Circular Shaft Torsion and Power

Method & assumptions

1. Uniform circular shaft with concentric bore, homogeneous isotropic material, constant torque and linear elastic small-strain response. Enter diameters, not radii. Inner diameter zero selects a solid shaft.
2. J = π(Do⁴ − Di⁴)/32; τmax = |T|Do/(2J). J is a second moment of area, not a mass moment of inertia.
3. Twist θ = TL/(GJ), stiffness = GJ/L, γmax = τmax/G, elastic energy = Tθ/2. θ is relative end rotation, not the total rotation of the running shaft.
4. Angular speed ω = 2πn/60 and mechanical power P = Tω. Use the same positive axis for torque and speed. Opposite signs give negative power; reversing torque reverses twist but preserves stress magnitude and elastic energy.
5. Does not assess keyways, stress concentrations, yielding, combined bending, fatigue, buckling, critical speed, bearings or losses. A noncircular section cannot use this J as its torsion constant. No safe torque or equipment rating is inferred.

Sources

• MIT OpenCourseWare, David Roylance: Shear and Torsion, equations 8 and 12–14 and strain energy ↗