Published since 1923
DOI: 10.33622/0869-7019
Russian Science Citation Index (RSCI) Web of Science
  • Catenoidal Shells
  • UDC 624.04.074.4:539.41:513.73
    Sergey N. KRIVOSHAPKO, e-mail:
    Vyacheslav N. IVANOV, e-mail:
    Peoples' Friendship University of Russia, ul. Miklucho-Maklaya, 6, Moscow 117198, Russian Federation
    Abstract. Architects, designing shells, use in their projects, as a rule, geometrical forms well proved themselves that constitute 5-10 % of the surfaces known to geometricians. But there is well known surface of revolution which has not a great popularity among architects and designers, practically, there are no examples of its application in building industry. It is a catenoidal surface. The mean curvature at all points of the catenoidal surface is equal to zero, hence, it is the minimal surface. The catenoid is formed by rotation of the catenary line around the Oz axis. The article gives an overview a review of the known methods of calculation of catenoidal shells and analysis the stress-strain state of five shells of rotation having the close geometrical parameters for the determination of optimal forms. A vast list of the used references containing 22 names helps to find the additional information.
    Key words: shell of revolution, catenoid, minimal surface, momentless theory of shells, variation-difference method of analysis, moment theory of shells.
    1. Krivoshapko S.N., Emeliyanova Yu. V. To a problem on surface of revolution with geometrically optimal camber of a shell. Montazhnye i spetsialnye raboty v stroitelstve, 2006, no. 2, pp. 11-14. (In Russian).
    2. Krivoshapko S. N. Developable products made by parabolic bending of thin metal blanks. Tehnologiya mashinostroeniya, 2008, no. 2, pp. 25-28 (In Russian).
    3. Kenneth Brecher. Mathematics, art and science of the pseudosphere. Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, pp. 469-472.
    4. Mamieva I. A., Razin A. D. Landmark spatial structures in the form of conic surfaces. Promyshlennoe i grazhdanskoe stroitel'stvo. 2017, no. 10, pp. 5-11. (In Russian).
    5. Krivoshapko S. N. Static, vibration, and buckling analyses and applications to one-sheet hyperboloidal shells of revolution. Applied Mechanics Reviews, vol. 55, no. 3, 2002, pp. 241-270.
    6. Krivoshapko S. N., Rynkovskaya M. Five types of ruled helical surfaces for helical conveyers, support anchors and screws. MATEC Web of Conferences, 2017, vol. 95. DOI:
    7. Ch. A. Bock Hyeng, Krivoshapko S. N. Umbrella-type surfaces in architecture of spatial structures. IOSR Journal of Engineering (IOSRJEN), 2013, vol. 3, iss. 3, pp. 43-53.
    8. Krivoshapko S. N., Mamieva I. A. Umbrella surfaces and surfaces of umbrella type in architecture. Promyshlennoe i grazhdanskoe stroitel'stvo, 2011, no. 7 (1), pp. 27-31. (In Russian).
    9. Podgorniy A. L., Grinko E. A., Solovey N. A. Research of new forms of surfaces as applied to structures of divorce purposes. Vestnik Rossiyskogo universiteta druzhby narodov. Seriya: Inzhenernie Issledovaniya. 2013, no. 1, pp. 140-145. (In Russian).
    10. Novozhilov V.V. Teoriya tonkih obolochek [Theory of thin shells]. St.Petersburg, 2010. 380 p. (In Russian).
    11. Krivoshapko S. N. Drop-shaped, catenoidal, and pseudospherical shells. Montazhnye i spetsialnye raboty v stroitelstve, 1998, no. 11-12, pp. 28-32. (In Russian).
    12. Ganeeva M. S., Skvortsova Z. V. Stress-strain state of a catenoidal shell of revolution from orthotropic material. Aktualnie problemy mehaniki sploshnoy sredy [Actual problems of continuum mechanics]. Proc., vol. II, Kazan, IMM KazNTs RAN Publ., 2011, pp. 153-160. (In Russian).
    13. Bernakevich I. E., Vagin P. P., Uzhegov S. O. Stress-strain state analyses of building structures on a base of accurate shell theory. Mistobuduvannya ta Teritorialne Planuvannya. Kiev, KNUBA Publ., 2014, vol. 54, pp. 42-49. (In Ukrainian).
    14. Kamoulakos A. A catenoidal patch test for the inextensional bending of thin shell finite elements. Computer Methods in Applied Mechanics and Engineering, 1991, vol. 92, iss. 1, pp. 1-32.
    15. Tornabene Fr., Fantuzzi N. Mechanics of laminated composite doubly-curved shell structures: the generalized differential quadrature method and the strong formulation finite element method. Societa Editrice Esculapio, 2014. 799 p.
    16. Thomas R. Powers, Greg Huber, Raymond E. Goldstein. Fluid-membrane tethers: minimal surfaces and elastic boundary layers. Available at: arXiv:cond-mat/0201290 [cond-mat.soft]. (accessed 16.01.2018).
    17. Aseev A. V., Makarov A. A. On visualization of elements of strengthened thin-walled shells. Kompyuternie instrumenty v obrazovanii, 2014, no. 2, pp. 35-45 (In Russian).
    18. Tornabene Fr., Fantuzzi N., Bacciocchi M., Viola Er. Laminated composite doubly-curved shell structures. Differential geometry higher-order structural theories. Societa Editrice Esculapio, 2016, vol. 1. 744 p.
    19. Filz G. H., Schiefer S. Rapid assembly of planar quadrangular, self-interlocking modules to anticlastically curved forms. Eco-Architecture, 2014, vol. 142, pp. 397-407.
    20. Jaime Horta-Rangel, Humberto Uehara-Guerrero, Teresa Lopez-Lara, et al. Optimal design of a fabric shell using a coupled fem-optimization procedure. Asian Journal of Science and Technology, 2014, vol. 5, iss. 11, pp. 722-726.
    21. Krivoshapko S. N. On mistaken in terminology on theory of surfaces and geometric modelling. Geometriya i grafika, 2017, vol. 5, no. 2, pp. 32-38. (In Russian).
  • For citation: Krivoshapko S. N., Ivanov V. N. Catenoidal Shells. Promyshlennoe i grazhdanskoe stroitel'stvo [Industrial and Civil Engineering], 2018, no. 12, pp. 7-13. (In Russian).