DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Basic functions and bilateral estimatesin the stability problems of elastic non-uniformly compressed rods expressed in terms of bending moments with additional conditions

Vestnik MGSU 2/2014
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Мoscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 39-46

The method of two-sided evaluations is extended to the problems of stability of an elastic non-uniformly compressed rod, the variation formulations of which may be presented in terms of internal bending moments with uniform integral conditions. The problems are considered, in which one rod end is fixed and the other rod end is either restraint or pivoted, or embedded into a support which may be shifted in a transversal direction.For the substantiation of the lower evaluations determination, a sequence of functionals is constructed, the minimum values of which are the lower evaluations for the minimum critical value of the loading parameter of the rod, and the calculation process is reduced to the determination of the maximum eigenvalues of modular matrices. The matrix elements are expressed in terms of integrals of basic functions depending on the type of fixation of the rod ends. The basic functions, with the accuracy up to a linear polynomial, are the same as the bending moments arising with the bifurcation of the equilibrium of a rod with a constant cross-section compressed by longitudinal forces at the rod ends. The calculation of the upper evaluation is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the elements of the modular matrices. It is noted that the obtained upper bound evaluation is not worse thanthe evaluation obtained by the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2014.2.39-46

References
  1. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formulations of the Problems of Elastic Rods Stability Using Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, vol. 3, no. 4, pp. 285—289.
  2. Alfutov N.A. Osnovy rascheta na ustoychivost' uprugikh sistem [Fundamentals of the Stability Analysis of the Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  3. Kupavtsev V.V. Dvustoronnie otsenki v zadachakh ustoychivosti uprugikh sterzhney, vyrazhennykh cherez izgibayushchie momenty [Bilateral Estimates in Elastic Rod Stability Problems Formulated through Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 2, pp. 47—54.
  4. Rektoris K. Variatsionnye metody v matematicheskoy fizike i tekhnike [Variational Methods in Mathematical Physics and Engineering]. Moscow, Mir Publ., 1985, 589 p.
  5. Doraiswamy Srikrishna, Narayanan Krishna R., Srinivasa Arun R. Finding Minimum Energy Configurations for Constrained Beam Buckling Problems Using the Viterbi Algorithm. International Journal of Solids and Structures. 2012, vol. 49, no. 2, pp. 289—297. DOI: 10.1016/j.ijsolstr.2011.10.003.
  6. Panteleev S.A. Dvustoronnie otsenki v zadachakh ob ustoychivosti szhatykh uprugikh blokov [Bilateral Assessments in the Stability Problem of Compressed Elastic Blocks]. Izvestiya RAN. MTT [News of the Russian Academy of Sciences. Mechanics of Solids]. 2010, no. 1, pp. 51—63.
  7. Santos H.A., Gao D.Y. Canonical Dual Finite Element Method for Solving Post-buckling Problems of a Large Deformation Elastic Beam. International Journal of Non-Linear Mechanics. 2012, vol. 47, no. 2, pp. 240—247. DOI: 10.1016/j.ijnonlinmec.2011.05.012.
  8. Selamet Serdar, Garlock Maria E. Predicting the Maximum Compressive Beam Axial Force During Fire Considering Local Buckling. Journal of Constructional Steel Research. 2012, vol. 71, pp. 189—201. DOI: 10.1016/j.jcsr.2011.09.014.
  9. Tamrazyan A.G. Dinamicheskaya ustoychivost' szhatogo zhelezobetonnogo elementa kak vyazkouprugogo sterzhnya [Dynamic Stability of the Compressed Reinforced Concrete Element as Viscoelastic Bar]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, vol. 2, no. 1, pp. 193—196.
  10. Manchenko M.M. Ustoychivost' i kinematicheskie uravneniya dvizheniya dinamicheski szhatogo sterzhnya [Dynamically Loaded Bar Stability and Kinematic Equations of Motion]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 71—76.

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