Model Explanation
1. Spring Index and Wahl Stress Correction Factor
C = D / d Kw = (4C − 1)/(4C − 4) + 0.615/C
C is the spring index (ratio of mean coil diameter to wire diameter). The Wahl factor Kw
corrects the simple torsion formula for the curvature and direct-shear effects present in a coiled
wire. Springs with low index (C < 4) experience significantly amplified stress concentrations.
2. Shear Stress
τ = Kw × (8 P D) / (π d³)
τmax and τmin are computed from the maximum and minimum operating loads
(Pmax, Pmin). These represent the cyclic stress range the spring experiences in service.
3. Stress Ratio
S = τmax / Ssu (Ssu ≈ 0.5 × Su)
The stress ratio compares peak operating shear stress to the material’s torsional ultimate shear
strength (approximated as half the ultimate tensile strength, Su). This ratio is the primary
driver of fatigue failure rate — as S approaches 1, the spring operates near its material limit and
failure probability rises sharply.
4. Base Failure Rate, λb
NSWC-11 provides λb as an empirically derived curve (in failures per 106 cycles)
that increases steeply with stress ratio. This calculator uses an exponential approximation of that curve:
λb = 0.0001 × e8(S − 0.3) for S ≥ 0.3, else λb = 0.0001
| Stress Ratio (S) | Approx. λb (per 106 cyc) | Interpretation |
|---|
| ≤ 0.3 | ~0.0001 (floor) | Very low fatigue risk |
| 0.5 | ~0.0005 | Low risk, typical conservative design |
| 0.7 | ~0.0024 | Moderate risk |
| 0.9 | ~0.012 | High risk — approaching fatigue limit |
| ≥ 1.0 | Rapidly increasing | Likely overstressed / short life |
5. Temperature Factor, πT
πT = e0.01(T − 70)
Elevated operating temperature reduces material fatigue strength and accelerates relaxation/creep in
spring materials. The factor is referenced to 70°F (room temperature).
6. Cycling Rate Factor, πCY
πCY = 1 + (cycling rate [cycles/min] / 1000)
Higher cyclic frequencies generate additional heating and dynamic stress effects (resonance, impact
loading) that can slightly elevate the effective failure rate.
7. Environment Factor, πE
| Environment | πE |
|---|
| Benign (clean, controlled) | 1.0 |
| Mild (occasional humidity/exposure) | 2.0 |
| Severe (corrosive, marine, chemical) | 4.0 |
8. Predicted Failure Rate
λp = λb × πT × πCY × πE
The combined failure rate λp (failures per 106 cycles) is then used to estimate
the expected number of failures over the spring’s design life:
Expected Failures = λp × (N / 106)
9. Time-Based Reliability Metrics (FPMH, FIT, MTBF)
Cycle-based failure rates are useful for mechanical fatigue analysis, but system-level FMECA
and reliability block diagrams require failure rates in time-based units — typically
FPMH (Failures Per Million Hours) or FIT (Failures in Time, per 109 hours)
— so the spring’s contribution can be combined with electronic and other component failure rates.
The input cycling rate bridges cycles and time.
Cycles per hour = Cycling Rate [cycles/min] × 60
Failures per hour = λp × (Cycles per hour / 106)
FPMH = Failures per hour × 106 · FIT = Failures per hour × 109
MTBF [hours] = 1 / (Failures per hour) · MTBF [years] = MTBF [hours] / (24 × 365.25)
MTBF is the inverse of the failure rate and assumes a constant (exponential) failure rate over the
spring’s life — a reasonable approximation away from the wear-out region but less accurate as the
stress ratio S approaches 1.
This calculator implements a simplified engineering-approximation of the NSWC-11 spring model for
educational and preliminary design-screening purposes. The original handbook derives the base failure rate
from empirical charts and includes additional correction factors (spring type, surface treatment, shot-peening).
For certified reliability predictions, designs near the fatigue limit, or safety-critical applications, consult
the full NSWC-11 handbook tables and/or perform physical fatigue testing.