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

The frequencies ω 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 = m π / L , where m is the mode number: ω 2 = A 4 q 4 + A 6 q 6 + ⋯ . 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 κ for any cross-section. The result agrees with the consensus best values in the literature, and provides a derivation of κ that does not rely on physical assumptions.