Abstract Timoshenko’s theory of flexural motions in an elastic beam takes into account both rotatory inertia and transverse-shear deformation and, accordingly, contains two dependent variables instead of the one transverse displacement of classical theory of flexure. For the case of forced motions, the solution involves complications not usually encountered. The difficulties may be surmounted in several ways, one of which is presented in this paper. The method described makes use of the property of orthogonality of the principal modes of free vibration and uses the procedure of R. D. Mindlin and L. E. Goodman in dealing with time-dependent boundary conditions. Thus the most general problem of forced motion is reduced to a free-vibration problem and a quadrature.