Analysis of Stress and Strain in a Thick-Walled Cylinder Report

This laboratory report details an experiment to determine stress and strain distributions in a thick-walled cylinder subjected to internal pressure. The experiment involved applying pressure to the cylinder's inner surface and analyzing stresses and strains at varying radii using strain gauges. The report includes an abstract summarizing the experiment's aim, procedure, and key findings, such as the highest radial strain recorded and its correlation with pressure and radial position. Background theory, including Lame's equations, explains the stress distribution within the cylinder. The report provides a detailed account of the equipment, experimental procedure, and results, including tables and graphs illustrating strain versus pressure and stress versus radius. The discussion section analyzes the results, comparing experimental and theoretical values and discussing the relationships between pressure, strain, and stress. The conclusion summarizes the key findings, and an applications section discusses the practical uses of the analysis. The report also includes a list of references and figures.

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Laboratory Report on Thick-Walled Cylinder
Contents
List of figures...............................................................................................................................................1
List of Tables................................................................................................................................................1
Abstract.......................................................................................................................................................2
Background Theory.....................................................................................................................................2
Equipment and Procedure...........................................................................................................................5
Materials and equipment........................................................................................................................5
Specifications of the set up......................................................................................................................6
Procedure................................................................................................................................................6
Results.........................................................................................................................................................6
Discussion..................................................................................................................................................14
Conclusion.................................................................................................................................................15
Application................................................................................................................................................15
List of figures
Figure 1: Diagram of a thick-walled cylinder ...............................................................................................2
Figure 2: Section of the thick-walled cylinder .............................................................................................3
Figure 3: Section of the element.................................................................................................................4
Figure 4: The apparatus set up ...................................................................................................................6
Figure 5: Strain against pressure for all gauges.........................................................................................10
Figure 6: Graph of radial and hoop stress values versus radius of the thick-walled cylinder.....................13
Figure 7: Graph of radial and hoop strain versus radius............................................................................14
List of Tables
Table 1: Gauge type, radius and position relative to radial axis for strain gauges.......................................6
Table 2: Values of strain obtained from the experiment.............................................................................7
Table 3: Radial and hoop stress values when P is 6 MPa...........................................................................12
Table 4: Values of radial and hoop strains.................................................................................................13

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Laboratory Report on Thick-Walled Cylinder
Abstract
The laboratory report aims at determining the stress and strain distributions in the walls of a thick
cylinder under internal pressure. The experiment involved application of pressure on the internal
surface of the thick cylinder and analyzing the stresses and strains at different radii. The largest
radial strain was -65.67 which was recorded by strain gauge 2 when pressure was set at 6.03
MN.m-2. Radial strain decreased with an increase in radial position. Strain increased with
increase in pressure. The highest value of strain recorded was 106 which was obtained at a
pressure of 6.03 MN.m-2. From Lame’s theory, the radial stress increased with increase in radius.
The hoop stress on the other hand decreased with increase in radius. To increase reliability on
data, parallel experiment on similar material at different pressure should be conducted.
Background Theory
Figure 1 below shows a thick-walled cylinder with both ends open. Two pressures are acting on
the cylinder. These pressures are external pressure P0 and internal pressure Pi. The inner radius a
is ri while the outer radius b is r0.
Figure 1: Diagram of a thick-walled cylinder (Vullo, 2014).

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Laboratory Report on Thick-Walled Cylinder
Figure 2: Section of the thick-walled cylinder (Vullo, 2014).
Take into consideration a section at radius r situated at an angle increment dθ. The section has a
radial increment dr. Applying circular symmetry, we get that the stresses σ H and σ r are functions
of the radius r only (Macherauch, 2014). The shear stress on the section is equal to zero. For a
unit thickness, the radial force equilibrium is as follows;
( σ r +d σr ) ( r+dr ) dθ=σr rdθ+σ H dθdr
Solving this equation by overlooking second order terms we get;
d σr
dr + σr +σ H
r =0
Consider that the system lacks body forces. Taking into consideration the st5rains in the section.
Applying symmetry, θ displacement v does not exist. Only radial displacement u described by
line aa’ is present (Younessi, 2012).

