Vibration analysis of a functionally graded beam under a moving mass by using different beam theories

Vibration of a functionally graded (FG) simply-supported beam due to a moving mass has been investigated by using Euler–Bernoulli, Timoshenko and the third order shear deformation beam theories. The material properties of the beam vary continuously in the thickness direction according to the power-law form. The system of equations of motion is derived by using Lagrange’s equations. Trial functions denoting the transverse, the axial deflections and the rotation of the cross-sections of the beam are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. In this study, the effects of the shear deformation, various material distributions, velocity of the moving mass, the inertia, Coriolis and the centripetal effects of the moving mass on the dynamic displacements and the stresses of the beam are discussed in detail. To validate the present results, the dynamic deflections of the beam under a moving mass are compared with those of the existing literature and a comparison study for free vibration of an FG beam is performed. Good agreement is observed. The results show that the above-mentioned effects play a very important role on the dynamic responses of the beam and it is believed that new results are presented for dynamics of FG beams under moving loads which are of interest to the scientific and engineering community in the area of FGM structures.