Worked Example

This page will work through a real life example to show how to apply the design and engineering equations we discussed in our introduction page.

Useful Equations

We begin from first principles:


And for our purposes here, Load is no different to Force.

The equations used for the calculations of Load, Deflection and Stress are derived from those for a simple beam and have the application of some corrective factors based on empirical measurements and experience.
These equations for loads & stresses, are not precise solutions, but they do provide very useful engineering estimates. As a measure of usefulness, they are more accurate and useful than any of the forecasting models Economists use to explain Recessions, but not nearly as accurate as Cosmologists employ to calculate the Hubble Constant.
Some skepticism should be expected if Forces are provided in Newtons with decimal places.
We can now derive various useful metrics, such as:

  1. Required Material Thickness - t, to support a Load:
  2. Deflection - s, expected from a Load:
  3. Induced stress - σ, created by a Load:
  4. Deflection - s, expected at stress threshold:

What should you use and when?


Single Turn 3 and 4 Wave Spring Washers, are the most common of the Wave Springs and have the widest range of dimensions which match the Single Row Deep Groove Ball Bearings designations.


The Single Turn 5, 6 and 7 Wave Springs with more contact points to distribute the load, provide a more balanced support than the lower wave washer springs and also can bear far higher loads, but are they always necessary?.

A Simple Worked Design Example

Our Task is to select and verify a wave washer which is needed to fit into an 80mm (3.15”) bore and over a 60mm (2.362”) shaft to support a load of between 500 and 550 Newtons (i.e. 112lbs to 124lbs). The Wave Washer Spring must accommodate at least a 1.8 mm (0.071”) deflection. Furthermore the application requires a steady load. The washer will operate in a benign environment where the ambient temperature will never exceed 30°C .


So, here we have a static application. Let uś assume a standard single wave washer spring will do the job to start off with.
Go to the menu, Our Products -> Wavy Washers -> Single Turn Wave Spring Sizes and see what there is that might cater for our needs.


You should find a 3 Wave Spring of dimension ID: 61.5mm x OD: 79.5mm x t:0.4mm and the Nominal Load stated as 30N at Working Height / HLoad of 3.0mm on Free Height of 6.5mm.
The diameter dimensions look OK, deflection is well over the minimum of 1.8mm, but the Design Load is way too low, can we make an easy change to get what we require? It looks like we will have to add a wave or two or change the material thickness.

Start by checking the material thickness t, if we changed it from 0.4mm to 0.6mm or 0.8mm, what would that do for us?

Using, the following equation, see if we get a reasonable material thickness, before considering adding waves:

We Substitute into the equation above, where:

Giving us:

and you should get:

Now verify that the initial Free Height, which was given as 6.5mm, will cater for this increase in material thickness. And with a Working Height of 3.2mm, this will give us an expected travel / deflection of 2.1mm. Perfectly adequate.

We better check that the stresses we induced do not lead to plastic deformation. The operating stress of a single turn wave spring should never exceed the minimum tensile strength of the base material from which it has been formed. Now with Wave Springs, unlike with Conical Disc Springs, we cannot create favourable Residual stresses within this profile, so we set our Maximum Stress at 75-80% of the base material's Tensile Strength. We are using Quench and Double Tempered Spring Steel, which at a Rockwell C hardness of 46-48, has as a Tensile Strength of 1650 MPa, giving us σmax of 1240 MPa.

So, at this threshold of stress, what deflection / compression would result?

Substitute into the equation above, where:

giving us:

Another approach might be just to check the induced stress at our Working Load value, using:

And you will confirm that at this Load, the induced stress is around 755 Mpa, which is perfectly acceptable.

It is a good operating principle to ensure that deflection/travel is maintained between 30% and 70% of Free Height and always make sure that there is adequate clearance between your constraining bore and the loaded Wave Spring. Remember that The diameter of a Wave Spring increases when compressed. The following formula provides its maximum achievable external diameter when compressed to its design/working load.

This gives us an increased D'e of 79.55mm, which gives us total clearance of 0.45mm, i.e. just over 0.2mm from with the edge of the Wave Spring and the 80mm bore, which is acceptable.

Let's estimate its Operational Life

Stress & Fatigue (just like with you and me!!) always catches up with and destroys Springs. The Fatigue Stress Ratio is used to estimate Fatigue life measured in Cycles. Any ratio over 0.7 or 70%, will provide excellent life (over 1 Million Cycles), anything below 0.4 or 40% and nothing more than 30 000 Cycles should be expected. For more on this, please contact Reliable Pressings, and we will gladly assist you.

Final Design Specifications