Structural Mechanics of Engineering Constructions and BuildingsStructural Mechanics of Engineering Constructions and BuildingsСтроительная механика инженерных конструкций и сооружений1815-52352587-8700Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)4270010.22363/1815-5235-2024-20-5-433-440CQAEAZAnalysis of thin elastic shellsРасчет тонких упругих оболочекResearch ArticleDynamic Response of Doubly-Curved Shallow Shells to Periodic External ActionДинамический отклик пологих оболочек двоякой кривизны на периодическое внешнее воздействиеhttps://orcid.org/0000-0001-9490-73649057-9882SemenovAlexey A.СеменовАлексей Александрович

Doctor of Technical Sciences, Professor of the Department of Information Systems and Technologies

доктор технических наук, профессор, кафедра информационных систем и технологий

sw.semenov@gmail.comSaint Petersburg State University of Architecture and Civil EngineeringСанкт-Петербургский государственный архитектурно-строительный университет15122024205VOL 20, NO5 (2024)ТОМ 20, №5 (2024)43344031012025Copyright ©; 2024, Semenov A.A.Copyright ©; 2024, Семенов А.А.2024Semenov A.A.Семенов А.А.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/structural-mechanics/article/view/42700

Shallow shells of double curvature are often used as elements of building structures and are subjected to various external effects, including dynamic periodic loads. The paper proposes to extend the previously proposed approach to modeling the process of deformation of thin shells to a class of problems with periodic effects. A mathematical model is used based on the Timoshenko - Reissner hypotheses, taking into account transverse shears, geometric nonlinearity and rotational inertia. The calculation algorithm is based on the method of L.V. Kantorovich and the Rosenbrock method for solving rigid ODE systems. The calculations are performed in Maple. Dynamic responses are obtained for an isotropic shallow shell of double curvature at different frequency values, and vertical displacement fields are shown at peak values of the oscillation amplitude.

Пологие оболочки двоякой кривизны часто используются как элементы строительных конструкций и подвергаются различным внешним воздействиям, в том числе динамическим периодическим нагрузкам. В работе предлагается расширение предложенного автором ранее подхода к моделированию процесса деформирования тонких оболочек на класс задач с периодическими воздействиями. Используется математическая модель на основе гипотез Тимошенко - Рейсснера, учитывающая поперечные сдвиги, геометрическую нелинейность и инерцию вращения. В расчетном алгоритме применяется в своей основе метод Л.В. Канторовича и метод Розенброка для решения жестких систем ОДУ. Расчеты выполнены в Maple. Получены динамические отклики для изотропной пологой оболочки двоякой кривизны при разных значениях частоты, показаны поля вертикальных перемещений при пиковых значениях амплитуды колебаний.

