Skip to main content
SpringerLink
Log in

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

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  Google Scholar 

  2. Timoshenko S., Young D.H.: Vibration problems in engineering. Van Nostrand, New Jersey (1955)

    Google Scholar 

  3. Gere, J.M., Lin, Y.K.: Coupled vibrations of thin-walled beams of open-cross section. J. Appl. Mech. 373–378 (1958)

  4. Vlasov V.Z.: Thin-walled elastic beams. Israel program for scientific translations, Jerusalem (1961)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  7. Bishop R.E.D., Price W.G.: Coupled bending and twisting of a Timoshenko beam. J. Sound Vib. 50, 469–477 (1977)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  9. Friberg P.O.: coupled vibration of beams—an exact dynamic element stiffness matrix. Int. J. Numer. Methods Eng. 19, 479–493 (1983)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Leung A.Y.T.: Natural shape functions of a compressed Vlasov element. Thin-walled Struct. 11, 431–438 (1991)

    Article  Google Scholar 

  14. Leung A.Y.T.: Dynamic stiffness analysis of twin-walled structures. Thin-walled Struct. 14, 209–222 (1992)

    Article  Google Scholar 

  15. Dvorkin E.N., Celentano D., Cuitino A., Gioia G.: A Vlasov beam element. Comput. Struct. 33, 187–196 (1989)

    Article  MATH  Google Scholar 

  16. Banerjee J.R., Williams F.W.: Coupled bending–torsional dynamic stiffness matrix for Timoshenko beam elements. Comput. Struct. 42, 301–310 (1992)

    Article  MATH  Google Scholar 

  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. Klausbruckner M.J., Pryputniewicz R.J.: Theoretical and experimental study of coupled vibrations of channel beams. J. Sound Vib. 183, 239–252 (1995)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  20. Bercin A.N., Tanaka M.: Coupled flexural–torsional vibrations of Timoshenko beams. J. Sound Vib. 207, 47–59 (1997)

    Article  Google Scholar 

  21. Tanaka M., Bercin A.N.: Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica. Comput. Struct. 71, 1–8 (1999)

    Article  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  26. Sapountzakis E.J., Tsiatas G.C.: Flexural–torsional vibrations of beams by BEM. Comput. Mech. 39, 409–417 (2007)

    Article  MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  28. Timoshenko S.P., Goodier J.N.: Theory of elasticity, 3rd edn. McGraw-Hill, New York (1984)

    Google Scholar 

  29. Cowper G.R.: The shear coefficient in Timoshenko’s beam theory. ASME J. Appl. Mech. 33, 335–340 (1966)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  32. Stephen N.G.: Timoshenko’s shear coefficient from a beam subjected to gravity loading. ASME J. Appl. Mech. 47, 121–127 (1980)

    MATH  Google Scholar 

  33. Hutchinson J.R.: Shear coefficients for Timoshenko beam theory. ASME J. Appl. Mech. 68, 87–92 (2001)

    Article  MATH  Google Scholar 

  34. Sapountzakis E.J., Mokos V.G.: Warping shear stresses in nonuniform torsion by BEM. Comput. Mech. 30, 131–142 (2003)

    Article  MATH  Google Scholar 

  35. Thomson W.T.: Theory of vibration with applications. Prentice Hall, Englewood Cliffs (1981)

    MATH  Google Scholar 

  36. Sapountzakis E.J., Mokos V.G.: A BEM solution to transverse shear loading of beams. Comput. Mech. 36, 384–397 (2005)

    Article  MATH  Google Scholar 

  37. Sapountzakis E.J.: Torsional vibrations of composite bars of variable cross section by BEM. Comput. Methods Appl. Mech. Eng. 194, 2127–2145 (2005)

    Article  MATH  Google Scholar 

  38. Banerjee P.K., Butterfield R.: Boundary element methods in engineering science. McGraw-Hill, New York (1981)

    MATH  Google Scholar 

  39. Sapountzakis E.J.: Solution of nonuniform torsion of bars by an integral equation method. Comput. Struct. 77, 659–667 (2000)

    Article  Google Scholar 

  40. MSC/NASTRAN for Windows. Finite element modeling and postprocessing system. Help System Index,Version 4.0, USA (1999)

Download references

Author information

Authors and Affiliations

  1. Institute of Structural Analysis, School of Civil Engineering, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece

    E. J. Sapountzakis & J. A. Dourakopoulos

Authors
  1. E. J. Sapountzakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Dourakopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. J. Sapountzakis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sapountzakis, E.J., Dourakopoulos, J.A. Shear deformation effect in flexural–torsional vibrations of beams by BEM. Acta Mech 203, 197–221 (2009). https://doi.org/10.1007/s00707-008-0041-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-008-0041-7

Keywords

Search

Navigation