Vibration analysis of shallow or deep, complete parabolic shells with variable thickness

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of shallow or deep, clamped, complete parabolic shells having variable thickness along the meridional direction by the Ritz method. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies from the present analysis are compared with those from the finite element method based 2-D thin shell theory. Nondimensional frequencies are presented for a variety of shallow or deep parabolic shells having variable thickness.