design of pressure vessel (thick and thin cylinders) may 2020

Design of Machine Elements
Pressure Vessel Design.
(Thin and Thick cylinders)
Gaurav Mistry
Assistant Professor
Diwaliba Polytechnic, UTU.

❑ Pressure Vessel Design Design of Machine Elements
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The pressure vessels are designed with
great care because rupture of a pressure
vessel means an explosion which may cause
loss of life and property. The material of
pressure vessels may be brittle such as cast
iron, or ductile such as mild steel.
It caused 58 deaths and 150 reported injuries.

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Based on different considerations, the pressure vessels are classified as follows:
1. According to ratio of Internal Diameter (d) to the Thickness of the shell (t) i.e.
𝒅
𝒕
ratio.
(i) When
𝑑
𝑡
ratio is equal to or more than 15.
𝑑
𝑡
≥ 15, then such vessel is known as thin cylinders.
(ii) When
𝑑
𝑡
ratio is less than 15.
𝑑
𝑡
< 15, then such vessel is known as thick cylinders
(It is also known as Based on Wall thickness)
Note: 1. In Some reference books the ratio may be considered between 10 to 20.
2. Both Thin and Thick cylinders are designed with complete separate procedure.
2. According to geometric shape: (i) Cylindrical, (ii) Spherical, (iii) Conical or (iv) Elliptical.
❑ Pressure Vessel Design

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3. According to end conditions of pressure vessels.
(i) Open ended pressure vessel. e.g. cylinder with piston (usually engine cylinders, hydraulic
cylinders used in automobiles and machineries, etc.)
(ii) Closed ended pressure vessels. E.g. receiver tank of an air compressor, boiler, containers, etc.
4. According to Shape of the end heads/ end covers:
(i) Flat headed, (ii) Convex headed or (iii) Dished headed.
5. According to direction of fluid pressure acting on the wall of vessel:
(i) Internal Pressure
(ii) External Pressure (Internal Vacuum).
6.

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7.
8.
9.

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10.
11. According to type of joint used for the fabrication of vessel:
(i) Riveted
(ii) Welded
(iii) Forged

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Main Consideration for the design of Pressure Vessel
Following considerations are made for the design of pressure vessel:
1. Material of the shell must be capable to resist the fluid pressure, thermal load and
corrosion.
2. Different types of stresses induced in the pressure vessel are determined and
suitable safety factor is to be adopted.
3. Based on the geometry of the vessels and various connections, stress concentration
take place which is required to be properly taken care.
4. To prevent leakage of the toxic acids, poisonous fluids, etc., efficient leak proof
joints are to be used.
5. Where ever necessary, to prevent the corrosion, the contact surface of the shell is
lined with rubber, glass fibre or appropriate anticorrosive materials.
6. To prevent the possible failure of the vessel, suitable safety devices like valves,
leakage indicating devices, etc., should be included in the design.

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Design of Thin Cylinders
Stresses in a Thin Cylindrical Shell due to an Internal Pressure
The analysis of stresses induced in a thin cylindrical shell are made on the following
assumptions:
1. The effect of curvature of the cylinder wall is neglected.
2. The tensile stresses are uniformly distributed over the section of the walls.
3. The effect of the restraining action of the heads at the end of the pressure vessel is neglected.
When a thin cylindrical shell is subjected to an
internal pressure, it is likely to fail in the
following two ways:
1. It may fail along the longitudinal section
(i.e. circumferentially) splitting the cylinder
into two troughs, as shown in Fig. (a).
2. It may fail across the transverse section (i.e.
longitudinally) splitting the cylinder into two
cylindrical shells, as shown in Fig. (b).

