I. G. Emelyanov

MECHANICS OF SHELLS AND ITS APPLICATIONS

The paper presents the results of applying shell theory methods to solving problems of determining the stress-strain state and durability of thin-walled structures. Some basic concepts and principles of the shell theory are given, as well as some information on its history. A relationship is shown between the construction of a mathematical model of a certain class of problems for shells and the development of a problem solution method. Frequently used shell models and numerical methods for solving applied problems are considered. A way of reducing problem dimensionality is considered; namely, the application of the method of reducing the equations of the theory of shells to a number of Cauchy problems with Godunov orthogonalization. It is shown that one type of problems for shells, which is often encountered in the analysis of various technical objects, is problems with moving boundaries, particularly, contact problems. The paper considers one of the effective ways of solving contact problems for shells of revolution, namely, the virtual element method. In contrast to the finite element method, an increase in the number of virtual elements does not increase the size of the structure stiffness matrix. This is a great advantage of the virtual element method, which allows one to save computational resources for many problems. The study presents scientific papers on the investigation of thin-walled structures and results obtained with allowance made for the use of application of recent methods of shell mechanics.

References:

1.Lyav A. Matematicheskaya teoriya uprugosti [Mathematical Theory of Elasticity]. Moscow, Leningrad, ONTI Publ., 1935, 674 p. (In Russian).

2.Vlasov V.Z. Obshchaya teoriya obolochek i ee prilozheniya v tekhnike [General Theory of Shells and its Application in Engineering]. Moscow, Leningrad, Gostekhizdat Publ., 1949, 784 p. (In Russian).

3.Goldenveyzer A.L. Teoriya uprugikh tonkikh obolochek [Theory of Elastic Thin Shells]. M., Nauka Publ., 1976, 512 p. (In Russian).

4.Lur’e A. I. Statics of thin-walled elastic shells, transl. from Russian, ed. [AEC-tr-3798, U.S. Atomic Energy Commission]. Tech. Info. Service, 1947.

5.Novozhilov V.V. Teoriya tonkikh obolochek [Theory of Thin-Walled Shells]. Leningrad, Sudostroenie Publ., 1962, 431p. (In Russian).

6.Timoshenko S.P., Woinowsky-Krieger S. Theory of Plates and Shells. McGraw-Hill Book Company Inc., 1959.

7.Flügge W. Statik und Dynamik der Schalen, 2-te neubearb. Aufl. Springer-Verl., 1957.

8.Chernykh K.F. Lineynaya teoriya obolochek [Linear Theory of Shells. Part I]. Leningrad, LGU Publ., 1962, 274 p. (In Russian).

9.Chernykh K.F. Lineynaya teoriya obolochek [Linear Theory of Shells. Part II]. Leningrad, LGU Publ., 1964, 396 p. (In Russian).

10.Grigorenko Ya.M., Vasilenko A.T. Metody rascheta obolochek. T. 4. Teoriya obolochek peremennoy zhestkosti [Shell Calculation Methods. Vol. 4. Theory of Variable-Stiffness Shells]. Kiev, Nauk. Dumka, 1981, 544 p. (In Russian).

11.Volmir A.S., Kuranov B.A., Turbaivsky A.T. Statika i dinamika slozhnykh struktur [Statics and Dynamics of Complex Structures]. Moscow, Mashinostroenie Publ., 1989, 248 p. (In Russian).

12.Grigorenko Ya.M. Izotropnye i anizotropnye sloistye obolochki vrashcheniya peremennoy zhestkosti [Isotropic and anisotropic layered shells of revolution with a variable stiffness]. Kiev, Naukova Dumka Publ., 1973, 228 p. (In Russian).

13.Rogalevich V.V. Kollokatsionnye metody. Sushchnost. Primery [Collocation Methods. Essence. Examples]. Ekaterinburg, AMB Publ., 2001, 298 p. (In Russian).

14.Crouch S. L., Starfield A. M. Boundary element methods in solid mechanics. George Allen & Unwin, London, 1983, 322 p.

15.Brebbia C.A., Walker S. Boundary Element Techniques in Engineering. Newnes-Butterworths, London, 1979.

16.Artyukhin Yu.P., Gribov A.P. Resheniye zadach nelineynogo deformirovaniya plastin i pologikh obolochek metodom granichnykh elementov [Solution of Nonlinear Deformation Problems by the Boundary Element Method]. Kazan, Fen Publ., 2002, 199 p. (In Russian).

