# Stress concentration in the region of a rec tangular hole on the side surface of a nonlinearly elastic orthotropic conical shell

 1Storozhuk, EA1Maksimyuk, VA1Chernyshenko, IS1S.P. Timoshenko Institute of Mechanics of the NAS of Ukraine, Kyiv Dopov. Nac. akad. nauk Ukr. 2019, 11:41-48 https://doi.org/10.15407/dopovidi2019.11.041 Section: Mathematics Language: Ukrainian Abstract:  The formulation and development of a numerical method for solving physically nonlinear static problems for a composite conical shell weakened by a rectangular hole is given. Geometric relationships are written in the vector form according to the theory of non-stray shells, in which Kirchhoff-Love’s hypotheses take place, and physical ones are based on the deformation theory of plasticity of anisotropic media. The inversion of physical relations with respect to stresses is performed numerically — by the Newton method. The system of resolving equations in displacements is obtained from the Lagrange variational equation using the method of additional stresses and the modified finite element method. A feature of the proposed version of the finite-element method is the implementation of the vector form of approximations of the unknown quantities and the implementation of the geometric part of the Kirchhoff—Love hypotheses at the nodes of the final element. Using the developed technique, the effect of nonlinear elasticity of the material and the location of a rectangular hole relative to the ends on the stress concentration in an orthotropic conical shell loaded with axial tensile forces is investigated. Keywords: composite conical shell, finiteelement method, nonlinear elastic state, rectangular hole, static load, stress concentration
References:

1. Guz, A. N., Lugovoy, P. Z. & Shulga, N. A. (1976). Conical shells weakened by holes. Kyiv: Naukova Dumka (in Russian).
2. Guz, A. N., Chernyshenko, I. S., Chekhov, Val. N., Chekhov, Vik. N. & Shnerenko, K. I. (1980). Calculation methods shells. (vol. 1) Theory of thin shells weakened by holes. Kyiv: Naukova Dumka (in Russian).
3. Mushchanov, V. F. & Demidov, A. I. (2008). Elastic-plastic stress state of circular conical shells of variable and constant thickness with aperture. Metal. constructions, 14, No. 3, pp. 125-142 (in Russian).
4. Troyak, E. N., Storozhuk, E. A. & Chernyshenko, I. S. (1988). Elastoplastic state of a conical shell with a cir cular hole on the lateral surface. Int. Appl. Mech., 24, No. 1, pp. 65-70. doi: https://doi.org/10.1007/BF00885078
5. Chernyshenko, I. S., Storozhuk, E.A. & Kharenko, S.B. (2007). Physically and geometrically nonlinear deformation of thin–walled conical shells with a curvilinear hole. Int. Appl. Mech., 43, No. 4, pp. 418-424. doi: https://doi.org/10.1007/s10778-007-0038-2
6. Chernyshenko, I. S., Storozhuk, E. A. & Kharenko, S. B. (2008). Physically and geometrically nonlinear deformation of conical shells with an elliptic hole. Int. Appl. Mech., 44, No. 2, pp. 174-181. doi: https://doi.org/10.1007/s10778-008-0032-3
7. Kharenko, S. B. (2010). Equilibrium of an inelastic conical shell with two circular holes. Problems of com putational mechanics and strength of structures. Dnipropetrovsk: Nauka i osvita, 14, pp. 340-346 (in Ukrainian).
8. Storozhuk, E. A., Chernyshenko I. S. & Kharenko, S. B. (2012). Elastoplastic deformation of conical shells with two circular holes. Int. Appl. Mech., 48, No. 3, pp. 343-348. doi: https://doi.org/10.1007/s10778-012-0525-y
9. Ermakovskaya, I. P. (1991). Effect of physical nonlinearity and orthotropy on the stress distribution around holes in a conical shell. Int. Appl. Mech., 27, No. 10, pp. 995-1000. doi: https://doi.org/10.1007/BF00887508
10. Lomakin, V. A. (1964). On the theory of anisotropic plasticity. Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, No. 4, pp. 49-53 (in Russian).
11. Guz, A. N., Kosmodamianskii, A. S. & Shevchenko, V. P. (Eds.). (1998). Mechanics of Composite Materials. (vol. 7) Stress Concentration. Kyiv: A.S.K. (in Russian).
12. Areias, P. M. A., Song, J.-H. & Belytschko, T. (2005). A finite-strain quadrilateral shell element based on discrete Kirchhoff–Love constraints. Int. J. Numer. Meth. Eng., 64, No. 9, pp. 1166-1206. doi: https://doi.org/10.1002/nme.1389