BIZOMAT MAKEOS · Strength
Shaft Diameter for Brittle Materials and Hollow Shafts
What diameter does this shaft need by the maximum-normal-stress theory, and how does a hollow shaft compare?
Shaft diameter for brittle materials (cast iron, hardened steel) by the maximum-normal-stress theory with shock factors, the hollow-shaft factor B, and the outside diameter of a hollow shaft equal in strength to a solid one.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Bending moment | N·m | Maximum bending moment at the section. |
| Torque | N·m | Torque at the section. |
| Shock factor, bending | Combined shock and fatigue factor for bending. | |
| Shock factor, torsion | Combined shock and fatigue factor for torsion. | |
| Allowable tensile stress | MPa | Allowable tensile stress of the brittle material (well below its tensile strength). |
| Bore ratio (inside ÷ outside, 0 = solid) | Ratio of bore to outside diameter for a hollow shaft; 0 for solid. |
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 acting on the shaft at the section being sized, from the beam analysis of the shaft and its loads.
- 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
- This is a moment in N·m, not a force; multiply the force by its lever arm to the section.
What this means: Torque
- What it is
- The twisting moment carried by the shaft at the same section, normally the transmitted power divided by angular 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
- Use the torque at the section being sized, which may be less than the motor torque after a pulley or gear takes some off.
What this means: Shock factor, bending
- What it is
- The multiplier applied to the bending moment to allow for shock and fatigue in service. It comes from the handbook’s shock-factor table or your design standard 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.
- Get it right
- Use the bending factor here and the torsion factor in its own field; they differ for the same machine.
What this means: Shock factor, torsion
- What it is
- The multiplier applied to the torque to allow for shock and fatigue in torsion, from the same table or design standard as the bending factor.
- 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
- A steady torque with gradual starts uses the lowest factor; sudden or reversing torque needs a higher one.
What this means: Allowable tensile stress
- What it is
- The tensile stress you allow in the brittle shaft material, set well below its tensile strength by your design standard. Brittle materials fail by fracture rather than yield, so tensile stress governs.
- 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 falls with the cube root of the allowable stress.
- Get it right
- Do not enter the tensile strength itself; use the reduced allowable stress from your design standard.
What this means: Bore ratio (inside ÷ outside, 0 = solid)
- What it is
- The bore diameter of a hollow shaft divided by its outside diameter. Zero means a solid shaft; larger ratios mean thinner walls.
- 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 ratio increases the hollow factor B, so the outside diameter grows to keep the same strength.
- Get it right
- This is a ratio of diameters, not of radii or areas, and not a percentage.
Method and formulas
- Solid: D = ∛(5.1 ÷ St × [Km M + √((Km M)² + (Kt T)²)]).
- Hollow factor B = ∛(1 ÷ (1 − k⁴)); outside diameter = B × solid diameter.
- Weight ratio = B²(1 − k²); torsional stiffness ratio = B⁴(1 − k⁴).
- Example: the default loads need 52.311 mm solid, or 54.788 mm outside × 32.873 mm bore hollow.
Limitations
- Brittle materials only (elongation under about 5%); ductile shafts use the maximum-shear method (shaft diameter tool).
- Stress concentrations and fatigue need separate checks.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size shafts of cast iron or hardened steel
- Replace a solid shaft with a lighter hollow one
- Compare weight and stiffness of hollow and solid shafts
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Solid shaft diameter (maximum normal stress)
- Hollow shaft outside and inside diameters
- Weight and stiffness compared with the solid shaft
Interpreting the result
- A hollow shaft with k = 0.6 is about 4% larger in diameter but roughly 30% lighter.
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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