In this paper we consider a dynamical system with boundary input and output describing the bending vibrations of a quasi-linear beam, where the nonlinearity comes from Hooke's law. First we derive an existence result for short-time solutions of the system of equations. Then we show that the structure of the boundary input and output forces the system to admit global solutions at least when the initial data and the boundary input are small in a certain sense. In particular, we prove that the norm of the state of the system decays exponentially if the input becomes zero after a finite time (the input being zero can be understood as a boundary feedback).