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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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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Two-sided evaluations based on the variational formulations of integral equations for the stability of elastic rods

Vestnik MGSU 10/2014
  • 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 41-47

The author considers the method of two-sided evaluations in solving the problems of stability of one-span elastic non-uniformly compressed rod with variable longitudinal bending rigidity in case of different classic conditions of fixation of the rod ends. The minimum critical value of the loading parameter for the rod is represented as a problem of calculating minimum value of the functional corresponding to the Euler equation, which is the same as the integral equation for the rod stability. Using the inequalities following from the problem of the best approximation of a Hilbert space element through the basic functions, the author constructs two sequences of functionals, the minimum values of which are the lower evaluations and the upper ones for the required value of the loading parameter. The basic functions here are the derivative forms of the stability loss for a rod with constant cross-section, compressed by longitudinal forces applied at the rod ends. The calculation of the lower bounds value is reduced to the determination of the maximum eigenvalues of block matrices. The elements of the aforesaid matrices are expressed through the integrals of basic functions depending on the type of the fixation of the rod ends. 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 by the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2014.10.41-47

References
  1. Kupavtsev V.V. Variatsionnye formulirovki integral'nogo uravneniya ustoychivosti uprugikh sterzhney [Variational Formulations of the Integral Equation of Stability of Elastic Bars]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 9, pp. 137—143. (in Russian)
  2. Rzhanitsyn A.R. Ustoychivost' ravnovesiya uprugikh system [Stability of Equilibrium of Elastic Systems]. Moscow, GITTL Publ., 1955, 475 p. (in Russian)
  3. Alfutov N.A. Osnovy rascheta na ustoychivost' uprugikh system [Fundamentals of the Stability Analysis of the Elastic Systems]. 2-nd edition. Moscow, Mashinostroenie Publ., 1991, 336 p. (in Russian)
  4. Kupavtsev V.V. Bazisnye funktsii metoda dvustoronnikh otsenok v zadachakh ustoychivosti uprugikh neodnorodno-szhatykh sterzhney [Basic Functions for the Method of Two-sided Evaluations in the Problems of Stability of Elastic Non-uniformly Compressed Rods]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 63—70. (in Russian)
  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. Mekhanika tverdogo tela [News of the Russian Academy of Sciences. Solid Body Mechanics]. 2010, no. 1, pp. 51—63. (in Russian)
  6. Santos H.A., Gao D.Y. Canonical Dual Finite Element Method for Solving Post-Buckling Problems of a Large Deformation Elastic Beam. International Journal Non-linear Mechanics. 2012, vol. 47, no. 2, pp. 240—247. DOI: http://dx.doi.org/10.1016/j.ijnonlinmec.2011.05.012.
  7. 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. (in Russian)
  8. Bogdanovich A.U., Kuznetsov I.L. Prodol'noe szhatie tonkostennogo sterzhnya peremennogo secheniya pri razlichnykh variantakh zakrepleniya tortsov. Soobshchenie 1 [Longitudinal Compression of a Thin-Walled Bar of Variable Cross Section with Different Variants of Ends Fastening (Information 1)]. Izvestiya vuzov. Stroitel'stvo [News of Institutions of Higher Education. Construction]. 2005, no. 10, pp. 19—25. (in Russian)
  9. Bogdanovich A.U., Kuznetsov I.L. Prodol'noe szhatie tonkostennogo sterzhnya peremennogo secheniya pri razlichnykh variantakh zakrepleniya tortsov. Soobshchenie 2 [Longitudinal Compression of a Thin-Walled Core of Variable Cross Section with Different Variants of Ends Fastening (Information 2)]. Izvestiya vuzov. Stroitel'stvo [News of Institutions of Higher Education. Construction]. 2005, no. 11, pp. 10—16. (in Russian)
  10. Selamet S., Garlock M.E. Predicting the Maximum Compressive Beam Axial During Fire Considering Local Buckling. Journal of Constructional Steel Research. 2012, vol. 71, pp. 189—201. DOI: http://dx.doi.org/10.1016/j.jcsr.2011.09.014.
  11. Vo Thuc P., Thai Huu-Tai. Vibration and Buckling Of Composite Beams Using Refined Shear Deformation Theory. International Journal of Mechanical Sciences. 2012, vol. 62, no. 1, pp. 67—76. DOI: http://dx.doi.org/10.1016/j.ijmecsci.2012.06.001.
  12. Kanno Yoshihiro, Ohsaki Makoto. Optimization-bazed Stability Analysis of Structures under Unilateral Constraints. International Journal for Numerical Methods in Engineering. 2009, vol. 77, no. 1, pp. 90—125.
  13. 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.
  14. Rektoris К. Variational methods in Mathematics, Science and Engineering. Prague, SNTL-Publ., Techn. Liter., 1980. (in Russian)
  15. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formuliations of Stability Problems of Elastic Rods Using Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 285—289. (in Russian)

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