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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Variational formulations of the integral equation of stability of elastic bars

Vestnik MGSU 9/2012
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics 8 (499) 183-46-74, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 137 - 143

The author considers the variational formulations of the problem of stability of non-uniformly
compressed rectilinear elastic bars that demonstrate their variable longitudinal bending rigidity in the
event of different classical conditions of fixation of bar ends.
Identification of the critical bar loading value is presented as a minimax problem with respect
to the loading parameter and to the transversal displacement of the bar axis accompanied by the
loss of stability. The author demonstrates that the critical value of the loading parameter may be formulated
as a solution to the dual minimax problem. Further, the minimax formulation is transformed
into the problem of identification of eigenvalues in the bilinear symmetric and continuous form, which
is equivalent to the identification of eigenvalues of a strictly positive, linear and completely continuous
operator. The operator kernel is presented in the form of symmetrization of the non-symmetric
kernel derived in an explicit form.
Within the framework of the problem considered by the author, the bar ends are fixed as follows:
(1) both ends are rigidly fixed, (2) one end is rigidly fixed, while the other one is pinned, (3) one
end is rigidly fixed, while the other one is attached to the support displaceable in the transverse direction,
(4) one end is rigidly fixed, while the other one is free, (5) one end is pinned, while the other
one is attached to the support displaceable in the transverse direction, (6) both ends are pinned.

DOI: 10.22227/1997-0935.2012.9.137 - 143

References
  1. Rzhanitsyn A.R. Ustoychivost’ ravnovesiya uprugikh system [Stability of the Equilibrium State of Elastic Systems]. Moscow, Gostekhizdat Publ., 1955, 475 p.
  2. Alfutov N.A. Osnovy rascheta na ustoychivost’ uprugikh system [Principles of the Stability Analysis of Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  3. Rektoris K. Variatsionnye metody v matematicheskoy fi zike i tekhnike [Variational Methods in Mathematical Physics and Engineering]. Moscow, Mir Publ., 1985, 589 p.
  4. Litvinov V.G. Optimizatsiya v ellipticheskikh granichnykh zadachakh s prilozheniyami k mekhanike [Optimization in Elliptic Boundary-value Problems Applicable to Mechanics]. Moscow, Mir Publ., 1985, 368 p.
  5. Litvinov S.V., Klimenko E.S., Kulinich I.I., Yazyeva S.B. Ustoychivost’ polimernykh sterzhney pri razlichnykh variantakh zakrepleniya [Stability of Polymer Bars in Case of Various Methods of Their Fixation]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 4, vol. 2, pp. 153—157.
  6. Il’yashenko A.V. Lokal’naya ustoychivost’ tavrovykh neideal’nykh sterzhney [Local Stability of Tshaped Imperfect Bars]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 162—166.
  7. Tamarzyan A.G. Dinamicheskaya ustoychivost’ szhatogo zhelezobetonnogo elementa kak vyazkouprugogo sterzhnya [Dynamic Stability of a Compressed Reinforced Concrete Element as a Viscoelastic Bar]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 1, vol. 2, pp. 193—196.
  8. Dudchenko A.V., Kupavtsev V.V. Dvustoronnie otsenki ustoychivosti uprugogo konsol’nogo sterzhnya, szhatogo polusledyashchey siloy [Two-way Estimates of Stability of an Elastic Cantilever Bar, Compressed by a Half-tracking Force]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 1, vol. 6, pp. 302—306.
  9. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formulations of Problems of Stability of Elastic Bars Derived by Using Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 285—289.
  10. Kupavtsev V.V. O variatsionnykh formulirovkakh zadach ustoychivosti sterzhney s uprugo zashchemlennymi i opertymi kontsami [About the Variational Formulations of Stability Problems for Bars with Elastic Fixation of Supported Bar Ends]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 4, pp. 283—287.

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