In this paper we discuss the well-posedness and the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a Kelvin-Voigt string-beam system which models a suspension bridge. For this model we obtain the existence and uniqueness of solutions and the exponential stability of the homogeneous system, provided that the constant axial force is smaller than a critical value. For a general the existence of the regular global attractor is proved when the external loads are independent of time.