UNIVERSITI TENAGA NASIONAL COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING
MEMB221 - MECHANICS & MATERIALS LAB Experiment title : Thin cylinder (6) Author
: Zaiful Fadly Bin Zawawi
Student ID
: ME086677
Section
: 02 (group 6)
Lecturer
: Siti Zubaidah Bte Othman
Performed Date
Due Date
Submitted Date
25/06/2012
09/07/2012
09/07/2012
Table of Content 1.0 Abstract…………………………………………………………………………………..2 2.0 Objective………………………………………………………………………………....3 3.0 Theory……………………………………………………………………………………4 4.0
Equipment……………………………………………………………………………….7
5.0
Procedure………………………………………………………………………………10
6.0
Data and Observations…………………………………………………………………11
7.0
Analysis and Results…………………………………………………………………...16
8.0
Discussion……………………………………………………………………………...18
9.0
Conclusions…………………………………………………………………………….20
10.0 References……………………………………………………………………………..21 11.0 Appendices…………………………………………………………………………….22
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1.0 Abstract To examine the stress and strain in a thin walled cylinder, students conduct the experiment by using thin cylinder apparatus (SM1007). The experiment clearly shows the principles, theories and analytical techniques and does help the student in studies. By using SM1007, student will be able to measure the strains of the cylinder in 2 ends condition. Open ends and closed ends. The difference between opened ends and closed ends is that, open ends does not have axial load and no direct axial stress, meanwhile in a closed ends there is axial load and axial stress. As the result of the experiment, the value of circumferential stress both under open condition and closed condition has been obtained. Analysis has been made and so the calculation. From the data collected in opened ends condition the values of Young’s modulus and Poisson’s ratio are calculated.
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2.0 Objective The objective of this experiment is to determine the circumferential stress under open and closed condition and to analyze the combined stress and circumferential stress.
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3.0 Theory Because this is a thin cylinder, i.e. the ratio of wall thickness to internal diameter is less than about 1/20, the value of σH and σL 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: σH =
………… (1)
and σL =
………… (2)
As previously stated, there are two possible conditions of stress obtainable; 'open end' and the 'closed ends'.
Figure 1: Stresses in a thin walled cylinder
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a) Open Ends Condition:
The cylinder in this condition has no end constraint and therefore the longitudinal component of stress σL will be zero, but there will be some strain in this direction due to the Poisson effect. Considering an element of material: σH will cause strains of:εH1 =
…………. (3)
and εL1 =
…………. (4)
These are the two principal strains. As can be seen from equation 4, in this condition εL will be negative quantity, i.e. the cylinder in the longitudinal direction will be in compression.
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: σH will cause strains of:εH1 =
…………. (5)
and εL1 =
…………. (6)
σL will cause strains of:εL =
…………. (7)
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and εH =
…………. (8)
The principal strains are a combination of these values which are: εH =
(σH -
) …………. (9)
εL =
(σL -
) …………. (10)
The principal of the strains may be evaluated and the Mohr Strain Circle constructed for each of the 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 :
Dividing equations 3 and 4 gives:=
…………… (11)
This equation is only applicable to the open ends condition.
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4.0 Equipment
Thin Cylinder SM1007
Figure 2: Thin Cylinder SM1007
Figure 2 shows a thin walled cylinder of aluminum containing a freely supported piston. The piston can be moved in or out to alter end conditions by use of the hand wheel. An operating range of 0 - 3.5 MN/m2 pressure gauge is fitted to the cylinder. 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. In purpose to release the pressure the return valve is unscrewed.
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Figure 3: Sectional plan of the thin cylinder
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 (hand wheel). When the hand wheel is screwed in, it forces the piston away from the end plate and the entire axial load is taken on the frame, thus relieving the cylinder of all longitudinal stress. This creates ‘open ends’. 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. When the hand wheel is screwed out, the pressurized oil in the cylinder forces the piston against caps at the end of the cylinder and become ‘closed ends’ of the cylinder. The cylinder wall then takes the axial stress.
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Figure 4: Strain gauges positions Six active strain gauges are cemented onto the cylinder in the position shown in Figure 4; these are selftemperature 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.
