Strength of Materials

, Volume 50, Issue 6, pp 852–858 | Cite as

Application of the Projection-Iterative Scheme of the Method of Local Variations to Solving Stability Problems for Thin-Walled Shell Structures Under Localized Actions

  • E. L. HartEmail author
  • V. S. Hudramovich

The tasks of the stability of reinforced spherical shells that are part of the extended inhomogeneous structures are investigated under localized actions based on the developed new projection-iterative version of the method of local variations. Stability under local loading implies the performance of thin-walled shell structures in rocket and space equipment, antenna and waveguide technology, as well as power engineering. Under localized actions, there is a significant concentration of stresses and strains. The methods for solving relevant tasks are rather complex, multiple-mesh variational-difference schemes are within the area of focus. For the tasks of stress-strain state and stability under considerably inhomogeneous stress states, the local variation method is effective, but it requires more computation time as compared with the Ritz-type finite difference and variation methods. Noteworthy is that this leads to the need to develop more adequate schemes for its implementation based on the concept of projection-iterative methods. This scheme of the method of local variations is analyzed, its convergence is investigated, and the practical efficiency associated with a significant decrease in computation time for numerical implementation is shown. The tasks of the stability of complex shell structures with spherical shells reinforced by support rings under the exposure of local loads are considered. The experimental investigation data are given. The calculated data of the critical forces and the configuration of the waveforms are corroborated experimentally.


projection-iterative scheme of the method of local variations tasks of contact interaction and stability under localized actions experimental investigations of the local stability of structures with spherical shells 


  1. 1.
    V. S. Gudramovich, E. M. Makeev, V. I. Mossakovskii, and P. I. Nikitin, “Contact interaction of shell structures with supporting bases under complex service conditions,” Strength Mater., 17, No. 10, 1463–1471 (1985).CrossRefGoogle Scholar
  2. 2.
    V. I. Mossakovskii, V. S. Gudramovich, and E. M. Makeev, Contact Interaction of Elements of Shell Structures [in Russian], Naukova Dumka, Kiev (1988).Google Scholar
  3. 3.
    V. S. Gudramovich, I. A. Diskovskii, and E. M. Makeev, Thin-Wall Components of Reflector Antennas [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
  4. 4.
    V. S. Hudramovych, “Contact mechanics of shell structures under local loading,” Int. Appl. Mech., 45, No. 7, 708–729 (2009).CrossRefGoogle Scholar
  5. 5.
    V. S. Hudramovych, “Contact interactions of elements of extended shell structures in the use of models of physical nonlinearity,” in: Proc. of the X Federal Meeting on Fundamental Problems of Theoretical and Applied Mechanics, NGU, Novgorod (2011), pp. 57–59.Google Scholar
  6. 6.
    V. S. Hudramovich, D. V. Klimenko, and E. L. Hart, “Influence of cutouts on strength of cylindrical compartments of launch vehicles in the case of inelastic deformation of the material,” Kosm. Nauka Tekhnol., No. 6, 12–20 (2016).Google Scholar
  7. 7.
    A. A. Il’yushyn, Works [in Russian], in 4 volumes, Vol. 2: Plasticity: 1946–1966, Fizmatlit, Moscow (2004).Google Scholar
  8. 8.
    V. S. Gudramovich, Stability of Elastoplastic Shells [in Russian], Naukova Dumka, Kiev (1987).Google Scholar
  9. 9.
    F. L. Chernous’ko, “A local variation method for the numerical solution of variational problems,” Zh. Vychisl. Matem. Matem. Fiz., 5, No. 4, 749–754 (1965).Google Scholar
  10. 10.
    F. L. Chernous’ko and N. V. Banichuk, Variational Problems of Mechanics and Control [in Russian], Nauka, Moscow (1973).Google Scholar
  11. 11.
    M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreiko, et al., Approximate Solution to Operator Equations [in Russian], Nauka, Moscow (1969).Google Scholar
  12. 12.
    H. Gajewski und R. Kluge, “Projections-Iterationsverfahren und nichtlineare Problemen mit monotonen Operatoren,” Monatsber. Deutsch. Akad. Wiss., 12, No. 23, 98–115 (1970).Google Scholar
  13. 13.
    G. I. Marchuk and V. I. Agoshkov, Introduction into Production-Grid Methods [in Russian], Nauka, Moscow (1981).Google Scholar
  14. 14.
    E. L. Hart, “Projection-iterative version of the pointwise relaxation method,” J. Math. Sci., 167, No. 1, 76–88 (2010).CrossRefGoogle Scholar
  15. 15.
    E. L. Hart and V. S. Hudramovich, “Application of the projective-iterative versions of FEM in damage problems for engineering structures,” in: Proc. of the 2nd Int. Conf. “Maintenance-2012” (University of Zenica, Bosnia and Herzegovina) (2012), pp. 157–164.Google Scholar
  16. 16.
    V. Hudramovich, E. Hart, and S. Rjabokon’, “Plastic deformation of nonhomogeneous plates,” J. Eng. Math., 78, No. 1, 181–197 (2013).CrossRefGoogle Scholar
  17. 17.
    V. S. Hudramovich and E. L. Hart, “Finite element analysis of non-localized fracture of plane deformed elastoplastic media with local stress concentrators,” in: Elasticity and Inelasticity (Proc. of the Int. Symp. on the Problems of Mechanics of Deformable Bodies, dedicated to the 105th Anniversary of A. A. Il’yushin, January 20–21, 2016, Moscow) [in Russian], Moscow University Publ., Moscow (2016), pp. 158–161.Google Scholar
  18. 18.
    E. Hart and V. Hudramovych, “Projection-iterative modifications of the method of local variations and aspects of their use in the tasks of local stability of shells,” in: Modern Problems of Mechanics and Mathematics [in Ukrainian], IPPMM NAS of Ukraine, Lviv (2008), pp. 18–20.Google Scholar
  19. 19.
    Sh. A. Mukhamediev, L. V. Nikitin, and S. L. Yunga, “Application of the modified method of local variations to tasks of nonlinear mechanics of fractures,” Izv. AN SSSR. Ser. Mekh. Tver. Tela, No. 1, 76–83 (1976).Google Scholar
  20. 20.
    G. N. Savin, Stress Distribution around Holes [in Russian], Naukova Dumka, Kiev (1968).Google Scholar
  21. 21.
    A. A. Lebedev (Ed.), B. I. Koval’chuk, F. F. Giginyak, and V. P. Lamashevskii, Mechanical Properties of Structural Materials in Complex Stress State [in Russian], In Yure, Kiev (2003).Google Scholar
  22. 22.
    E. L. Hart and V. S. Hudramovich, “Projection-iterative modification of the method of local variations for problems with a quadratic functional,” J. Appl. Math. Mech., 80, No. 2, 156–163 (2016).CrossRefGoogle Scholar
  23. 23.
    V. S. Gudramovich, “Critical state of inelastic shells under combined loading,” in: Stability in Mechanics of Deformed Solid [in Russian], Vol. 1, KGU, Kalinin (1981), pp. 61–87.Google Scholar
  24. 24.
    D. Bushnell, “Buckling of shells – pitfall for designers,” AIAA J., 19, No. 9, 1183–1226 (1981).CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Oles Honchar Dnipro National UniversityDneprUkraine
  2. 2.Institute of Technical Mechanics of the National Academy of Sciences of Ukraine and the State Space Agency of UkraineDneprUkraine

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