Treated herein is a class of large deflection beam problems where one end of the beam is being held while the other end portion is allowed to slide freely over a frictionless support fixed at a distance from this end. The elastic beam is subjected to a point load. This highly non-linear problem is solved using both the elliptic integral method and the shooting-optimization technique, thus providing independent checks on the solutions. This kind of problem features: (1) the possibility of two possible equilibrium solutions for a given load magnitude; and (2) a maximum (or critical) load and a maximum deformed arc-length for equilibrium.