BIZOMAT MAKEOS · Gears
Gear Tooth Bending Stress (AGMA Form)
What bending stress does this tangential load put on the gear tooth root, and is the safety factor enough?
Tooth-root bending stress by the AGMA fundamental rating formula in metric form, from tangential load, face width, module, the bending geometry factor and the application, dynamic, size, load-distribution and rim-thickness factors, compared with your allowable bending stress.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Tangential load at the pitch circle | N | Transmitted tangential force at the operating pitch diameter. |
| Face width | mm | Net face width of the narrower gear. |
| Transverse module | mm | Transverse module (normal module ÷ cos helix angle for helical gears). |
| Bending geometry factor J (Y_J) | Bending strength geometry factor from AGMA 908 or the gear design software. | |
| Overload (application) factor | Application factor for the driving and driven machinery. | |
| Dynamic factor | Dynamic factor for pitch-line velocity and gear accuracy. | |
| Size factor | Size factor (1 unless the standard calls for more). | |
| Load distribution factor | Face load distribution factor for alignment and face width. | |
| Rim thickness factor | Rim thickness factor (1 for solid gears). | |
| Allowable bending stress (adjusted for life and reliability) | MPa | Allowable bending stress for the material, already adjusted for stress cycles, temperature and reliability. |
| Required safety factor | Safety factor your design rule requires. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Tangential load at the pitch circle
- What it is
- The tangential force transmitted between the teeth at the operating pitch circle: the torque divided by the pitch radius.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The bending stress rises in proportion.
- Get it right
- Use the pitch radius, half the pitch diameter, when converting from torque.
What this means: Face width
- What it is
- The net face width of the teeth in contact, normally the face width of the narrower gear of the pair.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The bending stress is inversely proportional to the face width.
What this means: Transverse module
- What it is
- The transverse module of the gear, its pitch diameter divided by its number of teeth, taken in the plane of rotation.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The bending stress is inversely proportional to the module.
- Get it right
- For a helical gear, divide the normal module by the cosine of the helix angle.
What this means: Bending geometry factor J (Y_J)
- What it is
- The AGMA bending strength geometry factor, which accounts for the tooth form, root fillet and load sharing between teeth.
- Units and format
- Enter a plain number with no unit. The form accepts 0.05 to 1. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.35 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The bending stress is inversely proportional to it.
- Get it right
- Take it from AGMA 908 or gear design software for the actual tooth geometry and mating gear.
What this means: Overload (application) factor
- What it is
- The overload (application) factor for load peaks from the driving and driven machinery above the nominal tangential load.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.25 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The bending stress rises in proportion, as it does for each of the K factors.
- Get it right
- Take it from the AGMA standard or your design rules for the actual driver and driven machine.
What this means: Dynamic factor
- What it is
- The dynamic factor for the extra tooth load caused by transmission errors at the pitch-line speed, which depends on gear accuracy.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Size factor
- What it is
- The size factor for non-uniform material properties in large teeth; normally 1 unless the standard calls for more.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Load distribution factor
- What it is
- The face load distribution factor for uneven load across the face width caused by misalignment, deflection and lead errors.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Rim thickness factor
- What it is
- The rim thickness factor for thin-rimmed gears whose rim can bend under tooth load; 1 for solid gears.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Allowable bending stress (adjusted for life and…
- What it is
- The allowable tooth bending stress for the gear material and heat treatment, already adjusted for the number of load cycles, the temperature and the reliability required.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 5000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 350 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safety factor rises in proportion.
- Get it right
- Take it from the AGMA standard or the material supplier; it is not the tensile or yield strength.
What this means: Required safety factor
- What it is
- The minimum ratio of allowable to calculated bending stress that your design rules require; the verdict compares the calculated safety factor with it.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher requirement lowers the reported load capacity in proportion; the stress is unchanged.
Method and formulas
- σ_F = F_t K_o K_v K_s K_H K_B ÷ (b m_t Y_J) — the metric form of the AGMA fundamental bending equation.
- Safety factor = allowable ÷ σ_F; capacity = F_t × safety factor ÷ required factor.
- Example: 3000 N on a 30 mm face, module 2.5, J = 0.35 gives 222.86 MPa.
Limitations
- All factors and the allowable come from AGMA 2001 / 2101 or ISO 6336 data for your gears.
- Pitting resistance is a separate check.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check gear tooth bending strength for a new duty
- Compare face width or module options
- Rate an existing gear pair for a higher load
Who it is for
- Gear designers
- Machinists
- Maintenance engineers
What you get
- Tooth bending stress
- Safety factor
- Load capacity at the required factor
Interpreting the result
- Raising the module raises bending strength fastest; wider faces help only while alignment keeps K_H low.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 3304–3320. Topic reference; no source tables or recommended settings are reproduced.
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