Shaft Torsional Shear Stress and Angle of Twist
Polar moment J = pi(d^4-di^4)/32, max shear tau = T r/J, and twist theta = T L/(J G) for a solid or hollow circular shaft. A 1.5 in steel shaft at 1,000 lb-ft over 24 in -> 18,100 psi, 2.9 deg; a 2 in shaft cuts both far (the d^3/d^4 leverage). A design aid, not the engineer of record.
Formula and source
J = pi(d^4 - di^4)/32; tau = T (d/2)/J (= 16T/(pi d^3) solid); theta = T L/(J G).
The standard circular-shaft torsion relations (mechanics of materials), by name.
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