Stress, strain and Young's modulus calculator

What is Young’s modulus?

It is the stiffness of a material in tension or compression: Young’s modulus Y = stress ÷ strain. Stress is the force per unit area, F/A, and strain is the fractional change in length, ΔL/L₀ (OpenStax University Physics, Section 12.3). Stress and Y are in pascals (N/m²) or psi; strain has no units.

Example: stretching a steel rod

OpenStax Example 12.8: a 2.0 m steel rod with a cross-section of 0.30 cm² holds a 550 kg platform. The stress is 550 × 9.8 N ÷ 3.0 × 10⁻⁵ m² ≈ 1.8 × 10⁸ Pa, and with Y = 2.0 × 10¹¹ Pa the rod stretches ΔL = stress × L₀ ÷ Y ≈ 1.8 mm.

Young’s modulus of steel, aluminum, concrete and wood

Approximate values from OpenStax Table 12.1: steel 200 GPa (about 29,000 ksi), iron and nickel 210 GPa, copper 110 GPa, brass 90 GPa, aluminum 70 GPa (about 10,150 ksi), marble and crown glass 60 GPa, granite 45 GPa, concrete 20 GPa, brick and hardwood 15 GPa, lead 16 GPa. For design work use the value for your exact grade or species.

Does it work for compression?

Yes — compressive stress and strain use the same formulas with absolute values. OpenStax Example 12.7: 2.568 × 10⁴ N on a 0.20 m² granite pillar gives a stress of 128.4 kPa and, with Y = 4.5 × 10¹⁰ Pa, a strain of 2.85 × 10⁻⁶.

When does the formula stop working?

Stress is proportional to strain only in the linear, elastic range at low stress, where the part springs back when the load is removed. At higher stress, beyond the material’s proportionality limit, it deforms permanently and eventually breaks; OpenStax treats that range separately. This calculator assumes the linear range.

How can I find Young's modulus from a test?

Measure the force, the cross-sectional area, the original length and the elongation, then Y = (F/A) ÷ (ΔL/L₀). A rod of 1 m with a cross-section of 1 cm² (10⁻⁴ m²) that stretches by 0.5 mm under 10,000 N has a stress of 1.00 × 10^8 Pa and a strain of 5.00 × 10^-4, so Y = 2.00 × 10^11 Pa, which is 200 GPa, close to the value for iron or steel. Choose "Young's modulus" in the calculator for this.

How do I convert GPa to psi or ksi?

One pascal is 1/6,894.757 psi, so 1 GPa is about 145.0 ksi. Steel at 200 GPa is about 29,008 ksi, and aluminum at 70 GPa is about 10,153 ksi. The calculator switches between US and metric units for you.

Stress, strain and the modulus

Stress is the force divided by the cross-sectional area that carries it, σ = F/A, in pascals (newtons per square meter) or pounds per square inch. Strain is the fractional change in length, ε = ΔL/L₀, with no unit. In the linear range, the two are proportional, and the constant of proportionality is Young's modulus Y = σ/ε. A stiff material has a large Y: it takes a large stress to produce a small strain. Rearranged, the stretch is ΔL = F L₀ ÷ (A Y). The calculator solves for whichever one of the force, the elongation or the modulus you leave out, and it takes the modulus from a table of materials or from a value you enter.

Worked example: steel and aluminum rods

Take the OpenStax steel rod: 2.0 m long, 0.30 cm² in cross-section, carrying a 550 kg platform, so F = 550 × 9.8 = 5,390 N. The stress is 1.80 × 10^8 Pa, and with Y = 2.0 × 10^11 Pa the strain is 8.98 × 10^-4, so the rod stretches 1.80 mm. The same rod in aluminum (Y = 7.0 × 10^10 Pa) stretches 5.13 mm, 2.86 times as much, because the stress is the same and the modulus is lower. Doubling the cross-section of the steel rod halves the stretch to 0.90 mm, and doubling its length doubles the stretch to 3.59 mm. The rod behaves like a spring with a stiffness of Y A / L₀ = 3.00 × 10^6 N/m.

Where the formula stops working

The proportionality only holds below the proportional limit. Beyond that, the material may deform permanently, and finally it breaks. The values in the table are typical values for the materials named, from OpenStax, and real alloys, concretes and woods vary: wood differs along and across its grain, and the value for concrete depends on the mix. The calculation assumes a uniform bar loaded along its axis. It does not account for bending, buckling of slender columns in compression, stress concentrations at holes and notches, or changes with temperature. For design work, use the value and the safety rules for the exact grade.

Common mistakes

Mixing area units is the most frequent error: a cross-section of 0.30 cm² is 3.0 × 10⁻⁵ m², not 0.30 × 10⁻⁴. Using the diameter in place of the area, or the radius in place of the diameter, is the next one; the calculator can take a diameter for round bars. Confusing GPa with MPa or ksi with psi shifts the result by a factor of 1,000. Entering a force in kilograms instead of newtons is another: multiply by 9.8 (or use pounds-force in the US units). And a very small stretch compared with the length is normal for metals, so check the unit of the answer, which is shown in millimeters or inches.

Sources: OpenStax University Physics, Volume 1, section 12.3 (stress, strain and elastic modulus; equations 12.34 to 12.36, Table 12.1 of Young's moduli, Examples 12.7 and 12.8); conversions from NIST SP 811. Related: thermal expansion calculator, force calculator and pressure calculator.