---
title: "True stress; stress-strain curve of steel"
subject: "Civil Engineering Materials"
chapter: 1
key: "semester_3/civil_engineering_materials:true-stress-strain-curve-steel"
source: https://www.bctengineer.com/q/semester_3/civil_engineering_materials/true-stress-strain-curve-steel
site: BCT Engineering
---

# True stress; stress-strain curve of steel

Civil Engineering Materials · Chapter 1: Introduction to Civil Engineering Material

- Asked in: 2057 Chaitra (old course), 2+3+5 marks
- Video: [Understanding True Stress and True Strain](https://www.youtube.com/watch?v=AkX6JqlWRqc) — The Efficient Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/civil_engineering_materials/true-stress-strain-curve-steel
- Whole chapter: https://www.bctengineer.com/md/semester_3/civil_engineering_materials/chapter-1

## Question

Define true stress-strain. Draw a typical stress strain curve for structural steel and explain the significant points in the curve.

## Short answer

True stress is load divided by instantaneous area, and true strain is ln(L/L0). The engineering curve for structural steel shows elastic, yield (e.g. 250 MPa for Fe 250, or 0.2% proof for Fe 415), ultimate, necking and fracture. The true curve keeps rising because area decreases continuously.

## Answer

#### Definition of true stress and true strain
**True stress** is the load divided by the *instantaneous* (actual) cross-sectional area, $\sigma_t = P/A_i$. **True strain** is the sum of all small strains, each based on the instantaneous length:

$$
\varepsilon_t = \int_{L_0}^{L}\frac{dL}{L} = \ln\frac{L}{L_0}
$$

Up to the start of necking: $\sigma_t = \sigma(1+\varepsilon)$ and $\varepsilon_t = \ln(1+\varepsilon)$, where $\sigma$ and $\varepsilon$ are the engineering stress and strain.

#### Stress-strain curve of structural steel
```text
 stress
   |            C
   |        B ,-'''-. D        true curve rises
   |   A  ,.'/        \ - - - - -> T
   | ,-'  /             \  E
   |/    /               x  engineering curve
   +------------------------ strain
 A elastic limit, B yield, C ultimate, D necking, E fracture
```

#### Significant points
1. **A, proportional/elastic limit:** linear region where $\sigma = E\varepsilon$ and deformation is recoverable.
2. **B, yield point:** large plastic strain at almost constant stress. This stress is the yield strength (for example, about 250 MPa for Fe 250 mild/structural steel). High-strength bars such as Fe 415 show no clear yield point, so their 415 MPa is a 0.2% proof stress.
3. **Strain hardening:** the stress rises again with strain.
4. **C, ultimate tensile strength:** the highest engineering stress.
5. **D, necking:** local reduction of area; the engineering curve falls because the load decreases.
6. **E, fracture:** the specimen breaks. The true stress curve keeps rising up to fracture because the actual area keeps decreasing.

## Check yourself

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

1. True stress is load divided by
   A. original area
   B. final length
   C. instantaneous cross-sectional area
   D. gauge length
   Answer: C. sigma_t = P/A_i. (Key point: True stress)

2. True strain is
   A. (L-L0)
   B. L/L0
   C. L0/L
   D. ln(L/L0)
   Answer: D. Integral of dL/L. (Key point: True strain)

3. Up to necking, sigma_t equals
   A. sigma (1 + epsilon)
   B. sigma (1 - epsilon)
   C. sigma / epsilon
   D. sigma^2
   Answer: A. Given in the answer. (Key point: Relation)

4. After necking the true stress curve
   A. falls like the engineering curve
   B. keeps rising to fracture
   C. becomes flat at zero
   D. is undefined
   Answer: B. Actual area keeps decreasing. (Key point: After necking)

## Related in Civil Engineering Materials

- [Stress-strain curve of ductile material](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Astress-strain-curve-ductile) — Civil Engineering Materials, chapter 1
- [Stress, strain, elongation of steel rod](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Anumerical-steel-rod-stress-strain) — Civil Engineering Materials, chapter 1
- [Yield, strain hardening, stress-strain diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Ayield-strain-hardening-stress-strain) — Civil Engineering Materials, chapter 1, asked 2×
- [True stress and engineering stress](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Atrue-engineering-stress) — Civil Engineering Materials, chapter 1
- [Modulus of toughness of medium carbon steel](https://www.bctengineer.com/question.md?key=semester_3%2Fcivil_engineering_materials%3Anumerical-modulus-toughness-415) — Civil Engineering Materials, chapter 1

## Related in other subjects

- [Stress-strain diagram of mild steel](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Astress-strain-diagram-mild-steel) — Strength of Materials, chapter 2
- [True stress and true strain relations](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Atrue-stress-strain-relations) — Material Science, chapter 3
- [Stress-strain diagram for mild steel](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astress-strain-diagram-mild-steel) — Strength of Materials, chapter 3
- [Strain-hardening exponent and true stress](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Astrain-hardening-exponent-numerical) — Material Science, chapter 3