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Thus the wall of a cylindrical shell subjected to an internal pressure has to withstand tensile
stresses of the following two types:
(a) Circumferential or hoop stress (𝜎𝑡), and
(b) Longitudinal stress. (𝜎𝑙)
Circumferential or Hoop Stress (𝜎𝑡):
Consider a thin cylindrical shell subjected to an internal pressure as shown in Fig. (a), (b) and
(c)
(c)
hoop stress Longitudinal stress

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A tensile stress acting in a direction tangential to the circumference is called circumferential or
hoop stress. In other words, it is a tensile stress on longitudinal section (or on the cylindrical
walls).
Let
p = Intensity of internal pressure,
d = Internal diameter of the cylindrical
shell,
l = Length of the cylindrical shell,
t = Thickness of the cylindrical shell, and
𝜎𝑡 = Circumferential or hoop stress for the
material of the cylindrical shell.
We know that the total force acting on a longitudinal section (i.e. along the diameter) of the
Shell as shown in Fig (a)
= Intensity of pressure × Projected rectangular area = p × d × l ...(i)
and the total resisting force acting on the cylinder walls as shown in Fig (b)
= 𝜎𝑡 × 2t × l ...(∵ of two sections) ...(ii)
(b)(a)

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From equations (i) and (ii), we have
𝜎𝑡 × 2t × l = p × d × l or
The following points may be noted:
1. In the design of engine cylinders, a value of 6 mm to 12 mm is added in equation
To permit reboring after wear has taken place.
2. In constructing large pressure vessels like steam boilers, riveted joints or welded joints are
used in joining together the ends of steel plates. In case of riveted joints, the wall thickness
of the cylinder,
t =
p × d
2 𝝈 𝒕
t =
p × d
2 𝝈 𝒕
t =
p × d
2 𝝈 𝒕
+ (6 to 12) mm
t =
p × d
2 𝝈 𝒕 𝐱 𝜼 𝒍
Where, 𝜼𝒍 = 𝑬𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒄𝒚 𝒐𝒇 𝑳𝒐𝒏𝒈𝒊𝒕𝒖𝒅𝒊𝒏𝒂𝒍 𝒋𝒐𝒊𝒏𝒕
𝝈 𝒕 =
p × d
2 t
Here using hoop stress, the thickness (t) of
cylinder wall will be calculated

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Longitudinal Stress (𝜎𝑙):
Consider a closed thin cylindrical shell subjected to an internal pressure as shown in Fig (a) and
(b). A tensile stress acting in the direction of the axis is called longitudinal stress. In other
words, it is a tensile stress acting on the transverse or circumferential section (or on the ends of
the vessel).
We know that the total force acting on the
Transverse direction (i.e. at the end of the
vessel) as shown in Fig (a).
(b)
(a)= Intensity of pressure × Cross-sectional area
...(i)
and total resisting force acting on cylinder wall as shown in Fig (b).
= 𝜎𝑙 × 𝜋 d× t ...(ii)

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Longitudinal Stress (𝜎𝑙):
𝜎𝑙 × 𝜋 d× t 𝝈𝒍 =
p × d
4 t t =
p × d
4 𝝈 𝒍
t =
p × d
4 𝝈𝒍 𝐱 𝜼 𝒍
If 𝜼𝒍 𝒊𝒔 𝒕𝒉𝒆 𝒆𝒇𝒇𝒊cie𝒏𝒄𝒚 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒍 𝒋𝒐𝒊𝒏𝒕,then
From above we see that the longitudinal stress is half of the circumferential or hoop stress.
Therefore, the design of a pressure vessel must be based on the maximum stress i.e. hoop stress.
Further, the diameter d and length l of the cylindrical shell are determined based on the required
volume of the fluid to be stored in the vessel (maximum storage capacity V in litres)
Maximum shear stress
We know that according to maximum shear stress theory, the maximum shear stress is one-half
the algebraic difference of the maximum and minimum principal stress. Since the maximum
principal stress is the hoop stress (𝝈 𝒕) and minimum principal stress is the longitudinal stress (𝝈𝒍
), therefore maximum shear stress,
or
𝝈𝒍 =
p × d
4 t
=
p × d
2 . 2. t
=
𝝈 𝒕
2
𝝉 𝒎𝒂𝒙 =
𝝈 𝒕 −𝝈 𝒍
2
Here using longitudinal stress, the thickness (t) of
Cylinder cover plate will be calculated

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Design of Thin Spherical Shell:
Consider a thin spherical shell subjected to an internal pressure as
shown in Fig. 7.5.
Let V = Storage capacity of the shell,
p = Intensity of internal pressure,
d = Diameter of the shell,
t = Thickness of the shell,
𝝈 𝒕 = Permissible tensile (hoop) stress for the
shell material.
In designing thin spherical shells, we have to determine
1. Diameter of the shell, and 2. Thickness of the shell.

