---
title: "Spherical vessel, volume increase 300 cm3, find pressure"
subject: "Strength of Materials"
chapter: 5
key: "semester_3/strength_of_materials:spherical-vessel-300cm3"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/spherical-vessel-300cm3
site: BCT Engineering
---

# Spherical vessel, volume increase 300 cm3, find pressure

Strength of Materials · Chapter 5: Thin Walled Vessels

- Asked in: 2081 Baisakh, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/spherical-vessel-300cm3
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-5

## Question

A seamless spherical vessel of 2.5 m internal diameter and 8 mm thick is filled with a fluid under the pressure until its volume increases by $300\ \text{cm}^3$. Calculate the pressure exerted by the fluid on the vessel. Take $E = 2.10\times10^5\ \text{N/mm}^2$, $\nu = 0.3$.

## Short answer

For thin sphere, volumetric strain ΔV/V=3pd(1–ν)/(4tE). With V=πd³/6 and given data, solving gives fluid pressure p≈0.047 N/mm² (0.047 MPa).

## Answer

**Data:** $d = 2500$ mm, $t = 8$ mm, $\delta V = 300\ \text{cm}^3 = 3\times10^5\ \text{mm}^3$, $E = 2.1\times10^5\ \text{N/mm}^2$, $\nu = 0.3$.

#### Method
For a thin sphere the stress in every direction is $\sigma = \dfrac{pd}{4t}$. The strain in any direction is

$$
\varepsilon = \frac{\sigma}{E}(1-\nu) = \frac{pd}{4tE}(1-\nu)
$$

and the volumetric strain is three times this:

$$
\frac{\delta V}{V} = 3\varepsilon = \frac{3pd}{4tE}(1-\nu)
$$

#### Calculation
$$
V = \frac{\pi d^3}{6} = \frac{\pi(2500)^3}{6} = 8.1812\times10^{9}\ \text{mm}^3
$$

$$
\frac{3\times10^5}{8.1812\times10^{9}} = \frac{3p(2500)}{4(8)(2.1\times10^5)}(1-0.3)
$$

$$
3.6669\times10^{-5} = 7.8125\times10^{-4}\,p \Rightarrow p = 0.0469\ \text{N/mm}^2
$$

**Answer: $p \approx 0.047\ \text{N/mm}^2$ (0.047 MPa).**

## Check yourself

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

1. What is the stress in a thin spherical shell?
   A. pd/4t in every direction
   B. pd/2t in every direction
   C. pd/4t in hoop only, pd/2t in meridional
   D. pd/t in every direction
   Answer: A. Cutting a sphere through its centre gives the same balance as a cylinder's longitudinal cut. (Key point: Stress in a thin sphere)

2. How is volumetric strain related to linear strain in a sphere?
   A. delta V / V = 2 x linear strain
   B. delta V / V = 3 x linear strain
   C. delta V / V = linear strain
   D. delta V / V = 3 x sigma / nu
   Answer: B. V = pi d^3 / 6, so delta V / V = 3 delta d / d. (Key point: Volumetric strain)

3. Which formula gives strain in the sphere?
   A. epsilon = sigma/E
   B. epsilon = (sigma/E)(1 + nu)
   C. epsilon = (sigma/E)(1 - nu)
   D. epsilon = (sigma/E)(1 - 2nu)
   Answer: C. Each direction has stress sigma, with two biaxial lateral effects: sigma/E - nu sigma/E. (Key point: Linear strain)

## Close variants set in other exams

- [Spherical vessel 1.9 m, volume increase 400 cm3](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aspherical-vessel-1-9m-400cm3) — Strength of Materials, chapter 5, asked 2×

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- [Steel spherical vessel: circumferential stress and diameter change](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asteel-spherical-vessel-1000mm-1mpa) — Strength of Materials, chapter 5
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