Method of Solving Geometrically Nonlinear Bending Problems of Thin Shallow Shells of Complex Shape

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 25, no. 4, 2022 (December)
Pages 39-45
Cited by J. of Mech. Eng., 2022, vol. 25, no. 4, pp. 39-45



Serhii M. Sklepus, A. Pidhornyi Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-4119-4310



A new numerical analytical method for solving geometrically nonlinear bending problems of thin shallow shells and plates of complex shape is given in the paper. The problem statement is performed within the framework of the classic geometrically nonlinear formulation. The parameter continuation method was used to linearize the nonlinear bending problem of shallow shells and plates. An increasing parameter t related to the external load, which characterizes the shell loading process, is introduced. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible movement speeds, is constructed. To find the main unknowns of the problem of nonlinear bending of the shell (displacement, deformation, stress), the Cauchy problem was formulated by the parameter t for the system of ordinary differential equations, which was solved by the fourth order Runge-Kutta-Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometric linear deformation. The right-hand sides of the differential equations at fixed values of the parameter t, corresponding to the Runge-Kutta-Merson scheme, were found from the solution of the variational problem for the functional in the Lagrange form. Variational problems were solved using the Ritz method combined with the R-function method, which allows to accurately take into account the geometric information about the boundary value problem and provide an approximate solution in the form of a formula – a solution structure that exactly satisfies all (general structure) or part (partial structure) of boundary conditions. The test problem for the nonlinear bending of a square clamped plate under the action of a uniformly distributed load of different intensity is solved. The results for deflections and stresses obtained using the developed method are compared with the analytical solution and the solution obtained by the finite element method. The problem of bending of a clamped plate of complex shape is solved. The effect of the geometric shape on the stress-strain state is studied.


Keywords: flexible shallow shell, complex shape, R-function method, parameter continuation method.


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  1. Grigorenko, Ya. M. & Vasilenko, A. T. (1981). Metody rascheta obolochek [Methods for calculating shells]: in 5 vols. Vol. 4. Teoriya obolochek peremennoy zhestkosti [Theory of shells with variable stiffness]. Kyiv: Naukova dumka, 544 p. (in Russian).
  2. Grigorenko, Ya. M. & Gulyaev, V. I. (1991). Nonlinear problems of shell theory and their solution methods (review). International Applied Mechanics, vol. 27, pp. 929–947.
  3. Rasskazov, A. O., Sokolovskaya, I. I., Shulga, N. A. (1986). Teoriya i raschet sloistykh ortotropnykh plastin i obolochek [Theory and calculation of layered orthotropic plates and shells]. Kyiv: Vysshaya shkola, 191 p. (in Russian).
  4. Bathe, K. J. & Wilson, E. L. (1976). Numerical methods in finite element analysis. Prentice Hall.
  5. Sabir, A. B. & Djoudi, M. S. (1995). Shallow shell finite element for the large deflection geometrically nonlinear analysis of shells and plates. Thin-Walled Structures, vol. 21, iss. 3, pp. 253–267.
  6. Bucalem, M. L. & Bathe, K. J. (1997). Finite element analysis of shell structures. Archives of Computational Methods in Engineering, vol. 4, pp. 3–61.
  7. Rvachev, V. L. (1982). Teoriya R-funktsiy i nekotoryye yeye prilozheniya [R-functions theory of and some of its applications]. Kyiv: Naukova dumka, 552 p. (in Russian).
  8. Kurpa, L. V., Lyubitskaya, E. I., & Morachkovskaya, I. O. (2010). The R-function method used to solve nonlinear bending problems for orthotropic shallow shells on an elastic foundation. International Applied Mechanics, vol. 46, pp. 660–668.
  9. Smetankina, N., Merkulova, A., Merkulov, D., Postnyi, O. (2021). Dynamic response of laminate composite shells with complex shape under low-velocity impact. In: Nechyporuk M., Pavlikov V., Kritskiy D. (eds.) Integrated Computer Technologies in Mechanical Engineering-2020. ICTM 2020. Lecture Notes in Networks and Systems. Cham: Springer, vol. 188, pp. 267–276.
  10. Volmir, A. S. (1956). Gibkiye plastinki i obolochki [Flexible plates and shells]. Moscow: Gosteorizdat, 420 p. (in Russian).
  11. Grigolyuk, E. I. & Shalashilin, V. I. (1984). Metod prodolzheniya po parametru v zadachakh nelineynogo deformirovaniya sterzhney, plastin i obolochek [Method of continuation with respect to a parameter in problems of nonlinear deformation of rods, plates and shells]. Issledovaniya po teorii plastin i obolochekResearch on the theory of plates and shells, iss. 17, part 1, pp. 3–58 (in Russian).
  12. Washizu, K. (1982). Variational methods in elasticity and plasticity. Oxford: Pergamon Press, 630 p.
  13. Krylov, V. I, Bobkov, V. V., & Monastyrnyy, P. I. (1977). Vychislitelnyye metody [Computational methods]. Moscow: Nauka, 399 p. (in Russian).
  14. Urthaler, Y. & Reddy, J. N. (2008). A mixed finite element for a nonlinear bending analysis of laminated composite plates based on FSDT. Mechanics of Advanced Materials and Structures, vol. 15, pp. 335–354.
  15. Levy, S. (1942). Square plate with clamped edges under normal pressure producing large deflections. Tech. Report, National Advisory Committee for Aeronautics.


Received 02 September 2022

Published 30 December 2022