Isolation of type Timoshenko equations from spatial theory elasticity plate equations on the base contraction mapping principle

Mathematics. Physics. Mechanics


Zveryaev Е. M.

Keldysh Institute of Applied Mathematics (Russian Academy of Sciences), 4, Miusskaya sq., Moscow, 125047, Russia



A dynamic spatial theory elasticity equation for the plate is taken as initial ones. The equations are reduced to the dimensionless form and a small parameter equal to the ratio of the plate height to its character length is isolated from themes. The integration of the system of the 12 equations with the 12 unknowns is done with the help of the simple-iteration method permitting to establish the asymptotic small parameter development form for every unknown. The transversal displacement and shear stresses are chosen as the started functions permitting to select the prescribed stress-strain state and by means of them to determine the rest unknowns in the null approximation in turn, and at last the same ones. If the start approximation functions are chosen depending on middle plane coordinates only the problem solution is written by the quadrature. The same values being calculated in the first approximation give the correction of the null approximation values. The isolated plan equations have the hyperbolic type whereas bending equation are of hyperbolic type with respect to shear stresses and of parabolic type with respect to transversal displacement. In strength of this the obtained equations might be useful for the chock type solution taking into account the wave processes. If a load vary slowly along coordinates the problem solution is reduced to the slowly vary unknowns function of stresses and displacements. In this case the problem is reduced to the problems sequences in such a way that the output data of the one problem are the input data for the next one.


contraction mapping principle, Timoshenko theory, oscillations, plate theory, small parameter


  1. Tishkov V.V., Firsanov V.V. Aviakosmicheskoe priborostroenie, 2005, no. 1. pp. 10-17.
  2. Firsanov V.V., Tishkov V.V. Aviatsionnaya tekhnika, 2012, no. 4. pp. 30-33.
  3. Firsanov V.V., Tishkov V.V. Nauchnyi vestnik MGTU GA, 2010, no. 161, pp.74-84.
  4. Van-Daik. M. Metody vozmushchenii v mekhanike zhidkosti Perturbation methods in fluid mechanics), Moscow, Mir, 1967, 311 p.
  5. Zveryaev Ye.M., Makarov G.I. Stroitel’naya mekhanika i raschet sooruzhenii, 2013, no. 4, pp. 38-42.
  6. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow, Nauka, 1976, 544 p.
  7. Zveryaev Ye. M. Mekhanika kompozitsionnykh materialov i konstruktsii, 1997, vol. 3, no. 2, pp. 3-19.
  8. Zveryaev Ye. M. Prikladnaya matematika i mekhanika, 2003, vol. 67, no. 3, pp. 472-481.
  9. Zveryaev Ye. M., Makarov G.I. Prikladnaya matematika i mekhanika, 2008, vol. 72, no. 2, pp. 197-207.
  10. Timoshenko S.P., Yang D.Kh., Uiver U. Kolebaniya v inzhenernom dele (Fluctuations in engineering), Moscow, Mashinostroenie, 1984, 472 p.
  11. Grigolyuk E.I., Selezov I.T. Neklassicheskie teorii kolebanii sterzhnei, plastin i obolochek (No classic theory of bar, plate, shell vibration), Moscow, VINITI, 1973, 272 p.
  12.  Zveryaev E.M. Makarov G.I. Vestnik otdeleniya stroitel’nykh nauk RAASN, 2012, vol.16, pp. 84 — 91.

Download — informational site MAI

Copyright © 2000-2021 by MAI