---
title: "Material and geometrical non-linearity"
subject: "Theory of Structures I"
chapter: 1
key: "semester_4/theory_of_structures_i:material-geometric-nonlinearity"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/material-geometric-nonlinearity
site: BCT Engineering
---

# Material and geometrical non-linearity

Theory of Structures I · Chapter 1: Introduction

- Asked in: 2074 Bhadra, 4 marks
- Video: [Non Linearity in FEA | Sources | definition | Types | Geometric |material | Contact](https://www.youtube.com/watch?v=1Ep3SlOEP54) — Practical FEA
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/material-geometric-nonlinearity
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-1

## Question

Explain material non-linearity and geometrical non-linearity with neat sketches.

## Short answer

Material non-linearity: stress-strain relation is non-linear (e.g., steel yielding) so stiffness changes with stress, but equilibrium on undeformed shape. Geometric non-linearity: large displacements alter geometry, equilibrium must consider deformed shape (e.g., slender column P-delta effect).

## Answer

Non-linearity arises when displacement is not proportional to load. The two main sources are the material and the geometry.

#### Material non-linearity
The stress-strain relation of the material is not linear, so the stiffness changes with stress even when displacements are small.
- Examples: steel beyond the yield point, concrete in compression, cracked concrete, soil.
- Equilibrium is still written on the undeformed geometry.

```text
 stress
   |         ______________  yield (plastic)
   |       /
   |     /
   |   /   elastic
   | /
   +------------------------ strain
     elastic-plastic curve
```

#### Geometric non-linearity
Displacements are large enough to change the geometry of the structure, so equilibrium must be written on the **deformed** shape, even if the material is linear.
- Examples: slender columns (P-delta effect), cables and suspension bridges, thin shells, arches near buckling.

```text
        P                     Load
        |   |<- Delta ->|      |            ________ approaches Pcr
        v                      |        _.-'
        .   .                  |     .-'
        |    .                 |   /
        |    .  <- H           |  /   stiffness falls
        |   .                  | /    as load rises
        |  .                   |/
      __|__                    +-------------------- deflection
      /////
   slender column: base moment = H*h + P*Delta
```

Here the extra moment $P\Delta$ makes the load-deflection curve non-linear: the stiffness decreases as the axial load increases.

| Point | Material | Geometric |
|---|---|---|
| Cause | Stress-strain curve non-linear | Large change of geometry |
| Equilibrium on | Original shape | Deformed shape |
| Example | Yielding steel | Cable, slender column |

## Check yourself

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

1. Material non-linearity arises from:
   A. Large change of geometry
   B. Changing support conditions only
   C. Applying load statically
   D. Non-linear stress-strain relation even for small displacements
   Answer: D. Stiffness changes with stress. (Key point: Material non-linearity)

2. In geometric non-linearity equilibrium is written on:
   A. The deformed shape
   B. The undeformed shape
   C. Neither
   D. The plastic hinge only
   Answer: A. Large displacements change geometry. (Key point: Geometric non-linearity)

3. Which is an example of geometric non-linearity?
   A. Steel beyond yield
   B. Slender column with P-delta effect
   C. Cracked concrete
   D. Soil stress-strain curve
   Answer: B. Axial load on deflection adds moment P*delta. (Key point: Example)

4. Why does the P-delta effect make the load-deflection curve non-linear?
   A. It increases stiffness
   B. It makes the material yield
   C. Extra moment P*delta reduces stiffness as axial load increases
   D. It removes axial load
   Answer: C. Moment grows with deflection. (Key point: P-delta)

## Related in Theory of Structures I

- [Non-linearity in structural analysis](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Anonlinearity-structural-analysis) — Theory of Structures I, chapter 1

## Related in other subjects

- [Stress-strain curves: linear and non-linear cases](https://www.bctengineer.com/question.md?key=semester_5%2Fconcrete_technology_and_masonry_structure%3Astress-strain-nonlinear-cases) — Concrete Technology and Masonry Structure, chapter 4
- [Linearity of y[n]=x²[n]](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alinearity-check-square) — Digital Signal Analysis and Processing, chapter 1
- [Modular, semi-modular, non-modular outlets](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Aoutlets-modular-semimodular-nonmodular) — Irrigation and Drainage Engineering, chapter 7, asked 3×
- [Structural vs non-structural cracks and causes](https://www.bctengineer.com/question.md?key=semester_6%2Fbuilding_technology%3Acracks-struct-nonstruct-causes) — Building Technology, chapter 10