Re: How can you calculate the diameter of an dynamically loaded rod (axis)?

You are basically going to have to determine the total stress on the rod 
from all the loads that are on it, then compare this to the strength of 
the material that you're using for the rod to determine it's diameter.  
I'm not sure how much mechanical engineering education you've had, but 
I'll assume if you're trying this, that you've had a fair amount.  If you 
haven't, a good book on the subject is "Machine Design, An Integrated 
Approach" by Robert Norton.  It covers all sorts of mechanical design 
subjects including loading and material selection.

Anyway, first you have to determine the stress from the torsional load.  
The torsional stress will be greatest at the outer surface of the rod, 
given by:

t = T r / J

Where T is the torque (torsional load), r is the radius, and J is the 
polar area moment of inertia of the cross section.

I'm not sure what you mean for the second stress.  An axial force acts 
along the axis of the rod (the direction of the length).  For a tensile 
force, this is just the simple formula:

s = P / A

Where P is the loading force and A is the cross sectional area.  This 
stress will be uniform across the cross section of the rod.  If it's a 
compressive force, it's the same formula, as long as you don't cause 
buckling.  Buckling will depend on how the rod is supported, and it's best 
to refer to a textbook (like the one I mentioned above) to find all the 
proper formulas.

But, what you described sounds more like a tangential force, which would 
cause bending of the rod.  This is given by:

s = M c / I

Where M is the bending moment, c is the radius of the rod, and I is the 
area moment of inertia.  Again in  this case, the stress is at a maximum 
at the outer surface.  

Also, depending on exactly what type of loading you have, you may also 
have to compute the transverse shear stress from the force.  

If you do have a tangential force on a spinning rod, you'll probably also 
want to perform some fatigue calculations to determine if the rod will 
break over time from the fatigue (if you're bending it and turning it at 
the same time, this is will act like bending a paper clip back and forth 
and it will eventually fatigue and fracture).  

Once you've determined each individual stress, you'll have to find the 3D 
sum of the stresses.  This can also get tricky, and you may want to refer 
to a text book for help.  If this all sounds kind of complicated, that's 
because it is.  This is the kind of work that's usually covered in the 3rd 
or 4th year of a mechanical engineering program, so if you haven't had any 
education in this field, you may want to seek help from someone who has.  
Hope this help you out.  Good luck!

Your Mad Scientist,
Mike Scannell