Gear Strength Calculation Reference
ISO 6336 contact and root bending formulas with GB/T 3480, DIN 3990 and AGMA 2001: 20CrMnTi at HRC 58~62 gives σHlim 1500 MPa and σFlim 500 MPa, and t
Gear Strength Calculation Reference
Document number: GE-EN-001 | Version: V1.0 | Last updated: 2024-06
1. Overview#
Gear strength calculation is a core part of gear design and mainly consists of two parts: tooth flank contact strength calculation and tooth root bending strength calculation.
Reference standards:
- ISO 6336: 2006 — Calculation of load capacity of cylindrical gears (international standard)
- GB/T 3480 — Calculation of load capacity of involute cylindrical gears (Chinese standard)
- DIN 3990 — German standard
- AGMA 2001 — American standard
2. Tooth Flank Contact Strength Calculation#
2.1 Basic Formula#
ISO 6336 tooth flank contact stress formula:
$$ \sigma_H = Z_H \cdot Z_E \cdot Z_\varepsilon \cdot Z_\beta \cdot \sqrt{\frac{F_t}{b \cdot d_1} \cdot \frac{u+1}{u} \cdot K_A \cdot K_V \cdot K_{H\beta} \cdot K_{H\alpha}} $$
| Symbol | Meaning | Notes |
|---|---|---|
| $\sigma_H$ | Tooth flank contact stress (MPa) | Calculated value |
| $Z_H$ | Zone factor at the pitch point | Depends on pressure angle/helix angle |
| $Z_E$ | Elasticity factor (√(MPa)) | 189.8 (steel on steel) |
| $Z_\varepsilon$ | Contact ratio factor | Depends on contact ratio |
| $Z_\beta$ | Helix angle factor | Spur gear = 1, helical gear < 1 |
| $F_t$ | Nominal tangential force at the pitch circle (N) | $F_t = 2T/d$ |
| $b$ | Face width (mm) | |
| $d_1$ | Pinion pitch circle diameter (mm) | |
| $u$ | Gear ratio | $u = Z_2/Z_1 \ge 1$ |
| $K_A$ | Application factor | Load characteristics of prime mover and driven machine |
| $K_V$ | Dynamic factor | Influence of speed and accuracy |
| $K_{H\beta}$ | Face load factor | Contact strength |
| $K_{H\alpha}$ | Transverse load factor | Contact strength |
2.2 Allowable Contact Stress#
$$ \sigma_{HP} = \frac{\sigma_{Hlim} \cdot Z_N \cdot Z_L \cdot Z_V \cdot Z_R \cdot Z_W \cdot Z_X}{S_{Hmin}} $$
| Symbol | Meaning | Notes |
|---|---|---|
| $\sigma_{Hlim}$ | Contact fatigue limit of the reference gear (MPa) | Determined by material/heat treatment |
| $Z_N$ | Life factor | Depends on number of stress cycles |
| $Z_L$ | Lubricant factor | Depends on lubricant viscosity |
| $Z_V$ | Velocity factor | Depends on pitch line velocity |
| $Z_R$ | Roughness factor | Depends on tooth flank roughness |
| $Z_W$ | Work hardening factor | Depends on hardness difference of the mating gear |
| $Z_X$ | Size factor | Depends on module/size |
| $S_{Hmin}$ | Minimum safety factor | Contact strength |
2.3 Acceptance Condition#
$$ \sigma_H \le \sigma_{HP} $$
3. Tooth Root Bending Strength Calculation#
3.1 Basic Formula#
ISO 6336 tooth root bending stress formula:
$$ \sigma_F = \frac{F_t}{b \cdot m_n} \cdot Y_{Fa} \cdot Y_{Sa} \cdot Y_\varepsilon \cdot Y_\beta \cdot K_A \cdot K_V \cdot K_{F\beta} \cdot K_{F\alpha} $$
| Symbol | Meaning | Notes |
|---|---|---|
| $\sigma_F$ | Tooth root bending stress (MPa) | Calculated value |
| $m_n$ | Normal module (mm) | |
| $Y_{Fa}$ | Form factor | Depends on number of teeth/profile shift coefficient |
| $Y_{Sa}$ | Stress correction factor | Influence of the root fillet |
| $Y_\varepsilon$ | Contact ratio factor | |
| $Y_\beta$ | Helix angle factor | |
| $K_{F\beta}$ | Face load factor | Bending strength |
| $K_{F\alpha}$ | Transverse load factor | Bending strength |
3.2 Allowable Bending Stress#
$$ \sigma_{FP} = \frac{\sigma_{Flim} \cdot Y_N \cdot Y_{ST} \cdot Y_{\delta relT} \cdot Y_{RrelT} \cdot Y_X}{S_{Fmin}} $$
| Symbol | Meaning |
|---|---|
| $\sigma_{Flim}$ | Bending fatigue limit of the reference gear (MPa) |
| $Y_N$ | Life factor |
| $Y_{ST}$ | Stress correction factor (usually taken as 2.0) |
