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

•   ‘Longitudinal stress’ is given by

•  The Maximum shear stress is given by

•  Hoop strain is given by

•  Longitudinal strain is given by

•  Volumetric strain is given by

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

or


THIN SPHERICAL SHELLS 

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

s₁ =s₂ =  (tensile in nature)

•  The maximum shear stress,  

•  The strain in any direction is given by

•   The volumetric strain is given by

CYLINDERS WITH HEMISPHERICAL ENDS

Let t_c = thickness of the cylinder
t_s = thickness of the hemisphere

•   Hoop stress in cylindrical part

•   Hoop stress in hemispherical part

•  Longitudinal stress in cylindrical part

•  Longitudinal stress in hemispherical part

•   Circumferential strain in hemispherical part

•   Circumferential strain in cylindrical part

•   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.

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,

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

Hoop stress, is given by

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.