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

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Let us consider one elemental ring of thickness Î´r as displayed in above figure. Let us assume that radial stresses are Ïƒ

Constants A and B could be obtained from the boundary conditions

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Image
courtesy: Google

Today we will see here the basics of thick cylinder,
applications of thick cylinders, distribution of stresses in the thick
cylinders and thick cylinder lame's equation with the help of this post.

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**Thick
cylinder: Basics, Applications, Distribution of stresses and Lame's equation **

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**Basics
of thick cylinder **

Thick cylinders are basically those cylindrical
vessels that contain fluid under pressure and ratio of wall thickness to the
internal diameter of such cylindrical vessels will not be less than 1/15.

We can see the practical applications of thick
cylinders in various areas such as domestic gas cylinders, oxygen gas cylinders
used in medical field, barrel of a gun, water and gas pipelines, nozzle of a jet
etc.

Radial stress will be varied along the thickness and
it will be maximum at the inner radius and minimum at the outer radius.

We can obtain the variation of radial as well as
circumferential stress across the thickness with the help of Lame’s Theory.

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**So
let us now find out the Lame’s Theory or Lame’s equation **

In order to find out the distribution of stresses in
the thick cylinders, Lame’s theory or Lame’s equation will be applied.

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**Assumptions **

There are following assumptions are made as mentioned
below during the derivation of Lame’s equation.

- Material will be homogeneous and isotropic.
- Material will be stressed within the elastic limit as per hooks’ law.
- Plane transverse sections will remain plain even after their elongation under the action of internal pressure. Plane transverse sections will not be distorted.
- All the fibres of material will be stressed independently without being constrained by the adjacent fibres.

Let us consider a thick cylinder having length l,
internal radius a and external radius b subjected to internal and external
uniformly distributed pressure of intensities P

_{a}and P_{b}respectively as displayed here in following figure.
Let us consider the following terms and their
nomenclature, as mentioned below, which will be used in the derivation of
equation of Lame’s.

Ïƒ

_{r}= Radial stress at radius r
Ïƒ

_{Ï´}= Hoop stress at radius r
Ïƒ

_{L}= Longitudinal stress
a = Internal radius of thick cylinder

b = External radius of thick cylinder

r = Internal radius of the elemental ring

r + Î´r =
External radius of the elemental ring

P

_{a}= Pressure intensity at internal radius of thick cylinder
P

_{b}= Pressure intensity at external radius of thick cylinderLet us consider one elemental ring of thickness Î´r as displayed in above figure. Let us assume that radial stresses are Ïƒ

_{r}and Ïƒ

_{r}+ Î´r acting at the radius r and Î´r respectively.

Constants A and B could be obtained from the boundary conditions

At r = a, Ïƒ

_{r}= - P_{a}
At r = b, Ïƒ

_{r}= - P_{b }
After using the boundary conditions, we will have following
values for constants A and B as mentioned below.

Now we will use the values of constants A and B in the
equations 2 and 3 and we will have the following expression for the variation
of stresses.

Therefore, we have seen here the basic concepts of
fans and blowers, centrifugal fan and blowers, different types of impeller
blades and finally we have also seen the velocity triangles at the inlet and
outlet of different type of impeller blades.

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Further we will find out, in our next post,

### Reference:

Strength of materials, A. K Sirvastava and P.C. Gope

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