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

σ_{L}
= 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) d^{2}

Resisting
force due to longitudinal stress = σ_{L} 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) d^{2
}= σ_{L} x π d t

σ_{L}
= 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

Image
Courtesy: Google

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