This paper is concerned with the Bresse system that arises in the modeling of arched beams. It is given by a system of three coupled wave equations that reduces to the well-known Timoshenko model when the arch curvature is zero. In a context of nonlinear elastic foundation, we establish the existence of smooth finite-dimensional global attractors, by adding dissipation mechanism in only one of its equations. In addition, we study the uniform boundedness of longtime dynamics with respect to the curvature parameter. These results have not been considered for partially damped semilinear Bresse or Timoshenko systems.