Thin Pressure Vessels: Cylinders and Spheres

Definition

A thin pressure vessel is defined as a closed container, usually cylindrical or spherical, designed to hold gases or liquids at a pressure substantially different from the ambient pressure. In engineering mechanics, a vessel is classified as "thin" if the ratio of its wall thickness ($t$) to its internal radius ($r$) is less than 1/10 or 1/20 ($t/r < 0.1$). This is one of the important concepts often covered in the university syllabus for solid mechanics and machine design.

Main Content

1. Thin Cylindrical Shells

When a cylinder is subjected to internal fluid pressure, it experiences two primary stresses: Hoop (circumferential) stress and Longitudinal (axial) stress.
Hoop stress acts along the circumference, attempting to split the cylinder longitudinally, while longitudinal stress acts along the axis, attempting to blow off the cylinder ends.

       Internal Pressure (P)
      <--------------------->
    ___________________________
   |                           |
   |      ( Hoop Stress )      |
   |      /-------------\      |
   |     |               |     |
   |     |      (r)      |     |
   |     |               |     |
   |      \-------------/      |
   |___________________________|

2. Thin Spherical Shells

Due to symmetry, a spherical shell experiences a uniform stress distribution in all directions along the surface.
The stress developed in a sphere is significantly lower than the hoop stress in a cylinder of the same radius and thickness, making spheres more efficient for high-pressure storage.

3. Stress Formulas

For cylinders, Hoop Stress ($\sigma_h$) is $Pd / 2t$ and Longitudinal Stress ($\sigma_l$) is $Pd / 4t$.
For spheres, the stress ($\sigma$) is uniform throughout: $\sigma = Pd / 4t$, where $P$ is pressure, $d$ is diameter, and $t$ is thickness.

Working / Process

1. Identifying Shell Geometry

Determine the $t/d$ ratio to confirm if the vessel qualifies as a thin shell.
Identify the internal pressure $P$ and the material properties (like allowable yield strength) for exam preparation.

2. Calculating Principal Stresses

Apply the formulas for hoop and longitudinal stresses based on the geometry identified.
For cylindrical vessels, always calculate both hoop and longitudinal stresses, as the hoop stress is typically twice the longitudinal stress, making it the design-limiting factor.

3. Applying Factor of Safety

Calculate the resulting stress and compare it against the material's yield strength.
Divide by the factor of safety to ensure the vessel can withstand fluctuations in pressure, a common theme in interview questions for mechanical engineering roles.

Advantages / Applications

High volume-to-surface area efficiency: Spherical tanks are widely used for storing liquefied gases (LPG/LNG) as they minimize material usage for a given volume.
Safety: Thin-walled design allows for flexibility under pressure, provided the internal pressure does not exceed the design limit.
Applications: Found in boilers, air compressors, domestic hot water tanks, and large-scale industrial gas storage units.

Summary

Thin pressure vessels are critical structural components used to store fluids under pressure, classified by a wall thickness ratio of $t/r < 0.1$. Cylinders face both hoop and longitudinal stresses, whereas spheres encounter uniform membrane stress. Engineers use these important concepts to design safe storage solutions by balancing material thickness with internal pressure loads. Key terms include Hoop Stress, Longitudinal Stress, Internal Pressure, and Wall Thickness.