Elastic beams of minimum weight in the presence of several types of bending loads
Authors: Gurchenkov A.A., Vilisova N.T., German I.M., Romanenkov A.M.
Published in issue: #5(41)/2015
DOI: 10.18698/2308-6033-2015-5-1404
Category: Mechanics | Chapter: Mechanics of Deformable Solid Body
The article considers the problem of optimizing the loaded beam thickness, i.e. minimizing weight of the structure, for given boundary conditions and restrictions of strain capacity. It was found that the mathematical model in this case is the boundary value problem for ordinary differential equation of 4th order. Solving the optimization problem is based on two different approaches. The first one is the classical variational method based on studying the variation of the minimized functional and analyzing the stationary point of the functional. In the second method, the Pontryagin maximum principle is applied to the problem with fixed left and right ends. Numerical experiments carried out for different types of bending loads, are illustrated by graphs. Comparison of the results shows the equivalence of the two approaches. This significantly extends the range of optimization problems, for solution of which software with models of complex systems is developed.
References
[1] Banichuk N.V. Izvestiya AN SSSR. Mekhanika tverdogo tela - Proceedings of the USSR AS. Mechanics of Rigid Body, 1974, no. 4, pp. 44-51.
[2] Moiseev N.N., Ivanilov Yu.P., Stolyarova E.M. Metody optimizatsii [Optimization Techniques]. Moscow, Nauka Publ., 1978.
[3] Samarskiy A.A. Vvedenie v teoriyu raznostnykh skhem [Introduction to the Theory of Difference Schemes]. Moscow, Nauka Publ., 1971.
[4] Tsvey A.Yu. Balki i plity na uprugom osnovanii [Beams and Plates on Elastic Foundation]. Moscow, MADI Publ., 2014, 96 p.
[5] Vasserman N.N., et al. Soprotivlenie materialov [Strength of Materials]. Perm, Perm National Research Polytechnic University Publ., 2011, 365 p.
[6] Makarov E.G. Kursovaya rabotapo metodu konechnykh elementov [Term paper on the finite element method]. St. Petersburg, Baltic State Technical University "Voenmeh" Publ., 2011, 49 p.
[7] Sankin Yu.N., Yuganova N.A. Nestatsionarnye kolebaniya sterzhnevykh system pri soudarenii s prepyatstviem [Unsteady Oscillations of Rod Systems in a Collision with an Obstacle]. Ulyanovsk, Ulyanovsk State Technical University Publ., 2010, 174 p.
[8] Isaev V.I. Matematicheskie modeli sterzhney, balok i plit v zadachakh sosredotochennogo udara [Mathematical Models of Rods, Beams and Plates in Problems of the Centric Impact]. Ph.D. Thesis (Phys.-Math.). Moscow, 2007, 155 p.
[9] Atamuratov A.Zh. Molodoy uchenyy - Young Scientist, 2014, no. 1, pp. 13-18.
[10] Andreev V.I., Barmenkova E.V., Matveeva A.V. Vestnik MGSU - Herald of the National Research University Moscow State University of Civil Engineering, 2014, no. 1, pp. 25-32.
[11] Hjelmstad K.D. Fundamentals of the Structural Mechanics. Springer Science Media, 2005, XIV, 480 p.
[12] Andreev V.I. Optimization of thick-walled shells based on solutions 0f inverse problems of the elastic theory for inhomogeneous bodies. Computer Aided Optimum Design in Engineering, 2012, pp. 189-202.
[13] Kravanja S., Zlender B. Optimization of the underground gas storage in different rock environments. Computer Aided Optimum Design in Engineering, 2012, pp. 15-26.
[14] Issa H.K. Simplified structural analysis of steel portal frames developed from structural optimization. Computer Aided Optimum Design in Engineering, 2012, pp. 47-58.
[15] Syngellakis S. Longitudinal buckling of slender pressurized tubes. Fluid Structure Interaction XII, 2013, pp. 133-144.
[16] Gurchenkov A.A., Nosov M.V., Tsurkov V.I. Upravlenie vraschauschimisya tverdymi telami s zhidkostyu [Control of Fluid-Containing Rotating Rigid Bodies]. Moscow, Fismatlit Publ., 2011, 202 p.
[17] Gurchenkov A.A., Nosov M.V., Tsurkov V.I. Control of Fluid-Containing Rotating Rigid Bodies. CRC Press, 2013, p. 147.
[18] Chernousko F.L. Banichuk N.V. Variatsionnye zadachi mekhaniki i upravleniya [Variational Problems in Mechanics and Control]. Moscow, Nauka Publ., 1973.
[19] Gurchenkov A.A. Izvestiya vuzov. Priborostroenie - Proceedings of Universities. Instrument Engineering, 2001, vol. 44, No. 2, p. 44.
[20] Gurchenkov A.A. Stability of a fluid-filled gyroscope. J. of Engineering Physics and Thermo Physics, 2002, vol. 75, no. 3, p. 554.
[21] Gurchenkov A.A. Dinamika zavikhrennoy zhidkosti v polosti vraschayuschegosya tela [Dynamics of Swirling Liquid in the Cavity of a Rotating Body]. Moscow, Fismatlit Publ., 2010, 221 p.
[22] Gurchenkov A.A. Doklady RAN - Reports of RAS, 2002, vol. 382, no. 4, p. 476.