FIRST Team # 1164 - Project NEO

Paul Erdős

We’ve been examining the how and why parts fail in the last couple of articles. Right now, we’re just looking at the failures that occur in ductile materials in a situation of static loading. Last time, we explored this problem from a strain energy point of view and came to the conclusion that distortion energy - that is, strain energy associated with shear stress and a change in shape, rather than volume - is the primary culprit in this mechanism of failure. We came up with a fairly complex method of relating distortion energy to standard tensile yield strength that let us predict failure in a situation of complex loading.

But if we’re saying that shear stress is the cause of ductile failures in static loading, it seems like it would be easier to do this just by looking at shear stress in a part alone.

It is, in fact. It turns out that you can make a good guess at whether a ductile material in static loading will fail by comparing the maximum shear it experiences to the shear it experiences in pure tension at the point where it yields. For pure tension, the shear at this point is half of the tensile yield strength. So we’d be comparing the maximum shear stress in a part to this:

So how well does this work compared with the more complicated distortion energy approach? Well, take a look at this:

(Image from Machine Design: An Integrated Approach, 4th Ed., by Robert L. Norton, original source Mechanical Behavior of Materials, by N.E. Dowling.)

The axes of this graph are principal stresses, normalized to yield strengths. The ellipse is the failure envelope of the distortion energy theory - data points within it are considered safe according to this theory, and failures occur at its boundaries. The hexagon inscribed inside it is the failure envelope for the maximum shear theory. The data points show the points of failure for a number of materials.

You can clearly see that the distortion energy theory is pretty accurate - there’s a nice cluster of data points along its borders. The maximum shear theory is a little less accurate - the data isn’t a perfect match - but it’s also more conservative. For a ductile material in static loading, you’re not going to risk a failure by using maximum shear theory instead of distortion energy theory. If you want something really accurate, distortion energy theory is still your best bet, but if you need something quick and dirty, maximum shear needs less math.