Error
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering
  • Error loading Modules:Can't create/write to file '/var/lib/mysql/tmp/#sql_bd4_0.MYI' (Errcode: 30) SQL=SELECT id, title, module, position, content, showtitle, control, params FROM jos_modules AS m LEFT JOIN jos_modules_menu AS mm ON mm.moduleid = m.id WHERE m.published = 1 AND m.access <= 0 AND m.client_id = 0 AND ( mm.menuid = 8 OR mm.menuid = 0 ) ORDER BY position, ordering

DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

BILATERAL BOUNDS OF STABILITY OF AN ELASTIC CANTILEVER BAR COMPRESSED OVER A CONNECTING ROD

Vestnik MGSU 7/2012
  • Dudchenko Aleksandr Vladimirovich - Moscow State University of Civil Engineering (MSUCE) student, Moscow State University of Civil Engineering (MSUCE), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MSUCE) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics, +7 (499) 183-46-74, Moscow State University of Civil Engineering (MSUCE), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 75 - 81

The authors present both top and bottom limit values of loads within the two problems of stability of a rectilinear elastic cantilever bar that has a variable cross-section. In the first problem, a longitudinal compressive force applied to the bar end is transmitted through a connecting rod that has hinges on both ends, while the second problem is to be resolved in absence of any connecting rod.
The authors apply well-known expressions to identify the stability loss by a rectilinear elastic cantilever bar that has a constant cross-section compressed by a longitudinal force at its free end, with account for the inequalities generated by the best approximation problem in the Hilbert space. They constructed two series of functionals, the bottom bounds of which are the bilateral bounds of the unknown critical value of the load parameter. The calculation of the bottom bounds is reduced to determination of the biggest eigenvalues for the matrices presented in the form of second-order matrices with elements, expressed through the integrals of well-known forms of stability loss by a bar that has a constant cross-section. The calculation of the top bound is reduced to the determination of the biggest eigenvalue for the matrix which almost coincides with the one of the block matrices constructed for the determination of the bottom bound.
Bilateral bounds identified in accordance with the above method make it possible to assess the reduction of the critical load value in the first problem and to compare it to the one of the second problem.

DOI: 10.22227/1997-0935.2012.7.75 - 81

References
  1. Alfutov N.A. Osnovy rascheta na ustoychivost’ uprugikh sistem [Fundamentals of Stability Analysis of Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  2. Dudchenko A.V., Kupavtsev V.V. Dvustoronnie otsenki ustoychivosti uprugogo konsol’nogo sterzhnya, szhatogo polusledyashchey siloy [Bilateral Bounds of Stability of an Elastic Cantilever Bar, Compressed by the Half-Tracking Force]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 6, pp. 302—306.
  3. Klyushnikov V.D., Kupavtsev V.V. Dvustoronnie otsenki kriticheskikh nagruzok neodnorodno szhatykh sterzhney [Bilateral Evaluations of Values of the Critical Load Applicable to Non-Uniformly Compressed Elastic Rods]. Doklady akademii nauk SSSR [Reports of the Academy of Sciences of the USSR]. 1977, vol. 238, no. 3, pp. 561—564.
  4. Kupavtsev V.V. K dvustoronnim otsenkam kriticheskikh nagruzok neodnorodno szhatykh sterzhney [About Bilateral Assessments of Values of Critical Loads Applicable to Non-uniformly Compressed Elastic Rods]. Izvestiya VUZov. Stroitel’stvo i arkhitektura. [Proceedings of Higher Education Institutions. Construction and Architecture]. 1984, no. 8, pp. 24—29.

Download

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.

Download

Results 1 - 2 of 2