Timoshenko beam theory-based buckling of axially functionally graded columns on elastic foundations: a complementary functions method approach

Abstract The primary purpose of this research is to investigate the buckling behavior of nonuniform axially functionally graded (AFG) columns resting on a Winkler-type elastic foundation under axial compressive loading. To accurately capture the influence of shear deformation on the critical buckling load, the governing equations are formulated based on the Timoshenko beam theory (TBT). The resulting differential equation governing the buckling behavior is solved using the Complementary Functions Method (CFM). A major advantage of the CFM in this context is its ability to directly incorporate geometric nonuniformity and continuous material gradation without requiring predefined shape functions. This numerical technique transforms the two-point boundary value problem into an equivalent initial value problem, which is subsequently solved using the fourth- and fifth-order Runge–Kutta methods (RK4 and RK5) implemented in Python. Singularities arising from the formulation of the governing differential equation are analyzed in detail, and the valid solution domains of the CFM are identified. The numerical results demonstrate the effects of material gradation, column slenderness, boundary conditions, and foundation stiffness on the buckling response. Key findings reveal that an increase in foundation stiffness consistently enhances the critical buckling loads, whereas structural tapering reduces overall stability. Additionally, it is shown that as the column slenderness increases, the critical loads calculated using the TBT converge to those of the classical Euler–Bernoulli theory. The findings indicate that the proposed approach serves as an effective and consistent alternative for analyzing the buckling behavior of AFG columns resting on Winkler-type elastic foundations.