Thin Pressure vessels, Application of pressure vessels, stresses and strains acting on them, volumetric strain, maximum shear stress

1. Strength of Material THICK AND THIN PRESSURE
VESSELS
CYLINDRICAL & SPHERICAL PV
SURYAKANT KUMAR
ASSISTANT PROFESSOR
GAYA COLLEGE OF ENGINEERING, GAYA

2. STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 2
Course Outline
1. Introduction
2. Thin cylindrical Pressure vessels
3. Evaluation of stresses and strains in Thin PV
4. Thick Cylindrical Pressure Vessels
5. Spherical Pressure Vessels
4/14/2020

3. Introduction
We will cover these topics:
 Defining Pressure vessels
 Application of PV
 Various stresses acting in PV
STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 34/14/2020

4. STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 4
Defining Pressure vessels
4/14/2020
 A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the
ambient pressure.
 Pressure vessels can theoretically be almost any shape, but shapes made of sections of spheres, cylinders, and
cones are usually employed.
Spherical gas
container
Cylindrical pressure
vessel.
Fire Extinguisher with
rounded rectangle
pressure vessel
Aerosol spray can

5. STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 5
Application of Pressure vessels
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 Pressure vessels are used in a variety of applications in both industry and the private sector.
 They appear in these sectors as industrial compressed air receivers and domestic hot water storage tanks.
 Other examples of pressure vessels are diving cylinders, recompression chambers, distillation towers, pressure
reactors, autoclaves, and many other vessels in mining operations, oil refineries and petrochemical plants,
nuclear reactor vessels, submarine and space ship habitats, pneumatic reservoirs, hydraulic reservoirs under
pressure, rail vehicle airbrake reservoirs, road vehicle airbrake reservoirs, and storage vessels for liquefied
gases such as ammonia, chlorine, and LPG (propane, butane).

6. STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 6
Various Stresses in Pressure Vessels
4/14/2020
 These cylinders are subjected to internal fluid pressures.
 When a cylinder is subjected to a internal pressure, at any point on the cylinder wall, three types of stresses are
induced on three mutually perpendicular planes. They are:-
I. Hoop or Circumferential Stress (σC) – This is directed along the tangent to the circumference and tensile in
nature. Thus, there will be increase in diameter.
II. Longitudinal Stress (σL ) – This stress is directed along the length of the cylinder. This is also tensile in nature
and tends to increase the length.
III.Radial Stress (σR) – It is compressive in nature. Its magnitude is equal to fluid pressure on the inside wall and
zero on the outer wall if it is open to atmosphere.

7. STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 7
Various Stresses in Pressure Vessels
4/14/2020
p p
p
p
pp
σC
σCσC
σC
σL
σL
σL
σL
σR
Hoop or Circumferential Stress Longitudinal Stress Radial Stress

8. STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 8
Element on the cylinder wall subjected to these three stresses
4/14/2020
Hoop or Circumferential Stress Longitudinal Stress
Radial Stress
(σC)
(σC) (σL)
(σL) (σR)

9. Thin Cylindrical
Pressure Vessels
We will cover these topics:
 Defining Thin PV
 Stresses acting on thin PV
STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 94/14/2020

10. Defining Thin Pressure vessels
 A cylinder or spherical shell is considered to be thin when the metal thickness is small compared to internal
diameter.
 When the wall thickness, ‘t’ is equal to or less than ‘d/20’, where ‘d’ is the internal diameter of the cylinder
or shell, we consider the cylinder or shell to be thin, otherwise thick.
 Magnitude of radial pressure is very small compared to other two stresses in case of thin cylinders and
hence neglected.
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 10

11. Stresses acting on thin Pressure Vessels
Circumferential Stress
Longitudinal Stress
Longitudinal Axis
Thickness “t”
σC
σL
 The stress acting along the circumference of the cylinder is called circumferential stresses whereas
 The stress acting along the length of the cylinder (i.e., in the longitudinal direction ) is known as longitudinal
stress
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 11

12. Stresses acting on thin Pressure Vessels
 The bursting will take place if the force due to internal (fluid) pressure (acting vertically upwards and
downwards) is more than the resisting force due to circumferential stress set up in the material.
σC σC
p
σC Circumferential stress & p is
internal pressure
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 12
p

13. Evaluation of
stresses and strains
in Thin PV
We will cover these topics:
 Evaluation of Circumferential or
Hoop Stress
 Evaluation of Circumferential or
Hoop Strain
 Evaluation of Longitudinal Stress
 Evaluation of Longitudinal Strain
and Volumetric strain
STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 134/14/2020

14. Evaluation of Circumferential or Hoop Stress
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 14
p d
t
σC σC
p
d
dL
 Consider a thin cylinder closed at both ends and subjected to internal pressure ‘p’ as shown in the figure
above.
 Let d=Internal diameter, t = Thickness of the wall L = Length of the
cylinder.

