We perform an asymptotic static analysis on a simple two degrees of freedom structure, made up of a highly stiff bar and two springs, which shows two different linear behaviours and different finite stiffness depending on the magnitude of an applied static force. Both regimes as well as the non-linear transition between them occur in the framework of small forces and displacements. The problem gives physical insight to the relations between linearity and small displacement hypothesis and it is suitable for a first course in asymptotic analysis to students with basic skills in structural or rational mechanics.