Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load

In this paper, non-linear dynamic analysis of a functionally graded (FG) beam with pinned–pinned supports due to a moving harmonic load has been performed by using Timoshenko beam theory with the von-Kármán’s non-linear strain–displacement relationships. Material properties of the beam vary continuously in thickness direction according to a power-law form. The system of equations of motion is derived by using Lagrange’s equations. Trial functions denoting transverse, axial deflections and 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. The obtained non-linear equations of motion are solved with aid of Newmark-β method in conjunction with the direct iteration method. In this study, the effects of large deflection, material distribution, velocity of the moving load and excitation frequency on the beam displacements, bending moments and stresses have been examined in detail. Convergence and comparison studies are performed. Results indicate 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 non-linear dynamics of FG beams under moving loads which are of interest to the scientific and engineering community in the area of FGM structures.