Shear Stress, Area of Projection

[dlacombe13]

I am learning about shearing stress, and I am a little confused about the area of projection mentioned in my book. When it introduces it, it shows a plate with a rivet through it. The plate is of thickness t, and the diameter of the rivet, d. It shows the plate and the rivet cut in half by a plane, and reveals the rectangular area exposed as the rivet is cut, and claims that A = dt.

I am trying a practice problem where the same type of thing is going on, and asks me to find the shearing stress between the plate and the shaft. However, the solution states that A = pi*d*t. In this case, is it cutting straight through the shaft, or is it the area of half of the hole the shaft creates? Basically, why is pi included here?

 

[BvU]

It's the area between the plate and the shaft: the area of a cylinder with height t and diameter d

 

[Kaly]

It is the area on the plate in the shape of a through hole. You will have to imagine it without rivet in it.

 

[JBA]

If possible, post the figure and complete text for the practice problem.

 

[dlacombe13]

For some reason I was thinking of bearing stress, when it was actually asking for shearing stress. It just so happens that the area must include d and t. Either way, I am starting to understand this a little more. It is asking for the stress between the plate and the shaft, which is the area of the hole that touches the collar. My confusion now is about how to picture where this will shear. Is the shaft physically connected to the collar, or does it rotate within it? In either case, how exactly would this piece break under this shear stress? I have included a rough sketch of what my book has:

 

[JBA]

Given the limited information apparently limited provided to you for the practice problem the simplest assumption is to treat it as a one piece shaft with a collar section; in which case the shear area for the longitudinal joint between the shaft and it collar is, as described A = pi d t. (Given the longitudinal direction of the force, no rotation will occur).

The actual failure mode for this type of piece is a bit complex because it depends upon the thickness of the collar and at what diameter the collar is restrained and whether the shaft loading pushing force applied to the top end of the shaft or as a pulling force on the the bottom of the shaft.
As an example, if the shaft load is applied from the top and the support for the collar is directly at the diameter of the shaft , as though the shaft is passing through a hole, then the failure would be a direct shearing though the joint at the diameter of the shaft, which is the case with a bolt and nut connection. Alternatively, if the collar is thin and/or the collar support is at its outer diameter, then the collar is likely to deform downward in a conical manner, which will result in a combined radial tensile stress at the bottom face joint of the collar and shaft and direct shearing stress thru the collar at the diameter of the shaft.

 

[mle]

Is direction of force F correct? I don't see any shear for the rivet in that case. The only shear is the so called punching shear to the plate due the pressure of the rivet head but i don't think that's the case here. Is it a single plate or there are two of them connected with the rivet?