---
title: "Gear tooth failure and surface durability"
subject: "Machine Design I"
chapter: 9
key: "semester_6/machine_design_i:gear-tooth-failure-surface-durability"
source: https://www.bctengineer.com/q/semester_6/machine_design_i/gear-tooth-failure-surface-durability
site: BCT Engineering
---

# Gear tooth failure and surface durability

Machine Design I · Chapter 9: Gear design

- Asked in: year not given, 6 marks
- Video: [Surface Durability calculation of spur gear](https://www.youtube.com/watch?v=BX0xKB2JAGA) — Mohamed Essam
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/machine_design_i/gear-tooth-failure-surface-durability
- Whole chapter: https://www.bctengineer.com/md/semester_6/machine_design_i/chapter-9

## Question

Explain the common modes of failure of gear teeth and the remedies. Describe the surface fatigue (pitting) failure and write Buckingham's equation for the limiting wear load, explaining each factor.

## Short answer

Pitting is surface fatigue from repeated Hertz contact stress, forming pits near the pitch line. Buckingham's limiting wear load is Fw = d1 b Q K, with Q = 2z2/(z1+z2) for external gears and K = (σes^2 sinφ)/1.4 (1/E1 + 1/E2), based on surface endurance limit and elastic moduli.

## Answer

#### Modes of failure of gear teeth

| Failure | Cause | Remedy |
|---|---|---|
| Tooth breakage (bending fatigue or overload) | High bending stress at the root, shock, misalignment | Larger module and face width, tough core material, generous root fillet, shot peening |
| Pitting (surface fatigue) | Repeated high contact stress cracks the surface and pieces break out | Higher surface hardness, larger diameter, good lubrication, lower load |
| Scoring and scuffing | Breakdown of the oil film at high speed and load, local welding of the surfaces | EP additives, good surface finish, correct oil and cooling |
| Abrasive wear | Dirt and particles in the lubricant | Filtration, seals, clean oil |
| Corrosive wear | Chemical attack by lubricant additives or contaminants | Suitable oil, protective coating |
| Plastic flow | Surface deformation of soft material under heavy load | Harder material |

Gears are usually designed for **beam strength** (bending) and then checked for **wear strength** (surface durability) and **dynamic load**.

#### Surface fatigue (pitting)

The contact between two teeth is like two cylinders pressed together (Hertz contact). The maximum contact stress is high and repeats with every mesh, so subsurface shear stresses start a crack that grows to the surface, and a small pit forms. Pitting begins near the pitch line, where there is pure rolling and the oil film is thin.

#### Buckingham's equation for limiting wear load

$$
F_w = d_1\, b\, Q\, K
$$

where

- $F_w$ is the limiting (maximum) wear load, N;
- $d_1$ is the pitch diameter of the pinion, mm;
- $b$ is the face width, mm;
- $Q$ is the ratio factor: $Q = \dfrac{2 z_2}{z_1 + z_2}$ for external gears and $Q = \dfrac{2z_2}{z_2 - z_1}$ for internal gears ($z_1$ = pinion teeth, $z_2$ = gear teeth);
- $K$ is the load-stress factor (N/mm$^2$), obtained from the Hertz stress:

$$
K = \frac{\sigma_{es}^2 \sin\phi}{1.4}\left(\frac{1}{E_1} + \frac{1}{E_2}\right)
$$

with $\sigma_{es}$ the surface endurance limit (often taken as $2.75\,\text{BHN} - 70$ MPa for steel; the value depends on the textbook), $\phi$ the pressure angle, and $E_1$, $E_2$ the moduli of elasticity.

To be safe against pitting, the **wear load must be at least equal to the dynamic load**: $F_w \ge F_d$. For this reason the surface hardness is raised by through hardening, case hardening, or flame or induction hardening.

## Check yourself

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

1. What causes pitting of gear teeth?
   A. Dirt particles in the oil
   B. Repeated high Hertz contact stress causing subsurface cracks that reach the surface and break out pieces
   C. Bending at the root
   D. Chemical attack of the oil additives
   Answer: B. Pitting is surface fatigue from repeated contact stresses, not bending (breakage) or abrasive/corrosive wear. (Key point: Pitting mechanism)

2. Where does pitting usually begin?
   A. At the tip of the tooth
   B. At the root fillet
   C. Near the pitch line, where only rolling occurs and the oil film is thin
   D. At the face ends
   Answer: C. Near the pitch line there is pure rolling and a thin film, so pitting begins there. (Key point: Pitting location)

3. What is Buckingham's limiting wear load?
   A. F_w = d1 b Q K
   B. F_w = m b Y sigma
   C. F_w = b Q K / d1
   D. F_w = d1^2 Q K
   Answer: A. F_w = d1 b Q K with d1 the pinion pitch diameter, b the face width, Q the ratio factor and K the load-stress factor. (Key point: Buckingham equation)

4. What is required for safety against pitting and how can it be achieved?
   A. F_w must be at least F_d, by raising surface hardness
   B. F_w must be less than F_d, by softening the surface
   C. F_w must equal beam strength
   D. F_w must be zero
   Answer: A. The wear load must be at least the dynamic load; through, case, flame or induction hardening raises surface hardness and K. (Key point: Safe condition)

## Related in Machine Design I

- [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 equation for beam strength of a gear tooth](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Alewis-beam-strength-derivation) — Machine Design I, chapter 9
- [Compound gear train speeds, torques and tooth forces](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Acompound-gear-train-forces) — Machine Design I, chapter 9
- [Spur gear terminology and standard tooth system](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aspur-gear-terminology) — Machine Design I, chapter 9
- [AGMA bending stress equation and its factors](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Aagma-factors-stress-concentration-geometry) — Machine Design I, chapter 9

## Related in other subjects

- [Gear tooth vernier: chordal thickness](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Agear-tooth-vernier-chordal-thickness) — Metrology, chapter 11
- [Tooth forces on a spur gear pinion](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aspur-gear-tooth-forces-numerical) — Theory of Mechanism and Machine I, chapter 8
- [Force components on a helical gear tooth](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Ahelical-gear-force-components-axial-thrust) — Theory of Mechanism and Machine I, chapter 8
- [Straight bevel gear geometry and formative teeth](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Astraight-bevel-gear-geometry-formative-teeth) — Theory of Mechanism and Machine I, chapter 5