  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
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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), 106117 (2006) 
Year 
2006 
Volume 
41 
Number 
4 
Pages 
106117 
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 threedimensional 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 onedimensional 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 threedimensional
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 "shellfiller" 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. 525, Kazansk. Fiz.Mat. Int AN SSSR, 1971. 
2.  A. P. Varvak,
"Axially symmetric loss of stability of filled cylindrical shells,"
Prikl. Mekhanika, Vol. 3, No. 3, pp. 3341, 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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