Modal analysis of thin parabolic shells

Abstract

The modal analysis of parabolic shells of revolution is based on using the finite-element model of inhomogeneous shell. The shells can have complex-shaped midsurface, geometrical features throughout the thickness, or multilayer structure. To develop the finite-element shell model we approximate a thin shell by one spatial finite element throughout the thickness. The structural elements of an inhomogeneous shell require the finite element to be universal: it should be eccentrically arranged relative to the mid-surfaces of the casing, it should be possible to vary the thickness of the lateral edges of the finite element and ets. The universal finite element is based on an isoparametric spatial finite element with polylinear shape functions for coordinates and displacements. Additional variable parameters are introduced to enhance the capabilities of the modified finite element. Two hypotheses are used to describe the features of the stress–strain state of a thin inhomogeneous shell. The static hypothesis assumes that the compressive stresses in the fibers throughout the thickness are constant. The nonclassical kinematic hypothesis of deformed straight line is used: a straight segment along the thickness remains straight though stretched or shortened during deformation. This segment is not necessarily normal to the mid-surface of the shell.The stress–strain state of a shell and its structural elements is determined using the geometrically nonlinear equations of the three-dimensional theory of thermoelasticity. A linear elastic continuous medium with large displacements and small strains is used as a model whose properties correspond to the generalized Duhamel–Neumann law. To derive the governing finite-element equations for displacements the moment finite-element scheme is used. The moment finite-element scheme approximations of displacements and strains guarantee a correct description of the rigid-body displacements of finite elements, which enhances the convergence and accuracy of solutions on coarse meshes.The natural vibrations of parabolic shells with various heights have been investigated. The convergence of solutions has been studied and compared with the results obtained by other authors.During operation, realistic shell structures often undergo various changes in the temperature field. This can significantly affect their dynamic characteristics. Extension of this work to modal analysis of parabolic shells considering heating is currently being pursued.