DERIVATION OF HOOP STRESS IN THIN CYLINDER

We were discussing the basic concept of thin cylindrical and spherical shells and stresses in thin cylindrical shells in our previous posts.

Today we will derive the expression for circumferential stress or Hoop stress developed in the wall of cylindrical shell, with the help of this post.

Before going ahead, we will first remind here the fundamental of a thin cylindrical shell

Thin cylindrical shell is also termed as a pressure vessel and such vessels are usually used in various engineering applications such as for storing the fluid under pressure. Boilers, LPG cylinders, Air receiver tanks are the best examples of thin cylindrical shells.

A cylindrical or spherical shell will be considered as thin cylindrical or spherical shell, if the wall thickness of shell is very small as compared to the internal diameter of the shell.

Wall thickness of a thin cylindrical and spherical shell will be equal or less than the 1/20 of the internal diameter of shell.

Circumferential stress or Hoop stress

Stress acting along the circumference of thin cylinder will be termed as circumferential stress or hoop stress.

If fluid is stored under pressure inside the cylindrical shell, pressure will be acting vertically upward and downward over the cylindrical wall. Pressure vessel will tend to burst as displayed here in following figure and stresses developed in such failure of cylindrical shell will be termed as circumferential stress or Hoop stress.

Let us consider here following terms to derive the expression for circumferential stress or Hoop stress developed in the wall of cylindrical shell.

P = Internal fluid pressure

d = Internal diameter of thin cylindrical shell

t = Thickness of the wall of the cylinder
L = Length of the cylindrical shell

σ_H = Circumferential stress or hoop stress developed in the wall of the cylindrical shell

Cylindrical shell bursting will take place if force due to internal fluid pressure will be more than the resisting force due to circumferential stress or hoop stress developed in the wall of the cylindrical shell.

In order to secure the expression for circumferential stress or hoop stress developed in the wall of the cylindrical shell, we will have to consider the limiting case i.e. force due to internal fluid pressure should be equal to the resisting force due to circumferential stress or hoop stress.

Force due to internal fluid pressure = Internal fluid pressure x Area on which fluid pressure will be acting

Force due to internal fluid pressure = P x (d x L)

Force due to internal fluid pressure = P x d x L

Resisting force due to circumferential stress = σ_H x 2 L t

As we have seen above, we can write following equation as mentioned here.

Force due to internal fluid pressure = Resisting force due to circumferential stress

P x d x L = σ_H x 2 L t

σ_H = P x d / (2 t)