| $Y_{\delta relT}$ | Relative root fillet sensitivity factor |
| $Y_{RrelT}$ | Relative root surface condition factor |
| $Y_X$ | Size factor |
| $S_{Fmin}$ | Minimum safety factor (bending strength) |
3.3 Acceptance Condition#
$$ \sigma_F \le \sigma_{FP} $$
4. Fatigue Limits of Common Gear Materials#
| Material | Heat treatment | Hardness (HRC) | $\sigma_{Hlim}$ (MPa) | $\sigma_{Flim}$ (MPa) | Typical application |
|---|---|---|---|---|---|
| 20CrMnTi | Carburized and quenched | 58~62 | 1500 | 500 | Gearbox gears |
| 20CrMo | Carburized and quenched | 58~62 | 1500 | 500 | High-precision ground gears |
| 40Cr | Quenched and tempered + induction hardened | 48~55 | 1100 | 380 | Medium-load gears |
| 45# | Quenched and tempered | 35~45 | 800 | 300 | Low-speed light-load gears |
| 42CrMo | Quenched and tempered + nitrided | ≥50 (nitrided case) | 1200 | 420 | Large gears |
| 40CrNiMo | Quenched and tempered | 35~45 | 1000 | 350 | High-strength transmission shafts |
5. Recommended Safety Factors#
| Application | Contact strength $S_{Hmin}$ | Bending strength $S_{Fmin}$ |
|---|---|---|
| General drives | 1.0~1.1 | 1.25~1.5 |
| High reliability | 1.1~1.25 | 1.5~1.8 |
| Aerospace | 1.25~1.5 | 1.8~2.5 |
6. Load Factor Reference#
6.1 Application Factor $K_A$#
| Prime mover | Uniform load on driven machine | Moderate shock | Heavy shock |
|---|---|---|---|
| Electric motor/turbine | 1.00 | 1.25 | 1.75 |
| Multi-cylinder combustion engine | 1.25 | 1.50 | 2.00 |
| Single-cylinder combustion engine | 1.50 | 1.75 | 2.25 |
6.2 Dynamic Factor $K_V$#
$$ K_V = 1 + \left( \frac{K_1}{K_A \cdot F_t/b \cdot K_{H\beta}} + K_2 \right) \cdot \frac{z_1 \cdot v}{100} \cdot \sqrt{\frac{u^2}{1+u^2}} $$
Here $v$ is the pitch line velocity (m/s), and $K_1$, $K_2$ are accuracy coefficients.
7. Worked Example (Quick Calculation)#
Rapid Strength Estimation for a Cylindrical Gear#
| Known parameter | Value |
|---|---|
| Module $m_n$ | 3 mm |
| Number of teeth $Z_1$ / $Z_2$ | 23 / 74 |
| Face width $b$ | 40 mm |
| Input torque $T_1$ | 200 N·m |
| Material/heat treatment | 20CrMnTi carburized and quenched |
| Speed $n_1$ | 1450 rpm |
| Accuracy grade | DIN grade 6 |
Calculation steps:
- Tangential force: $F_t = 2T/d_1 = 2 × 200000 / (3×23) = 5797 N$
- Tooth flank contact stress: taking $Z_H=2.5,\ Z_E=189.8,\ Z_ε=0.87,\ Z_β=1.0,\ K_A=1.25,\ K_V=1.1,\ K_{Hβ}=1.15,\ K_{Hα}=1.1$ $σ_H = 2.5 × 189.8 × 0.87 × 1.0 × √(5797/(40×69) × (3.22+1)/3.22 × 1.25×1.1×1.15×1.1) ≈ 720\ MPa$
- Allowable contact stress: $σ_{HP} = 1500 / (1.1) ≈ 1364\ MPa$ (safe)
- Tooth root bending stress: taking $Y_{Fa}=2.7,\ Y_{Sa}=1.55,\ Y_ε=0.7,\ Y_β=1.0,\ K_{Fβ}=1.1,\ K_{Fα}=1.1$ $σ_F = 5797/(40×3) × 2.7×1.55×0.7×1.0 × 1.25×1.1×1.1×1.1 ≈ 258\ MPa$
- Allowable bending stress: $σ_{FP} = 500×2.0 / (1.5) ≈ 667\ MPa$ (safe)
8. Commonly Used Calculation Tools#
| Tool | Type | Notes |
|---|---|---|
| KISSsoft | Commercial software | Professional design and analysis of gear transmission systems |
| MASTA | Commercial software | Transmission system design and simulation |
| Romax | Commercial software | Gearbox/bearing system analysis |
| Gear calculation spreadsheet (Excel) | In-house | Geyontech internal gear strength check sheet |
| Online gear calculation | Web | www.geartechnology.com online calculation tool |
9. Notes#
- When tooth flank contact strength is insufficient, the priority measures are: increase the pitch circle diameter → increase the face width → increase hardness
- When tooth root bending strength is insufficient, the priority measures are: increase the module → increase the face width → apply positive profile shift
- In actual design, both strength criteria must be satisfied at the same time
- For high-speed gears, scuffing load capacity calculation is also required (flash temperature method/integral temperature method)