In this article, we will discuss two of the common failure theories for ductile materials: the maximum shear stress theory and the maximum distortion energy theory. We'll also discuss why the maximum normal stress theory should never be used with ductile materials.

When we have a situation with combined stress, we need to find an equivalent stress that can be compared with yield strength (usually from an axial tension test) to determine if a design has a proper safety factor. Failure theories allow for this to happen by correlating the stress state to a single value.

We'll limit the discussion to plane stress.

## Maximum Shear Stress Theory

The maximum shear stress (MSS) theory states that failure will occur when the maximum shear stress reaches 0.5 * yield strength. This is shown graphically below where any point outside the envelope suggests failure.

## Maximum Distortion Energy Theory

To use the maximum distortion energy theory (MDE), we need to calculate Von Mises stress. The Von Mises criterion is shown below where any point outside the envelope suggests failure.

## Comparison of MSS and MDE

MSS is more conservative than MDE as we can see when both failure envelopes are plotted together. For ductile materials, torsion tests show that the shear yield strength is about .57 of yield strength [1]. Since MSS predicts failure when maximum shear stress is .5 of yield strength, it is conservative. MDE is more lax than MSS. Also, MDE aligns better with empirical data [2].

## Why Maximum Normal Stress Theory Shouldn't be Used for Ductile Materials.

The maximum normal stress (MNS) theory states that the material will fail when the largest magnitude principal stress exceeds either the tensile or compressive strength. For ductile materials, we are usually concerned about yield strength. For this case, we can add the MNS failure envelope to the other theories.

The failure envelope is a square with +/- Sy in both principal stress directions. This assumes that the material's compressive yield strength is the same magnitude as its tensile yield strength.

In the first and third quadrants, the failure envelope for MSS and MNS are the same. The problem lies in the second and fourth quadrants. Let's look at an example in the second quadrant where the stress state is principal1 = Sy and principal2 = -Sy.

Since both principal stresses are equal to Sy, MNS suggests a safety factor of 1.

For MSS, maximum shear stress = (Sy - (-Sy))/2 = Sy. Therefore, effective stress = 2Sy and the safety factor is 0.5!

Likewise, for MDE the Von Mises stress is 1.73*Sy and the safety factor is 0.58.

This shows that MNS is completely unsuitable for ductile materials.

## References

[1] Riley, W., Zachary, L., *Introduction to Mechanics of Materials*, 1989, John Wiley & Sons, Inc., page 494.

[2] Budynas, Nisbett, *Shigley's Mechanical Engineering Design*, 8th Ed., 2006, McGraw-Hill, pages 222-223.

MEboost is an Excel® add-in that includes a combined stress tool to easily convert a combined stress state into effective stresses used by these failure theories.

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