The frequencies ω 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 = mπ/L, where m is the mode number: ω 2 = A 4 q 4 + A 6 q 6 + . . .. The coefficients A 4 and A 6 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.Timoshenko beam theory (TBT) provides shear deformation and rotatory inertia corrections 4 to the classic Euler-Bernoulli theory [1]; it predicts the natural frequency of bending vibrations 5 for long beams with remarkable accuracy if one employs the "best" value for the shear coefficient, 6 κ. Exact elastodynamic theory is available for beams of circular cross-section (Pochammer-7 Chree theory, see Love [2], article 202) and the thin (plane stress) rectangular section [3], and for 8 these cases the best coefficients are κ = 6(1+ν) 2 /(7+12ν+4ν 2 ) and κ = 5(1+ν)/(6+5ν), respec-9 tively, where ν is Poisson's ratio of the material. In turn, procedures have been developed for the 10 general cross-section which lead to an expression for the best κ in terms of the Saint-Venant flex-11 ure function, and which provide the above values when applied to these cross-sections. Stephen 12 and Levinson [4, 5] based their methods upon the static stress distribution for a beam subjected 13 to gravity loading, rather than the tip loading assumed in the method proposed by Cowper [6]. 14 More recently, Hutchinson [7] employed the Hellinger-Reissner variational principle to construct 15 a beam theory of Timoshenko type, which incorporated an expression for the shear coefficient 16 that was demonstrated to be equivalent to this best coefficient in the Discussion and Closure 17 section of Ref. [7]. Hutchinson [8] provided further results for thin-walled beams.18Despite these successes, all these works rely on ad hoc physical assumptions and are therefore 19 sometimes queried. It would be of some advantage to be able to dispense with these assumptions, 20 68Bernoulli theory, while A T 6 (κ) will allow κ to be determined, if A 6 can be found in an independent 69 way -which will be the task of the rest of this paper. 70 4 111 4. Order-by-order solution 112In this Section, we solve the equations of motion (15) subject to the boundary conditions 113 (16), order by order. 114 4.1. Zeroth order 115Suppose, as a matter of convention, that the lowest-order displacement is in the x-direction, 116 with the normalization set to unity. (Otherwise, all amplitudes will carry a factor u 0 and all 117 energies a factor u 2 0 , which will in the end cancel in Q = U/T .) Thuswith w 0 = 0 already assumed in (12). It is obvious that the equations of motion (15) are satisfied, 119 and since all...
