Stress and Strain Calculator
Find axial stress, strain, and the implied elastic modulus from force and dimensions.
σ = F ÷ A. 1 MPa = 1 N/mm² = 1×10⁶ Pa.
Normal stress is force over area, σ = F ÷ A, and strain is the fractional stretch, ε = ΔL ÷ L₀. A 10 kN load on a 100 mm² rod gives σ = 10000 ÷ 0.0001 = 100 MPa. If that 1 m rod stretches 0.5 mm, ε = 0.0005, so the implied modulus is E = σ ÷ ε = 200 GPa.
What stress and strain measure
When you pull on a bar, stress (σ) is how hard the internal material is working — the force spread over the cross-sectional area it acts on, measured in pascals or, more usefully for engineering, megapascals (MPa). Strain (ε) is the material’s response: how much it stretches relative to its original length. Because it is a length divided by a length, strain is a pure number with no units, often written as a percentage. Divide stress by strain and you recover the material’s stiffness — its Young’s modulus (E).
σ = stress (Pa), F = force (N), A = area (m²); ε = strain (unitless), ΔL = change in length, L₀ = original length
Worked example
A steel rod of 100 mm² cross-section, 1 m long, carries a 10 kN axial pull and stretches 0.5 mm. Find its stress, strain, and the implied modulus.
- 1 Convert the area to m². 100 mm² × 1×10⁻⁶ = 0.0001 m². Working in SI base units keeps the stress in pascals.
- 2 Compute the normal stress. σ = F ÷ A = 10000 N ÷ 0.0001 m² = 1×10⁸ Pa = 100 MPa.
- 3 Convert the elongation to metres. ΔL = 0.5 mm × 1×10⁻³ = 0.0005 m.
- 4 Compute the strain. ε = ΔL ÷ L₀ = 0.0005 ÷ 1 = 0.0005, or 0.05%. Strain is dimensionless.
- 5 Find the implied Young’s modulus. E = σ ÷ ε = 1×10⁸ ÷ 0.0005 = 2×10¹¹ Pa = 200 GPa — consistent with steel.
Stress units and strain
Stress has pressure units; strain is a bare ratio. These conversions let you move between them cleanly.
| Quantity | Value / relation | Notes |
|---|---|---|
| 1 MPa | 1 N/mm² = 1×10⁶ Pa | Most convenient unit for structural stress |
| 1 GPa | 1000 MPa = 1×10⁹ Pa | Typical scale for Young’s modulus |
| Strain ε | ΔL ÷ L₀ (unitless) | A ratio; 0.01 strain = 1% |
| Steel E | ≈ 200 GPa | Reference stiffness for structural steel |
| Aluminium E | ≈ 69 GPa | About a third as stiff as steel |
Engineering vs true stress, and the elastic limit
This calculator reports engineering stress — force divided by the original cross-sectional area. It is the standard quantity for design and matches published material properties. True stress instead uses the instantaneous area, which shrinks as a specimen necks near failure, so true stress climbs above engineering stress at large deformations. For the small strains of everyday design work the two are practically identical.
The relation E = σ ÷ ε only holds in the elastic region, where stress is proportional to strain (Hooke’s law, σ ∝ ε) and the material springs back on unloading. That linearity ends at the yield point: beyond it the material deforms permanently and a single modulus no longer describes it. Keep applied stress well below yield — and below the yield stress divided by your factor of safety — for a part to behave predictably.