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Laboratory Report on Thick-Walled Cylinder
Figure 3: Section of the element (Vullo, 2014).
c is dislodged outwardly by ( u+du ) represented by cc’. dr being the original radial length of the
section, the radial strain is;
ε r= u+du−u
dr = du
dr
The length of ab is rdθ while length of a’b’ is ( r +u ) dθ. Thus, the tangential strain will be given
by;
ε H = ( r +u ) dθ−rdθ
rdθ = u
r
Since both ends are open, σ z =σ3=0. This is the plane stress conditions. Applying Hooke’s law,
we obtain;
ε r= du
dr = 1
E ( σr −v σ H )
ε H = u
r = 1
E ( σH −σr )
Solving for the stresses;
σ r= E
1−v2 ( εr + v ε H )

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Laboratory Report on Thick-Walled Cylinder
σ H = E
1−v2 ( ε H +v ε r )
Substituting back into the equations above, we obtain;
d2 u
d r2 + 1
r
du
dr − u
r 2 =0
Solving this we get;
u=C1 r + C2
r
The stresses now become;
σ r= E
1−v2 [C1 ( 1+v ) −C2 ( 1−v
r2 ) ]
σ H = E
1−v2 [ C1 ( 1+ v ) −C2 ( 1−v
r2 ) ]
Expressing the stresses as a function of radius to obtain Lame’s equations as below (Zhang,
2011);
σ r= ri
2 Pi−ro
2 Po
( r0
2−ri
2
) − ( Pi−Po ) ri
2 ro
2
( ro
2−ri
2
) r2
σ H = ri
2 Pi−ro
2 Po
( r0
2−r i
2
) + ( Pi −Po ) ri
2 ro
2
( r o
2−ri
2
) r2
This can be reduced to get;
σ r=P ri
2
ro
2−r i
2 ( 1− r o
2
r 2 )
σ H =P ri
2
ro
2−ri
2 (1+ ro
2
r2 )

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Laboratory Report on Thick-Walled Cylinder
Equipment and Procedure
Materials and equipment
 Aluminum alloy type HE15
 Strain gauges – Electrical resistance
 Five Hoop strain
 Five Radial strain
 Two circumferential
 One Longitudinal
Figure 4: The apparatus set up (Zhang, 2011).
Specifications of the set up
 Apparatus net Weight – 30 kg
 Young’s Modulus (E) – 73 GN/m2
 Poisson’s Ratio (v) – 0.33
 Maximum Test Pressure – 7 MN/m2
Procedure
i) The current was passed through the gauges for 30 minutes with zero-gauge pressure
in order to obtain a balance reading for every gauge. Note that the more time the
system was permitted to stabilize, the more repeatable and precise the results became.
ii) The position and orientation of every strain gauge was taken and recorded.
iii) The internal pressure was increased at a constant rate of 1 MN/m2 until it reaches 7
MN/m2. At every increment reading were taken for every strain gauge. The data was
recorded in the appropriate data sheet. For every increment, the readings were to

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Laboratory Report on Thick-Walled Cylinder
taken only when they stabilize and were record in the results table. The cylinder
pressure did not exceed 7 MN/m2.
iv) The pressure was reduced to 0 and step iii repeated to verify the strain gauge data.
Results
The following data was obtained from the experiment;
Table 1: Gauge type, radius and position relative to radial axis for strain gauges
Strain Gauge
Layout
Gauge Number
Gauge
Radius
Gauge
Type
Position relative to radial
axis (°)
Strain 1 28 Hoop 90
Strain 2 28 Radial 0
Strain 3 36 Hoop 90
Strain 4 36 Radial 0
Strain 5 45 Hoop 90
Strain 6 45 Radial 0
Strain 7 56 Hoop 90
Strain 8 56 Radial 0
Strain 9 63 Hoop 90
Strain 10 63 Radial 0
Strain 11 18.5
Circumferent
ial N/A
Strain 12 75 Longitudinal N/A
Strain 13 75
Circumferent
ial N/A
Table 2: Values of strain obtained from the experiment.
Ti
me
Pre
ssu
re Hoop Strains
Circu
mfere
ntial
Strain
s Radial Strains
Longi
tudin
al
Strai
n
Cylinder
Dimensi
ons
Ti
me
Cyli
nde
r
Pre
ssu
re
Str
ain
1
S
tr
ai
n
3
S
t
r
a
i
n
S
tr
a
i
n
7
S
tr
ai
n
9
St
ra
in
1
1
St
ra
in
1
3
St
ra
in
2
St
rai
n
4
S
tr
ai
n
6
S
tr
ai
n
8
S
t
r
a
i
n
Strai
n 12
Oute
r
Dia
met
er
Inn
er
Dia
me
ter

