Journal of Engineering Mathematics

, Volume 78, Issue 1, pp 143–166 | Cite as

Equilibrium of an elastic finite cylinder under axisymmetric discontinuous normal loadings

  • V. V. Meleshko
  • Yu. V. TokovyyEmail author


This article presents an analytical technique for solving the axisymmetric elasticity problem for a finite solid cylinder subjected to discontinuous normal loadings on its ends and lateral surface. This technique is based on application of the method of crosswise superposition by representing the solution for stresses in the form of decompositions into Fourier and Bessel–Dini series. For determination of the coefficients in these series, the infinite systems of linear algebraic equations are obtained and solved by means of a modified algorithm of advanced reduction. The technique is numerically validated for typical cases of discontinuous loading. It is shown that the solution procedure is efficient for determination of the stresses in the cylinder including its edges and discontinuity points of normal loadings.


Axisymmetric problem of elasticity Discontinous normal loading Finite cylinder 


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  1. 1.
    Meleshko VV (2003) Equilibrium of an elastic finite cylinder: Filon’s problem revisited. J Eng Math 46: 355–376MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Filon LNG (1902) On the elastic equilibrium of circular cylinders under certain practical systems of load. Phil Trans R Soc Lond A 198: 147–233ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Föppl A, Föppl L (1928) Drang und Zwang. Eine höhere Festigkeitslehre für Ingenieure. Band 2. Oldenbourg, MünchenGoogle Scholar
  4. 4.
    Lur’e AI (1964) Three-dimensional problems of the theory of elasticity. Inter-science, LondonzbMATHGoogle Scholar
  5. 5.
    Barton MV (1941) The circular cylinder with a band of uniform pressure on a finite length of the surface. Trans ASME J Appl Mech 8: A97–A104MathSciNetzbMATHGoogle Scholar
  6. 6.
    Rankin AW (1944) Shrink-fit stresses and deformations. Trans ASME J Appl Mech 11: A77–A85Google Scholar
  7. 7.
    Tranter CJ, Craggs JW (1945) The stress distribution in a long circular cylinder when discontinuous pressure is applied to the curved surface. Philos Mag (Ser 7) 36: 241–250MathSciNetGoogle Scholar
  8. 8.
    Okubo H (1952) The stress distribution in a shaft press-fitted with a collar. Z Angew Math Mech 32: 178–186zbMATHCrossRefGoogle Scholar
  9. 9.
    Kogan BI (1956) Stress-state of an infinite cylinder clamped by a rigid semi-infinite cylindrical sleeve. Prikl Matem Mech 10:236–247 (in Russian)Google Scholar
  10. 10.
    Vihak VM, Yasinskyy AV, Tokovyy YuV, Rychahivskyy AV (2007) Exact solution of the axisymmetric thermoelasticity problem for a long cylinder subjected to varying with-respect-to-length loads. J Mech Behav Mater 18: 141–148CrossRefGoogle Scholar
  11. 11.
    Timoshenko S, Goodier JN (1951) Theory of elasticity. 2nd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  12. 12.
    Timoshenko S, Goodier JN (1970) Theory of elasticity. 3rd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  13. 13.
    Searle GFC (1920) Experimental elasticity: a manual for the laboratory. 3. Cambridge University Press, CambridgeGoogle Scholar
  14. 14.
    Ilyushin AA, Lensky VS (1967) Strength of materials. Pergamon, OxfordGoogle Scholar
  15. 15.
    Williams DK, Ranson WF (2003) Pipe-anchor discontinuity analysis utilizing power series solutions, Bessel functions, and Fourier series. Nucl Eng Des 220: 1–10CrossRefGoogle Scholar
  16. 16.
