Expert: Paul Soderman - 9/20/2008

Question

Hi sir, I am currently an undergraduate doing a bachelor in aeroengineering.
I was doing my lab report when I stumbled upon a question on torsion test.
The experiment basically involve doing a torsion test on 2 materials
namely mild steel and cast iron both hollow and circular solid.
After which we plot a graph of applied torque
against the angle of twist and using the result
from the graph plot a
graph of stress vs strain.

I stumbled for a long time on this particular question and therefore would like to consult your expertise on it.
It goes like this:

For the same allowable stress, determine the ratio of T/w
of the maximum allowable torque T and the weight per unit length w of the hollow circular shaft with inner diameter d1 and outer diameter d2. By denoting (T/w)0 the value of this ratio computed for a solid circular shaft of the same outer diameter d2, express
the ratio T/w for the hollow circular shaft in terms of
(T/w)0 and d1/D2

How do i actually go about using which formula and what value to
answer the question?

Your reply would be very much appreciated.

Answer
Jason
You are stretching my brain to recover information I learned in college 45 years ago.  But it is coming back to me.  The equations you want can be found at:

http://physics.uwstout.edu/StatStr/Statics/Torsion/tors52.htm

http://physics.uwstout.edu/StatStr/Statics/Torsion/tors51.htm

See also the link to example 1.  One set of equations show the relationship between angle of twist and torque, shaft length, polar moment of inertia, and modulus of rigidity.  Another set shows the relationship between stress and torque, radius, and polar moment of inertia.  I don't see weight of shaft entering the problem except being implicit in the modulus of rigidity.

So, if a torque is applied to a solid and hollow shaft of equal diameter, and stress is held constant if I understand your problem, then the ratio of torques will be equal to the inverse ratio of polar moments of inertia.  Do you agree ?
Paul