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.
Highlights
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
A noteworthy result shows that the nonlocal integral Equation (14c) can be replaced with an equivalent differential problem and foundation boundary conditions according to the proposition proved in Appendix B, starting from the results provided in [29]
We provide some numerical results of technical interest to illustrate the effectiveness of the proposed methodology for the analysis of Bernoulli–Euler beams on a nonlocal foundation
Summary
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 motivation of the present paper consists in formulating a well-posed nonlocal integral model of elastic foundation such that no fictitious forces are postulated at the end points of Bernoulli–Euler beams in order to solve the relevant structural problem. The nonlocal model of elastic foundation is cast in the framework of Eringen theory [31,32,33] requiring that reaction fields are outputs of convolutions between displacement fields of the elastic foundation and a suitable averaging kernel. No conflict is present if the elastostatic problem of a Bernoulli–Euler beam resting on elastic foundation is formulated by considering the displacement-driven nonlocal model of external elasticity. The effects of nonlocal parameters and the stiffness coefficient of the Winkler elastic soil on structural transverse displacements, reactions, bending, and shear forces are analytically evaluated and compared with outcomes in literature
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