We consider a hollow elastic cylinder of finite length subjected to the action of its own weight and an axisymmetric normal load applied to the upper base. The lower base of the cylinder is immovably fixed. The inner cylindrical surface is under the conditions of sliding restraint and its outer surface is immovably fixed. The problem is reduced to an integral equation of the first kind for normal stresses on the fixed lateral surface. We determine the character of singularity of the required function and propose an efficient algorithm for the solution of the obtained equation based on the expansion of the required function in a series in Jacobi polynomials. We present the results of calculations of normal stresses on the lateral surfaces of the cylinder, which show that, in the case of rigid fixing, the influence of the weight of the cylinder is much weaker than in the case of sliding restraint.
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References
B. L. Abramyan and A. Ya. Aleksandrov, “Axially symmetric problem of the theory of elasticity,” in: Proc. of the Second All-Union Congr. on Theoretical and Applied Mechanics [in Russian], Issue 3, Nauka, Moscow (1966), pp. 7–37.
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2: Bessel Functions, Parabolic Cylinder Functions, and Orthogonal Polynomials, McGraw-Hill, New York (1953).
G. M. Valov, “On the axisymmetric deformation of a solid circular cylinder of finite length,” Prikl. Mat. Mekh., 26, No. 4, 650–667 (1962).
V. M. Vihak and Yu. V. Tokovyi, “Exact solution of an axisymmetric problem of the theory of elasticity in stresses for a continuous cylinder of certain length,” Prykl. Probl. Mekh. Mat., Issue 1, pp. 55–60 (2003).
Ya. M. Grigorenko and L. S. Rozhok, “Applying discrete Fourier series to solve problems of the stress state of hollow noncircular cylinders,” Prikl. Mekh., 50, No. 2, 3–26 (2014); English translation: Int. Appl. Mech., 50, No. 2, 105–127 (2014).
M. A. Koltunov, Yu. N. Vasil’ev, and V. A. Chernykh, Elasticity and Strength of Cylindrical Bodies [in Russian], Vysshaya Shkola, Moscow (1975).
R. M. Kushnir, B. V. Protsyuk, and V. M. Synyuta, “Quasistatic temperature stresses in a multilayer thermally sensitive cylinder,” Fiz.-Khim. Mekh. Mater., 40, No. 4, 7–16 (2004); English translation: Mater. Sci., 40, No. 4, 433–445 (2004).
G. Ya. Popov, “New transforms for the resolving equations in elastic theory and new integral transforms, with applications to boundary-value problems of mechanics,” Prikl. Mekh., 39, No. 12, 46–73 (2003); English translation: Int. Appl. Mech., 39, No. 12, 1400–1424 (2003).
G. Ya. Popov, “Axisymmetric boundary-value problems of the theory of elasticity for cylinders and cones of finite length,” Dokl. Ros. Akad. Nauk, 439, No. 2, 192–197 (2011).
G. Ya. Popov and Yu. S. Protserov, “Axisymmetric problem for an elastic cylinder of finite length with fixed lateral surface with regard for its weight,” Mat. Met. Fiz.-Mekh. Polya, 57, No. 1, 57–68 (2014); English translation: J. Math. Sci., 212, No. 1, 67–82 (2016).
M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York (1972).
Ya. S. Uflyand, Survey of Articles on the Application of Integral Transforms in the Theory of Elasticity, North Carolina State Univ., Raleigh (1965).
K. T. Chau and X. X. Wei, “Finite solid circular cylinders subjected to arbitrary surface load. Part I. Analytic solution,” Int. J. Solids Struct., 37, No. 40, 5707–5732 (2000).
S. D. Conte, K. L. Miller, and C. B. Sensenig, “The numerical solution of axisymmetric problems in elasticity,” in: D. P. LeGalley (editor), Ballistic Missile and Space Technology. Proc. of the Fifth Symp. in Ballistic Missile and Space Technology, Vol. 4, Academic Press, New York–London (1960), pp. 173–202.
J. Zhou, Z. Deng, and X. Hou, “Transient thermal response in thick orthotropic hollow cylinders with finite length: High order shell theory,” Acta Mech. Solida Sinica, 23, No. 2, 156–166 (2010).
V. V. Meleshko, “Equilibrium of an elastic finite cylinder: Filon’s problem revisited,” J. Eng. Math., 46, No. 3-4, 355–376 (2003).
V. Radu, N. Taylor, and E. Paffumi, “Development of new analytical solutions for elastic thermal stress components in a hollow cylinder under sinusoidal transient thermal loading,” Int. J. Pres. Vessel. Pip., 85, No. 12, 885–893 (2008).
S. Z. Shao, “Mechanical and thermal stresses of a functionally graded circular hollow cylinder with finite length,” Int. J. Pres. Vessel. Pip., 82, No. 3, 155–163 (2005).
R. L. Sierakowski and C. T. Sun, “An exact solution to the elastic deformation of a finite length hollow cylinder,” J. Franklin Inst., 286, No. 2, 99–113 (1968).
J.-Q. Tarn, W.-D. Tseng, and H.-H. Chang, “A circular elastic cylinder under its own weight,” Int. J. Solids Struct., 46, No. 14-15, 2886–2896 (2009).
Yu. V. Tokovyy and C.-C. Ma, “Analysis of residual stresses in a long hollow cylinder,” Int. J. Pres. Vessel. Pip., 88, No. 5-7, 248–255 (2011).
X. X. Wei and K. T. Chau, “Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test,” Int. J. Solids Struct., 50, No. 14-15, 2395–2406 (2013).
J. Ying and H. M. Wang, “Axisymmetric thermoelastic analysis in a finite hollow cylinder due to nonuniform thermal shock,” Int. J. Pres. Vessel. Pip., 87, No. 12, 714–720 (2010).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 58, No. 3, pp. 128–138, July–September, 2015.
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Protserov, Y.S. Axisymmetric Problem of the Theory of Elasticity for a Hollow Cylinder of Finite Length with Regard for Its Weight. J Math Sci 226, 160–174 (2017). https://doi.org/10.1007/s10958-017-3527-9
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DOI: https://doi.org/10.1007/s10958-017-3527-9