Shear deformation effect in flexural–torsional vibrations of beams by BEM Authors E. J. Sapountzakis Institute of Structural Analysis, School of Civil Engineering National Technical University of Athens J. A. Dourakopoulos Institute of Structural Analysis, School of Civil Engineering National Technical University of Athens Article

First Online: 05 July 2008 Received: 24 September 2007 Revised: 10 March 2008 DOI :
10.1007/s00707-008-0041-7

Cite this article as: Sapountzakis, E.J. & Dourakopoulos, J.A. Acta Mech (2009) 203: 197. doi:10.1007/s00707-008-0041-7
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.

Download to read the full article text

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)

CrossRef 2.

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

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)

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)

MATH CrossRef 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)

MATH CrossRef 7.

Bishop R.E.D., Price W.G.: Coupled bending and twisting of a Timoshenko beam. J. Sound Vib.

50 , 469–477 (1977)

MATH CrossRef 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)

MATH CrossRef 9.

Friberg P.O.: coupled vibration of beams—an exact dynamic element stiffness matrix. Int. J. Numer. Methods Eng.

19 , 479–493 (1983)

MATH CrossRef 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)

CrossRef MathSciNet 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)

CrossRef 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)

MATH CrossRef MathSciNet 13.

Leung A.Y.T.: Natural shape functions of a compressed Vlasov element. Thin-walled Struct.

11 , 431–438 (1991)

CrossRef 14.

Leung A.Y.T.: Dynamic stiffness analysis of twin-walled structures. Thin-walled Struct.

14 , 209–222 (1992)

CrossRef 15.

Dvorkin E.N., Celentano D., Cuitino A., Gioia G.: A Vlasov beam element. Comput. Struct.

33 , 187–196 (1989)

MATH CrossRef 16.

Banerjee J.R., Williams F.W.: Coupled bending–torsional dynamic stiffness matrix for Timoshenko beam elements. Comput. Struct.

42 , 301–310 (1992)

MATH CrossRef 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)

18.

Klausbruckner M.J., Pryputniewicz R.J.: Theoretical and experimental study of coupled vibrations of channel beams. J. Sound Vib.

183 , 239–252 (1995)

MATH CrossRef 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)

CrossRef 20.

Bercin A.N., Tanaka M.: Coupled flexural–torsional vibrations of Timoshenko beams. J. Sound Vib.

207 , 47–59 (1997)

CrossRef 21.

Tanaka M., Bercin A.N.: Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica. Comput. Struct.

71 , 1–8 (1999)

CrossRef 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)

MATH CrossRef 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)

CrossRef 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)

CrossRef 26.

Sapountzakis E.J., Tsiatas G.C.: Flexural–torsional vibrations of beams by BEM. Comput. Mech.

39 , 409–417 (2007)

CrossRef MATH 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 28.

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

29.

Cowper G.R.: The shear coefficient in Timoshenko’s beam theory. ASME J. Appl. Mech.

33 , 335–340 (1966)

MATH 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)

MATH CrossRef 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)

CrossRef 32.

Stephen N.G.: Timoshenko’s shear coefficient from a beam subjected to gravity loading. ASME J. Appl. Mech.

47 , 121–127 (1980)

MATH 33.

Hutchinson J.R.: Shear coefficients for Timoshenko beam theory. ASME J. Appl. Mech.

68 , 87–92 (2001)

MATH CrossRef 34.

Sapountzakis E.J., Mokos V.G.: Warping shear stresses in nonuniform torsion by BEM. Comput. Mech.

30 , 131–142 (2003)

MATH CrossRef 35.

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

MATH 36.

Sapountzakis E.J., Mokos V.G.: A BEM solution to transverse shear loading of beams. Comput. Mech.

36 , 384–397 (2005)

MATH CrossRef 37.

Sapountzakis E.J.: Torsional vibrations of composite bars of variable cross section by BEM. Comput. Methods Appl. Mech. Eng.

194 , 2127–2145 (2005)

MATH CrossRef 38.

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

MATH 39.

Sapountzakis E.J.: Solution of nonuniform torsion of bars by an integral equation method. Comput. Struct.

77 , 659–667 (2000)

CrossRef 40.

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