Static analysis of functionally graded nanobeams based on Euler–Bernoulli and Timoshenko theories via complementary functions method

Abstract

This study investigates the static behavior of functionally graded (FG) nanobeams. Formulations are carried out using the Euler–Bernoulli and Timoshenko beam theories framework. The canonical governing equations are derived for the first time and solved using the Complementary Functions Method (CFM). The transverse deflections for a range of material gradient indices, nonlocal parameters, slenderness ratios, and different boundary conditions are obtained. Benchmark comparisons validate the accuracy of the proposed model, and new numerical data are presented for cantilever and clamped-supported beams. Results confirm that increased nonlocality and gradation amplify deflections, especially in thick beams where shear effects are significant. The findings affirm the CFM’s capability in accurately capturing scale-dependent mechanics, offering a reliable numerical platform for advanced nanoscale beam design.