The Nonlinear Deformation and Stability Study of the Non-circular Composite Cylindrical Shell at the External Pressure

Аuthors

Zheleznov L. P.

Siberian Aeronautical Research Institute named after S.A. Chaplygin, 21, Polzunov St., Novosibirsk, 630051, Russia

e-mail: zgeleznov@sibnia.ru

Abstract

Polymer composite materials (PCM) gained wide application in modern aircraft structures. The said materials employing reduces the weight of the structure while retaining its strength and stiffness characteristics. Despite the large number of published works on  such structures strength, the unsolved issues on strength and stability at their nonlinear deformation still exist. The latter is of especial necessity for the aircraft fuselage structures, for which buckling loss of the composite skin is inadmissible. The problem of the non-circular shells from the PCM stability is being solved in this article with regard to the momentness and nonlinearity of their subcritical stress-strain state. The finite elements of the composite cylindrical shells of natural curvature developed the by author earlier based on the Timoshenko hypothesis are being employed. Their rigid displacement are being accounted for in the approximation , which significantly improves convergence of the nonlinear problem solution. The nonlinear buckling problem was solved geometrically by the finite element and Newton-Kantorovich methods. Solution of a system of nonlinear algebraic equations with respect to nodal displacements of the shell is being found with the method of successive approximations and the step-by-step method of loading in the following way. A small value of the load parameter is set. The linear problem solution is being assumed herewith as a zero approximation. An iterative process, ensuring convergence of the solution with a given accuracy not exceeding 5% is being performed. Further, the loading increases, and the iterative process, in which the solution from the previous load step is assumed as the initial approximation is being performed again. Solution of the system of linear algebraic equations is being found by the Krauth method using the LTDL decomposition of the matrix into a diagonal D and two triangular matrices L at each iteration. The critical load is found either as a ultimate on the divergence of the iterative process, or as a bifurcation one by the energy stability criterion, according to which the equilibrium state is stable if the second variation of potential energy is greater than zero and unstable if it is less than zero. Critical loads are being determined in the process of solving a nonlinear problem. The stability of an oval, cantilevered cylindrical shell made of the PCM under external pressure is being studied. A shell with a length of L = 2000 mm, a thickness of h = 3.456 mm, and a radius of R0 = 2000 mm, made of 18-layer Torayca T700 PCM is being regarded. Five different layups were considered, including shells made of D16AT aluminum alloy for comparison. Assessment of the effect of monolayers stacking over the shell thickness, deformation nonlinearity and out-of-roundness parameter on the critical loadings causing the shell stability losses and weight efficiency of composite shells was performed. It was revealed that:
1. The critical values of the external pressure depend on both the stacking and the out-of-roundness parameter of the shell. The out-of-roundness of the shells reduces the critical values of the external load. The most optimal stacking options are considered to be those with pre-eminent installation angles of 90°. The nonlinearity reduces the critical values of loading parameter for all options of the monolayer stacking for all considered shells (up to 43%). 
2. The weight efficiency of composite shells depends on the stacking angle and slightly (within the limit of 5%) on the out-of-roundness parameter of the shell in the case of both linear and nonlinear initial stress-strain states. With the angle φ increase, the weight efficiency of composite shells increases. Nevertheless, for the angles φ < 40°, metal shells are more effective than the composite ones.

Keywords:

noncircular cylindrical composite shells, polymer composite materials, stability of an oval hinged cylindrical shell, nonlinear deformation of composite shells

