---
title: "Lewis equation for beam strength of a gear tooth"
subject: "Machine Design I"
chapter: 9
key: "semester_6/machine_design_i:lewis-beam-strength-derivation"
source: https://www.bctengineer.com/q/semester_6/machine_design_i/lewis-beam-strength-derivation
site: BCT Engineering
---

# Lewis equation for beam strength of a gear tooth

Machine Design I · Chapter 9: Gear design

- Asked in: year not given, 8 marks
- Video: [Lewis's beam strength equation(Full Derivation)(Machine design)](https://www.youtube.com/watch?v=gFQfTbz9970) — Unique Series - Civil Mantraa
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/machine_design_i/lewis-beam-strength-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_6/machine_design_i/chapter-9

## Question

Derive the Lewis equation for the beam strength of a spur gear tooth, stating the assumptions. Define the Lewis form factor and state its expression for 20 degree full-depth teeth.

## Short answer

Lewis treated the tooth as a cantilever beam with full tangential load at the tip, neglecting radial force and stress concentration. The permissible load is Wt = σb b π m y, where y, the form factor, is 0.154 – 0.912/z for 20° full-depth teeth.

## Answer

Lewis (1892) treated a gear tooth as a **cantilever beam** loaded at its tip by the full tangential load, and found its load-carrying strength in bending.

#### Assumptions

- The full load $W_t$ acts at the tip of one tooth only.
- The load is uniformly distributed across the face width $b$.
- The radial component (compression) and friction are neglected.
- The effect of stress concentration at the root is neglected.
- The tooth is treated as a parabola (a beam of uniform strength), and the section of maximum stress is where the parabola touches the tooth profile.

```text
         Wt
          |
          v
        __|__        <- tip
       /  |  \
      /   | h \  parabola
     /   _|_   \
    |<--- t --->|   critical section
```

#### Derivation

Let $t$ = tooth thickness at the critical section, $h$ = height of the load above that section, $b$ = face width.

Bending moment $M = W_t\,h$. Section modulus $Z = \dfrac{b t^2}{6}$.

$$
\sigma_b = \frac{M}{Z} = \frac{6\,W_t\,h}{b\,t^2}
$$

So the permissible tangential load is

$$
W_t = \frac{\sigma_b\, b\, t^2}{6h}
$$

The critical section is where the inscribed parabola touches the tooth profile. From the similar triangles of the parabola, $t^2 = 4hx$, where $x$ is a length fixed by the tooth profile. Substituting,

$$
W_t = \frac{\sigma_b\, b \cdot 4hx}{6h} = \sigma_b\, b\,\frac{2x}{3}
$$

Multiplying and dividing by the circular pitch $p_c = \pi m$:

$$
W_t = \sigma_b\, b\, \pi m\left(\frac{2x}{3 p_c}\right)
$$

The dimensionless term in brackets is the **Lewis form factor** $y$:

$$
\boxed{W_t = \sigma_b\, b\, \pi\, m\, y}
$$

where $\sigma_b$ is the allowable bending stress ($\sigma_o$ for static conditions, or $\sigma_o C_v$ with the velocity factor). Writing $Y = \pi y$ gives the other form, $W_t = \sigma_b b m Y$.

#### Form factor

$y$ depends on the tooth profile and the number of teeth $z$; it increases with $z$:

- 20 degree full depth: $y = 0.154 - \dfrac{0.912}{z}$
- 14.5 degree composite or full depth: $y = 0.124 - \dfrac{0.684}{z}$

For a given pair of gears, the design is based on the weaker of the pinion and the gear (the one with the smaller $\sigma_b y$). The pinion has fewer teeth, and so a smaller $y$, but it is often made of a stronger material.

## Check yourself

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

1. How does Lewis model a gear tooth?
   A. As a column in compression
   B. As a cantilever beam loaded at its tip by the full tangential load
   C. As a simply supported beam with load at the root
   D. As a shaft in torsion
   Answer: B. The tooth is a cantilever with the entire load W_t at the tip, and the root section is checked in bending. (Key point: Lewis model)

2. Which expression gives the bending stress at the critical section?
   A. sigma = W_t h / (b t)
   B. sigma = 6 W_t h / (b t^2)
   C. sigma = 3 W_t h / (b t^3)
   D. sigma = W_t t / (6 b h^2)
   Answer: B. M = W_t h and Z = b t^2 / 6 so sigma = M/Z = 6 W_t h / (b t^2). (Key point: Bending stress)

3. What step turns W_t = sigma b (2x/3) into the Lewis equation?
   A. Dividing by the module and calling 2x/(3 p_c) the form factor y, giving W_t = sigma b pi m y
   B. Taking x equal to the face width
   C. Setting t = h
   D. Ignoring the circular pitch
   Answer: A. Multiplying and dividing by p_c = pi m defines y = 2x/(3 p_c), a dimensionless factor depending on tooth profile. (Key point: Form factor)

4. For 20 degree full-depth teeth, how does y behave?
   A. y = 0.154 - 0.912/z and it increases with z
   B. y = 0.124 - 0.684/z and it decreases with z
   C. y is constant at 0.154
   D. y = 0.154 - 0.912/z and it decreases with z
   Answer: A. More teeth give a stronger, broader tooth shape so y rises with z; the pinion has the smaller y. (Key point: Form factor values)

## Related in Machine Design I

- [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
- [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
- [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
- [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
- [Diameter of a shaft carrying a spur gear](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Agear-loaded-shaft-diameter) — Machine Design I, chapter 5
- [Gear tooth failure and surface durability](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Agear-tooth-failure-surface-durability) — Machine Design I, chapter 9

## Related in other subjects

- [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
- [Gearbox shaft design with spur gear](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Agearbox-shaft-design-asme) — Machine Design II, chapter 5
- [Spur gear pair dimensions numerical](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aspur-gear-pair-geometry-numerical) — Theory of Mechanism and Machine I, chapter 4
- [Gear tooth vernier: chordal thickness](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Agear-tooth-vernier-chordal-thickness) — Metrology, chapter 11