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=mπ/L, where m is the mode number: ω2=A4q4A6 q6⋯. The coefficients A4 and A6 are obtained without further assumptions; the former captures EulerBernoulli 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.
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