BIZOMAT MAKEOS · Strength
Shaft Diameter for Bending and Torsion
What shaft diameter carries this bending moment and torque at the allowable shear stress?
Minimum solid shaft diameter for combined bending and torsion by the maximum-shear-stress theory with shock and fatigue factors, and the stress in a chosen diameter.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Maximum bending moment | N·m | Largest bending moment at the section, from the shaft’s bending moment diagram. |
| Torque | N·m | Torque transmitted through the section. |
| Shock and fatigue factor, bending (Km) | Combined shock and fatigue factor for bending, from your design method (the ASME code suggests about 1.5 for gradually applied loads on rotating shafts). | |
| Shock and fatigue factor, torsion (Kt) | Combined shock and fatigue factor for torsion (about 1.0 for gradually applied loads in the ASME code). | |
| Allowable shear stress | MPa | Allowable shear stress from your design method; the ASME code uses the lower of 0.3 × yield and 0.18 × tensile strength. |
| Keyway at the section | choice | A keyway at the section reduces the allowable stress by 25% in the ASME code method. |
| Actual diameter to check (0 = none) | mm | Diameter of an existing or proposed shaft to check. |
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: Maximum bending moment
- What it is
- The largest bending moment at the shaft section being sized, read from the shaft’s bending moment diagram.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0 N·m or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter N·m, not N·mm. Combine moments in two planes as the square root of the sum of their squares first.
What this means: Torque
- What it is
- The torque transmitted through the shaft at the section being sized, for example from motor power and speed.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0 N·m or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter N·m, not N·mm; use the torque that actually passes through this section.
What this means: Shock and fatigue factor, bending (Km)
- What it is
- A combined shock and fatigue factor that multiplies the bending moment, allowing for how suddenly and how often the bending 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.
- Get it right
- Take it from your design method or code for the type of loading; do not choose it to make a shaft pass.
What this means: Shock and fatigue factor, torsion (Kt)
- What it is
- A combined shock and fatigue factor that multiplies the torque, allowing for how suddenly and how often the torque 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 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from your design method or code for the type of loading.
What this means: Allowable shear stress
- What it is
- The allowable shear stress for the shaft material in your design method. The tool sizes the shaft so the combined stress does not exceed it.
- 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.
- Effect on the result
- The required diameter varies with the inverse cube root of this stress.
- Get it right
- Enter the allowable before any keyway reduction; the keyway option applies the reduction.
What this means: Keyway at the section
- What it is
- Whether a keyway is cut at the section being sized. With a keyway the tool reduces the allowable shear stress to 0.75 of the entered value.
- 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.
- Get it right
- Choose yes if any keyseat is at or near the section.
What this means: Actual diameter to check (0 = none)
- What it is
- The diameter of an existing or proposed solid shaft at this section, to check its shear stress against the requirement. Zero skips the check.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the smallest diameter at the section, for example at the bottom of a groove.
Method and formulas
- Equivalent torque Te = √((Km M)² + (Kt T)²).
- Required d = (16 Te ÷ (π τ_allow))^(1/3), with τ_allow × 0.75 when a keyway is present.
- Check: τ = 16 Te ÷ (π d³).
- Example: M 300 N·m and T 500 N·m need at least 48.516 mm; a 50 mm shaft runs at 27.407 MPa.
Limitations
- Solid circular shafts; for hollow shafts use the polar section modulus.
- Stress concentrations at shoulders and grooves need a fatigue check (see the Goodman tool); deflection and critical speed are separate checks.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size line shafts, pulley and gear shafts
- Check an existing shaft after a load change
- Compare the effect of a keyway
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Required diameter
- Equivalent torque
- Shear stress and safety ratio in the checked diameter
Interpreting the result
- Round up to a standard size and check bearing seats, key and deflection.
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
Shafts under bending and torque
In Metal Fatigue by the Numbers
What it covers: Sizing a shaft for combined bending and torque, then checking its shoulder and keyway for fatigue with von Mises alternating and mean stresses, and the error of counting torsion twice.
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