    Prasad SN, Dasgupta S (1977) Axisymmetric shrink fit problems of the elastic cylinder of finite length. J Elast 7: 225–242zbMATHCrossRefGoogle Scholar
  17. 17.
    Tranter CJ, Craggs JW (1947) Stresses near the end of a long cylindrical shaft under non-uniform pressure loading. Philos Mag (Ser 7) 38: 214–225MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dougall J (1904) An analytical theory of the equilibrium of an isotropic elastic plate. Trans R Soc Edinb 41: 129–228zbMATHCrossRefGoogle Scholar
  19. 19.
    Nelson CW (1962) Further consideration of the thick-plate problem with axially symmetric loading. Trans ASME J Appl Mech 29: 91–98zbMATHCrossRefGoogle Scholar
  20. 20.
    Fischer OF (1931) Näherungslösung zur Ermittlung der wirklichen Spannungsverteilung an konzentriert belasteten Zylinderenden. Ing Arch 2: 178–189CrossRefGoogle Scholar
  21. 21.
    Pickett G (1944) Application of the Fourier method to the solution of certain boundary problems in the theory of elasticity. Trans ASME J Appl Mech 11: 176–182MathSciNetGoogle Scholar
  22. 22.
    Saito H (1952) Axisymmetric strain of a finite circular cylinder and disk. Trans Jpn Soc Mech Eng 18: 58–63Google Scholar
  23. 23.
    Valov GM (1962) On the axially-symmetric deformations of a solid circular cylinder of finite length. J Appl Math Mech 26: 975–999MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Grinchenko VT (1966) Stressed state of a circular thick disk in a field of centrifugal forces. In: SM Durgar’yan (ed) Theory of shells and plates. Proceedings of the 4th all-union conference on shells and plates. Israel Program for Scientific Translations, Jerusalem, pp 383–388Google Scholar
  25. 25.
    Grinchenko VT (1978) Equilibrium and steady vibrations of elastic bodies of finite dimensions. Naukova dumka, Kiev (in Russian)Google Scholar
  26. 26.
    Chau KT, Wei XX (2000) Finite solid circular cylinders subjected to arbitrary surface load. Part I—analytic solution. Int J Solids Struct 37: 5707–5732zbMATHCrossRefGoogle Scholar
  27. 27.
    Iyengar KTSR, Chandrashekhara K (1966) Thermal stresses in a finite solid cylinder due to an axisymmetric temperature field at the end surface. Nucl Eng Des 3: 21–31CrossRefGoogle Scholar
  28. 28.
    Iyengar KTSR, Chandrashekhara K (1967) Thermal stresses in a finite solid cylinder due to steady temperature variation along the curved and end surfaces. Int J Eng Sci 5: 393–413zbMATHCrossRefGoogle Scholar
  29. 29.
    Kovalenko AD (1969) Thermoelasticity: basic theory and applications. Wolters-Noordhoff, GroningenzbMATHGoogle Scholar
  30. 30.
    Meleshko VV, Tokovyy YuV, Barber JR (2011) Axially symmetric temperature stresses in an elastic isotropic cylinder of finite length. J Math Sci 176: 646–669MathSciNetCrossRefGoogle Scholar
  31. 31.
    Love AEH (1927) A Treatise on the mathematical theory of elasticity. Cambridge University Press, CambridgezbMATHGoogle Scholar
  32. 32.
    Meleshko VV, Gomilko AM (1997) Infinite systems for a biharmonic problem in a rectangle. Proc R Soc Lond A453: 2139–2160MathSciNetADSGoogle Scholar
  33. 33.
    Kantorovich LV, Krylov VI (1958) Approximate methods of higher analysis. Wolters-Noordhoff, GroningenzbMATHGoogle Scholar
  34. 34.
    Macfarlane GG (1949) The application of Mellin transforms to the summation of slowly convergent series. Philos Mag (ser. 7) 40: 188–197MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Kiev National Taras Shevchenko UniversityKievUkraine
  2. 2.Pidstryhach Institute for Applied Problems of Mechanics and MathematicsUkrainian National Academy of SciencesLvivUkraine

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