What you are doing is not even close to a realistic analysis of an automobile tire under inflation loading or under contact loading. In a real tire, the tire cords carry most of the load, and the rubber is there just to glue the cord plies together and to prevent the air from escaping from between the cords. Have you looked into the literature on structural analysis of automobile tires, such as Tire Science and Technology. Early analyses of tires used membrane models which accounted for the effects of the tire cords. Later analyses used bending models, and finally, more recent models used detailed finite element. All these took into account the composite nature of the structure.