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Summary The quality of a wound roll is highly dependent upon the in-roll stress distribution, which is controlled by the operating parameters of the center-wind torque, nip and tension. With increasing demands for higher performance of paper winding systems in terms of higher winding speeds, wider web widths and larger wound roll diameters, it becomes of vital importance to determine and understand the optimum operating conditions of these machines. In this paper, a numerical formulation for estimating the in-roll stress of a wound roll is proposed with taking account of the effect of nonlinearity in web compressibility, air-entrainment and permeance. The in-roll stress is calculated as the superposition of the stress increments caused by each winding. The theory of elasto-hydrodynamic lubrication with the compressibility of air is introduced to evaluate the effect of air-entrainment at the roll-inlet. Permeance of air is newly incorporated into the winding model, which is expressed under the assumption that permeance is proportional to the pressure difference of both sides of a web. In order to assure the applicability of the proposed theory, winding tests were conducted using the dry-end section of the paper-making pilot machine. The numerical analysis and experimental observation show the significant effect of the air-entrainment and permeance upon the in-roll stress. [ABSTRACT FROM AUTHOR]