Library / Mechanical / MEC-86
Mechanical
Strain energy stored in a twisted shaft
Energy stored in a uniform circular shaft carrying a constant torque, within the elastic range.
- Ref
- MEC-86
- Rev
- v1
- Field
- Mechanical
- Result
- U [J]
A shaft twisted by a torque T stores elastic energy equal to the area under the torque–twist line: U = ½Tφ, with φ = TL/(GJ). Substituting gives U = T²L/(2GJ), where G is the shear modulus and J the polar second moment of area. The energy rises with the square of the torque and falls as the shaft is made stiffer. The inputs are T in N·m, L in m, G in GPa and J in cm⁴, with the result in joules.
Energy stored in a uniform circular shaft carrying a constant torque, within the elastic range. It equals half the torque times the angle of twist.
Calculator
- Angle of twist, TL/(GJ)0.0375 rad
- Angle of twist in degrees2.149 °
- Torsional stiffness, GJ/L13,330 N·m/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.
- A 1.5 m shaft carrying 500 N·m twists by 2.15° and stores 9.375 J, equal to ½ T φ.
- The stored energy is released when the shaft unloads, which is why torsional vibration is a concern in drivelines with long, light shafts.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Torque | 1 to 10,000 N·m | 3.75e-5 | 3,750 | 3,750 | +10% | +21 % |
| G Shear modulus | 1 to 200 GPa | 3.75 | 750 | 746.3 | +10% | −9.09 % |
| J Polar second moment of area | 1 to 2,000 cm⁴ | 0.1172 | 234.4 | 234.3 | +10% | −9.09 % |
| L Length | 0.1 to 10 m | 0.625 | 62.5 | 61.88 | +10% | +10 % |
Sweep of Torque
| Torque (N·m) | Strain energy (J) |
|---|---|
| 1 | 3.75e-5 |
| 3.162 | 3.75e-4 |
| 10 | 0.00375 |
| 31.62 | 0.0375 |
| 100 | 0.375 |
| 316.2 | 3.75 |
| 1,000 | 37.5 |
| 3,162 | 375 |
| 10,000 | 3,750 |
Sweep one variable
Variables and how each one contributes
Strain energy
What the equation gives you. Everything else feeds it.
Torque
The torque, in N·m. Raises the strain energy with its square: doubling the torque quadruples the energy. Raising it by 50 N·m (to 550 N·m) moves the result by +21 %, from 9.375 to 11.34 J.
Length
The length of the shaft carrying the torque, in m. Raises the strain energy in direct proportion. Raising it by 0.25 m (to 1.75 m) moves the result by +16.7 %, from 9.375 to 10.94 J.
Shear modulus
The shear modulus, in GPa (about 80 for steel). Lowers the strain energy in inverse proportion: a stiffer shaft twists less. Raising it by 5 GPa (to 85 GPa) moves the result by −5.88 %, from 9.375 to 8.824 J.
Polar second moment of area
The polar second moment of area of the cross-section, in cm⁴. For a solid shaft, J = πd⁴/32. Lowers the strain energy in inverse proportion: a thicker shaft twists less. Raising it by 2 cm⁴ (to 27 cm⁴) moves the result by −7.41 %, from 9.375 to 8.681 J.
Worked example with the values above
- Units: J = 25 cm⁴ = 2.5×10−7 m⁴ and G = 8×1010 Pa, so GJ = 20,000 N·m².
- Numerator T²L = 250,000 × 1.5 = 375,000; denominator 2GJ = 40,000.
- U = 9.375 J.
Assumptions and limits
- A linear elastic, homogeneous and isotropic material, with strains small.
- Stresses in MPa with moduli in GPa, giving strain in microstrain.
- A prismatic shaft carrying a constant torque along its whole length, with plane sections remaining plane.
Common mistakes
- Using the area moment of inertia (second moment) I for bending instead of the polar second moment J.
- Using cm⁴ without converting to m⁴.
Statistics and uncertainty
Energy rises with T², so a 2 % error in the torque gives about a 4 % error in the energy. J depends on the fourth power of the diameter, so a small diameter error has a large effect on J and therefore on U.
Where it is used
- Energy stored in a driveshaft, torsion bar or torsional spring.
- Torsional impact and the energy absorbed in sudden torque changes.
Maths you need
- Stress and strain: force per area and relative change in length
- Elastic modulus: stress divided by strain in the elastic range
- Polar second moment of area: J = πd⁴/32 for a solid shaft