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Laboratory Report on Thick-Walled Cylinder
5
1
0
(s)
(MN.
m-2)
(με
)
(μ
ε)
(
μ
ε
)
(
μ
ε)
(μ
ε)
(μ
ε)
(μ
ε)
(μ
ε)
(με
)
(μ
ε)
(μ
ε)
(
μ
ε
) (με)
(mm
)
(m
m)
Da
ta
Se
rie
s 1
0 0.08 -2 0
-
2 0 -1 -1 -1 -2 -1 0 -1
-
1 -2
150.
0
37.
0
2 0.08 -2 0
-
2 0 -1 -2 -1 -2 -2 0 -1
-
2 -2
150.
0
37.
0
4 0.08 -1 0
-
1 0 0 0 -1 -2 -1 0 0 0 -2
150.
0
37.
0
Da
ta
Se
rie
s 2
0 0.96 7 3 1 1 0 13 0
-
12 -6 -4 -3
-
3 -3
150.
0
37.
0
2 0.96 7 3 0 1 0 13 0
-
12 -8 -4 -3
-
3 -3
150.
0
37.
0
4 0.96 7 3 1 2 0 14 0
-
12 -7 -3 -3
-
2 -2
150.
0
37.
0
Da
ta
Se
rie
s 3
0 2.02 16 9 4 4 2 33 0
-
24 -14 -8 -6
-
5 -3
150.
0
37.
0
2 2.01 16
1
0 4 3 1 32 0
-
23 -13 -8 -6
-
6 -3
150.
0
37.
0
4 2.01 17
1
0 5 3 2 33 0
-
22 -13 -9 -7
-
5 -2
150.
0
37.
0
Da
ta
Se
rie
s 4
0 3.01 27
1
6 8 6 4 51 2
-
34 -18
-
1
1 -8
-
6 -2
150.
0
37.
0
2 3.01 27 1
6
9 6 3 50 2 -
33
-18 -
1
-9 -
7
-2 150.
0
37.
0

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Laboratory Report on Thick-Walled Cylinder
2
4 3.01 28
1
8
1
0 7 4 52 2
-
32 -18
-
1
0 -8
-
7 -1
150.
0
37.
0
Da
ta
Se
rie
s 5
0 4.02 38
2
3
1
3
1
0 7 69 4
-
44 -24
-
1
4 -9
-
9 -2
150.
0
37.
0
2 4.02 37
2
3
1
4
1
1 7 70 3
-
44 -24
-
1
4 -9
-
9 -2
150.
0
37.
0
4 4.02 38
2
1
1
3 9 5 69 3
-
44 -25
-
1
5
-
1
1
-
9 -3
150.
0
37.
0
Da
ta
Se
rie
s 6
0 5.03 46
2
7
1
4
1
0 6 87 4
-
57 -32
-
2
0
-
1
5
-
1
2 -3
150.
0
37.
0
2 5.03 46
2
7
1
5
1
0 7 86 4
-
56 -31
-
2
0
-
1
4
-
1
2 -3
150.
0
37.
0
4 5.03 46
2
7
1
5
1
0 7 87 4
-
57 -33
-
2
1
-
1
5
-
1
2 -3
150.
0
37.
0
Da
ta
Se
rie
s 7
0 6.03 58
3
4
2
1
1
4
1
0
10
7 7
-
66 -36
-
2
2
-
1
5
-
1
3 -2
150.
0
37.
0
2 6.03 58
3
4
2
1
1
4
1
0
10
6 6
-
65 -37
-
2
2
-
1
5
-
1
3 -3
150.
0
37.
0
4 6.03 58
3
3
2
0
1
4
1
0
10
5 6
-
66 -37
-
2
3
-
1
6
-
1
3 -3
150.
0
37.
0

