Exact and approximate shell geometry in the free vibration analysis of one-layered and multilayered structures

The present paper proposes the study of the approximation of the curvature terms in the three-dimensional (3D) equilibrium shell equations used for the free vibration analysis of one-layered and multilayered composite and sandwich structures. 3D equilibrium equations written for spherical shells degenerate into 3D equilibrium equations for cylindrical shells and plates considering one of the two radii of curvature or both as infinite, respectively. The approximation of curvature terms has been introduced in 3D equilibrium equations in order to study its effects in terms of frequency values. This study has been conducted by means of a comparison between 3D equilibrium equation results and 3D approximate curvature equilibrium equation results. These effects depend on the thickness and curvature of the considered structure, on the embedded material and lamination sequence, on the frequency order and vibration mode. The 3D equations have been considered in exact form for simply supported structures. The system of partial differential equations has been solved by means of the exponential matrix method. A layer-wise approach is considered for multilayered structures. The approximation of the curvature has been introduced in the 3D equilibrium shell equations and not in the interlaminar continuity conditions and in the top and bottom boundary and loading conditions. This choice has been made for numerical reasons. The investigation of curvature approximation effects in the equilibrium equations allows an exhaustive analysis to understand the importance of curvature terms in the free vibration problems.