Module 7 · Lesson 7.2
Hooke's law, stiffness and strength
The elastic relationship, and three ideas that are constantly confused.
Why this matters
Ask a room of students the difference between stiffness and strength and you will get a long silence. The two are unrelated properties, and confusing them leads to real design mistakes.
By the end of this lesson you should be able to
- Apply Hooke's law
- Calculate axial extension
- Separate material stiffness, member stiffness and strength
What it calculates: The elastic relationship between stress and strain, and the resulting change in length of an axially loaded member.
- E
- Young's modulus of the material (N/mm²)
- σ
- Direct stress (N/mm²)
- ε
- Direct strain
- δ
- Extension (positive) or shortening (mm)
- P
- Axial force (N)
- L
- Original length (mm)
- A
- Cross-sectional area (mm²)
This assumes
- The material is still behaving elastically — below the limit of proportionality
- The load, area and material are uniform along the member
In plain terms: E is the slope of the straight part of the stress–strain graph. A stiffer material has a steeper slope, so it needs more stress to reach the same strain. Steel is about 205 000 N/mm², aluminium about 70 000, timber roughly 11 000 along the grain.
Worked example
Stress, strain and extension in a steel tie
Given
- A steel tie carries a tensile force of 180 kN
- Cross-sectional area 1500 mm²
- Length 3.2 m
- E = 205 000 N/mm²
Find
The direct stress, the strain, and how much the tie stretches.
Assumptions
- The steel stays elastic
- The section is uniform and away from connections
Practice
A steel bar of area 800 mm² carries a tensile force of 96 kN. What is the direct stress?
Practice
A steel bar 4.0 m long with a cross-sectional area of 900 mm² carries 135 kN in tension. With E = 205 000 N/mm², how much does it stretch?
Practice
What is the axial stiffness AE/L of a member with A = 1200 mm², E = 205 000 N/mm² and L = 3000 mm? Give your answer in kN/mm.
Practice
Two bars carry the same load and are made of the same steel, but bar B is twice as long as bar A. How many times more does bar B stretch?
Summary
- σ = Eε while the material is elastic
- δ = PL/AE: extension depends on load, length, area and material
- Material stiffness E, member stiffness AE/L and strength are three different things
- Stress concentrations and yielding put a limit on simple theory
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint