based on the principle of virtual work. Based on the finite element formulas, an algorithm is established in MATLAB to find the column’s critical force. The results obtained using the proposed approach agree with the exact solution obtained with analytical methods. In many cases, the calculation results of the critical force are given to assess the effects of the foundation’s stiffness and boundary condition on the critical load.