First, stress stiffening occurs in membranes. Stress stiffening comes from the fact that the membrane is fixed on all sides which causes the membrane to change stiffness with deflection. This change in stiffness is due to the axial strain that needs to occur in the membrane to allow it to deflect. If the people at the ends of the string pull harder and put more tension on the string, then you will get a smaller deflection when you push with the same force as before. This requires a iterative solution and the problem is nonlinear.
Large deflections cause a similar geometric nonlinearity.
The stresses tend to be much higher because of the constraints involved and large axial strains are required to accomplish a small lateral displacement. I hope there is something there that helps. Any hint appreciated.