Vibration and buckling analysis of functionally graded beams using reproducing kernel particle method

This paper presents the vibration and buckling analysis of functionally graded beams with dierent boundary conditions, using the Reproducing Kernel Particle Method (RKPM). Vibration of simple Euler-Bernoulli beams using RKPM is already developed and reported in the literature. So far, the modeling of FGM beams using the theoretical method or the nite element technique has not evolved with accurate results for the power law form of FGM with a large power of \n" value. The accuracy of RKPM results is very good and is not sensitive to the n value. The system of equations of motion is derived using Lagrangeʹs method under the assumption of the Euler-Bernoulli beam theory. Boundary conditions of the beam are taken into account using Lagrange multipliers. It is assumed that material properties of the beam vary continuously in the thickness direction, according to the power-law form. RKPM is implemented to obtain the equation of motion and, consequently, natural frequencies and buckling loads of the FGM beam are evaluated. Results are veried for special cases reported in the literature. Considering the displacement of the neutral axis, buckling loads, with respect to length and material distribution, are evaluated. For special cases of homogenous beams, RKPM matches theoretical evaluation with less than one percent error.