Enter module, tooth count, face width, power, speed, and material to compute Lewis bending stress and Hertz contact stress. Evaluate safety factors against allowable limits.
Gear Geometry
Operating Conditions
Material
Results
Bending SF: — Contact SH: —
Theory
Results
Tangential Force Ft
—
N
Pitch Diameter d₁
—
mm
Bending Stress σF
—
MPa
Contact Stress σH
—
MPa
Stress vs Allowable Stress
Gear Mesh Sketch
KF = KH = 1.25 (dynamic load factor) ZE = 191 MPa0.5 (steel pair)
Lewis form factor Y approximated as Y ≈ 0.154 − 0.912/z (full-depth teeth)
Key Formulas
σF = Ft·KF / (b·m·Y)
σH = ZE·√(Ft·KH / (b·d₁·u))
What is Gear Tooth Stress?
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What exactly is "gear tooth stress"? When I see gears, they just seem to mesh together smoothly. Where does the stress come from?
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Basically, every time a gear tooth engages, it gets hit with two main forces. First, a bending force tries to snap the tooth off at its root. Second, a huge squeezing force at the contact point can cause surface pitting or wear. In practice, this is why gear design isn't just about shape, but about strength. Try moving the "Power" slider in the simulator above—you'll see how transmitting more power instantly increases the forces and stresses involved.
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Wait, really? So there are two different types of stress to calculate? Which one is more likely to cause failure?
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Good question! It depends on the material and operating conditions. Bending stress is the primary concern for brittle materials or high-impact loads—imagine a tooth breaking off. Contact stress is key for hard, surface-hardened gears where pitting (small surface craters) is the failure mode. A common case is in car transmissions: hardened steel gears often fail from pitting over time. In the simulator, try switching the "Material" selector. You'll see the allowable stress limits change, which determines which type of stress governs the design for that material.
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That makes sense. So the "Lewis Form Factor" in the calculator... what is that? It sounds complicated.
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Think of it as a "shape efficiency" number. It accounts for the fact that not all tooth shapes are equally strong—a thicker root is better at resisting bending. The factor is pre-calculated based on the number of teeth and pressure angle. For instance, a gear with 20 teeth has a different form factor than one with 50 teeth. When you adjust the "Number of Teeth" slider here, the Lewis Form Factor (Y) updates automatically, showing you how the geometry directly influences the bending stress result.
Physical Model & Key Equations
The primary model for predicting bending failure at the tooth root is the Lewis Bending Equation. It treats the tooth as a cantilever beam with the load applied at the tip.
$$ \sigma_F = \frac{F_t \cdot K_F}{b \cdot m \cdot Y}$$
Where:
$\sigma_F$ = Bending stress at the tooth root (MPa)
$F_t$ = Tangential force transmitted by the gear (N)
$K_F$ = Dynamic load factor (accounts for shock and vibration)
$b$ = Face width of the gear (mm)
$m$ = Module (mm) – a key measure of tooth size
$Y$ = Lewis Form Factor (dimensionless, based on tooth geometry)
For surface durability, the Hertzian Contact Stress model is used. It calculates the compressive stress at the point where two curved tooth surfaces are pressed together.
Where:
$\sigma_H$ = Maximum contact (Hertz) stress (MPa)
$Z_E$ = Elastic Coefficient ($\sqrt{\text{MPa}}$), based on materials' Young's modulus and Poisson's ratio (~191 for steel-on-steel)
$K_H$ = Dynamic factor for contact stress
$d_1$ = Pitch diameter of the pinion (mm)
$u$ = Gear ratio ($= z_2 / z_1$)
The square root shows this stress is less sensitive to load increase than bending stress.
Frequently Asked Questions
Both evaluations are necessary, but generally, Lewis bending stress is checked first to prevent tooth root breakage. However, Hertz contact stress is also important to prevent pitting (fatigue failure) on the tooth surface. If the safety factor is less than 1, it is recommended to review the material or specifications.
It is a dimensionless number determined by the number of teeth and the pressure angle (typically 20°). The smaller the number of teeth, the smaller Y becomes, resulting in larger bending stress. In this tool, it is automatically calculated from the input number of teeth and pressure angle, so manual setting is not required.
It is set based on the tensile strength or fatigue limit of the material. Generally, refer to standard values for gear steels (such as S45C, SCM440) or values listed in mechanical design handbooks. This tool automatically calculates the safety factor by comparing the input allowable stress with the calculated stress.
Ft is calculated as Ft = 60,000 × Power (kW) / (π × Pitch circle diameter (mm) × Rotational speed (rpm)). The pitch circle diameter is obtained from the module × number of teeth, so it is automatically calculated from the input values. Bending stress and contact stress are then calculated based on this Ft.
Real-World Applications
Automotive Transmissions: Every gear change subjects the gear teeth to high torque and shock loads. Engineers use these stress calculations to balance gear size (and weight) against required durability over hundreds of thousands of cycles, ensuring the transmission lasts the life of the vehicle.
Wind Turbine Gearboxes: These operate under highly variable and unpredictable loads from wind gusts. Accurate bending and contact stress analysis is critical to prevent catastrophic failure in these hard-to-access and expensive components, often pushing materials to their limits.
Industrial Robotics: Precision robotic arms use compact, high-torque gearboxes (like harmonic drives or planetary gears). Minimizing gear tooth stress allows for smaller, lighter gearboxes, which improves the robot's speed, accuracy, and payload capacity.
Aerospace Actuators: Landing gear or flight control surface actuators require extreme reliability. Gear design here uses high-strength alloys and precise stress calculations to ensure safety with an absolute minimum of weight, as every gram counts in aircraft design.
Common Misconceptions and Points to Note
First, the misconception that "increasing the module always makes the gear stronger". While a larger module does make the tooth thicker and increases bending strength, the pitch circle diameter also increases. This means the tangential force $F_t$ acting on the tooth face remains the same for a given torque, so the contact stress is not improved. The downsides of larger, heavier, and more expensive gears can often outweigh the benefits. For instance, instead of increasing the module from 2 to 3, widening the face width from 10mm to 15mm often reduces both stresses more effectively.
Next, the handling of input parameters like "Power" and "Rotational Speed". You input rated values into the tool, but in actual machinery, "shock loads" during startup or emergency stops can often reach 2-3 times the rated load. This is precisely the role of the "Load Factors $K_F$, $K_H$" – to account for these overloads. As a rule of thumb, for applications with severe load fluctuations like conveyor drives, if you don't set this factor to 1.5 or higher in your calculations, you might encounter abnormal noise shortly after starting operation.
Finally, a blind spot regarding material data. The label "S45C" in the tool is essentially just a material "type name". Even within S45C, the allowable stress can differ by nearly a factor of two between heat-treated and tempered "quenched and tempered steel" and untreated "raw material". When you select a material in the tool, always check the specifications to see what "heat treatment condition" or "surface hardness" is assumed behind it. Being vague here renders the calculated safety factor completely unreliable.
Material selection is critical: hardened steel (60 HRC) sustains 1600+ MPa contact stress vs. cast iron limited to 800 MPa; verify heat treatment certificates for critical applications
Contact stress typically governs gear life; keep below material fatigue limit or pitting occurs within 10⁷ cycles