17.Banerjee P. K., Butterfield R. Boundary Element Methods in Engineering Science. McGraw-Hill Book Company, London, 1981.

18.Kantorovich L.V., Krylov V.I. Approximate Methods of Higher Analysis. Mineola, New York, Dover Publications, Inc., 2018, 704 p.  

19.Grigorenko Ya.M., Vasilenko A.T., Emelyanov I.G., Kryukov N.N., Nemish Yu.N., Pankratova N.D., Pelekh B.L., Vlaikov G.G., Maksimuk A.V., Urusova G.P., A.N. Guz ed. Mekhanika kompozitov: v 12 t. T. 8. Statika elementov konstruktsiy [Composite Mechanics. In 12 vols. Vol. 8. Statics of Structural Components]. Kiev, Naukova Dumka Publ., 1999, 379 p. (In Russian).

20.Karmishin A.V., Lyaskovets V.A., Myachenkov V.I., Frolov A.N. Statika i dinamika tonkostennykh obolochechnykh konstruktsiy [Statics and dynamics of thin-walled shell structures]. Moscow, Mashinostroenie Publ., 1975, 376 p. (In Russian).

21.Godunov S.K. Numerical solution of boundary-value problems for a system of linear ordinary differential equations. Uspekhi Matematicheskikh Nauk, 1961, vol. 16, no. 3, pp. 171–174.

22.Grigorenko Ya.M., Vlaikov G.G., Grigorenko A.Ya. Chislenno-analiticheskoye resheniye zadach mekhaniki obolochek na osnove razlichnykh modeley [Numerical Analytical Solution of Shell Mechanics Problems on the Basis of Different Models]. Kiev, Akademperiodika Publ., 2006, 472 p. (In Russian).

23.Emelyanov I.G. Kontaktnye zadachi teorii obolochek [Contact Problems of the Shell Theory]. Ekaterinburg, UrO RAN Publ., 2009, 185 p. (In Russian).

24.Emelyanov I.G., Mironov V.I. Dolgovechnost obolochechnykh konstruktsiy [Durability of Shell Structures]. Ekaterinburg, UrO RAN Publ., 2012, 224 p. (In Russian).

25.Gallagher R.H. Finite Element Analysis Fundamentals. Prentice Hall, Englewood Cliffs, New Jersey, 1975.

26.Golovanov A.I., Tyuleneva O.N., Shigabutdinov A.F. Metod konechnykh elementov v statike i dinamike tonkostennykh konstruktsii [Finite Element Method in Thin-Walled Structure Statics and Dynamics]. Moscow, Fizmatlit Publ., 2006, 392 p. (In Russian).

27.Rikards R.B. Metod konechnykh elementov v teorii obolochek i plastin [Finite Element Analysis in Theory of Shells and Plates]. Riga, Zinatne Publ., 1988, 284 p. (In Russian).

28.Zenkevich O.K. Metod konechnykh elementov v tekhnike [Boundary-Element Method in Engineering]. Мoscow, Mir Publ., 1975, 541p. (In Russian).

29.Novozhilov I.I. Voprosy mekhaniki sploshnoy sredy [Issues of Continuum Mechanics].  Leningrad, Sudostroenie Publ., 1989, 400 p. (In Russian).

30.Emelyanov I.G., Efimov V.P., Kuznetsov A.V. A model of the stress-strain state of a tank car boiler with an improved scheme of support on the frame. Tyazheloe Mashinostroyenie, 2005, no. 8, pp. 44–49. (In Russian).

31.Emelyanov I.G., Mironov V.I., Kuznetsov A.V. Evaluation of the stress state and service life of shell structures. Problemy Mashinostroeniya i Nadezhnosti Mashin, 2007, no. 5, pp. 57–65. (In Russian).

32.Vasilenko A.T., Golub G.P., Grigorenko Ya.M. Calculating the parameters of the stress state of composite structural components on the basis of shell models. In: Raschety na prochnost, vyp. 30 [Strength Calculations, iss. 30]. Moscow, Mashinostroenie Publ., 1989, pp. 87–96. (In Russian).

33.Johnson K.L. Contact Mechanics. Cambridge University Press, Cambridge, 1985.

