BIZOMAT MAKEOS · Strength
Round Shaft under Axial Load, Bending and Torsion
What are the combined stresses in this round shaft or bar, and is it safe against yield?
Direct, bending and torsional stresses in a solid or hollow round shaft carrying axial load, bending moment and torque together, with principal stresses, von Mises and Tresca safety factors.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Outside diameter | mm | Outside diameter at the section. |
| Bore (0 = solid) | mm | Bore of a hollow shaft (0 if solid). |
| Axial force (+ tension, − compression) | N | Axial force at the section. |
| Bending moment | N·m | Resultant bending moment at the section. |
| Torque | N·m | Torque at the section. |
| Yield strength | MPa | Minimum yield strength from the material certificate or specification. |
| Required safety factor | Factor of safety required by your design code, customer specification or company standard. |
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: Outside diameter
- What it is
- The outside diameter of the shaft or bar at the cross-section being checked, usually the smallest section or the one with the largest loads.
- 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.
- Effect on the result
- For a solid shaft, bending and torsion stresses fall with the cube of the diameter.
- Get it right
- Check at the actual section, for example at a groove or shoulder, using its reduced diameter.
What this means: Bore (0 = solid)
- What it is
- The diameter of the bore through a hollow shaft at the same section. Leave it at 0 for a solid shaft.
- 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 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must be smaller than the outside diameter.
What this means: Axial force (+ tension, − compression)
- What it is
- The axial force along the shaft at this section: positive when it pulls (tension), negative when it pushes (compression).
- Units and format
- Enter a number in N: newtons. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the force at the section being checked. The bending stress is added on the side where it has the same sign as the axial stress.
What this means: Bending moment
- What it is
- The resultant bending moment at the section, combining bending in both planes, from a beam calculation or shaft 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
- Combine horizontal and vertical moments as √(Mh² + Mv²) before entering; the unit is N·m, not N·mm.
What this means: Torque
- What it is
- The twisting moment carried through the section, for example the torque transmitted from a motor or gearbox.
- 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. Torque can be found from power and speed with the shaft power tools.
What this means: Yield strength
- What it is
- The minimum yield strength of the shaft material in its delivered or heat-treated condition, used to give the safety factor against yielding.
- 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.
- Effect on the result
- The safety factor is directly proportional to the yield strength.
- Get it right
- Use the guaranteed minimum from the certificate or specification, not a typical value.
What this means: Required safety factor
- What it is
- The minimum safety factor against yield that the shaft must reach for the result to be marked adequate.
- 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.
- Get it right
- Set by your design code, customer or company standard; it does not change the stresses, only the verdict.
Method and formulas
- A = π(d² − dᵢ²)/4; Z = π(d⁴ − dᵢ⁴)/32d; Zp = 2Z.
- σ = P/A + M/Z (at the outer fibre where they add); τ = T/Zp.
- σ₁,₂ = σ/2 ± √((σ/2)² + τ²); von Mises √(σ² + 3τ²); Tresca √(σ² + 4τ²).
- Example: the default shaft has σ = 63.662 MPa, τ = 39.789 MPa and a von Mises stress of 93.82 MPa.
Limitations
- Static strength; stress concentrations and fatigue need separate checks (see the Goodman tool).
- Slender compressed shafts must also be checked for buckling.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a shaft with thrust, bending and torque
- Check a bolt or rod in tension and torsion
- Check a propeller or screw shaft
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Direct, bending and shear stresses
- Principal stresses
- von Mises and Tresca equivalent stresses and safety factors
Interpreting the result
- Use von Mises for ductile shafts; Tresca is slightly more conservative.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 212–218, 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
Stress cycles: amplitude, mean and ratio
In Metal Fatigue by the Numbers
What it covers: How to describe a repeated stress by its amplitude, mean and ratio, find the nominal stresses in a shaft and a lever, and avoid losing the sign of a stress in a von Mises check.
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Open the Round Shaft under Axial Load, Bending and Torsion calculator