1. Spring Index and Curvature Correction Factor
C = D / d Kb = (4C2 − C − 1) / (4C(C − 1))
C is the spring index (ratio of mean coil diameter to wire diameter). For torsion springs, the Wahl-type
curvature correction factor Kb corrects the simple bending formula for the additional stress
concentration caused by coil curvature. This differs from the shear-stress Wahl factor Kw used
for compression and extension springs. Springs with low index (C < 4) experience significantly amplified
stress concentrations.
2. Bending Stress
σ = Kb × (32 M) / (π d³)
σmax and σmin are computed from the maximum and minimum applied torques
(Mmax, Mmin) in lb·in. For a torsion spring the wire is loaded in
bending, not torsion — so the stress formula uses the section modulus of a circular
cross-section (32/πd³) multiplied by the curvature factor Kb.
If your load is specified as a force F at arm length L, compute M = F × L.
3. Stress Ratio
S = σmax / Su
The stress ratio compares peak operating bending stress to the material’s ultimate tensile strength
Su. Because torsion springs fail in bending (not shear), the full tensile strength is used
directly — unlike compression springs, which use Ssu ≈ 0.5×Su
for shear failure. This ratio is the primary driver of fatigue failure rate: as S approaches 1, the spring
operates at its material limit and failure probability rises sharply.
3b. Material Tensile Strength Factor, CY
CY = (190 000 / TS)3
CY is a dimensionless multiplying factor (NSWC-11 Table 4-3) that scales the failure rate
based on the material’s tensile strength relative to the reference material Copper-Beryllium
(TS = 190 000 psi, CY = 1.00). Softer materials (lower TS)
produce CY > 1; harder materials yield CY < 1.
Material strength enters implicitly through the stress ratio S = σ/Su, so
CY is shown here as a reference value.
Tensile-strength values are typical for a 0.1 in wire diameter; actual values vary with wire diameter.
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.
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 dynamic stress effects 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 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).
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)
This calculator implements a simplified engineering-approximation of the NSWC-11 torsion 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.