The paper presents an approach to the solution of a problem of stability of cylindrical anisotropic shells under the influence of axial compression. The approach is based on the application of the Bubnov-Galerkin procedure, taking into account boundary conditions on surfaces and end edges of cylindrical shells as well as on the application of a numerical method of discrete orthogonalization. It has been solved a problem of stability of cylindrical shells made of a material characterized by one plane of elastic symmetry. It has been investigated a dependence of values of critical stresses on the rotation angle of main directions of elasticity properties in respect to the main curvatures of the cylindrical shell. The results are presented in form of graphs and provided in the tables. Their analysis also has been carried out.

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