References
D. Alberg, E. Nielson, and L. Walsh, Theory of Splines and Its Application [Russian translation], Mir, Moscow (1972).
S. A. Ambartsumyan, Theory of Anisotropic Shells [in Russian], Fizmatgiz, Moscow (1961).
R. Bellman and R. Kalaba, Quasi-linearization and Nonlinear Boundary-Value Problems [Russian translation], Mir, Moscow (1968).
K. Z. Galimov, Principles of the Nonlinear Theory of Thin Shells [in Russian], Izd-vo Kazan. Un-ta, Kazan (1987).
S. K. Godunov, “Numerical solution of boundary-value problems for systems of linear ordinary differential equations,” Usp. Mat. Nauk,16, No. 3, 171–174 (1961).
A. L. Gol'denveizer, Theory of Thin Elastic Shells [in Russian], Nauka, Moscow (1976).
E. I. Grigolyuk and V. V. Kabanov, Stability of Shells [in Russian], Nauka, Moscow (1978).
Ya. M. Grigorenko, “Antisymmetric loading of a shell of variable thickness,” Prikl. Mekh.,6, No. 4, 385–392 (1960).
Ya. M. Grigorenko, “Cyclic deformation of shells of variable thickness,” Dopov. Akad. Nauk Ukr. RSR, No. 3, 317–322 (1965).
Ya. M. Grigorenko, “Application of numerical methods to the design of machine parts,” Din. Prochn. Mash., No. 5, 11–17(1967).
Ya. M. Grigorenko, Isotropic and Anisotropic Laminated Shells of Variable Thickness [in Russian], Nauk. Dumka, Kiev (1973).
Ya. M. Grigorenko, “Solution of problems of shell theory by the methods of numerical analysis,” Prikl. Mekh.,20, No. 10, 3–22 (1984).
Ya. M. Grigorenko, “Numerical methods of solving nonlinear boundary-value problems of the statics of flexible shells,” Transactions of an All-Union Conference in Tbilisi (1984), pp. 105–125.
Ya. M. Grigorenko, “Certain approaches to modeling and numerically solving problems on the deformation of flexible shells of revolution,” Prikl. Mekh.,29, No. 7, 3–9 (1993).
Ya. M. Grigorenko and M. N. Berenov, “Solution of two-dimensional problems on the bending of rectangular plates on the basis of spline-approximation,” Dopov. Akad. Nauk Ukr. RSR Ser. A, No. 8, 22–25 (1987).
Ya. M. Grigorenko and M. N. Berenov, “Numerical solution of problems of the statics of shallow shells on the basis of the spline-colocation method,” Prikl. Mekh.,24, No. 5, 32–36 (1988).
Ya. M. Grigorenko, E. I. Bespalova, A. B. Kitaigorodskii, and A. I. Shinkar', “Numerical solution of nonlinear boundary-value problems on the statics of flexible shells,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, No. 6, 44–48 (1980).
Ya. M. Grigorenko and A. T. Vasilenko, “Numerical solution on a computer of boundary-value problems on the stress state of shells of revolution,” Annotated Documents of the Fiftieth All-Union Conference on the Theory of Plates and Shells, Nauka, Moscow (1965), pp. 18–19.
Ya. M. Grigorenko and A. T. Vasilenko, “Nonsymmetric deformation of isotropic and anisotropic shells of revolution,” Prikl. Mekh.,4, No. 3, 19–28 (1968).
Ya. M. Grigorenko and A. T. Vasilenko, Theory of Shells of Variable Thickness, Vol. 4 of Methods of Shell Design [in Russian], Nauk. Dumka, Kiev (1981).
Ya. M. Grigorenko and A. T. Vasilenko, Problems of the Statics of Anisotropic Nonuniform Shells [in Russian], Nauka, Moscow (1992).
Ya. M. Grigorenko, A. T. Vasilenko, and G. P. Golub, Statics of Anisotropic Shells with a Finite Shear Stiffness [in Russian], Nauk. Dumka, Kiev (1987).
Ya. M. Grigorenko, A. T. Vasilenko, and N. N. Kryukov, “Numerical solution of nonlinear problems on the axisymmetric deformation of laminated anisotropic shells of revolution,” Mekh. Komp. Material., No. 6, 1023–1028 (1983).
Ya. M. Grigorenko, A. T. Vasilenko, and N. N. Kryukov, “Numerical study of the stress-strain state of nonuniform flexible shells of revolution made of composite materials,” Prikl. Mekh.,21, No. 6, 67–73 (1985).