References

1. Kabanov V.V. Ustoichivost' neodnorodnykh tsilindricheskikh obolochek (Stability of inhomogeneous cylindrical shells), Moscow, Mashinostroenie, 1982, 253 p.
2. Mushtari Kh.M. Trudy Kazanskogo aviatsionnogo instituta, 1935, no. 3-4, pp. 19–31.
3. Hutchinson J.W. Buckling and initial postbuckling behaviour of oval cylindrical shells under axial compression. Journal of Applied Mechanics, 1968, no. 35, pp. 66-72.
4. Inozemtsev B.Kh. Nekotorye zadachi ustoichivosti tsilindricheskoi obolochki oval'nogo secheniya (Some problems of stability of the cylindrical shell of oval section), Abstract of doctor’s thesis, Moscow, MISI im. V.V. Kuibysheva, 1967, 9 p.
5. Karmishin A.V., Lyaskovets V.A., Myachenkov V.I., Frolov A.N. Statika i dinamika obolochechnykh konstruktsii (Statics and dynamics of shell structures), Moscow, Mashinostroenie, 1975, 376 p.
6. Feinstein G., Chen Y.N., Kempner J. Buckling of clamped oval cylindrical shells under axial loads. AIAA Journal, 1971, vol. 9, no. 9, pp. 1733-1738. DOI: 10.2514/3.6423
7. Korobeinikov S.N. Dinamika sploshnoi sredy. Sbornik nauchnykh trudov. Novosibirsk, IGiL SO AN SSSR, 1987, no. 80, pp. 82–89.
8. Boiko D.V., Zheleznov L.P., Kabanov V.V. Prikladnaya matematika i mekhanika, 2003, vol. 67, no. 6, pp. 933- 939.
9. Tennyson R.C., Booton M., Caswell R.D. Buckling of imperfect elliptical cylindrical shells under axial compression. AIAA Journal, 1971, vol. 9, no. 2, pp. 250–255.
10. Mossakovskii V.I., Konokh V.I., Krasovskii V.P. Prikladnaya mekhanika, 1974, vol. 10, no. 3, pp. 3–8.
11. Vasiliev V.V., Morozov E.V. Advanced Mechanics of Composite Materials and Structures. 4th ed. Amsterdam, Elsevier, 2018, 882 p.
12. Levchenkov MD., Dubovikov E.A., Mirgorodskii Y.S., Fomin D.Y., Shanygin A.N. Weight efficiency of the design of a passenger aircraft barrel with a nonregular lattice structural layout. Aerospace MAI Journal, 2023, vol. 30, no 4, pp. 98–108. URL: https://vestnikmai.ru/publications.php?ID=177610
13. Akulin P.V., Gavrilov G.A. Multilayer composite material structure impact on the aircraft structure stiffness characteristics degradation. Aerospace MAI Journal, 2023, vol. 30, no. 3, pp. 78-84.
14. Alfutov N.A., Zinov'ev P.A., Popov B.G. Raschet mnogosloinykh plastin i obolochek iz kompozitsionnykh materialov (Calculation of multilayer plates and shells made of composite materials). Moscow, Mashinostroenie, 1984, 446 p.
15. Badrukhin Y.I., Terekhova E.S. Rational design of thin-walled load-bearing laminated composite panels under combined loading. Aerospace MAI Journal, 2023, vol. 30, no. 4, pp. 130–139. URL: https://vestnikmai.ru/publications.php?ID=177614
16. Vanin G.A., Semenyuk N.P., Emel'yanov R.F. Ustoichivost' obolochek iz armirovannykh materialov (Stability of shells made of reinforced materials). Kiev, Naukova Dumka, 1978, 211 p.
17. Zheleznov L.P. Izvestiya vysshikh uchebnykh zavedenii. Aviatsionnaya tekhnika, 2022, no. 4, pp. 44-50.
18. Kantorovich L.V., Akilov G.P. Funktsional'nyi analiz v normirovannykh prostranstvakh (Functional analysis in normalized spaces), Moscow, Fizmatgiz, 1959, 684 p.
19. Zheleznov L.P., Kabanov V.V., Boiko D.V. Polet. Obshcherossiiskii nauchno-tekhnicheskii zhurnal, 2013, no. 6, pp. 3-10
20. Zheleznov L.P., Ser'eznov A.N. Izvestiya vysshikh uchebnykh zavedenii. Aviatsionnaya tekhnika, 2021, no. 3, pp. 22-30.