Summary/ Abstract
Thin-walled pressure vessel provides an important application of the analysis of plane stress. Since their walls offer litter resistance to bending, it may be assumed that the internal forces exerted on a given portion of wall are tangent to the surface of the vessel. The resulting stresses on an element of the wall will thus be contained in a plane tangent to the surface of the vessel. In this experiment, we are going to determine the circumferential stress under open condition, and analysis of combined axial and circumferential stress. We propose to determine the stresses exerted on a small element of wall with sides respectively parallel and perpendicular to the axis of the cylinder. Because of axis symmetry of the vessel and its contents, it is clear that no shearing stress is exerted on the element. It can be seen that all strains are used to calculate stress. As it is not possible to measure all strains, these have to be computed on the basis of marginal conditions. With this experiment it is not possible to measure in particular longitudinal strain in the body and radian strain perpendicular to the surface. Initial stress at the surface must be zero, longitudinal stress is constant over the radius are the marginal conditions to obtain the solution. 2 The average E value we obtain from graph was 64 GN/m . with the help of the computer,
the Poisson’s ratio, v =0.35 and the principal strains for the thin cylinder were also calculated using formulas and the Mohr’s circle. For open ends conditions. The lower principal strains, -191 με, which the upper principal strains,
principal strains,
εL = 98.55
με,
εH1 =
579.71
με.
εL1 =
For closed ends conditions, the lower
which the upper principal strains
εH =
484.05
με.
We also
noted that there was errors occur during the experiment. The percentage error between the experiment result and the theoretical result is small for open ends condition but very large for closed ends condition .The possible reason for errors occur will be discuss at the discussion section.
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Statement of purpose/introduction/objec purpose/introduction/objective tive The objective of this experiment is:
1. To obtain a purely circumferential stress system which is the open ends condition and determine the value of Young’s Modulus Modulus when in this condition.
2. To satisfy a biaxial stress system that is the closed ends condition and calculate for the principle strains
The objectives of this can be fulfilled by using the Thin Cylinder device and the specially ram. The Young’s Modulus and the Hoop Strain Open Ends and Closed designed computer prog program. Ends conditions can be determined.
Theory a) Complex Stress System
The diagrams in Figure 1 represent (a) the stress and (b) the forces acting upon an element of material under the action of a two-dimensional stress system.
a) Stress Diagram
2
b) Force Diagram Figure 1: Stress and force diagrams for two dimensional stress systems
Assuming (b) to be a ‘wedge’ of material of unit depth and the side AB to be unit length:
Resolving along
σ σ
ө
ө
will give:
σ σ = σ y( cos² ө) + σ x( sin²ө) + 2τ(sinө cosө)
= ( y cosө) cosө + ( xsinө) sinө + (τ cosө) sinө+ (τ sinө) cosө
ө
σ
σ
ө
=
σ y (cosө2 cosө+1) + σ x (1- cos2ө) + τ sin2ө 2
σ
ө
σ
σ
2
σ σ
= 1 ( y + x) + 1 ( y - x) cos2ө+ τ sin2ө 2 2
(1)
3
Resolving along
τ
ө
will give:
τ =(σ y cosө) sinө- (σ xsinө) cosө+sin2ө (τ sinө) sinө- (τ cosө)cosө τ = σ ysin2ө-σ xsin2ө+ τ sin²ө- τ cos²ө ө
ө
2
τ
2
σ σ x) sin2ө- τ cosө
= 1 ( y 2
ө
(2)
τ
From equation 2 it can be seen that there are values for e for which ө is zero and the planes on which the shear component is z ero are called ‘Principal Planes’. For equation 2:
σ σ x) sin 2ө -
0 = 1 ( y 2
τ cos2ө
σ σ x) sin 2ө
τ cos 2ө=1 ( y 2
σ
σ
τ = 1( y – x) tan 2ө 2 (3)
ө
This will give two values of 2 ө differing by 180° and therefore, two values of differing by 90°,This shows that Principal Planes are two planes at right angles to each other.
