nanomaterials Article Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation Marzia Sara Vaccaro, Francesco Paolo Pinnola , Francesco Marotti de Sciarra and Raffaele Barretta *   Citation: 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. https://doi.org/10.3390/ 10.3390/nano11030573 Academic Editor: Yang-Tse Cheng Received: 3 February 2021 Accepted: 18 February 2021 Published: 25 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Structures for Engineering and Architecture, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy; marziasara.vaccaro@unina.it(M.S.V.); francescopaolo.pinnola@unina.it(F.P.P.); marotti@unina.it (F.M.d.S.) * Correspondence: rabarret@unina.it Abstract: The simplest elasticity model of the 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 that 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 a Wieghardt foundation is 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 obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the 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 integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a 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. Keywords: Wieghardt foundation; Bernoulli–Euler beams; nonlocal effects; integral nonlocal model 1. Introduction Structural models of beams on an elastic foundation have been widely exploited by the scientific community to describe engineering problems with numerous applications in geotechnics, road, railroad, marine engineering, and biomechanics; see e.g., [1]. The problem of a beam subjected to transverse distributed loading proportional to its deflection was considered by E. Winkler in the framework of the local theory of elasticity [2]. It was then considered to model railway tracks on continuous linear elastic foundations by H. Zimmermann in his handbook on railway constructions [3]. Winkler and Zimmermann’s theory quickly had followers due to its simplicity and easy mathematical treatment since the soil was modeled in terms of one parameter as a continuous bed of independent linear elastic one-dimensional springs with uniform stiffness. However, E. Wieghardt [4] remarked that, in spite of its intuitive nature, Winkler’s model was not physically fully reliable since it predicts sharp discontinuities in the beam- soil profile at beam ends that are not actually present in real phenomena. However, Nanomaterials 2021, 11, 573. https://doi.org/10.3390/nano11030573 https://www.mdpi.com/journal/nanomaterials