Analysis of the Problem of Stability of Thin Shells Compliant to Shear and Compression

The problem of stability of shells compliant to shear and compression is studied by the finite-element method. On the basis of relations of the geometrically nonlinear theory of thin shells compliant to shear and compression (six-mode version), we write the key equations for the determination of their initial postcritical state and formulate the corresponding variational problem. A numerical scheme of the finite-element method is constructed for the solution of the problems of stability of these shells. The order of the rate of convergence of the scheme proposed for the numerical solution of the problems of stability is investigated.

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

References

  1. 1.

    P. Vahin and I. Shot, “Analysis of stress-strain states of thin shells compliant to shear and compression,” Visn. L’viv. Univ. Ser. Prikl. Mat. Inf., Issue 11, 135–147 (2006).

  2. 2.

    A. S. Vol’mir, Stability of Deformable Systems [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  3. 3.

    É. I. Grigolyuk and V. V. Kabanov, Stability of Shells [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  4. 4.

    Ya. M. Grigorenko, G. G. Vlaikov, and A. Ya. Grigorenko, Numerical-Analytic Solutions of the Problems of Mechanics of Shells by Using Various Models [in Russian], Akademperiodika, Kiev (2006).

    Google Scholar 

  5. 5.

    Kh. M. Mushtari, “Some generalizations of the theory of thin shells with applications to the problem of stability of the elastic equilibrium,” Izv. Fiz.-Mat. Obshch. Kazan. Univ. Ser. 3, 9, 71–150 (1938).

    Google Scholar 

  6. 6.

    V. V. Novozhilov, Foundations of Nonlinear Elasticity Theory, Dover, New York (1999).

    Google Scholar 

  7. 7.

    R. B. Rikards, Finite-Element Method in the Theory of Shells and Plates [in Russian], Zinatne, Riga (1988).

    MATH  Google Scholar 

  8. 8.

    G. Strang and G. J. Fix, An Analysis of the Finite-Element Method, Prentice-Hall, Englewood Cliffs (1973).

    MATH  Google Scholar 

  9. 9.

    I. Ya. Shot, “Numerical solution of the problems of the theory of thin shells compliant to shear and compression,” Visn. Odes. Nats. Univ. Mat. Mekh., 18, Issue 1 (17), 132–141 (2013).

  10. 10.

    D. Bushnell, Stress, Stability, and Vibration of Complex Shells of Revolution: Analysis and User’s Manual for BOSOR 3, SAMSO TR 69-375. LMSC Rept. N-5J-69-1, Lockheed Missiles and Space Co. (1969).

  11. 11.

    A. Libai and J. G. Simmonds, The Nonlinear Theory of Elastic Shells, Cambridge Univ. Press, Cambridge (1998).

    Book  MATH  Google Scholar 

  12. 12.

    M. Stein, “Some recent advances in the investigation of shell buckling,” AIAA J., 6, No. 12, 2339–2345 (1968); https://doi.org/10.2514/3.4992.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to I. Ye. Bernakevych.

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, 59, No. 4, pp. 91–96, October–December, 2016.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bernakevych, I.Y., Vahin, P.P., Kozii, I.Y. et al. Analysis of the Problem of Stability of Thin Shells Compliant to Shear and Compression. J Math Sci 238, 108–115 (2019). https://doi.org/10.1007/s10958-019-04221-0

Download citation