The criterion for the appearance of a contact stress drop region at the entrance to the deformation center during cold rolling of the strip is established. Numerical integration of the Karman equation determines the effect of friction conditions, tension and hardening of the strip material on the size of the region. It is shown that neglecting the region of contact stress drop and the shape of the hardening curve in known solutions leads to an error in determining the rolling force reaching 20% or more. A new solution of the Karman equation is proposed, which significantly increases the accuracy of calculating contact stresses and rolling forces by taking into account the region of contact stress drop and the shape of the hardening curve of the strip material.