Calculation
φ = T·l/(G·Ip); Ip = π·d⁴/32
Angle of twist grows with torque and length and falls with the fourth power of diameter.
Dubbel Festigkeitslehre C 2.5.1, Gl. (45)/(47): φ = Mt·l/(G·Ip), Ip = π·d⁴/32
Solid-shaft angle of twist under torsion
Angle of twist grows with torque and length and falls with the fourth power of diameter.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Angle of twist φ — Elastic twist between the two ends of the considered shaft section.
- Torque T — Torque to be transmitted.
- Shaft length l — Length of the considered, uniformly loaded shaft section.
- Shear modulus G — Material property; structural steel about 81 GPa, aluminium alloys about 26 GPa.
- Shaft diameter d — Diameter of the solid shaft over the considered length.
Example
T=200 N·m, l=1 m, G=81 GPa and d=30 mm give Ip≈7.95×10⁻⁸ m⁴ and φ≈1.78°.
Assumptions and limits
Linear elasticity, constant cross-section and constant torque over length l; notch effects, shaft shoulders and plastic deformation are excluded.
Understand Solid-shaft angle of twist under torsion
This calculator complements torsional stress with the elastic deformation: the angle of twist between the two ends of a shaft section under torsion.
What does this quantity describe?
For a solid shaft with polar second moment of area Ip=π·d⁴/32, φ=T·l/(G·Ip). Twist angle is proportional to torque and length and falls with the fourth power of diameter.
Formula and variables
φ = T·l/(G·Ip); Ip = π·d⁴/32
φ = T·l/(G·Ip)Ip = π·d⁴/32
| Symbol / input | Meaning |
|---|---|
| Angle of twist φ | Elastic twist between the two ends of the considered shaft section. |
| Torque T | Torque to be transmitted. |
| Shaft length l | Length of the considered, uniformly loaded shaft section. |
| Shear modulus G | Material property; structural steel about 81 GPa, aluminium alloys about 26 GPa. |
| Shaft diameter d | Diameter of the solid shaft over the considered length. |
Choose the inputs correctly
T is the torque, l the length of the considered shaft section, G the material's shear modulus, and d the shaft diameter.
How to use the calculator
Use T and d as for torsional stress, enter l as the actual length between the considered cross-sections, and take G from the material.
Worked example
T=200 N·m, l=1 m, G=81 GPa and d=30 mm give Ip≈7.95×10⁻⁸ m⁴ and φ≈1.78°.
Understand the result and units
Excessive twist angle can impair positioning accuracy in machine tools or measurement shafts even when stress is uncritical.
T is a torque, l a length, G a stress (shear modulus) and d a length. φ is output as an angle.
Typical applications
Estimating torsional stiffness of drive shafts, torsion bars and measurement shafts in design work.
Assumptions, limits and common mistakes
Linear elasticity, constant cross-section and constant torque over length l; notch effects, shaft shoulders and plastic deformation are excluded.
Common mistake: Do not substitute Young's modulus E for shear modulus G; they differ by the factor 2(1+ν).
Frequently asked questions
What is “Solid-shaft angle of twist under torsion” used for?
Estimating torsional stiffness of drive shafts, torsion bars and measurement shafts in design work.
Where do the input values come from?
T is the torque, l the length of the considered shaft section, G the material's shear modulus, and d the shaft diameter.
What does the result not cover?
Linear elasticity, constant cross-section and constant torque over length l; notch effects, shaft shoulders and plastic deformation are excluded.