Short Notes: Thin Cylinders | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Thin Cylinders
Types of Pressure Vessels Pressure vessels are mainly of two type:
• Thin shells
If the thickness of the wall of the shell is less than 1/10 to 1/15 of its diameter, 
then shell is called thin shells.
• Thick shells
If the thickness of the wall of the shell is greater than 1/10 to 1/15 of its diameter, 
then shell is called shells.
where Nature of stress in thin cylindrical shell subjected to Internal pressure
1. Hoop stress/circumferential stress will be tensile in nature.
2. Longitudinal stress/axial stress will be tensile in nature
3. Radial stress will be compressive in nature.
Stresses in Thin Cylindrical Shell:
Page 2


Thin Cylinders
Types of Pressure Vessels Pressure vessels are mainly of two type:
• Thin shells
If the thickness of the wall of the shell is less than 1/10 to 1/15 of its diameter, 
then shell is called thin shells.
• Thick shells
If the thickness of the wall of the shell is greater than 1/10 to 1/15 of its diameter, 
then shell is called shells.
where Nature of stress in thin cylindrical shell subjected to Internal pressure
1. Hoop stress/circumferential stress will be tensile in nature.
2. Longitudinal stress/axial stress will be tensile in nature
3. Radial stress will be compressive in nature.
Stresses in Thin Cylindrical Shell:
In the figure we have shown a one half of the cylinder. This cylinder is subjected to 
an internal pressure p.
i.e. p = internal pressure
d = inside diametre 
L = Length of the cylinder 
t = thickness of the wall
Total force on one half of the cylinder owing to the internal pressure 'p'
= p x Projected Area 
= p x d x L
= p.d.L ------ (1)
The total resisting force owing to hoop stresses sH set up in the cylinder walls 
= 2 .oh .L.t --------(2)
Because s n-L.t. is the force in the one wall of the half cylinder.
the equations (1) & (2) we get
2 . 0h .L .t = p .d .L
oh = (p . d) / 2t
2t
=>
pd 
2 tn
q = Efficiency of joint 
• Longitudinal Stress
£ 7 ; =
pd pd
-— = > a =— — 
41 ¦ Atrj
Hoop Strain
* = 1
Page 3


Thin Cylinders
Types of Pressure Vessels Pressure vessels are mainly of two type:
• Thin shells
If the thickness of the wall of the shell is less than 1/10 to 1/15 of its diameter, 
then shell is called thin shells.
• Thick shells
If the thickness of the wall of the shell is greater than 1/10 to 1/15 of its diameter, 
then shell is called shells.
where Nature of stress in thin cylindrical shell subjected to Internal pressure
1. Hoop stress/circumferential stress will be tensile in nature.
2. Longitudinal stress/axial stress will be tensile in nature
3. Radial stress will be compressive in nature.
Stresses in Thin Cylindrical Shell:
In the figure we have shown a one half of the cylinder. This cylinder is subjected to 
an internal pressure p.
i.e. p = internal pressure
d = inside diametre 
L = Length of the cylinder 
t = thickness of the wall
Total force on one half of the cylinder owing to the internal pressure 'p'
= p x Projected Area 
= p x d x L
= p.d.L ------ (1)
The total resisting force owing to hoop stresses sH set up in the cylinder walls 
= 2 .oh .L.t --------(2)
Because s n-L.t. is the force in the one wall of the half cylinder.
the equations (1) & (2) we get
2 . 0h .L .t = p .d .L
oh = (p . d) / 2t
2t
=>
pd 
2 tn
q = Efficiency of joint 
• Longitudinal Stress
£ 7 ; =
pd pd
-— = > a =— — 
41 ¦ Atrj
Hoop Strain
* = 1
Longitudinal Strain
e L = — ( l- 2 ( j)
L 4 t E '
• Ratio of Hoop Strain to Longitudinal Strain
£ h _ 2 -\ l
si l - 2n
• Volumetric Strain of cylinder
£v = ^ ~ ( 5-4(J)
v 4 tE y
Stresses in Thin Spherical Shell:
• Hoop stress/longitudinal stress
Qt = %
p §
41
Stresses on 
spherical shell
• Hoop stress/longitudinal strain
- £» - ™ c1 - &
• Volumetric strain of sphere
H =
4 tE v
m
Cylindrical shell with hemisphere ends
Let us now consider the vessel with hemispherical ends. The wall thickness of the 
cylindrical and hemispherical portion is different. While the internal diameter of 
both the portions is assumed to be equal 
Let the cylindrical vassal is subjected to an internal pressure p.
