In the first part of the article 1, a physically and geometrically nonlinear boundary-value problem, that describes the compression of a fiber-reinforced plastic rod with 0 s layup, was formulated. The rod had a rectangular cross-section and thin elastic side tabs. The boundary-value problem was reduced to a system of integral-algebraic equilibrium equations containing Volterra integral operators of the second type. To find its numerical solution, the method of finite sums in the variant of integrating matrices was used. The advantage of the method is the possibility of a strong local thickening of the computational grid in the region of large gradients of solutions. Based on the algorithm constructed, an application software package was developed. The results of computational experiments showed that the test specimens under compression according to one of the most commonly used test schemes predominantly failed when the localized transverse shear stresses reached their ultimate values. Failure was also possible according to the shear buckling mode in stress concentration zones. The identification of such modes was possible by using a proposed refined geometrically and physically nonlinear deformation model built in the quadratic approximation with account of transverse shear strains and transverse compression. To verify the numerical method developed, physical experiments were carried out on unidirectional carbon-fiber-reinforced specimens with 0 s layup. They showed a good agreement between the theoretical and experimental results of the research.