| Abstract PDF ENG | Резюме PDF RUS | Full text PDF RUS |
Abstract. The design of multi-storey buildings is a natural trend in the development of a modern metropolis. Obtaining exact solutions when studying their own and forced oscillations within the framework of a continuous homogeneous medium model (continuum mechanics) with an infinite number of degrees of freedom is often difficult to implement. Therefore, in the article (as part of the modernization of the finite element method), the model of a multi-storey building is discretized and endowed with a finite number of degrees of freedom placed in the middle of the finite elements at the nodes (the mass of finite elements is also placed there), which elastically interact with the finite elements of the model that do not have mass. It is believed that the elements of a multi-storey building work only for bending, which is fully justified by comparing the frequencies of its bending and longitudinal oscillations. The resolving system of differential equations of oscillations of a multi-storey building, in which expressions for energies (potential, kinetic and Rayleigh) are written in quadratures, is obtained using Lagrange equations of the second kind. In the article, the problems of free oscillations of 3- and 100-storey buildings are solved using Green’s functions, stiffness, mass, compliance matrices, etc. The approximate results obtained in the article, when compared with the little-known approximate results obtained by other methods, as well as exact results (direct and indirect methods of boundary elements), showed a good correspondence.
Keywords:
oscillation of multi-storey buildings, oscillation frequencies, Green’s function, stiffness matrix, mass matrix, compliance matrix
For citation: Velikanov P.G., Artyukhin Yu.P. Research on the dynamics of multi-storey buildings. Geosistemy perehodnykh zon = Geosystems of Transition Zones, 2023, vol. 7, no. 3, pp. 304–315. (In Russ., abstr. in Engl.).
https://doi.org/10.30730/gtrz.2023.7.3.304-315, https://www.elibrary.ru/aotwnz
References
1. VelikanovP.G., ArtyukhinYu.P. 2023. Research on the dynamics of frame structures. Geosistemy perehodnykh zon = Geosystems of Transition Zones, 7(2): 180–195. (In Russ., abstr. in Engl.). https://doi.org/10.30730/gtrz.2023.7.2.180-195; https://www.elibrary.ru/llpxlp
2. Viktorova L.A. 2012. Tall buildings – the pros and cons of building. Architecture and construction of Russia, 10: 2–11. (In Russ.). EDN: PEJJDT
3. Khalikova A.S., Gamayunova O.S. 2021. Features of designing high-rise buildings in seismic areas. Inzhenernyye issledovaniya = Engineering Research, 5(5): 31–38. (In Russ.). EDN: PRJJEV; URL: http://eng-res.ru/archive/2021/5/31-38.pdf
4. Kurbatsky E.N., Mazur G.E., Mondrus V.L. 2017. [Critical analysis of the state of regulatory documentation for the calculation of structures for earthquakes]. Natural and Technological Risks. Building Safety, 2(28): 24–30. (In Russ.). EDN: XYDNSH
5. Magay A.A. 2015. [ Architectural design of high-rise buildings and complexes ]. Moscow: Izd-vo Assotsiatsii stroitel'nykh vuzov, 328 p. (In Russ.).
6. Artyukhin Yu.P., Gribov A.P. 2002. [ Solving problems of nonlinear deformation of plates and shallow shells by the boundary elements method ]. Kazan: Fen, 199 p. (In Russ.).
7. Katsikadelis J.T. 2007. Boundary elements: Theory and applications. Moscow: Izd-vo Assotsiatsii stroitel'nykh vuzov, 348 p. (In Russ.).
8. Velikanov P.G., Artyukhin Yu.P., Kukanov N.I. 2020. Bending of an anisotropic plate by the method of boundary elements. In: [ Actual problems of continuum mechanics ]. Kazan: Publ. House of the Academy of Sciences of the Republic of Tatarstan, p. 105–111. (In Russ.).
9. Harris C.M., Creed Ch.I. 1980. [ Handbook of shock loads ]. Leningrad: Sudostroenie, 358 p. (In Russ.). (Transl. from English: C.M. Harris, Ch.E. Crede (eds). 1976. Shock and vibration handbook. Second ed. New York: McGraw-Hill, 1218 p.)
10. Babakov I.M. 1958. [ Theory of oscillations ]: Textbook for higher tech. educational institutions. Moscow: Gostekhizdat, 628 p. (In Russ.).
11. Steklov V.A. 1927. [ Fundamentals of the integration theory of the ordinary differential equations ]. Moscow; Leningrad: Gosizdat, 419 p. (In Russ.).
12. Artyukhin Y.P., Guryanov N.G., Kotlyar L.M. 2002. [ The Mathematics 4.0 system and its applications in mechanics ]: Textbook. Kazan: Kazan Mathematical Society: Publ. House of CamPI, 415 p. (In Russ.).
13. Timoshenko S.P., Yang D.H., Weaver U. 1985. Vibration problems in engineering. Moscow: Mashinostroenie, 472 p. (In Russ.). http://www.physics.gov.az/book_K/Timoshenko.pdf
14. Clough R.W., Penzien J. 1975. Dynamics of structures. Moscow: Stroyizdat, 320 p.
15. Bishop R.E.D. 1965. Vibration. Cambridge: Cambridge Univ. Press, 156 p.