Date of Award


Document Type


Degree Name

Master of Science (MS)

Legacy Department

Mechanical Engineering


Joseph, Paul F

Committee Member

Rhyne , Timothy B

Committee Member

Li , Gang


The pneumatic tire has been studied extensively since its invention in 1888. With the advent of high-powered computers and the use of the finite element method, the understanding of the tire's complex non-linear behavior has grown tremendously. However, one weakness of finite element models is that parameter studies are difficult and time consuming to perform. In contrast, an analytical model can quickly and easily perform extensive parameter studies. To the knowledge of the author, all existing analytical models of the tire make assumptions concerning the tire's behavior and construction that while useful for obtaining some of the first-order characteristics, are limited since they cannot relate tire behavior such as force-deflection to individual tire stiffnesses. As such, an adequate two-dimensional model of a pneumatic tire, including a finite element model, does not exist.
Therefore, an analytical, two-dimensional model for a pneumatic tire in static contact with a rigid surface is developed and presented. The case of a non-pneumatic tire can be obtained as a special case. The quasi-static investigation concentrates on finding the relationships between the tire's size and stiffness and its deformation under loading. A total of seven stiffness parameters are accounted for. The belt of the tire is modeled using curved beam theory, developed by Gasmi, et al. (2011), which accounts for bending (EI), shearing (GA), and extensional (EA) deformations. The sidewall of the tire is modeled as a bi-linear spring (KrT, KrC) with pre-tensioning (FP*) in the radial direction and a linear torsional spring (Kθ) in the circumferential direction. Application of virtual work leads to a set of 6th order differential equations for the displacements in the belt that must be solved in three distinct regions. The first region is the region where the radial deformation is greater than the radial deformation of the inflated and unloaded tire. The second region is the region where the radial deformation of the sidewall is less than the inflated position but not in contact with the ground, and the third region is defined to be the region in contact with the ground.
The length of the contact patch is represented by the angle enclosed by the edges of contact, and analytical expressions of stress resultants and displacements at the centroids of cross-sections are expressed in terms of this angle. In order to improve the accuracy of the model for large deformations, a special inflation pressure was calculated that allowed the most accurate solution to the linear model to be obtained by minimizing the circumferential force in the region of the largest rotation of the curved beam. This solution was then modified to account for the true inflation pressure. This two-step solution procedure was validated with a geometrically nonlinear finite element model of a non-pneumatic tire.
Force vs. deflection and force vs. counter deflection results were compared to experimental data for a pneumatic tire for a range of inflation pressures from zero to four bar. From this, it is concluded that while it is clearly possible to match the data, more work needs to be done to determine the best method for determining parameters that match a real tire. Extensive sensitivity analysis was performed on all the stiffness parameters.