This research is concerned with the free vibration and buckling analysis of carbon nanotube-reinforced beams (CN-RBs) using a novel high-order shear deformation theory (HSDT). The current HSDT is modeled by a trigonometric function without a shear correction factor, and the displacement field has only four variables. Several different carbon nanotube distributions, including two new uneven CNT distributions (quadratic and exponential-cotangent), are considered. The mixture rule is applied to express the effective material properties of carbon nanotube-reinforced beams. The CNT beams are rested on two springs and a shear layer (Kerr foundation). Hamilton's principle is employed to derive the governing equations, which are then solved using the Navier technique. The current theory and several parameter effects are studied and validated in comparison to benchmark studies and theories.

The main purpose of this study is to enhance understanding of high-order shear theories, such as third order, sinusoidal, exponential, etc. In this context, our theory yields excellent results when compared to other theories. The difference between our theory and the exact solution is minimal, at just 0.054%, making it superior to other theories. The second part of the study focuses on investigating the distribution of carbon nanotubes to enhance understanding. This knowledge can assist panel manufacturers in determining the appropriate distribution shape. Our results indicate that the second distribution (exponential-cotangent) significantly influences the mechanical behavior, unlike the first distribution (quadratic).