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TECHNICAL PAPERS

Shear Coefficients for Timoshenko Beam Theory

[+] Author and Article Information
J. R. Hutchinson

Department of Civil and Environmental Engineering, University of California, Davis, CA 95616

J. Appl. Mech 68(1), 87-92 (Aug 15, 2000) (6 pages) doi:10.1115/1.1349417 History: Received June 07, 2000; Revised August 15, 2000
Copyright © 2001 by ASME
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References

Timoshenko,  S. P., 1921, “On the Correction for Shear of the Differential Equation for Transverse Vibrations of Bars of Prismatic Bars,” Philos. Mag., 41, pp. 744–746.
Kaneko,  T., 1975, “On Timoshenko’s Correction for Shear in Vibrating Beams,” J. Phys. D 8, pp. 1927–1936.
Timoshenko,  S. P., 1922, “On the Transverse Vibrations of Bars of Uniform Cross Section,” Philos. Mag., 43, pp. 125–131.
Cowper,  G. R., 1966, “The Shear Coefficient in Timoshenko’s Beam Theory,” ASME J. Appl. Mech., 33, pp. 335–340.
Hutchinson,  J. R., 1981, “Transverse Vibrations of Beams, Exact Versus Approximate Solutions,” ASME J. Appl. Mech., 48, pp. 923–928.
Leissa,  A. W., and So,  J., 1995, “Comparisons of Vibration Frequencies for Rods and Beams From One-Dimensional and Three-Dimensional Analyses,” J. Acoust. Soc. Am., 98, pp. 2122–2135.
Hutchinson,  J. R., 1996, comments on “Comparisons of Vibration Frequencies for Rods and Beams From One-Dimensional and Three-Dimensional Analyses,” J. Acoust. Soc. Am., 98, pp. 2122–2135100, pp. 1890–1893.
Spence,  G. B., and Seldin,  E. J., 1970, “Sonic Resonances of a Bar and Compound Torsional Oscillator,” J. Appl. Phys., 41, pp. 3383–3389.
Spinner,  S., Reichard,  T. W., and Tefft,  W. E., 1960, “A Comparison of Experimental and Theoretical Relations Between Young’s Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars,” J. Res. Natl. Bur. Stand., Sect. A, 64A, pp. 147–155.
Hutchinson,  J. R., and Zillmer,  S. D., 1986, “On the Transverse Vibration of Beams of Rectangular Cross-Section,” ASME J. Appl. Mech., 53, pp. 39–44.
Hutchinson,  J. R., and El-Azhari,  S. A., 1986, “Vibrations of Free Hollow Circular Cylinders,” ASME J. Appl. Mech., 53, pp. 641–646.
Armenàkas, A. E., Gazis, D. C., and Herrmann G., 1969, Free Vibrations of Circular Cylindrical Shells, Pergamon Press, Oxford, UK.
Leissa,  A. W., and So,  J., 1997, “Free Vibrations of Thick Hollow Circular Cylinders From Three-Dimensional Analysis,” ASME J. Vibr. Acoust., 119, pp. 89–95.
Reissner,  E., 1950, “On a Variational Theorem in Elasticity,” J. Math. Phys., 29, pp. 90–95.
Love, A. E. H., 1944, A Treatise on the Mathematical Theory of Elasticity, Dover, New York.

Figures

Grahic Jump Location
Coordinates and positive moment and shear sign convention for the uniform beam
Grahic Jump Location
Shear coefficient versus outer diameter to wavelength ratio for an infinitely long hollow cylinder. a is the ratio of the inner to outer radius. Straight horizontal lines are the new shear coefficient and the curved lines are the coefficient which is required to match the true solution.
Grahic Jump Location
Shear coefficient reciprocal versus width-to-depth ratio for an elliptical cross section for different values of Poisson’s ratio. (–) new coefficient; ([[dashed_line]]) Cowper’s coefficient.
Grahic Jump Location
Shear coefficient reciprocal versus width-to-depth ratio for a rectangular cross section for different values of Poisson’s ratio

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