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A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
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Online ISSN 1934-7936 |
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| Yu. I. Butenko, "A modified method of asymptotic integration: application to the construction of the theory of orthotropic rods. Part I," Mech. Solids. 36 (4), 70-82 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
4 |
Pages |
70-82 |
| Title |
A modified method of asymptotic integration: application to the construction of the theory of orthotropic rods. Part I |
| Author(s) |
Yu. I. Butenko (Kazan) |
| Abstract |
The ways of justifying the theory of plates is the subject matter of
current discussion in the literature. In this connection, the applicability
of the classical theory based on the Kirchhoff hypotheses is being
contrasted with that of the Reissner-Timoshenko theory. It is known that
a refinement of the basic stress state (called the penetrating solution, or
the solution of the internal problem, in terms of the asymptotic methods)
can be achieved by refining the equations of bending of plates and
a more precise statement of the boundary conditions. In the present paper,
we consider as a model
the plane problem for an orthotropic strip
and construct multi-term asymptotic expansions which yield a refinement
for the internal problem, as well as the boundary layer problem (rapid
decay away from the edge), and as a result, we obtain more precise boundary
conditions. The symmetric problem and the problem of bending are
constructed for the strip with the accuracy ε2 (ε=h/a is a small
parameter). It is shown that the Reissner-Timoshenko model for the rod
does not take into account the boundary layer solutions,
and therefore, the refinements of the basic stress state are due
only to more precise equations of equilibrium.
At the same time, the consideration of a more complex model of
rod bending (for instance, with the accuracy ε2 for all
parameters of the problem) leads to refinements in the
internal problem with the parameter ε under the
geometrical boundary conditions. This fact should be taken into
account when constructing a theory of plate design.
The longstanding problem of the construction
of the theory of beams, plates, and shells is, in essence,
an asymptotic problem [1-4]. In [3, 4],
an asymptotic method is proposed for the integration of
elasticity equations in order to obtain the corresponding
equations of the theory of plates and shells. In fact, this method
leads to the successive integration of the desired quantities with
respect to the transverse coordinate and their representation as power
series in that coordinate.
On the other hand, the desired quantities can be directly represented
as a series in powers of the transverse coordinate and the
corresponding differential equations can be obtained on the basis of
variational principles. The choice of the specific variational principle
depends on the choice of the basic unknown quantities.
Representation of the displacements as power series in the transverse coordinate
and subsequent application of variational principles for the construction of
the theory of plates and shells are described in [5-7].
However, this approach encounters substantial computational difficulties,
which are mainly due to the fact that the resulting differential equations and
boundary conditions involve the basic (internal) stress state (BSS) and the
boundary layer solutions (BLS). For theories capable of application it
is desirable to have separate (mutually independent) methods
for the determination of the internal solution (BSS) at the points of
the entire structure and the solution rapidly decaying away from the
edge (BLS).
When constructing
approximate theories (models), the difficulties of separating the stress states
can be overcome by asymptotic methods applied to
the system of equations obtained by variational methods [5-7],
as shown in the present paper for the theory of
orthotropic beams treated in geometrically and physically
linear formulations. The method proposed here is in a certain
sense close to the method of symbolic integration of elasticity equations
[8] and leads to results which, in many aspects, coincide with those
of [3, 9] obtained for the plane elasticity problem by the asymptotic
method. The representation of the displacements as asymptotic series with
respect to the transverse coordinate has been used in [10, 11] for the
construction of nonclassical theories of plates and shells. |
| References |
| 1. | I. I. Vorovich, "Some results and problems
of the asymptotic theory of plates and shells,"
Proceedings of the First All-Union Workshop on Numerical Methods
in the Theory of Shells and Plates [in Russian], pp. 51-149, Izd-vo Tbilisskogo
Un-ta, Tbilisi, 1975. |
| 2. | A. L. Goldenveizer,
Theory of Thin Elastic Shells [in Russian], Nauka, Moscow, 1976. |
| 3. | A. L. Goldenveizer,
"An approximate theory of plate bending obtained by the method of asymptotic integration
of elasticity equations," PMM [Applied Mathematics and Mechanics], Vol. 26, No. 4, pp. 668-686, 1962. |
| 4. | A. L. Goldenveizer, "An approximate theory of shells
obtained by the method of asymptotic integration of elasticity equations,"
PMM [Applied Mathematics and Mechanics], Vol. 27, No. 4, pp. 593-608, 1963. |
| 5. | Kh. M. Mushtari and I. G. Teregulov,
"On the theory of shells of medium thickness,"
Doklady AN SSSR, Vol. 128, No. 6, pp. 1144-1147, 1959. |
| 6. | I. G. Teregulov, "On the theory of plates of medium thickness,"
Proceedings of the Conference on the Theory of Plates and Shells,
pp. 367-375, Izd-vo Kazanskogo Un-ta, Kazan, 1961. |
| 7. | I. G. Teregulov, "On the construction of refined theory of plates
and shells," PMM [Applied Mathematics and Mechanics], Vol. 26, No. 2, pp. 346-350, 1962. |
| 8. | A. I. Lur'e, Three-Dimensional Problems in Elasticity [in Russian],
Gostekhizdat, Moscow, 1955. |
| 9. | L. A. Agalovyan, Asymptotic Theory of Anisotropic Plates and
Shells [in Russian], Nauka, Moscow, 1997. |
| 10. | V. V. Vasil'ev and S. A. Lur'e,
"On the construction of nonclassical theories of plates,"
Izv. AN SSSR. MTT [Mechanics of Solids], No. 2, pp. 148-167, 1990. |
| 11. | V. V. Vasil'ev and S. A. Lur'e,
"On the problem of refining the theory of shallow shells,"
Izv. AN SSSR. MTT, No. 6, pp. 139-146, 1990. |
| 12. | A. Nayfeh, Perturbation Methods [Russian translation], Mir,
Moscow, 1976. |
| 13. | M. I. Vishik and L. A. Lyusternik,
"Regular degeneration and boundary layer for linear
differential equations with a small parameter,"
Uspekhi Mat. Nauk, Vol. 12, No. 5, pp. 3-122, 1957. |
| 14. | S. P. Timoshenko and J. Goodier,
Theory of Elasticity [Russian translation], Nauka, Moscow, 1975. |
|
| Received |
05 February 1999 |
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