Cylindrical shell with a circular hole under various loads: Comparison of analytical and numerical solutions

S. V. Kashtanova ; A. V. Rzhonsnitskiy

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 2, pp. 195-206.

Résumé

In this paper, the authors present the results of calculations of the stress field of a cylindrical shell weakened by a circular hole and under the influence of various loads: uniaxial tension, internal pressure and torsion. Six simplified equations of the theory of cylindrical shells with a high variability index (coinciding with the equations of the theory of shallow shells) are reduced to an equation of mathematical physics with respect to the stress potential, which is solved by the Fourier method. The main obstacle to obtaining an answer is the need to search for coefficients in the decomposition of the solution into the sum of the basis functions for which this solution satisfies the boundary conditions. Also, this equation depends on the parameter $\beta$, which is responsible for the relationship between the geometric characteristics of the shell and the hole. From a mechanical point of view, for small and medium holes, this parameter has limitations of $\beta\leq 4$, because for large values, the hole is considered large, and the general equations of the theory of cylindrical shells are used to describe the stress-strain state. At the same time, a detailed study of classical works has led to the understanding that none of the previously proposed methods for finding coefficients can be considered definitively successful, and the results obtained by these methods vary. Among the variety of works by Soviet and Western scientists of the 1960-70s years of the twentieth century, the numerical results of engineer Van Dyke, which he obtained by collocation, stand out. Unlike his contemporaries, who lay out the solution in a row for a small parameter $\beta$ and therefore get results only close to the flat case, Van Dyke first published results for the entire working range of the parameter $\beta$ in the framework of considering small and medium holes. The authors proposed a new approach based on the decomposition of basic functions into a Fourier series. For the first time, it was possible to compose an infinite system of linearly independent equations for finding unknown coefficients. It is essential that the proposed method, unlike the well-known small parameter method, has no mathematical limitations and can be used not only for the values of the parameter $\beta$ close to zero, but for any values. Restrictions up to $\beta=4$ are imposed by the mechanical model. In this paper, systems for finding unknown coefficients for basic functions for three types of loads are compiled, and the results obtained by the authors are compared with the results obtained by the numerical method. At the same time, if in most sources only the results of calculating the circumferential stresses at the boundary of the hole are given, in the proposed work the stress field for the entire cylindrical shell is found, arising due to the presence of the hole, depending on the polar coordinates $(r,\theta)$.

Bibliographie

[1] Kashtanova S. V., Rzhonsnitskiy A. V., “An analytical approach to obtaining the stress field of a cylindrical shell with a circular hole under tension”, PNRPU Mechanics Bulletin, 2021, no. 2, 64–75 (in Russian) | DOI

[2] Kashtanova S. V., Rzhonsnitskiy A. V., “Investigation of systems of the stress field problem of a cylindrical shell with a circular cutout under various boundary conditions”, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 44 (2022), 259 | DOI

[3] Lurie A. I., “Concentration of stresses in the vicinity of an aperture in the surface of a circular cylinder”, Prikladnaya Matematica i Mekhanica, 10 (1946), 397–406

[4] New York University, Inst. of Math. Sci, 1960

[5] Guz A. N., Cylindrical Shells with Cutouts, Naukova dumka, Kiev, 1974, 271 pp. (in Russian)

[6] Lekkerkerker J. G., On the Stress Distribution in Cylindrical Shells Weakened by a Circular Hole, Uitgeverij Waltman, Delft, 1965, 99 pp.

[7] Naghdi A. K., Eringen A. C., “Stress distribution in a circular cylindrical shell with a circular cutout”, Archive of Applied Mechanics, 34:3 (1965), 161–172 | DOI

[8] Eringen A. C., Naghdi A. K., Thiel C. C., State of Stress in a Circular Cylindrical Shell With a Circular Hole, Welding Research Council Bulletin, 102, Welding Research Council, 1965, 21 pp.

[9] Foo S. S. B., “On the limit analysis of cylindrical shells with a single cutout”, International Journal of Pressure Vessels and Piping, 49:1 (1992), 1–16 | DOI | MR

[10] Kabanov V. V., Zheleznov L. P., “Application of the Finite-Elements method to the strength analysis of circular cylindrical shells with cutouts”, TsAGI Notes, 16:3 (1985), 92–99 (in Russian) | MR

[11] Shariati M., Akbarpour A., “Ultimate strength analysis of combined loaded stainless steel circular tubes with hole”, Journal of Basic and Applied Scientific Research, 2:8 (2012), 8457–8465

[12] Lee S. E., Sahin S., Rigo P., “Ultimate strength of cylindrical shells with cutouts”, Ships and Offshore Structures, 12:S1 (2017), 151–173 | DOI

[13] Kolodiazhnyi A., Mednikova M., “The Influence of the deformation nonlinearity on stress concentration in cylindrical shells with holes under torsion”, Materials Science Forum, 968 (2019), 548–559 | DOI

[14] Chawla K., Ray-Chaudhuri S., “Stress and strain concentration factors in orthotropic composites with hole under uniaxial tension”, Curved and Layered Structures, 5:1 (2018), 213–231 | DOI

[15] Silpa V. J. K., Raghu Vamsi B. V. S., Gowtham Kumar K., “Structural analysis of thin isotropic and orthotropic plates using finite element analysis”, Indian Journal of Medical Ethics, 4:6 (2017), 17–27 | DOI

[16] Kashtanova S. V., Rzhonsnitskiy A. V., Gruzdkov A. A., “On the issue of analytical derivation of stress state in a cylindrical shell with a circular hole under axial tension”, Materials Physics and Mechanics, 47 (2021), 186–195 | DOI | MR

[17] Van Dyke P., “Stresses about a circular hole in a cylindrical shell”, AIAA Journal, 3:9 (1965), 1733–1742 | DOI | MR