An investigation on the alternative methods for the bending analysis of nanobeams

In this study, the static behavior of nanoscale beams under different loading types and boundary conditions is investigated. The Complementary Functions Method (CFM) is employed to solve the boundary value problems formulated according to Eringen's nonlocal elasticity theory. The formulations are developed based on the Euler–Bernoulli and Timoshenko beam theories, assuming linear elastic and homogeneous material behavior. The considered problems are analyzed under uniform, trapezoidal, parabolic, and sinusoidal static loads. Analyses performed using the displacement, shear force, and moment expressions revealed that the beam size parameters, loading type, and boundary conditions play a decisive role in the overall structural response. The results indicate that uniform and trapezoidal loadings produced the highest displacement values, while the sinusoidal loading resulted in the lowest. Parabolic loading exhibits a more sensitive response, particularly to variations for μ coefficients. Furthermore, both Euler–Bernoulli and Timoshenko theories presented similar trends; however, higher displacement values are obtained with the Timoshenko theory due to the inclusion of shear deformation effects. The findings of this study are in good agreement with the results reported in the literature, thereby confirming the validity and accuracy of the proposed method.