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 developed in the wall of cylindrical shell in our previous posts.

Today we will derive here the expression for longitudinal stress developed in the wall of thin 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.

### Longitudinal stress

Stress acting along the length of thin cylinder will be termed as longitudinal stress.

If fluid is stored under pressure inside the cylindrical shell, pressure force will be acting along the length of the cylindrical shell at its two ends. Cylindrical shell will tend to burst as displayed here in following figure and stresses developed in such failure of cylindrical shell will be termed as longitudinal stress.
Let us consider here following terms to derive the expression for longitudinal 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
σL = Longitudinal stress developed in the wall of the cylindrical shell

Cylindrical shell bursting will take place if force due to internal fluid pressure, acting on the ends of the cylinder, will be more than the resisting force due to longitudinal stress developed in the wall of the cylindrical shell.

In order to secure the expression for longitudinal 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, acting on the ends of the cylinder, should be equal to the resisting force due to longitudinal stress developed in the wall of the cylindrical 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 longitudinal stress = σL 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 = σL x π d t
σL = P x d / (4 t)

Do you have suggestions? Please write in comment box.

We will now derive the expression for stress developed in the wall of thin spherical shells, in the category of strength of material, in our next post.

### Reference:

Strength of material, By R. K. Bansal

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