Modal analysis of the FGM beams with effect of axial force under longitudinal variable elastic Winkler foundation

A differential equation of the homogenized functionally graded material (FGM) beam deflection and its solution will be presented which has been used in free vibration analysis of the beams with polynomial continuous longitudinal and transversal variation of material properties. The FGM beams are considered to be resting on longitudinal variable (Winkler) elastic foundation. Homogenization of the varying material properties and calculation of other parameters have been done by the multilayer method and the direct integration method. The 2nd order beam theory has been used for establishing the kinematic and kinetic equations. Not only the shear force deformation effect and the effect of consistent mass distribution and mass moment of inertia have been taken into account, but also the effect of axial forces have been considered. Numerical experiments were performed to calculate the eigenfrequencies and corresponding eigenmodes of chosen FGM beams. The solution results are discussed and compared with those obtained using a very fine mesh of solid finite elements. The effects of material properties and elastic foundation variation, the layering fineness, the average shear force correction factor and the axial forces have been evaluated and discussed.