Free Stress & Strain Calculator
Calculate tensile stress (σ = F/A), strain (ε = ΔL/L₀), and Young's modulus (E = σ/ε) from force, cross-sectional area, original length, and elongation — with results in both Pa/MPa and Pa/GPa.
Enter the axial force, cross-sectional area, original length, and elongation to find stress, strain, and Young's modulus.
σ = F / A · ε = ΔL / L₀ · E = σ / ε. Use SI units: newtons, square metres, and metres. 1 MPa = 10⁶ Pa and 1 GPa = 10⁹ Pa. Results assume uniaxial loading within the elastic (linear) region.
Quick answer
Stress is force divided by cross-sectional area, σ = F/A, measured in pascals; strain is the change in length divided by the original length, ε = ΔL/L₀, which is dimensionless. Young's modulus is their ratio, E = σ/ε. For example, a 10,000 N force on a 0.0001 m² bar gives σ = 100 MPa, and stretching a 2 m bar by 0.001 m gives ε = 0.0005, so E = 100×10⁶ / 0.0005 = 200×10⁹ Pa = 200 GPa (typical of steel).
Formula & method
Tensile (normal) stress
σ = F / A
- σ — Normal stress (Pa = N/m²)
- F — Applied axial force (N)
- A — Original cross-sectional area (m²)
Stress is the internal force per unit area. 1 Pa = 1 N/m²; 1 MPa = 10⁶ Pa; 1 GPa = 10⁹ Pa.
Engineering strain
ε = ΔL / L₀
- ε — Strain (dimensionless, often shown as %)
- ΔL — Change in length / elongation (m)
- L₀ — Original (gauge) length (m)
Strain is fractional deformation and has no units. Multiply by 100 to express it as a percentage.
Young's modulus (Hooke's law)
E = σ / ε = (F · L₀) / (A · ΔL)
- E — Young's modulus / modulus of elasticity (Pa)
- σ — Stress (Pa)
- ε — Strain (dimensionless)
Valid only in the linear-elastic region (below the proportional limit), where stress is proportional to strain.
Examples
- Input
- F = 10,000 N, A = 0.0001 m², L₀ = 2 m, ΔL = 0.001 m
- Result
- σ = 100 MPa, ε = 0.0005 (0.05%), E = 200 GPa
- Why
- σ = 10,000 / 0.0001 = 1×10⁸ Pa = 100 MPa. ε = 0.001 / 2 = 0.0005 = 0.05%. E = 1×10⁸ / 0.0005 = 2×10¹¹ Pa = 200 GPa — the textbook value for structural steel.
- Input
- F = 5,000 N, A = 2×10⁻⁵ m², L₀ = 1 m, ΔL = 0.00125 m
- Result
- σ = 250 MPa, ε = 0.00125 (0.125%), E = 200 GPa
- Why
- σ = 5,000 / 0.00002 = 2.5×10⁸ Pa = 250 MPa. ε = 0.00125 / 1 = 0.00125 = 0.125%. E = 2.5×10⁸ / 0.00125 = 2×10¹¹ Pa = 200 GPa, confirming the rod is steel.
- Input
- F = 2,000 N, A = 0.0001 m², L₀ = 3 m, ΔL = 0.00087 m
- Result
- σ = 20 MPa, ε ≈ 0.00029 (0.029%), E ≈ 68.97 GPa
- Why
- σ = 2,000 / 0.0001 = 2×10⁷ Pa = 20 MPa. ε = 0.00087 / 3 = 0.00029 = 0.029%. E = 2×10⁷ / 0.00029 ≈ 6.897×10¹⁰ Pa ≈ 68.97 GPa — close to aluminium's ~69 GPa, so the bar is far softer than steel for the same stress.
When to use this tool
- Finding the stress, strain, or Young's modulus of a bar, rod, wire, or specimen loaded in simple tension or compression.
- Reducing raw tensile-test data (load, gauge length, elongation, specimen area) into σ, ε, and the elastic modulus.
- Checking whether a part stays within an allowable stress or deflection limit during mechanical or structural design.
- Verifying or identifying a material by comparing your computed Young's modulus against known values (steel ≈ 200 GPa, aluminium ≈ 69 GPa, copper ≈ 117 GPa).
Common mistakes
- Mixing units — entering area in mm² or cm² instead of m², or force in kN instead of N. Convert everything to SI first: 1 mm² = 10⁻⁶ m², 1 cm² = 10⁻⁴ m², 1 kN = 1,000 N.
- Confusing stress with force or strain with elongation. Stress and strain are per-unit quantities (force per area, length change per length); raw force in newtons and raw stretch in metres are not the same thing.
- Using the elongated (final) length instead of the original length L₀ in the strain denominator. Engineering strain always divides by the original gauge length.
- Applying E = σ/ε beyond the elastic limit. Once a material yields or necks, stress and strain are no longer proportional, so Young's modulus from this formula is no longer valid.
Frequently asked questions
What is the difference between stress and strain?
Stress is the internal force carried per unit of cross-sectional area (σ = F/A, in pascals), and it describes how hard the material is being pushed or pulled. Strain is the resulting fractional deformation (ε = ΔL/L₀, dimensionless), describing how much the material actually stretches. Stress is the cause; strain is the effect.
How do I calculate Young's modulus from this data?
Young's modulus is the slope of the stress–strain curve in the elastic region: E = σ/ε. With force, area, original length, and elongation you can write it directly as E = (F·L₀)/(A·ΔL). For the default values, E = (10,000 × 2)/(0.0001 × 0.001) = 2×10¹¹ Pa = 200 GPa.
Why is strain unitless?
Strain divides one length (the change ΔL) by another length (the original L₀), so the metres cancel and the result is a pure number. That is why it can be written as a plain decimal like 0.0005, as a percentage (0.05%), or in microstrain (1 microstrain = 10⁻⁶).
How do I convert pascals to MPa and GPa?
Divide by 10⁶ to go from pascals to megapascals and by 10⁹ to go from pascals to gigapascals. So 1×10⁸ Pa = 100 MPa, and 2×10¹¹ Pa = 200 GPa. Engineers usually report stress in MPa and stiffness (Young's modulus) in GPa.
Does this calculator use original or final cross-sectional area?
It uses the original cross-sectional area, giving engineering stress, which is the standard for most design work and tensile-test reporting. True stress, which uses the instantaneous (necked) area, differs only at large plastic strains and is not computed here.
What is a typical Young's modulus for common metals?
Approximate values are steel ≈ 200 GPa, copper ≈ 117 GPa, brass ≈ 100 GPa, titanium ≈ 116 GPa, and aluminium ≈ 69 GPa. If your computed E lands near one of these, it is a good sanity check that your units and inputs are correct.
Sources & references
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Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
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