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*

*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

Ïƒ
= 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 (Ï€/4) d

^{2}
Resisting
force due to hoop stress = Ïƒ x Ï€ 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 (Ï€/4) d

^{2 }= Ïƒ x Ï€ d t###
**Ïƒ = 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

Image
Courtesy: Google

How will you consider a wall is thin. what dimensions should wall have to be considered as thin ??

ReplyDeleteIf it's internal diameter is greater than or equal to 20 times it's wall thickness, it is considered thin

Deletethank you sir

ReplyDeleteWhat's the longitudinal stress in this case?

ReplyDelete