Library / Mechanical / MEC-03
Mechanical
Twist of a solid circular shaft
Angle of twist and peak shear stress in a loaded shaft.
- Ref
- MEC-03
- Rev
- v1
- Field
- Mechanical
- Result
- θ [°]
θ in radians (shown in degrees), T the torque in N·m, L the shaft length, G the shear modulus, J the polar second moment of area and d the shaft diameter. Peak shear stress is τ = 16T/(πd³).
A shaft twisted by a torque turns through an angle that grows with the torque and the length, and falls steeply with diameter: a thicker shaft is much stiffer in twist.
Calculator
- Peak shear stress 16T/(πd³)39.79 MPa
- Twist in radians0.0373 rad
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Diameter dominates: the fourth power makes a small increase in diameter the cheapest way to stiffen a shaft.
- The peak shear stress is at the outer surface and depends on d³, not d⁴, so a shaft can be strong enough yet still twist too much.
- Long drive shafts are often limited by allowable twist rather than by stress.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| d Shaft diameter | 10 to 150 mm | 0.01081 | 547.1 | 547.1 | +10% | −31.7 % |
| T Torque | 10 to 5,000 N·m | 0.04274 | 21.37 | 21.33 | +10% | +10 % |
| L Shaft length | 0.1 to 10 m | 0.1425 | 14.25 | 14.11 | +10% | +10 % |
| G Shear modulus | 20 to 100 GPa | 1.71 | 8.549 | 6.839 | +10% | −9.09 % |
Sweep of Shaft diameter
| Shaft diameter (mm) | Angle of twist (°) |
|---|---|
| 10 | 547.1 |
| 14.03 | 141.3 |
| 19.68 | 36.48 |
| 27.61 | 9.418 |
| 38.73 | 2.432 |
| 54.33 | 0.6279 |
| 76.22 | 0.1621 |
| 106.9 | 0.04186 |
| 150 | 0.01081 |
Sweep one variable
Variables and how each one contributes
Angle of twist
What the equation gives you. Everything else feeds it.
Torque
Twisting moment carried by the shaft. Directly proportional while elastic.
Shaft length
Length over which the twist accumulates. Directly proportional.
Shear modulus
Shear stiffness of the material (steel about 80 GPa). Inversely proportional.
Shaft diameter
Diameter of the solid shaft. The dominant driver: twist falls with d⁴, so +10% diameter cuts the twist by about 32%.
Worked example with the values above
- Polar second moment: J = πd⁴/32 = 2.513×10−7 m⁴.
- Twist in radians: θ = TL/(GJ) = 500 × 1.5 / (8×1010 × 2.513×10−7) = 0.0373 rad.
- Convert to degrees: 2.137 °.
Assumptions and limits
- Uniform solid circular shaft, constant torque.
- Linear elastic material, small twist.
- Hollow shafts need J = π(do⁴ − di⁴)/32.
Common mistakes
- Using diameter in mm for J in m⁴ without converting.
- Using Young’s modulus E instead of the shear modulus G.
- Reporting the angle in radians when degrees are asked for.
Statistics and uncertainty
The d⁴ dependence amplifies diameter errors by four: δθ/θ ≈ √[(δT/T)² + (δL/L)² + (δG/G)² + (4δd/d)²]. A 1% error in measured diameter becomes a 4% error in twist.
Where it is used
- Propeller and drive shaft design.
- Torsion bar springs.
- Checking torque on gun elevating gear shafts.
Maths you need
- Shear stress: force per area parallel to the surface
- Fourth powers: stiffness depends on d⁴