Torsion - Coupling connecting two shafts

[Tom McCurdy]

Homework Statement

There is a coupling attached two two shafts. The shafts have opposing and equal torques on them with a radius, r. Assuming the shear stress in the bolts used in the coupling is uniform, figure out how many bolts would be needed to make the max sheer stress in the shaft equal to the shear stress in the bolts.
Each bolt has a diameter (d)
There is a distance R between bolts.
(see picture attached)

Homework Equations

$$\tau_{max}=\frac{Tc}{J}$$
$$\frac{J}{c}=\frac{T}{\tau}$$
J for solid $$J= \frac{\pi}{2}r^4$$

The Attempt at a Solution

I tried to figure out the max sheer stress in the shaft which I got to be $$\tau_{max} = \frac{2T}{\pi r^3}$$

Then I have tried various things to get the sheer stress in the bolts.
I am not sure whether or not to consider the bolt a shaft and use d/2 to figure out sheer stress, or to figure out that the radius to the bolts would equal n(R+d)/(2 pi)

Basically I am not sure where to go from here.I know the answer should be $$\frac{2r^3}{Rd^2}$$

 

[Tom McCurdy]

Could someone maybe move this to the ME section?