Acta Mechanica

, Volume 203, Issue 3–4, pp 197–221 | Cite as

Shear deformation effect in flexural–torsional vibrations of beams by BEM



In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sapountzakis E.J., Mokos V.G.: Vibration analysis of 3-D composite beam elements including warping and shear deformation effects. J. Sound Vib. 306, 818–834 (2007)CrossRefGoogle Scholar
  2. 2.
    Timoshenko S., Young D.H.: Vibration problems in engineering. Van Nostrand, New Jersey (1955)Google Scholar
  3. 3.
    Gere, J.M., Lin, Y.K.: Coupled vibrations of thin-walled beams of open-cross section. J. Appl. Mech. 373–378 (1958)Google Scholar
  4. 4.
    Vlasov V.Z.: Thin-walled elastic beams. Israel program for scientific translations, Jerusalem (1961)Google Scholar
  5. 5.
    Rao J.S., Carnegie W.: Solution of the equations of motion of coupled-bending torsion vibrations of turbine blades by the method of Ritz-Galerkin. Int. J. Mech. Sci. 12, 875–882 (1970)MATHCrossRefGoogle Scholar
  6. 6.
    Mei C.: Coupled vibrations of thin-walled beams of open-section using the finite element method. Int. J. Mech. Sci. 12, 883–891 (1970)MATHCrossRefGoogle Scholar
  7. 7.
    Bishop R.E.D., Price W.G.: Coupled bending and twisting of a Timoshenko beam. J. Sound Vib. 50, 469–477 (1977)MATHCrossRefGoogle Scholar
  8. 8.
    Hallauer W.L., Liu R.Y.L.: Beam bending–torsion dynamic stiffness method for calculation of exact vibration modes. J. Sound Vib. 85, 105–113 (1982)MATHCrossRefGoogle Scholar
  9. 9.
    Friberg P.O.: coupled vibration of beams—an exact dynamic element stiffness matrix. Int. J. Numer. Methods Eng. 19, 479–493 (1983)MATHCrossRefGoogle Scholar
  10. 10.
    Dokumaci E.: An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry. J. Sound Vib. 119, 443–449 (1987)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Bishop R.E.D., Cannon S.M., Miao S.: On coupled bending and torsional vibration of uniform beams. J. Sound Vib. 131, 457–464 (1989)CrossRefGoogle Scholar
  12. 12.
    Friberg P.O.: Beam element matrices derived from Vlasov’s theory of open thin-walled elastic beams. Int. J. Numer. Methods Eng. 21, 1205–1228 (1985)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Leung A.Y.T.: Natural shape functions of a compressed Vlasov element. Thin-walled Struct. 11, 431–438 (1991)CrossRefGoogle Scholar
  14. 14.
    Leung A.Y.T.: Dynamic stiffness analysis of twin-walled structures. Thin-walled Struct. 14, 209–222 (1992)CrossRefGoogle Scholar
  15. 15.
    Dvorkin E.N., Celentano D., Cuitino A., Gioia G.: A Vlasov beam element. Comput. Struct. 33, 187–196 (1989)MATHCrossRefGoogle Scholar
  16. 16.
    Banerjee J.R., Williams F.W.: Coupled bending–torsional dynamic stiffness matrix for Timoshenko beam elements. Comput. Struct. 42, 301–310 (1992)MATHCrossRefGoogle Scholar
  17. 17.
    Banerjee J.R., Williams F.W.: Coupled bending–torsional stiffness matrix of an axially loaded Timoshenko beam element. Int. J. Solids Struct. 31, 743–762 (1994)Google Scholar
  18. 18.
    Klausbruckner M.J., Pryputniewicz R.J.: Theoretical and experimental study of coupled vibrations of channel beams. J. Sound Vib. 183, 239–252 (1995)MATHCrossRefGoogle Scholar
  19. 19.
    Banerjee J.R., Guo S., Howson W.P.: Exact dynamic stiffness matrix of a bending–torsion coupled beam including warping. Comput. Struct. 59, 612–621 (1996)CrossRefGoogle Scholar
  20. 20.
    Bercin A.N., Tanaka M.: Coupled flexural–torsional vibrations of Timoshenko beams. J. Sound Vib. 207, 47–59 (1997)CrossRefGoogle Scholar
