The influence of the mechanical characteristics of the workpiece material and the shape of the contact surface of the deformation zone on the change in the yield strength along the gripping arc during cold strip rolling is considered. The error in calculating the equivalent yield strength when replacing the hardening curve with linear and power dependencies is estimated. An approximation of the hardening curve is proposed, the parameters of which depend on the exponent n of the hardening curve, the initial degree of deformation and the equivalent yield strength. Using this approximation and replacing the capture arc with a parabola, analytical solutions of the Karman equation are obtained, contact stresses and rolling force are determined. A comparative analysis of the known and obtained solutions is fulfilled. In particular, it has been established that replacing the capture arc by a chord at n = 1 reduces the design rolling force by 20% and more.