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Tutorial: Nonlinear Structural Analysis of a Wheel

This article provides a step-by-step tutorial for the nonlinear structural analysis of a wheel. The objective of this simulation is to analyze the deformation and stress distribution across the wheel in operation mode, including nonlinear phenomena such as hyperelastic material, physical contact, and varying load.

von mises stress wheel and deformation wheel
Figure 1: Von Mises stress (left) and displacement (right) on the wheel

This tutorial teaches how to:

  • Set up and run a nonlinear structural analysis.
  • Assign boundary conditions, materials, and other models to the simulation.
  • Mesh the geometry with SimScale’s standard meshing algorithm.
  • Explore the results using SimScale’s online post-processor.

The typical SimScale workflow will be followed:

  1. Prepare the CAD model for the simulation.
  2. Set up the simulation.
  3. Create the mesh.
  4. Run the simulation.
  5. Analyze the results.

Learn with the video!

The following tutorial is also available in a video format with all steps described in equal details. Experience this interactive way of learning and let us know your thoughts in the comments section.

1. Prepare the CAD Model and Select Analysis Type

To start, you can import a copy of the project into your workbench to follow along the steps by clicking the button below:

The following picture shows what should be visible after importing the tutorial project. You can find the geometries Wheel and Wheel-Quarter and their 3D model displayed. You can interact with the model as with any CAD program.

simulation workbench nonlinear structural simulation wheel
Figure 2: CAD model displayed in the viewer

Did you know?

This simulation leverages the double symmetry of the model for two objectives:

  1. Save on computational resources.
  2. Be able to restrict the model from displacements on the (XY) plane directions, which otherwise would be very difficult to achieve.

It is always advisable to take advantage of symmetrical models in simulation. You can see the reduced model by selecting the Wheel-Quarter geometry from the tree on the left:

quarter model nonlinear structural simulation wheel
Figure 3: Quarter section of the wheel to be used for simulation

1.1 Create a Nonlinear Structural Analysis

Now the simulation setup can be started. Follow these steps to create a new simulation:

create simulation nonlinear structural simulation of a wheel
Figure 4: Create new simulation
  1. Select the Wheel-Quarter geometry element in the left panel.
  2. Click the ‘Create Simulation’ button in the pop-up window.

The simulation library window appears. Select the Static analysis type and click the blue ‘Create Simulation’ button at the bottom:

analysis types on SimScale, nonlinear structural simulation
Figure 5: Analysis types available on SimScale

As this structural simulation will include nonlinearities such as physical contact, hyperelastic materials, and varying load, the nonlinear analysis type must be activated. In the global settings panel that opens after creating the simulation, set the Nonlinear analysis toggle:

simulation parameters nonlinear analysis
Figure 6: Activate nonlinear analysis for the simulation.

Now a new tree will be automatically generated in the left panel with all the parameters and settings that are necessary to completely specify such an analysis. All parts that are completed are highlighted with a green check. Parts that need to be specified have a red circle, while the blue circle indicates an optional setting that does not need to be filled out. Check the picture for an example of this:

simulation tree nonlinear structural simulation wheel
Figure 7: Simulation tree

You can find more details about what characterizes a static analysis here.

2. Set Up the Nonlinear Structural Analysis

The analysis of the wheel will include the following operating conditions:

  1. The wheel rim is made out of polypropylene material.
  2. The wheel tyre is made out of rubber, with big deformations expected to occur.
  3. Maximum operating load of 1000 \(N\).
  4. Mean operating load of 500 \(N\).
  5. The tyre-ground pair must model the physical contact.

The following picture summarizes the nonlinear modelling conditions:

model nonlinearities overview for fea
Figure 8: Model nonlinearities overview

2.1 Contacts

Two contact conditions will be specifed:

  1. Interface between rim and tyre: A linear bonded contact is specified to get compliant deformation.
  2. Interface between ground and tyre: A nonlinear physical contact is specified to get realistic behavior.

The bonded contact (1) is automatically detected and created by SimScale. It can be found under the Contact items under the simulation tree as Bonded 1:

bonded contact nonlinear structural simulation wheel
Figure 9: Bonded contact between rim and tyre

Now, the physical contact (2) is created. Follow the instructions shown in the picture:

physical contact nonlinear simulation
Figure 10: Physical contact setup
  1. Click the ‘+’ icon next to Physical contacts
  2. Click on Master assignment box
  3. Select the corresponding face for the master assignment
  4. Click on Slave assignment box
  5. Select the corresponding face for the slave assignment

Click the check mark fixed value boundary condition checkbox to allow free movement to finish the setup.

