Scenario: A large cylindrical compressed air tank explodes in a manufacturing facility. Investigators must determine the cause of failure by analyzing the stresses that were present at the time of the rupture.

Analysis: In a cylindrical vessel under internal pressure, the tangential (hoop) stress is exactly twice the longitudinal stress ($\sigma_t = 2\sigma_l$). The investigators examine the rupture pattern and note that the tank split longitudinally (along its length). This pattern confirms that the hoop stress—which acts to pull the cylinder apart circumferentially—exceeded the material's ultimate strength. If the tank had split radially (like a sliced sausage), it would have implied a failure due to longitudinal stress, which is highly uncommon unless there was a specific circumferential flaw.

A cylindrical steel pressure vessel has an inner diameter of $1.2\text{ m}$ and a wall thickness of $12\text{ mm}$. The internal pressure is $1.5\text{ MPa}$. Determine the tangential (hoop) stress in the vessel wall.

A cylindrical water pipe of internal diameter $0.5\text{ m}$ and wall thickness $5\text{ mm}$ is subjected to an internal water pressure of $2.0\text{ MPa}$. Calculate both the longitudinal and tangential stresses.

A cylindrical tank is required to store propane gas at an internal pressure of $2.5\text{ MPa}$. The allowable tensile stress for the steel used is $140\text{ MPa}$. The inner diameter of the tank must be $2.0\text{ m}$. Determine the minimum required wall thickness of the tank.

A spherical gas storage tank has an inner diameter of $10\text{ m}$ and a wall thickness of $25\text{ mm}$. If the internal pressure is $1.0\text{ MPa}$, determine the normal stress in the wall of the tank.

A spherical pressure vessel is designed to hold a gas at $3.0\text{ MPa}$. The allowable stress of the material is $150\text{ MPa}$. If the tank must have an internal volume of $100\text{ m}^3$, determine the required wall thickness.

An engineer needs to design a tank to store gas at $2\text{ MPa}$. The volume must be $50\text{ m}^3$ and the allowable stress is $120\text{ MPa}$. Compare the required wall thickness if the tank is designed as a sphere versus a cylinder with a length equal to twice its diameter ($L=2D$).