Abstract: We studied the long-time dynamic behavior of solution to non-autonomous beam equation with  nonlocal structural damping by using the process theory in the time-dependent space. Firstly, we obtained the well-posedness of solution  by using Faedo-Galerkin approximation method. Secondly, the existence of pullback absorption set of the dynamical system  in the corresponding solution space was obtained by using energy estimation. Finally, we proved the existence of time-dependent pullback attractors by using the cocyclic technique and contraction function method.

Key words: structural damping, contraction function,  , time-dependent pullback attractor, beam equation