In a three-part series of papers, a generalized finite element analysis scheme is developed to handle the steady and transient response of moving/rolling nonlinear viscoelastic structure. This paper considers the development of the moving/rolling element strategy, including the effects of large deformation kinematics and viscoelasticity modeled by fractional integrodifferential operators. To improve the solution strategy, a special hierarchical constraint procedure is developed for the case of steady rolling/translating as well as a transient scheme involving the use of a Grunwaldian representation of the fractional operator. In the second and third papers [R. Kennedy and J. Padovan, Comput. Struct. 27, 259–273 (1987) and Y. Nakajima and J. Padovan, Comput. Struct. 27, 275–286 (1987)], 3-D extensions are developed along with transient contact strategies enabling the handling of impacts with obstructions. Overall the various developments are benchmarked via comprehensive 2- and 3-D simulations. These are correlated with experimental data to define modeling capabilities.