# Bias Truck Tire Deformation Analysis with Finite Element Modeling
## 3 Results and Discussion  
The described tire computational model was used to simulate the process of mounting on the rim and of inflating an actual 11.00-20 bias truck tire with nominal internal pressure of \(p_{\text{n}}\) = 725 kPa. The tire load-bearing members were: double bead wires, six nylon body plies (1400 dex/3), and reinforcement strips of bead fillers. In the area of beads that came into contact with rims, there were reinforcement plies known as chafers. All tire plies were made of textile cord. The ends of the first two body plies were fixed with the innermost bead wire, while the next two were held by the outer bead wire, and the ends of the two remaining ones--the outermost--reached the bead nose (Figure 1). Due to a large number of thin body plies, neighboring plies with angles of \(\pm\theta\) were connected in the model, thus one finite element had the thickness of two joined plies. Effective material parameters for plies connected in such a way were calculated with a user subroutine. There were seven types of rubber compounds in the tire (cord layers rubber, tread, underread, sidewall, bead filler, bead strip, protective strip). Their mechanical properties were described with a neo-Hookean material model, and the material constants necessary for calculations were determined based on tensile tests of adequate samples, as described in [18]. The \(C_{10}\) parameter of the neo-Hookean material determined for these subsequent rubbers had the values (MPa) of 0.63, 0.603, 0.72, 0.484, 2.12, 1.23 and 1.83, respectively. A starting point for tire deformation analysis was the discretized area of tire profile in a mold, right after the vulcanization process (Figure 5, item 1). The axial displacements of the nodes located in the tire profile plane of symmetry were blocked, whereas contact was defined between the outermost bead nodes and the rim. The mesh created for half the tire profile consisted of 600 finite elements and 421 nodes. All axisymmetric, isoparametric arbitrary quadrilateral elements used bilinear interpolation functions. The mesh consisted of 372 Herrmann elements with the properties of incompressible rubber materials, 208 orthotropic elements, and 20 isotropic elements. The orthotropic elements were overlaid on incompressible elements to model the properties of cord-rubber composite. Due to discretization error, the finite element mesh affected the accuracy of the obtained calculation results. In this case, the basic mesh was tested by generating a more dense one and comparing the results obtained using both meshes. Dense mesh was obtained by subdividing each element of the basic mesh into four elements. The results of the comparison were as follows: 4.0% and 2.3% for radial and axial displacement of points located at the tire crown and on the sidewall, respectively, and 3.2% for the maximum value of the cord tensile force. In each case, the values obtained for the basic mesh were smaller than for the dense one. The basic mesh was considered sufficient for the analyses carried out here. When the rubber was being vulcanized, the tire polyamide cord was heated and showed a tendency to shrink, which was restrained in this state by a diaphragm pressing the tire to mold walls. Tensile forces were generated in cords. When the vulcanization process was finished and the tire was removed from the mold, the cords were able to shorten owing to the compliance of the surrounding rubber. The tire radically changed its shape (Figure 5, item 2), especially within its crown area, which moved radially towards tire rotation axis. It was assumed that, in the tire, the shrinkage of polyamide cords amounted to 5%. The shrunk tire profile was determined by applying self-equilibrated initial stresses in cords equivalent with this shrinkage. The calculation of cord shrinkage stress and its transformation from principal axes of orthotropy for cord-rubber ply into an \(x,y,\varphi\) coordinate system was performed with the user subroutine.