# Finding maximum shearing stress in connected shafts

## Homework Statement

Shaft AB is fixed at point A, and has a flange attached at B. Shaft CD is fixed at D, and has a flange attached at C. The flanges at B and C are bolted together, but with bolts that are small enough to allow one shaft to rotate by 1.5$^{\circ}$ before the two begin to rotate together. Shaft AB has a diameter of 1.25 in. and a length of 2 ft., while shaft CD has a diameter of 1.5 in., and a length of 3 ft. The modulus of rigidity for both shafts is 11.2 $\times$ 10$^{6}$ psi. Find the maximum shearing stress in shafts AB and CD if a torque of 420 kip-ft (!) is applied to the flange at B.

To summarize:
L$_{AB}$ = 2 ft. = 24 in.
L$_{CD}$ = 3 ft. = 36 in.
$\phi_{AB}$ = 1.25 in.
$\phi_{CD}$ = 1.5 in.
$\theta_{C}$ = $\theta_{B}$ - 1.5$^{\circ}$
G = 11 $\times$ 10$^{6}$ psi.
T = 420 kip-ft = 5.04 $\times$ 10$^{6}$ lb-in.

## Homework Equations

$\tau_{max,AB}$ = $\frac{2T}{c^{3}\pi}$

## The Attempt at a Solution

For shaft AB, I just plugged in the numbers, since the torque was being applied directly to that member:

$\tau_{max,AB}$ = $\frac{2 * 5040000 psi}{0.625^{4}*\pi}$, which I won't even calculate, because I can tell you right now it isn't right.

That 420 kip-ft seems to me to be comically overlarge for an apparatus of these dimensions. It causes such a massive angle of twist between the two flanges that it would snap like a pretzel rod. That said, I had a plan of attack: Find the rotation of the flange at C by the relationship between the two angles, from there find the torsion in shaft CD, and use that to find the maximum shearing stress in the shaft. It's pretty much the same as gear problems where the angles are multiples of one another, except here the angles are related through a sum. With numbers this absurdly large, though, I have to wonder how I would even check the answer against the solutions manual! Am I at least on the right track in terms of my approach?