Ya. M. Grigorenko, A. T. Vasilenko, and N. D. Pankratova, Design of Noncircular Cylindrical Shells [in Russian], Nauk. Dumka, Kiev (1977).
Ya. M. Grigorenko and V. I. Gulyaev, “Nonlinear problems of the theory of shells and methods of solving them (survey),” Prikl. Mekh.,27, No. 10, 3–23 (1991).
Ya. M. Grigorenko, M. Y. Kodner, N. P. Andreeva, et al., “Automation of calculations of the strength of elements of turbine stators and rotors,” Probl. Prochn., No. 12, 39–42 (1975).
Ya. M. Grigorenko and N. N. Kryukov, “Solution of nonlinear boundary-value problems of the statics of flexible laminated shells in the supercritical region,” Prikl. Mekh.,19, No. 3, 35–40 (1983).
Ya. M. Grigorenko and N. N. Kryukov, “Determination of nonaxisymmetric solutions in the problem of the deformation of flexible cylindrical shells under axisymmetric loads,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 42–45 (1984).
Ya. M. Grigorenko and N. N. Kryukov, Numerical Solution of Problems of the Statics of Flexible Laminated Shells with Variable Parameters [in Russian], Nauk. Dumka, Kiev (1988).
Ya. M. Grigorenko and N. N. Kryukov, “Solution of problems of the theory of plates and shells with the use of spline-functions (survey),” Prikl. Mekh.,31, No. 6, 3–26 (1995).
Ya. M. Grigorenko, N. N. Kryukov, and T. V. Krizhanovskaya, “Numerical analysis of the deformation of flexible composite shell structures under combination loading,” Mekh. Komp. Material., No. 6, 1101–1105 (1990).
Ya. M. Grigorenko, B. I. Mitlin, G. A. Raer, and G. K. Sudavtsova, “Study of the stress state of the stators of centrifugal compressors with allowance for the discreteness of the blade location,” Prikl. Mekh.,14, No. 1, 116–119 (1978).
Ya. M. Grigorenko and A. P. Mukoed, Solution of Linear and Nonlinear Problems of Shell Theory on a Computer [Ukrainian], Libid', Kiev (1992).
Ya. M. Grigorenko, N. M. Stepanov, A. T. Vasilenko, et al., “Stress state of ring-reinforced glass-plastic tubes under local loads,” Prikl. Mekh.,12, No. 10, 110–113 (1976).
Ya. M. Grigorenko and G. K. Sudavtsova, “Solution of problems of the statics of flexible shells under local loads,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, No. 2, 139–142 (1971).
Ya. M. Grigorenko and A. M. Timonin, “Numerical solution of nonaxisymmetric problems of the nonlinear theory of laminated shells of revolution,” Prikl. Mekh.,18, No. 5, 43–48 (1982).
Ya. M. Grigorenko and A. M. Timonin, “One approach to the numerical solution of two-dimensional problems of the theory of plates and shells with variable parameters,” Prikl. Mekh.,23, No. 6, 54–61 (1987).
Ya. M. Grigorenko and A. M. Timonin, “One approach to the numerical solution of boundary-value problems of the theory of shells of complex geometry in a nonorthogonal curvilinear coordinate system,” Dopov. Akad. Nauk Ukr. SSR, No. 4, 42–45 (1991).
Ya. M. Grigorenko and A. M. Timonin, “Numerical solution of boundary-value problems of the mechanics of shells of complex geometry with the use of coordinate systems of general form,” Prikl. Mekh.,28, No. 7, 50–56 (1992).
Ya. M. Grigorenko and A. M. Timonin, “Numerical solution of nonlinear boundary-value problems of the theory of flexible shells of complex geometry,” Dopov. Akad. Nauk Ukr. SSR, No. 12, 3–10 (1992).
Ya. M. Grigorenko and A. M. Timonin, “Thermoelastic deformation of flexible shell systems of complex geometry,” Dopov. Akad. Nauk UKr.SSR, No. 9, 56–60 (1993).
Ya. M. Grigorenko and A. M. Timonin, “One approach to the numerical solution of boundary-value problems of shell theory in general coordinates,” Prikl. Mekh.,30, No. 4, 14–20 (1994).
Yu. S. Zav'yalov, Yu. I. Kvasov, and V. M. Miroshnichenko, Spline-Function Methods [in Russian], Nauka, Moscow (1980).
