spring-fatigue

name	spring-fatigue
description	Spring fatigue design — Wahl stress correction factor, mean and alternating shear stress, Goodman-Zimmerli diagram for springs, modified Goodman for helical springs, corrected endurance limit (Zimmerli data for spring wire), set removal (presetting), shot peening effect on fatigue, material fatigue properties (ASTM A228, chrome-silicon, chrome-vanadium wire), variable amplitude loading (Palmgren-Miner), relaxation under cyclic load, and SAE spring fatigue standards.
metadata	{"priority":7,"promptSignals":{"phrases":["spring fatigue","Goodman spring","Wahl factor","spring endurance","fatigue helical spring","spring stress"],"minScore":3}}

Spring Fatigue Design — Complete Skill

Spring Stress Analysis

Shear Stress in Helical Springs

Maximum shear stress at wire surface (inner coil): τ_max = K_W × (8 × F × D_m) / (π × D_w³) [MPa; inner fiber; critical for fatigue] [F = applied force [N]; D_m = mean coil diameter [mm]; D_w = wire diameter [mm]]

Wahl correction factor K_W: K_W = (4C - 1)/(4C - 4) + 0.615/C [Wahl 1963; accounts for curvature effect + direct shear] C = D_m / D_w [spring index; typical C = 4–12] For C = 6: K_W = (24-1)/(24-4) + 0.615/6 = 1.15 + 0.103 = 1.253

Helical index effect: Low C (< 4): K_W large → high stress concentration; coil difficult to manufacture High C (> 12): spring easily buckles; wire susceptible to tangling; K_W → 1.0 (less stress amplification) Typical optimal C = 6–9 for minimizing K_W while avoiding manufacturing issues

Alternative correction factor (Bergsträsser): K_B = (4C + 2)/(4C - 3) [slightly different; used in German DIN 2089; similar values to K_W for C > 4]

Torsional shear stress (Ks only, no curvature): K_S = 1 + 0.5/C [direct shear only; used in static analysis where curvature effect less important] τ_s = K_S × 8FD_m / (πD_w³) [use K_S for static; K_W for fatigue inner fiber]

Fatigue Stress Parameters

Mean and Alternating Stress

Variable load: F_min to F_max: τ_mean = K_W × 8 × F_mean × D_m / (π × D_w³) [F_mean = (F_max + F_min)/2] τ_alt = K_W × 8 × F_alt × D_m / (π × D_w³) [F_alt = (F_max - F_min)/2]

Note: K_W applied to both mean and alternating (conservative; inner fiber worst case for fatigue crack initiation) Some references apply K_W only to alternating (crack initiation driven by alternating stress)

Spring stress ratio: R = τ_min / τ_max [stress ratio; fully reversed = -1; zero-to-max = 0; typical spring R = 0.1–0.5] R = τ_min / τ_max = (K_W × 8 × F_min × D_m / πD_w³) / (K_W × 8 × F_max × D_m / πD_w³) = F_min / F_max

Goodman-Zimmerli Diagram for Springs

Zimmerli Fatigue Data

Zimmerli (1957) — universal spring wire fatigue data: For patented and cold-drawn spring wire (ASTM A228) and hardened spring wire, N = 10⁷ cycles: τ_alt = 310 MPa at any τ_mean (Zimmerli found τ_alt virtually independent of τ_mean for most spring wire) This is the Modified Goodman line for springs from Zimmerli data

SAE Modified Goodman for springs: τ_alt / τ_e + τ_mean / τ_u = 1 [standard Goodman form] For spring wire: τ_u_shear = 0.577 × S_u_tensile (von Mises); τ_e = shear endurance limit But Zimmerli data shows τ_alt virtually constant → horizontal line at τ_alt = 310 MPa

Goodman line (Zimmerli modified): Conservative: τ_alt_allow = 310 MPa (unpeened) or 465 MPa (shot peened) — constant regardless of τ_mean [For spring steel hardened wire: S_u ≈ 1,400–2,000 MPa depending on D_w] [τ_e_Zimmerli = 310 MPa is independent of τ_mean for typical spring steels]

Zimmerli endurance limits by wire type: ASTM A228 (music wire), A227 (hard drawn): τ_e = 310 MPa unpeened; 465 MPa peened Chrome-silicon (A401, SL, SM, DH): τ_e = 380 MPa unpeened; 540 MPa peened Chrome-vanadium (A232): τ_e = 380 MPa unpeened; 530 MPa peened Stainless 302 (A313): τ_e = 241 MPa unpeened; 310 MPa peened (lower than carbon steel)

Safety factor: SF_alt = τ_e / τ_alt_calculated ≥ 1.2 (minimum); recommend SF ≥ 1.5 for reliability

Design Example

Valve spring: D_w = 4 mm; D_m = 28 mm; F_min = 200 N; F_max = 500 N; ASTM A228 unpeened C = 28/4 = 7; K_W = (28-1)/(28-4) + 0.615/7 = 1.125 + 0.088 = 1.213 τ_mean = 1.213 × 8 × 350 × 28 / (π × 64) = 1.213 × 392.7 = 476 MPa τ_alt = 1.213 × 8 × 150 × 28 / (π × 64) = 1.213 × 168.3 = 204 MPa Compare: τ_alt = 204 MPa < τ_e = 310 MPa → SF = 310/204 = 1.52 ✓ (adequate) Check τ_max: τ_max = τ_mean + τ_alt = 476 + 204 = 680 MPa Material: S_u ≈ 1,650 MPa (D_w = 4 mm A228); S_sy = 0.45 × S_u = 742 MPa (yield in shear) Static: τ_max = 680 MPa < 742 MPa → SF_static = 742/680 = 1.09 (barely adequate — consider presetting)

Presetting (Set Removal)

Process and Effect

Presetting: compress spring beyond yield → deliberate partial plastic deformation → when released, residual compressive stress on inner coil surface Physical effect: inner coil has pre-compressive residual stress → reduces mean stress during operation

Procedure: Compress spring to 100% of available deflection (solid or near-solid); hold briefly; release Resulting: spring free length decreases by 0.5–2% (loss of length due to plastic set) Residual stress on inner fiber: τ_residual ≈ -(τ_yield - τ_operational_mean) [compressive; negative]

Effect on fatigue (Goodman): τ_mean_effective = τ_mean_applied - |τ_residual| → shifted left on Goodman diagram τ_alt_allow increases significantly (for same mean → more alternating allowable) Presetting can increase fatigue life 2–3× for springs operating at high mean stress

When to use presetting: τ_max / S_sy > 0.45 AND spring needs fatigue improvement → preset Not needed if mean stress is low (already generous fatigue margin)

Note: presetting changes free length → must design for post-set free length; account for loss

Shot Peening Effect

Shot peening: introduces compressive residual stress at inner coil surface (same mechanism as presetting but surface layer only) Improvement: τ_e increases from 310 to 465 MPa (A228) → 50% improvement in endurance limit Depth of compressive layer: 0.1–0.3 mm (sufficient for wire D_w ≥ 3 mm) Standard: SAE AMS 2430 or MIL-S-13165 for spring applications

Shot peening + presetting combined: maximum fatigue improvement; common for critical springs Sequence: shot peen first → then preset (presetting after peening preserves peened layer)

Variable Amplitude Fatigue (Palmgren-Miner)

For springs with variable load spectra: D = Σ (n_i / N_i) ≤ 1.0 [Miner's rule; n_i = cycles at τ_alt,i; N_i = life at τ_alt,i from S-N curve] S-N curve for spring wire (log-log): τ_alt × N^(1/b) = τ_e × C_0 For A228: b ≈ 9–12; τ_e = 310 MPa at N = 10⁷ cycles

Spectrum loading: Multi-level loading: heavy operating load (short duration) + light oscillation (long duration) Compute D for each level; sum to verify D < 1.0

Material Properties — Spring Wire

ASTM A228 (Music Wire) — most common: D_w = 1–6 mm: S_u = 2,170–1,650 MPa (decreases with wire diameter) G = 81.7 GPa; E = 206 GPa Price: highest; consistent quality; highest fatigue strength

ASTM A401 (Chrome-Silicon — Hard Draw): S_u = 1,700–1,950 MPa (D_w = 3–10 mm); better high-temperature performance than A228 T_max = 250°C (vs. 120°C for A228) Higher τ_e: 380 MPa unpeened

ASTM A232 (Chrome-Vanadium): S_u = 1,550–1,900 MPa; T_max = 230°C; impact resistance: better than chrome-silicon Automotive valve springs: chrome-silicon preferred (higher fatigue at operating temperature)

Stainless ASTM A313 (302/304/316): S_u = 1,100–1,700 MPa; corrosion resistant; lower τ_e (241 MPa) → use for corrosive environments only Nonmagnetic option: 316L for medical devices

Relaxation under Cyclic Loading

Stress relaxation: springs lose load over time at elevated temperature or under sustained stress τ_max → τ_max_final = τ_max × (1 - Δτ_relax%) Relaxation after 100 h at 100°C: A228: < 2%; Chrome-Si: < 1%; Stainless: 3–5%

Allowable maximum stress for relaxation: τ_max < 0.45 × S_y at operating temperature (creep resistance limit)

Standards and References

Standard	Scope
SAE HS 795	Design and Application of Helical and Spiral Springs
DIN 2089	Helical compression springs — calculation
ASTM A228	Music (carbon) spring wire
ASTM A401	Chrome-silicon spring wire
ASTM A232	Chrome-vanadium spring wire
Wahl "Mechanical Springs" (2nd ed.)	Spring fatigue Chapter 13
Shigley Chapter 10	Spring design with Goodman analysis

Output

Provide: spring geometry (D_w [mm]; D_m [mm]; C; n_a; material: ASTM grade; S_u [MPa]; G [GPa]), loading (F_min [N]; F_max [N]; R = F_min/F_max; operating temperature [°C]; expected life N_target [cycles]), Wahl factor (K_W; K_S; basis: inner fiber fatigue), stresses (τ_mean [MPa]; τ_alt [MPa]; τ_max [MPa]; τ_min [MPa]), Zimmerli endurance limit (τ_e [MPa] for material/condition: unpeened or peened), fatigue safety factor (SF_alt = τ_e/τ_alt ≥ 1.5 recommended; pass/fail), static safety factor (SF_static = S_sy/τ_max ≥ 1.2; S_sy = 0.45×S_u; pass/fail), presetting recommendation (τ_max/S_sy > 0.65? → preset recommended; estimated post-set free length [mm]; τ_residual benefit), shot peening (recommended: yes/no; new τ_e [MPa]; revised SF_alt), relaxation check (τ_max at temperature vs. 0.45×S_y_hot; relaxation Δτ [%] after design life), Miner's rule (if variable amplitude: D = Σn_i/N_i ≤ 1.0; result), and applicable standard (SAE HS 795; Wahl; Zimmerli data source).