Stiffness in one number
Pull on a steel rod and it stretches, very slightly. Pull twice as hard and it stretches twice as much, as long as you stay below the point where it starts to yield. The ratio between how hard you pull, per unit of area, and how much the rod stretches, per unit of length, is a property of the material itself: Young’s modulus, or the modulus of elasticity. The English scientist Thomas Young described it in 1807, and it is the first number engineers look up when they size a beam, a cable or a spring.
The formulas
- Stress σ = F ÷ A, the force divided by the cross-sectional area. For a round rod, A = π × d² ÷ 4.
- Strain ε = ΔL ÷ L, the elongation divided by the original length. It has no unit; it is often given in percent or in microstrain (µε, millionths).
- Young’s modulus E = σ ÷ ε.
Worked example
A steel rod 0.5 inch in diameter and 10 feet long carries a pull of 5,000 lbf and stretches 0.105 inch:
- Area: π × 0.5² ÷ 4 = 0.196 sq in (126.68 mm²).
- Stress: 5,000 ÷ 0.196 = about 25.5 ksi (175.6 MPa).
- Strain: 0.105 ÷ 120 = 0.000875, or 0.0875% and 875 µε.
- Young’s modulus: 25,465 ÷ 0.000875 = about 29.1 million psi (200.7 GPa), right where you would expect for steel.
Typical values
| Material | E (million psi) | E (GPa) |
|---|---|---|
| Rubber | 0.0001 to 0.015 | 0.001 to 0.1 |
| Nylon, PVC | 0.3 to 0.6 | 2 to 4 |
| Wood, along the grain | 1.3 to 2.3 | 9 to 16 |
| Concrete | 3.6 to 5.8 | 25 to 40 |
| Aluminum | 10 | 69 |
| Copper | 17 | 117 |
| Titanium | 16 | 110 |
| Steel | 28 to 30 | 190 to 210 |
| Tungsten | 58 | 400 |
Stiff is not the same as strong
Young’s modulus tells you how much a part deflects, not when it breaks. Most steels have nearly the same modulus, whether they are mild steel or high-strength alloy, yet their yield strengths differ by a factor of five or more. That is why swapping to a stronger steel does not make a beam sag less: only a deeper section or a stiffer material does.
Tips for an accurate measurement
- Stay in the elastic range: the rod must return to its original length when unloaded.
- Measure the elongation on the specimen itself with a dial gauge or extensometer. The travel of a testing machine includes slack in the grips.
- Take several loads and use the slope of the stress–strain line rather than a single point.
- Use the original diameter and length, not the values under load.