15. Evaluation of Circumferential or Hoop Stress
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 15
σC
p
σC

d
 To determine the Bursting force across the diameter:
 Consider a small length ‘dl’ of the cylinder and an
elementary area ‘dA’ as shown in the figure.
Force on the elementary area, dA
dF = p×dA = p×r×dl×dθ
= p×
𝑑
2
×dl×dθ
 Horizontal component of this force:
dFx = p×
𝑑
2
×dl×cosθ×dθ
 Vertical component of this force:
dFy = p×
𝑑
2
×dl×sin θ×dθ

16. Evaluation of Circumferential or Hoop Stress
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 16
 The horizontal components cancel out when integrated over semi-circular portion as there will be another equal and
opposite horizontal component on the other side of the vertical axis.
 Total diametrical bursting force = ‫׬‬0
𝜋
p×
𝑑
2
×dl×sin θ×dθ
= p×
𝑑
2
×dl×[−cos θ] 0
𝜋
×dθ
= p×d×dl
= p × projected area of the curved surface
∴ Resisting force (due to circumferential stress σc) =2 ×σc ×t×dl
Under equilibrium, Resisting force = Bursting force
i.e., 2×σc ×t×dl = p×d×dl
∴ Circumferential stress, σc =
𝒑𝒅
𝟐𝒕

17. Evaluation of Longitudinal Stress
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 17
A
B
The bursting of the cylinder takes
place along the section AB.
Thickness “t”
σL
The force, due to pressure of the
fluid, acting at the ends of the
thin cylinder, tends to burst the
cylinder as shown in figure.

18. Evaluation of Longitudinal Stress
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 18
Thickness “t”
σL
p
 Longitudinal bursting force (on the end of cylinder) = p×
𝜋
4
×d2
 Area of cross section resisting this force = π×d×t
Let σL = Longitudinal stress of the material of the cylinder.
∴Resisting force = σL ×π×d×t
Under equilibrium, bursting force = resisting force
i.e. p×
𝜋
4
×d2 = σL ×π×d×t
∴ Longitudinal stress, σL =
𝒑𝒅
𝟒𝒕

19. Evaluation of Strains
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 19
 The state of stress at a point on a thin cylinder can now easily be represented as follows:
σL =
𝒑𝒅
𝟒𝒕
σL =
𝒑𝒅
𝟒𝒕
σC =
𝒑𝒅
𝟐𝒕
σC =
𝒑𝒅
𝟐𝒕
 A point on the surface of thin cylinder is subjected to
biaxial stress system, (Hoop stress and Longitudinal
stress) mutually perpendicular to each other, as shown in
the figure.
 The strains due to these stresses i.e., circumferential and
longitudinal are obtained by applying Hooke’s law and
Poisson’s theory for elastic materials.

20. Circumferential, (εC) and Longitudinal Strains, (εL)
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 20
σL =
𝒑𝒅
𝟒𝒕
σL =
𝒑𝒅
𝟒𝒕
σC =
𝒑𝒅
𝟐𝒕
σC =
𝒑𝒅
𝟐𝒕
εC =
σC
𝐸
- µ×
σL
𝐸
= 2×
σL
𝐸
- µ×
σL
𝐸
=
σL
𝐸
×(2 - µ)
∴ εC =
𝜹𝒅
𝒅
=
𝒑𝒅
𝟒𝒕𝑬
×(𝟐 − µ)
εL =
σL
𝐸
- µ×
σC
𝐸
=
σL
𝐸
- µ×2×
σL
𝐸
=
σL
𝐸
×(1-2µ)
∴ εL =
𝛅𝒍
𝐥
=
𝒑𝒅
𝟒𝒕𝑬
×(𝟏 − 2µ)

21. Volumetric Strain, (εV)
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 21
Volumetric Strain, εV =
𝛅𝐯
𝐯
Where. Change in volume = δV = (final volume – original volume original)
Original volume = V = Area of cylindrical shell × length =
π ⅆ2
4
× L
Final volume = Final area of cross section ×Final length
=
π
4
× d + δd 2 × L + δL
=
π
4
× [d2 +(δd)2 +2dδd] × L + δL
=
π
4
× [d2L+(δd)2 L+2Ldδd +d2δL+(δd)2δL+2dδd δL]
Neglecting the smaller quantities such as (δd)2 L,(δd)2δL and 2dδd δL

22. Volumetric Strain, (εV)
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 22
Now, Final volume =
π
4
× [d2L+2Ldδd +d2δL]
Change in volume = δV = (final volume – original volume original)
=
π
4
× [d2L+2Ldδd +d2δL] -
π ⅆ2
4
× L
=
π
4
× [2Ldδd +d2δL]
εV =
δv
v
=
π
4
× [2Ldδd +d2δL]
π ⅆ2
4
×L
=
δL
L
+ 2×
δⅆ
ⅆ
= (εL + 2 × εC)
∴ εV =
𝐩𝐝
𝟒𝐭𝐄
×(𝟏 − 2µ) + 2×
𝐩𝐝
𝟒𝐭𝐄
×(𝟐 − µ) =
𝐩𝐝
𝟒𝐭𝐄
×(𝟓 − 4µ)

23. Maximum Shear Stress, (max)
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 23
Maximum Shear stress :
There are two principal stresses at any point, viz., Circumferential and longitudinal. Both these stresses are normal
and act perpendicular to each other.
∴ max =
σC−σC
2
=
pⅆ
2t
−
pⅆ
4t
2
i.e. max =
𝒑𝒅
𝟖𝒕

24. Thank You!
4/14/2020STRENGTH OF MATERIAL | THICK AND THIN PRESSURE VESSELS | SURYAKANT KUMAR | GCE GAYA 24