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Journal of Engineering Mathematics

, Volume 80, Issue 1, pp 91–104 | Cite as

Higher-order theory for bending and vibration of beams with circular cross section

  • Y. Huang
  • J. X. Wu
  • X. F. Li
  • L. E. Yang
Article

Abstract

This paper presents an efficient and simple higher-order theory for analyzing free vibration of cylindrical beams with circular cross section where the rotary inertia and shear deformation are taken into account simultaneously. Unlike the Timoshenko theory of beams, the present method does not require a shear correction factor. Similar to the Levinson theory for rectangular beams, this new model is a higher-order theory for beams with circular cross section. For transverse flexure of such cylindrical beams, based on the traction-free condition at the circumferential surface of the cylinder, two coupled governing equations for the deflection and rotation angle are first derived and then combined to yield a single governing equation. In the case of no warping of the cross section, our results are exact. A comparison is made of the natural frequencies with those using the Timoshenko and Euler–Bernoulli theories of beams and the finite element method. Our results are useful for precisely understanding the mechanical behavior and engineering design of circular cylindrical beams.

Keywords

Beam of circular cross section Higher-order theory  Free vibration Natural frequency Timoshenko beam 

Notes

Acknowledgments

This work was supported by the TianYuan Special Funds of the National Natural Science Foundation of China (Grant Nos. 11126340 and 11226303).

References

  1. 1.
    Weaver W Jr, Timoshenko SP, Young DH (1990) Vibration problems in engineering, 5th edn. Wiley, New YorkGoogle Scholar
  2. 2.
    Wong EW, Sheehan PE, Lieber CM (1997) Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277:1971–1975CrossRefGoogle Scholar
  3. 3.
    Kis A, Kasas S, Babic B, Kulik AJ, Benoit W, Briggs GAD, Schonenberger C, Catsicas S, Forro L (2002) Nanomechanics of microtubules. Phys Rev Lett 89:248101ADSCrossRefGoogle Scholar
  4. 4.
    Ru CQ (2003) Elastic models for carbon nanotubes. In: Nalwa HS (ed) Encyclopedia of nanoscience and nanotechnology. American Scientific Publishers, Valencia, pp 731–744Google Scholar
  5. 5.
    Love AEH (1944) A treatise of the mathematical theory of elasticity, 4th edn. Dover, New YorkMATHGoogle Scholar
  6. 6.
    Sokolnikoff IS (1956) Mathematical theory of elasticity, 2nd edn. McGraw-Hill, New YorkMATHGoogle Scholar
  7. 7.
    Cheung YK, Wu CI (1972) Free vibrations of thick, layered cylinders having finite length with various boundary conditions. J Sound Vib 24:189–200ADSMATHCrossRefGoogle Scholar
  8. 8.
    Hutchinson JR (1972) Axisymmetric vibrations of free finite-length rod. J Acous Soc Am 51:233–240ADSMATHCrossRefGoogle Scholar
  9. 9.
    Hutchinson JR (1981) Transverse vibrations of beams, exact versus approximate solutions. ASME J Appl Mech 48:923–928CrossRefGoogle Scholar
  10. 10.
    Leissa AW, So J (1995) Accurate vibration frequencies of circular cylinders from three-dimensional analysis. J Acous Soc Am 98:2136–2141ADSCrossRefGoogle Scholar
  11. 11.
    Leissa AW, So J (1995) Comparisons of vibration frequencies for rods and beams from one-dimensional and three-dimensional analyses. J Acous Soc Am 98:2122–2135ADSCrossRefGoogle Scholar
  12. 12.
    Karnovsky IA, Lebed OI (2000) Formulas for structural dynamics: tables, graphs and solutions. McGraw-Hill, New YorkGoogle Scholar
  13. 13.
    Han SM, Benarory HB, Wei T (1999) Dynamic of transversely vibrating beams using four engineering theories. J Sound Vib 225:935–988ADSMATHCrossRefGoogle Scholar
  14. 14.
    Li XF, Yu ZW, Zhang H (2011) Free vibration of shear beams with finite rotational inertia. J Constr Steel Res 67:1677–1683CrossRefGoogle Scholar
  15. 15.
    Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos Mag 41:744–746Google Scholar
  16. 16.
    Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross-section. Philos Mag 43:125–131Google Scholar
  17. 17.
    Esmailzadeh E, Ghorashi M (1997) Vibration analysis of a Timoshenko beam subjected to a travelling mass. J Sound Vib 199: 615–628Google Scholar
  18. 18.
    Krawczuk M, Palacz M, Ostachowicz W (2003) The dynamic analysis of a cracked Timoshenko beam by the spectral element method. J Sound Vib 264:1139–1153ADSCrossRefGoogle Scholar
  19. 19.
    Salarieh H, Ghorashi M (2006) Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. Int J Mech Sci 48:763–779MATHCrossRefGoogle Scholar
  20. 20.
    Li XF, Wang BL (2009) Vibrational modes of Timoshenko beams at small scales. Appl Phys Lett 94:101903ADSCrossRefGoogle Scholar
  21. 21.
    Blaauwendraad J (2010) Shear in structural stability: on the Engesser-Haringx discord. ASME J Appl Mech 77:031005CrossRefGoogle Scholar
  22. 22.
    Le Grognec P, Nguyen QH, Hjiaj M (2012) Exact buckling solution for two-layer Timoshenko beams with interlayer slip. Int J Solids Struct 49:143–150Google Scholar
  23. 23.
    Cowper GR (1966) The shear coefficient in Timoshenko’s beam theory. ASME J Appl Mech 33:335–340MATHCrossRefGoogle Scholar
  24. 24.
    Kaneko T (1975) On Timoshenko’s correction for shear in vibrating beams. J Phys D 8:1927–1936ADSCrossRefGoogle Scholar
  25. 25.
    Hutchinson JR (2001) Shear coefficients for Timoshenko beam theory. ASME J Appl Mech 68:87–92MATHCrossRefGoogle Scholar
  26. 26.
    Yu W, Hodges DH (2005) Generalized Timoshenko theory of the variational asymptotic beam sectional analysis. J Am Helic Soc 50:46–55CrossRefGoogle Scholar
  27. 27.
    Levinson M (1981) A new rectangular beam theory. J Sound Vib 74:81–87ADSMATHCrossRefGoogle Scholar
  28. 28.
    Reddy JN (1984) A simple higher-order theory for laminated composite plates. ASME J Appl Mech 51:745–752MATHCrossRefGoogle Scholar
  29. 29.
    Wadee MK, Wadee MA, Bassom AP, Aigner AA (2006) Longitudinally inhomogeneous deformation patterns in isotropic tubes under pure bending. Proc R Soc Lond A 462:817–838MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Asaro R, Lubarda V (2006) Mechanics of solids and materials. Cambridge University Press, New YorkCrossRefGoogle Scholar
  31. 31.
    Li XF (2008) A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams. J Sound Vib 318:1210–1229ADSCrossRefGoogle Scholar
  32. 32.
    Fuller CR, Elliott SJ, Nelson PA (1996) Active control of vibration. Academic Press, LondonGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsFoshan UniversityFoshanChina
  2. 2.School of Civil EngineeringCentral South UniversityChangshaChina

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