Stress due to internal pressure: Change in diameter and volume

Definition

When a thin-walled cylindrical or spherical pressure vessel is subjected to internal fluid pressure, the material of the shell experiences tensile stresses. These stresses cause the vessel to expand, resulting in a measurable change in its diameter (or radius) and internal volume. This phenomenon is critical for exam preparation and understanding the mechanical stability of industrial vessels.

Main Content

1. Hoop Stress (Circumferential Stress)

Hoop stress acts along the circumference of the cylinder and is responsible for the increase in diameter.
For a thin cylinder with internal pressure $P$, diameter $d$, and thickness $t$, the hoop stress ($\sigma_h$) is calculated as $\sigma_h = \frac{Pd}{2t}$.

2. Longitudinal Stress

Longitudinal stress acts along the axis of the cylinder and is responsible for the change in length.
For a closed-end cylinder, the longitudinal stress ($\sigma_L$) is $\sigma_L = \frac{Pd}{4t}$, which is exactly half of the hoop stress.

3. Volumetric Strain

Volumetric strain ($\epsilon_v$) is the total change in volume relative to the original volume, resulting from the combined effect of radial and longitudinal expansions.
It is a vital calculation in university syllabus topics related to thin shells, calculated as $\epsilon_v = \epsilon_L + 2\epsilon_h$.

Working / Process

1. Determining Stresses

First, identify the internal pressure $P$, the material's Young’s Modulus ($E$), and Poisson’s ratio ($\mu$).
Calculate the hoop stress ($\sigma_h$) and longitudinal stress ($\sigma_L$) using the thin-wall formulas.

2. Calculating Strains

Using Hooke's law, find the hoop strain ($\epsilon_h = \frac{1}{E}(\sigma_h - \mu\sigma_L)$) and longitudinal strain ($\epsilon_L = \frac{1}{E}(\sigma_L - \mu\sigma_h)$).
These strains represent the individual dimensional changes per unit length.

3. Calculating Change in Dimensions

Change in diameter ($\delta d = \epsilon_h \times d$) and change in volume ($\delta V = \epsilon_v \times V$).
This is often asked in interview questions to test a student's grasp of how pressure alters the physical capacity of a container.

       Internal Pressure (P)
          |     |     |
      ____v_____v_____v____
     |                     |
     |   (o) -------->     |  <-- Hoop Strain
     |    |                |
     |    |                |
     |    v Longitudinal |
     |      Strain         |
     |_____________________|

Visual representation of expansion in a cylindrical pressure vessel.

Advantages / Applications

Design and safety analysis of boilers and pipelines, which are important concepts for mechanical engineers.
Used in the aerospace industry for fuel tank design to ensure structural integrity under flight pressures.
Essential for calculating the burst pressure of storage tanks in chemical processing plants.

Summary

When thin shells are pressurized, internal stress induces dimensional changes; hoop and longitudinal strains combine to determine the total volumetric expansion. Mastering these calculations is fundamental for engineering design, safety assessment, and success in technical exams. Key terms include Hoop Stress, Longitudinal Stress, Volumetric Strain, Poisson’s Ratio, and Young’s Modulus.