Abstract
A nonlinear boundary-value problem for shells is formulated. The problem statement permits us to analyze the behavior of the postbuckling solutions in a finite-dimensional space with the dimension independent of the discretization technique. A branching pattern is established for the solutions of the problem in the case of a uniform external pressure. Four characteristic types of solution are recognized in the case of an arbitrary external pressure. The associated buckling loads are determined. It is found out that five significant parameters are sufficient for a qualitative analysis of a cylindrical shell
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L. V. Andreev and N. I. Obodan, “Experimental investigation of the stability of cylindrical shells under nonuniform pressure,” in: Proc. 6th All-Union Conf. on the Theory of Shells and Plates [in Russian], Nauka, Moscow (1966).
M. M. Vainberg and V. A. Trenogin, Theory of Branching of Nonlinear Equations [in Russian], Nauka, Moscow (1969).
A. S. Vol'mir, Stability of Elastic Systems [in Russian], Fizmatgiz, Moscow (1963).
Ya. M. Grigorenko and V. I. Gulyaev, “Nonlinear problems of shell theory and their solution methods (review),” Int. Appl. Mech., 27, No. 10, 929–947 (1991).
Ya. M. Grigorenko and Yu. B. Kas'yan, “Analysis of the effect of changed curvature and load distribution on the deformation of a flexible long cylindrical shell,” Int. Appl. Mech., 38, No. 3, 324–328 (2002).
Ya. M. Grigorenko and Yu. B. Kas'yan, “Analysis of the influence of a geometrical parameter on the deformation of a rigidly fixed, flexible, noncircular cylindrical shell,” Int. Appl. Mech., 39, No. 1, 64–69 (2003).
A. Yu. Evkin and V. L. Krasovskii, “Post-critical deformation and estimation of the stability of real cylindrical shells under external pressure,” Int. Appl. Mech., 27, No. 3, 290–296 (1991).
A. N. Kudinov and V. I. Muravitskii, “Experimental investigation of the stability of reinforced cylindrical shells under nonuniform external pressure and heat,” in: Proc. 7th All-Union Conf. on the Theory of Shells and Plates [in Russian], Nauka, Moscow (1970), pp. 318–322.
N. I. Obodan and A. G. Lebedev, “Secondary branching of solutions in the nonlinear theory of cylindrical shells,” Dokl. AN USSR, Ser. 2, No. 12, 38–41 (1980).
F. Fujii and H. Noguchi, “Symmetry breaking bifurcation and post buckling strength of a compressed circular cylinder,” in: Solid Mechanics and Fluid Mechanics: Computational Mechanics for the Next Millennium, Vol. 1, Pergamon, Amsterdam (1999), pp. 563–568.
F. Fujii, H. Noguchi, and E. Ramm, “Static path jumping to attain postbuckling equilibria of a compressed circular cylinder,” Comp. Mech., No. 26, 259–266 (2000).
E. A. Gotsulyak and A. I. Siyanov, “Stability and nonlinear deformation of cylindrical grids,” Int. Appl. Mech., 40, No. 4, 426–431 (2004).
Ya. M. Grigorenko and Yu. B. Kas'yan, “Deformation of a flexible noncircular long cylindrical shell under a nonuniform load,” Int. Appl. Mech., 37, No. 3, 346–351 (2001).
G. J. Lord, A. R. Champneys, and G. W. Hunt, “Computation of homoclinic orbits in partial differential equations: An application to cylindrical shell buckling,” SIAM J. Sci. Comp., 21, No. 2, 591–619 (1999).
L. V. Mol'chchenko, “A method for solving two-dimension nonlinear boundary-value problems of magnetoelasticity for thin shells,” Int. Appl. Mech., 41, No. 5, 490–495 (2005).
E. Riks, C. C. Rankin, and F. A. Brogan, “On the solution of mode jumping phenomena in thin-walled shell structures,” Comp. Meth. Appl. Mech. Eng., No. 36, 59–92 (1996).
J. L. Sanders, “Nonlinear theories for thin shells,” Quarterly of Applied Mathematics, 21, No. 1, 21–36 (1963).
E. A. Storozhuk and I. S. Chernyshenko, “Elastoplastic deformation of flexible cylindrical shells with two circular holes under axial tension,” Int. Appl. Mech., 41, No. 5, 506–511 (2005).
E. A. Storozhuk and I. S. Chernyshenko, “Physically and geometrically nonlinear deformation of spherical shells with an elliptic hole,” Int. Appl. Mech., 41, No. 6, 666–674 (2005).
J. C. Wohlever and T. J. Healey, “A group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell,” Comp. Meth. Appl. Mech. Eng., No. 122, 315–349 (1995).
W. Zhang, T. Hisada, and H. Noguchi, “Postbuckling analysis of shell and membrane structures by dynamic relaxation method,” Comp. Mech., No. 26, 267–272 (2000).
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Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 103–112, January 2006.
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Obodan, N.I., Gromov, V.A. Numerical analysis of the branching of solutions to nonlinear equations for cylindrical shells. Int Appl Mech 42, 90–97 (2006). https://doi.org/10.1007/s10778-006-0062-7
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DOI: https://doi.org/10.1007/s10778-006-0062-7