 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| V. I. Gorbachev and O. Yu. Tolstykh, "On an approach to the construction of engineering theory of inhomogeneous anisotropic beams," Mech. Solids. 40 (6), 101-106 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
6 |
Pages |
101-106 |
| Title |
On an approach to the construction of engineering theory of inhomogeneous anisotropic beams |
| Author(s) |
V. I. Gorbachev (Moscow)
O. Yu. Tolstykh (Moscow) |
| Abstract |
Engineering theory of a beam made of a homogeneous isotropic material is
based on kinematic hypotheses specifying the displacement distribution law
in the beam cross-section. In this way, the difficult three-dimensional
problem is replaced by a simpler one-dimensional problem, which permits one
to carry out approximate strength analysis promptly and estimate the
fitness of the structural member. Kinematic hypotheses are also
successfully applied to dynamic calculations as well as to the analysis of
beams working beyond the elastic limit. A major contribution to the
development of these trends in beam theory are due to Il'yushin and Lenskii
[1].
Various kinematic hypotheses result in engineering theories with
various numbers of unknowns. For example, the simplest theory for a
rectangular beam under plane strain is based on the hypothesis that the
normal to the midline remains straight and normal (this is known as the
Kirchhoff hypothesis in plate theory and the Kirchhoff-Love hypothesis in
shell theory [2]) and contains two unknown variables, namely, two
components of the displacement vector of the midline points. The theory
that takes into account not only the displacement vector of the midline
points but also the angle of deflection of the transverse fibers from the
normal to the deformed midline (Timoshenko's theory) already contains three
unknown kinematic variables. In the theory allowing for shear and
transverse compression of the fibers (Reissner theory), one has to seek
four kinematic characteristics, including two components of the
displacement vector, the angle of deflection of the transverse fibers from
the normal, and the compression of transverse fibers. Naturally, such
simplifications result in contradictions; in particular, in the simplest
beam theory one has to assume that the shear modulus and the transverse
Young modulus are infinite and the transverse Poisson ratio is zero. Thus
the material is anisotropic, which contradicts the original assumption.
If the beam material is inhomogeneous and anisotropic, then it is hardly
possible to construct a unified theory of such a beam on the basis of
kinematic hypotheses, since they should somehow take into account the
general character of inhomogeneity and anisotropy. Nevertheless, the
construction of a kinematic theory of inhomogeneous beams is possible in
special cases of inhomogeneity.
In the present paper, we suggest a different method for constructing
engineering beam theory equally suitable for homogeneous isotropic and the
inhomogeneous anisotropic beams. The method is based on the direct
integration of the equilibrium equations in the problem on a plane stressed
state of a long rectangular beam. Two transverse stress tensor components
are expressed from the equilibrium equations via the longitudinal stress.
The equilibrium equations of the classical beam theory are obtained as
necessary conditions for the transverse stresses to satisfy the boundary
conditions on two long sides of the beam. Next, all components of the
strain tensor are expressed from Hooke's law for an inhomogeneous
anisotropic material via the longitudinal stress in the beam. After that,
from the Cauchy relations we find the displacements at each point of the
beam and, moreover, obtain an integro-differential equation for the
longitudinal stress as well as the constitutive equations relating the
internal force factors to the strain and bending of the midline of the
beam.
Thus the original problem on a plane stressed state of a long strip with
integral conditions on the short sides is reduced to a system of two
ordinary differential equations for the two components of the displacement
vector of midline points coupled with one integro-differential equation for
the longitudinal stress. The boundary conditions for the displacements of
midline points follow from the boundary conditions in the original problem.
We suggest to solve the resulting system by the successive approximation
method, the simplest kinematic beam theory being used as the zero
approximation in the new theory. The new engineering beam theory contains
three unknown variables, namely, two components of the displacement vector
of midline points and one longitudinal stress tensor component at each
point of the beam. |
| References |
| 1. | A. A. Il'yushin and V. S. Lenskii, Strength of Materials [in
Russian], Fizmatgiz, Moscow, 1959. |
| 2. | A. G. Gorshkov, E. I. Starovoitov, and D.V. Tarlakovskii,
Elasticity and Plasticity [in Russian], Fizmatlit, Moscow, 2002. |
| 3. | V. A. Lomakin, Elasticity of Inhomogeneous Bodies [in Russian],
Izd-vo MGU, Moscow, 1976. |
| 4. | S. G. Lekhnitskii, Elasticity of Anisotropic Bodies [in Russian],
Nauka, Moscow, 1977. |
| 5. | B. E. Pobedrya, Lectures on Tensor Analysis [in Russian],
Izd-vo MGU, Moscow, 1979. |
| 6. | S. G. Lekhnitskii, Anisotropic Plates [in Russian],
Gostehizdat, Moscow, 1957. |
| 7. | A. G. Gorshkov, V. I. Shalashilin, and V. N. Troshin, Strength of
Materials [in Russian], Fizmatlit, Moscow, 2004. |
| 8. | A. V. Aleksandrov, V. D. Potapov, and B. P. Derzhavin, Strength of
Materials [in Russian], Vysshaya Skola, Moscow, 2001. |
| 9. | V. V. Vasilev, Mechanics of Composite Structures [in Russian],
Mashinostroenie, Moscow, 1988. |
| 10. | S. P. Timoshenko and S. Woinowski-Krieger, Plates and Shells
[Russian translation], Nauka, Moscow, 1966. |
|
| Received |
04 September 2005 |
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