BIZOMAT MAKEOS · Gears
Gear Tooth Bending Stress (Lewis)
What bending stress does the tooth load cause at the tooth root, by the Lewis equation?
Lewis tooth bending stress with the Barth velocity factor for the manufacturing method, compared with an allowable stress, as a first check before a full gear rating.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Tangential force | N | Tangential force at the pitch circle (see the gear tooth forces tool). |
| Module | mm | Module. |
| Face width | mm | Face width in contact. |
| Lewis form factor Y | Lewis form factor for the tooth count and pressure angle, from a form factor table (based on module, not diametral pitch). | |
| Pitch diameter | mm | Pitch diameter, for pitch-line velocity. |
| Speed | rpm | Gear speed. |
| Tooth manufacture | choice | How the teeth are produced; this sets the Barth velocity factor. |
| Allowable bending stress | MPa | Allowable bending stress for the gear material and heat treatment from your design data. |
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 force
- What it is
- The force acting tangentially at the pitch circle that bends the tooth, from the transmitted torque and the pitch diameter.
- 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
- Bending stress rises in proportion.
- Get it right
- Work it out with the gear tooth forces tool for the same gear and speed.
What this means: Module
- What it is
- The module of the gear: pitch diameter divided by the number of teeth, which sets the tooth size.
- 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 3 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Bending stress falls in proportion as module rises.
- Get it right
- Enter the module, not the diametral pitch; convert inch gears first.
What this means: Face width
- What it is
- The axial width of the tooth that is in contact with the mating gear.
- 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 35 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Bending stress falls in proportion as face width rises.
- Get it right
- Use the width actually in mesh, not the blank width of a wider gear.
What this means: Lewis form factor Y
- What it is
- The Lewis form factor, a dimensionless number describing the tooth’s bending strength for its tooth count and pressure angle, from a form factor table.
- 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 0.32 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Bending stress falls in proportion as Y rises.
- Get it right
- Take Y from a table for the module form of the Lewis equation that the method uses; factors defined for other forms of the equation give different numbers.
What this means: Pitch diameter
- What it is
- The pitch diameter of the gear, used with the speed to find the pitch-line velocity for the velocity factor.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the pitch diameter of the gear whose speed you enter.
What this means: Speed
- What it is
- The rotational speed of the same gear, used to find the pitch-line velocity that sets the velocity factor.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 1000000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1450 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Higher speed raises the velocity factor and the bending stress.
- Get it right
- Use the running speed of this gear, not of the mating gear.
What this means: Tooth manufacture
- What it is
- How the teeth are produced, which chooses the Barth velocity factor allowing for dynamic loading from tooth errors.
- Units and format
- Choose one of 4 options from the list: Cast teeth; Cut or milled; Hobbed or shaped; Shaved or ground.
- Example
- The form opens with “Hobbed or shaped” selected as a worked example. Choose the option that matches your case.
- Get it right
- Pick the final finishing method of the teeth, not the roughing method.
What this means: Allowable bending stress
- What it is
- The bending stress the gear material may carry in this application, for its material and heat treatment, from your design data or gear standard.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use a design allowable that already includes your safety margin.
Method and formulas
- V = π d n ÷ 60 000 (m/s).
- Velocity factor Kv: cast (3.05 + V)/3.05; cut (6.1 + V)/6.1; hobbed (3.56 + √V)/3.56; ground √((5.56 + √V)/5.56).
- σ = Kv Ft ÷ (b m Y).
- Example: 3000 N on a module 3, 35 mm wide tooth with Y = 0.32 gives σ = 142.81 MPa.
Limitations
- The Lewis equation ignores stress concentration, load sharing and many rating factors; use ISO 6336 or AGMA 2001 for design.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Quickly size the module and face width
- Screen a gear for tooth breakage
- Compare materials and widths
Who it is for
- Gear designers
- Machinists
- Maintenance engineers
What you get
- Pitch-line velocity and velocity factor
- Lewis bending stress
- Ratio to allowable
Interpreting the result
- Treat the result as a screening value; a full rating will usually give a higher stress.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2131–2160. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Design printed gears that mesh
In Printed Mechanisms That Run
What it covers: Choose module and tooth counts a printer can make, lay out a planetary stage that assembles, check tooth bending against measured coupon strength and add the backlash printing needs.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.