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*

*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

L = Length of the cylindrical shell

σ

_{H}= Circumferential stress or hoop stress developed in the wall of the cylindrical shellCylindrical 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)
Do you have suggestions? Please write in comment
box.

We will now derive the expression for longitudinal stress developed in the wall of cylindrical shell, in the category of strength of material, in our next post.

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google