The Limits of Linear Shear Stress Or When Shafts Fail (Credit for many illustrations is given to McGraw Hill publishers and an array of internet search results)

Parallel Reading Chapter 4 Section 4.11

Shear Stress Also Has An Elastic Limit (Where our lovely linear relationships and the material break-down) Exact behavior varies by material type but we have descriptions of “classic” behaviors. This is Elastoplastic. Material behaves linearly to the limit and then just starts to deform Some materials have more of a transition.

But Shear Stress in a Shaft is not Equally Distributed The Outer Edge of a Shaft may go plastic while the center remains elastic. Things like this get us into doing integration to decide what the torque transmission is. We will avoid it in this class beyond giving you qualitative descriptions of its existance. Why would you be designing such a thing anyway? You end up getting permanent twist in the shaft.

We Have Three Key Strengths Compression Most materials have their best limiting values for resisting compression. Actually getting a picture of something that plain and simple crushed out is pretty tricky (Mining Engineers can create crushing right around an explosive charge in rock). Tension Tension limits are almost never as high as compression. Things like metals with “ductility” tend to have tension as their second highest limit. Brittle things like rocks and concrete tend to have tension limits that may be only 1/10 th of their compressive limits. (I wonder why Civil Engineers like to be metal rebar in concrete)? Shear For more ductile materials the shear limit tends to be the first to go. Of course the loading scenario seldom pushes the compression, tension, and shear limits at the same rate.

So What Happens at the Break Point? Failure of a ductile Shaft – it sheared in two. Pattern of a Shear failure

What If Our Shaft is a bit Brittle That doesn’t look like a shear failure! Wait a Minute! Hold on – Back UP! I thought we were turning this shaft around and creating shear. We never put anything but a shear load on it. How did this happen?

Back to Moore’s Circle τ σ First We Plot Our Shear Stress There is no need to offset the dot one way or the other since we are not introducing any compressive or tensile forces.

Now We Draw the Circle τ σ At 45 degrees to the shear stress we will have Compression of equal magnitude (Of course compression is a tough old bird so its going to shear off before it fails in compression)

But at 45 degrees in the Other Direction! σ τ Oh Yowsa – I think We’ve got a problem. If the material is brittle tensile strength will be Less than shear strength. Yes that is a Tensile failure at 45 degrees.

The Dark Side of the Hollow Shaft What happens when there is no material at the center to stay in the Elastic shear range?