|Journal||Journal of Mechanical Engineering|
|Publisher||A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
|ISSN||0131-2928 (Print), 2411-0779 (Online)|
|Issue||Vol. 22, no. 2, 2019 (June)|
|Cited by||J. of Mech. Eng., 2019, vol. 22, no. 2, pp. 44-52|
This paper proposes an analytical-numerical approach to solving the spatial prob-lem of the theory of elasticity for the layer with a circular cylindrical tube. A cylin-drical empty thick-walled tube is located inside the layer parallel to its surfaces and is rigidly fixed to it. It is necessary to investigate the stress-strain state of the elastic bodies of both the layer and tube. Stresses are given on the inner surface of the tube, and displacements, on the boundaries of the layer. The solution to the spatial prob-lem of the theory of elasticity is obtained by the generalized Fourier method with respect to the system of Lamé’s equations in the cylindrical coordinates associated with the tube and the Cartesian coordinates associated with the boundaries of the layer. Infinite systems of linear algebraic equations obtained as a result of satisfying the boundary and conjugation conditions are solved by the truncation method. As a result, displacements and stresses are obtained at various points of the elastic layer and elastic tube. Due to the selected truncation parameter for the given geometrical characteristics, the satisfaction of boundary conditions has been brought to 10-3. An analysis of the stress-strain state for the elastic body at different thicknesses of the tube, as well as at different distances from the tube to the boundaries of the layer is conducted. Graphs of normal and tangential stresses at the boundary of the tube and layer, as well as normal stresses on the inner surface of the tube are presented. These stress graphs indicate that as the tube approaches the upper boundary of the layer, the stresses in the elastic bodies of both the layer and tube increase, and with decreasing tube thickness, the stresses in the elastic body of the layer decrease, growing in the elastic body of the tube. The proposed method can be used to calcu-late structures and parts, whose design schemes coincide with the formulation of the problem of this paper. The analysis of the stress state can be used to select the geo-metrical parameters of the designed structure, and the stress graph at the boundary of the tube and layer can be used to analyze the strength of the joint.
Keywords: thick-walled tube in a layer, Lamé’s equations, generalized Fourier method.
Full text: Download in PDF
- Vaysfeld, N., Popov, G., & Reut, V. (2015). The axisymmetric contact interaction of an infinite elastic plate with an absolutely rigid inclusion. Acta Mechanica, vol. 226, iss. 3, pp. 797–810. https://doi.org/10.1007/s00707-014-1229-7.
- Popov, G. Ya. & Vaysfeld, N. D. (2014). Solving an axisymmetric problem of elasticity for an infinite plate with a cylindrical inclusion with allowance for its specific weight. International Applied Mechanics, vol. 50, iss. 6, pp. 627–636. https://doi.org/10.1007/s10778-014-0661-7
- Guz, A. N., Kubenko, V. D., & Cherevko, M. A. (1978). Difraktsiya uprugikh voln [Diffraction of elastic waves]. Kiyev: Naukova Dumka, 307 p. (in Russian).
- Grinchenko, V. T. & Meleshko, V. V. (1981). Garmonicheskiye kolebaniya i volny v uprugikh telakh [Harmonic oscillations and waves in elastic bodies]. Kiyev: Naukova Dumka, 284 p. (in Russian).
- Grinchenko, V. T. & Ulitko, A. F. (1968). An exact solution of the problem of stress distribution close to a circular hole in an elastic layer. Soviet Applied Mechanics, vol. 4, iss. 10, pp. 31 – 37. https://doi.org/10.1007/BF00886618
- Grinchenko, V. T. & Ulitko, A. F. (1985). Prostranstvennyye zadachi teorii uprugosti i plastichnosti. Ravnovesiye uprugikh tel kanonicheskoy formy [Spatial problems of the theory of elasticity and plasticity. Equilibrium of elastic bodies of canonical form]. Kiyev: Naukova Dumka, 280 p. (in Russian).