Figure 2: Diagram Representation of Equation 3
4
From the diagram Sin2ө = ±
2τ
σ σ x)² + 4τ ²]1/2
[( y -
(4)
and
cos2ө = ±
σ σ x) 1/2 [(σ y - σ x)² + 4τ ²] ( y -
(5)
The stresses on the principal planes are normal to these planes and are called principal stresses. From equation 1 and substituting the above values:
σ
σ σ
σ σ
1/2
= 1 ( y - x) ± 1 [( y - x) ²+ 4τ ²] (6) 2 2 Principal stresses are the maximum and minimum values of normal stresses in the system.The sign will denote the type of stress. ө
i.e Negative sign-Compressive Stress Positive sign – Tensile Stress
Figure 3: Force Diagram for an Element
5
Assuming BC and AC are principal Planes i.e. τө=0, and σ1 and σ2 are the principal stresses
τ
ө
=1(σ2 2
σ1)sin2ө (7)
τ
ө
Now maximum shear stress ө will be seen to occur when sin2 =1 i.e. when ө=45° Therefore the maximum shear stress occurs on planes at 45° to the principal planes and
τ
=1(σ2 2
ө
σ1 ) (8)
or
τ
σ σ x) ²+ 4τ ²]1/2
= [( y -
ө
(9)
b) Two dimensional Stress System
Figure 4: Diagram of Principal Stresses on an Element
6
Strain in direction of σ1:
ε1= σ1 - ν σ2 E
E
(10)
Strain in direction of σ2:
ε2= σ2 - ν σ1 E
E
(11)
ε1 and ε2 are the values of the principal strains: A negative quantity denotes compressive strain. A positive quantity denotes tensile strain
These strains can be used to construct a ‘Mohr Strain Circle’ in the same way as stresses.
Figure 5: Representation of strain on a Mohr’s Circle 7
In the usual manner ,referring to figure 4.5 OR is the maximum principal strain. OP is the minimum principal strain at right angles to maximum Q is the centre of the strain circle From the diagram:
εm= ε2 +( ε1- ε2) +( ε1- ε2) cos2ө 2
2
εm= ( ε2- ε1) +( ε1- ε2) cos2ө 2
(12)
2
and
εn= ε2 +( ε1- ε2) -( ε1- ε2) cos2ө 3
2
εn= ( ε2+ ε1) +( ε2- ε1) cos2ө 2
2
(13)
Theory as Applied to the Thin Cylinder
In a thin cylinder, the ratio of wall thickness to internal diameter is less than 1/20, the value of H and L may be assumed reasonably constant over the area, i.e. throughout the wall thickness, and in all subsequent theory the radial stress, which is small, will be ignored. I symmetry the two principal stresses will be circumferential (hoop) and longitudinal and these, from elementary theory, will be given by:H = pd/2t and L = pd/4t a) Open ends condition The cylinder in this condition has no end constraint and therefore the longitudinal component of stress L will be zero, but there will be some strain in this direction due to the Poisson effect. Considering an element of material: H will cause strains of :H’ = H /E and L’ = -H /E
8
b) Closed ends condition By constraining the ends, a longitudinal as well as circumferential stress will be imposed upon the cylinder. Considering an element of material: H will cause strains of:H = H /E and L’ = -H /E L
will cause strains of:H
= 1/E*(H-L)
L
= 1/E*(L-H)
The principal strains may be evaluated and a Mohr Strain Circle constructed for each test condition. From this circle the strain at any position relative to the principal axes may be determined.
c)
To determine a value for Poisson’s Ratio L’ / H’
= -
This equation is only applicable to the open-end condition.
Apparatus
Figure 6: Thin Walled Cylinder 9
The thin walled cylinder is made of aluminum containing a freely supported piston. The piston can be moved in or out to alter end conditions by use of the adjustment screw. A 0-5.0 MN/m
2
pressure gauge is fitted to the cylinder. The cylinder unit, which is resting on four dowels, is supported in a frame and located axially by locking screw and the adjustment screw. When the adjustment screw is screwed out, the pressurized oil in the cylinder forces the piston against an end plate. When the adjustment screw is screwed in it forces the piston away from the end plate and all the axial load is taken on the frame, thus relieving the cylinder of all longitudinal stress. Pure axial load transmission from the cylinder to frame is ensured by the hardened steel rollers situated at the end of the locking and adjustment screws. Six active strain gauges are cemented onto the cylinder. These are self-temperature compensation gauges and are selected to match the thermal characteristics of the thin cylinder. Each gauge forms one arms of a bridge, the other three arms consisting of close tolerance high stability resistors mounted on a p.c.b. Shunt resistors are used to bring the bridge close to balance in its unstressed condition (this is done in factory test). The effect on gauge factor of this balancing process is negligible. Pressure is applied to the cylinder by closing the return valve, situated near the pump outlet, and operating the pump handle of the self contained hand pump unit. To release pressure, the return valve is unscrewed.
