A new boundary modeling scheme is introduced for the solution of thick beams on simple supports using the displacement-function elasticity approach. The performance of two sets of mixed boundary conditions at the opposing lateral ends of a quasi-infinitely long beam is investigated for the analytical solution of the thick beam. In this approach, a cropped solution strategy is implemented to obtain the elastic field from the central region of the long beam, in which the solution of the beam is obtained in terms of a single scalar function in the form of infinite trigonometric series. The appropriateness and accuracy of the solutions is verified through the comparison with corresponding solutions obtained by FEM and classical beam theory. The convergence characteristics of the series solution are analyzed in the perspective of required computational effort with varying load distribution factor, beam aspect ratio and far-field span ratio. Finally, the method is applied to analyze the local stress and displacement fields, especially in the immediate neighborhood of the discretely applied loads and supports of thick beams. Significant concentrations of bending and shear stresses are observed, especially at the terminal regions of the discrete loading, which are captured in details and verified to be highly sensitive to the loading span and beam aspect-ratio.