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

The present paper considers the analytical static bending response of a finite uniform free-free Euler-Bernoulli elastic beam resting on a Winkler elastic foundation with a spatially inhomogeneous stiffness coefficient. Specifically, a linear variation of the foundation coefficient is assumed. The beam-foundation system is loaded by a force and a moment applied at one beam's (top) end, a configuration which may resemble that of a foundation pile. The analytical solution of the governing Ordinary Differential Equation is derived and represented in explicit closed-form in terms of generalized hypergeometric functions. Through the derived solution, parametric analyses are carried out, by interpreting the parametric variation of the mechanical response of the beam-foundation system due to changes in its mechanical properties.