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The cylinder unit, which is resting on four dowels, is supported in a frame and located axially by the locking screw and the adjustment screw. When the hand wheel is screwed in, the pressurized oil in the cylinder forces the piston away from the end plate and all the axial load is taken on the frame, thus relieving the cylinder of all longitudinal stress. Pure axial load transmission from the cylinder to frame is ensured by the hardened steel rollers situated at the end of the locking screw and hand wheel.
Six active strain gauges are cemented onto the cylinder in the position shown in Figure 2.2; these are self-temperature compensation gauges and are selected to match the thermal characteristics of the thin cylinder. Each gauge forms one arm of a bridge, the other three arms consisting of close tolerance high stability resistors mounted on a p.c.b. Shunt resistors are used to bring the bridge close to balance in its unstressed condition (this is done on factory test). The effect on gauge factor of this balancing process is negligible.
Pressure is applied to the cylinder by closing the return valve, situated near the pump outlet, and operating the pump handle of the self-contained hand pump unit. To release pressure unscrew the return valve.
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thin cylinder technical information
Overall dimensions
715mm x 310mm x 380mm high
Total weight of unit
26 kg
Recommended oil
Shell Tellus 37 (or equivalent)
Cylinder & reservoir capacity
Approximately 2 litre
Pressure gauge
0 – 0.5 MN/m (Operating range 0 – 3.5
2
2
MN/m ) Nominal cylinder dimension Internal diameter
80 mm
Wall thickness
3 mm
Length
358.8 mm
Cylinder material
Aged aluminum alloy 606.3
Poisson’s Ratio
0.33
Strain gauge
Electrical resistance self temperature compensation type.
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The pump is fitted with a pressure relief valve adjacent to the pump handle pivot, which is set 2
operate at approximately 4 MN/m . A bleed nipple is fitted to the right hand end of the cylinder. Environmental conditions
Storage temperature range
-25 to +55°C (when packaged for transportation)
Operating temperature
+5 to +40°C
range Humidity 30-95% (non-condensing)
Mains supply
Operating environment
Laboratory environment
Voltage (nominal)
230V , 115V
Current (maximum)
0.58A, 0.62A
Frequency
50Hz, 60Hz
Type
TN-S (see IEC 264)
Fuse type and rating
20mm, T 1A
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Procedure Experiment 1 – Open Ends
1. The pressure control is opened and the hand wheel is screwed in to set up the Open Ends condition. 2. To set up the Open Ends condition, the piston is forced away from the end plate when the hand wheel is fully screwed in. 3. Next, the pressure control is closed and the strain gauge display reading is made zero by pressing the ‘press & hold to zero’ button. 2
2
4. The pressure is increased in 0.5 MN/m steps up to3.0 MN/m , allowing a few seconds for the pressure and strain readings to stabilize. The reading is recorded. 5. The pressure valve is then opened to relieve the pressure.
Experiment 2 – Closed Ends 1. The pressure control is opened and the hand wheel is carefully unscrewed to set up the Closed Ends condition. 2. The Closed Ends condition is set up by screwing in the hand wheel, forcing the piston against the cap at the end of the cylinder. 3. The pressure control is opened to release the pressure. 4. The pressure control is then closed and the strain gauge display readings are made zero by pressing the ‘press & hold to zero’ button. All the strain gau ge readings should read 0 (+/- 0.05). 2
2
5. The pressure is increased in 0.5 MN/m steps up to 3.0 MN/m , allowing a few seconds at each step for the pressure and strain readings to stabilize. The readings are recorded. 6. The pressure control is opened to relieve the pressure.