shellsdynamic loadKantorovich methodvibrationsfunctionalоболочкидинамическая нагрузкаметод КанторовичаколебанияфункционалThe article is published based on the results of the implementation of the 2024 SPbGASU grant.Статья публикуется по результатам исполнения гранта СПбГАСУ 2024 года.Bich D.H., Ninh D.G. Research on dynamical buckling of imperfect stiffened three-layered toroidal shell segments containing fluid under mechanical loads. Acta Mechanica. 2017;228(2):711-730. http://doi.org/10.1007/s00707-016-1724-0Gao K., Gao W., Wu D., Song C. Nonlinear dynamic stability of the orthotropic functionally graded cylindrical shell surrounded by Winkler-Pasternak elastic foundation subjected to a linearly increasing load. Journal of Sound and Vibration. 2018;415:147-168. http://doi.org/10.1016/j.jsv.2017.11.038Lavrenčič M., Brank B. Simulation of shell buckling by implicit dynamics and numerically dissipative schemes. Thin-Walled Structures. 2018;132:682-699. http://doi.org/10.1016/j.tws.2018.08.010Luo K., Liu C., Tian Q., Hu H. Nonlinear static and dynamic analysis of hyper-elastic thin shells via the absolute nodal coordinate formulation. Nonlinear Dynamics. 2016;85(2):949-971. http://doi.org/10.1007/s11071-016-2735-zRen S., Song Y., Zhang A.-M., Wang S., Li P. Experimental study on dynamic buckling of submerged grid-stiffened cylindrical shells under intermediate-velocity impact. Applied Ocean Research. 2018;74:237-245. http://doi.org/10.1016/j.apor.2018.02.018Sirivolu D., Hoo Fatt M.S. Dynamic stability of double-curvature composite shells under external blast. International Journal of Non-Linear Mechanics. 2015;77:281-290. http://doi.org/10.1016/j.ijnonlinmec.2015.09.005Sofiyev A.H., Kuruoglu N. Domains of dynamic instability of FGM conical shells under time dependent periodic loads. Composite Structures. 2016;136:139-148. http://doi.org/10.1016/j.compstruct.2015.09.060Amabili M., Paı̈doussis M.P. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Applied Mechanics Reviews. 2003;56(4):349-381. http://doi.org/10.1115/1.1565084Dey T., Ramachandra L.S. Dynamic stability of simply supported composite cylindrical shells under partial axial loading. Journal of Sound and Vibration. 2015;353:272-291. http://doi.org/10.1016/j.jsv.2015.05.021Dung D.V., Nam V.H. An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure. Part 2: Numerical results and discussion. Vietnam Journal of Mechanics. 2014;36(4):255-265. http://doi.org/10.15625/08667136/36/4/3986Kiani Y., Sadighi M., Eslami M.R. Dynamic analysis and active control of smart doubly curved FGM panels. Composite Structures. 2013;102:205-216. http://doi.org/10.1016/j.compstruct.2013.02.031Krysko V.A., Awrejcewicz J., Shchekaturova T.V. Chaotic vibrations of spherical and conical axially symmetric shells. Archive of Applied Mechanics. 2005;74(5-6):338-358. http://doi.org/10.1007/BF02637035Moussaoui F., Benamar R. Non-Linear Vibrations of Shell-Type Structures: A Review with Bibliography. Journal of Sound and Vibration. 2002;55(1):161-184. http://doi.org/10.1006/jsvi.2001.4146Qu Y., Wu S., Chen Y., Hua H. Vibration analysis of ring-stiffened conical-cylindrical-spherical shells based on a modified variational approach. International Journal of Mechanical Sciences. 2013;69:72-84. http://doi.org/10.1016/j.ijmecsci.2013.01.026Ungbhakorn V., Singhatanadgid P. A Scaling Law for Vibration Response of Laminated Doubly Curved Shallow Shells by Energy Approach. Mechanics of Advanced Materials and Structures. 2009;16(5):333-344. http://doi.org/10.1080/ 15376490902970430Abrosimov N.A., Novosel’tseva N.A. Computer Modeling of the Dynamic Strength of Metal-Plastic Cylindrical Shells Under Explosive Loading. Mechanics of Composite Materials. 2017;53(2):139-148. http://doi.org/10.1007/s11029017-9648-xKumar Y. The Rayleigh-Ritz method for linear dynamic, static and buckling behavior of beams, shells and plates: A literature review. Journal of Vibration and Control. 2017;24(7):1205-1227. http://doi.org/10.1177/1077546317694724Maksimyuk V.A., Storozhuk E.A., Chernyshenko I.S. Variational finite-difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review). International Applied Mechanics. 2012;48(6):613- 687. http://doi.org/10.1007/s10778-012-0544-8Dey T., Jansen E., Kumar R., Rolfes R. Instability characteristics of variable stiffness laminated composite curved panels under non-uniform periodic excitation. Thin-Walled Structures. 2022;171:108735. http://doi.org/10.1016/j.tws.2021. 108735Karpov V.V. The strength and stability of reinforced shells of revolution. In two parts. Part 1. Models and algorithms of research of the strength and stability of supported shells of revolution. Moscow: Fizmatlit Publ.; 2010. (In Russ.) EDN: UIRNWJКарпов В.В. Прочность и устойчивость подкрепленных оболочек вращения: в 2 ч. Ч. 1: Модели и алгоритмы исследования прочности и устойчивости подкрепленных оболочек вращения. М.: Физматлит, 2010. 288 с. EDN: UIRNWJNg T.Y., Lam K.Y., Reddy J.N. Dynamic stability of cylindrical panels with transverse shear effects. International Journal of Solids and Structures. 1999. Vol. 36. No. 23. P. 3483-3496. http://doi.org/10.1016/S0020-7683(98)00161-9Yu Y.-Y., Lai J.-L. Influence of Transverse Shear and Edge Condition on Nonlinear Vibration and Dynamic Buckling of Homogeneous and Sandwich Plates. Transactions of the ASME. 1966;33(4):934-936. http://doi.org/10.1115/1.3625205Bacciocchi M., Eisenberger M., Fantuzzi N., Tornabene F., Viola E. Vibration analysis of variable thickness plates and shells by the Generalized Differential Quadrature method. Composite Structures. 