The paper presents the results of studying the natural vibrations and stability of circular cylindrical shells made of functionally graded materials. The external surface of the structure
is heated and loaded by uniformly distributed pressure. The effective temperature-dependent properties of the material vary through the thickness of the shell according to a power law.
The distribution of temperature along the radial coordinate is defined by solving a quasi-linear one-dimensional equation of thermal conductivity. The relations of the classical shell theory
are reduced to the system of eight ordinary differential equations for new unknown quantities. Integration is carried out by applying Godunov’s orthogonal sweep method at each iteration
of the step-wise procedure used to calculate eigenfrequencies of vibrations. The dependences
of the lowest vibration frequencies on temperature and/or mechanical loads have been determined for circular cylindrical shells with variable consistency of functionally graded material under different boundary conditions.