Hong Life CalculatorsUnderstanding the axial deformation calculator — concept and principle
This calculator predicts how much a straight bar, rod, bolt or tie stretches (or shortens) under an axial load, using the classic δ = PL/(AE) formula. It also gives the axial stress and the lateral strain, the small contraction of the cross-section that accompanies stretching. Engineers use it to check bolt stretch, the elongation of tension rods and hangers, and the stiffness of struts.
How it is calculated
- Elongation: δ = P × L ÷ (A × E). P in kN is converted to N (×1,000), E in GPa to MPa (×1,000), L in mm and A in mm², so δ comes out in mm.
- Stress: σ = P ÷ A in MPa.
- Lateral strain: ε_lat = −ν × σ ÷ E, where ν is Poisson's ratio. The minus sign means the bar gets thinner when it is pulled.
The principle
The formula is Hooke's law applied to a whole bar. Stress σ = P/A produces strain ε = σ/E, and strain is elongation per unit length, so the total elongation is εL = PL/(AE). The product AE is the axial rigidity: doubling the area or using a material twice as stiff halves the stretch, while doubling the length doubles it. Poisson's ratio describes the sideways effect: for most metals ν is about 0.3, so the width shrinks by roughly 30% of the axial strain.
Worked example
- Inputs: P = 20 kN = 20,000 N, L = 2,000 mm, A = 314 mm² (about a 20 mm diameter rod), E = 200 GPa = 200,000 MPa, ν = 0.3.
- Elongation: 20,000 × 2,000 ÷ (314 × 200,000) = 40,000,000 ÷ 62,800,000 = 0.636943 mm.
- Stress: 20,000 ÷ 314 = 63.69427 MPa.
- Axial strain: 63.69427 ÷ 200,000 = 0.000318; lateral strain: −0.3 × 0.000318 = −0.0000955414.
| Area (mm²) | Elongation (mm) |
|---|---|
| 78.5 | 2.547770701 |
| 157 | 1.27388535 |
| 236 | 0.8492569002 |
| 314 | 0.6369426752 |
| 392 | 0.5095541401 |
| 471 | 0.4246284501 |
| 628 | 0.3184713376 |
How to read the result
Sub-millimetre stretch is typical for steel members in service. The stress should stay well below the yield strength, otherwise the linear formula no longer applies. For a 20 mm rod the lateral strain of about −0.0000955 means the diameter shrinks by roughly 0.002 mm, usually negligible.
Common mistakes
- Entering E in MPa into the GPa field (200,000 instead of 200).
- Using the gross diameter of a threaded rod instead of its tensile stress area.
- Applying the formula to a bar whose area or load changes along its length; split it into segments and add the δ values.
Frequently asked questions
What Poisson's ratio should I use?
About 0.3 for steel, about 0.33 for aluminium, close to 0.5 for rubber, and around 0.2 for concrete.
Does the formula work in compression?
Yes, with a negative load the bar shortens, but long slender struts may buckle first, which this formula does not check.
How does temperature change compare?
Thermal elongation is αΔTL; for steel (α ≈ 12 × 10⁻⁶ /°C) a 2 m bar grows about 0.024 mm per °C.
Sources: J. M. Gere and B. J. Goodno, Mechanics of Materials, Cengage · R. C. Hibbeler, Mechanics of Materials, Pearson