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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

BASIC FUNCTIONS FOR THE METHOD OF TWO-SIDED EVALUATIONS IN THE PROBLEMS OF STABILITY OF ELASTICNON-UNIFORMLY COMPRESSED RODS

Vestnik MGSU 6/2013
  • 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 63-70

The author considers the method of two-sided evaluations in the problems of stability of a one-span elastic non-uniformly compressed rod under various conditions of fixation of its ends.The required minimum critical value of the loading parameter for the rod is the minimum value of the functional equal to the ratio of the norms of Hilbert space elements squared. Using the inequalities following from the problem of the best approximation of a Hilbert space element through the basic functions, it is possible to construct two sequences of functionals, the minimum values of which are the lower evaluations and the upper ones. The basic functions here are the orthonormal forms of the stability loss for a rod with constant cross-section, compressed by longitudinal forces at the ends, which are fixed just so like the ends of the non-uniformly compressed rod.Having used the Riesz theorem about the representation of a bounded linear functional in the Hilbert space, the author obtains the additional functions from the domain of definition of the initial functional, which correspond to the basic functions. Using these additional functions, the calculation of the lower bounds is reduced to the determination of the maximum eigenvalues of the matrices represented in the form of second order modular matrices with the elements expressed in the form of integrals of basic and additional functions. The calculation of the upper bound value is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the modular matrices. It is noted that the obtained upper bound evaluations are not worse than the evaluations obtained through the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2013.6.63-70

References
  1. Kupavtsev V.V. K dvustoronnim ocenkam kriticheskih nagruzok neodnorodno szhatyh uprugih sterzhnej. [On Bilateral Evaluations of Critical Loading Values in Respect of Non-uniformly Compressed Elastic Rods]. Izvestija vuzov. Stroitel’stvo I arhitektura. [News of Institutions of Higher Education. Construction and Architecture]. 1984, no. 8, pp. 24—29.
  2. Alfutov N.A. Osnovy rascheta na ustojchivost’ uprugih sistem. [Fundamentals of Stability Analysis of Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  3. Rektoris K. Variatsionnye metody v matematicheskoy fizike I tekhnike. [Variational Metods in Mathematical Physics and Engineering]. Moscow, Mir Publ., 1985, 589 p.
  4. Panteleev S.A. Dvustoronie otsenki v zadache ob ustojchivosti szhatyh uprugih blokov. [Bilateral Assessments in the Stability Problem of Compressed Elastic Blocks]. Izvestyja RAN. MTT. [News of Russian Academy of Sciences. Mechanics of Solids]. 2010, no. 1, pp. 51—63.
  5. Bogdanovich A.U., Kuznetsov I.L. Prodol’noe szhatie tonkostennogo sterzhnja peremennogo sechenija pri razlichnyh variantah zakreplenija torcov [Longitudinal Compression of a Thin-Walled Bar of Variable Cross Section with Different Variants of Ends Fastening (Informftion 1)]. Izvestija vuzov. Stroitel’stvo [News of Institutions of Higher Education. Construction]. 2005, no. 10, pp. 19—25.
  6. Bogdanovich A.U., Kuznetsov I.L. Prodol’noe szhatie tonkostennogo sterzhnja peremennogo sechenija pri razlichnyh variantah zakreplenija torcov [Longitudinal Compression of a Thin-Walled Core of Variable Cross Section with Different Variants of Ends Fastening (Informftion 2)]. Izvestija vuzov. Stroitel’stvo [News of Institutions of Higher Education. Construction]. 2005, no. 11-12, pp. 10—16.
  7. Nicot Francois, Challamel Noel, Lerbet Jean, Prunier Frorent, Darve Felix. Some in-sights into structure instability and the second-order work criterion. International Journal of Solids and Structures. 2012. Vol. 49, no. 1. pp. 132—142.
  8. Aristizabal-Ocha J. Dario. Matrix method for stability and second rigid connections. Engineering Structures. 2012. Vol. 34. pp. 289—302.
  9. TemisYu.M.,Fedorov I.M. Sravnenie metodov analiza ustojchivosti sterzhnej peremennogo sechenija pri nekonservativnom nagruzhenii [Comparing the Methods for Analysing the Stability of Rods of a Variable Cross-section under Non-conservative Loading]. Problems of strength and plasticity [Proceeding sof Nizhni Novgorod University]. 2006, no. 68, pp. 95—106.
  10. Le Grognec Philippe, Nguyen Quang-Hay, Hjiaj Mohammed. Exat buckling solution for two-layer Timoshenko beams with interlayer. International Journal of Solids and Structures. 2012. Vol. 49, ¹ 1. pp. 143—150.
  11. Chepurnenko A.S., Andreev V.I., Yazyev B.M. Energeticheskiy metod pri raschete na ustoychivost’ szhatykh sterzhney s uchetom polzuchesti. [Energy Method of Analysis of Stability of Compressed Rods with Regard for Creeping]. Vestnik MGSU. [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 1, pp.101—108.

