## Thursday, 16 November 2017

We were discussing the basic concept of thin cylindrical and spherical shells, stresses in thin cylindrical shells and derivation of expression for circumferential stress or Hoop stress and longitudinal stress developed in the wall of cylindrical shell in our previous posts.

Today we will derive here the expression for stress developed in the wall of thin spherical shell, with the help of this post.

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

Thin spherical 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. Air receiver tank is one of the best examples of thin spherical shells.

A spherical shell will be considered as thin 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 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 spherical 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 thin spherical shell.

P = Internal fluid pressure
d = Internal diameter of thin spherical shell
t = Thickness of the wall of thin spherical shell
σ = Circumferential stress or hoop stress developed in the wall of thin spherical shell

Thin spherical shell bursting will take place if force due to internal fluid pressure, acting on the wall of thin spherical shell, will be more than the resisting force due to circumferential stress or hoop stress developed in the wall of thin spherical shell.

In order to secure the expression for circumferential stress or hoop stress developed in the wall of thin spherical 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 hoop stress developed in the wall of thin spherical shell.

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 (π/4) d2
Resisting force due to hoop stress = σ x π d t

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

Force due to internal fluid pressure = Resisting force due to longitudinal stress
P x (π/4) d2 = σ x π d t

### σ = P x d / (4 t)

Do you have suggestions? Please write in comment box.

We will now discuss the basic concept of thick cylindrical and spherical shell, in the category of strength of material, in our next post.

### Reference:

Strength of material, By R. K. Bansal

1. 1. 2. 