A steady state formulation of the rolling contact problem with friction that allows the analysis of free rolling, cornering, acceleration, and braking is presented. This formulation is applied to the finite element analysis of tires. A layered shell finite element with shear deformation that allows for large deflection and rotation is developed. In each layer, orthotropic Hookean materials or Mooney-Rivlin type materials with fiber reinforcements can be used and the incompressibility constraint is enforced with Lagrange multipliers. The contact constraint is enforced with a penalty and the friction term, instead of the usual Coulomb friction, is regularized by a differentiable form that makes it more suitable for numerical analysis. A numerical example for a typical tire is also given.