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Belleville Spring Washers
All belleville spring washers work like compression springs but offer some advantages. They can work in areas where space requirements offer very little range of motion. They can also handle high stresses over this short range of motion. Compression springs would fail in these cases. Not to mention, they would be impossible to design for most of these applications.
Common uses in these high load/short deflection areas would be as buffers, cushions, and stripper springs. These belleville spring washers produce a nonlinear load/deflection curve. This makes them useful where a constant thrust is needed in spite of wear. The performance of the washer is based on the ratio of height to thickness. Typical load/deflection curves for various ratios are shown in this chart- Load/Deflection Characteristics in Special Spring Washers. When this ratio is small the curve is almost a straight line and springs of this type will carry high loads at small deflections. Note how the chart shows with any ratio above 1.41 the load decreases after reaching a peak. Springs of this ratio will give a relatively constant or declining load. The design formulas for these parts have many variables. This makes it difficult for the designer. Use of the charts and formulas here will help in manipulating the variables. Therefore, requiring less time and cost with the spring manufacturers running samples for trial and error.
We will be dealing with loads and stresses at the flat position. You can determine intermediate loads using the previous chart-
Load/Deflection Characteristics in Special Spring Washers.
The graph we use for determining compressive stress in the flat position can be found on this page- Tensile-stress constants for special spring washers. The stress at intermediate stages is roughly in proportion to the deflection. When designing for critical applications, close tolerances, or other unusual cases always check the stresses according to the formulas before finalizing your design. 
In static applications stress is judged according to the formula for Sc. This formula is the one most commonly used because it gives the highest numerical value. It computes the compressive stress at the point marked Sc in the sketch at the right.
These spring washers need to be designed so they can be compressed to flat without taking a set. This can be accomplished in two ways-
These values should be reduced for plated parts, elevated operating temperatures, or for other specific material. For life cycle(fatigue) you need to consider the tensile stresses at the points marked ST1 and ST2 in the sketch. The higher value can occur at either the OD or the ID. It depends on the dimensions of the spring. You need to consider both values because of this.
Fatigue life depends on stress range as well as the maximum stress. It also is affected by material imperfections such as nicks or scratches on the edges. It can be improved by manufacturing techniques such as shot peening. The chart below shows a diagram for various tihcknesses and stress ranges based on the tensile stress ST1 and ST2 whichever is higher-
Stacked Belleville Spring Washers The deflection in a single belleville spring washer is small. By combining a number of washers and stacking them in series you can increase the deflection in proportion to the number of washers and still keep the load of each single washer the same. To increase the load, the washers can be stacked parallel. Theoretically, this increases the load in proportion to the number of washers. Please note, friction between the washers causes an apparent "hysteresis" in the load/deflection curve as shown in the graph.
A combination of increased load and increased deflection can be attained by stacking the belleville washers in a series-parallel arrangement. These stacked washers can be installed either on a rod or in a cylinder. For a dynamic application these guides need to be hardened to at least a Rockwell C48. Hardened end plates will prevent indenting adjacent parts. Take note: Allowance must be made for a slight decrease in ID and a slight increase in OD as the springs deflect. Now it's time to review a good design example. This page
Belleville Spring Design Techniques
has both a logic diagram showing the proper train of thought and a good design example that goes with it.
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This shows test curves made on laboratory equipment without vibration. The width of the hysteresis loop is in the order of 6% for each washer added in parallel. This may vary with lubrication and surface finish. It may also change during the life of the mechanism because of burnishing of adjacent surfaces. The test curve for five in parrallel also illustrates the extra deflection which occurs from imperfect nesting of the washers.
