A three-dimensional linear elasticty theory problem is considered for an anisotropic body bounded by two coaxial surfaces of revolution and two planes normal to the axe of the cylindrical frame. The dimensional reduction method is used to reduce the three-dimensional problem to the system of two-dimensional boundary value problems. The Legendre polynomials are used as the expansion system. The constructed finite equations system corresponds to the approximated shel theory of N-th order [3-7]. The exact and approximate solutions for the test problem for the sylinder are constructed and the convergence of the approximated solution to the exact one is shown. The solution of the boundary value problem of second kind for the arbitrary-shaped body of revolution is obtained on the basis of the proposed approach.