Journal of Engineering Mathematics

, Volume 78, Issue 1, pp 143–166

# Equilibrium of an elastic finite cylinder under axisymmetric discontinuous normal loadings

• V. V. Meleshko
• Yu. V. Tokovyy
Article

## Abstract

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.

## References

1. 1.
Meleshko VV (2003) Equilibrium of an elastic finite cylinder: Filon’s problem revisited. J Eng Math 46: 355–376
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: 147–233
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, London
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–A104
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: 241–250
8. 8.
Okubo H (1952) The stress distribution in a shaft press-fitted with a collar. Z Angew Math Mech 32: 178–186
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: 141–148
11. 11.
Timoshenko S, Goodier JN (1951) Theory of elasticity. 2nd edn. McGraw-Hill, New York
12. 12.
Timoshenko S, Goodier JN (1970) Theory of elasticity. 3rd edn. McGraw-Hill, New York
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: 1–10
16. 16.
Prasad SN, Dasgupta S (1977) Axisymmetric shrink fit problems of the elastic cylinder of finite length. J Elast 7: 225–242
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: 214–225
18. 18.
Dougall J (1904) An analytical theory of the equilibrium of an isotropic elastic plate. Trans R Soc Edinb 41: 129–228
19. 19.
Nelson CW (1962) Further consideration of the thick-plate problem with axially symmetric loading. Trans ASME J Appl Mech 29: 91–98
20. 20.
Fischer OF (1931) Näherungslösung zur Ermittlung der wirklichen Spannungsverteilung an konzentriert belasteten Zylinderenden. Ing Arch 2: 178–189
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: 176–182
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: 975–999
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: 5707–5732
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: 21–31
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: 393–413
29. 29.
Kovalenko AD (1969) Thermoelasticity: basic theory and applications. Wolters-Noordhoff, Groningen
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: 646–669
31. 31.
Love AEH (1927) A Treatise on the mathematical theory of elasticity. Cambridge University Press, Cambridge
32. 32.
Meleshko VV, Gomilko AM (1997) Infinite systems for a biharmonic problem in a rectangle. Proc R Soc Lond A453: 2139–2160
33. 33.
Kantorovich LV, Krylov VI (1958) Approximate methods of higher analysis. Wolters-Noordhoff, Groningen
34. 34.
Macfarlane GG (1949) The application of Mellin transforms to the summation of slowly convergent series. Philos Mag (ser. 7) 40: 188–197