Abstract: The decomposition theorem for an isotropic elastic plate is extended to an isotropic rectangular beam, and the decomposition theorem for elastic beam bending is presented. It is shown that the stress states of the beam without transverse surface loadings can be decomposed into two parts: the interior state and the Papkovich-Fadle state (P-F state, for short). With the introduction and proof of two lemmas, a rigorous mathematical proof of the decomposition theorem is given. The proof is simple, concise and independentof the Papkovich-Fadle eigenfunction expansion of bi-harmonic functions. Only some basic mathematical operations are used in the process.