In this paper we scrutinize the asymptotic behavior of a nonlinear problem which models the vertical vibrations of a suspension bridge. The single-span road-bed is modeled as an extensible viscoelastic beam which is simply supported at the ends. It is suspended to a rigid and immovable frame by means of a distributed system of vertical one-sided elastic springs. A constant axial force p is applied at one end of the deck, and time-independent vertical loads are allowed. For this model we obtain original results, including the existence of a regular global attractor for all In spite of the extremely weak dissipation due to the convolution term, this result is achieved by exploiting the exponential decay of the memory kernel.