34.Hertz H. Über die Berührung fester elastischer Körper. Journal für die reine und angewandte Mathematik, 1881, num. 92, ss. 156–171.      

35.Grigolyuk E.I., Tolkachev V.M. Kontaktnyye zadachi teorii plastin i obolochek [Contact Problems in Plates and Shells Theory]. Moscow, Mashinostroeniye Publ., 1980, 411p. (In Russian).

36.Kantor B.Ya. Kontaktnye zadachi nelineynoy teorii obolochek vrashcheniya [Contact Problems in Nonlinear Theory of Rotated Shells]. Kiev, Naukova dumka Publ., 1990, 136 p. (In Russian).

37.Artyukhin Ju.P., Malkin S.A. Artjuhin Ju.P., Malkin S.A. Analiticheskie i chislennye metody resheniya integralnykh uravneniy v zadachakh uprugogo vozdeistviya tel [Analytical and Numerical Methods for Solving Integral Equations in Problems of Elastic Action of Bodies]. Kazan, Kazanskiy Gos. Un-t Publ., 2007, 292 p. (In Russian).

38.Mossakovskii, V.I., Gudramovich, B.C., and Makeev, E.M. Kontaktnoe vzaimodeistvie elementov obolochechnykh konstruktsii [Contact Interaction of Shell Elements]. Kiev, Naukova dumka Publ., 288 p. (In Russian).

39.Obraztsov I.F., Nerubailo B.V., Olshansky V.P. Obolochki pri lokalizovannykh vozdeistviyakh (obzor rabot, osnovnye rezultaty i napravleniya issledovaniy). Dep. v VINITI 12.02.88 [Shells under Local Effects. Survey, Main Results and Lines of Research]. Moscow, VINITI Publ., 1988, 192 p. (In Russian).

40.Pelekh B.L., Maksimuk A.V., Korovaichuk I.M. Kontaknye zadachi dlya sloistykh elementov konstruktsiy i tel s pokrytiyami [Contact Problems for Layered Structural Components and Coated Bodies]. Kiev, Naukova Dumka Publ., 1988, 280 p. (In Russian).

41.Emelyanov I.G., Kuznetsov A.V. Application of Virtual Elements to the Determination of the Stress State of Shells of Revolution. Vychislitelnaya Mekhanika Sploshnykh Sred, 2014, vol. 7, no. 3, pp. 245–252. (In Russian).

42.Emel’yanov I.G. Investigation into the contact interaction between shell and base with notches. Journal of Machinery Manufacture and Reliability, 2015, vol. 44, no. 3, pp. 263–270. DOI: 10.3103/S1052618815030048.

43.Emel’yanov I.G., Mironov V.I., Kuznetsov A.V. Evaluation of the life of a shell construction lying on supports. Journal of Machinery Manufacture and Reliability, 2010, vol. 39, iss. 1, pp. 83–88. DOI: 10.3103/S1052618810010139.

44.Mironov V.I., Yemel'yanov I.G. Complete diagram method for fatigue resistance calculation. In: Trudy mezhdunarodnoy nauchno-tekhnicheskoy konferentsii “Prochnost materialov i elementov konstruktsiy” [Strength of materials structure elements: Proceedings of International Scientific and Technical Conference, Kiev, September 28–30, 2010]. Kiev, IPP NANU, 2011, pp. 697–704.

45.Emel’yanov I.G., Mironov V.I., Kuznetsov A.V. Cyclic life of tank car shell. In: Trudy mezhdunarodnoy nauchno-tekhnicheskoy konferentsii “Prochnost materialov i elementov konstruktsiy” [Strength of materials structure elements: Proceedings of International Scientific and Technical Conference, Kiev, September 28–30, 2010]. Kiev, IPP NANU, 2011, pp. 836–843.

46.Yemelyanov I.G., Mironov V.I., Kuznetsov A.V., Yakushev A.V. Contact interaction of boiler at the tank-truck with the tracks bearings. Izvestiya Samarskogo Nauchnogo Tsentra RAN, vol. 13, nos. 1–2, 2011, pp. 436–439. (In Russian).

47.Yemelyanov I.G. Mironov V.I., Yakushev A.V. Contact problem in the fatigue strength calculation of tank wagon elements. Transport Urala, 2011, no. 3 (30), pp. 49–55.

