Young's Modulus Calculator
Calculate Young's modulus of elasticity from stress and strain, or from force, cross-sectional area and length change. Results in GPa, MPa, Pa and psi.
Find Young's Modulus from Stress and Strain
Young's Modulus Formula
Where:
- \(E\) — Young's modulus of elasticity (Pa)
- \(\sigma\) — stress, the force per unit area \(\sigma = F/A\) (Pa)
- \(\varepsilon\) — strain, the relative deformation \(\varepsilon = \Delta L / L_0\) (dimensionless)
- \(F\) — applied tensile or compressive force (N)
- \(A\) — original cross-sectional area of the specimen (m²)
- \(L_0\) — original length of the specimen (m)
- \(\Delta L\) — change in length under the load (m)
The formula expresses Hooke's law in its ratio form: Young's modulus is the constant ratio between stress and strain in the elastic region, i.e. the slope of the straight-line portion of the stress–strain curve.
Young's Modulus – Calculation Example
Example: a steel rod is loaded so that the stress in it is 200 MPa and it stretches by 0.1% of its original length.
\[ \varepsilon = 0.1\% = 0.001 \]
\[ E = \frac{\sigma}{\varepsilon} = \frac{200\,000\,000\ \text{Pa}}{0.001} = 200\,000\,000\,000\ \text{Pa} \]
Result: \(E = 200\ \text{GPa}\) — a typical value for structural steel.
How to Use the Calculator
- Open the Calculation method dropdown and choose From stress & strain if you already know these two values, or From force, area & lengths if you want the calculator to derive stress and strain for you.
- For the first method, enter Stress (σ) with its unit (Pa, kPa, MPa, GPa or psi) and Strain (ε) as a decimal fraction — 0.001 means 0.1% extension.
- For the second method, enter the Force (F) (N, kN, lbf or kgf), the Cross-sectional area (A) (mm², cm², m² or in²), the Original length (L₀) and the Length change (ΔL) in matching length units.
- Read the result: Young's modulus is shown in GPa, MPa, Pa, psi, ksi and psf, together with a summary of the stress and strain used in the calculation.
- Check the interpretation note, which compares your result with typical values for common materials such as steel, aluminium, concrete and rubber.
What Is Young's Modulus?
Young's modulus (also called the elastic modulus or modulus of elasticity) is a material property that describes how stiff a material is — how much it resists being stretched or compressed. It is defined as the ratio of stress to strain in the elastic region of deformation, where the material returns to its original shape once the load is removed. The higher the modulus, the more force is needed to produce the same amount of deformation.
Stress \(\sigma = F/A\) measures the intensity of the load — the force spread over the cross-sectional area — while strain \(\varepsilon = \Delta L / L_0\) measures the response — the extension or compression relative to the original length. Because strain is dimensionless, Young's modulus has the same units as stress, usually gigapascals (GPa) in metric practice or pounds per square inch (psi) in imperial units.
The result tells you the slope of the linear part of the stress–strain curve. A value of 200 GPa means that a 1-metre bar of the material stretches by only 5 micrometres for every megapascal of applied stress. Typical values range from roughly 0.01 GPa for soft rubber to about 1100 GPa for diamond; structural steel sits near 200 GPa, aluminium alloys around 70 GPa, and concrete around 30 GPa. Knowing a material's modulus is essential in structural engineering for predicting deflections, in materials science for characterising new materials, and in any situation where elastic deformation matters — from bridges to springs.
Assumptions, Limitations and Practical Tips
- Linear elastic range only. The formula \(E = \sigma/\varepsilon\) is valid only below the proportional limit, where stress and strain are proportional. Beyond it, the material yields or deforms nonlinearly and a single modulus no longer describes its behaviour.
- Homogeneous and isotropic material. The calculation assumes the stiffness is the same everywhere and in every direction. Wood, composites and fibre-reinforced materials are anisotropic — their modulus along the grain can be several times higher than across it.
- Small deformations. Use the original cross-sectional area \(A\) and original length \(L_0\), not the values measured under load. For large strains, the engineering stress–strain definition becomes inaccurate.
- Positive magnitudes. Enter the magnitude of the extension or compression; the calculator rejects zero or negative values. In practice, tension and compression moduli of the same material are usually close, but some materials (e.g. concrete) behave very differently in compression and tension.