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Laboratory Report on Thick-Walled Cylinder
The graph in was developed;
0.08 0.96 2.01 3.01 4.02 5.03 6.03
-80
-60
-40
-20
0
20
40
60
80
100
120
strain 1 strain 2 strain 3 strain 4 strain 5 strain 6 strain 7
strain 8 strain 9 strain 10 strain 11 strain 12 strain 13
Pressure in MN/m^2
Strain
Figure 5: Strain against pressure for all gauges.
From figure 7, the value of E for radial strain which is equivalent to the inverse of the gradient of
the average strain is -0.1990 MPa. When P = 4.5 MPa; the strain will be given by dividing
pressure by the Young’s modulus as below
ε = P
E
ε = 4.5
−0.1990 =−22.62
For hoop strain, the value of Young’s modulus E is 0.2121 MPa. Thus, the hoop strain will be
given by dividing pressure by the Young’s modulus as below
ε = 4.5
0.2121 =21.21
For circumferential strain, the value of Young’s modulus E is 0.1042 MPa. Thus, the
circumferential strain will be given by dividing pressure by the Young’s modulus as below

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Laboratory Report on Thick-Walled Cylinder
ε = 4.5
0.1042 =43.19
For longitudinal strain, the value of Young’s modulus E is 0.1042 MPa. Thus, the longitudinal
strain will be given by dividing pressure by the Young’s modulus as below
ε = 4.5
12.02=0.37
The stresses and strains at various radii are calculated using the Lame’s equation as below;
σ r=P ri
2
ro
2−r i
2 ( 1− r o
2
r 2 )
σ H =P ri
2
ro
2−ri
2 (1+ ro
2
r2 )
Substituting P=4.5 MPa ,r o=0.75 m ,r i=0.0185 m we get
σ r=0.2915 ( 1− 0.005625
r2 )
σ H =0.2915 (1+ 0.005625
r2 )
The strains are given by the formula;
ε r= du
dr = 1
E ( σr −v σ H )
ε H = u
r = 1
E ( σH −σr )
ε L= v
E ( σ H +σ r )
For every value of radius, we will apply the above formulas to obtain the radial, hoop and
longitudinal stresses and strain.
For radius r = 28 mm, the stresses and strain will be as follows;

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Laboratory Report on Thick-Walled Cylinder
σ r=−1.80 MPa and σ H =2.383 MPa
ε r=−0.035 , εH=0.041 and ε L=0.0026
For radius r = 36 mm, the stresses and strain will be as follows;
σ r=−0.974 MPa and σ H =1.557 MPa
ε r=−0.032, ε H=0.026 and ε L=0.0026
For radius r = 45 mm, the stresses and strain will be as follows;
σ r=−0.518 MPa and σ H =1.101 MPa
ε r=−0.012, ε H =0.017 and ε L=0.0026
For radius r = 56 mm, the stresses and strain will be as follows;
σ r=−0.231 MPa and σ H =0.814 MPa
ε r=−0.007 , εH =0.012 and ε L=0.0026
For radius r = 63 mm, the stresses and strain will be as follows;
σ r=−0.122 MPa and σ H =0.705 MPa
ε r=−0.005 , εH=0.01 and ε L=0.0026
At P = 6 MPa, the following values are obtained using Lame’s formula;
Table 3: Radial and hoop stress values when P is 6 MPa.
Radius
radial
stress
hoop
stress
28 -2.4 3.176
36 -1.296 2.076
45 -0.692 1.468
56 -0.308 1.084
63 -0.164 0.94