V. I. Korolev, Laminated Anisotropic Plates and Shells Made of Reinforced Plastics [in Russian], Mashinostroenie, Moscow (1965).
Kh. M. Mushtari and K. Z. Galimov, Nonlinear Theory of Elastic Shells [in Russian], Tatknigoizdat, Kazan (1957).
V. V. Novozhilov, Principles of the Nonlinear Theory of Elasticity [in Russian], Gostekhizdat, Moscow-Leningrad (1948).
V. V. Novozhilov, Theory of Thin Shells [in Russian], Sudostroenie, Leningrad (1962).
K. Tsiao, “Relations between the strains and the displacements in the theory of large deflections of shells,” AIAA J., No. 11, 236–238 (1964).
D. Keller and S. Entman (eds.), Branching Theory and Nonlinear Eigenvalue Problems [Russian translation], Mir, Moscow (1974), pp. 19–34.
S. P. Timoshenko, Course in the Theory of Elasticity [Russian translation], Nauk. Dumka, Kiev (1972).
S. P. Timoshenko and S. Woinowsky-Krieger, Plates and Shells [Russian translation], Fizmatgiz, Moscow (1963).
V. E. Shamanskii, Methods of Numerically Solving Boundary-Value Problems on a Computer [in Russian], Nauk. Dumka. Kiev (1966).
L. A. Shapovalov, “Simple variant of equations of the geometrically nonlinear theory of thin shells,” Izv. Akad. Nauk SSSR Ser. Mekh. Tverd. Tela, No. 1, 56–62 (1968).
B. Budiansky and P. P. Radkowski, “Numerical analysis of unsymmetrical bending of shells of revolution,” AIAA J.,1, No. 8, 1833–1842 (1963).
G. A. Cohen, “Computer analysis of asymmetrical deformation of orthotropic shells of revolution,” AIAA J.,2, No. 5, 932–934 (1964).
Ya. M. Grigorenko, “On some approaches to numerical solution of linear and nonlinear boundary-value problems of the theory of layered anisotropic shells,” Computational Mechanics '86: Proceedings of International Conference on Computational Mechanics, May 25–29, 1986, Tokyo, Springer-Verlag, pp. 197–202.
Ya. M. Grigorenko, “Some approaches to modeling and numerical solution of the deformation problems of flexible shells of revolution,” European Mechanics Colloquium 292 “Modeling of Shells with Nonlinear Behavior,” Sept. 2–4, 1992. Techn. University of Munich, Germany, pp. 121–122.
Ya. M. Grigorenko, “Influence of anisotropy and non-homogeneity on the deformation of flexible shells,” Euromech Collogium 317 “Buckling-Strength of Imperfection-Sensitive Shells,” March 21–23, 1994, The University of Liverpool, England, p. 44.
Ya. M. Grigorenko, N. N. Kryukov, and J. J. Ivanova, “The analysis of geometrically nonlinear deformation of tne shallow shell using the spline-functions,” Dopov. Akad. Nauk Ukr. RSR, No. 11, 40–43 (1992).
A. Kalnins, “Analysis of shells of revolution subjected to symmetrical and nonsymmetrical loads,” Trans. ASME,E31, No. 3, 467–476 (1964).
A. K. Noor and W. S. Burton, “Assessment of computational models for multilayered composite shells,” App. Mech. Rev.,43, No. 4, 67–97 (1990).
A. Rittweger, Statik, Stabilitat under Eigenschwingungen anisotroper Rotationsschalen beliebigen Meridians transmit der Ubertragungsmatrizen — Methode. R-W Techn. Hochschule, Aachen (1992).
W. K. Sepetoski et al., “A digital computer program for the general axially symmetric thin shells problem,” Trans. ASME,E29, No. 4, 655–661 (1962).
W. Wunderlich, “Zur Berechnung variation Rotationsschalen transmit Ubertragungs-matrizen,” Ing. Arch.,36, 262–279 (1967).
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Translated from Prikladnaya Mekhanika, Vol. 32, No. 6, pp. 3–39, June, 1996.
This article was written using a survey read at a seminar on mechanics at Berlin Technical University (Berlin, Germany, February 6, 1995).
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Grigorenko, Y.M. Approaches to the numerical solution of linear and nonlinear problems in shell theory in classical and refined formulations. Int Appl Mech 32, 409–442 (1996). https://doi.org/10.1007/BF02088409
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DOI: https://doi.org/10.1007/BF02088409