Free Vibration Analysis of Functionally Graded Nanobeams Via Complementary Functions Method in The Laplace Domain

Abstract

This study presents a unified framework for the free vibration analysis of functionally graded (FG) nanobeams within Eringen’s nonlocal elasticity, consistently formulated under Euler–Bernoulli (EBT) and Timoshenko (TBT) beam theories. The canonical first-order governing equations are derived in a unified closed-form manner and solved using the Complementary Functions Method (CFM) in the Laplace domain. A comprehensive parametric study addresses four boundary conditions with variations in slenderness ratios, gradation indices, and nonlocal parameters. The primary contribution of this work lies in providing a unified closed-form canonical state-space formulation for nonlocal FG nanobeams under both EBT and TBT, and in demonstrating that the Laplace–CFM implementation offers a stable and efficient eigen-solver with consistent boundary enforcement across multiple support conditions. The resulting benchmark frequencies can serve as a reliable reference for verifying of future refined or multi-physics nanobeam models. The results confirm monotonic frequency softening with increasing nonlocal parameter and with grading toward the softer constituent. The EBT–TBT discrepancy is most pronounced for low-slenderness ratios, highlighting the role of shear deformation and rotary inertia in short/thick nanobeams, while the two theories converge as slenderness increases.