Model of a spatial dome cover. Deformations and oscillation frequency

Model of a spatial dome cover. Deformations and oscillation frequency

Автор: Kirsanov Mikhail Nikolaevich

Журнал: Строительство уникальных зданий и сооружений @unistroy

Статья в выпуске: 1 (99), 2022 года.

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The object of research. A new scheme of a statically determinate spatial truss is considered. The design has a hexagonal dome resting on two belts. The belts are supported by vertical racks. Two corner supports have spherical and cylindrical hinges. The outer support contra consists of 6n horizontal rods, the inner one consists of 6(n-1) rods. The contours are connected by skews. Formulas are derived for the deflection of the vertex and the angular hinge depending on n. The upper and lower analytical estimates of the first frequency of natural oscillations of the structure are found. Method. Calculation of the forces in the rods is carried out by cutting out the nodes from the solution of the system of equilibrium equations for all nodes in the projection on the coordinate axes. To derive formulas for the dependence of breakdowns, forces, and the frequency of free oscillations, an inductive generalization of the sequence of solutions for structures with a different number of panels is used. The structural stiffness matrix and deflection are calculated using the Maxwell - Mohr formula in analytical form. To find estimates of the lowest frequency of vibrations of nodes endowed with masses, the Dunkerley and Rayleigh methods are used. Results. The vertical load distributed over the nodes and the concentrated load applied to the top are considered. Formulas for the forces in the characteristic bars of the structure are derived. A picture of the distribution of forces throughout the structure is presented. The resulting formulas for the deflection and frequency estimates have a compact form. The upper estimate of the first oscillation frequency of nodes under the assumption of vertical displacements of points has fairly high accuracy. The analytical solution is compared with the lowest oscillation frequency obtained numerically. All analytical transformations are performed in the Maple symbolic mathematics system. Some asymptotics of solutions is found.

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Spatial truss, vibrations frequency, maple, analytical solution, deflection, induction, rayleigh method, dunkerley method, asymptotics, maxwell-mohr formula

Короткий адрес: https://sciup.org/143178766

IDR: 143178766   |   DOI: 10.4123/CUBS.99.4

Список литературы Model of a spatial dome cover. Deformations and oscillation frequency

