Skip to main content

Analysis of a spherical tank under a local action


A spherical tank, being perfect as far as weight is concerned, is used in spacecraft, where the thin-walled elements (shells) are united by frames. Obviously, local actions on the shell and hence the stress concentration in the shell cannot be avoided. Attempts to make weight structure of the spacecraft perfect inevitably decrease the safetymargin of the components, which is possible only if the stress-strain state of the components is determined with a controlled error. A mathematical model of shell deformation mechanics is proposed for this purpose, and its linear differential equations are obtained with an error that does not exceed the error of Kirchhoff assumptions in the theory of shells. The algorithm for solving these equations contains procedures for estimating the convergence of the Fourier series and the series of the hypergeometric function with a prescribed error, and the problem can be solved analytically.

This is a preview of subscription content, access via your institution.


  1. Ya.M. Grigorenko, L. A. Il’in, and A. D. Kovalenko, Theory of Thin Conical Shells and Their Applications in Machine Engineering (Nauka, Moscow, 1992) [in Russian].

    Google Scholar 

  2. G. B. Men’kov, Solution of Problems of Mechanics of Shell Deformation by the Method of Functional Normalization, Candidate’s Dissertation in Mathematics and Physics (Kazan, 1999) [in Russian].

    Google Scholar 

  3. Yu. N. Vinogradov and G. B. Men’kov, A Method for Functional Normalization for Boundary Value Problems in the Theory of Shells (Editorial URSS,Moscow, 2001) [in Russian].

    MATH  Google Scholar 

  4. A. P. Filin, Elements of the Theory of Shells (Stroiizdat, Leningrad Division, Leningrad, 1975) [in Russian].

    MATH  Google Scholar 

  5. V. V. Novozhilov, Theory of Thin Shells (St. Petersburg Univ., St. Petersburg, 2010) [in Russian].

    MATH  Google Scholar 

  6. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd. ed. (Chapman & Hall/CRC, Boca Raton, 2003).

    MATH  Google Scholar 

  7. Yu. Luke, Mathematical Functions and Their Approximations (Academic Press, New York–San Francisco–London, 1975;Mir,Moscow, 1980).

    MATH  Google Scholar 

  8. K. Aomoto and M. Kita, Theory of Hypergeometric Functions (Springer, 2011).

    Book  MATH  Google Scholar 

  9. E. Madenci and I. Guven, The Finite Element Method and Applications in Engineering Using ANSYS (Springer, 2006).

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yu. N. Vinogradov.

Additional information

Original Russian Text © Yu.N. Vinogradov, M.V. Konstantinov, 2016, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2016, No. 2, pp. 109–120.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vinogradov, Y.N., Konstantinov, M.V. Analysis of a spherical tank under a local action. Mech. Solids 51, 223–233 (2016).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: