The equivalence of the refined theory and the decomposition theorem of rectangular beams

This paper presents the decomposition theorem of rectangular beams and indicates that the general state of stress of beams can be decomposed into two parts: the interior state and the Papkovich–Fadle state (shortened form the P–F state). The refined theory of beams is derived by using Papkovich–Neuber solution (shortened form the P–N solution) and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. Based on the refined beam theory, the exact equations for the beam without transverse surface loadings are derived and consist of two governing differential equations: the fourth-order equation and the transcendental equation. It is then proved that the refined beam theory and the decomposition beam theorem are equivalent, i.e., the fourth-order equation and the transcendental equation are equivalent to the interior state and the P–F state, respectively.