BIZOMAT MAKEOS · Strength
Hollow Shaft Conversion (Brittle Material, Size, Weight, Twist)
If I make this cast-iron (brittle) shaft hollow, what outside and bore diameters keep it as strong, how much weight do I save, and does it still fit and stay stiff enough?
Conversion of a solid shaft in a brittle material to a hollow one of equal strength: the solid diameter by the maximum-normal-stress method with shock factors and the hollow outside and bore diameters for the bore ratio (as the brittle and hollow shaft tool), wall thickness, mass per metre of both, angle of twist per metre for the torque, and checks against the largest outside diameter that fits, the least wall and your twist limit.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Bending moment | N·m | Bending moment on the shaft. |
| Torque | N·m | Torque on the shaft. |
| Shock factor, bending | Shock and fatigue factor on bending. | |
| Shock factor, torsion | Shock and fatigue factor on torsion. | |
| Allowable tensile stress | MPa | Allowable tensile stress of the brittle material. |
| Bore ratio (inside ÷ outside) | Bore diameter as a fraction of the outside diameter. | |
| Material and limits | ||
| Density | kg/m³ | Density of the shaft material. |
| Shear modulus | GPa | Shear modulus of the material (cast iron about 40 GPa). |
| Largest outside diameter that fits | mm | Largest outside diameter the bearings or housing allow. |
| Least wall thickness | mm | Thinnest wall you accept for casting or machining. |
| Largest twist | °/m | Largest angle of twist per metre you accept. |
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: Bending moment
- What it is
- The largest bending moment on the shaft at the section being sized, from a beam analysis of the shaft’s loads and bearings.
- 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.
- Effect on the result
- A larger moment increases the shaft diameter, which grows with roughly the cube root of the combined load.
- Get it right
- Enter newton-metres, not N·mm, and use the moment at the most heavily loaded section.
What this means: Torque
- What it is
- The torque carried by the shaft at that section, from the power and speed it transmits.
- 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.
- Effect on the result
- More torque increases the diameter and the angle of twist.
- Get it right
- Enter newton-metres. The twist check uses this torque without the shock factor.
What this means: Shock factor, bending
- What it is
- The shock and fatigue factor applied to the bending moment for the type of loading, from your shaft design code or handbook.
- 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.
- Effect on the result
- It multiplies the bending moment before sizing, so a larger factor gives a larger shaft.
- Get it right
- Take it from your design code for the loading type (steady, sudden or heavy shock).
What this means: Shock factor, torsion
- What it is
- The shock and fatigue factor applied to the torque for the type of loading, from your shaft design code or handbook.
- 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.
- Effect on the result
- It multiplies the torque before sizing, so a larger factor gives a larger shaft.
- Get it right
- It is used only for strength sizing; it does not affect the twist check.
What this means: Allowable tensile stress
- What it is
- The allowable tensile stress of the brittle shaft material, such as cast iron, from the material standard and 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.
- Effect on the result
- The solid diameter varies with the inverse cube root of this stress.
- Get it right
- Use a tensile allowable for the brittle material, not a shear value.
What this means: Bore ratio (inside ÷ outside)
- What it is
- The bore diameter of the hollow shaft as a fraction of its outside diameter; zero gives a solid shaft.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.95. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger bore ratio saves more weight but needs a larger outside diameter and a thinner wall.
- Get it right
- Type a fraction, not a percentage: a bore half the outside diameter is 0.5.
Material and limits
What this means: Density
- What it is
- The density of the shaft material, used to compare the mass per metre of the solid and hollow shafts.
- 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 7200 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Both masses per metre rise in direct proportion to density; the weight saving does not change.
What this means: Shear modulus
- What it is
- The shear modulus of the shaft material, used to work out the angle of twist, from the material data sheet.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 500 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The angle of twist falls in inverse proportion to the shear modulus.
- Get it right
- Enter gigapascals, not megapascals.
What this means: Largest outside diameter that fits
- What it is
- The largest outside diameter that fits the bearings, housing or hub the shaft passes through.
- 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 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 smallest diameter limit along the shaft, such as a bearing bore.
What this means: Least wall thickness
- What it is
- The thinnest wall of the hollow shaft you accept for casting soundness, machining or handling.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Base it on the foundry’s or machine shop’s capability for the material.
What this means: Largest twist
- What it is
- The largest angle of twist per metre of shaft length you accept, set by the accuracy or stiffness the drive needs.
- Units and format
- Enter a number in °/m: °/m. The form accepts 0.01 °/m to 100 °/m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 °/m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter degrees per metre, not radians or degrees over the whole shaft.
Method and formulas
- Solid diameter D³ = 5.1 ÷ S_t × [K_m M + √((K_m M)² + (K_t T)²)]; hollow outside = B × solid with B = ∛[1 ÷ (1 − k⁴)] (as the brittle and hollow shaft tool).
- Wall = (outside − bore) ÷ 2; mass per metre = area × density.
- Twist per metre = T ÷ (G J) with J = π (D⁴ − d⁴) ÷ 32.
- Example: solid 52.311 mm or hollow 54.788/32.873 mm; the hollow shaft weighs 10.863 kg/m against 15.474 kg/m and twists 0.93022°/m.
Limitations
- Maximum-normal-stress theory for brittle materials; fatigue and stress raisers need separate checks.
- Equal strength in combined bending and torsion.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Lighten a cast-iron or brittle shaft
- Check a hollow shaft fits its bearings
- Compare twist of solid and hollow shafts
Who it is for
- Design engineers
- Machine designers
- Students
What you get
- Solid and hollow diameters
- Wall, mass per metre and saving
- Twist per metre of each
Interpreting the result
- A moderate bore (k about 0.5 to 0.6) saves a quarter to a third of the weight for a small increase in diameter.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 295–303. Topic reference; no source tables or recommended settings are reproduced.
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Open the Hollow Shaft Conversion (Brittle Material, Size, Weight, Twist) calculator