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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 ‘s_a’ 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 s_a

Hence

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

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 | EduRev CYLINDERS WITH HEMISPHERICAL ENDS

Let t_c = thickness of the cylinder
t_s = 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

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

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 (p_x) is also not negligible.

The following three types of stresses are existing in thick cylinders :

(i) The radial pressure ‘p_x’ (compressive)
(ii) The hoop stress f_x (tensile)
(iii) The longitudinal tensile stress p_o (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

Equation (ii) and (iii) are called Lame’s equation.
r_o = outer radius of shell
r_i = 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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