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SWINBURNE UNIVERSITY OF TECHNOLOGY (SARAWAK CAMPUS)
FACULTY OF ENGINEERING AND INDUSTRIAL SCIENCE
HES5320 Solid Mechanics Semester 2, 2011
Lecturer: Dr. Saad A. Mutasher
Lab Supervisor: Tay Chen Chiang
Laboratory Report: Thick-walled Pressure Vessels
By
Stephen Bong (4209168)
Ngui Yong Zit (4201205)
Ling Wang Soon (4203364)
Date Performed Experiment: 28th
October 2011
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 2 of 17
1. INTRODUCTION
Thick-walled cylinders are now extensively applicable in wide range of industries such as the pressure
vessels utilized in nuclear and steam power plants for fluids transmission. Such applications may touch
upon the existence of high pressures and temperatures exerted by the working fluids which might leads to
stress corrosion cracking (A. B. Ayob, et al., 2009; J. M Kihiu, et al., n. d. pp. 370). Hence, in order to
diminish the probability of disruptive failures, the development of rightful comprehending on the analysis
of stresses and strains distributed in the thick-walled cylinders from solid mechanics is significant.
The conversion of strains in the strain gauges obtained from this experiment to experimental stresses by
utilizing the theoretical equations and compare with theoretical stresses are the primary objectives of this
experiment. The equipment used in this experiment is the LS-22014 THICK CYLINDER APPARATUS
which comprises of a thick wall cylinder, pressure gauge, relief valve, ON/OFF switch, digital strain
meter, strain reading selector switch, hydraulic pump, and an oil refill port as shown in Fig. 1 and Fig. 2
below:
Fig. 1: LS-22014 THICK CYLINDER APPARATUS
Parts Name
A Thick wall cylinder
B Pressure gauge
C Relief valve
D ON/OFF switch
E Digital strain meter
F Strain reading selector switch
G Oil refill port (smaller screw)
Table 1: Parts labeled in Fig. 1 with their names
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 3 of 17
Fig. 2: Strain gauges in LS-22014 THICK CYLINDER APPARATUS
Gauge No. Radius (mm) Strain
1 29.50 Hoop
2 29.50 Radial
3 38.00 Hoop
4 38.00 Radial
5 47.50 Hoop
6 47.50 Radial
7 62.50 Hoop
8 62.50 Radial
9 19.00 Longitudinal
10 19.00 Circumferential
11 74.50 Circumferential
12 74.50 Longitudinal
Table 2: Description of strains contained in the strain gauges in Fig. 2
2. THEORY AND ANALYSIS
In the case of thick cylinder, the stresses distributed along the longitudinal direction, σz can be neglected
and only a biaxial stresses need to be taken into account. Fig. 3 below shows an element of a material at
some radius, r and contained within the elemental cylinder. The cylinder is subjected to an internal
pressure of pi. Due to the biaxial stress distribution based on the assumption made above, the principal
stresses σθ and σr are acting on this element where the principal strains (εr, εθ, and εz) set up by these
stresses can be determined by using the following consecutive equations:
( )
( )
( )zr
r
zr
r
zr
r
EE
EE
EE
σσυσ
ε
σσυσ
ε
σσυσ
ε
θ
θ
θ
+−=
+−=
+−=
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 4 of 17
Fig. 3: Distribution of biaxial stresses in the cylinder (P. P. Benham, et al., 1996, pp. 383)
The equations of equilibrium for the element is given by
0=−
+rdr
d rr θσσσ
which leads to the general solutions:
2r
BAr −=σ and
2r
BA +=θσ Eq. [1]
where A and B are constants which can be determined by utilizing the boundary conditions.
Consider the cylinder with piston as shown in Fig. 4 below:
Fig. 4: Cylinder with piston (P. P. Benham, et al., 1996, pp. 383)
The boundary conditions for the cylinder with piston as shown in Fig. 4 above are:
At r = ri, σr(r = ri) = – pi and At r = ro, σr(r = ro) = – po
Thus, 2
i
ir
BAp −=− and
2
o
or
BAp −=− .
