SKILL DEVELOPMENT STUDY #$2$

Due date: December $22,2024$

Material Specification for a Pressurized Cylindrical Tank

Consider a thin$-$walled cylindrical tank of radius $0\mathrm{.}5m(500mm)$ and wall thickness of $8\mathrm{.}0$ mm that is to be used as a pressure vessel to contain a fluid at a pressure of $2\mathrm{.}0$ MPa $.$ Assume a crack exists within the tank$\text{'}$s wall that is propagating from the inside to the outside as shown in Figure $1.($Note: Crack propagation may occur due to cyclic loading associated with fluctuations in pressure, or as a result of aggressive chemical attack of the wall material.$)$

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Figure $1.$ Schematic diagram showing the cross section of a cylindrical pressure vessel subjected to an internal pressure $p$ that has a radial crack of length a located on the inside wall.

Regarding the likelihood of failure of this pressure vessel, two scenarios are possible:

Leak$-$before$-$break. Using principles of fracture mechanics, allowance is made for the growth of the crack through the thickness of the vessel wall prior to rapid propagation. Thus, the crack will completely penetrate the wall without catastrophic failure, allowing for its detection by the leaking of pressurized fluid.

Brittle fracture. When the advancing crack reaches a critical length, which is shorter than for leak$-$before$-$break, fracture occurs by its rapid propagation through the entirety of the wall. This event typically results in the explosive expulsion of the vessel's fluid contents.

Obviously, leak$-$before$-$break is almost always the preferred scenario.

For a cylindrical pressure vessel, the circumferential $($or hoop$)$ stress ${}_h$ on the wall is a function of the pressure $p$ in the vessel and the radius $r$ and wall thickness $t$ according to the following expression:

${}_h=\frac{pr}{t}$

Using values of $p,r,$ and t provided earlier, we compute the hoop stress for this vessel as follows:

SKILL DEVELOPMENT STUDY #$2$

Due date: December $22,2024$

Material Specification for a Pressurized Cylindrical Tank

Consider a thin$-$walled cylindrical tank of radius $0\mathrm{.}5m(500mm)$ and wall thickness of $8\mathrm{.}0$ mm that is to be used as a pressure vessel to contain a fluid at a pressure of $2\mathrm{.}0$ MPa $.$ Assume a crack exists within the tank$\text{'}$s wall that is propagating from the inside to the outside as shown in Figure $1.($Note: Crack propagation may occur due to cyclic loading associated with fluctuations in pressure, or as a result of aggressive chemical attack of the wall material.$)$

$$

Figure $1.$ Schematic diagram showing the cross section of a cylindrical pressure vessel subjected to an internal pressure $p$ that has a radial crack of length a located on the inside wall.

Regarding the likelihood of failure of this pressure vessel, two scenarios are possible:

Leak$-$before$-$break. Using principles of fracture mechanics, allowance is made for the growth of the crack through the thickness of the vessel wall prior to rapid propagation. Thus, the crack will completely penetrate the wall without catastrophic failure, allowing for its detection by the leaking of pressurized fluid.

Brittle fracture. When the advancing crack reaches a critical length, which is shorter than for leak$-$before$-$break, fracture occurs by its rapid propagation through the entirety of the wall. This event typically results in the explosive expulsion of the vessel's fluid contents.

Obviously, leak$-$before$-$break is almost always the preferred scenario.

For a cylindrical pressure vessel, the circumferential $($or hoop$)$ stress ${}_h$ on the wall is a function of the pressure $p$ in the vessel and the radius $r$ and wall thickness $t$ according to the following expression:

${}_h=\frac{pr}{t}$

Using values of $p,r,$ and t provided earlier, we compute the hoop stress for this vessel as follows: