Free Vibration Analysis of Functionally Graded Nanocomposite Beams on Elastic Foundation Using a Mesh-Free Method

Abstract The use of Carbon Nanotubes as the reinforcing constituent for polymer matrix composites in place of conventional fibers has led to the emergence of a new generation of advanced composite materials. In this paper, the free vibration of functionally graded nanocomposite beams on elastic foundations are studied. Three different types of Carbon Nanotubes distributions in the polymer matrix material are studied; Uniform distribution, symmetrically functionally graded distribution and unsymmetrically functionally graded distribution. The analysis is carried out by a mesh-free method using the two-dimensional theory of elasticity. The Moving Least Square shape functions are implemented to approximate the displacement field. Due to the absence of the Kronecker delta property of the shape functions, a transformation technique is used to apply the essential boundary conditions. After validation, the effects of different design parameters such as Carbon Nanotubes distribution, slenderness ratios, boundary conditions and foundation stiffness on the vibrational behavior of the structure are investigated. It can be seen that from a design perspective, the vibrational response of a FG structure may be controlled in two ways; one way is through changing the distribution of the CNT’s in the matrix material and the other way is by changing stiffness of the elastic foundation on which it is resting. A notable observation is that increasing the stiffness of the foundation will move the neutral axis away from the foundation support of the beam. The current approach can serve as a benchmark against which other semi-analytical and numerical methods based on classical beam theories can be compared.