An improved mathematical model was constructed to describe the geometrically and physically nonlinear deformation of specimens of a fiber-reinforced plastic with a rectangular cross-section. The specimens had thin elastic side tabs on the clamping ends in the test fixture to transfer the external load to the specimens in kinematic loadings (by the friction forces that arise between the tabs and rigid elements of the test fixture). The specimens had the form of a three-layered rod. For the tabs, the S. P. Timoshenko shear model taking into account the transverse compression was used. For the middle layer across the thickness, a linear approximation for the transverse displacement and a cubic approximation for the axial displacement were accepted. The kinematic relations and equilibrium equations of the theory were obtained based on geometrically nonlinear relations of elasticity theory in a simplified quadratic approximation. They contained geometrically nonlinear terms that, having the necessary degree of accuracy and content, make it possible to identify the classical bending and nonclassical transverse shear buckling modes of the specimens during their compression tests. For unidirectional fiber-reinforced plastics, the physical nonlinearity was taken into account only in the relationship between the transverse shear stress and the corresponding shear strain. When compressing a ±45 fiber-reinforced plastic, the physical nonlinearity was also taken into account in the relation between the normal stress in the specimen cross-section and the corresponding axial strain.