Model Explanation
1. Inner-Edge Bending Stress (Eq. 4-17)
When a Belleville washer is loaded it tends to flatten, causing radial and circumferential strains. Stress is not
distributed uniformly; the highest stress occurs at the top inner edge and is estimated by NSWC-11 as:
S = EM · f · R · t / [ (1 − μ²) · a² ] a = OD / 2
where S is the bending stress (lb/in²), EM the modulus of elasticity, f the deflection, μ Poisson’s
ratio, R a dimension factor, t the material thickness and a = OD/2. This stress is shown for reference; the failure-rate
model below is driven by the dimensionless multiplying factors rather than by S directly. The diameter ratio R = OD/ID
is used as the dimension factor in this estimate.
2. Failure Rate Model (Eq. 4-18)
The failure rate of a Belleville washer is obtained by scaling a base rate by a product of multiplying factors, each capturing
one design or operating influence on the base rate:
λSP = λSP,B · CE · Ct · CD · Cf · CY · CS · CCS · CR · CM
with the base failure rate λSP,B = 2.6 failures / 106 hours. Because the
base rate is already per-hour, λSP is reported in FPMH directly.
| Factor | Equation | Effect on base failure rate |
|---|
| CE | (EM / 28.5×106)3 | Modulus of elasticity (Table 4-2). Reference: Hard Drawn Steel, EM = 28.5×106. |
| Ct | (t / 0.025)3 | Material thickness (Fig 4.16). Thicker stock raises stress and the failure rate. |
| CD | (1.20 / OD)6 | Washer size (Fig 4.17). Larger outside diameter lowers the factor steeply. |
| Cf | (f / 0.055)3 | Washer deflection under load (Fig 4.18). More deflection raises the factor. |
| CY | (190 000 / TS)3 | Tensile strength (Table 4-3). Reference: Copper-Beryllium, TS = 190 000 psi. |
| CS | f(R), R = OD/ID | Compressive-stress geometry (Fig 4.19). See equation below. |
| CCS | piecewise in CR | Spring cycle rate (Fig 4.20). See equation below. |
| CR | 1.0 (protected) | Corrosive environment (Section 4.3.10). 1.0 when protection is applied; higher from experience. |
| CM | 1.0 (good QC) | Manufacturing process (Section 4.3.11). 1.0 for acceptable QC; higher from experience. |
3. Compressive-Stress Factor, CS (Fig. 4.19)
CS = (6 / (π ln R))3 · [ (R − 1)/ln R − 1 ] · [ (R − 1)/2 ] · [ (R − 1)/R ]2 R = OD / ID
CS depends only on the diameter ratio R = outside diameter / inside diameter. It rises from
roughly 0.16 at R = 1.1 to about 3.2 at R = 5.1, mirroring how the inner-edge compressive stress concentration
grows with a wider washer. The model is valid over approximately 1.1 ≤ R ≤ 5.1.
4. Cycle-Rate Factor, CCS (Fig. 4.20)
CCS = 0.100 for CR ≤ 30 cyc/min
CCS = CR / 300 for 30 < CR ≤ 300 cyc/min
CCS = (CR / 300)3 for CR > 300 cyc/min
Higher cyclic frequencies generate additional heating and dynamic-stress effects that elevate the effective failure rate. At
low cycle rates the factor floors at 0.100, reflecting the near-static duty of many washer applications.
5. Time-Based Reliability Metrics (FPMH, FIT, MTBF)
λSP [FPMH] = λSP (already per 106 h)
FIT = λSP × 103 · MTBF [hours] = 106 / λSP
MTBF [years] = MTBF [hours] / (24 × 365.25)
MTBF is the inverse of the failure rate and assumes a constant (exponential) failure rate, a reasonable approximation away from
the wear-out region.
This calculator implements the NSWC-11 Belleville washer multiplying-factor model (Section 4.4.6, Eq. 4-18) directly
from the handbook equations and the factor curves of Figures 4.16–4.20. The inner-edge stress (Eq. 4-17) is shown
for reference only and is not used in the failure-rate product. For certified reliability predictions, designs near material
limits, or safety-critical applications, consult the full NSWC-11 handbook tables and/or perform physical testing.