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IssuesArchive of Issues2006-4pp.106-117

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V. P. Georgievskii and P. B. Pilipenko, "Vibrations and stability of orthotropic cylindrical shells attached to an elastic cylinder of finite length," Mech. Solids. 41 (4), 106-117 (2006)
Year 2006 Volume 41 Number 4 Pages 106-117
Title Vibrations and stability of orthotropic cylindrical shells attached to an elastic cylinder of finite length
Author(s) V. P. Georgievskii (Moscow)
P. B. Pilipenko (Moscow)
Abstract Papers dealing with problems of vibrations and stability of shells [1] attached to an elastic cylindrical body (filler) can be divided into three groups depending on the computation scheme used for the filler.

In the first group, the filler is modeled by a cylinder with elastic base with one (the Winkler base) or two (the Pasternak base) moduli of subgrade reaction. The second group contains papers in which the filler is considered as an elastic three-dimensional body and is described by either the elasticity equations or the equations obtained from these equations by reducing the spatial problem to a two- or one-dimensional problem without taking into account the subcritical stress state of the filler. The last, third group contains the papers in which the subcritical state of the filler is taken into account. In these papers, the three-dimensional linearized equations of elastic stability obtained by linearizing the nonlinear equations of elasticity are used.

In solving the problems based on the first model of the filler, one can find the basic laws of loss of stability of filled shells. The papers in the second group contain more rigorous statements of stability problems for filled shells. In these problems, the mixed boundary conditions (stating that the axial stresses and the radial displacements are zero) are posed at the ends of the filler, which corresponds to the case of an infinitely long cylinder. The case of a filler with free ends was studied approximately in [2], where the problem was solved by the energy method and only the radial interaction between the shell and the filler was taken into account. The stability of filled shells with the subcritical stress state of the filler taken into account was studied by Vlasov, Ivanov, German, and Forrestal [1]. They showed that, in a wide range of rigidity and geometric parameters, the subcritical state of the filler can be neglected and its operation can be described by the linear Lamé equations. Although there are numerous papers dealing with vibrations and stability of orthotropic shells attached to an elastic filler, many problems have not been studied yet:

There is no rigorous solution of the problem on the vibrations and stability of a compressed cylindrical shell attached to an elastic filler under different boundary conditions on its ends, including the ends free of stresses; experimental verification of theoretical solutions has been insufficient.

In the present paper, the filler is considered as an elastic isotropic body of finite length, which corresponds to the case in which four possible versions of the boundary conditions are considered at the ends of the filler.

The mathematical statement of the problem consists of differential equations that describe the behavior of the "shell-filler" system in question and boundary conditions at the ends of the filler and the shell as well as on the boundary surface of the filler.
References
1.  V. A. Ivanov, "Survey of the literature in stability of filled shells," in Proceedings of Seminar in Theory of Shells [in Russian], No. 2, pp. 5-25, Kazansk. Fiz.-Mat. In-t AN SSSR, 1971.
2.  A. P. Varvak, "Axially symmetric loss of stability of filled cylindrical shells," Prikl. Mekhanika, Vol. 3, No. 3, pp. 33-41, 1967.
3.  A. I. Lur'e, Spatial Problems in Elasticity Theory [in Russian], Gostekhizdat, Moscow, 1955.
4.  K. D. Tranter, Integral Transforms in Mathematical Physics [in Russian], Gostekhizdat, Moscow, 1956.
Received 31 March 2004
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