Gear Tooth Bending Stress Calculator (Lewis Equation) for Spur Gears

A first strength check for a spur gear tooth in bending.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the number of teeth on the gear and its mate, the module and the face width.

Enter the power, the speed, how the teeth are made and the allowable stress.

The calculator gives the force, the Lewis stress and the factor of safety, and the speed and torque of the mating gear.

Formulas used

  • W_t = \frac{P}{v}
  • \sigma = \frac{K_vW_t}{FmY}
  • v = \frac{\pi dn}{60000}

Worked example

With the default numbers in the calculator above:

  1. Pitch circle diameter and pitch line velocity: 100 mm and 5.236 m/s
  2. Tangential force at the pitch circle: 954.93 N
  3. Velocity (dynamic) factor for that way of making the teeth: 1.643
  4. Lewis form factor for the number of teeth (20 degree full depth teeth): 0.322
  5. Bending stress at the tooth root (Lewis equation): 24.359 MPa
  6. Factor of safety on the allowable stress: 5.747
  7. Most power this gear can carry at that stress (the Kv factor held fixed): 28.737 kW
  8. Speed and torque of the mating gear: 333.33 rev/min and 143.24 N m (the first gear carries 47.746 N m)
  9. Centre distance: 200 mm

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is the Lewis equation?

It treats a gear tooth as a cantilever beam loaded at its tip, to find the bending stress at the root.

What is the Lewis form factor?

A number that depends on the tooth shape and the number of teeth. It is larger for gears with more teeth.

Is the Lewis check enough?

No. It is a bending check only. Surface fatigue (pitting) often governs, so a full rating to AGMA or ISO 6336 is needed.

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Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any machine design text (spur gear tooth strength, Lewis equation and Barth velocity factors).

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