Module 3: Thin-Walled Pressure Vessels

Pressure vessels are closed structures containing liquids or gases under pressure. Examples include water pipes, compressed air tanks, boilers, and submarine hulls. In this module, we analyze the stresses developed in the walls of these vessels due to internal pressure.

When a fluid is pressurized inside a vessel, it exerts a uniform normal force against all interior surfaces. This internal pressure tends to burst the vessel outwards, creating tensile stresses within the material forming the vessel's walls.

A cylindrical pressure vessel (like a typical pipe or a standard hot water tank) experiences two primary types of normal stress when subjected to internal pressure: tangential (hoop) stress and longitudinal stress.

Hoop stress acts around the circumference of the cylinder. It is the stress that resists the bursting of the cylinder along its longitudinal axis (like a hot dog splitting lengthwise).

To find the hoop stress, imagine slicing the cylinder in half longitudinally. The internal gauge pressure ($p$) pushes against the projected rectangular area ($Diameter \times Length$), creating a bursting force. This force is resisted by the tensile force in the two cut surfaces of the vessel wall.

The formula for tangential (hoop) stress is:

Longitudinal stress acts parallel to the longitudinal axis of the cylinder. It is the stress that resists the vessel being pulled apart end-to-end (like a soda can popping its top off).

To find the longitudinal stress, imagine slicing the cylinder transversely. The internal pressure acts against the circular area of the end cap, creating a longitudinal bursting force. This force is resisted by the tensile force in the circular ring area of the vessel wall.

The formula for longitudinal stress is:

Many cylindrical pressure vessels are constructed by rolling steel plates and welding or riveting the longitudinal and circumferential seams. These joints are often the weakest point in the structure.

To account for this, engineers introduce a Joint Efficiency factor ($\eta$, a value $\le 1$) into the stress equations. The modified formulas become:

Spherical pressure vessels (like natural gas storage spheres or deep-sea submersibles) are highly efficient structural forms. Due to perfect geometric symmetry, a spherical vessel subjected to internal pressure experiences the exact same normal stress in all tangential directions along its surface.

If we slice a sphere exactly in half, the internal pressure acts on the projected circular area ($\frac{\pi}{4}D^2$), creating a bursting force. This force is resisted by the tensile force in the circular ring of the vessel wall ($\pi D t$). This derivation is mathematically identical to the longitudinal stress derivation for a cylinder.

The formula for uniform stress in a spherical vessel is:

The formulas for tangential ($\sigma_t = pr/t$) and longitudinal ($\sigma_l = pr/2t$) stresses in pressure vessels assume that the stress distribution across the wall thickness is uniform. This assumption is only valid if the wall is sufficiently thin compared to the radius of the vessel.

A pressure vessel is generally considered thin-walled if the ratio of its wall thickness ($t$) to its inner radius ($r$) is less than or equal to $0.1$ ($t/r \le 0.1$). If $t/r > 0.1$, the vessel is thick-walled, the stress distribution becomes non-linear, and more complex equations (like Lame's Equations) must be used.