Abstract <p>In the paper, the Saint-Venant for a naturally twisted rod with a rectangular cross section has been considered. The problems have been studied based on a method of homogeneous solutions in conjunction with the finite element method. The general solution has been constructed as a linear combination of elementary solutions corresponding to three four-roots eigenvalues of spectral problem on the cross section. Elementary solutions determining the stress-strain state of Saint-Venant type contain unknown eigenvectors and associated vectors. To determine of unknown solutions, we have previously formulated boundary value problems and their variational formulations. They correspond to problems of stretching-torsion, pure bending and bending of the lateral force. Variational problems have been solved using a finite element method. The stress-strain state of the rod has numerically been studied, while non-zero elements of the stiffness matrix have been found in the case of square and rectangular cross sections of the rod for the different values of twist. The numerical results for a wide range of change of the twist parameter τ have graphically been shown. Calculations have shown that the identified patterns are consistent with the corresponding behavior untwisted rods (for small twist parameter), and, at the growth of the twist, new effects, which confirm the hypothesis proposed earlier.</p>