I am partly in charge of our drive train, myself being included in the construction for 3 years (this being my fourth) but only my second year with design. In an interest to save time, we have opted to use the kit gear box as stage one of a cut down four speed gearbox converted to a two speed. We initially planned on using 3/8 shafts in the “2nd stage” two speed addition, however what crossed my mind was, "Will that be strong enough to put up with the riggers of a FIRST match. I did a little research, and I found a chart with recommended loads on shafts, and it recommended only 15 inch-pounds (it actually had it written lb.in.), with an output torque at 40 amps being 17.7 Newton-meters, I get about 156.7 inch pounds of force, and that puts me at a shaft size of 3/4 inch. Now I know the output shaft on the kit gear box is a large diameter, just a little larger than 5/8 (evil metric sizes), which the chart recommends only 125 inch pounds of torque. now, I don’t think that small difference will be a problem, in fact I’m sure the chart I’m referring to is using a safety factor, but my question is, 3/8 inch shaft, is it worth the weight saved, my own personal feelings are that the shafts may not hold up, and repairing an old 3/8 axle shaft on one of our older robots, with a weaker drive train leads me to believe that even if the distances are kept short, that I may be taking a bit of a risk. Or, my lack of sleep and coffee could be throwing my numbers off, and maybe I’m just paranoid, but I really like to have 100% confidence in the drive train, we’ve won matches just being able to drive around when we lost our arm functionality last year, so I maybe taking an “if all else fails” approach, but I would rather be able to rely on one of the more basic functions a robot should have.
With something like this it can be a mistake to make it too strong. I like to have a well defined point which will break first when things go wrong - in my case I typically use taper pins to lock a gear to the shaft, and when the drive locks up (chain loose is a typical cause) the pin shears. In the absence of such a feature, you may find your gear teeth in a loose pile 
Specifically, to address your concerns about the torsional strength of varying shaft sizes, I have found a couple of references that might help you. First, at the bottom of this page (http://www.ecs.umass.edu/mie/labs/mda/dlib/peter/statload.htm) is a calculator that will give you the torsional stress for a beam of any cross section. You will need to input the torque applied, the distance of the “fiber from the neutral axis” (the shaft radius), and the polar moment of inertia of the section (a formula/definition/discussion of “moment of inertia” for a circle can be found here: http://www.efunda.com/math/areas/MomentOfInertia.cfm?Moment=zc).
Then you need to know the strength of your material, and what type of safety factor you want to apply … and don’t forget to keep track of your units …
I am sure others can elaborate more …
Good luck!!
edit: Sorry … I did not realize that efunda site required membership, however I believe you can sign up for a limited free membership to access this information.
I think we have solved the problem, we plan on going to a stronger material, and a slightly larger shaft diameter. The new material we are planning on using is 4140 steel, and we think that it will give us the needed strength to allow us to keep the shafts small enough to use the gears we have, otherwise regearing would be necessary.
Thanks Dave, I’ll keep that thought in mind, I certainly do not want to start breaking the gear teeth, however I am after a very strong gearbox, I think the gears will hold up well, the gears are wide, I would only be concerned about chipping the teeth during shifting - crash box style.
Stu, both links really helped, thanks. I did not need to sign up for a membership to view the page, I’m not sure why, but I had no problems at all.
Again, thanks guys!