The subject of this paper is a simply supported functionally graded beam in three variants: (A) without rigid partitions, (B) with two rigid partitions, and (C) with four rigid partitions. The mechanical properties, specifically Young’s modulus, vary symmetrically in the depth direction of these beams. An analytical model of the beam, considering the individual nonlinear shear deformation of a plane cross-section, has been developed. Two differential equations of equilibrium for the beam, based on the principle of stationary total potential energy, were derived. This system of two equations was approximately solved for each beam variant, and the critical forces were determined with the distinguished shear effect coefficient. Detailed calculations were performed for exemplary beams with selected dimensionless sizes and mechanical properties. The results of these calculations are presented in tables and figures. The main goal of the research is to develop an analytical model of this beam and investigate the impact of rigid partitions placed within the beam on the critical force.