In the present work, the behavior of a combined bridge-vehicle system in which, the bridge is modeled as a Timoshenko beam and the vehicle is considered as a half car model has been investigated using the finite element method. Responses of the beam and the vehicle model have been obtained and then validated with those reported in literature. The limitation of considering the vehicle as a quarter car model and also the effect of taking into account the rotatory inertia and shear deformation i.e. using the Timoshenko beam model has also been investigated.  The finite element formulation of the Timoshenko with the attached Tuned Mass Dampers (TMDs) has been derived. Then the general equation of motion of a Timoshenko beam element with the attached TMDs traversed by a moving half car model has been obtained by the combination of two previously derived finite element equations of motion; for the beam with attached TMDs and the beam under the moving vehicle.  Finally, a design optimization algorithm has been developed in which the derived finite element analysis module has been combined with the optimization procedure. The algorithm is based on the Sequential Programming Technique (SQP), to determine the optimum values of the parameters (frequency and damping ratios) of one TMD, for minimization of the maximum frequency response of the beam midspan under the moving vehicle.  The obtained results show that by adding an optimally tuned mass damper to the system a significant faster damping can be achieved