This article investigates dynamic responses of symmetric and sigmoid FG Timoshenko beam rested on elastic foundation and subjected to moving mass. The gradation is described by symmetrical and sigmoidal law functions. The Hamilton principle is exploited to derive the system equations of motion. Finite element has been developed to convert the partial differential equations to ordinary equations in time domain, those will be solved incrementally by Newmark method. The model is verified. Parametric studies are presented to investigate influences of gradation type, gradation index, elastic foundation stiffnesses, inertia, and variable velocity of the moving mass on the dynamic response.