Internal constraints, reactive stresses, and the Timoshenko beam theory

It is shown that the kinematical assumptions of the Timoshenko theory for shearable beams can be regarded as internal constraints, some involving the first, others the second deformation gradient; such constrains are thought of as maintained by reaction stresses and hyperstresses of the type occurring in non-simple materials of grade 2. It is discussed how these reactions can be used to better the first approximation of the unknown equilibrium stress field in the three-dimensional body modelled by the Timoshenko beam, an appoximation which is based on the solution of the one-dimensional Timoshenko problem.