# All Things Springs: Stress Formula – The Final Factor of Design Integrity

In the article titled “Rate Formula – For Those With Calipers,” I gave the example of the math needed to determine the rate of a submitted spring sample. With a few physical dimensions, we were able to calculate the rate.

Our example had the following dimensions:

**d (wire size) = 0.250″**

**Mean diameter = 0.980″**

**Total coils = 9.4**** **

**Free length = 3.300″**

**Material = Chrome Silicon per ASTM A401**

**Modulus of Elasticity = 11,500,000 PSI**

**Rate = 806 #/in**

With this data, we can now determine stress. Stress is by far the most important parameter in spring design because it determines at what point a compression spring is overstressed, which can result in both safety and life issues.

The stress formula for a compression spring is:

**S = 8PD / ****p****d ^{3}**

With:

**8 = a constant derived from the originating deflection formulae**

**P = a force/load (stress is based on the force produced by a given deflection)**

**D = mean diameter**

**p****= 3.14159**

**d = wire size**

To determine the maximum stress the design may see, we will need the force at solid height. To do this, we will need to know, first, what the solid height is. If the spring is a standard design with each end being ground, the formula for solid height is:

**Solid Height (SH) = total coils * d or 9.4 * 0.250″ = 2.350″**

With solid height known, we can now calculate the total deflection of the spring.

**Total Deflection to Solid Height (Fsh) = FH – SH or 3.300″ – 2.350″ = 0.950″**

We can now use the total deflection to determine the load at solid height.

**Load at Solid Height (Psh) = Fsh * R = 0.950″ * 806 #/in = 765.7#**

*(This is a calculated load and does NOT represent the true solid load)*

Now we can calculate the stress at solid height.

**Stress at Solid Height (Ssh) = (8 * 765.7# * 0.980) / f3.14159 * 0.250(3))**

**or Ssh = 6003.1 / 0.0491 = 122.263 PSI**

Now that we know the value, what do we compare it to? Well, that would be the tensile strength of the given size of the given material. Per ASTM A401, the low range of Chrome Silicon wire is **250,000 PSI**. And the yield/set point for a compression spring made from Chrome Silicon is **50%**.

**Therefore, the set point for this design is 50% (0.5) * 250,000 or 125.000 PSI**

__Comparing the 125.000 PSI set point to a solid height stress of 122.263 PSI. this spring should not take a set when pressed to solid height since the solid height stress is LESS than that set point.__

This design is safely stressed and no presetting is needed. Also, the spring should not take a set in use.

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