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

Area of Science:

• Civil Engineering
• Mechanical Engineering
• Applied Mathematics

Background:

• Winkler's model assumes local soil reaction-beam deflection proportionality, leading to unrealistic elastic responses.
• Wieghardt's model uses convolution but results in ill-posed problems due to compatibility conflicts.
• Previous modifications to Wieghardt's model introduced physically questionable elements.

Purpose of the Study:

• To develop a robust and accurate model for beams on foundations.
• To overcome the limitations of existing elasticity models for soil-structure interaction.
• To provide a theoretically sound and practically applicable framework for structural analysis.

Main Methods:

• A displacement-driven nonlocal integral strategy is proposed, reversing Wieghardt's formulation.
• The model utilizes integral convolutions of beam deflection with an averaging kernel.
• A bi-exponential averaging kernel is employed, leading to a nonlocal differential problem with non-standard boundary conditions.

Main Results:

• The new approach replaces complex integrodifferential equations with simpler differential equations.
• The model incorporates kinematic, static, and novel constitutive boundary conditions.
• Analytical solutions for exemplar structural engineering problems are derived, providing benchmark data.

Conclusions:

• The displacement-driven nonlocal integral foundation model offers a significant improvement over existing methods.
• This research provides a more accurate and computationally tractable approach to analyzing beams on foundations.
• The findings have implications for structural design and numerical analysis in engineering applications.