Stresses in Thin, Thick, Spherical PVs
Stresses in Thin, Thick, Spherical PVs
Stresses in Thin, Thick, Spherical PVs
) t (length) = ( p ) (d
i
) (length)
E F
x
= 0
o
=
p
d
i
2 t
Max. Hoop stress
o
l
= o
Stress element
Stresses in Pressurized Cylinders
Cylindrical pressure vessels, hydraulic cylinders, shafts with components
mounted on (gears, pulleys, and bearings), gun barrels, pipes carrying fluids
at high pressure,.. develop tangential, longitudinal, and radial stresses.
Wall
thickness
t
Radial stress
o
r
Longitudinal stress
o
l
(closed ends)
< 10
r
t
Thin-walled pressure vessel
Tangential
stress o
Hoop stress
Thin wall refers to a vessel having an inner-radius-thickness ratio of 10
or more (r/t 10)
Thick walled Pressure Vessels
< 10
r
t
Thick-walled pressure vessel
Stresses in a Thick-Walled Pressure Vessels
For thick-walled pressure veessels, the radial stress, o
r
, cannot be neglected.
Assumption longitudinal elongation
is constant around the plane of cross
section, there is very little warping of
the cross section,
l
= constant
dr
o
o
o
r
+ do
r
o
r
2(o
)(dr)(l) + o
r
(2rl) (o
r
+ do
r
) [2(r + dr)l] = 0
l = length of cylinder
E F = 0
(do
r
) (dr) is very small compared to
other terms 0
o
o
r
r
dr
do
r
= 0
(1)
Stresses in a Thick-Walled Pressure Vessels
l
=
o
E
o
r
E
Deformation in the longitudinal direction
o
+ o
r
= 2C
1
=
l
E
constant
(2)
Consider,
d (o
r
r
2
)
dr
=
r
2
do
r
dr
+ 2r o
r
Subtract equation (1) from (2),
o
r
+ o
r
+ r
dr
do
r
= 2C
1
o
o
r
r
dr
do
r
= 0
(1)
2ro
r
+ r
2
dr
do
r
= 2rC
1
Multiply the above equation by r
d (o
r
r
2
)
dr
=
2rC
1
o
r
r
2
= r
2
C
1
+ C
2
o
r
= C
1
C
2
r
2
+
o
= C
1
C
2
r
2
Stresses in a Thick-Walled Pressure Vessels
Boundary conditions
o
r
= - p
i
at r = r
i
o
r
= - p
o
at r = r
o
o
=
p
i
r
i
2
- p
o
r
o
2
r
i
2
r
o
2
(p
o
p
i
) / r
2
r
o
2
- r
i
2
Hoop stress
o
r
=
p
i
r
i
2
- p
o
r
o
2
+ r
i
2
r
o
2
(p
o
p
i
) / r
2
r
o
2
- r
i
2
Radial stress
p
i
r
i
2
- p
o
r
o
2
o
l
=
r
o
2
- r
i
2
Longitudinal stress
Stresses in a Thick-Walled Pressure Vessels
Special case, p
o
(external pressure) = 0
o
=
p
i
r
i
2
r
o
2
- r
i
2
(1 +
r
o
2
r
2
)
o
r
=
r
o
2
- r
i
2
(1 -
r
o
2
r
2
)
p
i
r
i
2
Hoop stress distribution, Radial stress distribution,
maximum at the inner surface
Spherical pressure vessels
If pressure vessels have walls that are thin in comparison to their
overall dimensions they are known as shell structures
In this section we consider thin walled (r/t>10) pressure vessels of
spherical shape
Spherical pressure vessel
Cross section of spherical pressure vessel
showing inner radius r, wall thickness t, and
internal pressure p
Spherical pressure vessels
FI G. 8-3 Tensile stresses in the wall of a
spherical pressure vessel
t
pr
2
= o
Formula for calculating the tensile
stresses in the wall of a spherical shell
The wall of a pressurized spherical
vessel is subjected to uniform tensile
stresses in all directions (because of
spherical symmetry)
Stresses that act tangentially to the
curved surface of a shell are known as
membrane stresses. The name arises
from the fact that these are the only
stresses that exist in true membranes,
such as thin polymer films, soap films
etc
Stresses at the outer surface
Stresses in a spherical pressure vessel at (a) the outer surface and (b) the inner surface
Element in fig (a) is in biaxial stress. No in-plane shear stresses acting on the
thin element
Every plane is a principal plane and every direction is a principal direction
0 ,
2
3 2 1
= = = o o o
t
pr
Principal stresses for the element
t
pr
4 2
max
= =
o
t
Out of plane maximum shear
stresses
Problem 1
A certain cylindrical vessel with a spherical upper head and a semi-elliptical lower
head, its ratio of major semi-axis to short semi-axis is a / b = 2. The average
diameter D=420mm, thickness of all cylindrical shell and heads are 8mm. The
working pressure P=4MPa.
