A HIERARCHY OF TIMOSHENKO BEAM THEORIES

Abstract. This paper shows that conventional Timoshenko theory for bending waves is a member of a two-parameter family (m; n) of approximations to the exact equations of linear elasticity. Higher members of the family are shown to represent the exact dispersion relation with extraordinary accuracy; in particular, an arbitrary number of branches can be captured accurately over their entire length, i.e. up to arbitrarily high frequencies and wavenumbers. The theory admits a rational accuracy analysis, and resolves certain controversies about the validity of higher-branch approximations. The paper demonstrates conclusively that Ti-moshenko theory is a completely rational theory, thus ending decades of doubt on the matter. The standard Euler-Bernoulli theory is a lower member of the two-parameter family. Espe-cially useful is Timoshenko (1; 2) theory, which extends conventional Timoshenko (0; 1) the-ory by capturing the first four branches of the exact dispersion relation rather than merely the first two