Closed-form analytical solutions for the buckling and vibration of Euler-Bernoulli nanobeams within Doublet Mechanics incorporating Higher-Order Boundary Conditions

This study presents a unified closed-form analytical framework for the buckling and vibration behavior of Euler–Bernoulli nanobeams within the Doublet Mechanics theory while consistently incorporating all variationally admissible higher-order boundary conditions. Using Hamilton's principle, the sixth-order governing equations and associated boundary conditions are derived, enabling a general analytical formulation that is not available in existing numerical-based studies. The results show that higher-order boundary conditions have negligible influence on simply supported nanobeams, whereas their effects become significant for clamped–clamped and clamped–simply supported configurations. The dimensionless buckling load and natural frequency decrease as the scale parameter increases, indicating a clear size-dependent softening effect. Conversely, extending the beam length causes these parameters to approach the corresponding predictions of the classical Euler–Bernoulli theory. Furthermore, the influence of higher-order boundary conditions diminishes with increasing beam length, demonstrating convergence toward classical Euler–Bernoulli predictions. The proposed analytical solutions provide reliable benchmark data for the validation of numerical models and support the modelling, design, and optimisation of thin-walled nanoscale structural components such as carbon nanotube-based resonators and sensors.