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Mechanical response of Bernoulli Euler beams on fractional order elastic foundation

Some models of elastic foundations are provided by supposing that they are composed by elastic columns with kind of interactions such as contact forces. That yield a differential equation involving gradients of the displacement field. Recent models of elastic foundation are proposed introducing into the constitutive equation of the foundation soil forces depending on the relative vertical displacements and distance-decaying functions rule the amount of interactions. The distance-decaying function correspond to different kind of interactions and foundation behavior. It is relevant however to consider also the presence and the interaction of foundation structures over the soil. The use of an power law decay distance-decaying function yields a fractional model of elastic beam and soil. It is shown that in the case of power law decaying function represents a model in which all the gradients of the displacement function appear, while the fractional model is an enriched model between integral and gradient approaches. A fully equivalent discrete point-spring model of long-range interactions is used for the numerical solution. The reported results highlight the effects of long-range forces and the governing parameters of the linear elastic beam over a foundation soil proposed.