| 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.
