Lewis Bending Strength Calculator
AGMA Bending Stress Calculator
Contact Stress (Hertzian) Calculator
Calculating gear tooth strength involves determining how much load a gear tooth can handle before it fails through bending or surface contact stress. You’ll need to evaluate both the bending stress at the tooth root and the contact stress at the tooth surface, then compare these values against the material’s allowable limits.
Two Types of Gear Tooth Failure
Gear teeth fail in two primary ways: bending failure at the root and surface failure from contact stress. Bending failure happens when the tooth snaps off at its base like a cantilever beam under excessive load. Surface failure occurs when repeated contact creates pitting and wear on the tooth face.
You must check both failure modes because gears can fail either way. A gear with thick, stubby teeth might handle bending stress well but fail from surface pitting. Conversely, thin teeth might break before showing any surface wear.
The Lewis Bending Strength Equation
The Lewis equation calculates the bending stress at the tooth root using this formula: σ = Wt / (F × m × Y)
Here’s what each variable means:
- Wt = tangential load on the tooth (in Newtons or pounds)
- F = face width of the gear (in mm or inches)
- m = module of the gear (pitch diameter divided by number of teeth)
- Y = Lewis form factor (depends on tooth shape and number of teeth)
The Lewis form factor accounts for the tooth geometry and stress concentration at the root. You can find Y values in standard gear design tables – they typically range from 0.3 to 0.5 for common tooth counts.
To use this equation, first calculate the tangential load from your torque and pitch radius. Then look up the Lewis factor for your specific gear geometry. Finally, plug everything into the formula and compare the result to your material’s allowable bending stress.
The AGMA Bending Stress Method
The American Gear Manufacturers Association (AGMA) method provides a more accurate calculation by including additional correction factors: σ = Wt × Kv × Ko × Ks × KH × KB / (F × m × J)
The correction factors account for real-world conditions:
- Ko = overload factor (1.0 for uniform loading, up to 2.0 for heavy shock)
- Kv = dynamic factor (accounts for speed and manufacturing quality)
- Ks = size factor (typically 1.0 for most gears)
- KH = load distribution factor (considers shaft deflection and misalignment)
- KB = rim thickness factor (1.0 for solid gears)
- J = geometry factor (replaces the Lewis Y factor)
Start with the base stress calculation, then multiply by each factor to get the actual stress. Most design software includes these factors automatically, but understanding them helps you make better design decisions.
Calculating Contact Stress (Hertzian Stress)
Contact stress determines surface durability using the Hertzian stress equation: σc = Cp × √(Wt × Ko × Kv × Ks × KH / (F × dp × I))
The new variables here are:
- Cp = elastic coefficient (depends on the materials of both gears)
- dp = pitch diameter of the pinion
- I = geometry factor for surface stress
The elastic coefficient accounts for how the materials deform under load. Steel-on-steel gears typically use C_p = 190 √MPa (or 2290 √psi). Different material combinations require different values.
Contact stress creates surface fatigue over millions of cycles. Even if the stress seems acceptable for a single load, repeated cycling can cause pitting and eventual failure.
Safety Factors and Material Selection
Apply safety factors of 1.5 to 2.0 for bending stress and 1.0 to 1.4 for contact stress. These factors account for uncertainties in loading, material properties, and manufacturing variations.
Choose materials based on both strength requirements:
- For bending strength: Look at the material’s yield strength or endurance limit
- For contact stress: Consider the material’s surface hardness and fatigue resistance
Case-hardened steels provide excellent surface durability while maintaining a tough core. Through-hardened steels offer consistent properties but may have lower surface ratings.
Practical Calculation Steps
Follow this systematic approach for any gear strength calculation:
- Determine the tangential load from your torque and gear geometry: Wt = 2T/dp
- Calculate bending stress using either the Lewis or AGMA method
- Calculate contact stress using the Hertzian equation
- Look up allowable stresses for your chosen material from design tables
- Compare calculated stresses to allowable values including safety factors
- Iterate your design if stresses exceed limits – adjust face width, module, or material
FAQs
What’s the difference between module and diametral pitch?
Module is the metric system’s way of defining gear tooth size (pitch diameter divided by number of teeth, in mm). Diametral pitch is the imperial equivalent (number of teeth divided by pitch diameter, in inches). They’re reciprocals when converting between systems.
How do I find the Lewis form factor for my gear?
Lewis form factors are tabulated in gear design handbooks based on the number of teeth and pressure angle. For 20° pressure angle gears, Y ranges from about 0.32 for 15 teeth to 0.47 for 300+ teeth.
Can I use the same calculation for helical gears?
Helical gears require modified calculations that account for the helix angle. The virtual number of teeth (actual teeth divided by cos³ of helix angle) replaces the actual tooth count in form factor lookups.
What safety factor should I use?
Use 1.5-2.0 for bending and 1.0-1.4 for contact stress in normal applications. Critical applications (aerospace, medical) may require factors of 3.0 or higher. Consider the consequences of failure when selecting factors.
How does gear quality affect strength calculations?
Higher quality gears (AGMA Q10-12) have lower dynamic factors because they run smoother. Lower quality gears (Q5-7) need higher dynamic factors to account for manufacturing variations and resulting impact loads during meshing.