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BILATERAL ESTIMATES IN ELASTIC ROD STABILITY PROBLEMS FORMULATED THROUGH BENDING MOMENTS

Vestnik MGSU 2/2013
  • 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 47-54

In the article, the author proposes an original method of identification of upper and lower bounds of critical values of loading parameters in respect of three stability problems for a non-uniformly compressed rectilinear one-span elastic rod with a varying longitudinal bending stiffness value.Initial variational formulations of stability problems under consideration are presented through internal bending moments that emerge at the moment of the rod stability loss and that satisfy uniform boundary conditions rather than additional integral conditions. The author has obtained forms of the bending moment and respective loading parameter values in case of the rod equilibrium bifurcation in the basic problem of stability of an elastic rectilinear rod with a constant cross section, compressed by longitudinal forces at the rod ends.The calculation of the lower bound is reduced to determination of the greatest eigenvalues for the matrices presented in the form of modular matrices of the second order with the elements expressed through the integrals of available forms of bending moments. The calculation of the upper bound is reduced to determination of the greatest eigenvalue for the matrix that almost coincides with one of modular matrices.

DOI: 10.22227/1997-0935.2013.2.47-54

References
  1. Rzhanitsyn A.R Ustoychivost’ ravnovesiya uprugikh system [Stability of Equilibrium of Elastic Systems]. Moscow, Gostekhizdat Publ., 1955, 475 p.
  2. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formulations of Problems of Stability of Elastic Rods Using Bending Moments]. Vestnik MGSU. [Proceedings of Ìoscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 285—289.
  3. Kupavtsev V.V. O variatsionnykh formulirovkakh zadach ustoychivosti sterzhney s uprugo zashchemlennymi i opertymi kontsami [Variational Formulations of Stability Problems for Rods That Have Elastically Fixed and Supported Ends]. Vestnik MGSU [Proceedings of Ìoscow State University of Civil Engineering]. 2011, vol. 3, no. 4, pp. 283—287.
  4. Kupavtsev V.V. K dvustoronnim otsenkam kriticheskikh nagruzok neodnorodno szhatykh sterzhney [On Bilateral Evaluations of Critical Loading Values in Respect of Non-uniformly Compressed Elastic Rods]. Izvestiya vuzov. Stroitel’stvo i arkhitektura. [News of Institutions of Higher Education. Construction and Architecture]. 1984, no. 8, pp. 24—29.
  5. 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.
  6. Izhendeev A.V. Otsenka vnutrennikh usiliy tonkostennogo sterzhnya otkrytogo profilya [Assessment of Internal Forces of a Thin-walled Rod with an Open Profile]. Izvestiya vuzov. Stroitel’stvo. [News of Institutions of Higher Education. Construction]. 2004, no. 3, pp. 100—103.
  7. Chanyshev A.I., Igonina E.A. O potere ustoychivosti beskonechno dlinnoy polosy za predelom uprugosti pri szhatii [On the Loss of Stability of an Indefinitely Long Strip beyond the Elasticity in Compression]. Fizicheskaya mezomekhanika [Physical Mesomechanics]. 2010, vol. 13, no. 51, pp. 89—95.
  8. Paymushin V.N., Gyunal I.Sh., Lukankin S.A. Issledovanie kachestva nelineynykh uravneniy teorii uprugosti na zadachakh ustoychivosti ploskikh krivolineynykh sterzhney sloistoy struktury (postanovka zadachi) [Research into the Quality of Non-linear Equations of the Theory of Elasticity Exemplified by the Problems of Stability of Flat Curvilinear Rods That Have a Layered Structure (Problem Definition)]. Izvestiya vuzov. Aviatsionnaya tekhnika. [News of Institutions of Higher Education. Aeronautical Engineering]. 2010, no. 2, pp. 34—37.
  9. Dudchenko A.V., Kupavtsev V.V. Dvustoronnie otsenki ustoychivosti uprugogo konsol’nogo sterzhnya, szhatogo polusledyashchey siloy [Bilateral Estimates of Stability of an Elastic Cantilever Rod, Compressed by the Half-tracking Force]. Vestnik MGSU [Proceedings of Ìoscow State University of Civil Engineering]. 2011, no. 6, pp. 302—306.
  10. Dudchenko A.V., Kupavtsev V.V. Dvustoronnie otsenki ustoychivosti uprugogo konsol'nogo sterzhnya, szhatogo cherez shatun [Bilateral Estimates of Stability of an Elastic Cantilever Rod, Compressed over the Connecting Rod]. Vestnik MGSU [Proceedings of Ìoscow State University of Civil Engineering]. 2012, no. 7, pp. 75—81.

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