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1. Diameter of the shell (d)
Internal diameter of the shell is determined based on its maximum storage capacity.
We know that the storage capacity of the shell,
2. Thickness of the shell wall (t)
As a result of the internal pressure, the shell is likely to rupture along the centre of the sphere.
Therefore force tending to rupture the shell along the centre of the sphere or bursting force, Fig(a)
,
...(i)
...(ii)
and resisting force of the shell wall, Fig (b),

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2. Thickness of the shell wall (t)

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1. A thin cylinder with 150 mm inside diameter and 15 mm plate thickness is subjected to
internal pressure of 5 N/ 𝒎𝒎 𝟐
. Determine the hoop stress and longitudinal stress and
maximum shear stress in the cylinder.
Solution: Given data, d = 150 mm, t = 15 mm and p = 5 N/ 𝑚𝑚2,
Now, we know Hoop stress = =
5 × 150
2 x 15
= 25 N/𝑚𝑚2.
Longitudinal stress = =
5 × 150
4 x 15
= 12.5 N/ 𝑚𝑚2.
Maximum shear stress ==
𝟐𝟓 − 𝟏𝟐.𝟓
2
= 6.25 N/ 𝑚𝑚2.
Numerical: Design of Thin cylinder
𝝈 𝒕 =
p × d
2 t
𝝈𝒍 =
p × d
4 t
𝝉 𝒎𝒂𝒙 =
𝝈 𝒕 −𝝈 𝒍
2

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2. A spherical shell having capacity of 4000 litres is subjected to internal pressure of 1.2
N/mm2. Determine the thickness of shell wall. Permissible stress for shell material is 60 N/
𝒎𝒎 𝟐
and joint efficiency is 75%.
Solution: Given data, p = 1.2 N/ 𝑚𝑚2, capacity V = 4000 litres = 4000 x 106 𝑚𝑚3, 𝜂 = 0.75
Now (i) Diameter of shell can be calculated from
d = (
6 x 4000 x 106
𝜋
)
1
3 = 1969.49 mm = 1970 mm = 2000 mm
Now,
= = 13.33 mm = 16 mm.
Therefore, the diameter of shell is 2000 mm and the thickness is t = 16 mm
Numerical: Design of Thin Spherical Shell.
1.2 × 2000
4 𝐱 𝟔𝟎 𝐱 𝟎.𝟕𝟓

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Design of Thick Cylinders
When a cylindrical shell of a pressure vessel, hydraulic cylinder, Gunbarrel and a pipe is
subjected to a very high internal fluid pressure, then the walls of the cylinder must be made
extremely heavy or thick.
In thin cylindrical shells, we have assumed that the tensile stresses are uniformly distributed over
the section of the walls. But in the case of thick wall cylinders as shown in Fig. (a), the stress
over the section of the walls cannot be assumed to be uniformly distributed.
They develop both tangential and radial stresses with values which are dependent upon the radius
of the element under consideration.
The distribution of stress in a thick cylindrical shell is shown in Fig. (b) and (c)

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In the design of thick cylindrical shells, the following equations are mostly used:
1. Lame’s equation; 2. Birnie’s equation; 3. Clavarino’s equation; and 4. Barlow’s equation.
We will particularly study about Lame’s theory and equation only.
LAME’s THEORY:
Lame’s theory is commonly used for the stress analysis across the walls of the thick cylinders
which is based on MAXIMUM NORMAL STRESSES THEORY OR MAXIMUM PRINCIPLE
STRESS THEORY.
According to Lame’s theory,
1. Hoop stress or Tangential stresses are tensile in
nature and it is maximum at the inner surface of
the cylinder wall and minimum at the outer
surface of the cylinder wall.
(The equation for distribution of Maximum tangential
stress and Minimum tangential stress across the wall
of thick cylinder will be obtained later.) Tangential stress distribution

