Abstract Damping properties of flat layered plates and beams have been studied previously by many researchers. In a limited sense, the properties of curved structures have also been analyzed by the present, and other, authors. In such studies, the influence of shear deformation and thickness-wise extensional deformation on damping were neglected. In the present paper, the fundamental equations for symmetric and non-symmetric vibrations have been derived from Flügge's theory, the Herrman-Mirsky theory and the three-dimensional elasticity theory. The damping properties in bending and extensional vibrations, as well as the effects of shear deformation and thickness-wise extensional deformations have been included in the present work. Numerical data has been presented in this paper for a simply supported cylindrical shell.