Find o
m
and o
u
of the shell body. (2)Find the maximum stress on the both the
heads and their position respectively.
Solution:
(1)o
m
and o
u
:
D
S
P
) MPa ( 5 . 52
8 4
420 4
4
=
= =
S
pD
m
o
(MPa) 105 2
2
= = =
m
S
pD
o o
u
(MPa) 5 . 52
4
= = =
S
pD
m u
o o
(2)Upper head spherical
(3)Lower head- elliptical (a/b = 2: a = D/2 = 210 mm b = a/2 =
105 mm)
(MPa) 105
2 2
= = =
|
.
|
\
|
= =
S
pD
S
pa
b
a
S
pa
m u
o o
(MPa) 5 . 52
4 2
= = =
S
pD
S
pa
m
o
(MPa) 105
2
2
2
2
2
= = =
|
|
.
|
\
|
=
S
pD
S
pa
b
a
S
pa
u
o
x=a (Bottom):
17
PROBLEM 2
A cylindrical pressure vessel has an inner diameter of 1.2 m and
thickness of 12 mm. Determine the maximum internal pressure it can
sustain so that its maximum stress does not exceed 140 MPa. Under
the same conditions, what is the maximum internal pressure that a
similar-size spherical vessel can withstand?
Cylindrical pressure vessel
Maximum stress occurs in the circumferential direction. The stress in the
longitudinal direction will be o
2
= 0.5(140 MPa) = 70 MPa.
o
1
= pr/t
140 N/mm
2
= P x(600 mm) / 12 mm
Hence p = 2.8 N/mm
2
18
EXAMPLE 8.1 (SOLN)
The maximum radial stress at the inner wall will be (o
3
)
max
= p = 2.8 Mpa, 50 times smaller than the
circumferential stress (140 Mpa)
Spherical pressure vessel
Here, the maximum stress occurs in any two perpendicular directions on an element of the vessel.
From Eqn, we have
140 N/mm
2
= p(600 mm) / 2 (12mm)
So, p = 5.6 N/mm
2
Although it is more difficult to fabricate, the spherical pressure vessel will carry
twice as much internal pressure as a cylindrical vessel.
o
2
= pr/2t
Concept of Boundary Stress
Boundary The point of discontinuity and its vicinity is where two parts with
different geometry, shape, load, material and physical conditions exist. Boundary
stresses are a group of internal forces with the same value but opposite directions
occur between the two parts which are forced into harmony
Characteristics of boundary stress:
i. Non-even Distribution
ii. Local stress large at the boundary, diminishing as the point
moves away from the boundary
iii. Can be 3~5 times that of membrane stress
Handling boundary stress
Assure quality of weld at the boundary
Thro heat treatment, decrease/ eliminate the
remnant stress
Use Materials of sufficient plasticity
Improve the structure of joint boundary
Strengthen the boundary locally