Helical Torsion Spring Rate and Torque

A torsion spring loads its wire in BENDING, so its rate uses the Young modulus E, not the shear modulus G: k = d^4 E/(10.8 D Na) in-lbf per turn, torque T = k(deg/360), bending stress sigma = Kb 32 T/(pi d^3) with the Wahl factor Kb = (4C^2 - C - 1)/(4C(C - 1)). A 0.1875 in music-wire, 1.5 in coil, 30-coil spring rates 75 in-lbf/turn (0.208 in-lbf/deg); wound 90 degrees it pushes back 18.8 in-lbf at 32 ksi, and a full turn quadruples both. Torque and stress scale linearly with wind-up, so wind only as far as the material allowable permits. The rate rise as the coil tightens, end-arm bending, and fatigue life are separate. A design aid; Machinery Handbook / Shigley and the spring maker govern.

k' = d^4 E/(10.8 D Na) in-lbf/turn; T = k'(deg/360); C = D/d; Kb = (4C^2 - C - 1)/(4C(C - 1)); sigma = Kb 32 T/(pi d^3). E by material (music wire 29.5e6, phosphor bronze 15.0e6 psi).

The helical torsion-spring rate k' = d^4 E/(10.8 D Na) and the Wahl round-wire bending correction (Shigley, Mechanical Engineering Design, Ch. 10; Machinery's Handbook), by name; the wire is in bending so the Young's modulus E is used.

The torsion-spring rate and Wahl factor are standard published spring-design results; the per-material Young's modulus is a published constant. Wire and coil dimensions, deflection, and material are the user's inputs.

Estimate. AHJ and licensed professional govern.

Field names used by the API: wire_diameter_in, mean_coil_diameter_in, active_coils, deflection_deg, material, rate_in_lb_per_turn, torque_in_lb, bending_stress_psi

• Rate k' = d^4 E/(10.8 D Na) in-lbf/turn, wire in bending (Young's modulus E)Shigley Ch. 10
• Bending stress sigma = Kb 32 T/(pi d^3), Kb the Wahl round-wire bending factorShigley / Machinery's Handbook
• Scope rate/torque/stress; coil tightening, end-arm bending, fatigue are separatescope of this tile