48.Yemelyanov I.G., Mironov V.I., Yakushev A.V., Lukashuk O.A. Development of rapid method for car steel quality. Transport Urala, 2012, no. 2 (33), pp .13–17. (In Russian).

49.Barashkova E., Emelyanov I. Stress State of Shells under Arbitrary Load. In: Proceedings of the 5th WSEAS International Conference on Finite Differences - Finite Elements – Finite Volumes - Boundary Elements (F-and-B'12), Prague, Czech Republic, September 24-26, 2012. Prague, WSEAS Press, 2012, pp.33–37.

50.Emelyanov I. G., Mironov V. I. Contact problem for a shell considering the transverse load. Journal of Machinery Manufacture and Reliability, 2013, vol. 42, no. 1, pp. 36–40. DOI: 10.3103/S1052618813010056.

51.Barashkova Ye.A., Yemel'yanov I.G. Determining stress state of shells at local loading. In: Matematicheskiye metody i metody optimizatsii v mashinostroyenii: materialy I Mezhdunarodnoy konferentsii po metodam optimizatsii v tekhnike (OTENG '13) [Mathematical methods and optimization methods in mechanical engineering: Proceedings of the Ist International Conference on Optimization Methods in Engineering, Antaliya, Turkey, October 8-10, 2013, pp.109–113.

52.Yemelyanov I.G., Kuznetsov A.V., Mironov V.I. Mathematical model describing the stress state of gas turbine locomotive cabin upon collision with obstacle. Transport Urala, 2013, no. 4 (39), pp. 71–74. (In Russian).

53.Emelyanov I. G., Kuznetsov A. V. The stressed state of shell structures under local loads.  Journal of Machinery Manufacture and Reliability, 2014, vol. 43, no. 1, pp. 42–47. DOI: 10.3103/S1052618814010051.

54.Yemel'yanov I.G., Kuznetsov A.V., Mironov V.I. Determination of lifetime of tank car structural elements. Reliability and life of machines and structures, 2014, iss. 38, pp. 45–54.

55.Mironov V.I., Kuznetsov A.V., Emel’yanov I.G. Consideration of cyclic degradation of the material and abnormality of the surface layer mechanical properties in calculating the life of a plate with an opening. Strength of Materials, 2014, vol. 46, no. 5, pp. 638–643. DOI: 10.1007/s11223-014-9594-y.

56.Yemel'yanov I.G. Application of discrete Fourier series to the stress analysis of shell structures. Computational Continuum Mechanics, 2015, vol. 8, no. 3, pp. 245–253. DOI: 10.7242/1999-6691/2015.8.3.20.

57.Emel’yanov I.G. Mironov V.I. Kuznetsov A.V. Estimation of the fracture strength of a spatial beam–rod structure notches. Journal of Machinery Manufacture and Reliability, 2015, vol. 44, no. 5, pp. 449–454. DOI: 10.3103/S1052618815050076.

58.Emel’yanov I. G. Mironov V.I. Kuznetsov A.V. On an approach to the evaluation of the strength of a spatial rod system under impact loading. Diagnostics, Resource and Mechanics of materials and structures, 2015, iss. 2, pp. 16–23. Available at http://dream-journal.org/issues/2015-2/2015-2_24.html (accessed 05.05.2015).

59.Emel’yanov I.G., Mironov V.I., Kuznetsov A.V. The definition of the resistance area boundaries for a locomotive operator cab under non-regulated loads. In: AIP Conf. Proc., 2016, 1785, 040014. DOI: 10.1063/1.4967071.

60.Emel’yanov I.G. Mironov V.I. Kuznetsov A.V. Determination of safe operating conditions for supporting structures under abnormal loads. Journal of Machinery Manufacture and Reliability, 2017, vol. 46, no. 5, pp. 136–142. DOI: 10.3103/S1052618817020042.

61.Emel’yanov I.G., Mironov V.I., Kuznetsov A.V. Vehicle Survivability Calculation. Russian Journal of Construction Science and Technology, 2017, vol. 3, no. 1, pp. 40–44.  DOI: 10.15826/rjcst.2017.1.005.

62.Yemel'yanov I.G., Kuznetsov A.V. Opredeleniye napryazhennogo sostoyaniya tonkostennykh konstruktsiy s ispol'zovaniyem metodov teorii obolochek. Transportnye Sistemy i Tekhnologii, 2017, no. 3, pp. 64–78.

                                 

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