---
title: "Pure bending, neutral layer, neutral axis"
subject: "Strength of Materials"
chapter: 3
key: "semester_4/strength_of_materials_mechanical:pure-bending-vs-simple-bending"
source: https://www.bctengineer.com/q/semester_4/strength_of_materials_mechanical/pure-bending-vs-simple-bending
site: BCT Engineering
---

# Pure bending, neutral layer, neutral axis

Strength of Materials · Chapter 3: Pure Bending

- Asked in: year not given, 5 marks
- Video: [Neutral axis](https://www.youtube.com/watch?v=C1OW_irFFxs) — Mechanics Of Materials - IITM
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/strength_of_materials_mechanical/pure-bending-vs-simple-bending
- Whole chapter: https://www.bctengineer.com/md/semester_4/strength_of_materials_mechanical/chapter-3

## Question

Differentiate between pure bending and simple (non-uniform) bending. Define neutral layer and neutral axis, and state where the neutral axis lies in elastic bending.

## Short answer

Pure bending: constant bending moment, zero shear, only bending stress, constant curvature. Simple bending: varying moment and shear, both bending and shear stresses. Neutral layer is unstressed; neutral axis passes through centroid in elastic bending.

## Answer

#### Pure and simple bending
**Pure bending** is bending under a constant bending moment with zero shear force, for example the central portion of a beam loaded by two equal loads symmetrically placed. **Simple bending** is general bending in which the bending moment varies along the beam and shear force is also present.

| Point | Pure bending | Simple bending |
|---|---|---|
| Bending moment | Constant along the length | Varies along the length |
| Shear force | Zero | Present ($V = dM/dx$) |
| Stresses | Only bending (normal) stress | Bending stress and shear stress |
| Example | Middle third of a beam with two equal loads at the third points | Cantilever with end load; SS beam with a central load |
| Curvature | Constant (arc of a circle) | Varies |

```text
 P              P      pure bending between the loads
 |              |
 v______________v
 A  a   |   L   | a  B    SF = 0, BM = Pa between loads
```

#### Neutral layer and neutral axis
- **Neutral layer (surface):** the layer of fibres that is neither stretched nor shortened during bending, so its stress and strain are zero. The fibres above it are in compression and those below it in tension for a sagging beam.
- **Neutral axis (NA):** the line of intersection of the neutral layer with a cross-section of the beam. Bending stress varies linearly from zero at the NA to the maximum at the extreme fibres.

#### Position of the neutral axis
For linearly elastic material the net axial force over the section is zero, $\int \sigma\,dA = 0$, hence $\int y\,dA = 0$. The neutral axis therefore passes through the **centroid** of the cross-section. For unsymmetrical sections (such as a T-section) it is not at mid-depth, so the maximum tensile and compressive stresses differ.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which describes pure bending?
   A. Constant bending moment with zero shear force
   B. Varying bending moment with shear force present
   C. Zero bending moment with constant shear
   D. Only shear stress present
   Answer: A. Pure bending has constant M and V = dM/dx = 0, e.g. between two equal symmetric loads. (Key point: Pure bending definition)

2. What is the neutral axis?
   A. The top fibre of the beam
   B. The line where the neutral layer meets a cross-section
   C. The line of maximum stress
   D. The line through the load
   Answer: B. The neutral layer is neither stretched nor shortened; its intersection with a section is the neutral axis, where bending stress is zero. (Key point: Neutral layer and axis)

3. Where does the neutral axis lie in elastic bending?
   A. At mid-depth always
   B. Through the centroid of the section
   C. At the extreme fibre in tension
   D. At the point of maximum shear
   Answer: B. Net axial force is zero, so integral of y dA = 0 and the NA is at the centroid, which for a T-section is not at mid-depth. (Key point: Position of the NA)

## Related in Strength of Materials

- [Bending stresses in a T-section beam](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Acast-iron-t-beam-bending-stresses) — Strength of Materials, chapter 3
- [Differential equation of the elastic curve](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Abeam-deflection-differential-equation) — Strength of Materials, chapter 7
- [Derivation of the flexure formula](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Aflexure-formula-derivation-assumptions) — Strength of Materials, chapter 3

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