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Monday, 4 November 2019

THICK CYLINDER LAME'S EQUATION

We were discussing the various basic concept of thin cylinders such as thin cylindrical and spherical shellsstresses 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. 

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. 

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


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. 

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. 

Assumptions 


There are following assumptions are made as mentioned below during the derivation of Lame’s equation. 
  1. Material will be homogeneous and isotropic. 
  2. Material will be stressed within the elastic limit as per hooks’ law. 
  3. Plane transverse sections will remain plain even after their elongation under the action of internal pressure. Plane transverse sections will not be distorted. 
  4. 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 Pa and Pb 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. 

σr = Radial stress at radius r
σϴ = Hoop stress at radius r
σL = Longitudinal stress
a = Internal radius of thick cylinder
b = External radius of thick cylinder
r = Internal radius of the elemental ring
r + δr = External radius of the elemental ring  
Pa = Pressure intensity at internal radius of thick cylinder
Pb = Pressure intensity at external radius of thick cylinder 

Let us consider one elemental ring of thickness δr as displayed in above figure. Let us assume that radial stresses are σr and σr δr acting at the radius r and δr respectively. 



Constants A and B could be obtained from the boundary conditions
At r = a, σr = - Pa
At r = b, σr = - P

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 

Image courtesy: Google  


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