Effect of Axial Functional Gradation and Periodic Square Perforations on the Vibration of Nanobeams with Elastic Foundation

This study introduces a model capable of investigating the vibrational response of nanobeams that simultaneously incorporate structural perforations and axially functionally graded materials while being supported by elastic foundations. By using Euler–Bernoulli beam theory with the nonlocal elasticity framework, the model captures nanoscale phenomena often neglected in classical analysis and governing equation of motion is derived. The objective of this study is to examine the coupled effects of axial material gradation, periodic perforations, and elastic foundation parameters on the vibrational behavior of nanobeams, highlighting how variations in perforation geometry and material gradation jointly influence the mode shape and frequency response. The modified equivalent modeling strategy is employed to account for the periodic perforations. The Galerkin method is utilized to handle and extract natural frequencies and mode shapes. The proposed numerical scheme exhibits high accuracy, as verified by comparison with existing results from the literature, with the maximum deviation limited to just 0.081%. Perforation and axial material gradation significantly impact the vibrational characteristics, governed by the nonlocal effects and geometric configuration. Numerical findings highlight the sensitivity of dynamic response to filling ratio, perforation geometry, and foundation interaction across different boundary conditions.