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

[tex] \tau_{max}=\frac{Tc}{J} [/tex]
[tex] \frac{J}{c}=\frac{T}{\tau} [/tex]
J for solid [tex] J= \frac{\pi}{2}r^4 [/tex]

The Attempt at a Solution

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

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 [tex] \frac{2r^3}{Rd^2} [/tex]

 

[Tom McCurdy]

Could someone maybe move this to the ME section?

 

[thomate1]

but I am not sure how to get there.

I would suggest approaching this problem by first considering the forces and moments acting on the system. The opposing torques on the shafts will create a shear force on the bolts in the coupling. This shear force will be distributed among the bolts, with each bolt carrying a portion of the total force.

To determine the number of bolts needed, we need to find the maximum shear stress in the bolts and compare it to the maximum shear stress in the shaft. This will ensure that the bolts are strong enough to handle the forces acting on them.

To find the maximum shear stress in the bolts, we can use the equation \tau_{max}=\frac{Tc}{J}, where T is the applied torque, c is the distance from the center of the bolt to the edge of the shaft, and J is the polar moment of inertia of the bolt. Since the bolts are assumed to have a uniform shear stress, we can use the equation \frac{J}{c}=\frac{T}{\tau} to solve for J in terms of T and \tau_{max}.

Next, we can use the equation J=\frac{\pi}{2}r^4 to find the polar moment of inertia for a solid bolt. Plugging this into the equation we solved for earlier, we can find the maximum shear stress in the bolts.

To find the maximum shear stress in the shaft, we can use the equation \tau_{max} = \frac{2T}{\pi r^3}, as you did in your attempt. Since the shear stress in the bolts should be equal to the shear stress in the shaft, we can set these two equations equal to each other and solve for the number of bolts needed.

Once you have solved for the number of bolts, it may be helpful to check your answer by considering the distribution of forces among the bolts and ensuring that each bolt is carrying a reasonable amount of force. Additionally, you may want to consider the effect of the distance R between bolts on the maximum shear stress in the bolts and adjust your answer accordingly.

Overall, it is important to approach this problem systematically and consider all the relevant equations and factors in order to arrive at the correct answer.