Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

The document Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Civil Engineering SSC JE (Technical).
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 THIN CYLINDERS

  • If the thickness of the cylinder is less thanThin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev of the diameter of the cylinder, it is treated as the thin cylinder.
  • It is assumed that the stresses are uniformly distributed through out the thickness of the wall.
  • ‘Hoop stress’ or ‘Circumferential stress’ is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • ‘Longitudinal stress’ is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • The Maximum shear stress is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Hoop strain is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Longitudinal strain is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Volumetric strain is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

Where p = internal pressure
d = diameter of cylinder
t = thickness of the cylinder
µ = Poisson's ratio

  • If ‘σa’ be the permissible tensile stress for the shell material, then from strength point of view, the major principal stress ( σ ) should be less than or equal to σa Hence

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

THIN SPHERICAL SHELLS

  • In case of thin spherical shells, longitudinal stress and circumferential stress are equal and are given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev (tensile in nature)

  • The maximum shear stress,

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • The strain in any direction is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • The volumetric strain is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

CYLINDERS WITH HEMISPHERICAL ENDS
Let tc = thickness of the cylinder ts = thickness of the hemisphere

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Hoop stress in cylindrical part

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Hoop stress in hemispherical part

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Longitudinal stress in cylindrical part

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Longitudinal stress in hemispherical part

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Circumferential strain in hemispherical part

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Circumferential strain in cylindrical part

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • For the condition of no distortion of the junction

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRevThin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev 

  • This means thickness of cylindrical part should be more than the hemispherical part.
  • For the condition of same maximum stress in cylindrical and hemispherical parts,

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

THICK CYLINDRICAL SHELL

If the thickness of shell is greater than Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRevof  its diameter then it is called thick shell.

In thick cylinders the circumferential stress no longer remains constant, but varies along the thickness and the radial pressure (px) is also not negligible.
The following three types of stresses are existing in thick cylinders :

(i) The radial pressure ‘px’ (compressive)
(ii) The hoop stress fx (tensile)
(iii) The longitudinal tensile stress po (tensile)

  • The longitudinal stress may be given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Hoop stress, is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

  • Radial pressure is given by

 Thin and Thick Cylinders and Spheres Civil Engineering (CE) Notes | EduRev

Equation (ii) and (iii) are called Lame’s equation.

ro = outer radius of shell 

ri = inner radius of shell A and B are Lame’s constant

Note:-

1. Longitudinal tension is uniform across the thickness.

2. Hoop tension vary form maximum at inner face to minimum at outer face Hyperbolically.

3. Radial compression varies from maximum at inner face to zero at outer face. Hyperbolically.

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