Keywords: laminated hollow cylinder, vibration, eigenfrequencies, general plane strain, finite element approximation.

In various structures cylinders of circular shape are exposed to impact loading. Such structures involve the linings of tunnels of different kinds, aircrafts structures, submersibles, etc. The problem is solved as pseudo three-dimensional, i.e. generalized plane strain is considered. This assumption is in very good compliance with natural behaviour of the above mentioned structures and moreover, it enables us to describe mathematically several phenomena such as dissipation layers inside the cylinders, optimal distribution of reinforcement, and so on. In the radial direction a linear finite element like approach is introduced. This approach is introduced so that the layers are considered thin enough and because the explicit solution leads us to the Bessel functions and the work with them is not easy and transparent. Eigenfrequencies are calculated using the standard approach for generalized plane strain and the finite element approximation.

This paper is concerned with a pilot study of vibration of cylindrical laminated cylinders or arches. Hamilton's principle is the starting point for the formulation of axisymmetric problem. Introducing a very important simplification: generalized plane strain, and deriving a weak formulation (involving the above mentioned simplification) in cylindrical coordinates, boundary conditions are obtained and the finite element method is employed in radial direction. In former papers of the first author it was shown that the pseudo 3D formulation utilizing the idea of the generalized plane strain condition is usable and delivers results which are in reasonable agreement with the 3D formulation. Assuming generalized plane strain, the only problem occurs in creating relations connected with the axial direction, as considering thick layers the stresses are not in compliance with requirements on their regularity. This is overcome by engaging definition of average stress, which is calculated in cylindrical coordinates. Then a sum of these averages is required to be in equilibrium with the pressure (stress) employed at the face of the cylinder.

Simple examples are studied in the end of this paper. The eigenfrequencies are aggregated with mass density and the arisen coefficient lambda linearly depends on the length of the cylinder. The other eigenfrequencies does not depend on the length, only on the thickness and radius of the structure. If the ratio L/b<1, the coefficient lambda increases exponentially.

The approach in this paper develops ideas presented in [1,2]. Similar problems to that solved in this paper can be found in other papers, such as [3].

1

P. Prochazka, "Optimal eigenstress fields in layered structures", Journal of Computational and Applied Mathematics, 63(1-3), 475-480, 1995. doi:10.1016/0377-0427(95)00093-3

2

G.J. Dvorak, P. Prochazka, "Thick-walled composite cylinders with optimal fiber prestress", Composites Part B: Engineering Volume 27(6), 643-649, 1996. doi:10.1016/S1359-8368(96)00001-7

3

G.A. Kardomateas, "Koiter-based solution for the initial post buckling behavior of moderately thick orthotropic and shear deformable cylindrical shells under external pressure", J. Appl. Mech. 64, 885-896, 1996. doi:10.1115/1.2788996

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