The focus of this paper is to study and test a Lie group multisymplectic integrator (Part 1) for the particular case of a geometrically exact beam. We exploit the multisymplectic character of the integrator to analyze the energy and momentum map conservations associated to the temporal and spatial discrete evolutions. This allows us to explore the temporal motion of the beam and the spatial evolution of the wave motion through the beam. We discuss the necessary conditions to obtain a stable displacement in space versus time.