---
title: "Elastic, inelastic, linear and non-linear behaviour"
subject: "Theory of Structures I"
chapter: 1
key: "semester_4/theory_of_structures_i:elastic-inelastic-linear-nonlinear"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/elastic-inelastic-linear-nonlinear
site: BCT Engineering
---

# Elastic, inelastic, linear and non-linear behaviour

Theory of Structures I · Chapter 1: Introduction

- Asked in: 2071 Magh, 4 marks
- Video: [Is every Elastic Material linear?](https://www.youtube.com/watch?v=U2rVNpuMXRE) — Viewd Mechanical
- 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/elastic-inelastic-linear-nonlinear
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-1

## Question

Explain with suitable force-displacement diagrams the elastic, inelastic, linear and non-linear behaviour of structure.

## Short answer

Linear elastic: straight force-displacement line through origin, unloading returns on same path, no permanent deformation. Non-linear elastic: curved path but unloading returns to origin. Inelastic: unloading parallel to initial slope leaves residual deformation, e.g. steel after yield. Non-linear: stiffness or slope changes with load, so superposition is not valid.

## Answer

The behaviour of a structure is described by its force-displacement (load-deflection) curve during loading and unloading.

```text
 F |      /             F |         __--
   |     /                |      _-'
   |    /  loading =      |    /   loading =
   |   /   unloading      |   /    unloading
   |  /                   |  /     (same curve)
   | /                    | /
   |/                     |/
   +----------- D         +------------ D
  (a) Linear elastic     (b) Non-linear elastic

 F |      _____         F |         __--
   |     /    /           |      _-'
   |    /    / unloading  |    /    stiffness
   |   /    /  parallel   |   /     (slope) changes
   |  /    /   to initial |  /      with load
   | /    /    slope      | /
   |/    /                |/
   +----+------ D         +------------ D
     residual
  (c) Inelastic          (d) Non-linear
```

**1. Linear elastic:** The load-deflection curve is a straight line through the origin. On unloading, the structure returns to its original shape along the same line. Example: a steel beam under working loads.

**2. Non-linear elastic:** The curve is not straight, but unloading follows the same curve back to the origin, so there is no permanent deformation. Example: a rubber band, or a cable whose sag changes under load.

**3. Inelastic:** Beyond the elastic limit, the loading and unloading paths differ. Unloading follows a line parallel to the initial elastic slope, leaving a **residual (permanent) deformation**. Example: steel stressed beyond yield.

**4. Non-linear:** The stiffness (slope of the curve) changes with load. This arises from material non-linearity (yielding, cracking) or geometric non-linearity (large displacements). The principle of superposition is not valid.

Linear and non-linear describe the shape of the curve; elastic and inelastic describe whether the deformation is recovered on unloading.

## Check yourself

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

1. In linear elastic behaviour unloading:
   A. Leaves residual deformation
   B. Follows a parallel line
   C. Follows a curve with higher stiffness
   D. Follows the same straight line back to origin
   Answer: D. There is no permanent deformation. (Key point: Linear elastic)

2. In inelastic behaviour, unloading follows:
   A. A line parallel to the initial elastic slope, leaving permanent deformation
   B. The same loading curve
   C. A horizontal line
   D. A line steeper than the initial slope that returns to origin
   Answer: A. Residual deformation remains. (Key point: Inelastic)

3. A non-linear elastic structure (e.g. rubber band):
   A. Leaves residual deformation
   B. Returns along the same curve to the origin with no permanent deformation
   C. Has constant stiffness
   D. Obeys superposition
   Answer: B. Curved but fully recoverable. (Key point: Non-linear elastic)

4. Linear/non-linear versus elastic/inelastic differ in that:
   A. Both describe curve shape only
   B. Both describe recovery only
   C. Linear/non-linear describes curve shape; elastic/inelastic describes recovery on unloading
   D. Linear means elastic always
   Answer: C. They describe different aspects. (Key point: Distinction)

## Related in Theory of Structures I

- [Linear and non-linear behaviour of structures](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Alinear-nonlinear-behaviour) — Theory of Structures I, chapter 1, asked 3×

## Related in other subjects

- [Analog and digital system block diagrams; linear and non-linear distortion](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-digital-block-linear-nonlinear-distortion) — Communication Systems, chapter 1
- [Hardened concrete properties; elastic behaviour](https://www.bctengineer.com/question.md?key=semester_5%2Fconcrete_technology_and_masonry_structure%3Ahardened-properties-elastic-behaviour) — Concrete Technology and Masonry Structure, chapter 4
- [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
- [Short note: elastic and plastic behaviour](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Aelastic-plastic-behaviour) — Civil Engineering Materials, chapter 1