On the free vibration problem of thin shallow and non-shallow shells using tensor formulation

Abstract

Linear free vibration of thin shells with rectangular base such as parabolic conoids, elliptic paraboloids and hyperbolic paraboloids are investigated. These shells are described by non-orthogonal curvilinear coordinates hence, Koiter theory, which uses a tensor formulation, is applied to accurately describe the strain–displacement relations. Four different boundary conditions are considered: simply supported with movable edges, simply supported with immovable edges; fully clamped with movable edges and fully clamped with immovable edges. Using an energy approach, Rayleigh–Ritz method is applied to obtain a set of ordinary differential equations of motion and Fourier series and Legendre polynomials are used to describe the displacements fields of the shell. Both, shallow and non-shallow shells were studied and obtained results of natural frequencies and vibration modes were compared with those from Maguerre theory and FEM using Abaqus, as can be seen, a good agreement was observed with FEM showing the efficacy of the applied theory. When comparing natural frequencies to those from Marguerre theory, there are substantial differences, indicating that lack of use of the tensor formulation which is not appropriate to shells with non-orthogonal coordinates.