I use Roarks 6th Ed. This formula is just for the portion of the stem that is in the double-D (opposing flat sides) shape. The section below the double D is (probably) circular in section and needs to be evaluated separately. The below is the formula for non-circular section under pure torsion.
Formula as follows:
Table 20, Figure 9, page 350 of 6th Ed.
Θ = angle of twist
T = twisting moment (input torque)
L = length of the bar (length of the double-D)
K = non-circular polar moment of inertia
G = modulus of rigidity of the material
r = radius of the bar (the curved part of the double-D)
w = centerline of bar to edge of the flat
h = r - w
Θ = TL/KG where:
K = 2Cr4, where C varies with h/r as follows:
For two flat sides where 0≤(h/r)≤0.8:
C = 0.7854 - 0.4053(h/r) - 3.5810(h/r)2 + 5.2708(h/r)3 - 2.0772(h/r)4
Maximum stress, τ, at the surface is:
τ = (TB)/r3, where B varies with (h/r) as follows:
0≤(h/r)≤0.6:
B = 0.6366 + 2.5303(h/r) - 11.157(h/r)2 + 49.568(h/r)3 - 85.886(h/r)4 + 69.849(h/r)5
If you want to be sure your stem doesn't twist past an allowable point (pulling a disc off of the seat on a butterfly valve for example), you want to run the first calculation. If you're just concerned about the stress, run the second calculation. After you have the result of the second calculation, compare the result to the maximum allowable stress of your design code.
To calculate your MAST...lower your input torque, T in τ = (TB)/r3 until you have an acceptable stress.
