BIZOMAT MAKEOS · Strength
Shaft Design: Strength, Twist and Critical Speed
What shaft diameter satisfies strength, torsional stiffness and critical speed together for this drive?
Shaft sizing that checks three criteria at once: diameter for combined bending and torsion with shock factors (maximum-shear theory, with the handbook’s 75% working stress allowance for a keyway), diameter for a twist limit per metre, and diameter that keeps the first critical speed of the shaft with a central mass a margin above running speed — reporting which governs.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Duty | ||
| Power | kW | Power transmitted. |
| Speed | rpm | Shaft speed. |
| Maximum bending moment | N·m | Largest bending moment on the shaft from belt pull, gear forces and weights. |
| Shock factor, bending | Combined shock and fatigue factor for bending (1.5 for gradually applied, rotating loads). | |
| Shock factor, torsion | Combined shock and fatigue factor for torsion. | |
| Material and limits | ||
| Allowable shear stress | MPa | Allowable shear stress for the shaft material. |
| Keyway at the critical section | choice | Whether the critical section has a standard keyway. |
| Twist limit | °/m | Allowable angle of twist per metre of length. |
| Shear modulus | GPa | Shear modulus. |
| Elastic modulus | GPa | Elastic modulus. |
| Density | kg/m³ | Density of the shaft material. |
| Critical speed | ||
| Bearing span | mm | Distance between bearings. |
| Central mass (pulley, rotor) | kg | Mass carried at mid-span. |
| Required critical ÷ running speed | How far above running speed the first critical speed must be. | |
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.
Duty
What this means: Power
- What it is
- The power the shaft transmits through the critical section.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW to 100000 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 rises in proportion to power.
What this means: Speed
- What it is
- The rotational speed of the shaft, which sets the torque for the power and is compared with the critical speed.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 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
- For the same power a higher speed lowers the torque.
What this means: Maximum bending moment
- What it is
- The largest bending moment on the shaft from belt pull, gear forces and weights, read from a bending moment diagram.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0 N·m to 100000000 N·m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A moment, not a force: multiply each load by its lever arm.
What this means: Shock factor, bending
- What it is
- The combined shock and fatigue factor applied to the bending moment, reflecting how suddenly and how often the load is applied.
- 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 factor, torsion
- What it is
- The combined shock and fatigue factor applied to the torque in the strength check.
- 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.
Material and limits
What this means: Allowable shear stress
- What it is
- The allowable shear stress for the shaft material, from your design code or material data.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 1000 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.
- Effect on the result
- The diameter for strength varies with the inverse cube root of this stress.
What this means: Keyway at the critical section
- What it is
- Whether there is a standard keyway at the critical section; if so the tool reduces the allowable shear stress to 75 % of the value entered.
- Units and format
- Choose one of 2 options from the list: Yes (use 75% of the allowable); No.
- Example
- The form opens with “Yes (use 75% of the allowable)” selected as a worked example. Choose the option that matches your case.
What this means: Twist limit
- What it is
- The largest angle of twist you allow per metre of shaft length under the transmitted torque.
- Units and format
- Enter a number in °/m: °/m. The form accepts 0.000001 °/m to 10 °/m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.25 °/m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The diameter for twist varies with the inverse fourth root of this limit.
- Get it right
- Degrees per metre, not degrees over the whole shaft.
What this means: Shear modulus
- What it is
- The shear modulus of the shaft material, used in the twist check.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 300 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 79.3 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- In GPa, not MPa.
What this means: Elastic modulus
- What it is
- The elastic modulus of the shaft material, used for the static deflection in the critical speed check.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 206 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- In GPa, not MPa.
What this means: Density
- What it is
- The density of the shaft material, used to include the shaft’s own weight in the deflection.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 20000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Critical speed
What this means: Bearing span
- What it is
- The distance between the two bearings that support the shaft as a simple span.
- 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 400 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer span lowers the critical speed, so a larger diameter is needed for the same margin.
What this means: Central mass (pulley, rotor)
- What it is
- The mass of the pulley, rotor or other part carried at mid-span between the bearings.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 1000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Mass in kg, not weight in N; 0 for the shaft alone.
What this means: Required critical ÷ running speed
- What it is
- How many times the running speed the first critical speed must be, as a margin against whirling.
- 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.
Method and formulas
- Torque T = P ÷ ω; strength: d = [16 ÷ (π τ) √((K_m M)² + (K_t T)²)]^(1/3), with τ × 0.75 for a keyway.
- Twist: J ≥ T ÷ (G θ), d = (32 J ÷ π)^(1/4).
- Critical speed ≈ 946 ÷ √δ rpm (δ in mm), with δ the static deflection of the simply supported shaft under the central mass plus its own weight; solved for d.
- Example: 15 kW at 1450 rpm with 150 N·m of bending needs a 41.295 mm shaft; twist governs.
Limitations
- Simply supported span with one central mass; overhung loads and several masses need a fuller analysis.
- Stress concentrations other than a standard keyway need their own factors.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a drive shaft for a motor, pulley or fan
- Check whether stiffness or whirling governs rather than strength
- Decide a shaft size before detailed design
Who it is for
- Design engineers
- Mechanical engineers
- Engineering managers
What you get
- Diameter for strength
- Diameter for twist
- Diameter for critical speed and the governing size
Interpreting the result
- When twist governs, a slightly larger diameter is often cheaper than a stronger material.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 295–303. 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
Sizing a shaft for strength, twist and whirl
In Whirling Shafts by the Numbers
What it covers: Sizing a rotating shaft against three criteria at once, combined bending and torsion, twist and critical speed, and why on fast, heavy rotors the critical speed usually sets the diameter.
Useful alongside · BIZOMAT original book
Stress waves in bars: speed, impedance and reflection
In Vibrating Machine Elements by the Numbers
What it covers: Estimate impact stress from wave speed and impedance, follow a stress wave through a shaft shoulder and a free end, and see why bearings must never be hammered on.
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Open the Shaft Design: Strength, Twist and Critical Speed calculator