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5.0 Procedure The power of the thin cylinder is switched on and it leaves for at least 5 minutes before the experiment is conducted. This allows the strain gauges to reach a stable temperature and to give the accurate readings. Two conditions of stress may be achieved in the cylinder during test: (i) A purely circumferential stress system which is the 'open ends' condition (ii) A biaxial stress system which is the 'closed ends' condition.
To obtain the circumferential condition of stress, Ensure that the return valve on the pump is fully unscrewed so that oil can return to the oil reservoir. The hand wheel is screwed in until it reaches the stop. This moves the piston away from the left-hand end plate and thus the longitudinal load is transmitted onto the frame. When in this condition, the value of the Young's Modulus for the cylinder material may be determined and also the value for Poisson's Ratio can also be determined.
To obtain the biaxial stress system, Ensure that the return valve on the pump is fully unscrewed. The hand wheel is unscrewed and the crosspiece is pushed to the left until it contacts the frame end plate. The return valve is closed and the hand pump is operated to pump oil into the cylinder and push the piston to the end of the cylinder. Thus, when the cylinder is pressurized, both longitudinal and circumferential stresses are set up in the cylinder. Before any test being made, and at zero pressure, each strain gauge channel should be brought to zero or the initial strain readings recorded as appropriate.
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6.0 Data and Observation Table 1: Open Ends Results
Cylinder Condition: OPEN ENDS Direct
Reading
Strain (με)
Pressure
Hoop
Gauge
Gauge
Gauge
(MN.m-2)
Stress
1
2
3
4
5
6
Gauge
Gauge
Gauge
(MN.m-2) 1
0.03
0.40
0
0
0
1
0
0
2
0.51
6.80
94
-33
-3
30
62
96
3
1.02
13.60
200
-72
-6
64
130
204
4
1.50
20.00
297
-110
-11
96
193
305
5
2.00
26.67
400
-146
-13
130
261
410
6
2.50
33.33
502
-181
-16
165
328
518
7
3.00
40.00
605
-217
-17
202
394
621
-
-217
-9
200
405
-
580
-191
2
195
388
580
Values from actual Mohr’s Circle (at 3 MN.m-2) Values from theoretical Mohr’s Circle (at 3 MN.m-2)
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Sample Calculations for Open Ends (Theoretical values) : Thickness = 3mm Internal Diameter = 80mm Poisson’s ratio = 0.33 Young’s Modulus = 69 × 109 N.m-2
σH =
………… (1)
= 40 MN.m-2
σH =
εH1 = εH1 =
εL1 =
…………. (3) = 580με (for 1,6)
…………. (4)
εL1 =
= -191με (for 2)
ε1 = 580με , ε2 = -191με εn = (
)+(
εn = 2με (for 3)
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) cos2θ (θ=30)
εm = (
)+(
) cos2θ
εm = 388με (for 5) When θ is equal 45° ε= (
) = 195με (for 4)
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Table 2: Closed Ends Results
Cylinder Condition: CLOSED ENDS Direct
Reading
Strain (με)
Pressure
Hoop
Gauge
Gauge
Gauge
(MN.m-2)
Stress
1
2
3
4
5
6
Gauge
Gauge
Gauge
(MN.m-2) 1
0.01
0.13
0
0
0
0
1
1
2
0.50
6.67
78
15
32
50
64
78
3
1.00
13.33
164
33
67
102
133
167
4
1.51
20.13
248
48
99
152
199
254
5
1.99
26.53
329
63
131
203
334
425
6
2.50
33.33
414
82
167
257
334
425
7
3.01
40.13
499
99
199
310
401
512
-
99
199
203
400
-
484
99
195
292
388
484
Values from actual Mohr’s Circle (at 3 MN.m-2) Values from theoretical Mohr’s Circle -2
(at 3 MN.m )
Sample Calculations for Closed Ends (Theoretical Values): Thickness = 3mm Internal Diameter = 80mm Poisson’s ratio = 0.33 Young’s Modulus = 69 × 109 N.m-2
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σH = 40 MN.m-2 σL = 20 MN.m-2
εH =
(σH -
) …………. (9)
εH = 484με (for 1,6) εL =
(σL -
) …………. (10)
εL = 99με (for 2)
ε1 = 484με , ε2 = 99με εn = (
)+(
) cos2θ (θ=30)
εn = 195με (for 3)
εm = (
)+(
) cos2θ
εm = 388με (for 5) When θ is equal 45° ε= (
