Stress-strain state of elliptic cross-section cylindrical shells with beveled cuts

TitleStress-strain state of elliptic cross-section cylindrical shells with beveled cuts
Publication TypeJournal Article
Year of Publication2020
AuthorsGrigorenko, Ya.M, Grigorenko, AYa., Kryukov, NN, Yaremchenko, SN
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published6/2020

The solution to the problem on the bending of cylindrical shells with an elliptic cross-section with beveled cuts is given. The relations of the theory of shells based on the straight line hypothesis is used. Systems of partial differential equations are obtained from the equilibrium equations of shell theory for determining displace ments and total angles of rotation of a non-circular cylindrical shell. The initial relations are written for an orthogonal coordinate system, the coordinate lines of which are the generatrix and directrix of the cylinder. A new non-orthogonal coordinate system is chosen and associated with the original one. The non-rectangular region of the old system for a non-circular shell with beveled cuts is transformed into a rectangular one in the new coordinates by the substitution of the coordinates. This allowed us to use the spline-collocation method to reduce two-dimensional boundary problem, which describes the stress-strain state of the shell, to a one-dimensional one. The one-dimensional boundary-value problem is solved by a stable numerical method of discrete orthogonalization. Using the described approach, problems of the stress-strain state of closed shells with an elliptic cross-section under the action of a uniformly distributed internal pressure with clamped beveled cuts are solved. To assess the reliability of the approach using the described methodology, problems for non-circular shells without beveled cuts, as well as for circular shells with beveled cuts, which are special cases, are solved. The displacements of the mid-surface of the shells are compared depending on the cut angles for circular and elliptic cross-section shells.

Keywordsbeveled cuts, noncircular cylindrical shells, refined shell theory, stress-strain state

1. Soldatos, K. P. (1999). Mechanics of cylindrical shells with non-circular cross section. A survey. Appl. Mech. Rev., 52, No. 8, pp. 237-274.
2. Grigorenko, Ya. M., Kryukov, N. N. & Krizhanovskaya, T. V. (1992). Improved calculation of the stress-strain state of orthotropic noncircular cylindrical shells. Int. Appl. Mech., 28, No. 1, pp. 54-60.
3. Grigorenko, Ya. M. & Yaremchenko, S. N. (2004). Stress analysis of orthotropic noncircular cylindrical shells of variable thickness in a refined formulation. Int. Appl. Mech., 40, No. 3, pp. 266-247.
4. Grigorenko, Ya. M., Kryukov, N. N. & Kholkina, N. S. (2009). Spline-approximation solution of stress-strain problems for beveled cylindrical shells. Int. Appl. Mech., 45, No. 12, pp. 1357-1364.
5. Grigorenko, Ya. M., Grigorenko, А. Ya., Kryukov, N. N. & Yaremchenko, S. N. (2020). Calculation of cylindrical shells with beveled cuts in refined formulation on the basis of spline-approximation. Int. Appl. Mech., 56, No. 3 (in Russian).
6. Kornishin, M. S., Paimushin, V. N. & Snigirev, V. F (1989). Computational geometry in shell mechanics problems. Мoscow: Nauka (in Russian).
7. Kryukov, N.N., Lyashko, O.V., Shutovskyi, O.M., Andreytsev, A.Yu. (2017). Using В-splines for solution of the problem on deformation of noncircular cylindrical shell with beveled cuts. 18 International Kravchuk scientific conference materials, Kyiv, 7-10 October, 2017. Kyiv: NTUU “KPI”, pp. 95-98 (in Ukrainan).
8. Grigorenko, Ya.M ., Vasilenko, А. Т. & Golub, G. P. (1987). Static of anisotropic shells with finite shear stiffness. Кyiv: Naukova dumka (in Russian).
9. Zavyalov, Yu. S., Kvasov, Yu. I. & Miroshnichenko, V. L. (1980). Spline-functions methods. Мoscow: Nauka (in Russian).
10. Grigorenko, Ya.M., Kryukov, N.N. (1995). Solution of problems of the theory of plates and shells with spline functions (survey). Int. Appl. Mech. 31, No. 6, pp. 413-434.
11. Grigorenko, Ya.M., Shevchenko, Yu.N., Kryukov, N.N. at al. (2002). Numerical methods. Кiev: “А.S.К”. (Mechanics of composites: in 12 vol.; Vol. 11) (in Russian).
12. Grigorenko, Ya.M., Grigorenko, A.Ya., Vlaikov, G.G. (2009). Problems of mechanics for anisotropic inhomogeneous shells on the basis of different models. Kyiv: Akademperiodyka.