Solid Shaft Diameter for an Allowable Torsion
The inverse of the shaft-torsion tile: the minimum SOLID-shaft diameter that keeps the max surface shear stress within an allowable, d = (16 T / (pi tau_allow))^(1/3). A 12,000 lb-in torque at an 8,000 psi allowable needs about a 1.97 in shaft (round up to stock). Sizes for stress only; check the angle of twist (shown with a length and G) against the service limit. Pure torsion, no keyway concentration. A design aid, not the engineer of record.
Formula and source
d = (16 T / (pi tau_allow))^(1/3) (solid shaft), the inverse of tau = 16 T / (pi d^3); J = pi d^4 / 32; theta = T L / (J G).
The standard circular-shaft torsion relations (mechanics of materials), solid shaft, solved for the diameter, by name.
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