- Temperature and rate effects. Young's modulus decreases with temperature and can depend on how quickly the load is applied, especially in polymers. Values in reference tables are usually measured at room temperature under slow loading.
Typical Young's Modulus Values
| Material | Young's modulus (GPa) |
|---|---|
| Rubber (natural) | 0.01 – 0.1 |
| Nylon | 2 – 4 |
| Wood (along grain) | 9 – 16 |
| Concrete | 30 – 40 |
| Aluminium alloy | 69 – 79 |
| Brass | 100 – 110 |
| Copper | 110 – 130 |
| Steel (structural) | 190 – 210 |
| Tungsten | 400 – 411 |
| Diamond | 1050 – 1200 |
If you are analysing the forces in a loaded structure, you may also find our tension in ropes calculator and friction force calculator useful. For the energy stored in a deformed elastic body, see the elastic potential energy calculator.
Frequently Asked Questions
How do I calculate Young's modulus by hand without a calculator?
Divide the stress by the strain. First compute stress \( \sigma = F/A \) and strain \( \varepsilon = \Delta L / L_0 \), then apply \( E = \sigma / \varepsilon \). Example: a 5000 N force on a bar of 50 mm² cross-section gives \( \sigma = \) 100 MPa; if a 200 mm gauge length stretches by 0.1 mm, \( \varepsilon = 0.0005 \), so \( E = 100\,\text{MPa} / 0.0005 = 200\,\text{GPa} \). Make sure force and area units are consistent before dividing.
What is a good or normal Young's modulus value?
It depends entirely on the material — there is no universal 'good' number. Rubber sits around 0.01–0.1 GPa, wood along the grain 9–16 GPa, concrete 30–40 GPa, aluminium alloys about 70 GPa, structural steel 190–210 GPa, and diamond over 1000 GPa. If your result falls far outside the expected range for your material, double-check the units, especially the cross-sectional area (mm² vs m²) and the strain (0.001 vs 0.1).
How do I convert Young's modulus between GPa, MPa and psi?
The conversions are: 1 GPa = 1000 MPa = 1,000,000,000 Pa, and 1 psi = 6,894.757 Pa. So 1 GPa ≈ 145,038 psi, and 1 ksi (a thousand psi) ≈ 6.895 MPa. For example, steel at 200 GPa equals 200,000 MPa or about 29,000,000 psi (29 Msi). The calculator shows your result in all these units automatically, so you can read off any of them.
How is Young's modulus different from yield strength or stiffness?
Young's modulus measures stiffness — resistance to elastic deformation — while yield strength measures the stress a material can take before it permanently deforms. A material can be very stiff but weak (glass) or flexible but strong (some steels have similar modulus but very different strength). Modulus is the slope of the elastic part of the stress–strain curve; yield strength is the stress at which that curve stops being linear.
Can Young's modulus be negative or zero?
No. For ordinary engineering materials, Young's modulus is always a positive value, typically between about 0.001 GPa (soft gels) and 1200 GPa (diamond). If your hand calculation gives a negative or zero result, you made a sign or unit error. If it gives a huge number like 1,000,000 GPa, you probably entered the strain as a percentage (0.1) instead of a decimal fraction (0.001), or confused mm² with m².
Why did I get a wrong result — what are common mistakes?
The three most common errors are: (1) entering strain as a percentage — 0.1% must be typed as 0.001, not 0.1; (2) mixing units, such as force in N with area in mm² — the calculator handles conversions, but manual calculations require consistent SI units (N and m² give Pa); (3) using diameter instead of cross-sectional area — a round bar of diameter \( d \) has \( A = \pi d^2/4 \). Also make sure you use the original length and area, not the values measured under load.
How does Young's modulus relate to shear modulus and Poisson's ratio?
For an isotropic material, the three elastic constants are linked: \( E = 2G(1+\nu) \), where \( G \) is the shear modulus and \( \nu \) is Poisson's ratio. Knowing any two lets you compute the third. For steel with \( E = 200 \) GPa and \( \nu \approx 0.3 \), the shear modulus is about 77 GPa. This relation does not hold for anisotropic materials like wood or carbon-fibre composites, which have different moduli in different directions.