The order-three, integro differential, non-linear equations of motion for an inextensional beam, derived by the authors in a previous publication, are analyzed to investigate non-linear resonant coupling effects between the non-planar free oscillation modes of a fixed-free beam with asymmetric support conditions. The transition curves that separate non-linear resonant and non-resonant types of motions for the beam, and the main characteristics of the non-linear motions, are determined analytically.