Exact solution of the bending vibration problem of FGM beams with variation of material properties

In the theoretical part of this contribution we deal with deriving a fourth-order differential equation of the functionally graded material (FGM) beam deflection with variation of material properties. The variation of the effective elasticity modulus and effective mass density can be caused by variation of both the volume fraction and material properties of the FGM constituents in a one-layer beam or in the layers of a multilayered sandwich beam. Homogenization of the varying material properties of the beams is achieved by extended mixture rules and laminate theory. The linear beam theory has been used for establishing the equilibrium and kinematic equations of the FGM beam. The shear force deformation effect and the effect of consistent mass distribution and mass inertia moment have been taken into account too. Numerical experiments were performed to calculate the eigenfrequencies and corresponding eigenmodes of chosen one-layer beams and multilayered FGM sandwich beams. The solution results are discussed and compared with those obtained using a very fine mesh of two-dimensional solid elements of a commercial finite element model (FEM) code.