Modelling the Dynamic Response of Clay Nanoparticle-Modified Concrete Beams Resting on Two-Parameter Elastic Foundations

This study presents an analytical investigation of the dynamic behavior of concrete beams reinforced with different types of nano-clay (NC) particles and resting on a Winkler–Pasternak elastic foundation. The equivalent elastic properties of the nanocomposite were determined using an Eshelby-based homogenization model. An improved quasi-three-dimensional beam theory was applied to formulate the governing equations of motion, which were subsequently then analytically solved using Navier’s method. The analysis shows that NC reinforcement systematically elevates the natural frequencies of the beam, with the magnitude of improvement varying by particle type and concentration. Increasing the NC volume fraction to 30% leads to a significant rise in the fundamental frequency, reaching about 30% for hectorite (SHca-1) compared with the unreinforced beam, whereas montmorillonite (SWy-1) produces a more moderate increase of approximately 13%. This reinforcing effect remains consistent across different span-to-depth ratios, indicating that the influence of nano-clay content on the dynamic response is largely independent of beam slenderness. Furthermore, increasing the Winkler foundation stiffness results in an almost linear rise in frequency of approximately 18–22%, whereas the Pasternak shear parameter produces a stronger effect, reaching around 25% enhancement depending on the reinforcement type. These results indicate that incorporating nano-clay platelets can be an effective strategy for enhancing the vibrational stiffness of concrete beams and improving their dynamic performance when interacting with supporting soil media.