What this calculator does
Elongation is how much a material stretches when a tensile force is applied along its length, and for a bar or rod within its elastic limit it follows directly from Hooke's law: ΔL = FL₀/(AE), where F is the applied force, L₀ the original length, A the cross-section area and E the material's Young's modulus. Get the units right and this formula is exact within the elastic range; get one wrong, such as area in the wrong scale, and the answer is out by orders of magnitude.
This is a different situation from a spring, where the restoring force depends on a spring constant rather than a cross-section and a material property. It is also the reverse of what a Young's modulus test measures: there, elongation is measured and used to work out E; here, E is known and elongation is what you are solving for.
The formula
Convert the cross-section area to square metres and Young's modulus to pascals, then divide the applied force by the product of area and modulus, and multiply by the original length. The result is the elongation in metres, shown here in millimetres as well since that is the more usual scale for this kind of stretch.
| Term | Meaning |
|---|---|
| ΔL | Elongation: the increase in length caused by the applied force. |
| F | The applied axial (pulling) force, in newtons. |
| L₀ | The original, unstretched length of the bar or rod. |
| A | The cross-section area the force acts through. |
| E | Young's modulus, the material's stiffness: force per unit strain per unit area. |
The inputs explained
| Field | What to enter |
|---|---|
| Applied axial force (N) | The tensile (pulling) force applied along the length of the bar or rod. |
| Original length (m) | The original length before the force is applied. |
| Cross-section area (mm²) | The cross-section area, in square millimetres. |
| Young's modulus (steel ≈ 200) (GPa) | Young's modulus of the material. Steel is about 200 GPa, aluminium about 69 GPa, copper about 117 GPa. |
When to use it
Checking a tie rod or bolt does not stretch too far
Structural tie rods, turnbuckles and long bolts under load all stretch a small but sometimes significant amount, and this figure matters for clearances and pre-tension calculations.
Comparing how different materials perform under the same load
Swapping a steel component for aluminium at the same dimensions and load changes the elongation roughly in proportion to the ratio of their Young's moduli, since the other terms in the formula stay the same.
Sanity-checking a Young's modulus test setup
Before running a tensile test, working out the expected elongation for a known material and load shows roughly what deflection the test rig or extensometer needs to be able to resolve.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How elongation changes with applied force
The same rod and material, under a range of applied forces.
| Applied force | Elongation | Tensile stress |
|---|---|---|
| 1,000 N | 0.100 mm | 10.00 MPa |
| 2,000 N | 0.200 mm | 20.00 MPa |
| 5,000 N | 0.500 mm | 50.00 MPa |
| 10,000 N | 1.000 mm | 100.00 MPa |
| 20,000 N | 2.000 mm | 200.00 MPa |
| 50,000 N | 5.000 mm | 500.00 MPa |
How elongation changes with material
The same load, length and cross-section, across several common structural materials.
| Young's modulus (material) | Elongation |
|---|---|
| Steel (200 GPa) | 0.500 mm |
| Aluminium (69 GPa) | 1.449 mm |
| Copper (117 GPa) | 0.855 mm |
| Titanium (114 GPa) | 0.877 mm |
| Concrete (30 GPa) | 3.333 mm |
| Timber (11 GPa) | 9.091 mm |
Questions
What is the elongation formula?
ΔL = FL₀/(AE): the applied force multiplied by the original length, divided by the cross-section area multiplied by Young's modulus. It holds as long as the material stays within its elastic limit and does not yield.
How do I calculate elongation without knowing Young's modulus?
You need it, or an equivalent stiffness figure, because it is what links stress to strain for the material. Published values exist for common materials such as steel, aluminium and copper; for anything unusual it needs to be measured or looked up from a materials datasheet.
Is this the same as a spring stretching?
No. A spring's stretch follows F = kx using a spring constant that already bakes in the material, geometry and coil shape. This calculator is for a solid bar or rod under direct tension, using the material's Young's modulus and the bar's own cross-section and length.
What happens past the elastic limit?
This formula only applies while the material is still behaving elastically, meaning it would spring back fully if the load were removed. Beyond the yield point the material deforms permanently and this linear relationship no longer holds.
To go the other way and find Young's modulus from a measured elongation, see the Young's modulus calculator. For a spring rather than a solid bar, see Hooke's law and spring energy.