Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and deflection. Such an approach leads to well-posed elastostatic elastodynamic problems, but as highlighted Wieghardt, it provides elastic responses that are not technically significant for wide variety engineering applications. Thus, Winkler’s replaced Wieghardt himself assuming deflection is convolution integral reaction field averaging kernel. Due conflict constitutive kinematic compatibility requirements, corresponding problem inflected resting on ill-posed. Modifications original were proposed introducing fictitious boundary concentrated forces type, which physically questionable, being significantly influenced prescribed conditions. Inherent difficulties issues overcome in present research using displacement-driven nonlocal strategy obtained swapping input output fields involved Wieghardt’s formulation. That is, convolutions with Equipping law bi-exponential kernel, equivalent differential problem, supplemented non-standard conditions involving reactions, established. As key implication, integrodifferential equations governing much simpler kinematic, static, new illustrated examining analytically solving exemplar problems structural engineering. Benchmark solutions numerical analyses also detected.