The problem of stresses in a rectangular bar is considered in three statements: 1) with assignment on all boundaries of displacements, 2) stresses and 3) with mixed boundary conditions. The solution is represented by a fast expansion whose coefficients were determined by fast trigonometric interpolation. The solution of the boundary value problem with Dirichlet conditions is the most accurate of the three considered boundary value problems. Compared with this problem, the accuracy of determining the components of the stress tensor and the residual of the Lamé equations in the other two boundary value problems drops by an order of magnitude. The largest residual of the Lamé equilibrium equations is observed in the boundary value problem with given stresses on all sides of the rectangle. Computational experiments showed that the aspect ratio of the rectangle affects the qualitative form of the stress intensity distribution and the location of points with the maximum stress intensity. Among all rectangular sections with different overall dimensions, but the same sectional area, the smallest value of   is observed in a bar with a square section.