BIZOMAT MAKEOS · Power Transmission
Overhung Gear Shaft Sizing (Gear Forces, Bending, Shaft Diameter, Stress)
For a spur or helical gear overhung on a shaft, what forces does the gear put on the shaft, what diameter does the shaft need, and is my chosen diameter within stress?
Sizing of a shaft carrying an overhung spur or helical gear: tangential, radial and axial gear forces from the power, speed and pitch diameter (as the gear forces tool), the resultant bending moment at the bearing from the overhang and the axial-force couple, the diameter needed by the ASME-style shock-factor method with a keyway allowance (as the shaft diameter tool), and the combined axial, bending and torsional stress and safety factor of the chosen diameter (as the combined shaft stress tool).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Power transmitted | kW | Power through the gear. |
| Shaft speed | rpm | Speed of the shaft. |
| Gear pitch diameter | mm | Pitch diameter of the gear on this shaft. |
| Normal pressure angle | ° | Normal pressure angle of the teeth. |
| Helix angle (0 = spur) | ° | Helix angle; 0 for a spur gear. |
| Overhang (bearing to gear centre) | mm | Distance from the centre of the nearest bearing to the gear’s mid-face. |
| Shaft | ||
| Shaft diameter at the bearing | mm | Diameter of the shaft at the bearing (the critical section). |
| Shock and fatigue factor, bending | Combined shock and fatigue factor on bending. | |
| Shock and fatigue factor, torsion | Combined shock and fatigue factor on torsion. | |
| Allowable shear stress | MPa | Allowable shear stress for the shaft method. |
| Keyway at the section | choice | Whether a keyway is cut at the section. |
| Yield strength | MPa | Yield strength of the shaft. |
| Required safety factor on yield | Safety factor on yield you need for the combined stress. | |
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: Power transmitted
- What it is
- The power transmitted through the gear on this shaft at the operating condition being checked.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW to 1000000 kW. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Torque and all three gear forces rise in proportion to power.
What this means: Shaft speed
- What it is
- The rotational speed of the shaft carrying the gear under the load being checked.
- 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
- At a set power, higher speed means lower torque and lower gear forces.
What this means: Gear pitch diameter
- What it is
- The pitch diameter of the gear mounted on this shaft, where the tooth forces are taken to act.
- 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 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger pitch diameter lowers the tangential force for the same torque.
What this means: Normal pressure angle
- What it is
- The pressure angle of the teeth measured in the plane normal to the tooth, from the gear drawing; it sets the radial force.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 10° to 30°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a helical gear use the normal pressure angle, not the transverse pressure angle.
What this means: Helix angle (0 = spur)
- What it is
- The angle between the teeth and the gear axis; zero for a spur gear. On a helical gear it produces an axial force on the shaft.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 45°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Overhang (bearing to gear centre)
- What it is
- The distance along the shaft from the centre of the nearest bearing to the middle of the gear face, the lever arm of the gear forces.
- 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.
Shaft
What this means: Shaft diameter at the bearing
- What it is
- The shaft diameter at the bearing next to the gear, the most heavily bent section, checked for combined stress against the diameter needed.
- 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 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Shock and fatigue factor, bending
- What it is
- A multiplier on the bending moment for shock and fatigue in the ASME-style shaft formula, chosen from a design table for the type of loading.
- 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 1.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Shock and fatigue factor, torsion
- What it is
- A multiplier on the torque for shock and fatigue in the ASME-style shaft formula, chosen from a design table for the type of loading.
- 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.
- Get it right
- This is the shock factor on torsion in the shaft formula, not a stress concentration factor.
What this means: Allowable shear stress
- What it is
- The allowable shear stress for the shaft material used in the diameter formula, from your design code.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 10000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter it without the keyway reduction; the tool applies 0.75 itself when you select a keyway.
What this means: Keyway at the section
- What it is
- Whether a keyway is cut at the checked section; if so, the tool reduces the allowable shear stress used in the diameter formula.
- Units and format
- Choose one of 2 options from the list: Yes (allowable × 0.75); No.
- Example
- The form opens with “Yes (allowable × 0.75)” selected as a worked example. Choose the option that matches your case.
What this means: Yield strength
- What it is
- The yield strength of the shaft material, from the material certificate; it is compared with the von Mises stress at the chosen diameter.
- 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 350 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Required safety factor on yield
- What it is
- The least ratio of yield strength to combined von Mises stress you accept at the chosen shaft diameter.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- T = 9549 P ÷ n; F_t = 2T ÷ d; F_r = F_t tan α_n ÷ cos β; F_a = F_t tan β (as the gear forces tool).
- Moment at the bearing M = √[(F_t a)² + (F_r a + F_a d ÷ 2)²].
- Diameter d = ∛[16 √((K_m M)² + (K_t T)²) ÷ π τ] with τ × 0.75 for a keyway (as the shaft diameter tool); combined stress σ = F ÷ A + M ÷ Z, τ = T ÷ 2Z, von Mises (as the combined shaft stress tool).
- Example: the gear pushes 2,469.6 N tangentially and 930.59 N radially; the shaft needs 35.91 mm, and at 40 mm the stress gives a factor of 11.405.
Limitations
- Moment taken at the bearing nearest the gear; check other sections and deflection separately.
- Shock-factor method; detailed fatigue design uses endurance limits and notch factors.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a pinion shaft on a gearbox or motor
- Check a shaft after a power increase
- Show the effect of overhang on shaft size
Who it is for
- Design engineers
- Machine designers
- Students
What you get
- Gear forces and torque
- Bending moment at the bearing
- Diameter needed and the stress in the chosen diameter
Interpreting the result
- Halving the overhang halves the bending moment: keep gears close to their bearings.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2131–2160, 2195–2200, 295–303, 212–218. Topic reference; no source tables or recommended settings are reproduced.
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Open the Overhung Gear Shaft Sizing (Gear Forces, Bending, Shaft Diameter, Stress) calculator