When carrying out a stress analysis, it’s important that the boundary conditions are accurately modeled. For wheels that are fitted with pneumatic tires, it isn’t obvious what these loading conditions are.
Forces act on the rim of the wheel due to both the air pressure in the tire and the reaction force of the ground on the tire. The way that these forces are transferred through the tire into the rim have a significant impact on the stress in the wheel. Although it is possible to directly model the tire, this is generally unnecessary and will significantly increase the complexity of the model. There are, however, established analytical and empirical ways of simplifying the tire into a few boundary conditions that can be applied directly to the rim. The theory behind these is explained by Stearns et. al.
The Complexity of Modelling Tire-Rim Interaction
Firstly, let’s look at why we don’t want to model the actual tire using finite elements. There are a few reasons for this.
Firstly, as the toroidal shape of the tire makes contact with the planar ground surface, it must deform significantly to form a planar contact patch. This involves large deformations, which requires non-linear modeling. Secondly, a tire is not made up of a homogeneous isotropic solid material. Rather, a tire is a composite structure with a rubber matrix surrounding anisotropic textile casings and bead wires. Modeling all of this would require considerable pre-processing and solution times. Although tire interactions are modeled academically, it doesn’t make sense to do this type of work when designing a wheel.
Identifying the Forces Acting on the Rim
Before identifying the forces acting on the rim, the terminology used to refer to parts of the rim should be explained. The key parts of a rim and a tire are labeled below.
The tread is the part of the tire which contacts with the ground, the bead is a wire running around the edges of the tire which contact with the rim, and the sidewall is the vertical section of the tire connecting the bead to the tread.
The bead seats are the sections of the rim where the beads rest on the tire and vertical forces are transferred, and the rim flanges extend vertically to resist horizontal movement of the bead seat.
Forces act on the rim due to two primary sources: the air pressure within the tire, and the ground reaction forces. The air within the tire exerts a uniform pressure on all internal faces of the tire and rim; this is the inflation pressure, P. Where this pressure acts on the inside of the tread, it is contained by the tire casing and bead, causing internal hoop stresses in the tire but no reaction forces on the rim. However, where the inflation pressure acts on the side wall of the tire, it causes the beads to splay outwards. These sideways forces, Fs, are contained by the rim flange.
Calculating the force of the sidewall on the rim flange involves integrating the pressure over the area of the tire. The integration is quite simple because we’re only interested in the area of the sidewall projected onto the vertical plane. This area is given by π(r22 – r12). The area is multiplied by the pressure P to give the total force acting on each side wall of the tire. The force acting on the rim flange is half of this because the bottom of the side wall is constrained by the rim while the top of the side wall is constrained by the tread of the tire itself. Therefore, these forces are given by the equation:
The other forces acting on the rim result from the ground reaction forces. These may be classified as vertical forces supporting the weight of the vehicle, torque resulting from acceleration and braking forces, and axial forces caused by cornering. These forces are transferred through the bead seat and rim flange, but they are not constant over the circumference of the rim. Instead, these forces act over a section of the rim, related to the tires contact patch and stiffness and given by the angle of loading, θ. The forces are distributed according to a cosine function over this region.
The loading angle depends on the combination of tire and rim, the tire pressure and the ground reaction force. In practice, it is not possible to determine this angle analytically, and an empirical method must be used. One approach is to run the simulation with several different loading angles and observe how this affects the stress in the wheel. It may then be possible to use a worst-case value. Alternatively, the results can be compared with experimental measurements to determine the actual loading angle.
Due to the rotation of the wheel and the periodic nature of the spokes, the stress in the wheel will cycle between two extreme states. In one state, the vertical ground reaction will be directly centered over the spoke and in the other it will fall halfway between the spokes. For every revolution of the wheel, each spoke will experience one cycle which should be taken into account for fatigue calculations.
Additional ground reaction forces also occur when cornering, braking or accelerating. Cornering results in an axial force which is transferred through the flange. This can also be expected to be sinusoidally distributed and act over a similar loading angle to the vertical reaction force.
In summary, there are five forces acting on the rim:
- Inflation pressure, P, acting uniformly on the internal faces of the rim not in contact with the tire.
- Side wall pressure reaction, Fs, acting on both rim flanges.
- Vertical ground reaction, Fv, distributed sinusoidally over both bead seats.
- Axial cornering reaction, FA, distributed sinusoidally over one of the rim flanges depending on cornering direction.
- Tangential braking or cornering reaction, FT, acts over the same region of the bead seats as the vertical ground reaction and is also sinusoidally distributed, with the tangential load transfer related to the normal force.
Simulating the Tire Loads in SOLIDWORKS
Before attempting to apply the tire forces to the rim, a few changes should be made to the solid model to simplify the analysis. Firstly, split lines must be added to the bead seats, so that the vertical ground reaction can be applied over the load angle. It also makes sense to cut the wheel in half so that symmetry can be used to simplify the model. Further defeaturing and the creation of surfaces for shell elements may also be desirable.
Here you can see the simple sketch containing a single line used with the Split Line command to split the bead seats, followed by the split lines in the two bead seats.
A static stress analysis is given as an example here. The following fixtures were used:
- Symmetry fixture on the three cut surfaces. This simply constrains all nodes on the surface so that they are able to move tangential to the surface, but no normal motion is allowed.
- Roller/Slider fixture on the inner face in contact with the hub.
- Foundation Bolts through the bolt holes to ground.
Next, the air pressure was applied to the rim. First, a uniform 50 psi pressure to the internal faces of the rim which were not in contact with the tire.
Next the side wall reaction force was calculated, using the equation for Fs described above. The radius of the inner face of the tire tread is 268 mm and the bead seat radius is 163 mm, resulting in an area of 142,173 mm2. The inflation pressure of 50 psi is equal to 0.345 N/mm2 halving the force on the sidewall gives a reaction force of 24,525 N, a surprisingly large force. Because of the symmetry in the model, this force is halved again and then separately applied to each sidewall, setting the direction normal to a reference plane.
Finally, the ground reaction force is added. Because this is sinusoidally distributed, the easiest way to apply it is using a Bearing Load. Before creating the bearing load, a coordinate system must be created with its z-axis on the axis of rotation for the wheel and the x axis through the center of the distribution.
The model can now be meshed and solved. A coarse mesh is shown, with local refinement after adaptive meshing. Although the polynomial solid elements in SOLIDWORKS Simulation cope reasonably well with thin walled sections, there is still an argument for meshing regions of this model with shell elements to efficiently obtain an accurate solution.
The simulation shows that the maximum von Mises stress is 146 MPa, occurring on the inner radius of the rim flange. This stress is almost entirely caused by the sideways force on the rim flange as the sidewalls attempt to spread outwards as a result of the inflation pressure. In fact, suppressing the ground reaction force produces no visible change in the stress distribution and only reduces the maximum stress by 3%. This shows the critical importance of properly considering boundary conditions when setting up a simulation.
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