This paper deals with an axially moving viscoelastic beam with a boundary non- linear term. Our purpose is to extend the class of the relaxation functions Z that guaranteeing a general deacy. For this, we consider the following general condition Z′() ≤ -(t)(Z(t)) where is an increasing and convex function and is a non- increasing function on the whole ℝ+,This type of functions was developed by Conti and Pata in 6 and improved by Kelleche and Feng in 11. The obtained result improves the previous ones12,13.