Stress, strain and Hooke's law: the starting point of SOM

Every question in Strength of Materials comes back to two ideas: how hard a material is being pushed or pulled (stress), and how much it deforms as a result (strain).

Stress

Stress is internal force per unit area. For a bar carrying an axial load P over a cross-section of area A:

σ = PA

Units are N/mm² (= MPa). Tension is taken as positive, compression as negative.

Strain

Strain is the change in length divided by the original length. It has no units.

ε = δLL

Hooke's law

Up to the limit of proportionality, stress is proportional to strain. The constant is Young's modulus, E.

σ = E · ε

For mild steel, E ≈ 2 × 10⁵ N/mm² (200 GPa). For concrete it is much lower, about 2.5 × 10⁴ to 3.5 × 10⁴ N/mm² depending on grade.

Elongation of a bar

Putting the three equations together gives the formula you will use most:

δ = P LA E

Example. A steel rod 2 m long with A = 500 mm² carries P = 100 kN.

δ = 100 × 10³ × 2000500 × 2 × 10⁵ = 2 mm

Poisson's ratio

A bar that stretches also gets thinner. The ratio of lateral strain to longitudinal strain is Poisson's ratio, μ (about 0.3 for steel, 0.15 to 0.2 for concrete).

μ = − lateral strainlongitudinal strain

The three elastic constants

E, the shear modulus G and the bulk modulus K are linked:

E = 2G(1 + μ) = 3K(1 − 2μ)

Exam tip: from the second relation, μ can never exceed 0.5, because K would become negative.