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Data and Observations:
Cylinder Condition: OPEN ENDS Reading
Pressure
Direct
(MN/m2)
Hoop
Strain
Stress (MN/m2)
Gauge 1
Gauge 2
Gauge 3
Gauge 4
Gauge 5
Gauge 6
Hoop strain
1
0.03
0.4
0
2
2
2
2
1
0.5
2
0.5
6.67
95
-31
0
32
63
99
97
3
1
13.33
199
-69
-2
66
130
206
202.5
4
1.5
20
300
-104
-6
100
197
310
305
5
2
26.67
400
-141
-8
135
203
415
407.5
6
2.5
33.33
502
-148
-12
169
330
519
510.5
7
3
40
604
-212
-13
204
398
626
615
values from actual mohr's circle at (3 MN.m²) values from theoretical mohr's
604
-212
-195
5
0
205
195
555
626
525
circle at (3 MN.m²)
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Cylinder Condition: CLOSED ENDS Reading
Pressure
Direct
(MN/m2)
Hoop
Strain
Stress (MN/m2)
Gauge 1
Gauge 2
Gauge 3
Gauge 4
Gauge 5
Gauge 6
Hoop strain
1
0
0
0
1
2
1
1
0
0
2
0.5
6.5
96
14
36
61
80
100
98
3
1
13.6
175
27
66
109
145
185
180
4
1.5
19.7
261
46
101
161
209
269
265
5
2
26.1
344
61
133
210
275
356
350
6
2.5
33.2
429
77
167
263
344
444
436.5
7
3
39.1
515
94
201
318
415
536
525.5
values from actual mohr's cirlce at (3MN.m²) values from theoretical
515
94
200
309.5
495
98.55
195
292
460
536
mohr's cirlce at (3MN.m²)
Observation:
1. The stress-strain relationship produces a linear graph for both the Gauge 1 and Gauge 2. 2. From the Mohr Strain Cycle produced for Open Ends condition, the experimental values obtained were a bit off from the theoretical values. It is also observed that the error at the beginning is larger than the end part of the experiment. 3. From the Mohr Strain Cycle produced for Open Ends condition, the experimental values obtained were also a little bit different from the theoretical values. Only at the middle part of the experiment the experimental value is almost the same as the theoretical value.
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hope stress against hope strain 45 40 35 ) ² m / 30 N M25 ( s s e r t 20 s p o 15 o h
y = 0.0645x + 0.3582
10 5 0 0
100
200
300
400
500
600
700
hoop strain
longitudinal strain vs hoop strain 50 0 0 n i a r t s l a n i d u t i g n o l
100
200
300
400
500
600
700
-50 -100
y = -0.3505x + 2.3531
-150 -200 -250 hoop strain
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Analysis:
Calculation of young modulus Using equation 18 from the SM1007 manual:
E=
H/ H
6.67 / 97 0.069 (MN/m2)
For the average value of young’s modulus obtain from graph which is 0.064 (MN/m2) and when convert it 64(GN/m2)
Percentage error:
Calculation of Poisson’s Ratio
Using equation 24 from the SM1007 manual: - =
L1
/
H1
= 0.5/2
=0.25
For the average value of poison’s ratio from graph which is 0.35.
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Calculation of Principal Strains (Open Ends Condition) From equation 16 in the SM1007 manual: H=pd/2t 0.003(0.08)/2(0.003) =0.04
From equation 16 in the SM1007 manual: H1
=
H
/E
where, H
= 0.04
E = 69 GN / m
2
Therefore, H1
= 579.71 (in graph make it as 580)
From equation 17 in the SM1007 manual: L1
=-
H
/E
where,
= -0.33
Therefore, L1
= -191
(in graph make it as 190)
Principles strains error:
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Calculation of Principal Strains (Closed Ends Condition) From equation 15 in the SM1007 manual:
L=pd/4t 0.003(0.08)/4(0.003) =0.02
From equation 22 in the SM1007 manual: H
=(
H
-
L)
/E
where, H
= 0.04
L
= 0.02
Therefore, H
= 484.05 (in graph make it as 485)
From equation 23 in the SM1007 manual: L
=(
L
-
H)
/E
Therefore, L
= 98.55 (in graph make it as 100)
Principles strains error:
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Discussion
From this experiment, it is determined that the relationship of hoop stress and hoop strain is use to find the Young’s modulus. The graph produced is linear and the Young’s Modulus value for 2
average value is 64 GN / m .
