Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Mechanical Engineering: Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

The document Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
All you need of Mechanical Engineering at this link: Mechanical Engineering


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

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   ‘Longitudinal stress’ is given by

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  The Maximum shear stress is given by

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Hoop strain is given by

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Longitudinal strain is given by

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Volumetric strain is given by

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
or
Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

THIN SPHERICAL SHELLS 

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

s1 =s2 = Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering (tensile in nature)

  •  The maximum shear stress,  

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  The strain in any direction is given by

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   The volumetric strain is given by

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical EngineeringCYLINDERS WITH HEMISPHERICAL ENDS

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

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   Hoop stress in cylindrical part

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   Hoop stress in hemispherical part

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Longitudinal stress in cylindrical part

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Longitudinal stress in hemispherical part

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   Circumferential strain in hemispherical part

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   Circumferential strain in cylindrical part

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   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.

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
THICK CYLINDRICAL SHELL

  •   If the thickness of shell is greater than of
    Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

 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, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Hoop stress, is given by
Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Radial pressure is given by
Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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.

The document Thin, Thick Cylinders & Spheres Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
All you need of Mechanical Engineering at this link: Mechanical Engineering

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