Vibration studies on moderately thick doubly-curved elliptic shallow shells

Based on a refined first-order shear deformation theory, the vibratory characteristics of doubly-curved shallow shells of elliptical planform are investigated. Integral expressions incorporating the effects of shear deformation and rotary inertia for the strain and kinetic energies are derived. The transverse shear strain components are obtained as linear functions of thickness in contrast to constant strains available in the current literature. With the use of the extremum energy principle, a governing eigen-matrix equation is formulated which is subsequently solved to extract the frequencies and mode shapes. The shallow shells considered here are the spherical, cylindrical and hyperbolic paraboloidal shells. The effects of various geometric parameters and boundary constraints on the resonant frequencies and mode shapes are examined. The solution method employs displacement functions comprising of a set of two-dimensional orthogonal polynomials and a basic function for each degree of freedom: three orthogonal displacement components and two rotations. The convergence of eigenvalues is examined. The validity of the present results is verified by comparing, if possible, with the values from the literature.