All Things Springs: Rate Formula – for those with Calipers

Many times, I have received samples instead of a blueprint for quoting purposes. It is common to slip out to the QA department and have the spring tested for rate.

However, there is another way to calculate the spring rate if you have a set of dial calipers handy. And… you don’t have to be a spring expert to use this method.

The rate formula is as shown.

R = Gd4/ 8NaD3

The components of the formula are basically the spring dimensions.

d = round wire diameter

Na = active coils (on a standard compression spring, the total coils minus 2)

D = mean diameter (outside diameter minus a wire size, or inside diameter plus a wire size)

“8” is a constant, a remnant from the numbers used in the original deflection formula—a nice round number

G = modulus of elasticity in torsion (for carbon steels and chrome alloys use 11,500,000 PSI—For 300 series stainless steel use 10,000,000 PSI)

So now, let us assume that you have a compression spring in front of you. You wield your calipers and yield the following measurements:

d = 0.250″

O.D. = 1.230″
(Therefore, D = 1.230″ n 0.250″ = 0.980″)

Total coils = 9.4
(Therefore, Na = 9.4 – 2 = 7.4)

Material is Chrome Silicon alloy
(Therefore, G = 11,500,000)

From all this data, we can now calculate the rate of the spring in pounds/inch.

Rate = 11,500,000 * 0.2504” / 8 * 7.4 * 0.9803

Rate = 44921 / 55.719

Rate = 806.222 pound for every inch of travel

Now that the spring rate is known, virtually any needed deflection for a given load can be determined, or a load for a given deflection can be found.

And, although any spring company can test the spring for rate in a common load tester, this calculation requires only a few measurements and a little keypad manipulation if you have calipers and a calculator.

What we do not know at this point is what is a safe deflection for this design. Any spring can be manufactured, but the stresses at a load height tells the story if the spring will take a set and require further processing. We discuss this step in the article “The Material Mix – What if I change to….”