Belleville Springs (aka “Belleville Washers”, “Spring Washers”, or “Disk Springs”) are wonderfully useful mechanical devices, but with one minor drawback – their properties are difficult to calculate – until now! Almen and Laszlo were the first to develop a workable calculation method, but it can only loosely be described as “workable.” (This is the method used in the DIN 2092 standard.)
The most common approach to designing with disk springs is to use a manufacturers catalog (e.g. Schnorr, Key Bellevilles, McMaster-Carr) and choose from the available options using the published values for load at maximum deflection. This method is insufficient for many applications though because of the non-linear nature of the load-deflection curve, and it is precisely this non-linear behavior that is one of the most useful features of Belleville Springs!
The figure below describes the variety of spring characteristic curves that can be obtained by varying the ratio of formed height (ho) to material thickness (t). At a ratio of of ho/t=0.5 the spring curve is nearly linear but if the ratio is ~1.5, the spring will provide nearly it’s full load at 50% of its stroke and then continue to deliver approximately that same load for the second half of the stroke. (This property is sometimes used to retain bearings in the presence of large amounts of axial shaft play.)
The Excel workbook can be used to calculate the load-deflection curves of any Belleville Spring. Start by entering your desired spring measurements in the yellow-highlighted boxes and then select one of the material options from the dropdown menu in D12. The Excel workbook is unprotected so it’s easy to add, edit, or tweak the material options as well as include these calculations in a larger design workbook.
I should note that the terminology used in this tool follows the pattern set in the Schnorr Disk Spring handbook (which can be found here). There are a few different conventions in use, but the small screenshot images in the worksheet should be sufficient explanation.
If this calculator is useful to you, please send me a note or leave a comment below. I should also point out that this implementation of the calculation method is licensed under the Creative Commons 3.0 CC-BY-SA.
EDIT (2015_06_30): The spreadsheet has been updated with cell highlighting (for inputs and outputs) as well as comments regarding spreadsheet usage. Please let me know in the comments section if you have improvements to offer to the community at large.