thin wall pressure vessels
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NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 1
THIN CYLINDERS & SPHERES
Introduction: -
The vessels such as boilers, compressed air receivers
etc are of cylindrical and spherical forms. The vessels are generally used
for storing fluids (liquids or gases) under pressure. The walls of such
vessels are thin as compared to their diameters. If the thickness of the
wall of the cylindrical vessel is less than
of its internal diameter,
the cylindrical vessel is known as a thin cylinder. In case of thin
cylinders, the stress distribution is assumed uniform over the thickness
of the wall.
CIRCUMFERENTIAL STRESS OR HOOP STRESS OR MAXIMUM
PERMISSIBLE STRESS OR TENSILE STRESS
Consider a thin cylindrical vessel subjected to an internal fluid pressure.
The circumferential stress will be set up in the material of the cylinder,
if the bursting of the cylinder takes place as shown.
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COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 2
The bursting will take place if the force due to fluid pressure is more
than the resisting force due to circumferential stress set up in the
material. In the limiting case, the two forces should be equal.
( )
( )
Equating the two equating
( )
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COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 3
This stress is tensile as shown in the figure.
LONGITUDINAL STRESS
Consider a thin cylindrical vessel subjected to an internal fluid pressure.
The longitudinal stress will be set up in the material of the cylinder, if
the bursting of the cylinder takes place as shown.
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COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 4
The bursting will take place if the force due to fluid pressure is more
than the resisting force due to longitudinal stress ( ) set up in the
material. In the limiting case, the two forces should be equal.
Hence in the limiting case
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COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 5
Maximum shear stress: -
EFFICIENCY OF A JOINT
(Joint efficiency means efficiency of longitudinal joint)
The cylindrical shells such as boilers are having two types of joints
namely longitudinal joint and circumferential joint. In case of a joint,
holes are made in the material of the shell for the rivets. Due to the
holes, the area offering resistance decreases. Due to decrease in area,
the stress developed in the material of the shell will be more.
Hence in case of the riveted shell the circumferential and longitudinal
stresses are greater. If the efficiency of a longitudinal joint and
circumferential joint are given then the circumferential and longitudinal
stresses are obtained as:
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 6
EFFECT OF INTERNAL PRESSURE ON THE DIMENSION OF A THIN
CYLINDRICAL SHELL
[
]
[
]
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 7
[
]
[
]
[
]
[
]
( ) ( )
[ ]
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 8
[
]
[
]
(
)
(
)
A THIN CYLINDERICAL VESSEL SUBJECTED TO INTERNAL FLUID
PRESSURE AND A TORQUE
When a thin cylindrical vessel is subjected to internal fluid pressure (p),
the stresses set up in the material of the vessel are circumferential
and longitudinal stress . These two stresses are tensile and are acting
perpendicular to each other. If the cylindrical vessel is subjected to
torque, shear stresses will also be set up in the material of the vessel.
Hence at any point in the material of the cylindrical vessel, there will be
two tensile stresses mutually perpendicular to each other accompanied
by a shear stress.
( )
( )
√(
)
√(
)
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 9
√(
)
[ ]
THIN SPHERICAL SHELL
Figure shows a thin spherical shell of internal diameter d and thickness
t and subjected to an internal fluid pressure p. The fluid inside the shell
has a tendency to split the shell into two hemispheres along x-x axis.
Circumferential Stress developed in hemispherical portion
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 10
Change in dimension of a thin spherical shell due to an internal
pressure.
We know that the stresses and at any point are equal and like.
There is no shear stress at any point in the shell. Maximum shear
stress
, the stresses and are acting at right angle to each
other.
Therefore the strain in any one direction is given by
Circumferential strain
( )
( )
( )
Volumetric strain
Volume of a sphere
Taking the differential of the above equation
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( )
CYLINDRICAL SHELL WITH HEMISPHERICAL ENDS
d= Internal diameter of the cylinder,
t1= Wall thickness of cylindrical portion, and
t2= Wall thickness of hemispherical portion.
Circumferential stress developed in cylindrical portion,
Longitudinal stress developed in cylindrical portion,
Circumferential strain in cylindrical portion
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 12
[
]
Circumferential stress developed in hemispherical portion,
Circumferential strain developed in hemispherical portion,
( )
In order that there is no distortion at the junction of cylindrical and
hemispherical portions the circumferential strains in the two have to be
equal
[
]
( )
Total change in volume of cylindrical shell with hemispherical
end=
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COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 13
(
)
( )
WIRE WINDING OF THIN CYLINDERS
A tube can be strengthened against the internal pressure by winding it
with wire under tension and putting the tube wall in compression. As
the pressure is applied, the resultant hoop stress produced is much less
as it would have been in the absence of the wire. The maximum stress
will be in the wire which is made of a high-tensile material.
The analysis of wire wounded cylinders is made on the assumption that
one layer of wire of diameter is closely wound on the tube with an
initial tension T. The procedure is as follows:
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COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 14
Initial tensile stress in wire ,
( )
Initial tensile force in wire for length L (
)
Where n= Numbers of turns in length L
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Initial compressive stress in cylinder
Initial compressive force in the cylinder for length L
We know that L=nd
So
As we know that
Initial tensile force in wire for length L (
)
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Initial tensile force in wire for length L
(
)
Initial tensile force in wire for length L (
)
And
Initial compressive force exerted by wire on cylinder for length
L
For Equilibrium
Initial tensile force in wire = Compressive force on cylinder
(
)
CASE 2nd:-
Circumferential stress developed in the cylinder due to fluid
pressure only (tensile)
Stress developed in the wire due to fluid pressure only (tensile)
The resultant stress in the cylinder ( )
The resultant stress in the wire ( )
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 17
WIRE WINDING OF THE THIN CYLINDER IS USED
To increase the pressure carrying capacity of the cylinder.
To reduce the chances of bursting of the cylinder in the
longitudinal direction.
Bursting force due to fluid along longitudinal section per cm
length
Resisting force of cylinder along longitudinal section per cm length
due to fluid pressure
Resisting force of wire per cm length due to fluid pressure
(
)
(
)
(
)
Bursting force due to fluid pressure = Resisting force of cylinder
+ Resisting force of wire
(
)
(
)
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 18
The circumferential strain in the pipe is also equal to the strain in the
steel wire. Since the wire and cylinder remain in contact, the
circumferential strain in the cylinder should be equal to the strain in the
steel wire. Due to fluid pressure, the stresses set up in the cylinder are
circumferential stress and longitudinal stress. But in the wire there is
only one stress.
Circumferential strain in cylinder = Strain in wire
(
)
NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)
COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 19
ROTATIONAL STRESSES IN THIN CYLINDER
( )
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COMPILED BY AMIT SINGH 9827740442 (amit28niist@gmail.com) Page 20
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