Abstract
The paper proposes a three-dimensional elastic analysis of the free vibration problem of one-layered spherical, cylindrical, and flat panels. The exact solution is developed for the differential equations of equilibrium written in orthogonal curvilinear coordinates for the free vibrations of simply supported structures. These equations consider an exact geometry for shells without simplifications. The main novelty is the possibility of a general formulation for different geometries. The equations written in general orthogonal curvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panel, cylindrical closed shell, and plate cases. Results are proposed for isotropic and orthotropic structures. An exhaustive overview is given of the vibration modes for a number of thickness ratios, imposed wave numbers, geometries, embedded materials, and angles of orthotropy. These results can also be used as reference solutions to validate two-dimensional models for plates and shells in both analytical and numerical form (e.g., closed solutions, finite element method, differential quadrature method, and global collocation method).
Highlights
Exact solutions for the three-dimensional analysis of plates and shells have been developed by several researchers
The present paper aims to fill this gap by proposing a general formulation for the equations of motion in orthogonal curvilinear coordinates that is valid for square and rectangular plates, cylindrical shell panels, spherical shell panels, and cylinders
Equilibrium equations are for spherical shell panels; they automatically degenerate into equilibrium equations for cylindrical closed/open shell panels when Rα or Rβ is infinite and into equilibrium equations for plates when Rα and Rβ are infinite
Summary
Exact solutions for the three-dimensional analysis of plates and shells have been developed by several researchers. Exact closed form solutions (based on three-dimensional elasticity theory) were carried out in [33] for both inplane and out-of-plane free vibration of thick homogeneous supported rectangular plates coated by a functionally graded (FG) layer. Xu and Zhou [35] used the three-dimensional elasticity theory to derive the general expressions for the displacements and stresses of the plate under static loads, which exactly satisfy the governing differential equations and the supported boundary conditions at four edges of the plate. The equations of motion written in orthogonal curvilinear coordinates allow general exact solutions for supported plate and shell geometries with constant radii of curvature These equations are exactly solved, for the first time in this paper, by means of the exponential matrix method
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