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2. Radial stress is compressive in nature and it is
maximum at inner surface and zero at the outer
surface.
(The equation for distribution of maximum radial
stress and minimum radial stress across the wall of
thick cylinder will be obtained later.)
______________________________________
Assumptions made in design of thick cylinders:
1. The material of the cylinder is homogeneous and
isotropic and it follows the Hook’s Law.
2. The cross section of the cylinder is symmetrical to
its longitudinal axis and load acting in the
cylinder is also symmetrical to its longitudinal
axis.
3. Cylinder is open at both the ends and the
longitudinal stresses induced in the cylinder is
negligible.
Radial stress distribution

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LAME’s Expressions for Hoop Stress and Radial Stress on Thick Cylinder:
As shown in Figure, for thick cylinder subjected to internal and external pressures the basic
equations for tangential (hoop) and radial stresses at any radius R given by Lame are a shown
below:
Tangential or hoop stress 𝝈 𝒕:
𝝈 𝒕 = 𝐴 +
𝐵
𝑅2 …(1)
Radial stress 𝝈 𝒓:
𝝈 𝒓 = 𝐴 −
𝐵
𝑅2 …(2)
Where,
A and B are Lame’s constants and R = Radius of cylinder in consideration (shown by Orange
color at radius R from centre of Cylinder.
R1
R2R2
R1
R2
R1
R2

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Both equations (1) and (2) shown above are known as Lame’s General Equations. With the
appropriate boundary conditions, constants A and B for both the equations can be determined to
get the values of the hoop and radial stresses.
Let,
R1 = Inner radius of the cylinder
R2 = Outer radius of the cylinder
p1 = Internal pressure. (As it produces a compression on inner cylinder walls, it is considered as
negative, maximum radial stress “- p1” .)
p2 = External pressure. (Neglecting the external pressure, it will be considered “0”)
t = Thickness of cylinder wall = R2 – R1.
Now, applying the boundary conditions of radial stress to Lame’s general equation (2) of radial
stress, we can determine the constant A and B. And further substituting back these constants in
equation (1) and (2)

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Design of Thick CylindersDesign of Thick Cylinders
Therefore substituting the condition
At R = R1, 𝝈 𝒓 = - p1 and
At R = R2, 𝝈 𝒓 = 0. in equation (2) i.e. 𝝈 𝒓 = 𝐴 −
𝐵
𝑅2 , we get the value of A and B as
𝐴 = p1 x
R1
2
(R2
2 − R1
2)
…(3) and B= p1 x
R1
2. R2
2
(R2
2 − R1
2)
…(4)
Now again substituting the value of A and B from equation (3) and (4) in equation (1) and
equation (2) we get
𝑻𝒂𝒏𝒈𝒆𝒏𝒕𝒊𝒂𝒍 𝒉𝒐𝒐𝒑 𝒔𝒕𝒓𝒆𝒔𝒔 = 𝝈 𝒕 = p1 x
R1
2
(R2
2 − R1
2)
[ 1 +
R2
2
R2 ] …(5)
𝑹𝒂𝒅𝒊𝒂𝒍 𝒔𝒕𝒓𝒆𝒔𝒔 = 𝝈 𝒓 = p1 x
R1
2
(R2
2 − R1
2)
[ 1 −
R2
2
R2 ] …(6)
Here the equation of tangential and radial stress at any radius R, for thick cylinder
subjected to internal pressure only. External pressure is neglected. Here we can see
that, 𝝈 𝒕 > 𝝈 𝒓

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Tangential stress distribution
The equation for distribution of Maximum tangential
stress and Minimum tangential stress across the wall of
thick cylinder subjected to internal pressure only:
(a) Hoop stress will be maximum when, R = R1.
Therefore, Substituting R with R1 in equation (5)
𝝈 𝒕 𝒎𝒂𝒙 = p1 x
R1
2
(R2
2 − R1
2)
[ 1 +
R2
2
R1
2 ]
𝝈 𝒕 𝒎𝒂𝒙 = p1 x
R2
2 + R1
2
(R2
2 − R1
2)
…(7) (at internal surface)
(b) Hoop stress will be minimum when, R = R2.
𝝈 𝒕 𝒎𝒊𝒏 = p1 x
R1
2
(R2
2 − R1
2)
[ 1 +
R2
2
R2
2 ]
𝝈 𝒕 𝒎𝒊𝒏 = p1 x
2 R1
2
(R2
2 − R1
2)
…(8) (at outer surface)

