Refined methodology for the analysis of beams on elastic foundations

The study proposes a refined method for analyzing beams on a two-parameter elastic foundation, overcoming the limitations of the classical Winkler model. Unlike the traditional approach, which considers soil deformation only in the applied load zone, the proposed methods introduce an additional parameter of bending stiffness, providing a more accurate description of beam-foundation interaction. A governing differential equation was derived, and its analytical solutions are presented for various boundary conditions and loading types. The numerical analysis results show that the distribution of vertical displacement, bending moment, and shear force along the normalized length of the beam is symmetric with respect to the midspan. It has been established that the maximum values of vertical displacement and bending moment are observed at the midspan: the vertical displacement reaches 0.000999157, while the bending moment attains 0.124892. At the same time, the shear force reaches its maximum value near the beam supports, amounting to 0.49966. The results indicate that the stress-strain state critical points of the beam on an elastic foundation are concentrated at the midspan (for displacement and bending moment) and at the supports (for shear force). The analysis demonstrates that the maximum shear stresses occur near the fixed end of the beam (x = 0, z = 0), gradually decrease to zero at midspan, and reach negative values at the opposite end (x = 1, z = 0). The normal stresses vary linearly along the cross-sectional height, from negative in the lower zone (x=1/2, z = −1/2) to positive in the upper zone (x=1/2, z = 1/2), with values close to zero near the neutral axis (x=1//2, z = 0). Comparison with the classical Winkler model shows close agreement in displacements, bending moments, and shear forces, while the proposed method provides improved accuracy in predicting normal and shear stress distributions.