Numerical solution of nonaxisymmetrical problems in the nonlinear theory of laminated shells of revolution

Abstract

LITERATURE CITED I. B.A. Boley and J. H. Weiner, Theory of Thermal Stresses, Wiley (1960). 2. M.A. Lavrent'ev, "Mechanics and scientific--technical progress," in: Two-Hundred and Fifty Years of the Academy of Sciences of the USSR [in Russian], Nauka, Moscow (1977), pp. 240-253. 3. A.I. Lur'e, Three-Dimensional Problems of the Theory of Elasticity [in Russian], Gos- tekhteorizdat, Moscow (1955). 4. Yu. N. Nemish, "Boundary-value problems of heat conduction and thermoelasticity for nearly spherical deformable bodies," Prikl. Mekh., 17, No. i0, 42-50 (1981). 5. Yu. N. Nemish, "One method of solving three-dimensional problems of the mechanics of deformable bodies bounded by arbitrary surfaces," Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. i, 48-52 (1976). 6. Yu. N. Nemish, "Boundary-value problems of heat conduction and thermoelasticity for nearly cylindrical deformable bodies," Prikl. Mekh., 18, No. i, 28-35 (1982). 7. M.A. Lavrent'ev, A. A. Deribas, E. I. Bichenkov, et al., in: Basic Investigations: Physical-Mathematical Sciences [in Russian], Nauka, Novosibirsk (1977), pp. 259-262. 8. A.E. Andreev (ed.), Bellows. Planning and Design [in Russian], Mashinostroenie, Mos- cow (1975). NUMERICAL SOLUTION OF NONAXISYMMETRICAL PROBLEMS IN THE NONLINEAR THEORY OF LAMINATED SHELLS OF REVOLUTION Ya. M. Grigorenko and A. M. Timonin UDC 539.3 A great many papers (see [i, 7-10] and others) have been devoted to calculating the geometrically nonlinear axisymmetrical deformations of circular plates and shells of revo- lution on the basis of the Kirchhoff--Love hypotheses. The nonlinear axisymmetrical deforma- tions of multilayer shells have been investigated with allowance for transverse shear [5]. Methods for calculating the linear nonaxisymmetrical deformations of shells have been described earlier [3, 6]. Valishvili [i] decomposes the nonlinear nonaxisymmetrical defor- mation problem approximately into two constituents: nonlinear axisymmetrical and linear nonaxisymmetrical. A similar approach has been used [8] in determining the critical loads and natural frequencies of shell structures. The method of lines has been used [4] to solve the problem of nonlinear nonaxisymmetrical deformation of a circular plate whose thickness varies in the circumferential direction. We now propose a method for calculating the geometrically nonlinear deformations of multilayered shells of revolution closed in the circumferential direction and characterized by a small shear stiffness under the action of nonaxlsymmetrical and local loads. Transverse shear is included in accordance with a Timoshenko-type model. The complete system Of equations for the stated problem has the following form [5]: equations for the tangential and flexural strains of the coordinate surface of the shell :