Tires are made of rubber composites which normally consist of several plies. Each ply consists of a thin layer of rubber matrix and a family of unidirectional steel or polyester cords for reinforcement. Such tire structures enable tires to endure severe road condition, but cause separation of reinforced layers due to stress concentration between plies. Since the separation of reinforced layers not only shortens the life of tires but results in an unexpected accident, the stress analysis of reinforced tires is indispensable for the tire design.
The stress analysis of reinforced tires is usually performed by finite element methods in which material properties of each element should be carefully considered and determined. The material properties of reinforced layers or other composite parts have been determined by several well-known equations for composite materials, such as Halpin-Tsai equations, Gough-Tangorra equations, and Akasaka-Hirano equations. However, such equations do not have the bending effect of reinforced cords during deformation under consideration. As a result, the stress analysis with such equations often fails in describing realistic stress state of tires since tires are constructed as a shell-like structure in which the bending effect is sometimes dominant.
In this paper, the equations for the material properties of each part of tires are developed for application to finite element methods. The equations take the bending effect of reinforced cords into consideration, laying emphasis on the bending effect during shear deformation of elements. A finite element formulation is then derived from the equilibrium eqations by the principle of virtual work.
An updated Lagrangian formulation is derived for incremental analysis with respect to the updated reference coordinates. Then, a contact formulation is added to the finite element formulation to calculate the stress state of tires in contact with flat rigid road under the load due to the self-weight of a vehicle.
The finite element code developed produces interesting results for the stress state in the cross section of inflated tires and tires in contact with road. The results with the pro-posed constitutive equations show the stress concentration around the reinforced layers, while others not. Various stress state of tires in con-tact with road versus different loads and reinforced belt angles in presented. Also, the corresponding contact pressure between tires and road is plotted. Finally, discussion and comment on the methods and result is given.