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 5 of 17
By solving these two equations simultaneously gives:
22
22
io
ooii
rr
rprpA
−
−= and
22
22)(
io
oioi
rr
rrppB
−
−=
Substitute A and B into Eq. [1] gives the following Lamé’s formulations:
+−
+
−=
−
−+
−
−=
−−
−
−=
−
−−
−
−=
2
2
2
222
22
22
22
2
2
2
222
22
22
22
111
1)(
111
1)(
r
rkp
r
rp
krr
rrpp
rr
rprp
r
rkp
r
rp
krr
rrpp
rr
rprp
i
o
o
i
io
oioi
io
ooii
i
o
o
i
io
oioi
io
ooii
r
θσ
σ
Eq. [2]
where σr = Radial stress (MPa)
σθ = Circumferential stress (Hoop stress) (MPa)
ri = Inner radius (m)
ro = Outer radius (m)
k = ro/ri (Radius ratio)
For the case which the cylinder only exposed to internal pressure (po = 0), the radial and hoop stresses are
given by:
+
−=
+
−=
−
−=
−
−=
2
2
2
22
2
2
2
2
22
2
11
1
11
1
r
r
k
p
r
r
rr
rp
r
r
k
p
r
r
rr
rp
oio
io
ii
oio
io
ii
r
θσ
σ
Eq. [3]
At the inner surface, r = ri,
Radial stress, ir p−=σ (radial compressive stress)
Circumferential or hoop stress, ip
k
k
1
12
2
−
+=θσ
At the outer surface, r = ro,
Radial stress, 0=rσ
Circumferential or hoop stress, 1
22
−=
k
pi
θσ
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 6 of 17
Fig. 5: Stress distribution for σr and σθ for the thick cylinder which subjected to internal pressure only
with a k ratio of 3 (P. P. Benham, et al., 1996, pp. 387)
Since the thick cylinder is only subjected to internal pressure, pi, and due to the biaxial stress distribution,
by the consecutive equations, the principal strains set up the principal stress σr and σθ are:
( )
( )
( )
+=
−=
−=
θ
θθ
θ
σσυ
ε
υσσε
υσσε
rz
r
rr
E
E
E
1
1
Eq. [4]
Substitute Eq. [3] into Eq. [4] yields
−−
+
−=
+−
−
−=
22
2
22
2
11)1(
11)1(
r
r
r
r
kE
p
r
r
r
r
kE
p
ooi
ooi
r
υε
υε
θ
Eq. [5]
where E = 6.895 × 1010
Pa (Young modulus for Aluminium)
33.0=υ (Poisson ratio)
Cylinder diameter: 150 mm (ro = 75 mm) for this experiment take (ro = 74.5 mm)
Cylinder length: 320 mm
Wall thickness: 55 mm (ri = 20 mm) for this experiment take (ri = 18.625 mm)
Radius ratio = ro/ri = 74.5/18.625 = 4
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 7 of 17
Substitute these values into Eq. [5] gives:
−−
+×=
+−
−×=
−
−
22
13
22
13
5500133.0
5500110524.9
5500133.0
5500110524.9
rrp
rrp
i
ir
θε
ε
Eq. [6]
The relationship which used for the computation of stresses based on the value of strains obtained can be
derived from Eq. [4].
From Eq. [4], the radial and hoop stresses can be expressed as:
[B] Eq. ----------
[A] Eq. ----------
r
rr
E
E
υσεσ
υσεσ
θθ
θ
+=
+=
By substituting Eq. [B] into Eq. [A] gives:
−
+=
−
+=
2
2
1
)(
1
)(
υ
υεεσ
υ
υεεσ
θ
θ
θ
r
r
r
E
E
Eq. [7]
3. PROCEDURE
(1) The main power supply (D) was switched ON and current was allowed to pass through the
gauges about five to eight minutes in order to ensure all the strain gauges at steady state
temperature condition.
(2) The relief valve (C) was relieved to ensure the pressure reading is zero.
(3) All the initial readings for 12 strain gauges with zero pressure in the system were recorded from
the strain meter (E).
(4) The relief valve was tightened.
(5) Some pressure was applied slowly into the cylinder by means of the hand pump. The pressure
was pumped up to approximately 10 bar.
(6) All the readings for 12 strain gauges were recorded from the strain meter.
(7) The experiment was repeated until 50 bar with an increment of 10 bar.
