Every time a gear tooth comes into mesh, it takes a sharp push from its partner and then lets go a fraction of a second later. Run the gearbox at a few thousand rpm and each tooth absorbs that pulse millions of times an hour. The tooth does not snap from a single overload; it fails slowly, from a fatigue crack growing at the root fillet where the bending stress is highest. Predicting that root stress is the heart of gear strength design.
This article explains how to model a gear tooth as a short cantilever beam, how the classic Lewis equation converts transmitted power into a bending stress, and how to run the numbers with a worked example.
Why this calculation matters
Gears fail in two main ways, and bending is one of them. The other is surface pitting from contact pressure, but tooth breakage at the root is the more dramatic failure — it can shed a tooth and wreck the gearbox in one revolution. A bending-stress check is how you keep the root stress comfortably below the material's fatigue limit.
The check also drives sizing decisions. Module, face width, and tooth count all appear in the stress equation, so a bending calculation tells you directly whether a gear is too small, whether widening the face will rescue it, or whether you need a stronger material. Doing this early, before cutting metal, is far cheaper than discovering a weak tooth on a test rig.
The core method
Wilfred Lewis proposed the foundational model in 1892: treat the gear tooth as a cantilever beam, fixed at the root and loaded at the tip by the tangential component of the mesh force. The bending stress at the root then follows ordinary beam theory, repackaged into gear terms.
First you need the force. Power transmitted through a gear is the product of the tangential load and the pitch-line velocity, so the tangential load is:
W_t = P / v
The pitch-line velocity itself comes from the pitch diameter and the rotational speed:
d = m * N
v = pi * d * n / 60
Here m is the module, N the tooth count, n the speed in rpm, and d the pitch diameter. With the tangential load known, the Lewis bending stress is:
sigma = W_t / (F * m * Y)
F is the face width, m the module again, and Y is the Lewis form factor — a dimensionless number that captures the tooth's shape. A tooth with more teeth, or a fuller profile, has a larger Y and so a lower stress. Typical values run from about 0.25 for a small pinion to past 0.4 for a gear with many teeth.
The structure of the equation is worth reading. Stress falls as face width grows, as module grows, and as the form factor grows. The cheapest levers are usually module and face width; changing the form factor means changing the tooth count or the cutter.
The plain Lewis equation is a static, baseline estimate. Real designs multiply in a velocity factor for dynamic loading and other correction factors, but the Lewis stress is the honest starting point every refinement builds on.
A worked example
Check the pinion of a spur pair. It transmits power P = 5 kW, has N = 20 teeth, a module m = 4 mm, and runs at n = 1000 rpm. The face width is F = 40 mm, and for 20 teeth the Lewis form factor is Y = 0.32.
Step 1 — pitch diameter.
d = m * N = 0.004 * 20 = 0.080 m
Step 2 — pitch-line velocity.
v = pi * d * n / 60 = pi * 0.080 * 1000 / 60 = 4.19 m/s
Step 3 — tangential load.
W_t = P / v = 5000 / 4.19 = 1194 N
Step 4 — Lewis bending stress.
sigma = W_t / (F * m * Y) = 1194 / (0.040 * 0.004 * 0.32) = 23.3 MPa
So this pinion carries about 23.3 MPa of bending stress at the tooth root under steady load. That is a low number for a steel gear — typical gear steels tolerate root stresses an order of magnitude higher — which tells you the static bending case has plenty of margin. The remaining margin is what dynamic and overload factors eat into, and it is also why surface contact stress, not bending, often becomes the governing limit for this kind of gear.
Common mistakes
Mixing up module and diametral pitch. Module is a length in millimetres; diametral pitch is its imperial inverse, in teeth per inch. The Lewis equation in the form above expects module. Plugging in the wrong one throws the stress off by a large factor.
Using one form factor for both gears. The pinion and the gear in a pair usually have different tooth counts, so they have different Y values and different bending stresses. The pinion, with fewer teeth, is often the weaker member — check it first.
Forgetting the velocity factor. The plain Lewis stress assumes a smooth, static load. At high pitch-line speeds, dynamic effects raise the effective load noticeably. A baseline Lewis number that looks safe can still fail once dynamic loading is included.
Ignoring stress concentration at the fillet. The Lewis model uses a nominal beam stress. The actual root fillet concentrates stress further, which is why refined methods add a geometry factor. Treat the Lewis value as a floor, not the final word.
Assuming bending governs. For many hardened gears, surface pitting from contact pressure limits the design before bending does. A complete check looks at both failure modes, not just the root.
Try the interactive NovaSolver calculator
Stepping through the velocity, the tangential load, and the stress by hand is good practice once, but design iteration calls for speed. The Gear Tooth Stress Calculator on NovaSolver lets you enter the module, tooth counts, face width, transmitted power, speed, and material, then returns the tangential force, pitch diameter, Lewis bending stress, and Hertz contact stress together — so you can see both failure modes against allowable limits at once.
Related calculators
- Gear tooth calculator — for tooth geometry: addendum, dedendum, and the profile dimensions behind the form factor.
- Gear ratio calculator — to set the speed and torque each gear in the pair must carry.
- Gear train calculator — for multi-stage drives, where each stage needs its own strength check.
The complete gear and machine-element toolkit is in the mechanical engineering tools hub.
Closing note
The Lewis equation endures because it captures the right idea in a single line: a gear tooth is a small cantilever, and the stress at its root depends on the load it carries and the geometry that resists it. Convert power to a tangential force, divide by face width, module, and form factor, and you have a defensible baseline stress. Treat it as the starting point, layer on dynamic and geometry corrections, and always check contact stress alongside it. A tooth that survives both checks is a tooth that lasts.
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