Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions

This paper investigates free and forced vibration of bi-directional functionally graded (BDFG) Timoshenko beam under the action of a moving load. The material properties of the beam vary exponentially in both axial and thickness directions. The equations of motion are derived by means of Lagrange equations based on Timoshenko beam theory (TBT) as well as Euler-Bernoulli beam (EBBT) theory. In order to obtain free and forced vibration responses, the trial functions for axial, transverse deflections and rotation of the cross-sections are expressed in polynomial forms. Various boundary conditions considered in the study are satisfied by adding auxiliary functions to the displacement functions. The resulting time-dependent equations of motion are solved with the help of the implicit time integration Newmark-β method. At the same time, free vibration frequencies are presented. Some numerical results are provided in tabular form and in figures to examine the effects of the material gradation, moving load velocity, aspect ratio and different boundary conditions on the dynamic responses of BDFG beam.