Vibration Analysis of Functionally Graded Timoshenko Beams

The vibration behavior of a functionally graded Timoshenko beam is investigated by applying the transformed-section method. The material properties of a functionally graded (FG) beam are assumed to vary across the thickness according to a simple power law. The cross section of FG beam with two constituents is first transformed into an equivalent cross section of the material on the top. Then, the lateral and longitudinal vibration equations of a homogeneous Timoshenko beam are separately applied to the beam with the transformed section. The bending natural frequencies of FG beam are evaluated using the Chebyshev collocation method, and the longitudinal natural frequencies are also obtained from the known closed-form solutions. Some of the analytical results are compared with the existing numerical data to validate the present model accuracy. Good agreement has been observed between the analytical and numerical data. The effects of aspect ratio, volume fraction, and boundary conditions on the free-vibration behavior of FG beam are discussed. The present analytical solutions provide an insight to the effects of various parameters on the vibration behavior of the beam. They also serve as benchmarks for testing the vibration results obtained by other analytical or approximate methods.