- Volchkov, V. V., Vukolov, D. S., & Storozhev, V. I. (2016). Difraktsiya voln sdviga na vnutrennikh tunnel’nykh tsilindricheskikh neodnorodnostyakh v vide polosti i vklyucheniya v uprugom sloye so svobodnymi granyami [Diffraction of shear waves on internal tunnel cylindrical inhomogeneities in the form of a cavity and inclusion in the elastic layer with free face]. Mekhanika tverdogo tela – Mechanics of Rigid Bodies, vol. 46, pp. 119 – 133 (in Russian).
- Nikolayev, A. G. & Protsenko, V. S. (2011). Obobshchennyy metod Furye v prostranstvennykh zadachakh teorii uprugosti [The generalized Fourier method in spatial problems of the theory of elasticity]. Kharkov: Nats. aerokosm. universitet im. N. Ye. Zhukovskogo «KHAI», 344 p. (in Russian).
- Nikolayev, A. G. & Orlov, Ye. M. (2012). Resheniye pervoy osesimmetrichnoy termouprugoy krayevoy zadachi dlya transversalno-izotropnogo poluprostranstva so sferoidalnoy polostyu [Solution of the first axisymmetric thermoelastic boundary value problem for a transversely isotropic half-space with a spheroidal cavity]. Problemy obchyslyuvalnoyi mekhaniky i mitsnosti konstruktsiy – Problems of computational mechanics and strength of structures, vol. 20, pp. 253-259 (in Russian).
- Miroshnikov, V. Yu. (2018). First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities. Journal of Mechanical Engineering, vol. 21, no. 2, pp. 12–18. https://doi.org/10.15407/pmach2018.02.012
- Protsenko, V. & Miroshnikov, V. (2018). Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned. Eastern-European Journal of Enterprise Technologies, vol. 4, no. 7, pp. 43–50. https://doi.org/10.15587/1729-4061.2018.139567
- Nikolayev, A. G., Shcherbakova, A. Yu., & Yukhno, A. I. (2006). Deystviye sosredotochennoy sily na transversalno-izotropnoye poluprostranstvo s paraboloidalnym vklyucheniyem [Action of concentrated force on a transversely-isotropic half-space with paraboloidal inclusion]. Voprosy proyektirovaniya i proizvodstva konstruktsiy letatelnykh apparatov – Questions of design and production of aircraft structures, vol. 2, pp. 47–51 (in Russian).
- Miroshnikov, V. Yu. (2018). Evaluation of the stress-strain state of half-space with cylindrical cavities. Visnyk Dniprovskoho universytetu. Seriya: Mekhanika – Bulletin of the Dnipro University. Series: Mechanics, vol. 26, no. 5, pp. 109 – 118.
- Nikolayev, A. G. & Tanchik, Ye. A. (2013). Raspredeleniye napryazheniy v yacheyke odnonapravlennogo kompozitsionnogo materiala, obrazovannogo chetyrmya tsilindricheskimi voloknami [Stress distribution in a cell of a unidirectional composite material formed by four cylindrical fibers]. Visnyk Odeskoho natsionalnoho universytetu. Matematyka. Mekhanika – Odesa National University Mathematics and Mechanics, vol. 4, pp. 101-111. (in Russian).
- Protsenko, V. S. & Ukrainets, N. A. (2015) Primeneniye obobshchennogo metoda Fur’ye k resheniyu pervoy osnovnoy zadachi teorii uprugosti v poluprostranstve s tsilindricheskoy polostyu [Application of the generalized Fourier method to the solution of the first main problem of the theory of elasticity in a half-space with a cylindrical cavity]. Visnyk Zaporizkoho natsionalnoho universytetu – Visnyk of Zaporizhzhya National University, 2, pp. 193–202 (in Russian).
- Solyanik-Krasa, K. V. (1987). Osesimmetrichnaya zadacha teorii uprugosti [Axisymmetric problem of the theory of elasticity]. Moscow: Stroyizdat, 336 p. (in Russian).
Received 21 March 2019