Next, the difference between Open Ends condition and Closed Ends condition was determined and the Mohr Cycles were produced. For the Open Ends condition, the lowest error recorded for the actual strain value is from Gauge 5 with only 5.71
differences between the
actual and theoretical value. The highest error between the actual and theoretical strain value is recorded from Gauge 2 with 8.7
differences. From the Mohr Cycle produced it can be clearly
seen that the earlier part of the cycle recorded higher error compare to the end part of the cycle.
For the Closed Ends condition, the lowest error recorded for the actual strain value is from Gauge 3 with only 2.6
differences between the actual and theoretical value. The highest
error between the actual and theoretical strain value is recorded from Gauge 5 with 7.6
differences. From the Mohr Cycle produced it can be clearly seen that the only part of the cycle which has the lowest error is at the middle part of the cycle.
The difference between Open Ends and Closed Ends is when in the Open Ends condition the cylinder has no end constrain and therefore the longitudinal component of stress will be zero. Still, there is some strain in this direction due to the Poisson effect and this brings the cylinder in the longitudinal direction to be in compression. Closed Ends condition is due by constraining the ends. A longitudinal as well as circumferential stress will be imposed upon the cylinder. That makes it to have both strain of longitudinal and horizontal and the principal strains are combination of the values emitted by the stresses. Therefore, compression and tension are the effects.
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For the Poisson’s ratio, this equation is only applicable to Open Ends condition and the value obtained is 0.35. Calculating the Principle of Strains for both Open Ends and Closed Ends condition, it is calculated that the Principle of Strains is -191
for Open Ends and 98
for
Closed Ends. These strains are from their longitudinal strain and it can be seen that the value for Closed Ends are larger. Error Analysis:
There may be errors in this experiment. For the Open Ends condition it is supposed to take the 2
strain values at 3.00 MN / m cylinder pressure but during the experiment the cylinder pressure 2
was 3.03 MN / m . This will obviously affect the strain values obtained. The reading on the pressure gauge was also slightly off from the value displayed on the computer monitor screen. This could also cause the errors.
Some of the error is: a. Constructive errors These are result of tolerance in the dimensions of component and the values of the electrical components used and are inherited in the manufacture of the instrument. There can be error to due to the accuracy with which the manufacturer of an instrument has calibrated it. b. Operating error It can occur for variety of reasons such as in reading the position of a pointer in scale. It is so called parallax errors. It also can occur due to the limited resolution of an instrument and taking a reading. Beside that, operating error can also arise when the instrument has made contact with the object being measured. c. Human error When adjustment the screw for open and closed ends condition is not fully in or out, the reading will give out differently. d. Insertion errors In some measurements, the insertion of the instrument into the position in order to measure a quantity can affect its value.
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Conclusion
The experiment performed has successfully fulfilled the given objectives. The principles of thin cylinder testing by using the Thin Cylinder device have been successfully understood. The Young’s Modulus, Principle of Strains and Poisson’ Ratio values were successfully obtained.
It is determined that the Hoop Stress-Hoop Strain relationship produces a linear graph. 2
The average Young’s Modulus is 64 GN / m . The Principle of Strains for the Open Ends condition is -191
and for the Closed Ends condition is 98
.
The Poisson’s Ration is 0.35.
The errors that occur in this experiment can be reduced if the experiment is done carefully with extreme caution and taking every little consideration into account.
References
1.
Ferdinand P. Beer, E. Russell Johnston, John T. DeWolf. 2004. Mechanics of Material. rd
3 edition. Mc Graw Hill. Page 209-217. 2.
William D. Callister Jr. 2003. Material Science and Engineering an Introduction. 6
th
edition. John Wiley and Sons Pte Ltd. Page 118-120. 3.
Rosdi Abd Manan, 2000, Static and Dynamics testing of Materials Using Instron Universal Testing Machine Model 8516; B.Eng Thesis, University Tenaga Nasional. Page 43-58.
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