---
title: "Percent cold work of a drawn wire"
subject: "Material Science"
chapter: 4
key: "semester_3/material_science:percent-cold-work-numerical"
source: https://www.bctengineer.com/q/semester_3/material_science/percent-cold-work-numerical
site: BCT Engineering
---

# Percent cold work of a drawn wire

Material Science · Chapter 4: Deforming process for materials

- Asked in: year not given, 5 marks
- Video: [MSE 201 S21 Lecture 25 - Module 4 - Cold Work & Annealing Examples](https://www.youtube.com/watch?v=RuXeD7fzUjc) — Thom Cochell
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/material_science/percent-cold-work-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_3/material_science/chapter-4

## Question

(a) A copper wire of diameter 12.7 mm is cold drawn to 10.2 mm diameter. Calculate the percentage cold work. (b) A 15 mm diameter rod is to be given 30% cold work by drawing. Find the final diameter. (c) State what happens to its strength and ductility.

## Short answer

%CW = (1 - d_d²/d_0²)×100. (a) d0=12.7, dd=10.2 → 35.5% CW. (b) Desired 30% CW, d0=15 mm, solve d_d = 15√(1-0.30) = 12.55 mm. (c) Strength and hardness increase, ductility decreases.

## Answer

**Formula:** with $A_0$ the original and $A_d$ the deformed cross-section area,

$$
\%\,CW = \frac{A_0 - A_d}{A_0}\times 100 = \left(1 - \frac{d_d^{\,2}}{d_0^{\,2}}\right)\times 100
$$

#### (a)
$$
\%\,CW = \left(1 - \frac{(10.2)^2}{(12.7)^2}\right)\times 100 = \left(1 - \frac{104.04}{161.29}\right)\times 100 = 35.5\%
$$

#### (b)
$$
0.30 = 1 - \frac{d_d^{\,2}}{15^2} \ \Rightarrow\ d_d^{\,2} = 0.70 \times 225 = 157.5
$$

$$
d_d = \sqrt{157.5} = 12.55\ \text{mm}
$$

#### (c)
Cold drawing makes the wire **stronger and harder** (higher yield strength and UTS) but **less ductile**, because dislocation density increases.

**Answer: (a) 35.5% cold work; (b) final diameter 12.55 mm.**

## Check yourself

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

1. The percentage cold work formula is:
   A. (A0 - Ad)/A0 x 100
   B. (Ad - A0)/Ad x 100
   C. (d0 - dd)/d0 x 100
   D. (A0/Ad) x 100
   Answer: A. %CW is based on the change in cross-sectional area. (Key point: Formula)

2. Cu wire 12.7 to 10.2 mm gives %CW =
   A. 20.0%
   B. 35.5%
   C. 64.5%
   D. 80.3%
   Answer: B. 1 - (10.2/12.7)^2 = 0.355. (Key point: Part a)

3. 30% CW from 15 mm: dd =
   A. 10.5 mm
   B. 11.0 mm
   C. 12.55 mm
   D. 13.1 mm
   Answer: C. dd^2 = 0.7 x 225 = 157.5, dd = 12.55 mm. (Key point: Part b)

4. After cold drawing, the wire is:
   A. Weaker and more ductile
   B. Stronger, harder, less ductile
   C. Stronger and more ductile
   D. Unchanged
   Answer: B. Dislocation density rises. (Key point: Effect)

## Related in Material Science

- [Cold working and strain hardening](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Acold-working-and-strain-hardening) — Material Science, chapter 4
- [Critical nucleus radius for copper](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Acritical-nucleus-radius-copper) — Material Science, chapter 5
- [Tensile test data of a steel bar](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Atensile-test-numerical-steel-bar) — Material Science, chapter 3

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