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Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Mechanical Engineering SSC JE (Technical)

Mechanical Engineering : Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

The document Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
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THIN & THICK CYLINDERS AND SPHERES

THIN CYLINDERS

  •  If the thickness of the cylinder is less than 1/10 to 1/15 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 ofthe wall.
  •  'Hoop stress’ or ‘Circumferential stress’ is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   ‘Longitudinal stress’ is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  The Maximum shear stress is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Hoop strain is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Longitudinal strain is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Volumetric strain is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

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

Hence

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
or
Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

THIN SPHERICAL SHELLS 

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

s1 =s2 = Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev (tensile in nature)

  •  The maximum shear stress,  

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  The strain in any direction is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   The volumetric strain is given by

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRevCYLINDERS WITH HEMISPHERICAL ENDS

Let tc = thickness of the cylinder
ts = thickness of the hemisphere

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   Hoop stress in cylindrical part

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   Hoop stress in hemispherical part

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Longitudinal stress in cylindrical part

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Longitudinal stress in hemispherical part

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   Circumferential strain in hemispherical part

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   Circumferential strain in cylindrical part

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   For the condition of no distortion of the junction  

Thus, equating the two strains in order that there shall be no distortion of the junction.

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering 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,

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
THICK CYLINDRICAL SHELL

  •   If the thickness of shell is greater than of
    Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

 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

Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Hoop stress, is given by
Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Radial pressure is given by
Chapter 3 Thin and Thick Cylinders and Spheres - Strength of Material, Mechanical Engineering Mechanical Engineering 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 (atm.) hyperbolically.

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