On Timoshenko’s Beams Coefficient of Sensibility to Shear Effect

F. Pietrobon Costa


Classical beams theory usually neglect shear contribution to deformation. Timoshenko´s Beam Theory (TBT) corrects this negligence. This work is a step in the research to check accuracy of TBT in piezoactuators and in nano beams, particularly in Carbon nanotubes (CNT). Before immerse into the nanoscale problem, the superior limit of applicability of TBT must be investigated. This work introduces a proposition to the range of validation of TBT. A sensibility coefficient related to free vibratory beams is proposed, in terms of a ratio of deflected geometry to aspect ratio of the beams. Finite difference and Galerkin finite element were used to formulate the computational model. This approach was checked with experimental results to propose a top validation of TBT. Convergence in solution process was verified with precedent works. Results are related to symmetric dynamic bending of beams considering shear deformability and rotatory inertia, at small change of configuration. Timoshenko’s beam models show a displacement field greater than those obtained with Euler-Bernoulli Theory (EBT). The magnitude of the difference is of 6 to 34 % greater in TBT in relation to EBT, for tree kinds of boundary conditions, for the same beam geometry and load pattern.


[1] H. Antes, M. Schanz, S. Alvermann, Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams, Journal of Sound and Vibrations, 276, No. 3-5 (2004), 807 - 836.

D.N. Arnold, A.L. Madureira, S. Zhang, On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models, Journal of Elasticity, 67, No. 3 (2002), 171 - 185.

M.J. Beran, The use of classical beams theory for micro-beams composed of polycristals, International Journal of Solids and Structures, 35, No. 19 (1998), 2407-2412.

G.R. Cowper, The shear coefficient in Timoshenko’s beam theory, Journal of Applied Mechanics 33, No. 2 (1966) (Trans. ASME 88 E.)

J. Donea, A. Huerta, “Finite Element Methods for Flow Problems”, JohnWiley and Sons, 2003.

L. Formmagia, J.F. Gerbeau, F. Nobili, A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Compute Methods in Applies Mech. Engineering, 191 (2001), 561-582.

Frankland et al., “The Stress-strain Behavior of Polymer-nanotube Composites from Molecular Dynamics Simulations”, Langley Research Center, ICASE Report No. 2002-41, 2002.

L. Garcia, F. Taborda, S.F. Villa¸ca, “Teoria da Elasticidade”, COPPE-UFRJ, 2000.

V.M. Harik, Ranges of applicability for the continuum beam model in the constitutive analysis of carbon nanotubes: anotubes or nano-bems?, NASA - ICASE Thecnical Report, No. 2001-16, 2001.

J.T.R. Hughes, “The Finite Element Method”, Prentice-Hall Inc., 1995.

H. Katori, Considerations of the problem of shearing and torsion of thin-walled beams with arbitrary cross section, Thin-Walled Structures, 39 (2001), 671-684.

M. Krommer, H. Irschik, On the influence of coupling between electrical and mechanical fields upon the flexural behavior of smart composite beams, in “Annals of ECCOMAS”, 2000.

M. Levinson, D.W. Cook, Thick rectangular plates - I: The generalized Navier solution, International Journal of Mechanics Science, 25, No. 3 (1983).

M. Levinson, D.W. Cook, Thick rectangular plates - II: The generalized L´evy solution, International Journal of Mechanics Science, 25, No. 3 (1983).

R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates, Journal of Applied Mechanics, 3 (1951).

F.C. Pietrobon, Impacto do coeficiente de cisalhamento na flexão dinâmica de vigas, Revista Engenharia Ciência Tecnologia, UFES, 6, No. 1 (2003), 21-30.

F.C. Pietrobon, “Análise Numérica da Flexão Dinâmica de Vigas com a Consideração da Deformabilidade por Cortante e da Inércia de Rotação”, Tese de Mestrado, PEC/COPPE/UFRJ, Rio de Janeiro, RJ, 1998.

S.S. Rao, “Mechanical Vibrations”, Addison-Wesley Publishing Co. 3rd ed.,1995.

E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics, 6 (1945).

I.H. Shames, “Mechanics of Deformable Solids”, Prentice Hall Inc., 1964.

C.W. Silva, Dynamic beam model with internal damping rotatory inertia and shear deformation, AIAA Journal, 12, No. 5 (1976).

D. Sun, L. Tong, S.N. Atturi, Effects of piezoletectric sensors/actuators debonging on vibration control of smart beams, International Journal of Solids and Structures, 38, No. 50-51 (2001), 9033-9051.

S. Timoshenko, On the vibration of bars of uniform cross section, Philosophycal Magazine, 43, No. 6 (1922), 125-131.

C.M. Wang, V.B.C. Tan, Y.Y. Zhang, Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes, Journal of Sound and Vibration, 294 (2006), 1060-1072.

DOI: https://doi.org/10.5540/tema.2008.09.03.0447

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