 | | 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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| F. M. Detinko, "Follower load and the lateral buckling stability of a beam," Mech. Solids. 37 (5), 115-121 (2002) |
| Year |
2002 |
Volume |
37 |
Number |
5 |
Pages |
115-121 |
| Title |
Follower load and the lateral buckling stability of a beam |
| Author(s) |
F. M. Detinko (USA) |
| Abstract |
The lateral buckling stability of a beam has been the
subject matter of numerous studies (e.g., [1-3]). In the present paper,
the lateral buckling stability is investigated for a
beam of a solid cross-section under the action of a follower
force in the plane of its greatest rigidity. For example, of
this type is the load created by a jet engine fixed to the wing of an
aircraft.
A solution of the problem of plane bending is represented in closed form, and
then equations of the perturbed motion of the beam are constructed.
The linearized equation of the perturbed motion can be divided into
two groups, one governing flexural-and-torsional vibrations leading
the beam out of the plane of the undeformed state and the other governing
flexural vibrations in that plane.
The eigenvalues of these equations with variable coefficients are found by
the finite-element method.
It is found that the behavior of eigenvalues corresponding to
flexural-and-torsional vibrations of a beam with rectangular cross-section
can be of two different types. If the ratio of the sides of the rectangle
satisfies the inequality β≤0.65, the variation in the eigenvalues
as the load changes is the same as in the problem of [4, 5]. Namely,
if there is no damping and the load is small, the eigenvalues are purely
imaginary; when the load p reaches a value p=p*, the two lowest
eigenvalues joint together, and for p>p*, there appear eigenvalues
with small real parts. The value p* is the critical load upper bound.
For β>0.65, a large positive eigenvalue appears on the lateral
buckling stability boundary. As the load increases beyond the critical
value, this eigenvalue decreases, then
attains a minimum and increases again.
For β<0.48, the critical value for the load of constant direction
is less than the critical value for the follower load, however, for β>0.48 the former critical value exceeds the latter one.
Everywhere in this paper, we deal with Lyapunov stability in small. |
| References |
| 1. | S. P. Timoshenko, Stability of Elastic Systems [Russian translation],
Gostekhizdat, Moscow, 1955. |
| 2. | D. H. Hodges and D. A. Peters,
"On the lateral buckling of uniform slender cantilever beams,"
Intern J. Solids and Structures, Vol. 11, No. 12, pp. 1269-1280, 1975. |
| 3. | Yong Lin Pi and N. S. Trahair,
"Prebuckling deflections and lateral buckling, I. Theory,"
J. Struct. Eng., Vol. 118, No. 11, pp. 2949-2966, 1992. |
| 4. | V. V. Bolotin,
Nonconservative Problems in the Theory of Elastic Stability [in Russian],
Fizmatgiz, Moscow, 1961. |
| 5. | Ya. G. Panovko and I. I. Gubanova,
Stability and Vibrations of Elastic Systems [in Russian], Nauka, Moscow, 1987. |
| 6. | K. F. Chernykh,
Nonlinear Theory of Elasticity in Machine Design [in Russian],
Mashinostroenie, Leningrad, 1986. |
| 7. | A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity
[Russian translation], ONTI, Moscow, Leningrad, 1935. |
| 8. | G. G. Denisova and V. V. Novikov,
"On the stability of a beam loaded by a "follower" force,"
Izv. AN SSSR. MTT [Mechanics of Solids], No. 1, pp. 150-154, 1975. |
|
| Received |
11 May 2000 |
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