Structural Analysis Fundamentals / Stress and strain
Stress and strain
Why engineers divide by area, and what strain actually measures.
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Read the lesson register and outstanding review notesWhy this matters
Stress relates an internal force to the area carrying it. It helps compare sections, although the stress distribution and material resistance are also needed to assess the member.
By the end of this lesson you should be able to
- Calculate direct and shear stress
- Calculate direct strain
- Use the right area
Direct stress is the force divided by the area it is spread over, acting perpendicular to that area. Tension is normally taken as positive, compression negative.
Strain is the response: how much the member changes length compared with the length it started at. Because it is a length divided by a length, strain has no units. Typical structural strains are tiny — often a few parts in ten thousand.
What it calculates: The intensity of force in the material, and the fractional change in length it causes.
- σ
- Direct stress (N/mm²)
- P
- Axial force (N)
- A
- Cross-sectional area resisting the force (mm²)
- ε
- Direct strain (no units)
- δ
- Change in length (mm)
- L
- Original length (mm)
This assumes
- The force is spread evenly over the area
- The section is away from load points and holes, where stress concentrates
In plain terms: Note the units. Working in N and mm² gives N/mm², which is the same as MPa. Mixing kN with mm² is the most common source of answers that are wrong by a factor of a thousand.
Try it
Stretch a bar
Stress depends on the load and the area. Extension also depends on the length and the material.
kN (tension)
mm
mm²
Material
- Stress σ = P/A
- 100.0 N/mm²
- Strain ε = σ/E
- 4.878e-4
- Extension δ = PL/AE
- 1.220 mm
- Member stiffness AE/L
- 98.4 kN/mm
A stiffer material gives a steeper line: the same strain needs more stress.
Predict first
Two steel bars carry the same load. Bar B has twice the cross-sectional area of bar A. How do their stresses compare?
Practice
A circular steel bar of diameter 25 mm carries a tensile force of 60 kN. What is the direct stress?
Practice
A steel plate 200 mm wide and 12 mm thick has a 22 mm diameter bolt hole through it. It carries 180 kN in tension. What is the stress on the net section through the hole?
Practice
A bar 2000 mm long stretches by 1.6 mm under load. What is the direct strain?
Summary
- Stress = force ÷ area; strain = extension ÷ original length
- Strain has no units and is usually very small
- Use net area where the section is reduced
- Work in N and mm² to get N/mm²