| name | belleville-washer |
| description | Belleville disc spring (washer) — Almen-Laszlo load-deflection, stacking, fatigue, DIN 2093, flat-load characteristic, progressive rate design. |
| metadata | {"priority":7,"promptSignals":{"phrases":["belleville washer","belleville spring","disc spring","conical washer","Belleville"],"minScore":3}} |
Belleville Disc Spring — Complete Skill
Geometry
D_e = outside diameter [mm]
D_i = inside diameter [mm]
t = thickness [mm]
h_0 = cone height (unloaded) [mm]
δ = deflection [mm] (positive = flattening)
Key ratio: h_0/t determines load characteristic shape
Load-Deflection (Almen-Laszlo Equation)
F = (4E × δ) / ((1-ν²) × K₁ × D_e²) × [( h_0 - δ/2)(h_0 - δ)t + t³]
Simplified constants:
K₁ = (6/π) × [(R-1)² / (R²×ln R)] where R = D_e/D_i
Load at flat (δ = h_0):
F_flat = 4E h_0 t³ / ((1-ν²) K₁ D_e²)
Spring rate (instantaneous):
k = dF/dδ = (4E) / ((1-ν²) K₁ D_e²) × [h_0(h_0 - 3δ/2 + δ) + t³ - partial terms]
(Non-linear — use finite difference from F-δ curve for k at operating point)
h_0/t Ratio and Characteristic Behavior
| h_0/t | Load Characteristic |
|---|
| < 0.4 | Approximately linear (like flat washer) |
| 0.4–1.4 | Progressive (increasing stiffness) |
| ~1.41 | Inflection — load plateau region |
| 1.5–2.8 | S-curve (decreases then increases) |
| > 2.8 | Negative rate region (snap-through) |
Design for constant load (flat characteristic): h_0/t ≈ √2 ≈ 1.41
High-energy absorption: h_0/t ≈ 2.0–2.5
Stacking Configurations
Series stacking (same orientation, one on top of other):
Load same as single spring; deflection multiplies by n
k_series = k_single / n
Parallel stacking (alternating orientation):
Load multiplies by n; deflection same as single
k_parallel = n × k_single
Combined: mix series groups of parallel stacks
Stress Analysis (DIN 2093)
Maximum compressive stress at inner edge, upper surface:
σ_I = 4E δ / ((1-ν²) K₁ D_e²) × [K₂(h_0 - δ/2) + K₃ t]
K₂ = (6/π ln R) × ((R-1)/R - 1/ln R)
K₃ = 3(R-1)/(π ln R)
DIN 2093 Group A, B, C:
Group A: t ≤ 1.25 mm; Group B: 1.25 < t ≤ 6 mm; Group C: t > 6 mm
Static allowable (Group B): σ_allow = 1.6–2.0 × S_y
Fatigue: from DIN 2093 S-N curves based on stress ratio σ_under/σ_upper
Friction and Hysteresis
Contact friction (between washer edge and seat):
Hysteresis ≈ 3–10% of load range for guided stacks
Larger for taller stacks or dry metal-metal contact
Reduce by: Lubricating (molybdenum disulfide grease), proper surface finish
DIN 2093 Standards
Preferred sizes: D_e = 8, 10, 12.5, 14, 16, 18, 20, 22.4, 25, 28, 31.5, 35.5, 40, 45, 50, 56, 63, 71, 80, 90, 100, 112, 125 mm
Standard materials: 51CrV4 (spring steel), 1.4310 (stainless), 17-7PH
Applications
- Bolted joints: maintain preload through thermal cycles
- Vibration isolation stacks
- Constant-force mechanisms (h_0/t ≈ √2)
- Clutch packs, valve springs, safety devices
- High-force, short-travel applications
Design Procedure
- Establish required F at δ_working; check h_0/t for desired characteristic
- Select D_e, D_i, t from DIN preferred sizes
- Compute K₁, K₂, K₃; calculate F at δ_working and δ_max
- Check σ_I vs. allowable (static or fatigue)
- Determine stacking: series for more travel, parallel for more load
- Account for hysteresis if tight load control needed
Output
Provide: D_e × D_i × t × h_0 [mm], h_0/t, F at working deflection [N], k [N/mm] at operating point, σ_max [MPa] vs. allowable, stacking configuration, hysteresis estimate [%].
