An analytical solution for vibration in a functionally graded sandwich beam by using the refined zigzag theory

In this paper, an analytical solution for vibration in cantilevered functionally graded (FG) sandwich beams based on the refined zigzag theory (RZT) is presented. In RZT, the zigzag term in the kinematics of axial displacements is considered by introducing a zigzag function that is related to the shear modulus of each layer in the sandwich beam. The face layers with FGM are modeled by infinite sublayers, and the RZT formulations can be applied to the FG sandwich beam. The natural frequency equations, mode shapes, orthogonal relations, and frequency responses are presented in exact forms. In the modal analysis, the differences of the first two natural frequencies between RZT and FEM are less than 2.81%. For the third vibration mode, the shear deformations are more severe than for the first two modes and the differences of natural frequencies between RZT and FEM are slightly increased to 6.95%. For the mode shape and frequency response comparisons, the RZT results also agree well with the FEM results. It is concluded that the RZT is quite applicable to calculate the natural frequencies, mode shapes, and frequency responses of on FG sandwich beams.