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Torsion spring

Moment and bending stress at the installed (preload) and working angles, plus the fatigue safety factor for cycling between them. The wire is in bending, so this uses Young's modulus; it also tracks how far the coil tightens, which is what binds it on the arbor.

Helical torsion spring M = —

Wire diameter d mm

Mean coil diameter D mm

Active coils n_a turns

Preload angle θ₁ deg

Working angle θ₂ deg

Material

Surface finish

Engineering notes

Two operating points

Rate per turn k' = E·d⁴ / (10.8·D·n_a); moment M = k'·(θ/360) at preload (θ₁) and working (θ₂).
Bending stress σ = K_b · 32·M / (π·d³), K_b = (4C²−C−1)/(4C(C−1)) (inner fibre).
S_ut = A / d^m; static bending allowable ≈ 0.78·S_ut.
Wind-up D' = D·n_a/(n_a + θ/360) — leave arbor clearance.

Fatigue

Alternating / mean bending between θ₁ and θ₂; Goodman with bending endurance ≈ (peened 0.5 : as-wound 0.4)·S_ut and S_ut.
n_f = 1/(σ_a/S_e + σ_m/S_ut); n_f ≥ 1 ≈ infinite life. Always load in the direction that winds the coil tighter.

Result

Live