Correctly defining boundary conditions is of vital importance when using finite element analysis (FEA). As with any simulation or calculation, if the input is rubbish, the output will also be rubbish. For structural analysis, the important boundary conditions are the fixtures preventing movement of the structure and the external forces acting on it.

Dr. Jody Muelaner, PhD CEng MIMechE

For wheels fitted with pneumatic tires, defining the boundary conditions properly can be particularly challenging. The forces acting on the rim of the wheel are caused by air pressure within the tire and by the ground reaction force transferred through the tire. Defining these forces in a realistic way requires some thought and without experience of this type of problem, the most significant force could easily be overlooked. Luckily, considerable work has been done in this area and so we can apply standard methods to correctly define the boundary conditions. This means that it’s not necessary to include the tire in the FEA and it can be represented by a few simple boundary conditions, determined by a combination of analytical and empirical methods.

**Detailed models of tire-ground and tire-rim interactions**

Before we get into the practical methods for applying boundary conditions to a wheel, let’s take a moment to consider the alternative. If we don’t make any assumptions about how the tire will impart forces on the rim, we would need to first model how the tire makes contact with the ground. This might be idealized as a toroidal or cylindrical face of the tire making contact with a planar ground surface. This would respectively result in a point or line contact initially, with corresponding infinite force. The tire then deforms to produce a contact patch. This type of contact analysis always requires an iterative solution but is relatively straightforward when the contact is between two linear elastic solids. However, although the compression of the air within the tire is elastic, the casing of the tire will undergo large deformations and there are significant damping effects. It is this damping that often accounts for most of the rolling resistance when a wheel rolls over a firm surface. Modeling these effects requires non-linear material models, adding significantly to the complexity of the analysis. The complexity doesn’t end there as the tire is not made up of a homogeneous isotropic solid material. Tires have anisotropic textile casings and bead wires, encased in a rubber matrix. Modeling this type of composite structure becomes extremely complex. It is only when all of this has been simulated that the deformation into a contact patch and the transfer of force from the ground into the rim can be determined.

**Analysis of the forces acting on rim**

A typical wheel consists of a central hub with a disk that supports the rim. The rim has flanges at each side which prevent sideways movement of the tire and bead seats which hold the tire radially. It is through the bead seats that the ground reaction force is transferred. The parts of a rim are labeled in the photo below.

The part of a tire which contacts with the ground is the tread. The tread generally has an approximately cylindrical outer surface, although it may be toroidal, especially for tires used on bikes, which lean into corners. In common usage, tread may only refer to the outer textured surface which makes direct contact with the ground but the term is used here to refer to the entire thickness of the horizontal part of the tire. The sides of the tire, which extend vertically to meet the tread at each size is known as the sidewall. The inner circumference of each sidewall is referred to as the bead. The bead is supported radially by the bead seat of the rim and is contained axially by the rim flange. The bead may contain wire to help it resist the radial force caused by the air pressure within the tire. The parts of a tire are shown below.

The air within the tire exerts uniform pressure on all internal faces of the tire and rim, the inflation pressure, P. The simplest boundary condition for the wheel is where any surfaces of the rim not covered by the tire are exposed to this pressure. Where the inflation 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. Where the inflation pressure acts on the sidewall of the tire, it is resisted at the outer circumference of the sidewall by the tread and the inner circumference by the rim flange. The resulting reaction force, Fs, is an important boundary condition for the wheel. Note that in the below diagram, the force Fs is shown acting on the tire, the force acts on the wheel in the opposite direction.

To calculate the reaction force, *Fs*, we need to consider the axial component of the inflation pressure, and the area over which it acts. It acts over the area of the sidewall, projected onto the vertical plane, which is given by:

If we assume that the force on the sidewall is divided equally between the tread and the rim flange, the force is given by:

Ground reaction forces have three components: Radial force due to the vehicle’s weight, tangential force caused by acceleration and braking, and axial force caused by cornering. These forces are distributed according to a cosine function over a region of the bead seat and rim flange related to the tire’s contact patch. The range of this distributed load is given by an angle of loading, θ. There are no analytical methods to determine the loading angle. It depends on the shape of the rim, the shape and stiffness of the tire, the tire pressure, and the ground reaction force. The loading angle may be determined experimentally or a worst-case value may be used. To use the worst-case value, multiple simulations are carried out with different loading angles to determine how the angle affects the stress in the wheel. It can be assumed that it will not be smaller than the contact patch of the tire which can be easily observed. The ground reaction may create a cyclic loading in the wheel if the wheel contains periodic spokes. This is because the stress state when the ground reaction is centered on a single spoke is different from the stress state when the ground reaction is between two spokes. For every revolution of the wheel, each spoke will experience one cycle which should be taken into account for fatigue calculations.

There can be up to five forces acting on the rim, although axial and tangential reactions are not always present:

• **Inflation pressure, P**, acting uniformly on the internal faces of the rim, not in contact with the tire.

• **Sidewall pressure reaction, Fs**, acting on both rim flanges.

• **Radial 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.

Practical issues of applying boundary conditions in FEA software

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, the faces of the bead seats must be split at the extents of the load angles. It is also a good idea to cut the wheel in half so that symmetry can be used to simplify the model. It may also be useful to de-feature the model to further reduce meshing and solution times. For example, by removing external fillets and other small features that won’t affect the stress significantly. It may even be worth extracting mid surfaces and meshing using shell elements.

In addition to the forces applied to the rim, fixtures will be needed at the hub. This is probably best achieved using a frictionless support on the inner face in contact with the hub and bolt connectors at the holes.

In this example, an inflation pressure of 0.345 N/mm2 (50 psi) was simulated. With a bead seat radius of 163 mm and a tread inner radius of 268 mm this results in a sidewall gives a reaction force of 24,525 N. The ground reaction force is 2 kN and the sinusoidal distribution can be applied using a bearing load. Because of the symmetry in the model these forces are halved.

The final von Mises stress results, following H-adaptive meshing, are shown below. The greatest stress occurs at the radius between the bead seat and the rim flange. This is a result of the bending stress caused by the sidewall reaction force. The ground reaction force has surprisingly little effect on the stress field, only increasing the peak stress by 3%. This really highlights how important it is to fully consider how the air pressure is transferred through the tire.

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