  21. 21.
    Tanaka M., Bercin A.N.: Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica. Comput. Struct. 71, 1–8 (1999)CrossRefGoogle Scholar
  22. 22.
    Hashemi, S.M., Richard, M.J.: Free vibrational analysis of axially loaded bending–torsion coupled beams: a dynamic finite element. Comput. Struct. 711–724 (2000)Google Scholar
  23. 23.
    Li J., Shen R., Hua H., Jin X.: Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Int. J. Mech. Sci. 46, 299–320 (2004)MATHCrossRefGoogle Scholar
  24. 24.
    Li J., Shen R., Hua H., Jin X.: Coupled bending and torsional vibration of axially loaded Bernoulli–Euler beams including warping effects. Appl. Acoust. 65, 153–170 (2004)CrossRefGoogle Scholar
  25. 25.
    Kim N.I., Kim M.Y.: Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects. Thin-walled Struct. 43, 701–734 (2005)CrossRefGoogle Scholar
  26. 26.
    Sapountzakis E.J., Tsiatas G.C.: Flexural–torsional vibrations of beams by BEM. Comput. Mech. 39, 409–417 (2007)CrossRefMATHGoogle Scholar
  27. 27.
    Katsikadelis J.T.: The Analog Equation Method, a boundary-only integral equation method for nonlinear static and dynamic problems in general bodies. Theor. Appl. Mech. 27, 13–38 (2002)MATHMathSciNetGoogle Scholar
  28. 28.
    Timoshenko S.P., Goodier J.N.: Theory of elasticity, 3rd edn. McGraw-Hill, New York (1984)Google Scholar
  29. 29.
    Cowper G.R.: The shear coefficient in Timoshenko’s beam theory. ASME J. Appl. Mech. 33, 335–340 (1966)MATHGoogle Scholar
  30. 30.
    Schramm U., Kitis L., Kang W., Pilkey W.D.: On the shear deformation coefficient in beam theory. Finite Elem. Anal. Des. 16, 141–162 (1994)MATHCrossRefGoogle Scholar
  31. 31.
    Schramm U., Rubenchik V., Pilkey W.D.: Beam stiffness matrix based on the elasticity equations. Int. J. Numer. Methods Eng. 40, 211–232 (1997)CrossRefGoogle Scholar
  32. 32.
    Stephen N.G.: Timoshenko’s shear coefficient from a beam subjected to gravity loading. ASME J. Appl. Mech. 47, 121–127 (1980)MATHGoogle Scholar
  33. 33.
    Hutchinson J.R.: Shear coefficients for Timoshenko beam theory. ASME J. Appl. Mech. 68, 87–92 (2001)MATHCrossRefGoogle Scholar
  34. 34.
    Sapountzakis E.J., Mokos V.G.: Warping shear stresses in nonuniform torsion by BEM. Comput. Mech. 30, 131–142 (2003)MATHCrossRefGoogle Scholar
  35. 35.
    Thomson W.T.: Theory of vibration with applications. Prentice Hall, Englewood Cliffs (1981)MATHGoogle Scholar
  36. 36.
    Sapountzakis E.J., Mokos V.G.: A BEM solution to transverse shear loading of beams. Comput. Mech. 36, 384–397 (2005)MATHCrossRefGoogle Scholar
  37. 37.
    Sapountzakis E.J.: Torsional vibrations of composite bars of variable cross section by BEM. Comput. Methods Appl. Mech. Eng. 194, 2127–2145 (2005)MATHCrossRefGoogle Scholar
  38. 38.
    Banerjee P.K., Butterfield R.: Boundary element methods in engineering science. McGraw-Hill, New York (1981)MATHGoogle Scholar
  39. 39.
    Sapountzakis E.J.: Solution of nonuniform torsion of bars by an integral equation method. Comput. Struct. 77, 659–667 (2000)CrossRefGoogle Scholar
  40. 40.
    MSC/NASTRAN for Windows. Finite element modeling and postprocessing system. Help System Index,Version 4.0, USA (1999)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute of Structural Analysis, School of Civil EngineeringNational Technical University of AthensAthensGreece

Personalised recommendations