A new method to determine the shear coefficient of Timoshenko beam theory

The frequencies ωω of flexural vibrations in a uniform beam of arbitrary cross-section and length L   are analysed by expanding the exact elastodynamics equations in powers of the wavenumber q=mπ/Lq=mπ/L, where m   is the mode number: ω2=A4q4+A6q6+⋯ω2=A4q4+A6q6+⋯. The coefficients A4 and A6 are obtained without further assumptions; the former captures Euler–Bernoulli theory while the latter, when compared with Timoshenko beam theory rendered into the same form, unambiguously yields the shear coefficient κκ for any cross-section. The result agrees with the consensus best values in the literature, and provides a derivation of κκ that does not rely on physical assumptions.