Stress and Strain

In plain English

When you pull on a steel bar, the material inside resists. Stress describes how hard each part of the material is being pulled or pushed — the force shared over the area. Strain describes how much the material stretches compared with its length. For most metals, while the load is small, stress and strain are proportional: double the force, double the stretch.

Technical definition

Normal stress σ is the internal force acting perpendicular to a cross-section divided by its area. Shear stress τ is the force acting parallel to the section divided by its area. Normal strain ε is the change in length divided by the original length (dimensionless). In the linear-elastic range, Hooke’s law gives σ = E ε, where E is Young’s modulus (modulus of elasticity).

How it works and where it is used

Stress lets engineers compare a part’s loading with the material’s strength regardless of its size: a large and a small bar at the same stress are equally close to yielding. Strain is measured with extensometers or strain gauges and, through E, gives the stress in real structures. Above the yield strength, Hooke’s law no longer applies and the material deforms permanently.

Formula

σ = F ÷ A    τ = V ÷ A    ε = ΔL ÷ L₀    σ = E × ε    ΔL = F L₀ ÷ (A E)

F = axial force (N), V = shear force (N), A = cross-sectional area (m²), ΔL = change in length, L₀ = original length, E = Young’s modulus (Pa). 1 MPa = 1 N/mm².

Worked example

A steel rod of 10 mm diameter and 1 m length carries a tensile force of 10 kN. Take E = 200 GPa (a typical published value for steel; use your material data for design).

• A = π × 10² ÷ 4 = 78.54 mm²; σ = 10 000 ÷ 78.54 = 127.3 MPa
• ε = 127.3 ÷ 200 000 = 0.000637 (637 microstrain)
• ΔL = 0.000637 × 1000 mm = 0.64 mm

In the workshop and in practice

Designers keep the working stress below an allowable value — the material strength divided by a factor of safety chosen from experience or a design code. Stress calculations size shafts, keys, bolts, brackets and lifting lugs; strain calculations predict deflection and spring rates, and strain gauges are used to measure loads in testing.

Common mistakes and misconceptions

• Using the wrong area. Use the smallest cross-section actually carrying the load, for example the tensile stress area of a thread, not the nominal diameter.
• Ignoring stress concentrations. Holes, grooves, keyways and sharp corners raise local stress well above F ÷ A.
• Applying Hooke’s law beyond yield. σ = Eε only holds in the elastic range.
• Mixing MPa and GPa. E for steel is in the order of 200 GPa = 200 000 MPa.

Related calculators

• Shaft Torsional Stress Calculator — Torsional shear stress and angle of twist of a round shaft, or the diameter needed.
• Parallel Key Length Estimator — Minimum parallel-key length for shear and crushing, or the stresses in a given key.
• Spring Rate Calculator (Hooke’s Law) — Spring rate, force or deflection using Hooke’s law — plus stored energy.

Related topics

• Torque — The turning effect of a force about an axis — force times lever arm.
• Yield Strength — The stress at which a material starts to deform permanently.
• Tensile Strength — The maximum stress a material withstands in a tensile test before it necks and breaks.

References

• Hibbeler, Mechanics of Materials (stress, strain, Hooke’s law)
• Gere and Goodno, Mechanics of Materials
• ISO 6892-1: Metallic materials — Tensile testing — Method of test at room temperature