For the cylindrical portion
Page 4


Thin Cylinders
Types of Pressure Vessels Pressure vessels are mainly of two type:
• Thin shells
If the thickness of the wall of the shell is less than 1/10 to 1/15 of its diameter, 
then shell is called thin shells.
• Thick shells
If the thickness of the wall of the shell is greater than 1/10 to 1/15 of its diameter, 
then shell is called shells.
where Nature of stress in thin cylindrical shell subjected to Internal pressure
1. Hoop stress/circumferential stress will be tensile in nature.
2. Longitudinal stress/axial stress will be tensile in nature
3. Radial stress will be compressive in nature.
Stresses in Thin Cylindrical Shell:
In the figure we have shown a one half of the cylinder. This cylinder is subjected to 
an internal pressure p.
i.e. p = internal pressure
d = inside diametre 
L = Length of the cylinder 
t = thickness of the wall
Total force on one half of the cylinder owing to the internal pressure 'p'
= p x Projected Area 
= p x d x L
= p.d.L ------ (1)
The total resisting force owing to hoop stresses sH set up in the cylinder walls 
= 2 .oh .L.t --------(2)
Because s n-L.t. is the force in the one wall of the half cylinder.
the equations (1) & (2) we get
2 . 0h .L .t = p .d .L
oh = (p . d) / 2t
2t
=>
pd 
2 tn
q = Efficiency of joint 
• Longitudinal Stress
£ 7 ; =
pd pd
-— = > a =— — 
41 ¦ Atrj
Hoop Strain
* = 1
Longitudinal Strain
e L = — ( l- 2 ( j)
L 4 t E '
• Ratio of Hoop Strain to Longitudinal Strain
£ h _ 2 -\ l
si l - 2n
• Volumetric Strain of cylinder
£v = ^ ~ ( 5-4(J)
v 4 tE y
Stresses in Thin Spherical Shell:
• Hoop stress/longitudinal stress
Qt = %
p §
41
Stresses on 
spherical shell
• Hoop stress/longitudinal strain
- £» - ™ c1 - &
• Volumetric strain of sphere
H =
4 tE v
m
Cylindrical shell with hemisphere ends
Let us now consider the vessel with hemispherical ends. The wall thickness of the 
cylindrical and hemispherical portion is different. While the internal diameter of 
both the portions is assumed to be equal 
Let the cylindrical vassal is subjected to an internal pressure p.
For the cylindrical portion
'c' here synifiesthe cylindrical portion. hoop or circumferential stress = oH C 
= p d 
2t,
I? ri gitud na I st re ss = uL C 
=
4t,
hoop or circumferential strain e 2 =
- P d r
4t,E
v= poisson's ratio
For the hemispherical ends
Because of the symmetry of the sphere the stresses set up owing to internal 
pressure will be two mutually perpendicular hoops or circumferential stresses of 
equal values. Again the radial stresses are neglected in comparison to the hoop 
stresses as with this cylinder having thickness to diametre less than1:20.
Consider the equilibrium of the half - sphere
Force on half-sphere owing to internal pressure = pressure x projected Area 
= p. TTd2/4
Re sisting force = uH . w . d.t2 
w .d2
P - — = < 7 H-H d.t2 
=¥ aH (for sphere) =
sirnilarly the hoop strain = 1 Jo, - v.c„ J = -^-[1 - y] = " v ] or e^-BJLri-
4t2 EL
Thus equating the two strains in order that there shall be no distortion of the 
junction
_ P ^ _ r2-w l =-^-ri-
4t,EL J 4t2E L
1 -v 
2-v
Lame's Theory
Lame's theory is based on the following assumptions
Assumptions
1. Homogeneous material.
2. Plane section of cylinder, perpendicular to longitudinal axis remains under 
plane and pressure. Hoop stress at any section
b
<7, = —+ a
Page 5


Thin Cylinders
Types of Pressure Vessels Pressure vessels are mainly of two type:
• Thin shells
If the thickness of the wall of the shell is less than 1/10 to 1/15 of its diameter, 
then shell is called thin shells.
