# Bias Truck Tire Deformation Analysis with Finite Element Modeling
## 2 Description of Bias Tire Model
It was assumed that bead wire material is isotropic and linear elastic with high elasticity limit. The reinforcement area was discretized with quadrilateral, four-noded axisymmetric elements, and the rubber area was discretized with incompressible Herrmann elements with the same number of corner nodes. The latter had the fifth node with one degree of freedom, defining the hydrostatic pressure mean value. The textile cord used for bias tire building is usually made of a polyamide (nylon). It is compliant in tension, and the curve of tensile stress-strain is non-linear (Figure 4, curve 1). The example graph shows mean strain values obtained experimentally for five samples of polyamide body ply cord with a measurement base length of 500 mm. Taking into account the non-linear characteristic of textile cord, the computational model accounted for the variability of its stiffness modulus in function of strain (Figure 4, curve 2). This curve was obtained by differentiation of the analytical form of curve 1. Figure 3: Cord parameter variations along body ply in a bias tire; 1—cord ends per decimeter (_epdm_), 2—cord angle \(\theta\). Figure 2: Changes in cords angle and density in a ply during tire forming stages: a—initial on the tire building drum, b—intermediate, c—final (\(\theta_{2}<\theta_{1}<\theta_{0}\)). The initial value of the stiffness modulus used in textile cord bias tire was 2232 MPa. Then, with the increase in strain, it dropped to 1272 MPa, and later it increased monotonically to almost 3565 MPa with a strain of 7.5%. Due to the negative coefficient of textile cord thermal expansion, when a tire is taken out of a mold, the cord is subject to shrinkage, and the tire changes its shape [21]. Therefore, before the mounting simulation, and with the use of a user subroutine, stress induced by cord shrinkage was applied, and a tire profile was obtained, which was shrunk compared with the form in the vulcanization mold. Once the tire was mounted on the rim in the simulation process, the tire loading was applied incrementally in the form of internal follower pressure, and subsequent system equilibriums were iteratively determined. A major issue in bias tire computational model creation is caused by variable cord density and variable cord angle in a ply. The literature describes the use of reverse engineering methods to determine those parameters that are based on actual tire scanning [22; 23; 24; 25]. Although this method requires laborious measurements, in the case of lack or inaccessibility of data for raw plies set on the tire building drum, it is the only way to obtain such parameters. For this paper, a more effective method of determining composite parameters in a vulcanized tire was used based on the reference to known and constant values of these parameters in ply set on the tire building drum (Table 1) and with the use of the so-called pantographing rule [2]. The cord-rubber ply stiffness matrix was then arranged with global coordinates \(x\), \(y\), \(\varphi\), in which the FEM equation system was created and resolved. For this purpose, adequate transformations of the stiffness matrix were performed for the ply described in the system of its principal axes of orthotropy [26]. The calculations were executed with user subroutine coupled to the code of MSC.Marc. Similarly, to determine the forces in ply cords with the use of the user subroutine, the cord-rubber composite strain was calculated in the direction of the reinforcement cords.