Nonlinear Axisymmetric Analysis
- John Parsons
- Jun 28, 2025
- 4 min read
Updated: Jul 6, 2025
Today, we are going to explore the power of an axisymmetric analysis using ANSYS and a Belleville washer, otherwise known as a conical spring washer.
A Belleville washer is a type of spring that is shaped like a cone. Belleville washers can carry high loads and have nonlinear stiffness. These unique nonlinear springs can also be stacked in parallel or series to create a desired stiffness. All of these characteristics make Belleville washers ideal for use in dynamic loading environments.
Belleville washers can be difficult to investigate using the Finite Element Method, as the nonlinear stiffness of these unique springs requires a powerful nonlinear solver. In an axisymmetric FE model, a meshed 2-D geometry mathematically represents a more complex 3-D geometry that is revolved around the Y-axis centerline. The real beauty of an axisymmetric analysis is that a low-element-count, low-node-count 2-D model can accurately represent complex revolved 3-D geometry, allowing users to solve complex models expeditiously. Because the Belleville washer is a highly nonlinear spring that is symmetric relative to its centerline, it represents an ideal combination of geometry and analysis technique.
In ANSYS, an axisymmetric analysis is created as a 2-D finite element model that exists solely in the +X, +Y plane. This 2-D finite element model represents a 3-D geometry that is axisymmetric with respect to the Y-axis centerline.
To show you the true power of an axisymmetric analysis, we will investigate a parallel stack of five Belleville washers. These stacked washers add further complexity to the FE model, not only because of the increased element count but also because of the nonlinear contact that must be included between each washer.
For this example, I have created a bottom plate that will be constrained in the Y-axis and a top plate that will be displaced -0.1 inches. This simple geometry has been well meshed with four axisymmetric elements through the thickness of each washer.
When defining element behavior, remember to specify the behavior type as "Axisymmetric"; otherwise, you will not receive the desired output.
Because we are investigating five Belleville washers stacked in parallel, contact surfaces need to be created between all entities. We will use the ANSYS contact search algorithm to find all of the contacts. We will then change the contact "Type" to "Frictionless" and add a Contact Stabilization Factor of 0.1 for all surfaces. Again, this problem is highly nonlinear, and this stabilization will aid in model convergence.
Belleville washers may exhibit snap-through buckling and general instability, depending upon their geometry and thickness. Snap-through buckling occurs when an object exhibits an abrupt change in stiffness during the component's deformation. Nonlinear solvers incrementally change the applied load on a model to converge on a solution. If the stiffness in a model changes abruptly, it can be very difficult for the solver to converge on a solution. These particular washers, as modeled, will not experience snap-through buckling; however, they do exhibit a small amount of instability. We will therefore need to give the ANSYS solver the ability to decrease and increase the number of substeps based on model convergence during the solution process. I know you are interested in a real snap-through buckling problem, and I will make this a topic for a future blog post.
In Analysis Settings > Step Control, set Minimum Substeps to 25, Initial Substeps to 25, and Maximum Substeps to 100,000. Set Large Deflection to "On" in Solver Controls.
This model is set up and ready to run. Let's take a moment to review what we have created.
We have an FE model that is finely meshed with nearly perfect mesh quality.
We have a highly nonlinear model that will exhibit a small amount of snap-through buckling.
We have an FE model that has seven independent nonlinear contacts.
By all accounts, this should be an extremely difficult FE model to solve, and we have every right to be a little scared. However, because we chose to model these five Belleville washers using an axisymmetric analysis, we have only 5,748 elements and can solve this highly nonlinear model in under four minutes.
Let's look at the results:
If we extract the reaction force of the displacement constraint on the top plate and plot this force as a function of deflection, we can see the true nonlinearity of these washers.
Reaction Force as a Function of Top Plate Displacement
In summary, an axisymmetric analysis allows us to efficiently investigate any geometry and load case that is symmetric relative to a centerline—basically, any geometry and load that can be revolved around a line. ANSYS has an extremely powerful nonlinear solver, and it made short work of this complex nonlinear problem. Please remember that, when using ANSYS, all axisymmetric analyses need to be created in the +X, +Y plane. If you are interested in downloading this FE model or CAD geometry, please let me know, and I'll upload it to www.MESim.us.
Thank you for taking the time to read this article and don't hesitate to submit topics for future articles.
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