) = 292με (for 4)
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7.0 Analysis and Result
Graph of Hoop Stress against Hoop Strain Hoop Stress (MN.mˉ²)
50 y = 0.0653x + 0.5373 R² = 1
40 30 20 10 0 0
100
200
300
400
500
600
Hoop Strain (με)
Graph 1: Graph of Hoop Stress against Hoop Strain
From the Graph, we know that the value of the Young’s Modulus is 65.3GPa. (Gradient of graph is 0.0653TPa) The actual value of Young’s Modulus is 69GPa. Percentage Error = (69-65.3)/(69) = 5.36%
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700
Graph of Longitudinal Strain against Hoop Strain Longitudinal Strain (με)
0 0
100
200
300
400
500
600
-50 -100 -150 -200
y = -0.3606x - 0.3457 R² = 0.9997
-250
Hoop Strain (με)
Graph 2: Graph of Longitudinal Strain against Hoop Strain From the graph, we know that the Poisson’s ratio is 0.33 (Gradient of the graph is –0.3606) The actual value of the Poisson’s ratio given is also 0.33 The Percentage error = (0.33-0.3606)/(0.33) = 9.27%
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700
8.0 Discussions Table 3: Open Ends Condition – at a cylinder pressure of 3MN.m-2 Gauge no
Actual Strain
Theoretical Strain
Error
(με)
(με)
(%)
1
-
580
-
2
-217
-191
13.6
3
-9
2
550
4
200
195
2.6
5
405
388
4.4
6
-
580
-
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Table 4: Closed Ends Condition – at a cylinder pressure of 3MN.m-2 Gauge no
Actual Strain
Theoretical Strain
Error
(με)
(με)
(%)
1
-
484
-
2
99
99
0
3
199
195
2.05
4
203
292
30.47
5
400
388
3.1
6
-
484
-
From the Graph 1, Graph of Hoop Stress against Hoop Strain, we know that the value of the Young’s Modulus is 65.3GPa. (Gradient of graph is 0.0653TPa). The actual value of Young’s Modulus is 69GPa. The Percentage Error = 5.36%.
From the Graph 2, Graph of Longitudinal Strain against Hoop Strain we know that the Poisson’s ratio is 0.3606 (Gradient of the graph is –0.3606). The actual value of the Poisson’s ratio given is also 0.33. The Percentage error = 9.27%
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9.0 Conclusion From the experiment we can determine the circumferential stress under open condition and under the closed condition. We are being able to analyses the theoretical values of each condition by using the formula which is given from the theory parts. We are also being able to analyses the theoretical value which the actual values by self-drawing of Mohr Strain Circles. By using details from the open condition, we are also being able to get the values of Young’s Modulus and the Poisson’s ratio. For this experiment, we get the value of Young’s Modulus is 65.3GPa and the value of Poisson’s ratio is 0.3606.
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10.0
References
1. Laboratory Manual Mechanics & Materials Lab 2012. 2. Mechanics of Materials.2009.5th edtion.Singapore.McGraw-Hill.pp423 3. Material Testing.2012 http://www.tecquipment.com/Materials-Testing/Stress-Strain/SM1007.aspx 4. Thin Wall Cylinder.2012 http://homepage.mac.com/sami_ashhab/courses/strength/subjects/thin_wall_cylinder/thin _wall_cylinder.html 5. Thin Cylinder.2012 http://www.tech.plym.ac.uk/sme/mech226/Thincylinders/thincyl.pdf
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11.0
Appendices
Table 5: Technical Details Items
Details
Dimensions
370mm high × 700mm long × 380 front to back
Nett Weight
30kg
Electrical Supply
85VAC to 264VAC 50Hz to 60 Hz
Fuse
20mm 6.3A Type F
Max. Cylinder Pressure
3.5MN.m-2
Strain Gauges
Electrical resistance self-temperature compensation type
Cylinder oil
Shell Tellus 37
Total oil capacity
App. 2 liters
Cylinder Dimensions
80mm internal diameter 3mm wall thickness 359mm length
Cylinder Material
Aged Aluminium Alloy 6063
Young’s Modulus (E)
69GN.m-2
Poisson’s Ratio
0.33
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