Library / Mechanical / MEC-85
Mechanical
Strain energy stored in an axially loaded member
Energy stored in a uniform bar carrying a constant axial force, within the elastic range.
- Ref
- MEC-85
- Rev
- v1
- Field
- Mechanical
- Result
- U [J]
A bar stretched by an axial force F stores elastic energy equal to the area under the force–extension line: U = ½Fδ, with δ = FL/(EA). Substituting gives U = F²L/(2EA), so the energy rises with the square of the force. It is the energy returned if the load is removed, and the basis of energy methods for deflection. The inputs are F in kN, L in m, E in GPa and A in mm², with the result in joules.
Energy stored in a uniform bar carrying a constant axial force, within the elastic range. It equals half the force times the extension.
Calculator
- Axial stress, F/A100 MPa
- Extension, FL/(EA)1 mm
- Strain energy per unit volume25 kJ/m³
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 2 m bar of 500 mm² loaded with 50 kN stretches 1 mm and stores 25 J, which is the same as ½ F δ.
- Doubling the load quadruples the stored energy (to 100 J) while doubling the area halves it: a thicker member stores less for the same load.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J) | Local step | Change in result |
|---|---|---|---|---|---|---|
| F Axial force | 0.1 to 1,000 kN | 1e-4 | 10,000 | 10,000 | +10% | +21 % |
| E Young’s modulus | 1 to 500 GPa | 10 | 5,000 | 4,990 | +10% | −9.09 % |
| A Cross-sectional area | 10 to 10,000 mm² | 1.25 | 1,250 | 1,249 | +10% | −9.09 % |
| L Length | 0.1 to 20 m | 1.25 | 250 | 248.8 | +10% | +10 % |
Sweep of Axial force
| Axial force (kN) | Strain energy (J) |
|---|---|
| 0.1 | 1e-4 |
| 0.3162 | 0.001 |
| 1 | 0.01 |
| 3.162 | 0.1 |
| 10 | 1 |
| 31.62 | 10 |
| 100 | 100 |
| 316.2 | 1,000 |
| 1,000 | 10,000 |
Sweep one variable
Variables and how each one contributes
Strain energy
What the equation gives you. Everything else feeds it.
Axial force
The axial force, in kN. Raises the strain energy with its square: doubling the force quadruples the energy. Raising it by 5 kN (to 55 kN) moves the result by +21 %, from 25 to 30.25 J.
Length
The length of the member, in m. Raises the strain energy in direct proportion: a longer bar stores more energy at the same force. Raising it by 0.25 m (to 2.25 m) moves the result by +12.5 %, from 25 to 28.13 J.
Young’s modulus
Young’s modulus of the material, in GPa. Lowers the strain energy in inverse proportion: a stiffer bar stretches less and stores less energy. Raising it by 10 GPa (to 210 GPa) moves the result by −4.76 %, from 25 to 23.81 J.
Cross-sectional area
The cross-section area, in mm². Lowers the strain energy in inverse proportion: a thicker bar stretches less and stores less energy. Raising it by 50 mm² (to 550 mm²) moves the result by −9.09 %, from 25 to 22.73 J.
Worked example with the values above
- Force in newtons: F = 50 kN = 50,000 N; area 500 mm² = 5×10−4 m²; E = 2×1011 Pa.
- Numerator F²L = 2.5×109 × 2 = 5×109; denominator 2EA = 2×108.
- U = 25 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.
- Uniform axial stress over the length and a prismatic member, with no buckling.
Common mistakes
- Mixing units: kN with mm² and GPa without conversion.
- Using it above the elastic limit.
- Adding energies of different load cases directly: energies from separate loads do not add, but forces do.
Statistics and uncertainty
Because energy rises with F², a 2 % error in the force gives a 4 % error in the energy, so the load is the most important input to know accurately.
Where it is used
- Energy stored in a bolt, tie or spring-like bar, and impact loading.
- Energy methods for deflection and for sizing a member to absorb energy.
Maths you need
- Stress and strain: force per area and relative change in length
- Elastic modulus: stress divided by strain in the elastic range