Free Vibration of Agglomerated SiO₂ Nanoparticle-Reinforced Concrete Beams on Elastic Foundations

Nano-silica can stiffen cementitious composites, but its benefit depends on how well the nanoparticles are dispersed, and agglomeration is common in the alkaline pore solution of cement paste. This study presents an analytical framework for the free vibration of SiO₂ nanoparticle-reinforced concrete beams resting on a Winkler–Pasternak elastic foundation, with nanoparticle agglomeration explicitly incorporated into the effective properties. Agglomeration is described by two parameters: the volume fraction of nanoparticle-rich inclusions, ξ, and the fraction of nanoparticles located within them, ζ. These parameters enter a two-stage homogenization scheme that combines a Voigt–Reuss estimate for each region with Paul’s model for the composite, and their admissible range is derived from the volume-fraction constraints. The resulting properties are embedded in a three-unknown higher-order shear deformation theory that satisfies the traction-free surface conditions without a shear correction factor. The governing equations are derived from Hamilton’s principle and solved analytically for simply supported, clamped–clamped and clamped–simply supported beams. The beam formulation agrees with published benchmarks for functionally graded beams to within 0.03% without a foundation and 0.6% on a Winkler–Pasternak foundation. For a fixed matrix and hydration state, increasing the SiO₂ volume fraction from 0 to 0.20 raises the fundamental frequency by about 14% for all three boundary conditions. This gain comes almost entirely from stiffening, since the change in density contributes less than 1%. When half of the nano-silica is agglomerated and does not contribute to load transfer, this gain falls to about 8%, so roughly half of the dynamic benefit is lost. Over the investigated range of dimensionless foundation stiffness, the Pasternak parameter raises the frequency more than the Winkler parameter, and clamped–clamped beams give the highest frequencies, followed by clamped–simply supported and simply supported beams. The framework links dispersion quality, boundary restraint and foundation support to the dynamic response of nano-engineered concrete members.