The problem of bending vibrations and Euler stability of transversely isotropic beams of circular cross-section with variable radius is considered. The problem is considered within the framework of the refined plate theory, taking into account transverse effects, in the presence of axial force. The obtained equations allow us to determine both the frequencies of natural vibrations and the critical values of the axial force at which the beam loses stability. The problem is solved by the collocation method for various values of geometric and physical parameters characterizing a beam of variable radius. A numerical analysis of the problem is carried out.