---
title: "Upsetting diameter and force calculation"
subject: "Workshop Technology"
chapter: 9
key: "semester_2/workshop_technology:upsetting-diameter-force-numerical"
source: https://www.bctengineer.com/q/semester_2/workshop_technology/upsetting-diameter-force-numerical
site: BCT Engineering
---

# Upsetting diameter and force calculation

Workshop Technology · Chapter 9: Forging Practice

- Asked in: year not given, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/workshop_technology/upsetting-diameter-force-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_2/workshop_technology/chapter-9

## Question

A steel billet 40 mm in diameter and 120 mm long is upset hot between flat dies to a height of 50 mm. Assuming volume remains constant and ignoring friction and barrelling, find (a) the final diameter, (b) the true strain, and (c) the upsetting force if the flow stress at the forging temperature is 100 MPa. Comment on whether the billet will buckle during upsetting.

## Short answer

Final diameter d₂ = 40√(120/50) = 61.97 mm. True strain ε = ln(120/50) = 0.875. Upsetting force = 100 MPa × (π/4 × 61.97² mm²) ≈ 302 kN. The initial height/diameter ratio is 3, borderline for buckling; use a die or light blows.

## Answer

**Given:** $d_1 = 40$ mm, $h_1 = 120$ mm, $h_2 = 50$ mm, $\sigma_f = 100$ MPa.

#### (a) Final diameter
Volume is constant:

$$
\frac{\pi}{4}d_1^2 h_1 = \frac{\pi}{4}d_2^2 h_2 \Rightarrow d_2 = d_1\sqrt{\frac{h_1}{h_2}}
$$

$$
d_2 = 40\sqrt{\frac{120}{50}} = 40\times 1.549 = 61.97\ \text{mm}
$$

#### (b) True strain
$$
\varepsilon = \ln\frac{h_1}{h_2} = \ln\frac{120}{50} = 0.875
$$

#### (c) Force at the end of the stroke
Final cross-section:

$$
A_2 = \frac{\pi}{4}(61.97)^2 = 3016\ \text{mm}^2
$$

$$
F = \sigma_f A_2 = 100\times 3016 = 301{,}600\ \text{N} \approx 302\ \text{kN}
$$

The force is highest at the end of the stroke as the area is largest. With friction the real force would be higher.

#### Buckling check
$$
\frac{h_1}{d_1} = \frac{120}{40} = 3
$$

This is at the limit of the rule that the unsupported length should not exceed 3 times the diameter in a single blow. The upsetting should be done in a die or by a series of light blows, or the free length should be reduced.

**Answer: d₂ = 61.97 mm; true strain = 0.875; F = 302 kN; h/d = 3 is at the safe limit, so use a die or light blows.**

## Check yourself

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

1. Which relation gives the final diameter d2 after upsetting?
   A. d2 = d1 (h1/h2)
   B. d2 = d1 sqrt(h2/h1)
   C. d2 = d1 sqrt(h1/h2)
   D. d2 = d1 (h2/h1)^2
   Answer: C. Equal volumes give d1^2 h1 = d2^2 h2, so d2 = d1 sqrt(h1/h2) = 61.97 mm. (Key point: Constant volume principle)

2. What is the true strain for h1 = 120 mm to h2 = 50 mm?
   A. 0.875
   B. 0.583
   C. 1.4
   D. 2.4
   Answer: A. ε = ln(h1/h2) = ln 2.4 = 0.875. 2.4 is the ratio itself, not its logarithm. (Key point: True strain)

3. Why is the force calculated with the final area A2 and what is it?
   A. Area is largest at end of stroke, so force is highest: about 302 kN
   B. Area is smallest at end, F about 126 kN
   C. Force is constant; F about 126 kN using A1
   D. Friction is ignored so force is zero at the end
   Answer: A. F = σf A2 = 100 x 3016 = about 302 kN, the peak force when area is greatest. (Key point: Upsetting force)

4. h1/d1 = 3. What is the conclusion?
   A. Buckling is impossible
   B. It is at the safe limit, so use a die or light blows
   C. It must buckle, billet is unusable
   D. The billet is too short to upset
   Answer: B. A ratio of 3 is at the limit of the rule, so upset in a die or by light blows. (Key point: Buckling check)

## Related in Workshop Technology

- [Shaper machining time calculation](https://www.bctengineer.com/question.md?key=semester_2%2Fworkshop_technology%3Ashaper-machining-time-numerical) — Workshop Technology, chapter 6
- [Blanking force and press capacity for washer](https://www.bctengineer.com/question.md?key=semester_2%2Fworkshop_technology%3Ablanking-force-press-capacity) — Workshop Technology, chapter 8
- [Turning time and metal removal rate](https://www.bctengineer.com/question.md?key=semester_2%2Fworkshop_technology%3Alathe-turning-time-mrr-numerical) — Workshop Technology, chapter 6
- [Basic forging operations](https://www.bctengineer.com/question.md?key=semester_2%2Fworkshop_technology%3Aforging-operations-explained) — Workshop Technology, chapter 9

## Related in other subjects

- [Cold upsetting force with friction](https://www.bctengineer.com/question.md?key=semester_4%2Fmanufacturing_and_production_processes%3Aopen-die-forging-friction-numerical) — Manufacturing and Production Processes, chapter 5
- [Cutting volume, 120 m, upward slope 1 in 40](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-cutting-120m-formation-380) — Surveying I, chapter 9, asked 8×
- [Three reservoirs 120 m, 70 m and 50 m](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Athree-reservoir-a120-b70-c50) — Hydraulics, chapter 3
- [Force on 120° pipe bend, 0.23 m³/s](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Apipe-bend-120-force-0085) — Fluid Mechanics, chapter 7, asked 2×