Vaccaro, M.S.; Pinnola, F.P.; Marotti de Sciarra, F.; Barretta, R. Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation. Nanomaterials2021, 11, 573.
Vaccaro, M.S.; Pinnola, F.P.; Marotti de Sciarra, F.; Barretta, R. Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation. Nanomaterials 2021, 11, 573.
The simplest elasticity model of foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but, as highlighted by Wieghardt, it provides elastic responses which are not technically significant for a wide variety of engineering applications. Thus, Winkler's model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on Wieghardt foundation results to be ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy got by swapping input and output fields involved in Wieghardt's original formulation. That is, nonlocal soil reaction fields are output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integro-differential equations governing the elastostatic problem of an inflected elastic slender beam resting on displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.
Wieghardt foundation; Bernoulli-Euler beams; nonlocal effects; integral nonlocal model
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