lewis formula calculator

What is the Lewis formula?

The Lewis formula is a classic mechanical design equation used to estimate bending strength of a gear tooth. It treats one tooth like a cantilever beam and estimates how much tangential load that tooth can carry before bending stress becomes critical.

W_T = σ × b × y × p × (C_v/K_s)

• W_T = allowable tangential tooth load (N)
• σ = allowable bending stress (MPa = N/mm²)
• b = face width (mm)
• y = Lewis form factor (dimensionless)
• p = circular pitch = πm (mm)
• C_v = velocity/dynamic factor
• K_s = service factor

How this calculator works

This tool calculates circular pitch from module, estimates (or accepts) the Lewis form factor, and computes allowable tangential load. It also reports pitch diameter and estimates torque capacity. If RPM is entered, it adds a power estimate.

Lewis factor selection

For quick design checks, many engineers use empirical formulas for y based on tooth count and profile. This calculator includes three common approximations and a custom mode for standards-specific values.

Tooth profile	Approximation	Typical use
20° full-depth involute	y = 0.154 − 0.912/z	General-purpose modern spur gears
14.5° full-depth involute	y = 0.124 − 0.684/z	Legacy or older designs
20° stub involute	y = 0.175 − 0.95/z	Higher root strength preference

When to use this result (and when not to)

Good use cases

• Preliminary gear sizing
• Sanity-checking a hand calculation
• Comparing alternate tooth counts or modules

Use caution for final design

• Lewis formula mainly addresses bending at the tooth root
• It does not fully capture contact stress (pitting), micro-geometry, and full dynamic loading behavior
• Final validation should follow AGMA/ISO methods and application-specific fatigue criteria

Step-by-step input guidance

1. Enter material allowable stress in MPa.
2. Enter module and face width in mm.
3. Provide the tooth count of the gear being checked.
4. Select tooth system or custom y value.
5. Adjust C_v and K_s if you have better design data.
6. Click Calculate to get load, torque, and optional power.

Practical design tips

• Very low tooth count can produce undercut and weak roots; watch for non-physical y values.
• Face width often starts near 8m to 12m for preliminary sizing, then refined by standards.
• If vibration, shock, or misalignment is significant, use conservative service factors.
• Always confirm units before comparing with torque and power requirements.

Summary

The Lewis formula remains one of the fastest ways to estimate gear tooth bending capacity during early design. Use this calculator as a first-pass engineering tool, then refine with detailed standards-based calculations and safety factors appropriate for your application.