Library / Mechanical / MEC-33
Mechanical
Hooke’s law
In the elastic range, stress is proportional to strain.
- Ref
- MEC-33
- Rev
- v1
- Field
- Mechanical
- Result
- σ [MPa]
E is Young’s modulus in GPa (about 200 for steel, 70 for aluminium) and ε the strain, the extension divided by the original length. Because E is in GPa and the answer in MPa, the formula multiplies by 1000 internally. It holds only up to the proportional limit.
In the elastic range, stress is proportional to strain. The constant is Young’s modulus E, a measure of stiffness.
Calculator
- Strain in microstrain500 µε
- Extension of a 1 m length0.5 mm
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 strain gauge reading of 500 microstrain on steel means a stress of 100 MPa. A 1 m bar stretches only 0.5 mm.
- Steel’s yield strain is only about 0.1 to 0.2 %, so the elastic range is tiny, and the formula should not be used beyond it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ε Strain | 1e-6 to 0.01 | 0.2 | 2,000 | 2,000 | +10% | +10 % |
| E Young’s modulus | 1 to 1,000 GPa | 0.5 | 500 | 499.5 | +10% | +10 % |
Sweep of Strain
| Strain | Stress (MPa) |
|---|---|
| 1e-6 | 0.2 |
| 3.162e-6 | 0.6325 |
| 1e-5 | 2 |
| 3.162e-5 | 6.325 |
| 1e-4 | 20 |
| 3.162e-4 | 63.25 |
| 0.001 | 200 |
| 0.003162 | 632.5 |
| 0.01 | 2,000 |
Sweep one variable
Variables and how each one contributes
Stress
What the equation gives you. Everything else feeds it.
Young’s modulus
Young’s modulus of the material, in GPa. Directly proportional: at the same strain a stiffer material carries proportionally more stress.
Strain
The strain, a plain ratio (0.0005 = 500 microstrain). Directly proportional while the material stays elastic: doubling the strain doubles the stress.
Worked example with the values above
- Convert E: 200 GPa = 200,000 MPa.
- Multiply by the strain: 200,000 × 5×10−4.
- σ = 100 MPa.
Assumptions and limits
- The material is within its elastic limit, so stress and strain are proportional.
- The material is isotropic and the stress is uniaxial.
Common mistakes
- Using E in the wrong unit: mixing GPa and MPa gives answers that are out by 1000.
- Applying the formula above the yield point, where it overestimates the stress.
Statistics and uncertainty
σ = Eε, so the stress is as uncertain as the product of the two inputs. Strain gauges are good to about 1 %, but E varies by 3 to 5 % between grades and falls with temperature, so E is often the larger source of error.
Where it is used
- Converting a strain gauge reading into stress.
- Finding the stress in a member from its measured extension.
Maths you need
- Strain: extension divided by length
- Units: GPa to MPa