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The Timoshenko-Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest early in the 20th century. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams, or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter (in principle comparable to the height of the beam or shorter), and thus the distance between opposing shear forces decreases. Rotary inertia effect was introduced by Bresse and Rayleigh. If the shear modulus of the beam material approaches infinity—and thus the beam becomes rigid in shear—and if rotational inertia effects are neglected, Timoshenko beam theory converges towards ordinary beam theory.
References
- Timoshenko's Beam Equations http://ccrma.stanford.edu/~bilbao/master/node163.html
- Thomson, W. T., 1981, Theory of Vibration with Applications, second edition. Prentice-Hall, New Jersey.
- Rosinger, H. E. and Ritchie, I. G., 1977, On Timoshenko's correction for shear in vibrating isotropic beams, J. Phys. D: Appl. Phys., vol. 10, pp. 1461-1466.
- "Experimental study of the Timoshenko beam theory predictions", A. Díaz-de-Anda, J. Flores, L. Gutiérrez, R.A. Méndez-Sánchez, G. Monsivais, and A. Morales, Journal of Sound and Vibration, Volume 331, Issue 26, 17 December 2012, pp. 5732–5744.
- Timoshenko, Stephen P., 1932, Schwingungsprobleme der Technik, Julius Springer.
- Mindlin, R. D., Deresiewicz, H., 1953, Timoshenko's Shear Coefficient for Flexural Vibrations of Beams, Technical Report No. 10, ONR Project NR064-388, Department of Civil Engineering, Columbia University, New York, N.Y.
- Cowper, G. R., 1966, "The Shear Coefficient in Timoshenko’s Beam Theory", J. Appl. Mech., Vol. 33, No.2, pp. 335–340.
- Stephen, N. G., 1980. "Timoshenko’s shear coefficient from a beam subjected to gravity loading", Journal of Applied Mechanics, Vol. 47, No. 1, pp. 121–127.
- Hutchinson, J. R., 1981, "Transverse vibration of beams, exact versus approximate solutions", Journal of Applied Mechanics, Vol. 48, No. 12, pp. 923–928.
- Le, Khanh C., 1999, Vibrations of shells and rods, Springer.
- Stephen Timoshenko, James M. Gere. Mechanics of Materials. Van Nostrand Reinhold Co., 1972. pages 207.
- Kaneko, T., 1975, "On Timoshenko's correction for shear in vibrating beams", J. Phys. D: Appl. Phys., Vol. 8, pp. 1927–1936.
- "Experimental check on the accuracy of Timoshenko’s beam theory", R. A. Méndez-Sáchez, A. Morales, J. Flores, Journal of Sound and Vibration 279 (2005) 508–512.
- "On the Accuracy of the Timoshenko Beam Theory Above the Critical Frequency: Best Shear Coefficient", J. A. Franco-Villafañe and R. A. Méndez-Sánchez, Journal of Mechanics, January 2016, pp. 1–4. DOI: 10.1017/jmech.2015.104.