• Thick shells
If the thickness of the wall of the shell is greater than 1/10 to 1/15 of its diameter, 
then shell is called shells.
where Nature of stress in thin cylindrical shell subjected to Internal pressure
1. Hoop stress/circumferential stress will be tensile in nature.
2. Longitudinal stress/axial stress will be tensile in nature
3. Radial stress will be compressive in nature.
Stresses in Thin Cylindrical Shell:
In the figure we have shown a one half of the cylinder. This cylinder is subjected to 
an internal pressure p.
i.e. p = internal pressure
d = inside diametre 
L = Length of the cylinder 
t = thickness of the wall
Total force on one half of the cylinder owing to the internal pressure 'p'
= p x Projected Area 
= p x d x L
= p.d.L ------ (1)
The total resisting force owing to hoop stresses sH set up in the cylinder walls 
= 2 .oh .L.t --------(2)
Because s n-L.t. is the force in the one wall of the half cylinder.
the equations (1) & (2) we get
2 . 0h .L .t = p .d .L
oh = (p . d) / 2t
2t
=>
pd 
2 tn
q = Efficiency of joint 
• Longitudinal Stress
£ 7 ; =
pd pd
-— = > a =— — 
41 ¦ Atrj
Hoop Strain
* = 1
Longitudinal Strain
e L = — ( l- 2 ( j)
L 4 t E '
• Ratio of Hoop Strain to Longitudinal Strain
£ h _ 2 -\ l
si l - 2n
• Volumetric Strain of cylinder
£v = ^ ~ ( 5-4(J)
v 4 tE y
Stresses in Thin Spherical Shell:
• Hoop stress/longitudinal stress
Qt = %
p §
41
Stresses on 
spherical shell
• Hoop stress/longitudinal strain
- £» - ™ c1 - &
• Volumetric strain of sphere
H =
4 tE v
m
Cylindrical shell with hemisphere ends
Let us now consider the vessel with hemispherical ends. The wall thickness of the 
cylindrical and hemispherical portion is different. While the internal diameter of 
both the portions is assumed to be equal 
Let the cylindrical vassal is subjected to an internal pressure p.
For the cylindrical portion
'c' here synifiesthe cylindrical portion. hoop or circumferential stress = oH C 
= p d 
2t,
I? ri gitud na I st re ss = uL C 
=
4t,
hoop or circumferential strain e 2 =
- P d r
4t,E
v= poisson's ratio
For the hemispherical ends
Because of the symmetry of the sphere the stresses set up owing to internal 
pressure will be two mutually perpendicular hoops or circumferential stresses of 
equal values. Again the radial stresses are neglected in comparison to the hoop 
stresses as with this cylinder having thickness to diametre less than1:20.
Consider the equilibrium of the half - sphere
Force on half-sphere owing to internal pressure = pressure x projected Area 
= p. TTd2/4
Re sisting force = uH . w . d.t2 
w .d2
P - — = < 7 H-H d.t2 
=¥ aH (for sphere) =
sirnilarly the hoop strain = 1 Jo, - v.c„ J = -^-[1 - y] = " v ] or e^-BJLri-
4t2 EL
Thus equating the two strains in order that there shall be no distortion of the 
junction
_ P ^ _ r2-w l =-^-ri-
4t,EL J 4t2E L
1 -v 
2-v
Lame's Theory
Lame's theory is based on the following assumptions
Assumptions
1. Homogeneous material.
2. Plane section of cylinder, perpendicular to longitudinal axis remains under 
plane and pressure. Hoop stress at any section
b
<7, = —+ a
Radial pressure
b
p,= 7 - a
Subjected to Internal Pressure (p) 
• At
r = rt> v * = P 
At
r ,n -r.
Subjected to External Pressure (p) 
• At
r. < ? . . =
• At
r = r o > = P
- r ,
Note: Radial and hoop stresses vary hyperbolically.