The selection of faces to be assigned the master/slave is performed according to the key points explained in the following article:

2.2 Material Model

Next, add the material models from the material library. For this, we start by clicking the ‘+’ button next to the Materials node in the simulation tree. This opens a material library from which we select the adequate material and click on the ‘Apply’ button. This will then load the standard properties for the selected material.

materials library structural simulation  simscale
Figure 11: SimScale materials library

In the settings panel, the target body that will have the material properties applied is assigned. Accept the selection with the blue checkmark fixed value boundary condition checkbox to allow free movement .

Use these instructions to assign the following material to each body in the CAD model:

A. Ground

For the ground body, select the Concrete material from the material library, and assign the ground body (selected from the viewer or the Scene tree on the right), as shown in the picture:

ground material assignment nonlinear structural simulation wheel
Figure 12: Material assignment for the ground

B. Rim

For the wheel rim, select the PP (polypropylene) material from the material library, and assign the Rim body, as shown in the picture:

rim material assignment nonlinear analysis
Figure 13: Material assignment for the wheel rim

C. Tyre

For the wheel tyre, select the Rubber material from the material library, and assign the mount+tyre body. As this body is so soft and will undergo large deformations due to the load and the physical contact with the ground, a hyperelastic material model is specified. Change the Material behavior to Hyperelastic, and set up the parameters as shown in the picture:

rubber material assignment nonlinear structural simulation of a wheel
Figure 14: Material assignment for the wheel tyre

The input values are:

  • Hyperelastic model: Mooney-Rivlin
  • \(C_{10} = \) 7.36e6 \(Pa\)
  • \(C_{01} = \) 1.84e6 \(Pa\)
  • \(D_{1} = \) 1e-4 \(1/Pa\)
  • \(\rho = \) 930 \(kg/m^3\)

2.3 Boundary Conditions

Now, we define the boundary conditions. To create a boundary condition, click on the ‘+’ button next to the Boundary conditions node in the simulation tree, and select the required boundary condition type from the drop-down menu, as shown in Figure 15.

create select boundary condition in simscale
Figure 15: Selecting a boundary condition

A. Ground Fixed Support

Our first boundary condition will be to fix the ground. Select the Fixed support boundary condition from the boundary conditions drop-down menu. Assign the ground body by first activating Assign Volume from the top bar, as shown on figure 16. Give an appropriate name to the boundary condition, such as ‘Fixed support ground’.

assign volume nonlinear structural simulation wheel simscale
Figure 16: Activating Assign Volume for a boundary condition assignment involving the whole part
ground fixed support boundary condition nonlinear structural simulation wheel
Figure 17: Fixed support boundary condition for ground

B. Symmetry Plane Normal to X

The second boundary condition specifies the restricted normal displacement in the symmetry plane through which the model was cut. Select a Fixed value boundary condition, assign all faces belonging to the symmetry plane and set up the values as shown in the picture:

x symmetry boundary condition nonlinear structural simulation wheel
Figure 18: Symmetry X boundary condition

C. Symmetry Plane Normal to Y

The second boundary condition specifies the symmetry around the second plane through which the model was cut. Use the same procedure as before and set up the condition as shown in the picture, assigning the corresponding faces:

y symmetry boundary condition
Figure 19: Symmetry Y boundary condition

Fixed Value vs. Symmetry Plane Boundary Conditions

You might have noticed that there is a boundary condition that is called Symmetry Plane, and be wondering, why didn’t we use this instead of a fixed value? There are actually two reasons:
First is that the Symmetry Plane works by creating some additional equations, to be able to restrict the normal deformation in slanted planes. These equations can create a ‘Singular matrix’ error, if there are shared nodes between the contacts and the symmetry plane assignment. Second is for performance reasons. Due to the mentioned additional equations, the computing cost and time increases. Thus, when possible, the Fixed Value should always be preferred.

D. Load

For the operating load supported by the wheel, select a Force boundary condition. Assign the central cylindrical faces as shown in Figure 8. Then, click the varying load icon as shown in the picture:

load force boundary condition nonlinear
Figure 20: Load boundary condition

The Specify value window will appear for us to input the load curve. In this case, we will use a table input, with the setup as shown in the picture:

load curve table input nonlinear, structural simulation
Figure 21: Load curve specification
Table 1: Load curve table

Notice that our load force starts at zero, maxes out at \(t = 0.5\ s\) and then goes back down to the mean value at \(t = 1.0\ s\). The negative value indicates the force direction with respect to the Z axis. The load curve can be visualized in the following plot:

load history plot
Figure 22: Load history plot

Did you know?

This type of load curve allows us to study the history of the deformations, and the amplitude of the operating stresses, which are useful in fatigue analysis and other failure assesments.

Time in nonlinear static analysis

As this is a pseudo-static simulation, the time units are expressed in seconds, but actually do not have physical meaning. It just indicates the sequence of the events. There are no velocity or acceleration effects taken into account in the model, and all phenomena are assumed to happen slowly.

