Key ideas
Materials and structures are described by stress and strain. Stress is force per area, strain is relative deformation, and for a given material the ratio between them is a constant called the modulus.
- Stress is σ = F ÷ A, in pascals. Use the cross-sectional area that carries the load.
- Strain is ε = ΔL ÷ L₀. It has no units and is often given as a percentage.
- Young’s modulus E = σ ÷ ε tells how stiff a material is. Within the elastic range, deformation is proportional to the load.
- A bar that is heated or cooled changes length by ΔL = αL₀ΔT, and a structure that is held in place builds stress instead.
Key equations
σ = F ÷ Aε = ΔL ÷ L₀E = σ ÷ εΔL = αL₀ΔTgear ratio = N₂ ÷ N₁
Worked example
A steel rod 2.0 m long and 10 mm in diameter carries 20 kN (E = 200 GPa). A = π × (0.005 m)² = 7.85 × 10⁻⁵ m². σ = 20 000 ÷ 7.85 × 10⁻⁵ = 2.5 × 10⁸ Pa. ε = σ ÷ E = 2.5 × 10⁸ ÷ 2.0 × 10¹¹ = 1.3 × 10⁻³, so the rod stretches ΔL = 1.3 × 10⁻³ × 2.0 m = 2.5 mm.
Common mistakes
- Using the diameter instead of the radius when computing the area of a round bar.
- Mixing mm² with m² or MPa with Pa. Convert everything to SI units first.
- Applying Young’s modulus beyond the elastic limit, where the material deforms permanently.
Practice
Test yourself with practice questions on this topic: Stress, strain and structures