# Is the Spring Rate for a Belleville Spring Linear?

Belleville disc springs are usually spec'd with a free height, loaded height, and loaded force. From this information you can calculate the average spring rate of a belleville spring. However, what I would like to know if if the force is actually linear (or nearly linear).

I assume it is not, but don't know what curve the force might follow. Is there an equation that can predict the force-curve (plot of force versus height) of a belleville spring given ID, OD, free height, and thickness?

• From anecdotal experience, the rate gradually rises as you compress it – Jonathan R Swift Dec 4 '19 at 15:46

The curve is not linear.

$$s =$$ actual deflection (mm)
$$h_o =$$ total possible deflection (mm)
$$F =$$ actual load (N)
$$F_c =$$ designed limit load (N)

Here you can see the original archive with the ploted graph from one industry (I usualy use this material for reference about these type of spring).

I don't know the exact formula. But the reasoning should be (this is my guess):

• From 0% Force to ~70% force should be affected mostly by the compression between the inner diameter and outer diameter.
• From ~70% Force to 100% should be affected mostly by bending.