3. Numerics and Simulation Control

The Numerics section of the simulation tree can be left with the default values, as they are good enough to solve this case. In most cases, you will not have the need to mess with these parameters, due to the carefully tuned default parameters.

For Simulation Control you also do not need to change any parameter. As we assumed the simulation interval to be the default value of 0 to 1, we can proceed without needing to touch this section.

4. Mesh Setup

Select the mesh option and change the Fineness setting to 10 as shown in the figure below. You do not need to click the Generate button at this step, as the mesh will be computed as part of the simulation run.

mesh parameters in simscale
Figure 23: Mesh setup

5. Start the Simulation

The last thing to do for running this simulation is to create a run. The new run is created by clicking on the ‘+’ button next to Simulation Runs:

In the pop-up window, you can give a meaningful name to the run, then click the ‘Start’ button to start.

create simulation run nonlinear structural simulation wheel
Figure 24: New simulation run

The Job status item below the simulation tree updates the status of the run. Also, a Solver log is provided after a few seconds which shows the exact output of the actual computing algorithm. The simulation run should take a few minutes to be carried out. Once the simulation run is Finished, we can post-process the results.

6. Post-Processing

Once the simulation is finished, you can post-process the wheel load results on the platform in one of two ways:

access online post-processor simscale
Figure 24: Accessing the online post-processor
  1. Click the ‘Post-process results’ button in the simulation run dialog, or
  2. Click ‘Solution Fields’ under the simulation run tree item.

6.1 Visualizing Stress

In order to examine the stress results on the wheel, we should select the field assigned to the Parts Color as ‘Von Mises Stress’:

setting stress parts color online post-processor simscale
Figure 25: Selecting ‘Von Mises Stress’ as the option for Coloring

The parts are now colored according to the levels of stress, but we will perform some tweaking to better visualize the results:

tweaking stress plot online post-processor simscale
Figure 26: Tweaking the stress visualization to get comprehensive results
  1. Change the time step to ‘0.5 \(s\)’, which corresponds to the point of maximum loading.
  2. Change the stress units to ‘\(MPa\)’.
  3. Right-click on the legend bar and select ‘Use continuous scale’ from the contextual menu.

You should end up with the following visualization:

stress contour plot nonlinear analysis structural simulation of a wheel online post-processor simscale
Figure 27: Stress contour plot on the wheel at maximum load of 38.8 \(MPa\). Maximum stress can be seen around the rim and the contact patch between tire and ground.

Figure 27 shows the stress distribution on the model of the wheel at the time of maximum load. It can be seen that maximum stress levels of around 38.8 \(MPa\) occur at the rim radius element and at the contact patch between the tire and the ground.

6.2 Deformed Shape

  1. In the Parts Color panel, select the ‘Displacement Magnitude’ as Coloring.
  2. Right-click on the legend bar and select ‘Use continuous scale’ from the contextual menu.
displacement magnitude using continuous scale
Figure 28: Changing the Parts Color settings to visualiSe the displacement along the model, before adding the deformation of its shape.

In order to visualize the deformed shape of the wheel, create a Displacement plot:

create displacement plot online post-processor simscale
Figure 29: Create a Displacement filter to plot the deformation.

The deformed shape along with the coloring can then be inspected:

deformation detail online post-processor simscale
Figure 30: Detail of the wheel deformation under maximum load (time 0.5 \(s\)) at the contact patch by observing the displacement magnitude plot

The details of the deformation of the tire in the region of contact with the ground is well appreciated.

6.3 Animation of the Results

We can visualize the loading and unloading process and the evolution of the deformation by creating an Animation filter, the same way we added the Displacement filter (see Figure 29):

create results animation online post-processor simscale
Figure 31: Parameters for the Animation filter

The default parameters are enough for our purpose. Click the ‘Play’ button to start the animation and you should see something as shown:

deformation in rim and tyre of wheel online post-processor simscale
Animation 1: Deformation process of the wheel with the stress contour plot

Animation 1 shows the deformation process and the corresponding contour plot for the von Mises stress. The load curve effect and the points of maximum deformation and stress can be appreciated.

Finally, we can take a look at the load vs. deformation plot, created locally by measuring the displacement of a point at the center of the wheel in post-processing tool Paraview, and using the known force curve:

load vs. deformation plot
Figure 32: Load vs. deformation plot showing non-linear behavior of the model

Figure 32 shows the vertical displacement magnitude of a point at the load application face, versus the applied load magnitude. Here, the nonlinear behavior of the model can be clearly seen, with the curvature of the load vs displacement curve following the hyperelastic behavior on the loading and unloading steps.

If you want to learn more about SimScale’s online post-processor, you can have a look at our dedicated guide:

Congratulations, you finished the differential casing tutorial!


If you have questions or suggestions, please reach out either via the forum or contact us directly.

Last updated: July 20th, 2022