In the present thesis, the dynamic behavior of functionally graded (FG) inclined beams with simply supported and clamped boundary conditions under moving mass with constant velocity is investigated based on First-order Shear Deformation Theory (FSDT). The effects of Coriolis and centrifugal accelerations caused by the moving mass, and the fraction force between the beam and moving mass assuming rolling and sliding simultaneously are considered in addition to the effects of the weight force and inertia acceleration of the moving mass. The material properties of the beam vary continuously and based on power law distribution along the thickness of the beam. The mass is considered at the top face of the beam instead of assuming it on the mid-plane of the beam. The effects of the moving mass are applied using the contact force method on the element that the mass is on it at any moment. In order to develop a general solution procedure, which can be employed for FG beams, the finite element method (FEM) is adopted by using seven degree of freedom elements that three of them are related to transverse displacement, two of them are related to axial displacement and two of them are related to rotation. After deriving the governing equations for an arbitrary element, assembling and imposing the geometrical boundary conditions, the corresponding equations of motion are integrated numerically in order to obtain the responses of the system in each time step by applying the average acceleration scheme from the Newmark's time integration family procedures. Firstly, the numerical results for the free vibration and the dynamic responses of the beam under a moving load or mass in the limit cases are presented in order to check the accuracy and convergence of the present method and formulation. Then, parametric study is carried out for a vast range of velocities and material indices, and the effects of material indices, Coriolis and centrifugal acceleration, considering the moving mass