Acta Mechanica

, Volume 203, Issue 3–4, pp 197–221

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

Article

## 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.

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### References

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)
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)
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)
7. 7.
Bishop R.E.D., Price W.G.: Coupled bending and twisting of a Timoshenko beam. J. Sound Vib. 50, 469–477 (1977)
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)
9. 9.
Friberg P.O.: coupled vibration of beams—an exact dynamic element stiffness matrix. Int. J. Numer. Methods Eng. 19, 479–493 (1983)
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)
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)
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)
13. 13.
Leung A.Y.T.: Natural shape functions of a compressed Vlasov element. Thin-walled Struct. 11, 431–438 (1991)
14. 14.
Leung A.Y.T.: Dynamic stiffness analysis of twin-walled structures. Thin-walled Struct. 14, 209–222 (1992)
15. 15.
Dvorkin E.N., Celentano D., Cuitino A., Gioia G.: A Vlasov beam element. Comput. Struct. 33, 187–196 (1989)
16. 16.
Banerjee J.R., Williams F.W.: Coupled bending–torsional dynamic stiffness matrix for Timoshenko beam elements. Comput. Struct. 42, 301–310 (1992)
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)
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)
20. 20.
Bercin A.N., Tanaka M.: Coupled flexural–torsional vibrations of Timoshenko beams. J. Sound Vib. 207, 47–59 (1997)
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)
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)
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)
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)
26. 26.
Sapountzakis E.J., Tsiatas G.C.: Flexural–torsional vibrations of beams by BEM. Comput. Mech. 39, 409–417 (2007)
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)
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)
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)
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)
32. 32.
Stephen N.G.: Timoshenko’s shear coefficient from a beam subjected to gravity loading. ASME J. Appl. Mech. 47, 121–127 (1980)
33. 33.
Hutchinson J.R.: Shear coefficients for Timoshenko beam theory. ASME J. Appl. Mech. 68, 87–92 (2001)
34. 34.
Sapountzakis E.J., Mokos V.G.: Warping shear stresses in nonuniform torsion by BEM. Comput. Mech. 30, 131–142 (2003)
35. 35.
Thomson W.T.: Theory of vibration with applications. Prentice Hall, Englewood Cliffs (1981)
36. 36.
Sapountzakis E.J., Mokos V.G.: A BEM solution to transverse shear loading of beams. Comput. Mech. 36, 384–397 (2005)
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)
38. 38.
Banerjee P.K., Butterfield R.: Boundary element methods in engineering science. McGraw-Hill, New York (1981)
39. 39.
Sapountzakis E.J.: Solution of nonuniform torsion of bars by an integral equation method. Comput. Struct. 77, 659–667 (2000)
40. 40.
MSC/NASTRAN for Windows. Finite element modeling and postprocessing system. Help System Index,Version 4.0, USA (1999)Google Scholar