# Static and Dynamic Analysis of Non-Pneumatic Tires Based on Experimental and Numerical Methods
## 3 Static Analysis Using FE Method
### FE Modeling  
Starting from the outcome of the laser scanning operation, the FE model of the Michelin X-Tweel UTV was developed. In the modeling phase, five elements were considered: spokes, inner ring, shear band, belt plies, and tread. The spokes modeling was a critical step, since they strongly affect both the radial behavior of the tire and the elements number. They can be modeled using solid elements starting from the geometry acquired with the laser scanner, realizing a 3D model. However, to minimize the elements number and thus, the computational time, a 2D-3D mixed model was studied. The mixed model is made of 4-node linear shell elements placed on the  
Figure 6: Contact patch analysis: (**a**) Influence of load; (**b**) influence of camber angle. mid-surface of the spokes for thinner zone and 8-node linear solid elements for the thicker one, which is the interface of the spokes with the inner ring and the shear band. The connection between 3D and 2D elements was realized with shell elements located inside the solid elements, with the nodes equivalenced. This transition zone is required to ensure the continuity of the component because solid elements nodes have 3 degrees of freedom while shell elements nodes have 6 degrees of freedom. The model of the spoke is shown in Figure 7a whilst a focus on the transition zone is shown in Figure 7b. In the mentioned figure, some solid elements were hidden to allow the view of the internal shell elements. The inner ring and the shear band were modeled with 8-node linear solid elements with the hybrid formulation or Hermann formulation, which consists of the separated integration of pressure and displacement field in order to prevent volumetric locking that may occur during compression of hyperelastic materials, which may cause convergence problems. To recreate the chemical linking existing between the top and bottom surfaces of the spokes and, respectively, the inner ring and shear band, and between the shear band and the tread, a tie surface to surface constraint was used. The belt plies were reproduced using a simplifying technique that consists of modeling a belt ply with surface elements [23]. Each surface element is integrated into the three-dimensional structure using the embedded elements technique while the mechanical behavior of the plies is simulated through the introduction of the rebar layers. The information about the location of the surfaces that reproduce the belt plies can be deduced from the cross-section of the shear band obtained in the sectioning step previously described. The adopted rebar properties were 1.5 mm\({}^{2}\) as resistant area of the single fiber with a distance between two bar equal to 1.33 mm and an orientation angle equal to 0\({}^{\circ}\). The orientation angle is defined as the angle between the direction of the fibers and the direction of the wheel travel. The tread is composed by the repetition, about the axial direction of the tire, of a pattern constituted by six blocks that repeats itself every 20\({}^{\circ}\). Each element was modeled with a 4-node linear tetrahedron with hybrid formulation. Since the tread elements were meshed only in the contact zone in order to not increase excessively the number of elements and because they do not work, outside the contact zone, during the contact patch simulation, a slightly smaller element size was used to better describe the contact between the tread and the road. Finally, the road was modeled with a rigid quadrilateral element with 4 nodes, placed in a one-point contact position with the tread. The road was assumed to be rigid and modeled as a discrete rigid body. In Figure 8a, the entire FE model of the NPT is presented,  
Figure 7: (**a**) Spoke 2D–3D mixed model with shell elements (green) and solid elements (blue), (**b**) focus on the transition zone, with solid elements hidden. whilst in Figure 8b, some elements are hidden to allow the view of the surfaces representing the belt plies embedded in the shear band.