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

机译：伯努利 - 欧拉梁围绕位移驱动的非本地基础的弹性学

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摘要

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.
机译：由Winkler构思弯曲下纤维梁下方的基础的最简单弹性模型，需要局部反应与梁偏转之间的局部比例。这种方法导致良好的弹性动力和弹性动力学问题，而是通过Wieggardt突出，它提供了针对各种工程应用的技术性重要性的弹性响应。因此，通过假设光束偏转是土壤反应场和平均核之间的卷积整体，Wieghardt本身被Wieghardt自己所取代。由于组成型和运动兼容性要求之间的冲突，在Wieghardt基金会上休息的相应弹性问题均不呈现。提出了原始Wieghardt模型的修改，提出了本构类类型的虚拟边界集中力，它们是物理上可疑的，在规定的运动边界条件下显着影响。使用通过交换Wieghardt原始配方中涉及的输入和输出字段获得的位移驱动的非局部积分策略，在本研究中克服了固有的困难和问题。也就是说，非局部土壤反应场是具有平均核的梁偏转场积分卷曲的输出。建立了用双指数平均核，等效非局部差异问题的位移驱动的非局部整体法，建立了补充有涉及非局部土壤反应的非标准本构体反应的非标准本构关系。作为一个关键含义，控制在位移驱动的非局部积分基础上搁置在位移驱动的非局部整体基础上的弹性纤维梁的集成电化方程被补充有运动，静态和新的构成边界条件的更简单的微分方程。通过检查结构工程的示例性问题和分析求解和分析拟议的非局部方法。还检测了用于数值分析的基准解决方案。

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