This paper investigates the buckling of a Timoshenko beam with a variable cross-section on an elastic foundation using the Ritz energy method and considering spring boundary conditions. To evaluate the effects of shear deformation in the energy equations, two approaches are presented: an indirect approach based on the angle (θ), and a direct approach based on the angle (β). During the derivation of the longitudinal displacement field (u), a new term associated with the change in angle is obtained. Cosine functions and Bézier polynomials are employed to approximate the angular and transverse displacement fields, respectively. Finite element results obtained using Abaqus are used to validate the analytical results. The convergence results show that both approaches converge to the same solution as the number of approximation terms increases. However, for low shear coefficients (GI<100), the indirect approach achieves satisfactory convergence with fewer terms, whereas for high shear coefficients (GI>500), the direct approach exhibits faster convergence with fewer terms. Comparison of the analytical, finite element, and reference results shows that, in most cases, the differences between the results are less than 15%. For shear coefficients below 100, the buckling loads obtained from the finite element analysis are slightly higher than the analytical and reference results, with the smallest discrepancy observed for the simply supported boundary condition. Furthermore, for beams with variable cross-sections, the formulation that does not account for the strain induced by cross-sectional variations provides better agreement with the finite element results. Finally, increasing the elastic foundation stiffness increases the buckling load for all boundary conditions, although this increase becomes less pronounced as the beam slope increases.