  • Kaveh, A., Hosseini, S.M., Zaerreza, A. Size, Layout, and Topology Optimization of Skeletal Structures Using Plasma Generation Optimization. Iranian Journal of Science and Technology, Transactions of Civil Engineering 2020 45:2. 2020. 45(2). Pp. 513–543. DOI:10.1007/S40996-020-00527-1. URL: https://link.springer.com/article/10.1007/s40996-020-00527-1 (date of application: 4.03.2022).
  • Deshpande, V.S., Fleck, N.A., Ashby, M.F. Effective properties of the octet-truss lattice material. Journal of the Mechanics and Physics of Solids. 2001. 49(8). Pp. 1747–1769. DOI:10.1016/S0022-5096(01)00010-2.
  • Dou, C., Guo, Y.F., Jiang, Z.Q., Gao, W., Pi, Y.L. In-plane buckling and design of steel tubular truss arches. Thin-Walled Structures. 2018. 130(July). Pp. 613–621. DOI:10.1016/j.tws.2018.06.024. URL: https://doi.org/10.1016/j.tws.2018.06.024.
  • Tejani, G.G., Savsani, V.J., Patel, V.K., Mirjalili, S. Truss optimization with natural frequency bounds using improved symbiotic organisms search. Knowledge-Based Systems. 2018. 143. Pp. 162–178. DOI:10.1016/j.knosys.2017.12.012. URL: https://doi.org/10.1016/j.knosys.2017.12.012.
  • Shao, X., He, G., Shen, X., Zhu, P., Chen, Y. Conceptual design of 1000 m scale steel-UHPFRC composite truss arch bridge. Engineering Structures. 2021. 226. Pp. 111430. DOI:10.1016/j.engstruct.2020.111430.
  • Queheillalt, D.T., Wadley, H.N.G. Pyramidal lattice truss structures with hollow trusses. Materials Science and Engineering: A. 2005. 397(1–2). Pp. 132–137. DOI:10.1016/J.MSEA.2005.02.048.
  • Hutchinson, R.G., Fleck, N.A. Microarchitectured cellular solids - The hunt for statically determinate periodic trusses. ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik. 2005. 85(9). Pp. 607–617. DOI:10.1002/zamm.200410208.
  • Hutchinson, R.G., Fleck, N.A. The structural performance of the periodic truss. Journal of the Mechanics and Physics of Solids. 2006. 54(4). Pp. 756–782. DOI:10.1016/j.jmps.2005.10.008.
  • Kaveh, A. Optimal analysis of structures by concepts of symmetry and regularity. Optimal Analysis of Structures by Concepts of Symmetry and Regularity. 2013. 9783709115657. Pp. 1–463. DOI:10.1007/978-3-7091-1565-7.
  • Kaveh, A., Rahami, H., Shojaei, I. Swift Analysis of Civil Engineering Structures Using Graph Theory Methods. 2020. 290. DOI:10.1007/978-3-030-45549-1. URL: http://link.springer.com/10.1007/978-3-030-45549-1 (date of application: 11.03.2022).
  • Goloskokov, D.P., Matrosov, A. V. Approximate analytical approach in analyzing an orthotropic rectangular plate with a crack. Materials Physics and Mechanics. 2018. 36(1). Pp. 137–141. DOI:10.18720/MPM.3612018_15.
  • Matrosov, A. V. Computational Peculiarities of the Method of Initial Functions. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). 2019. 11619 LNCS. Pp. 37–51. DOI:10.1007/978-3-030-24289-3_4.
  • Galileev, S.M., Matrosov, A. V. Method of initial functions: Stable algorithms in the analysis of thick laminated composite structures. Composite Structures. 1997. 39(3–4). Pp. 255–262. DOI:10.1016/S0263-8223(97)00108-6.
  • Kirsanov, M. Planar Trusses: Schemes and Formulas. Cambridge Scholars Publishing Lady Stephenson Library. Newcastle upon Tyne, GB, 2019.
  • Kirsanov, M. Trussed Frames and Arches: Schemes and Formulas. Cambridge Scholars Publishing Lady Stephenson Library. Newcastle upon Tyne, GB, 2020.
  • Ovsyannikova, V.M. Dependence of deformations of a trapezous truss beam on the number of panels. Structural Mechanics and Structures. 2020. 26(3). Pp. 13–20. URL: https://www.elibrary.ru/item.asp?id=44110286 (date of application: 11.03.2021).
  • Kazmiruk, I.Y. On the arch truss deformation under the action of lateral load. Science Almanac. 2016. 17(3–3). Pp. 75–78. DOI:10.17117/na.2016.03.03.075. URL: http://ucom.ru/doc/na.2016.03.03.075.pdf (date of application: 9.05.2021).
  • Rakhmatulina, A.R., Smirnova, A.A. The dependence of the deflection of the arched truss loaded on the upper belt, on the number of panels. Science Almanace. 2017. 28(2–3). Pp. 268–271. DOI:10.17117/na.2017.02.03.268. URL: http://ucom.ru/doc/na.2017.02.03.268.pdf (date of application: 9.05.2021).
  • Tinkov, D. V. Comparative analysis of analytical solutions to the problem of truss structure deflection. Magazine of Civil Engineering. 2015. 57(5). DOI:10.5862/MCE.57.6.
  • Belyankin, N.A.; Boyko, A.Y. Formula for deflection of a girder with an arbitrary number of panels under the uniform load. Structural Mechanics and Structures. 2019. 1(20). Pp. 21–29. URL: https://www.elibrary.ru/download/elibrary_37105069_21945931.pdf.
  • Ilyushin, A. The formula for calculating the deflection of a compound externally statically indeterminate frame. Structural mechanics and structures. 2019. 3(22). Pp. 29–38. URL: https://www.elibrary.ru/download/elibrary_41201106_54181191.pdf.
  • Kirsanov, M. Deformations of the Rod Pyramid: An Analytical Solution. Construction of Unique Buildings and Structures. 2021. 95. Pp. 9501. DOI:10.4123/CUBS.95.1.
  • Petrenko, V.F. The natural frequency of a two-span truss. AlfaBuild. 2021. (20). Pp. 2001. DOI:10.34910/ALF.20.1.
  • Vorobev, O.V. Bilateral Analytical Estimation of the First Frequency of a Plane Truss. Construction of Unique Buildings and Structures. 2020. 92(7). Pp. 9204–9204. DOI:10.18720/CUBS.92.4. URL: https://unistroy.spbstu.ru/article/2020.92.4 (date of application: 17.04.2021).
  • Vorobev, O.V. On methods of obtaining an analytical solution for the problem of natural frequencies of hinged structures. Structural mechanics and structures. 2020. 24(1). Pp. 25–38. URL: http://vuz.exponenta.ru/pdf/NAUKA/elibrary_42591122_21834695.pdf.
  • Buka-Vaivade, K., Kirsanov, M.N., Serdjuks, D.O. Calculation of deformations of a cantilever-frame planar truss model with an arbitrary number of panels. Vestnik MGSU. 2020. (4). Pp. 510–517. DOI:10.22227/1997-0935.2020.4.510-517.
  • Liu, M., Cao, D., Zhang, X., Wei, J., Zhu, D. Nonlinear dynamic responses of beamlike truss based on the equivalent nonlinear beam model. International Journal of Mechanical Sciences. 2021. 194. Pp. 106197. DOI:10.1016/J.IJMECSCI.2020.106197.
  • Pollino, M., Bruneau, M. Dynamic seismic response of controlled rocking bridge steel-truss piers. Engineering Structures. 2008. 30(6). Pp. 1667–1676. DOI:10.1016/J.ENGSTRUCT.2007.10.016.
  • Rybakov, L. S., Mishustin, I. V. Small elastic vibrations of planar trusses of orthogonal structure. Mechanics of composite materials and structures. 2003. 9(1). Pp. 42–58. URL: https://elibrary.ru/item.asp?id=11724233 (date of application: 5.07.2021).
  • Low, K.H. Natural frequencies of a beam-mass system in transverse vibration: Rayleigh estimation versus eigenanalysis solutions. International Journal of Mechanical Sciences. 2003. 45(6–7). Pp. 981–993. DOI:10.1016/j.ijmecsci.2003.09.009.
  • Low, K.H. A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses. International Journal of Mechanical Sciences. 2000. 42(7). Pp. 1287–1305. DOI:10.1016/S0020-7403(99)00049-1.
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