A new method to determine the shear coe cient of Timoshenko beam theory

The frequencies ! of flexural vibrations in a uniform beam of arbirary cross-section and length L are analyzed by expanding the exact elastodynamics equations in powers of the wavenumber q = m= L, where m is the mode number: ! 2 = A4q 4 + A6q 6 +:::. The coe cients 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 coe cient for any cross-section. The result agrees with the consensus best values in the literature, and provides a derivation of that does not rely on physical assumptions.