Stress-Strain Calculator (from a Known Modulus)

Background.

Hang 50 kN from a 3-metre steel rod with a 500 mm² cross-section and it gets longer by one and a half millimetres. You will not see that, but a designer setting a turnbuckle's adjustment range, checking that a shaft still clears its housing under load, or deciding whether a cable will sag too far needs exactly that number, along with how hard the steel is working to carry the load.

Three linked quantities give both answers. Stress is force divided by the area resisting it, measured in pascals or, more usefully, megapascals. Strain is the fractional change in length, a pure ratio. Within the elastic range the two are tied together by the material's Young's modulus, E, so if you know E you can go from load to stress to strain to an actual change in length in three short steps.

That is the direction worked here: force, area, length and a modulus looked up from a materials table go in; stress, strain and the change in length come out. The reverse problem, measuring a specimen's stretch to find E, belongs to the Young's modulus calculator.

The straight-line relationship only holds while the material stays elastic, meaning it springs back fully once unloaded. Mild structural steel stops doing that at about 250 MPa. Nothing here knows your material's yield strength, so it will compute a tidy-looking stretch for a stress that would in reality bend the bar permanently. Comparing the stress result with a published yield strength for your exact grade is the check that makes the deformation figure worth using.

What is stress-strain calculator?

Axial stress, σ, is the internal force per unit area in a member loaded along its length: σ = F/A, in pascals (N/m²). Strain, ε, is the change in length divided by the original length, a dimensionless ratio often quoted as a percentage or in microstrain. Young's modulus, E, is a material's stiffness in tension and compression, the ratio σ/ε in the elastic region, and is tabulated in gigapascals. Deformation, ΔL, is the actual change in length, positive for stretching and negative for shortening.

How to use this calculator.

1. 01Enter the axial force in newtons: positive for tension, negative for compression. A load given as a mass in kilograms is converted by multiplying by 9.81, so a 5,100 kg hanging load is about 50,000 N.
2. 02Enter the cross-sectional area in square metres, perpendicular to the force. Divide square millimetres by 1,000,000, so 500 mm² is 0.0005 m²; for a round bar of diameter d in mm, the area in mm² is π × d² ÷ 4.
3. 03Enter the unloaded length in metres. The change in length comes back in metres too, so 0.0015 means 1.5 mm.
4. 04Enter Young's modulus in GPa from a materials table: about 200 for steel, 69 for aluminium, 11 for wood along the grain and 25 for concrete. Use the value for your actual alloy or grade where you can find it.
5. 05Compare the stress in MPa with the yield strength of your material and grade before relying on the strain and deformation figures.

The formula.

Three equations are applied in order. Stress: σ = F ÷ A. Strain: ε = σ ÷ E, with E converted from gigapascals to pascals by multiplying by 10⁹. Deformation: ΔL = ε × L₀. Substituting the first two into the third gives the familiar single form ΔL = F·L₀ ÷ (A·E), which shows at a glance what controls the stretch: it grows in proportion to load and length and shrinks in proportion to area and stiffness. Swap a steel member (200 GPa) for an aluminium one (69 GPa) of the same size and the stretch goes up by a factor of 200 ÷ 69, about 2.9.

The model behind these lines is Hooke's law applied to a prismatic bar: a uniform cross-section along the whole length, a force acting through the centroid along the axis, a homogeneous material, and stresses below the proportional limit. The area used is the original one, the engineering-stress convention found in mechanics-of-materials texts; the slight narrowing of a bar under tension is ignored, which is accurate at elastic strains. Bending, self-weight, temperature change and stress concentrations at holes or shoulders are all outside it. A bar with steps in its section can still be handled by running each length separately and adding the deformations.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax. University Physics Volume 1, Section 12.3 'Stress, Strain, and Elastic Modulus' — defines stress σ = F/A, strain ε = ΔL/L₀, and Young's modulus E = σ/ε, rearranged here to predict σ, ε, and ΔL from a known E.
2. [2]National Institute of Standards and Technology (NIST). SI unit definitions — the pascal (Pa = N/m²) as the SI-derived unit of stress.
3. [3]Gere, J. M., & Goodno, B. J. (2013). Mechanics of Materials, 8th edition, Chapter 1 (Tension, Compression, and Shear) — the elastic-range stress-strain-deformation relationship and the engineering-stress convention. Cengage Learning.