Large-amplitude vibrations of a clamped-free beam are considered. Reduced-order models (ROMs) are derived for this problem, within the framework of non-linear normal modes (NNMs), defined as invariant manifolds in phase space. The method of real normal form theory, which allows computation of all NNMs in a single operation, is used. A specific development enables to handle the non-linear inertia terms stemming from the large rotation beam model. The dynamics onto the manifold is derived up to order five. Non-linear mode shapes are exhibited, as well as frequency-amplitude relationships. Finally, the case of a harmonic base-excitation is considered.