extension-spring

name	extension-spring
description	Extension spring design — initial tension, hook stress, rate calculation, extended length, fatigue, material selection, ASME tolerances.
metadata	{"priority":7,"promptSignals":{"phrases":["extension spring","tension spring","initial tension","hook stress"],"minScore":3}}

Extension Spring Design — Complete Skill

Geometry and Parameters

d = wire diameter [mm] D = mean coil diameter [mm] C = spring index = D/d (ideal: 4-12) N_a = active coils (all coils, since ends are fixed) L_free = free length = 2(hook inside radius) + (N_a + 1)d + 2d_hook [approx]

Initial Tension

Extension springs are wound with initial tension F_i (pre-load before extension begins) F_i must be overcome before spring deflects

Initial tension stress: τ_i = F_i × K_w / (πd³/8D) = 8 F_i D K_w / (πd³)

Typical τ_i: 4-45 MPa (depends on wire size and spring index)

Initial tension range (Wahl's recommended): τ_i = G×d / (8πD²N_a) × δ_0 where δ_0 is residual deflection due to winding

Rule of thumb: τ_i ≈ 0.05–0.3 × S_y (wire shear yield)

Spring Rate

k = Gd⁴ / (8D³N_a) [N/mm] — same formula as compression spring

Force-deflection: F = F_i + k×δ (δ = extension beyond free length)

Hook Stress

Hooks are stress concentration points — often fatigue critical

Hook bending stress (curved section at hook): σ_b = K_b × F × D / (d³/32 × π) = 16 F D K_b / (πd³)

Correction factor: K_b = (4C-1)/(4C-4) (similar to Wahl, ignoring 0.615/C)

Hook torsional stress: τ_hook = K_h × 8FD/(πd³) K_h = (4C²-C-1) / (4C(C-1))

Design rule: hook stress typically governs fatigue life; avoid sharp hooks

Allowable Stress

Same as compression spring — from Shigley wire tables: Music wire (A228): S_u = 2061/d^0.163 [MPa]

Static: τ_allow ≈ 0.45 S_u Fatigue: τ_allow ≈ 0.36 S_u (Zimmerli, 10⁷ cycles, unpeened) Hook bending: σ_allow ≈ 0.75 S_u static; 0.60 S_u fatigue

Extended Length

L_ext = L_free + δ_max Maximum extension: F_max / k + F_i/k → but must check stress

Deflection limit: δ ≤ δ_max such that τ ≤ τ_allow

Loop/Hook Types

Type	Stress Concentration	Notes
Standard (full) loop	High K_b	Simple to make
Half loop	Moderate	Reduced bending
Expanded loop	Lower	More room for attachment
Threaded insert	Lowest	Best for fatigue; higher cost

Fatigue Considerations

Hooks carry full bending + torsion — multiaxial state Goodman for shear: τ_a/τ_e + τ_m/S_su ≤ 1 Life improvement: machine hooks, remove sharp transitions, shot peen hook radius

Tolerances (ASME B27.1 / commercial)

Free length: ±1-2% (commercial); ±0.5% (precision) Load at specified length: ±10% (commercial); ±5% (precision) Initial tension: ±35% (commercial) Rate: ±10% (commercial)

Material Reference

Same as compression springs:

Material	G [GPa]	Max temp	Notes
Music wire (A228)	79	120°C	Best fatigue
Hard drawn (A227)	79	120°C	Static loads
302 SS (A313)	68	260°C	Corrosion
Phosphor bronze	44	95°C	Non-magnetic

Output

Provide: d [mm], D [mm], C, N_a, k [N/mm], F_i [N], hook type, τ_max [MPa] vs. τ_allow, σ_b,hook [MPa] vs. σ_allow, L_free [mm], fatigue safety factor.