torsion-spring

name	torsion-spring
description	Torsion spring design — angular rate, bending stress (not torsion), coil diameter change on deflection, end types, fatigue, Wahl correction.
metadata	{"priority":7,"promptSignals":{"phrases":["torsion spring","torsional spring","angular spring rate","spring moment"],"minScore":3}}

Torsion Spring Design — Complete Skill

Key Distinction

Torsion springs carry bending stress (not torsional), despite the name. The wire bends as the spring winds/unwinds under applied moment.

Geometry and Parameters

d = wire diameter [mm] D = mean coil diameter [mm] C = spring index = D/d N_a = active coils θ = angular deflection [rad or degrees] Body length = N_a × d (coils touching free state typically; or with gaps)

Angular Spring Rate

Rate: k_θ = Ed⁴ / (10.8 DN_a) [N·mm/deg] or k_θ = Ed⁴ / (64DN_a) × (π/180) [N·mm/deg]

In SI consistent units: k_θ = Ed⁴ / (64DN_a) [N·mm/rad]

E = elastic modulus (steel: 207 GPa) Note: E not G — because wire bends, not twists

Moment: M = k_θ × θ [N·mm]

Stress — Wahl Correction for Curvature

Bending stress (inner fiber is critical):

σ = K_i × 32M / (πd³)

Inner surface correction (Wahl): K_i = (4C²-C-1) / (4C(C-1))

Outer surface correction: K_o = (4C²+C-1) / (4C(C+1))

K_i > K_o → inner fiber governs for static loading For fatigue: use K_i (stress concentration is real)

Coil Diameter Change Under Load

As torsion spring deflects, coil diameter changes: D_wound = D × N_a / (N_a + θ/360) (coiling direction, N_a in turns, θ in degrees) D_unwound = D × N_a / (N_a - θ/360)

Design implication: must check clearance to arbor or housing at maximum deflection

Body Length Under Load

L_loaded = (N_a × d) when coils are in contact Active length may be constrained by arbor + end tang geometry

End Types

Type	Stress	Notes
Straight torsion (tang)	High at bend	Simple; bending at angle
Hinged end (hook)	Moderate	Easier attachment
Short hook	Lower	Reduced stress raise
Straight offset	Variable	Depends on geometry

Tang stress: σ_tang = 6Fa/d³ × 32/π where a = tang arm length

Allowable Stress

Static: σ_allow = 1.0 S_y (spring set occurs above yield; typical allowable = 0.78 S_u) Per Shigley: S_y ≈ 0.78 S_u for cold drawn wire

Fatigue (inner fiber, N = 10⁷ cycles, music wire): σ_allow,fatigue ≈ 0.50 S_u (estimated endurance limit for bending) Use Goodman: σ_a/σ_e + σ_m/S_u ≤ 1

Wire S_u: same table as compression spring Music wire (A228): S_u = 2061/d^0.163 [MPa, d in mm]

Design Procedure

1. Set C (spring index), d, material
2. Calculate k_θ, N_a from required rate and envelope
3. Compute K_i, σ_max at maximum moment M_max
4. Check σ_max vs. allowable
5. Check coil diameter change vs. arbor clearance
6. Check body length and tang geometry
7. Calculate fatigue safety factor if dynamic

Fatigue Procedure

Mean moment M_m = (M_max + M_min)/2 Alternating M_a = (M_max - M_min)/2 σ_m = K_i × 32M_m/(πd³); σ_a = K_i × 32M_a/(πd³) Goodman: σ_a/σ_e + σ_m/S_u ≤ 1

Material Reference

Same as compression springs; however E governs:

Material	E [GPa]	G [GPa]
Music wire (A228)	207	79
Stainless 302	193	68
Phosphor bronze	103	44

Output

Provide: d [mm], D [mm], C, N_a, k_θ [N·mm/deg], σ_max [MPa] vs. σ_allow, coil diameter at max deflection [mm], arbor clearance, body length [mm], fatigue safety factor.