How can I visualize shear stress concentration?

Question

I have seen plenty of diagrams showing parallel axial stress lines to aid in understanding the stress concentration around corners and other things. But this doesn't seem to be a good visualization for shear stress concentration.

For example, in the below diagram, there is shear stress concentration around the fillet. But in my head, I visualize the part's parallel planes being twisted against each other, with the "lines" not crossing between the planes. In that case, how am I able to visualize the shear stress concentration occurring here? I am having trouble understanding the concept since I can't picture (a model of) what's really going on.

Answer 1

There's nothing stopping you from drawing "shear stress lines" whose density corresponds to the stress magnitude. These will have as much meaning as the "axial stress lines": useful for broad visualization and intuition but not really quantitatively predictive from a sketch.

See here for a previous discussion on the meaning of such "stress flow lines." I don't see any indications that these lines are relied upon for quantitative calculation.

Answer 2

This can be very simple when you look at an alternative sign convention for torsional moments.

Figure 1:Alternative sign convention for torque (double arrow and curved arrow) (source:ENGR2140 usu.instructure.com)

Both representations above represent the same loading condition on the beam. Of course the most usual is the later (with the curved arrow), which shows more readily the direction of the rotation on the plane.

The double curve sign notation is the vector of the torsional torque. That vector is normal to the plane of rotation. This is essentially an outcome of the right hand rule.

Under that perspective, you can appreciate that the following two representations are equal.

IF you use the second one you can visualise the shear stresses the same way you can for axial forces. i.e.

Answer 3

In the part, you have sketched it is easy: the torque will cause tangential shear stress in the two shafts. But because at the fillet the there is a large step up in the shear stress moving from left to the right along the shaft, as the graph shows the smaller the fillet radius, r the less ductility to transit gradually from high shear in smaller diameter shaft to the low shear in the larger shaft, the more stress concentration.

Ultimately the larger D section can act as a rigid wise and just twist the small D part to failure.

Imagine if there was no fillet that intersection would be so fragile it could easily break due to stress concentration as has been seen in many moving mechanical parts.