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Queensferry

Module 9 · Lesson 9.1

Why bending causes stress

Curvature, the neutral axis, and the linear stress distribution.

Why this matters

Bending is how most beams work, and bending stress is what usually decides the size of the section. Understanding where the stress comes from makes section choice obvious rather than mysterious.

By the end of this lesson you should be able to

  • Explain the strain distribution in a bent beam
  • Locate the neutral axis
  • Describe the stress distribution

Take a rubber beam with lines drawn on its side and bend it into a sag. The lines near the bottom get further apart — that face is stretching. The lines near the top get closer together — that face is shortening. Somewhere in between is a layer whose length has not changed at all.

That layer is the neutral axis. It has zero strain, and therefore zero bending stress. For an elastic material with a symmetrical section it passes through the centroid.

The key assumption is that plane sections remain plane: a flat cross-section before bending stays flat afterwards, simply rotating. If that holds, strain must vary linearly with distance from the neutral axis — and since σ = Eε, so must stress. That is why the bending stress diagram is always a pair of triangles.

Bending stress varying linearly from compression at the top face through zero at the neutral axis to equal tension at the bottom face.-13.3 N/mm²compression13.3 N/mm²tensionN.A.
Linear bending stress. Maximum at the extreme fibres, zero at the neutral axis — which is why material near the middle does relatively little.

Predict first

What is the bending stress at the neutral axis of a beam carrying a bending moment?

Practice

A beam 400 mm deep bends so that the strain at the top face is 500 microstrain. What is the strain at a point 100 mm above the neutral axis? (Give your answer in microstrain.)

Practice

At a point 150 mm from the neutral axis the bending strain is 300 microstrain. If E = 205 000 N/mm², what is the bending stress there?

Practice

A symmetric beam section is 500 mm deep and carries a bending moment. What is the bending stress exactly at mid-depth?

Summary

  • Bending stretches one face and shortens the other
  • The neutral axis has zero strain and zero bending stress
  • Plane sections remaining plane gives a linear stress distribution
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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