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IssuesArchive of Issues2003-1pp.133-138

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I. S. Astapov, N. S. Astapov, and E. L. Vasil'eva, "Quadratic approximation of large displacements of a flexible compressed beam," Mech. Solids. 38 (1), 133-138 (2003)
Year 2003 Volume 38 Number 1 Pages 133-138
Title Quadratic approximation of large displacements of a flexible compressed beam
Author(s) I. S. Astapov (Moscow)
N. S. Astapov (Moscow)
E. L. Vasil'eva (Novosibirsk)
Abstract We consider examples of approximation of elliptic integrals which demonstrate a method of quadratic approximation of functions defined in terms of power series. This method is a modification of the Padé rational approximation method. It is shown that the known formulas for the approximation of the maximal deflection of a pinned flexible beam subjected to longitudinal compression can be obtained by means of the generalized Padé approximation of a complete elliptic integral of the first kind. New simple formulas are obtained for the maximal deflection and the shape of a flexible beam subject to plane bending.
References
1.  G. A. Baker and P. R. Graves-Morris, Padé Approximants [Russian translation], Mir, Moscow, 1986.
2.  I. F. Obraztsov, B. V. Nerubailo, and I. V. Andrianov, Asymptotic Methods in Structural Mechanics of Thin-walled Structures, Mashinostroenie, Moscow, 1991.
3.  A. H. Nayfeh, Introduction to Perturbation Techniques [Russian translation], Mir, Moscow, 1984.
4.  A. N. Krylov, "On buckling equilibrium shapes of compressed columns subjected to longitudinal bending," in Selected Works [in Russian], pp. 486-538, Izd-vo AN SSSR, Moscow, 1958.
5.  Yu. S. Sikorskii, Elements of the Theory of Elliptic Functions with Applications to Mechanics [in Russian], OGIZ, Moscow, Leningrad, 1936.
6.  N. S. Astapov, "Approximate formulas for the deflection of compressed flexible beams," Zh. Prikl. Mekhaniki i Tekhn. Fiziki, Vol. 37, No. 4, pp. 135-138, 1996.
7.  A. R. Rzhanitsyn, Equilibrium Stability of Elastic Systems [in Russian], Gostekhizdat, Moscow, 1955.
8.  I. A. Birger and R. R. Mavlyutov, Strength of Materials [in Russian], Nauka, Moscow, 1986.
9.  M. M. Filonenko-Borodich, S. M. Izyumov, I. N. Kudryavtsev, et al., Strength of Materials [in Russian], Gosstroiizdat, Moscow, Leningrad, 1940.
10.  M. M. Mostkov, Refined Solutions of Stability and Bending Problems[in Russian], Gosizdat Beloruss., Minsk, 1936.
11.  V. A. Kiselev, Structural Mechanics: A Special Course. Dynamics and Stability of Structures [in Russian], Stroiizdat, Moscow, 1980.
12.  A. N. Dinnik, Stability of Elastic Systems [in Russian], ONTI, Moscow, 1935.
13.  E. L. Nikolai, "On Euler's works on buckling theory," in E. L. Nikolai, Works in Mechanics [in Russian], pp. 436-454, Gostekhteorizdat, Moscow, 1955.
14.  A. S. Vol'mir, Stability of Deformable Systems [in Russian], Nauka, Moscow, 1967.
15.  J. M. T. Thompson, Instability and Catastrophes in Science and Technology [Russian translation], Mir, Moscow, 1985.
16.  N. A. Alfutov, Fundamentals of Stability Analysis of Elastic Systems [in Russian], Mashinostroenie, Moscow, 1991.
17.  R. von Mises, "Zur Steuermathematik," ZAMM, Bd. 4, H. 5, S. 436-438, 1924.
18.  S. P. Timoshenko, Stability of Elastic Systems [in Russian], Gostekhizdat, Moscow, Leningrad, 1946.
Received 28 December 2000
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