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Radial stress distribution
The equation for distribution of maximum radial
stress and minimum radial stress across the wall of
thick cylinder subjected to internal pressure only:
(a) Radial stress will be maximum when, R = R1.
𝝈 𝒓 𝒎𝒂𝒙 = p1 x
R1
2
(R2
2 − R1
2)
[ 1 −
R2
2
R1
2
]
𝝈 𝒓 𝒎𝒂𝒙 = p1 x
(R1
2−R2
2)
(R2
2 − R1
2)
𝝈 𝒓 𝒎𝒂𝒙 = − p1 …(9) (at internal surface)
(b) Radial stress will be minimum when, R = R2.
𝝈 𝒓 𝒎𝒊𝒏 = p1 x
R1
2
(R2
2 − R1
2)
[ 1 −
R2
2
R2
2 ]
𝝈 𝒓 𝒎𝒊𝒏 = p1 x
R1
2
(R2
2 − R1
2)
[0]
𝝈 𝒓 𝒎𝒊𝒏 = 0 …(10) (at outer surface)

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For the design of thick cylinder subjected to internal pressure only, the maximum shear
condition is considered.
Taking t = R2 – 𝑅1, from equation (7) , for 𝝈 𝒕 𝒎𝒂𝒙 > p1 , we get
𝑡 = 𝑅1 [
𝝈 𝒕 𝒎𝒂𝒙 + p1
𝝈 𝒕 𝒎𝒂𝒙 − p1
- 1] or …(11)
Considering permissible value of 𝝈 𝒕 𝒎𝒂𝒙 for the cylinder material, the thickness of the cylinder
can be obtained from above equation.

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Numerical: Design of Thick Cylinders
A cast steel hydraulic cylinder having internal diameter of 200 mm is subjected to oil
pressure of 31.5 N/ 𝒎𝒎 𝟐. Determine the thickness of the shell for the maximum permissible
hoop stress of 70 N/ 𝒎𝒎 𝟐
.
Solution:
Given data, D = 200 mm therefore R = 100 mm , p1 = 31.5 N/ 𝑚𝑚2
and 𝝈 𝒕 𝒎𝒂𝒙 = 70 N/ 𝑚𝑚2
Based on Lame’s Equation:
𝑡 = 𝑅1 [
𝝈 𝒕 𝒎𝒂𝒙 + p1
𝝈 𝒕 𝒎𝒂𝒙 − p1
- 1]
𝑡 = 100 [
70 + 31.5
70 − 31.5
- 1]
t = 100 [ 2.637 - 1] = 100 [1.6237 – 1] = 100 [0.6237] = 62.37 mm = 62.5 mm
Therefore, the thickness of cylinder is t = 62.5 mm.

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Main Applications of Pressure Vessel is as listed below:
1. To store and process water, gas, acid and various chemicals in the liquid or gaseous
form.
2. In boilers, heat exchanger and condenser.
3. As a receiver tank for compressed air in air compressor.
4. As a cylinder in case of steam engine, automobile engine, air compressor, etc.
5. As a cylinder and accumulator in case of hydraulic and pneumatic systems.
6. As a pipe to carry steam, fluid, gas or air in the piping system.
7. As a cylinder to carry pressurized, oxygen, acetylene, hydrogen, argon, LPG, CNG,
etc.
automobile engine
air compressor
heat exchanger
condenser

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(Here, S stands for allowable stress and E stands for efficiency)
Just for Information Slide 1

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Just for Information Slide 2

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REFERENCES:
1. A textbook of Machine design, R. S. Khurmi, S. Chand.
2. Design of Machine Elements, S. B. Soni, Atul prakashan.
3. www.google.com
4. Fundamental concepts of PVD, by Sandip G. Patel, former
Assistant Professor of Mech/Auto Dept, CGPIT, UTU.

design of pressure vessel (thick and thin cylinders) may 2020

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