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 8 of 17
4. RESULTS
The experimental strains obtained from the strain meter are recorded and tabulated in Table 3 & 4 below:
Radius (mm) Hoop Strain, εθ (µm) Experimental
0 10 20 30 40 50
29.5 0 5E-06 3E-06 6E-06 5E-06 6E-06
38.0 0 3E-06 2E-06 3E-06 2E-06 4E-06
47.5 0 2E-06 2E-06 1E-06 3E-06 1E-06
62.5 0 1E-06 2E-06 1E-06 1E-06 1E-06
Radius (mm) Radial Strain, εr (µm) Experimental
0 10 20 30 40 50
29.5 0 -5E-06 -6E-06 -6E-06 -6E-06 -4E-06
38.0 0 -2E-06 -4E-06 -3E-06 -2E-06 -3E-06
47.5 0 -1E-06 -2E-06 -3E-06 -1E-06 -2E-06
62.5 0 1E-06 -0.000002 -1E-06 -0.000001 -0.000002
Table 3 & 4: Experimental strains obtained from the strain meter
The theoretical strains are calculated from Eq. [5] by using Microsoft Excel Spreadsheet and the results
are tabulated in Table 5 & 6 below:
Radius (mm) Hoop Strain, εθ (µm) Theoretical
0 10 20 30 40 50
29.5 0 8.64363E-06 1.73E-05 2.59E-05 3.46E-05 4.32E-05
38.0 0 5.46277E-06 1.09E-05 1.64E-05 2.19E-05 2.73E-05
47.5 0 3.72589E-06 7.45E-06 1.12E-05 1.49E-05 1.86E-05
62.5 0 2.42161E-06 4.84E-06 7.26E-06 9.69E-06 1.21E-05
Radius (mm) Radial Strain, εr (µm) Theoretical
0 10 20 30 40 50
29.5 0 -0.000007367 -1.47E-05 -2.21E-05 -2.95E-05 -3.68E-05
38.0 0 -4.18655E-06 -8.37E-06 -1.26E-05 -1.67E-05 -2.09E-05
47.5 0 -2.44967E-06 -4.9E-06 -7.35E-06 -9.8E-06 -1.22E-05
62.5 0 -1.14539E-06 -2.29E-06 -3.44E-06 -4.58E-06 -5.73E-06
Table 5 & 6: Theoretical strains calculated by using Eq. [5]
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 9 of 17
The graphs of hoop strain, εθ (µm) vs. radius, r (mm) and radial strain, εr (µm) vs. radius, r (mm) for
pressures of 10 bar and 30 bar are plotted by using Microsoft Excel as shown in Fig. 6, 7, 8 & 9 below
respectively.
Fig. 6: Plot of hoop strain, εθ (µm) vs. radius, r (mm) for pi = 10 bar
Fig. 7: Plot of hoop strain, εθ (µm) vs. radius, r (mm) for pi = 30 bar
1.00E-06
2.00E-06
3.00E-06
4.00E-06
5.00E-06
6.00E-06
7.00E-06
8.00E-06
9.00E-06
25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
Ho
op
Str
ain
, εθ (µ
m)
Radius, r (mm)
Hoop Strain, εθ (µm) vs. Radius, r (mm) (pi = 10 bar)
Experimental
Theoretical
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
Ho
op
Str
ain
, εθ (µ
m)
Radius, r (mm)
Hoop Strain, εθ (µm) vs. Radius, r (mm) (pi = 30 bar)
Experimental
Theoretical
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 10 of 17
Fig. 8: Plot of radial strain, εr (µm) vs. radius, r (mm) for pi = 10 bar
Fig. 9: Plot of radial strain, εr (µm) vs. radius, r (mm) for pi = 30 bar
-8.00E-06
-7.00E-06
-6.00E-06
-5.00E-06
-4.00E-06
-3.00E-06
-2.00E-06
-1.00E-06
0.00E+00
1.00E-06
25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
Rad
ial
Str
ain
, ε
r (µ
m)
Radius, r (mm)
Radial Strain, εr (µm) vs. Radius, r (mm) (pi = 10 bar)
Experimental
Theoretical
-2.50E-05
-2.00E-05
-1.50E-05
-1.00E-05
-5.00E-06
0.00E+00
5.00E-06
25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
Rad
ial
Str
ain
, ε
r (µ
m)
Radius, r (mm)
Radial Strain, εr (µm) vs. Radius, r (mm) (pi = 30 bar)
Experimental
Theoretical
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 11 of 17
The experimental and theoretical stresses are computed by using Eq. [7] and the results are tabulated in
Table 7, 8, 9 & 10 below:
Radius (mm)
Hoop Stress, σθ (MPa) Experimental
0 10 20 30 40 50
29.5 0 0.259211 0.078924 0.311053 0.233676 0.362121
38.0 0 0.18106 0.052616 0.155526 0.103684 0.232903
47.5 0 0.129218 0.103684 0.000774 0.206595 0.026308
62.5 0 0.10291 0.103684 0.051842 0.051842 0.026308
Radius (mm)
Hoop Stress, σθ (MPa) Theoretical
0 10 20 30 40 50
29.5 0 0.480691123 0.961382 1.442073 1.922764 2.403456
