Asymptotic behaviour of solutions to the nonlinear variational equations of the two basic types of cable stayed bridges is studied. The equations describe vertical and torsional oscillations of the center span, which is the part of road bed between pylons. Due to damping, amplitude and velocity of oscillations remain bounded if wind force is bounded as well. Moreover, the solution to the nonlinear equations converges to the solution of the steady state problem if wind drops. If the center span has certain aerodynamical properties and the initial state is near to the solution of the steady state problem, then the solution converges to the steady state solution even if the forces representing wind do not vanish.