Free vibration analysis of carbon nanotube RC nanobeams with variational approaches

There is not enough mixed finite element method (MFEM) model developed for dynamic analysis of carbon nanotube reinforced (CNTRC) composite beams in the literature. In the present study, free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) nanobeams is carried out in the framework of variational formulations. The rule of mixture is employed to estimate the effective material properties of single-walled CNT reinforced nanobeams. Four kinds of CNT distribution of un-axially aligned reinforcement material are investigated in the through-thickness direction of the nanobeams. There are the uniform distribution (UD) and functionally graded distributions FG-O, FG-X and FG-Ʌ of CNTs in the thickness direction of the nanobeams (z axis direction) are assumed here for the analysis. The Hamilton