Analytical solution for the elastic bending of beams lying on a variable Winkler support

The interaction between structures and supporting media is crucial for several engineering applications. Among existing models, the elastic one originally attributed to Emil Winkler is rather well-known to both researchers and engineers, with main reference to a constant foundation modulus. The present paper reviews first the state of the art on this model and then considers, analytically, the little-explored case of a space-dependent stiffness coefficient. The analytical solution of the ordinary differential equation describing the static deflection of a simply-supported Euler–Bernoulli elastic beam lying on a variable Winkler elastic foundation is sought. Specifically, the case of a nonlinear, minus four power variation of the stiffness coefficient is considered, allowing for closed-form representations. These are derived and examined in view of interpreting their parametric variations due to changes in the mechanical properties of the beam-foundation system. In the end, a consistent validation comparison between analytical solution and alternative reimplemented numerical treatments is produced.