The discussion focuses on two numerical algorithms for solving the nonlinear static problems of multilayer composite shells of revolution, namely the algorithm based on the discrete orthogonalization method and the algorithm based on the finite element method with a local linear approximation in the meridian direction. The material of each layer of the shell is assumed to be linearly elastic and anisotropic ( nonorthotropic ).A feature of this approach is that the displacements of the face surfaces of the shell are chosen as unhnown functions, i.e., the functions which allows us to formulate the kinematic boundary conditions on these surfdces. As an example, a cross-ply cylindrical shell subjected to uniform axisymmetric tension is considered.It is shown that the algorithms elaborated correctly describe the local distribution of the stress tensor over the shell thicl~ness without an expensive software based on the 3D anisotropic theory of elasticity.