---
title: "AGMA bending stress equation and its factors"
subject: "Machine Design I"
chapter: 9
key: "semester_6/machine_design_i:agma-factors-stress-concentration-geometry"
source: https://www.bctengineer.com/q/semester_6/machine_design_i/agma-factors-stress-concentration-geometry
site: BCT Engineering
---

# AGMA bending stress equation and its factors

Machine Design I · Chapter 9: Gear design

- Asked in: year not given, 6 marks
- Video: [Machine Design: AGMA Gear Design (from Shigley's)](https://www.youtube.com/watch?v=dp2evrIfjCo) — The_Mechanical_Pencil
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/machine_design_i/agma-factors-stress-concentration-geometry
- Whole chapter: https://www.bctengineer.com/md/semester_6/machine_design_i/chapter-9

## Question

Explain the AGMA bending stress equation for spur gears. Describe the geometry factor J (including the effect of stress concentration at the tooth root), the dynamic factor Kv, overload factor Ko and load distribution factor Km, and explain how the factor of safety is obtained from the bending fatigue strength.

## Short answer

AGMA bending stress σ = Wt Ko Kv Ks Km Kb / (b m J). J replaces the Lewis factor, including root fillet stress concentration and load sharing. Ko covers overload shocks, Kv dynamic effects, Km misalignment. Safety factor n = St YN / (KT KR σ).

## Answer

The AGMA method calculates the actual bending stress at the tooth root using the Lewis idea, but adds factors for the real operating conditions. In SI units, for spur gears,

$$
\sigma = \frac{W_t\, K_o\, K_v\, K_s\, K_m\, K_b}{b\, m\, J}
$$

where $W_t$ is the tangential load (N), $b$ the face width (mm), $m$ the module (mm) and $J$ the geometry factor.

#### Geometry factor $J$

$J$ replaces the Lewis form factor. It accounts for:

- the **shape of the tooth** (number of teeth, pressure angle, addendum), as the Lewis factor does;
- the **stress concentration** at the root fillet ($K_f$), because the fillet raises the local stress above the nominal beam stress;
- the **load sharing** between teeth (for spur gears the critical load acts not at the tip but at the highest point of single-tooth contact, because the load is shared in the double-contact zone).

$J$ is read from charts for the given numbers of teeth on the pinion and gear; it is lower for small pinions. Because the AGMA equation uses the module rather than the circular pitch, $J$ is comparable to $Y = \pi y$ in the Lewis form $W_t = \sigma b m Y$, not to $y$ itself. For a 20-tooth pinion, $J$ is about 0.3 to 0.35, similar to $\pi y \approx 0.34$: the root stress concentration lowers it, while the higher load position (load sharing) raises it.

#### Other factors

| Factor | Name | What it covers |
|---|---|---|
| $K_o$ | Overload factor | Shocks from the driving and driven machines: uniform 1.0, moderate 1.25, heavy 1.75 or more |
| $K_v$ | Dynamic factor | Tooth errors and vibration at speed; it falls with a higher quality number $Q_v$ and rises with pitch line velocity |
| $K_s$ | Size factor | Larger teeth are weaker per unit size; usually 1 for small gears |
| $K_m$ | Load distribution factor | Misalignment and face width effects; 1.3 to 1.6 for ordinary mountings |
| $K_b$ | Rim thickness factor | 1 for solid gears; more if the rim is thin |

#### Strength and factor of safety

The allowable bending stress depends on the material, heat treatment and hardness; for example, for Grade 1 through-hardened steel $S_t = 0.533\,\text{HB} + 88.3$ MPa. It is corrected for the required life and the operating conditions by the life factor $Y_N$, temperature factor $K_T$ and reliability factor $K_R$:

$$
n = \frac{S_t\, Y_N}{K_T\, K_R\, \sigma}
$$

$Y_N$ is about 1 at $10^7$ cycles and falls as the number of cycles increases ($Y_N = 1.3558\,N^{-0.0178}$ for $N > 10^7$). The safety factor $n$ must be above 1; values of 1.5 to 3 are commonly used.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the geometry factor J account for?
   A. Only the number of teeth
   B. Tooth shape, stress concentration at the root fillet, and load sharing between teeth
   C. Shock from the driven machine
   D. Tooth errors at high speed
   Answer: B. J replaces Lewis y and includes root fillet stress concentration and load sharing; it is lower than y. (Key point: Geometry factor)

2. What does Kv cover?
   A. Misalignment of shafts
   B. Tooth errors and vibration, rising with pitch line velocity and falling with higher quality number
   C. Shocks from prime mover
   D. Thin rim thickness
   Answer: B. Kv reflects dynamic effects from tooth errors at speed; better quality number lowers it. (Key point: Dynamic factor)

3. Which pairing is correct?
   A. Ko covers misalignment; Km covers shock from machines
   B. Ko covers shock from driving and driven machines; Km covers misalignment and face width effects
   C. Ko is rim thickness; Km is size
   D. Both cover tooth errors
   Answer: B. Ko is the overload factor (1.0 uniform to 1.75 or more heavy shock); Km is the load distribution factor (1.3 to 1.6 typical). (Key point: Overload and load distribution)

4. How is the bending factor of safety obtained?
   A. n = sigma / (S_t Y_N)
   B. n = S_t Y_N / (K_T K_R sigma)
   C. n = S_t K_T K_R / sigma
   D. n = sigma K_T / S_t
   Answer: B. The allowable stress is corrected by life, temperature and reliability factors and divided by the actual stress; n should exceed 1. (Key point: Factor of safety)

## Related in Machine Design I

- [Bending stress and safety factor of a spur pinion](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aagma-bending-stress-safety-factor) — Machine Design I, chapter 9
- [Dynamic and wear load check, spur pair](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Abuckingham-dynamic-wear-check) — Machine Design I, chapter 9
- [Lewis method module for spur gears](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Alewis-spur-gear-module-design) — Machine Design I, chapter 9

## Related in other subjects

- [Stress concentration and its effect in brittle material](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astress-concentration-brittle-effect) — Strength of Materials, chapter 3
- [Normal, shear, bearing stress and factor of safety](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Astress-types-factor-of-safety) — Strength of Materials, chapter 1