38.0 0 0.315788402 0.631577 0.947365 1.263154 1.578942
47.5 0 0.22574505 0.45149 0.677235 0.90298 1.128725
62.5 0 0.158128496 0.316257 0.474385 0.632514 0.790642
Radius (mm) Radial Stress, σr (MPa) Experimental
0 10 20 30 40 50
29.5 0 -0.259211 -0.387655 -0.311053 -0.336587 -0.1563
38.0 0 -0.07815 -0.258437 -0.155526 -0.103684 -0.129992
47.5 0 -0.026308 -0.103684 -0.206595 -0.000774 -0.129218
62.5 0 0.10291 -0.103684 -0.051842 -0.051842 -0.129218
Radius (mm) Radial Stress, σr (MPa) Theoretical
0 10 20 30 40 50
29.5 0 -0.349355163 -0.69871 -1.048065 -1.397421 -1.746776
38.0 0 -0.184452442 -0.368905 -0.553357 -0.73781 -0.922262
47.5 0 -0.09440909 -0.188818 -0.283227 -0.377636 -0.472045
62.5 0 -0.026792536 -0.053585 -0.080378 -0.10717 -0.133963
Table 7, 8, 9 & 10: Theoretical and experimental stresses computed by using Eq, [7]
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 12 of 17
The plots of hoop and radial stresses vs. radius for pressure of 10 bar and 30 bar are shown in Fig. 10, 11,
12 & 13 below:
Fig. 9: Plot of hoop stress, σθ (µm) vs. radius, r (mm) for pi = 10 bar
Fig. 10: Plot of hoop stress, σθ (µm) vs. radius, r (mm) for pi = 30 bar
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
Ho
op
Str
ess
, σθ (
MP
a)
Radius, r (mm)
Hoop Stress, σθ (MPa) vs. Radius, r (mm) (pi = 10 bar)
Experimental
Theoretical
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
Ho
op
Str
ess
, σθ (
MP
a)
Radius, r (mm)
Hoop Stress, σθ (MPa) vs. Radius, r (mm) (pi = 30 bar)
Experimental
Theoretical
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 13 of 17
Fig. 11: Plot of hoop stress, σr (µm) vs. radius, r (mm) for pi = 10 bar
Fig. 12: Plot of hoop stress, σr (µm) vs. radius, r (mm) for pi = 30 bar
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0R
ad
ial
Str
ess
, σr (
MP
a)
Radius, r (mm)
Radial Stress, σr (MPa) vs. Radius, r (mm) (pi = 10 bar)
Experimental
Theoretical
-1.05
-0.85
-0.65
-0.45
-0.25
-0.0525.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
Ra
dia
l S
tre
ss, σr (
MP
a)
Radius, r (mm)
Radial Stress, σr (MPa) vs. Radius, r (mm) (pi = 30 bar)
Experimental
Theoretical
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 14 of 17
5. DISCUSSION AND CONCLUSION
The hoop strain and radial strain at various locations (strain gauges) with the increment of pressure are
measured in this experiment. The principal stresses distributed along the LS-22014 THICK CYLINDER
APPARATUS are the hoop (circumferential) stress, σθ and radial (longitudinal) stress, σr due to the
biaxial stress distribution property. The primary objectives of this experiment are to convert both the
experimental and theoretical strains to stresses and compare the deviations of the experimental and
theoretical stresses.
Based on the results obtained from numerical computations by using Microsoft Excel Spreadsheet which
are tabulated in tables above, the deviations between the values of strains obtained from experiment and
theoretical calculations are quite small. Even the distinctions between the results obtained by utilizing
both the two methodologies are small, but there is still the existence of errors. Among the reasons which
leads to the occurrence of errors are accuracy of data obtained from the experiment or rounding of
decimal places during the numerical computations by using Microsoft Excel, human error which occurs
during the experiment as a result of fluctuation occurs on the strain meter especially when the selector
switch (F) is controlled in order to obtained the appropriate internal pressure which required in this
experiment. Apart from that, machine error or calibrating error will occurs in conjunction with the
experiment as well.
