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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.

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The boundary value problemfor one-dimensional fractional differential equationof advection-diffusion

Vestnik MGSU 6/2015
  • Isaeva Leyla Magametovna - Moscow State University of Civil Engineering (MGSU) postgraduate student, Department of Higher Mathematics, 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 16-22

The use of fractional derivatives for describing and studying the physical processes of stochastic transport has become one of the most popular fields of physics in the recent years, many of the problems of fluid flow in highly-porous (fractal) environments also lead to the need to study boundary value problems for the equations of fractional order.The paper considers one of the boundary value problems for one-dimensional differential equation of fractional order. Using the Fourier method, the solution to this problem was explicitly written. The author also studied the qualitative properties of the solutions of the boundary value problem. It was proved that, in the case of going to infinity, the limit of the decisions recorded in the form of the function and the limit of the derivative of this solution tend to zero.The results can find application in the theory of fluid flow in a fractal environment and in order to simulate changes in temperature.Fractional integrals and derivatives of fractional integral-differential equations find wide application in contemporary studies of theoretical physics, mechanics and applied mathematics. Fractional calculus is a very powerful tool for describing the physical systems, which have memory and are non-local. Many processes in complex systems are non-locality and have long-term memory. The fractional integral operators and the fractional differential operators allow describing some of these properties. The use of fractional calculus will be useful for obtaining the dynamic models, in which integraldifferential operators describe the power of long-term memory and time coordinate and three-dimensional nonlocality for medium and complex processes.

DOI: 10.22227/1997-0935.2015.6.16-22

References
  1. Nakhushev A.M. Drobnoe ischislenie i ego primenenie [Fractional Calculus and its Application]. Moscow, Fizmatlit Publ., 2003, 272 p. (In Russian)
  2. Aleroev T.S. Kraevye zadachi dlya differentsial’nykh uravneniy drobnogo poryadka [Boundary Value Problems for Differential Equations of Fractional Order]. Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic Mathematical Reports]. 2013, no. 10, pp. 41—55. (In Russian)
  3. Aleroev T.S., Kirane M., Malik S.A. Determination of a Source Term for a Time Fractional Diffusion Equation with an Integral Type Over-Determining Condition. Electronic Journal of Differential Equations. 2013, vol. 2013, no. 270, pp. 1—16. Available at: http://ejde.math.txstate.edu/. Date of access: 10.03.2015.
  4. Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of Mathematical Physics]. 6th edition. Moscow, MGU Publ., 1999, 799 p. (In Russian)
  5. Samko S.G., Kilbas A.A., Marichev O.I. Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya [Integrals and Derivatives of Fractional Order and Some of Their Applications]. Minsk, Nauka i tekhnika Publ., 1987, 688 p. (In Russian)
  6. Dzhrbashchyan M.M. Kraevaya zadacha dlya differentsial’nogo operatora tipa Shturma-Liuvillya drobnogo poryadka [Boundary Value Problem for the Differential Operator of Sturm-Liouville of Fractional Order]. Izvestiya AN Armyanskoy SSR. Seriya Matematika [News of the Academy of Sciences of the Armenian SSR. Series: Mathematics]. 1970, issue 5, no. 2, pp. 71—96. (In Russian)
  7. Aleroev T.S., Aleroeva Kh.T. Ob odnom klasse nesamosopryazhennykh operatorov, soputstvuyushchikh differentsial’nym uravneniyam drobnogo poryadka [On a Class of Self-Adjoint Operators Associated with Differential Equations of Fractional Order]. Izvestiya vysshikh uchebnykh zavedeniy. Matematika [Russian Mathematics (Izvestiya VUZ)]. 2014, no. 10, pp. 3—12. (In Russian)
  8. Khasambiev M.V., Aleroev T.S. Kraevaya zadacha dlya odnomernogo drobnogo differentsial’nogo uravneniya advektsii-diffuzii [Boundary Value Problem for One-Dimensional Differential Advection-Dispersion Equation]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2014, no. 6, pp. 71—76. (In Russian)
  9. Aleroev T.S., Aleroeva H.T. A Problem on the Zeros of the Mittag-Leffler Function and the Spectrum of a Fractional-Order Differential Operator. Electron. J. Qual. Theory Diff. Equ. 2009, no. 25, 18 p. Available at: https://zbmath.org/?q=an:1183.34004. Date of access: 10.03.2015.
  10. Aleroev T.S., Kirane M., Tang Y.-F. Boundary-Value Problems for Differential Equations of Fractional Order. Journal of Mathematical Sciences. Nov. 2013, vol. 194, no. 5, pp. 499—512. DOI: http://dx.doi.org/10.1007/s10958-013-1543-y.
  11. Popov A.Yu., Sedletskiy A.M. Raspredelenie korney funktsiy Mittag-Lefflera [Distribution of Zeros of the Mittag-Leffler Functions]. Sovremennaya matematika. Fundamental’nye napravleniya [Contemporary Mathematics. Fundamental Directions]. 2011, vol. 40, pp. 3—171. (In Russian)
  12. Płociniczak L. Eigenvalue Asymptotics for a Fractional Boundary-Value Problem. Applied Mathematics and Computation. 15 August 2014, vol. 241, pp. 125—128. DOI: http://dx.doi.org/10.1016/j.amc.2014.05.029.
  13. Ushkov V.A., Abramov V.V., Lalayan V.M., Kir'yanova L.V. Slabogoryuchie epoksidnye polimerrastvory, ispol'zuemye dlya vosstanovleniya i remonta stroitel'nykh konstruktsiy [Low-Flamnable Epoxy Polyner Mortars Used for Reconstruction and Repair of Building Structures]. Pozharovzryvobezopasnost' [Fire and Explosion Safety]. 2012, vol. 21, no. 10, pp. 36—40.
  14. Ushkov V.A., Abramov V.V., Grigor'eva L.S., Kir'yanova L.V. Termostoykost' i pozharnaya opasnost' epoksidnykh polimerrastvorov [Thermal Resistanse and Fire Hazard of Epoxy Polimer Mortars]. Stroitel'nye materialy [Construction Materials]. 2011, no. 12, pp. 68-71.

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