Gear Contact & Bending Stress Calculator — σH / σF Tooth Load Validator
Calculate contact stress and bending stress using torque, gear geometry, load factors, and material stress limits.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Gear geometry
Load
Load factors
Material
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
Contact stress formula:
αrad = α · π / 180
d₁ = m · z₁
Ft = 2 · T / d₁
Ktotal = KA · KV · Kβ
ZH = √((2 · cos(αrad)) / sin(αrad))
σH = ZE · ZH · √((Ft · Ktotal) / (b · d₁)) · 1000
Bending stress formula:
Y = 0.484 − 2.87 / z₁
σF = (Ft · Ktotal) / (b · m · Y)
where:
- σH — Contact stress
- σF — Bending stress
- α — Pressure angle [deg]
- αrad — Pressure angle [rad]
- m — Module
- z₁ — Number of teeth (pinion)
- d₁ — Pitch diameter d₁ [m]
- T — Torque
- Ft — Tangential force Ft [N]
- KA — Application factor
- KV — Dynamic factor
- Kβ — Load distribution factor
- Ktotal — Total load factor
- ZE — Elastic coefficient
- ZH — Zone factor
- b — Face width
- Y — Lewis factor
When to use this calculator
When to use this calculator:
- Check contact stress and bending stress for a spur gear pair using torque, module, face width, pressure angle, and tooth count.
- Compare calculated tooth stresses against contact and bending stress limits from the selected gear material.
- Evaluate how application factor, dynamic factor, and load distribution factor increase the calculated gear tooth load.
- Verify whether the pinion tooth count and gear tooth count avoid the undercut condition for the selected pressure angle.
- Estimate gear tooth stress when material data and load factors are provided manually or through the material and factor providers.
How to interpret the result
Contact stress is defined as the calculated tooth flank stress from tangential force, load factors, face width, pitch diameter, pressure angle, and elastic coefficient.
Bending stress is defined as the calculated tooth root stress from tangential force, load factors, face width, module, and Lewis factor.
Contact stress depends on torque, module, pinion tooth count, face width, pressure angle, load factors, and elastic coefficient. Increasing torque increases contact stress. Increasing face width decreases contact stress. Increasing application factor, dynamic factor, or load distribution factor increases contact stress.
Bending stress depends on torque, module, pinion tooth count, face width, and load factors. Increasing torque increases bending stress. Increasing face width decreases bending stress. Increasing application factor, dynamic factor, or load distribution factor increases bending stress.
- Safe — contact safety factor is at least 1.3 and bending safety factor is at least 1.3.
- Warning — contact safety factor or bending safety factor is below 1.3, but both are at least 1.0.
- Unsafe — contact safety factor is below 1.0 or bending safety factor is below 1.0.
- Invalid — at least one required input is not a positive finite value, the tooth count is below the undercut limit, the Lewis factor is not positive, or the stress calculation does not return a positive finite result.
The result is used to compare calculated gear tooth stress against the selected contact and bending stress limits.
Calculation example
Example:
A user wants to check whether a spur gear pair can transmit torque without exceeding contact and bending stress limits.
- m — Module: 0.005 m
- z₁ — Number of teeth (pinion): 24
- z₂ — Number of teeth (gear): 48
- b — Face width: 0.03 m
- α — Pressure angle: 20 deg
- T — Torque: 120 N·m
- KA — Application factor: 1.25
- KV — Dynamic factor: 1.10
- Kβ — Load distribution factor: 1.15
- σH,lim — Contact stress limit: 900 MPa
- σFlim — Bending stress limit: 300 MPa
- ZE — Elastic coefficient: 189.8
Result: σH = 896.6 MPa and σF = 141.2 MPa.
Assumptions and limitations
- The pressure angle is entered in degrees and converted to radians before trigonometric calculations.
- The pitch diameter is calculated as module multiplied by tooth count.
- The tangential force is calculated from torque and pinion pitch diameter.
- The total load factor is calculated as the product of application factor, dynamic factor, and load distribution factor.
- The bending calculation uses the Lewis approximation Y = 0.484 − 2.87 / z₁.
- The undercut check uses zmin = 2 / sin²(α).