A unilateral linearized contact problem in a linear elastic beam is considered (boundary value problem with Robin boundary condition). The small relative thickness is parameterized and an asymptotic approximation of the three dimensional displacement is constructed, using an asymptotic expansion with respect to the relative thickness.The main effort in the construction of the asymptotic approximation is the derivation of a recursive one dimensional (1D) limit system (as the relative thickness tends to zero) for the tension, bending and torsion components. A mechanical interpretation of the 1D system is provided and the asymptotic approximation error estimate is proven and the high precision of the approximation is illustrated with several numerical examples.