# What is Young's modulus?

Young's modulus calculator: E = stress ÷ strain = F × L₀ ÷ (A × ΔL) from a load test, or the stretch or force for a material, with the stress and strain.

- Page: https://www.acalculator.org/physics/youngs-modulus-calculator
- JSON spec: https://www.acalculator.org/physics/youngs-modulus-calculator.json
- Version: 25032d749ec9

## Default answer

Example with the default inputs (Find Young's modulus, Force (F) 1,211.72020330744 lbf, Cross-section Area, Cross-section area (A) 0.046500093000186 in², Original length (L₀) 6.56167979002625 ft, Change in length (ΔL) 0.0708661417322835 in): E = 199.63 GPa (28,953,800 psi), with a stress of 179.667 MPa and a strain of 0.0009.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| find | Find | Which value to work out from E = F × L₀ ÷ (A × ΔL), or E = σ ÷ ε from the stress and strain. |
| F | Force (F) | The pulling or pushing force along the bar, square to its cross-section. |
| sec | Cross-section | Type the cross-section area, or the diameter of a round bar or wire. |
| A | Cross-section area (A) | The area of the cut across the bar, square to the force. |
| d | Diameter (d) | The diameter of a round bar or wire; the area is π × d² ÷ 4. |
| L | Original length (L₀) | The length of the bar before the force acts. |
| dL | Change in length (ΔL) | How much the bar stretches (or shortens under a push). |
| s | Stress (σ) | The force per unit of area, in the unit chosen below. |
| su | Stress unit | The unit of the stress you type. |
| e | Strain (ε) | The change in length divided by the original length, a pure number (0.001 is 0.1%). |
| mat | Material | The material of the bar, which sets Young's modulus (OpenStax Table 12.1). |
| E | Young's modulus (E) | The material's Young's modulus, in the unit chosen below. |
| Eu | Modulus unit | The unit of the Young's modulus you type. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Answer | The value worked out, to 6 significant figures, in SI and US units. |
| modulus | Young's modulus (Pa) | E = stress ÷ strain, in pascals. |
| gpa | Young's modulus (GPa) | E in gigapascals: 1 GPa = 10⁹ Pa. |
| psi | Young's modulus (psi) | E in pounds-force per square inch. |
| stress | Stress (MPa) | σ = F ÷ A, in megapascals. |
| stressPsi | Stress (psi) | σ in pounds-force per square inch. |
| strain | Strain | ε = ΔL ÷ L₀, a pure number. |
| strainPct | Strain (%) | ε × 100. |
| stretch | Change in length (mm) | ΔL in millimetres. |
| stretchIn | Change in length (in) | ΔL in inches. |
| force | Force (N) | F in newtons. |
| lbf | Force (lbf) | F in pounds-force. |
| area | Cross-section area (mm²) | A in square millimetres; π × d² ÷ 4 for a round bar. |

## Method

σ = F ÷ A; ε = ΔL ÷ L₀; E = σ ÷ ε = F × L₀ ÷ (A × ΔL); ΔL = F × L₀ ÷ (A × E); F = E × A × ΔL ÷ L₀; a round bar's A = π × d² ÷ 4.

## Assumptions

- The bar obeys Hooke’s law: the stress stays in the straight-line (elastic) part of the stress–strain curve, below the proportional limit.
- The force acts along the bar and spreads evenly over the cross-section; a push (compression) uses the same formulas with sizes, not signs.

## Worked examples

1. find = stretch, mat = steel, sec = area, F = 5,390, A = 0.00003, L = 2 gives dL = 1.796667, s = 179.666667, result = ΔL = 1.79667 mm (0.0707349 in). Source: OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19).
2. find = modulus, sec = area, F = 5,390, A = 0.00003, L = 2, dL = 0.0018 gives gpa = 199.62963, e = 0.0009, result = E = 199.63 GPa (28,953,800 psi). Source: OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19).
3. find = stress, s = 180, su = MPa, e = 0.0009 gives gpa = 200, E = 200,000,000,000. Source: OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19), Equation 12.36: E = σ ÷ ε = 180 MPa ÷ 0.0009 = 200 GPa, the steel value of Table 12.1.
4. find = force, mat = steel, sec = area, A = 0.00003, L = 2, dL = 0.0018 gives F = 5,400, s = 180, result = F = 5,400 N (1,213.97 lbf). Source: OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19).
5. find = stretch, mat = aluminum, sec = round, d = 0.01, L = 1, F = 10,000 gives dL = 1.818914, A = 78.539816. Source: OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus (Equations 12.34 to 12.36; Table 12.1; Example 12.8: a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm at 1.8 × 10⁸ Pa), https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus (retrieved 2026-10-19), Table 12.1 (aluminum 7.0 × 10¹⁰ Pa).

## FAQ

### How do you calculate Young's modulus?

Divide the stress by the strain: E = σ ÷ ε, with σ = F ÷ A (force over cross-section area) and ε = ΔL ÷ L₀ (change in length over original length). A 2 m rod of 0.30 cm² that stretches 1.8 mm under 5,390 N has E = 5,390 × 2 ÷ (0.00003 × 0.0018) = 1.996 × 10¹¹ Pa, about 200 GPa.

### What is Young's modulus of steel?

About 200 GPa (2.0 × 10¹¹ Pa, or 29 million psi), the value in OpenStax Table 12.1. Iron and nickel are 210 GPa, copper 110 GPa, brass 90 GPa and aluminum 70 GPa.

### How much does a steel rod stretch under a load?

ΔL = F × L₀ ÷ (A × E). A 2.0 m steel rod with a 0.30 cm² cross-section holding 550 kg (5,390 N) stretches 5,390 × 2 ÷ (0.00003 × 2 × 10¹¹) = 0.0018 m, or 1.8 mm.

### What are the units of Young's modulus?

The same as stress: force per area. In SI that is the pascal (1 Pa = 1 N/m²), and metals are quoted in gigapascals (1 GPa = 10⁹ Pa). In US units it is psi, ksi (1,000 psi) or Msi (10⁶ psi). Strain has no unit.

### What is the difference between stress and strain?

Stress is the cause: the force on each square metre of the cross-section, σ = F ÷ A. Strain is the effect: the fraction by which the length changes, ε = ΔL ÷ L₀. Young’s modulus is the ratio of the two in the elastic range.

### Is Young's modulus the same in tension and compression?

For most metals, yes, so the same E gives the stretch under a pull and the shortening under a push. Some materials differ: OpenStax lists bone at 16 GPa in tension and 9 GPa in compression.

## Sources

- OpenStax, University Physics Volume 1, §12.3 Stress, Strain, and Elastic Modulus: Equation 12.34 (tensile stress F⊥ ÷ A), 12.35 (tensile strain ΔL ÷ L₀), 12.36 (Y = F⊥L₀ ÷ (AΔL)); Table 12.1 (approximate elastic moduli); Example 12.8 (a 2.0 m steel rod of 0.30 cm² holding 550 kg stretches 1.8 mm), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus
- OpenStax, College Physics 2e, §5.3 Elasticity: Stress and Strain: Table 5.3, elastic moduli in 10⁹ N/m² (polystyrene 3, silk 6, spider thread 3), retrieved 2026-10-19. https://openstax.org/books/college-physics-2e/pages/5-3-elasticity-stress-and-strain
- NIST Special Publication 811 (2008), Appendix B.8: 1 in = 0.0254 m, 1 ft = 0.3048 m, 1 lbf = 4.4482216152605 N, 1 psi = 1 lbf/in², retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811