Investigating of free vibration behavior of bidirectional FG beams resting on variable elastic foundation

In the present study, the free vibration of bidirectional functionally graded (FG) beams resting on variable elastic foundation are comprehensively investigated. The beam's behavior is modeled using 2D displacement field that contain undetermined integral terms and involves a reduced unknown functions. The material properties of the FG beam are assumed to be graded in both the thickness and longitudinal directions according to a power law. The beams are considered simply supported and resting on variable elastic foundation. The differential equation system governing the free vibration behavior of bidirectional beams is derived based on the Hamilton principle. The problem is then solved using the Navier solution for a simply supported beam. The accuracy of the used model can be noticed by comparing it with other solutions available in the literature where a good conformance was obtained. A detailed parametric study is conducted to explore the influences of material composition and variable elastic parameters on the vibration characteristics of the beams. The results reveal that the grading indexes in one or both directions as well as the parameters of the elastic foundation strongly impact the fundamental frequencies.