thin wall pressure vessels

NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL (M.P)

COMPILED BY AMIT SINGH 9827740442 ([email protected])

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.



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

( )



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.



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



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:



EFFECT OF INTERNAL PRESSURE ON THE DIMENSION OF A THIN

CYLINDRICAL SHELL

[

]

[

]



[

]

[

]

[

]

[

]

( ) ( )

[ ]



[

]

[

]

(

)

(

)

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.

( )

( )

√(

)

√(

)



√(

)

[ ]

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



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



( )

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



[

]

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=



(

)

( )

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:



Initial tensile stress in wire ,

( )

Initial tensile force in wire for length L (

)

Where n= Numbers of turns in length L



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

(

)


)

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 ( )



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

(

)

(

)



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

(

)



ROTATIONAL STRESSES IN THIN CYLINDER

( )

thin wall pressure vessels

Documents