Buckling analysis of functionally graded nonlocal nanobeams using the initial value method and approximate transfer matrix approach

In the current study, the buckling behavior of functionally graded (FG) nano-sized beams is investigated using a semi-analytical method. The beams are modeled using the Euler–Bernoulli beam theory combined with Eringen’s nonlocal elasticity theory, under various classical boundary conditions including simply supported (SS), clamped–clamped (CC), clamped–simply supported (CS), and clamped–free (CF). The governing equations are derived via the principle of minimum total potential energy and reformulated as a system of first-order differential equations. The key contribution of this study is the integration of the Initial Value Method (IVM) with the Approximate Transfer Matrix (ATM), yielding a compact and numerically stable solution framework. This enables efficient computation of critical buckling loads without symbolic operations or global matrix assembly. Parametric analyses are carried out to evaluate the influence of power-law exponent, nonlocal parameter, and slenderness ratio on buckling behavior. To the authors’ knowledge, this is the first application of the IVM–ATM combination to nonlocal FG beam models. The approach is validated against reference results and offers a robust alternative for nanoscale structural design.