Replacing a hollow shaft with a solid shaft

[stainton1]

1. The problem statement, all variables and given/known data

" determine the diameter of a replacement solid shaft manufactured from the same material and subjected to the same maximum shear stress and torque"

Ok so I previously calculated the following:

Maximum shear stress= 35.8 x 106
Torque= 450.117 x 103
Hollow shaft external and internal diameters= 400mm and 200mm

2. Relevant equations

T/J = shear stress/r

J= Pi x d4/32

r= D/2

3. The attempt at a solution

Basically I attemted it in a very long winded way which was found to be wrong when I went through it with my tutor. What happened was I went off an example we done in class which was similar but for replacing a solid shaft with a hollow one whereas I am wanting to replace a hollow shaft with a solid one. I've read the rules on here and I know its not a place to get your assignments done for you but I was hoping someone would be able to give me an explanation and point me in the right direction as my tutor isn't being to helpful. Cheers.
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

[nvn]

stainton1: Hint: Look up or write the formula for polar moment of inertia, J1, of a round tube. Set shear stress tau1 = tau2. Substitute.

[DannoXYZ]

Isn't there a difference in the equations for a thin versus thick-walled hollow shaft?

[nvn]

DannoXYZ: In the limit, yes, you can obtain a different, approximate formula for a thin tube. But why bother with an approximate formula, when you can instead use the exact formula. Besides, stainton1 does not have a thin tube.