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Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method

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A generalized Kantorovich–Vlasov method is used to solve stationary problems for shallow shells with rectangular planform and arbitrary boundary conditions. The efficiency of the approach is illustrated by examples

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References

  1. V. L. Berdichevskii, Variational Principles in Continuum Mechanics [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  2. E. I. Bespalova, “Solving problems of elasticity by methods of complete systems,” Zh. Vych. Mat. Mat. Fiz., 29, No. 9, 1346–1353 (1989).

    MATH  MathSciNet  Google Scholar 

  3. E. I. Bespalova and I. F. Latsinnik, “Stressed state in nonclosed cylindrical shells of arbitrary profile,” Int. Appl. Mech., 5, No. 5, 477–481 (1969).

    Google Scholar 

  4. A. I. Vaindiner, “A generalization of the Bubnov–Galerkin–Kantorovich method for approximate solution of boundary-value problems,” Vestnik MGU, Mat. Mekh., 2, 101–109 (1967).

    MathSciNet  Google Scholar 

  5. V. Z. Vlasov, “A new practical method to design folded-plate structures and shells,” Stroit. Promyshlen., No. 11, 33–38; No. 12, 21–26 (1932).

  6. Ya. M. Grigorenko, E. I. Bespalova, A. B. Kitaigorodskii, and A. I. Shinkar’, Free Vibrations of Elements of Shell Structures [in Russian], Naukova Dumka, Kyiv (1986).

    Google Scholar 

  7. Ya. M. Grigorenko and A. T. Vasilenko, Theory of Shells of Variable Rigidity [in Russian], Vol. 4 of the five-volume series Methods of Shell Design, Naukova Dumka, Kyiv (1981).

    Google Scholar 

  8. Ya. M. Grigorenko, A. T. Vasilenko, and G. P. Golub, Statics of Anisotropic Shells with Finite Shear Stiffness [in Russian], Naukova Dumka, Kyiv (1987).

    Google Scholar 

  9. Ya. M. Grigorenko, A. T. Vasilenko, and N. D. Pankratova, Design of Noncircular Cylindrical Shells [in Russian], Naukova Dumka, Kyiv (1977).

    Google Scholar 

  10. V. F. Kirichenko and V. A. Krys’ko, “Substantiation of the variational iteration method in the theory of plates,” Int. Appl. Mech., 17, No. 4, 366–370 (1981).

    MATH  MathSciNet  Google Scholar 

  11. Von L. Collatz, Eigenvalue Problems with Engineering Applications [in German], Akad. Verlagsges., Leipzig (1963).

    Google Scholar 

  12. N. N. Leont’ev, “A generalized Vlasov–Kantorovich variational method and its application to solve two-dimensional problems of the theory of plates,” in: Problems of Design of Spatial Structures [in Russian], Moscow (1980), pp. 65–78.

  13. V. G. Piskunov, Yu. M. Fedorenko, and A.E., Stepanova, “Variational-difference method in the problem of the vibrations of laminated plates,” Int. Appl. Mech., 28, No. 8, 506–511 (1992).

    Article  Google Scholar 

  14. V. G. Prokopov, E. I. Bespalova, and Yu. V. Sherenkovskii, “Kantorovich’s method of reduction to ordinary differential equations and a general method of solving multidimensional heat-transfer problems,” Inzh.-Fiz. Zh., 42, No. 6, 93–97 (1982).

    Google Scholar 

  15. L. A. Rozin, Variational Formulations of Problems for Elastic Systems [in Russian], Izd. Leningr. Univ., Leningrad (1978).

    Google Scholar 

  16. P. M. Varvak and A. F. Ryabov (eds.), A Handbook of Elasticity Theory [in Russian], Budivel’nyk, Kyiv (1971).

    Google Scholar 

  17. E. I. Bespalova, “Vibrations of polygonal plates with various boundary conditions,” Int. Appl. Mech., 43, No. 5, 526–533 (2007).

    Article  Google Scholar 

  18. E. I. Bespalova and A. B. Kitaygorodskii, “Advanced Kantorovich’s method for biharmonic problems,” J. Eng. Math., 46, 213–226 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  19. E. I. Bespalova and G. P. Urusova, “Determining the natural frequencies of highly inhomogeneous shells of revolution with transverse strain,” Int. Appl. Mech., 43, No. 9, 980–987 (2007).

    Article  Google Scholar 

  20. Ya. M. Grigorenko, “Nonconventional approaches to static problems for noncircular cylindrical shells in different formulations,” Int. Appl. Mech., 43, No. 1, 35–53 (2007).

    Article  MathSciNet  Google Scholar 

  21. Ya. M. Grigorenko and S. N. Yaremchenko, “Influence of orthotropy on displacements and stresses in nonthin cylindrical shells with elliptic cross section,” Int. Appl. Mech., 43, No. 6, 654–661 (2007).

    Article  Google Scholar 

  22. M. K. Huang and H. D. Conway, “Bending of a uniformly loaded rectangular plate with two adjacent edges clamped the others either simply supported or free,” J. Appl. Mech., Dec., 451–460 (1952).

  23. A. W. Leissa and F. W. Niedenfuhr, “Bending of a square plate with two adjacent edges free and the others clamped or simply supported,” AIAA J., 1, No. 1, 116–120 (1963).

    Article  Google Scholar 

  24. J. P. H. Weber, “On extension of the Kantorovich method,” Aeronautical J., 74, No. 710, 32–35 (1970).

    Google Scholar 

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Correspondence to E. I. Bespalova.

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Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 99–111, November 2008.

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Bespalova, E.I. Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method. Int Appl Mech 44, 1283–1293 (2008). https://doi.org/10.1007/s10778-009-0138-2

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  • DOI: https://doi.org/10.1007/s10778-009-0138-2

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