Vibration Analysis of Thick Functionally Graded Beam under Axial Load Based on Two-Dimensional Elasticity Theory and Generalized Differential Quadrature

In this paper, vibration analysis of thick functionally graded beam with simply supported boundary condition under constant axial load is studied. The beam has a uniform cross-sectional area and the mechanical properties of the fungtionally graded beam are assumed to be vary through the thickness of the beam. Fundamental relations, the equilibrium and stability equations based on the displacement components are derived using the two-dimensional elasticity theory and hamilton's principle. Generalized differential quadrature (GDQ) method is used to solve the system of coupled differential equations at equilibrium and moving condition. In this paper, the influences of axial loads, dimensionless geometric parameter, functionally graded index and ratio of thickness to length on the vibration of beam is presented. To study the accuracy of the present analysis, a compression is carried out between the present results and published results and also results obtained from ABAQUSE program. Results showed that the generalized differential quadrature method is quite good. Based on the results obtained by increasing the volume fraction of fibers in the functional graded beam, the natural frequency of the beam increases and for high volume fraction, it is not possible to see much change in the natural frequency. Also, by increasing the ratio of thickness to length in the absence of the critical load the natural frequency decreased.