A new method for calculating rolling resistance is developed. This method includes a simplified structural model of a wheel in the form of a rigid cylinder with a thin viscoelastic rim. The wheel loading mode is considered in the absence of shearing (horizontal) force. In the entire contact area, it is assumed that the wheel is gripped with a non-deformable support surface. Deformation of the viscoelastic layer is analyzed on the basis of the Winkle model. In contrast to the previously known ones, the proposed method implies a description of the symmetry breaking in the distribution of contact pressure when a small torque is applied to the wheel, not exceeding the rolling resistance at rest. Contact pressure diagrams are obtained at different values of the torque applied to the wheel. It is shown that as the torque increases, the maximum value of the contact pressure increases, and the position of this maximum shifts in the direction of possible rolling. From the condition of violating the contact area dimensions at the beginning of rolling, an analytical expression is derived for the moment of rolling resistance at rest. The calculated dependence of this moment on the wheel draft is compared with known experimental data. When describing stationary free rolling, the asymmetry of the contact pressure corresponding to the beginning of rolling and the loss of energy during deformation of the viscoelastic material of the wheel rim are taken into account. The dependences of the coefficient of rolling resistance and wheel draft on the speed of the wheel center of mass at a fixed value of the vertical load are obtained. It is shown that taking into account the initial asymmetry of the contact pressure leads to a more intensive increase in the calculated estimates of the resistance coefficient and a less marked decrease in the wheel draft with an increase in rolling speed.