Free vibration analysis of axially functionally graded beams using Fredholm integral equations

This article outlines the use of Fredholm integral equations (also known as Fredholm transformation approach) for free vibration analysis of non-uniform and stepped axially functionally graded (AFG) beams. The method is shown to be capable of dealing with beams of arbitrary variations of both cross section dimensions and material properties. Tabulated results of free vibration analysis for beams with various classical boundary conditions are presented. The governing equation with varying coefficients is transformed to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence, and obtained numerical results have high accuracy. Effects of axial force and shear deformation are investigated on the natural frequencies of AFG beams. The accuracy of obtained results is verified with those obtained in other available references. The present results are of benefit to optimum design of non-homogeneous tapered beam structures and graded beams of special polynomial non-homogeneity.