Free Vibration Analysis of Bidirectional Functionally Graded Beams Using a Novel Perturbation Theory–Based Approach

Bidirectional (2D) functionally graded material (FGM), which is characterized by a continuous spatial gradient in two directions in both its microstructure and material properties, exhibits numerous benefits, including interfacial delamination prevention, frequency regulation, and stress concentration elimination. Because they are load-bearing structures, the natural frequencies of bidirectional functionally graded (2D FG) beams play a crucial role in structural design, and these frequencies are usually calculated using the numerical generalized differential quadrature method (GDQM) based on interpolation. Instead of the GDQM, this study pioneered a perturbation approach (PA) as an alternative for the first time. Unlike traditional perturbation theory, our approach used a homogeneous beam as an approximate reference model, whereas the gradients of the material properties were selected as the perturbation parameters to approximate the intricacies of the 2D FG beam. Through analysis of the PA, it was found that the solution could be simplified, resulting in a 22% reduction in calculation time. First, a free vibration model of a 2D FG beam was proposed based the rule of mixtures, Timoshenko beam theory, and Hamilton’s principle. Second, by applying the PA, the 2D FG beam was equivalent to a homogeneous beam and a small material perturbation. The high-precision results showed that the PA is adaptive and widely applicable. Finally, analyses of the results were conducted. It was found that the truncation of higher-order terms constituted the primary source of error and the inaccuracy of the PA, and also that the relative error of the frequency decreased with an increase in the power-law exponent. The PA-calculated frequencies were mostly slightly higher than those calculated using the GDQM.