According the strains and stresses versus radius plots as shown in previous section, the graphs of hoop
strain of the experimental values follows the same trend and acts consistently to the series of theoretical
values. Likewise, both the graphs of the hoop stress and the radial stress behave alike except for the plot
of radial stress vs. radius for an internal pressure of 30 bar as shown in Fig. 12 above. As the radial
distance increased, the stresses distribution obtained from experiment and theoretical calculations seems
like to overlap each other when the radial distance approach the outer diameter of the thick-walled
cylinder.
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 15 of 17
6. APPENDICES
6.1 Sample Calculations
The sample calculations shown below are the calculations for experimental and theoretical strains and
experimental and theoretical stresses with an internal pressure of pi = 10 bar and a radius of r = 29.5 mm.
Given that E = 6.985 × 1010 Pa
33.0=υ
ro = 74.5 mm
k = 4
Theoretical Hoop Strain
( )
( ) ( )( )
6108.7164
−
−
−
×=
−−
+××=
−−
+×=
−−
+
−=
22
513
22
13
2
2
2
2
2
)5.29(
5550133.0
)5.29(
55501101010524.9mm 29.5 bar, 10
5550133.0
5550110524.9
11)1(
,
θ
θ
ε
υε
rrp
r
r
r
r
kE
prp
i
ooi
i
Theoretical Radial Strain
( )
( ) ( )( )
6107.4402
−
−
−
×−=
+−
−××=
+−
−×=
+−
−
−=
22
513
22
13
2
2
2
2
2
)5.29(
5550133.0
)5.29(
55501101010524.9mm 29.5 bar, 10
5550133.0
5550110524.9
11)1(
,
r
i
ooi
ir
rrp
r
r
r
r
kE
prp
ε
υε
Experimental Hoop Stress
Based on the tables above, the experimental hoop strain and radial strain at r = 29.5 mm with an internal
pressure of pi = 10 bar are εθ = 5E-06 and εr = -5E-06.
( )
( ) ( ) ( )[ ]( )
MPa 0.259210=
=
−
×−+××=××−
−
+=
−−
−−
Pa 53.259210
33.01
10533.0105)10895.6(105,105
1
)(,
2
661066
2
θ
θθθ
σ
υ
υεεεεσ r
r
E
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 16 of 17
Experimental Radial Stress
( )
( ) ( ) ( )[ ]( )
MPa 0.259210−=
−=
−
×+×−×=××−
−
+=
−−
−−
Pa 53.259210
33.01
10533.0105)10895.6(105,105
1
)(,
2
661066
2
r
r
rr
E
σ
υ
υεεεεσ θ
θ
Theoretical Hoop Stress
Based on the table above, the theoretical hoop strain and radial strain at pi = 10 bar and r = 29.5 mm are εθ
= 8.64363E-06 and εr = -7.367E-06.
( )
( ) ( ) ( )[ ]( )
MPa 0.48070167=
=
−
×−+××=××−
−
+=
−−
−−
Pa 67.480701
33.01
10367.733.01064363.8)10895.6(1064363.8,10367.7
1
)(,
2
661066
2
θ
θθθ
σ
υ
υεεεεσ r
r
E
Theoretical Radial Stress
( )
( ) ( ) ( )[ ]( )
MPa 0.3493231−=
−=
−
×+×−×=××−
−
+=
−−
−−
Pa 10.349323
33.01
1064363.833.010367.7)10895.6(1064363.8,10367.7
1
)(,
2
661066
2
r
r
rr
E
σ
υ
υεεεεσ θ
θ
Thick-walled Pressure Vessels 4209168; 4201205; 4203364
HES5320 Solid Mechanics, Semester 2, 2011 Page 17 of 17
6.2 References
Ayob, A. B., Tamin, M. N. & M. Kabashi Elbasheer, ‘Pressure Limits of Thick-Walled Cylinders’,
Proceedings of the International MutiConference of Engineers and Computer Scientists 2009 Vol.
II, IMECS: 2009, March 18, Hong Kong.
J. M. Kihiu, S. M. Mutuli & G. O. Rading, n. d., Stress characterization of autofrettaged thick-walled
cylinders, pp. 370, International Journal of Mechanical Engineering Education, 31/4, Department of
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