Abstract: To study the influence of non- linear relationship of foundation reaction force and beam deflection on deformation and internal force of long，medium and short these three kinds of foundation beam. Based on experimental data，the reaction force of foundation and the beam deflection were fitted as linear relationship and cubic polynomial，and then did program with finite difference method and Newton iteration method，to solve the curve of beams′ length and deflection，slope，shearing force，bending moment of non- linear elastic foundation beams and classical linear elastic foundation beams. The numerical case showed the deformation and internal force of short non-linear and linear elastic Winkler beams are the same；the relative errors could reach 10%-20% for medium and long beams，so the nonlinear relationship of foundation reaction force and settlement should be considered in real engineering；the relative error of the deformation and internal force of medium non-linear and linear elastic beams changed with the length of beams，but to long beams，the relative error did not changed with the length.

Key words: non-linear elastic foundation beams, finite difference method, deflection, slope, shearing