Gear Strength Calculator

Gear Strength Analysis — Lewis Bending and AGMA Contact Stress

Gear strength analysis predicts whether gear teeth survive the loads of power transmission. Engineers check two failure modes — tooth bending fatigue and surface contact pitting — using the Lewis equation and the AGMA method. This simulator computes both stresses and safety factors live for spur and helical gears.

A small confession from the workshop: I used to drill students on Lewis numbers and then watch them sketch a gear with a perfectly square tooth root. The root fillet is where fatigue cracks actually start — the AGMA J-factor exists because Lewis's straight-cantilever assumption misses the geometry that matters most. If you ever see a snapped tooth, look at the root: the fracture face will tell you whether bending fatigue or shock overload killed it.

The Lewis bending stress equation, developed by Wilfred Lewis in 1892, treats the gear tooth as a cantilever beam: σ = W_t / (b × m × Y), where W_t is the tangential load, b is the face width, m is the module, and Y is the Lewis form factor. The form factor depends on the number of teeth and the pressure angle, and it accounts for the tooth geometry at the weakest cross-section. While the Lewis equation provides a useful baseline, it does not account for dynamic loads, load distribution across the face width, or manufacturing tolerances.

AGMA Stress Analysis Method

The American Gear Manufacturers Association (AGMA) extended the Lewis approach by introducing correction factors for real-world operating conditions. The AGMA bending stress equation is σ = W_t × K_o × K_v × K_s × K_m × K_B / (b × m × J), where K_o is the overload factor (accounts for shock loads), K_v is the dynamic factor (accounts for tooth-to-tooth speed variations), K_s is the size factor, K_m is the load distribution factor, K_B is the rim thickness factor, and J is the AGMA geometry factor. For contact stress, the AGMA equation is σ_c = C_p × √(W_t × K_o × K_v × K_s × K_m × C_f / (b × d × I)), where C_p is the elastic coefficient and I is the geometry factor for pitting resistance.

Safety Factors in Gear Design

The factor of safety (FOS) is the ratio of allowable stress to actual calculated stress. For bending: FOS = σ_allowable / σ_actual. Material selection is crucial — hardened steel gears can withstand bending stresses of 250–400 MPa and contact stresses of 1000–1500 MPa, while cast iron and bronze have significantly lower limits. A minimum FOS of 1.5 for bending and 1.0 for contact is typical for industrial applications. The gear module, face width, and number of teeth are the primary design variables that engineers adjust to achieve adequate safety margins.

Who Uses This Simulator?

This gear strength simulator is designed for mechanical engineering students studying machine design, power transmission trainees, workshop instructors teaching gear analysis, and practising engineers performing preliminary gear sizing. The SI / Imperial unit toggle, presets, undo/redo, custom-material modal, and CSV/PNG exports make it equally useful for classroom demonstrations, design assignments, and quick "back-of-envelope" sizing checks before moving to FEA.

Helical Gears — Normal vs Transverse, and Why They Are Stronger

A helical gear’s teeth are cut at a helix angle ψ to the axis, so a tooth enters mesh gradually at one end and rolls across the face instead of striking along its whole length at once. Two or more tooth pairs share the load at any instant, which is why a helical gear of the same normal module and face width carries more torque than a spur gear — and runs far quieter.

Watch what happens when you move the helix-angle slider, because it is the single most misunderstood point in helical design. This simulator holds the transverse module fixed, so the pitch diameter stays put as ψ increases and the teeth get smaller in the normal plane (m_n = m_t cos ψ). You will see contact stress fall steadily — the helix genuinely spreads the contact. Bending stress, though, barely moves: two effects nearly cancel. The tooth gets thinner in the normal plane (which raises stress) while the virtual tooth count rises and with it the form factor (which lowers it). Which wins depends on the tooth count — a 20-tooth pinion stiffens slightly, a 30-tooth one softens slightly, and neither moves more than a few per cent. Helical gears are decisively stronger at equal normal module, where raising ψ grows the gear; they are not automatically stronger at equal transverse module.

Normal and transverse planes. Helical gears are specified in two planes and confusing them is the most common error in a first gear calculation:

• Module: m_n = m_t · cos ψ — the normal module is always the smaller of the two.
• Pressure angle: tan φ_n = tan φ_t · cos ψ
• Virtual tooth count: z_v = z / cos³ψ — look the Lewis form factor up against z_v, not z. A 20-tooth gear at a 25° helix behaves in bending like a 27-tooth spur gear, so Y rises from 0.322 to 0.349.

The cost: axial thrust. The helix that spreads the load also pushes the gear along its shaft, with F_a = F_t · tan ψ. At 25° that is 47 % of the tangential force arriving as thrust the bearings must carry. It is why helix angles above about 30° are rare in single-helical drives, and why double-helical (herringbone) gears exist — the two opposing hands cancel the thrust.

What this simulator models. Switching to helical carries the normal-module relationship into the stress equations, looks the Lewis form factor up against the virtual tooth count z_v rather than z, and takes the contact geometry factor from the transverse pressure angle φ_t. Each of those reduces exactly to the spur case at ψ = 0, so a 0° helical gear and a spur gear return identical answers — the helix benefit is earned from geometry rather than applied as a flat bonus.

What is not modelled is the AGMA load-sharing ratio m_N, which raises both J and I further for a real helical mesh because more than one tooth pair is in contact. The tool is therefore conservative on helical gears: it will report a slightly higher stress and lower safety factor than AGMA 2101 would. A production design should take J from the AGMA J-factor charts for the actual helix angle and tooth count, where the value depends on the mating gear as well.

Gear Strength Formulas — Lewis & AGMA Quick Reference

Lewis Form Factor (Y) Table — 20° Full-Depth Spur Gears

Lewis form factor Y for 20° full-depth involute teeth, from 12 teeth to a 150-tooth gear. Use it with the Lewis bending equation σ = W_t / (b · m · Y), where W_t is the tangential tooth load, b the face width and m the module. These are the exact values the simulator above interpolates between, so a hand calculation from this table reproduces the tool’s answer.

Helical gears: look Y up against the virtual (formative) tooth count z_v = z / cos³ψ, not the actual tooth count. A 20-tooth gear cut at a 25° helix behaves like a 27-tooth spur gear in bending.

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