2016;156:218-237. http://doi.org/10.1016/j.compstruct.2015.12.004Patel S.N., Datta P.K., Sheikh A.H. Buckling and dynamic instability analysis of stiffened shell panels. Thin- Walled Structures. 2006;44(3):321-333. http://doi.org/10.1016/j.tws.2006.03.004Zhang J., van Campen D.H. Stability and bifurcation of doubly curved shallow panels under quasi-static uniform load. International Journal of Non-Linear Mechanics. 2003;38(4):457-466. http://doi.org/10.1016/S0020-7462(01)00069-5Azarboni H.R., Ansari R., Nazarinezhad A. Chaotic dynamics and stability of functionally graded material doubly curved shallow shells. Chaos, Solitons & Fractals. 2018;109:14-25. http://doi.org/10.1016/j.chaos.2018.02.011Khudayarov B.A., Ruzmetov K.Sh., Turaev F.Zh., Vaxobov V.V., Hidoyatova M.A., Mirzaev S.S., Abdikarimov R. Numerical modeling of nonlinear vibrations of viscoelastic shallow shells. Engineering Solid Mechanics. 2020:199-204. http://doi.org/10.5267/j.esm.2020.1.004Bazhenov V.G., Baranova M.S., Kibets A.I., Lomunov V.K., Pavlenkova E.V. Buckling of elastic-plastic cylindrical and conical shells under axial impact loading. Scientific Notes of Kazan University. Series: Physical and Mathematical Sciences, 2010;152(4):86-105. EDN: NPULEB Баженов В.Г., Баранова М.С., Кибец А.И., Ломунов В.К., Павленкова Е.В. Выпучивание упругопластических цилиндрических и конических оболочек при осевом ударном нагружении // Ученые записки Казанского университета. Серия: Физико-математические науки. 2010. Т. 152. № 4. С. 86-105. EDN: NPULEBWei Z.G., Yu J.L., Batra R.C. Dynamic buckling of thin cylindrical shells under axial impact. International Journal of Impact Engineering. 2005;32(1-4):575-592. http://doi.org/10.1016/j.ijimpeng.2005.07.008Zhang J., Li S. Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load. Composite Structures. 2010;92(12):2979-2983. http://doi.org/10.1016/j.compstruct.2010.05.009Eshmatov B., Abdikarimov R., Amabili M., Vatin N. Nonlinear vibrations and dynamic stability of viscoelastic anisotropic fiber reinforced plates. Magazine of Civil Engineering. 2023;118(1):11811. http://doi.org/10.34910/MCE.118.11Phu K.V., Bich D.H., Doan L.X. Nonlinear Forced Vibration and Dynamic Buckling Analysis for Functionally Graded Cylindrical Shells with Variable Thickness Subjected to Mechanical Load. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering. 2022;46:649-665. http://doi.org/10.1007/s40997-021-00429-1Keshav V., Patel S.N., Kumar R., Watts G. Effect of Cutout on the Stability and Failure of Laminated Composite Cylindrical Panels Subjected to In-Plane Pulse Loads. International Journal of Structural Stability and Dynamics. 2022; 22(08):2250087. http://doi.org/10.1142/S0219455422500870Kogan E.A., Yurchenko A.A. Nonlinear Oscillations of a Three-Layer and Multi-Layer Plates and Shells During Periodic Impacts (Survey). News of MSTU “MAMI”: Natural Sciences. 2014;4(1):55-70. (In Russ.) EDN: SKYBIXКоган Е.А., Юрченко А.А. Нелинейные колебания трехслойных и многослойных пластин и оболочек при периодических воздействиях (обзор) // Известия МГТУ «МАМИ»: Естественные науки. 2014. Т. 4. № 1 (19). С. 55-70. EDN: SKYBIXAlijani F., Amabili M. Non-linear vibrations of shells: A literature review from 2003 to 2013. International Journal of Non-Linear Mechanics. 2014;58:233-257. http://doi.org/10.1016/j.ijnonlinmec.2013.09.012Krivoshapko S.N. Research on General and Axisymmetric Ellipsoidal Shells Used as Domes, Pressure Vessels, and Tanks. Applied Mechanics Reviews. 2007;60(6):336-355. http://doi.org/10.1115/1.2806278Qatu M.S., Sullivan R.W., Wang W. Recent research advances on the dynamic analysis of composite shells: 2000-2009. Composite Structures. 2010;93(1):14-31. http://doi.org/10.1016/j.compstruct.2010.05.014Sahu S.K., Datta P.K. Research Advances in the Dynamic Stability Behavior of Plates and Shells: 1987-2005 - Part I: Conservative Systems. Applied Mechanics Reviews. 2007;60(2):65-75. http://doi.org/10.1115/1.2515580Prado Z. del, Gonçalves P.B., Païdoussis M.P. Non-linear vibrations and instabilities of orthotropic cylindrical shells with internal flowing fluid. International Journal of Mechanical Sciences. 2010;52(11):1437-1457. http://doi.org/10.1016/j.ijmecsci.2010.03.016Volmir A.S. Stability of deformable systems. Moscow: Nauka Publ.; 1967. (In Russ.)Вольмир А.С. Устойчивость деформируемых систем. М.: Наука, 1967. 984 с.Karpov V.V., Aristov D.I., Ovcharov A.A. Features of the stress-strain state of panels of ribbed shells of revolution under dynamic loading. Vestnik TGASU. 2007;(1):94-101. (In Russ.) EDN: JUCZANКарпов В.В., Аристов Д.И., Овчаров А.А. Особенности напряженно-деформированного состояния панелей ребристых оболочек вращения при динамическом нагружении // Вестник ТГАСУ. 2007. № 1. С. 94-101. EDN: JUCZAN.Semenov A. Dynamic Buckling of Stiffened Shell Structures with Transverse Shears under Linearly Increasing Load. Journal of Applied and Computational Mechanics. 2022;8(4):1343-1357. http://doi.org/10.22055/jacm.2022.39718.3452Hairer E., Wanner G. Solving Ordinary Differential Equations II : Springer Series in Computational Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg; 1991. http://doi.org/10.1007/978-3-662-09947-6Shampine L.F., Corless R.M. Initial value problems for ODEs in problem solving environments. Journal of Computational and Applied Mathematics. 2000;125(1-2):31-40. http://doi.org/10.1016/S0377-0427(00)00456-8