The axisymmetric problems of geometrically nonlinear deformation and stability of a sandwich cylindrical shell with contour reinforcing beams

I.B. Badriev ^a∗ , M.V. Makarov ^a∗∗ , V.N. Paimushin ^a,b∗∗∗ , S.A. Kholmogorov ^{b∗∗∗∗}

^a Kazan Federal University, Kazan, 420008 Russia
^b Tupolev Kazan National Research Technical University, Kazan, 420111 Russia

E-mail: ^∗ildar.badriev1@mail.ru,  ^∗∗makarovmaksim@mail.ru,  ^∗∗∗vpajmushin@mail.ru, ^∗∗∗∗hkazan@yandex.ru

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Received April 11, 2017

Abstract

A numerical investigation of the problem of geometrically nonlinear axisymmetric deformation of a sandwich cylindrical shell with a transversally soft core reinforced in the end sections by elastic rods has been carried out. To describe the process of deformation, we have used the previously derived equations of the refined geometrically nonlinear theory that allow to both study the subcritical behavior of the shell and to reveal all possible buckling forms of the carrier layers. These equations are based on the introduction, as unknown variables, of the contact force of the interaction of the outer layers with the core, as well as of the outer layers and the filler with the reinforcing bodies at all points on the surfaces of their conjugation. Numerical methods for solving the formulated problems have been developed. They are based on the preliminary reduction of the original problems to a system of integro-algebraic equations, for the solving of which the finite sum method is used. A method has been pro- posed for investigating the subcritical and supercritical geometrically nonlinear behavior of a shell with its end compression through contour reinforcing rods, according to which unstable equilibrium positions are determined by the method of continuation of the solution with respect to the parameter when the external forces are selected as a parameter. A method has been proposed for finding the critical load (the bifurcation point) at which the shell buckling occurs. This method is based on the linearization of the initial geometrically nonlinear problem in the neighborhood of its nonlinear solution, followed by the formulation of the eigenvalue problem with a nonlinear presence of the parameter. The results of the numerical experiments have been discussed. The results of the experiments have been analyzed.

Keywords: sandwich cylindrical shell, transversally soft filler, contour reinforcing beam, geometric nonlinearity, contact stresses, axial compression, axisymmetric deformation, finite sum method, subcritical and supercritical behavior, bifurcation point, linearized problem, buckling forms

Acknowledgments. The study was supported by the Russian Science Foundation (project no. 16-11-10299) and in part by the Russian Foundation for Basic Research (project no. 16-08-00316) in the framework of the state task of the Ministry of Education of the Russian Federation (task no. 9.5762.2017/VU, project no. 9.1395.2017/PCh).

Figure Captions

Fig. 1. a) The scheme of connection of a sandwich shell with a frame; b) the scheme of the reinforcing beam.

Fig. 2. Sandwich plate shell with contour reinforcing beams.

Fig. 3. Deflections of the middle surfaces of the carrier layers w^(k), cm.

Fig. 4. Axial displacements of the middle surfaces of the carrier layers u^(k)₁, cm.

Fig. 5. Axial membrane normal stresses in the carrier layers σ^(k) ₁₁ = T^(k) ₁₁ / (2h^(k)), MPa.

Fig. 6. Bending moments in the carrier layers M^(k) ₁₁ , N.

Fig. 7. Transverse tangential stresses in the core q₁, MPa.

Fig. 8. Circumferential membrane stresses in the carrier layers σ^(k) ₂₂ = T^(k)₂₂ / (2h^(k)), MPa.

Fig. 9. Generalized shear forces of the carrier layers N^(k)₁, kN/m.

Fig. 10. Shear forces of the carrier layers Q^(k)₁, kN/m.

Fig. 11. The deformed state of the middle surface of the upper carrier layer at E = E₂^(k) = 20 * 10³ MPa.

Fig. 12. The deformed state of the middle surface of the upper carrier layer at E = E₂^(k) = 80 * 10³ MPa.

Fig. 13. The dependence of the running load T_ς⁽⁺⁾ on the maximum modulus of the deflection w⁽²⁾ of the upper layer.

Fig. 14. Deflections of the middle surfaces of the carrier layers w^(k), cm.

Fig. 15. Transverse tangential stresses in the core q₁, MPa.

Fig. 16. Axial membrane normal stresses in the carrier layers σ^(k)₁₁ = T^(k)₁₁ / (2h^(k)), MPa.

Fig. 17. Circumferential membrane stresses in the carrier layers σ^(k) ₂₂ = T^(k)₂₂/ (2h^(k)), MPa..

Fig. 18. The dependence of the running load T_ς⁽⁺⁾ on the maximum modulus of the deflection w⁽²⁾ of the upper layer.

Fig. 19. The deformed state of the middle surface of the upper carrier layer at the bifurcation point.

Fig. 20. Deflections of the middle surface of the upper carrier layer w⁽²⁾, cm.

Fig. 21. Transverse tangential stresses in the core q₁ , MPa.

Fig. 22. Membrane axial stresses in the upper carrier layer σ⁽²⁾ ₁₁ = T⁽²⁾₁₁/ (2h⁽²⁾), MPa.

Fig. 23. Circumferential membrane stresses in the upper carrier layer σ⁽²⁾ ₂₂ = T⁽²⁾₂₂/ (2h⁽²⁾), MPa.

Fig. 24. Dependence of the eigenvalue λ on the load valueT_ς⁽⁺⁾.

Fig. 25. Increments in the de?ections of the points of the middle surfaces of the outer layers at the transition to the disturbed state.

Fig. 26. Increments of the tangential stresses in the core at the transition to the disturbed state.

Fig. 27. Increments in axial membrane stresses in the carrier layers at the transition to the disturbed state.

Fig. 28. Increments of circumferential membrane stresses in the carrier layers at the transition to the disturbed state.

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For citation: Badriev I.B., Makarov M.V., Paimushin V.N., Kholmogorov S.A. The axisymmetric problems of geometrically nonlinear deformation and stability of a sandwich cylindrical shell with contour reinforcing beams. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 395–428. (In Russian)

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