Non-linear Axisymmetric Analysis
- John Parsons
- Jun 28
- 4 min read
Updated: 7 days ago
Today we are going to explore the power of an axisymmetric analysis using ANSYS and a Belleville washer otherwise know as a conical spring washer.
Belleville washer is a type of spring that is shaped like a cone. Belleville washers can carry high loads, and have a non-linear stiffness. These unique non-linear spring 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 non-linear stiffness of these unique springs requires a power non-linear solver. In a Axisymmetric FE model a meshed 2-D geometry is mathematically representative of 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/node count 2-D model can accurately represent complex revolved 3-D geometry, allowing users to solve complex models expeditiously. Given that the Belleville washer is highly non-linear spring that is symmetric relative to its centerline, it represents an ideal geometry and analysis technique combination.
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 will represent a 3-D geometry that is axisymmetric with respect to the Y-Axis centerline.
In order to show you the true power of an Axisymmetric analysis we will be investigating a parallel stack of five Belleville washers. Theses stacked washers will add further complexity to FE-Model not only as a result of an increased element count but also the non-linear 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 4 Axisymmetric Elements thru 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.
As we are investigation five Belleville washers stacked in parallel we will need contacts surfaces to be created between all entities. We will use the ANSYS contact search algorithm to find all of the contacts. We will than change the contact "Type" to "Frictionless", and add Contact Stabilization Factor of 0.1 for all surfaces. Again this problem is highly non-linear and this stabilization will aid in model convergence.
Bellville washers may exhibit snap through buckling and general instability depending upon geometry and the thickness of the individual washers. Snap though buckling occurs when an object exhibits an abrupt change in stiffness during the components deformation. Non-linear solvers incrementally change the applied load on a model to converge on a solution, if the stiffness in a model changes abruptly it can make it very difficult for the solver to converge on a solution. These particular washers as modeled will not experience snap though buckling however they do exhibit a small quantity of instability thus we will need provide the ANSYS solver with the ability to decrease and increase the number of sub steps based upon model convergence during the solving process. I know you are interested in a real snap though buckling problem, and I will make this a topic for the future blog posts.
In Analysis Setting -> Step Control, set Minimum Substeps to 25, Initial Substeps to 25 and Maximum Substeps to 100,000. Large Deflection to "On" in the Solver Controls.
This model is set-up and ready to run, lets take a moment to review what we have created.
We have FE-model that is finely meshed with nearly perfect mesh quality
We have a highly non-linear model that will exhibit a small amount of snap though buckling.
We have a FE-model that has seven independent non-linear 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 Bellville washers as an axisymmetric analysis we only have 5,748 elements and we can solve this highly non-linear model in under four minutes.
Let look at the results:
If we pull the reaction force of the displacement constraint on the top plate and plot this force as a function of the deflection we can see the true non-linearity of these washers.
Reaction for 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 non-linear solver and it made short work of this complex non-linear problem. Please remember 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 I'll up load 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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