A new three-dimensional tire ring on elastic foundation model in rolling contact has been proposed and verified. The motivation is to apply this new physical model to the three-dimensional forces’ estimation for intelligent tires, a current research challenge in the automotive industry. The core contribution is the analytical solution of the three-dimensional tire deformation, stress, and forces in complex rolling contact. This new physical model is universal and efficient in linking intelligent tire signals with the solution results, thereby providing a powerful theoretical tool for intelligent tire modeling. Furthermore, the results have been validated by sensitivity analysis of key structural parameters on tire performance. In summary, through modeling, verification, and application in case studies, it has been shown that the proposed methodology can be used for real, practical three-dimensional forces estimation, offering the potential to simplify the vehicle state observer and benefit autonomous vehicles.