Buckling analysis of nonlocal Timoshenko beams via an efficient semi-analytical approach

This study presents a semi-analytical method for analyzing the buckling behavior of nonlocal Timoshenko beams using Eringen’s nonlocal elasticity theory. The method combines the initial value method (IVM) with a segment-wise approximate transfer matrix (ATM) approach, enabling accurate and efficient computation of critical buckling loads under various boundary conditions. The IVM calculates displacements and stress resultants from initial conditions, while the ATM constructs the principal matrix required by the IVM through segment-wise integration, ensuring numerical stability. The IVM–ATM framework offers a practical alternative to traditional analytical and numerical methods, especially for size-dependent problems. The results show excellent agreement with existing solutions, validating the method’s accuracy. The method’s accuracy is further supported by detailed convergence analyses. Parametric studies highlight the effects of length-to-diameter ratio, nonlocal parameter, and boundary conditions on buckling behavior. The proposed method provides a reliable and efficient tool for nanoscale beam structures.