Calculator
Example data table
These sample values show a typical torsion scenario.
| Case | Shaft | T | L | Do | Di | G | θ (deg) |
|---|---|---|---|---|---|---|---|
| 1 | Solid | 1200 N·m | 1.8 m | 40 mm | 0 mm | 79.3 GPa | 2.94 |
| 2 | Hollow | 900 N·m | 2.0 m | 50 mm | 30 mm | 26 GPa | 6.78 |
| 3 | Hollow | 1500 N·m | 1.2 m | 60 mm | 20 mm | 44 GPa | 2.21 |
Formula used
For a circular shaft in elastic torsion, the twist angle is:
Where T is torque, L is length, G is shear modulus, and J is the polar moment.
Polar moment of inertia for common circular sections:
- Solid: J = π · D4 / 32
- Hollow: J = π · (Do4 − Di4) / 32
This tool also reports maximum shear stress using τmax = T·c/J, where c is the outer radius.
How to use this calculator
- Pick solid or hollow shaft geometry.
- Enter torque and select its unit.
- Enter shaft length and choose its unit.
- Enter outer diameter, then inner diameter if hollow.
- Select a material preset or type your shear modulus.
- Optional: enter allowable shear stress for a quick check.
- Press Calculate to view results above the form.
- Use CSV or PDF buttons to save your report.
Notes for engineering use
- Assumes linear elastic behavior and uniform circular sections.
- Use consistent design codes for allowable stresses and safety factors.
- For stepped shafts or noncircular sections, a more detailed model is needed.
What angle of twist measures
Angle of twist (θ) describes how much a shaft rotates under torque. It is reported in radians or degrees, and it depends on load, length, and torsional stiffness. A small θ means a stiff drivetrain, while a large θ can cause backlash, misalignment, and timing errors in couplings.
Core torsion relationship
For circular shafts in elastic torsion, the calculator uses θ = (T·L)/(J·G). Here T is applied torque, L is shaft length, G is shear modulus, and J is polar moment of inertia. Increasing torque or length increases θ linearly, while larger J or G reduces twist.
Why diameter dominates stiffness
Polar moment scales with the fourth power of diameter. For a solid shaft, J = πD⁴/32, so a 10% diameter increase raises J by 1.1⁴ = 1.464, about 46% more stiffness. This is why modest diameter changes can greatly reduce twist and stress.
Solid versus hollow efficiency
Hollow shafts often keep stiffness while saving mass. A hollow shaft with Dᵢ = 0.6Dₒ has J = (1 − 0.6⁴) = 0.870 of a solid shaft’s J, yet the cross‑sectional area is only (1 − 0.6²) = 0.64 of solid. That saves about 36% material.
Typical shear modulus values
Material choice changes G strongly. Common presets used here include steel ≈ 79.3 GPa, aluminum ≈ 26 GPa, brass ≈ 39 GPa, titanium ≈ 44 GPa, cast iron ≈ 41 GPa, and magnesium ≈ 16.5 GPa. Lower G means more twist for identical geometry and torque.
Linking twist to shear stress
The tool also reports τmax using τmax = T·c/J, where c is outer radius. For solid circular shafts, this simplifies to τmax = 16T/(πD³), so stress drops with the cube of diameter. Doubling torque doubles τmax, and increasing diameter is a powerful stress reducer.
Reading twist rate for long shafts
Twist rate is θ/L, useful when shafts span long distances. If θ = 2° over 2 m, the average twist rate is 1°/m (≈0.01745 rad/m). Designers often compare this rate to allowable angular deflection for gears, pumps, or timing drives, based on application needs.
Using results in a workflow
Enter torque, length, and diameters, then choose a material or type G. The output provides θ (deg and rad), J, τmax, and stiffness k = JG/L. Use mixed units: 1 lbf·ft = 1.35582 N·m, and 1 in = 25.4 mm for checks.
FAQs
1) What inputs are required for a correct twist result?
Provide torque, shaft length, outer diameter, and shear modulus. If the shaft is hollow, add the inner diameter. Select the correct units so values are converted consistently before calculation.
2) Why does a small diameter change affect results so much?
Because polar moment J scales with diameter to the fourth power. A 10% diameter increase raises J about 46%, which reduces twist θ and also reduces surface shear stress for the same torque.
3) Can I use this for noncircular shafts?
Not accurately. The equations here assume circular shafts. Noncircular sections need a torsion constant and warping considerations. Use a dedicated formula or FEA for rectangular, thin‑walled, or keyed shapes.
4) What does torsional stiffness k mean?
k = JG/L tells how much torque is needed per radian of twist. Higher k means less rotation under the same torque, which helps keep couplings aligned and reduces timing errors in drive systems.
5) How is maximum shear stress computed?
The calculator uses τmax = T·c/J at the outer surface, with c = D/2. For solid shafts, τmax reduces with D³, so larger diameters quickly lower stress.
6) What is a good allowable twist angle?
It depends on your application and alignment tolerance. Many designs aim for only a few degrees over the full shaft length. Compare the reported θ and twist rate to your machine’s deflection limits or standards.
7) How do the export buttons work?
After you calculate, Download CSV saves the result table as a spreadsheet‑friendly file. Download PDF generates a printable report with key values and the results table for documentation and sharing.