Vibratory Characteristics of Cantilevered Rectangular Shallow Shells of Variable Thickness

Abstract

In this paper, a comprehensive study on the modeling of vibratory response of variable thickness cantilevered shallow cylindrical shells of rectangular planform is carried out. Through the principle of minimum total energy, the equations of stretching and bending strain energies, kinetic energy, and the associated boundary conditions of the shells are derived. The present model is developed based on the Ritz formulation with the assumption of sets of mathematicall y complete admissible two-dimensional polynomials to approximate the in-plane and transverse displacement amplitude functions. A basic function is introduced in the approximate method to enforce automatic satisfaction of the kinematic boundary conditions. Sets of comprehensive reasonably accurate vibration frequencies of the shallow shells are presented for wide ranges of aspect ratio, thickness variation ratios, and shallowness ratio. These results, where possible, are verified by comparing with other established experimental and theoretical solutions. Few selected contour plots for the first known mode shapes of these cylindrical shells of variable thickness are included.