Abstract: Response analyses of narrow straight beams of uniform rectangular cross-section with different end conditions subject to various loads are the main content of plane problems in the course of elasticity. For the cases of the beam subject to polynomial loads on the main boundaries, a general rational solution strategy is introduced here. The form of stress function is solely determined according to the form of load, based on which an analytical elasticity solution in polynomial form may be derived step-by-step by strict mathematical derivations. For a straight beam subject to a linearly distributed load, with one end simply-supported and the other fixed, the analytical expression for the stress function is presented, with its correctness verified by comparing the theoretical stress results with the finite element modeling.