UNIVERSITI TEKNIKAL MALAYSIA MELAKA
No Dokumen: No Isu./Tarikh SB/MMSB2/BMCS2333/6 1/12-12-2007
SOLID MECHANICS 2 Thin & Thick Cylinders Analysis
No Semakan/Tarikh 3/28-06-2010
Jumlah Mukasurat
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OBJECTIVES OF EXPERIMENTAL WORK 1. To inve invest stig igat ate e and and anal analyz yze e the the stre stress ss sy syst stem ems s in thin thin and and thic thick k cylinders 2. To compar compare e stress stress systems systems or or distribu distributions tions between between thin and and thick thick cylinders
LEARNING OUTCOMES OUTCOMES (N.B Students should not include these as part of their report) At the end of this laboratory session students should should be able to 1. apply the the thin and thick thick cylinder cylinder formulatio formulations ns to obtain obtain the principle principle stresses due to internal pressures. 2. determ determine ine the magni magnitud tude e of stresses stresses in thin thin cylind cylinder er under under closed closed and open end conditions. 3. analyze analyze the stress stress distribut distributions ions of the thick thick cylinder cylinder with respect respect to the radial dimension or its wall thickness. 4. understan understanding ding of basic laborato laboratory ry practice, practice, including including design design of experiments, write a clear and well-presented technical report, data acquisition, interpretation and analysis, and the relationship between experiments and theory.
THEORY
The analysis of the stress distribution in a thin or thick walled cylinder is of considerab considerable le practical practical importanc importance e in pressure pressure vessels and gun barrels. barrels. Strain Strain gauges gauges mount mounted ed on variou various s radiu radius s and at differ different ent alignm alignment ents s throughout the cylinder wall provide the measurement of the strains. Thus stre stress ss dist distri ribu buti tion on thro throug ugho hout ut the the wall wall of a cyli cylind nder er su subj bjec ecte ted d to an internal pressure could be analyzed. Theory of Thick Cylinder σH
r
σH
σL
σR
Material Element at radius ‘r’ 1
Figure 1
Cylinder under Internal
Figure 1 shows a hollow cylinder, which is subjected to a uniformly distributed internal pressure P . The figure details an element of material at some radius r, contained within an elemental cylinder. Due to the design of the SM1011 Thick Cylinder the longitudinal stress σ L may be ignored (i.e σL = 0) and only a bi-axial stress system be considered. Hence the stress formulas are shown below and Figure 2 shows the variations of radial stress σR and hoop stress σH throughout the cylinder wall. Maximum Pressure )
σ R
occurs at the inner radius (R1) i.e.
Minimum σ R occurs at the outer radius (R2) i.e. Maximum σ H occurs at the inner radius (R1) i.e.
σ R
σ R
= -P ( where P = Internal
=0
σ
H
( K = P ⋅ ( K
2
+ 1)
2
− 1)
(1) Minimum σ H occurs at the outer radius (R2) i.e.
σ
H
=
( 2 P )
( K
2
− 1)
(2) where
K =
R2 R1
P
( K ( K
) − 1)
2
P ×
σH
( 2 P )
0 - P
Figure 2
+1
2
( K
2
)
−1
σR
Stressses variation throughout a cylinder thickness
Now for a cylinder under internal pressure P (MPa) and free from axial loading (σL = 0), the maximum shear stress will occur at the inner radius. i.e. Maximum shear stress,τ max = ½ × (difference of the two principal stresses). =
σ
R
− σ H
2
2
(3)
Substituting we get:-
=
P ⋅ R2
( R
2
2
2
(4)
− R12 )
2
Therefore:
τ max
=
P ⋅ K
( K
2
(5)
− 1)
In the case of the TQ cylinder:
K =4.054
and therefore τmax =1.065 P .
Note: The theoretical development of thin cylinder theory may be found from any reference books for Mechanics of Solids. Students are required to include this thin cylinder theory as part of their formal report.
APPARATUS Thin Cylinder Cylinder
Mechanical Pressure Gauge
Hand Wheel
Pump Socket for Communicatio n Cable
Diagram, Indicating Gauge Factor Figure 3 Layout of the SM1007
1.
2.
Figure 3 shows the SM1007 Thin Cylinder apparatus. It consists of a thin walled aluminum cylinder of 80 mm inside diameter and 3 mm wall thickness. Operating the hydraulic pump pressurizes the cylinder with oil. The cylinder has six sensors on its surface that measure strain. A mechanical gauge and electronic sensor measure the hydraulic pressure in the cylinder. The cylinder is held in sturdy frame in which it is free to move along its axis. The strain (and thus the stress) can be measured with the cylinder in two configurations: a. “Open” ends – where the axial loads are taken by the frame (not the cylinder), therefore there is no direct axial stress b. “Closed” ends – where the axial loads are taken by the cylinder, therefore there must be direct axial stress 3
The two configurations are achieved using the large hand wheel at the end of the frame. 3.
In the “open” ends condition the hand wheel is screwed fully in. This pushes the two pistons away from the cylinder end caps so that there is no contact between them. Therefore, the axial force is transmitted from the pressurized oil into the frame rather than the cylinder. See Figure 4.
Pistons (touching frame) End cap
Pistons (touching end cap) End cap End cap
End cap
Oil under pressure
Oil under pressure
Gap
Handwheel wound in
wound out
FrameGap
Frame Figure 5 Closed Ends Condition
Path of load Figure 4 Open Ends Condition 4.
5.
6.
Gap Handwheel
In the “closed” ends condition the hand wheel is wound out. This allows the pistons to move outward against the cylinder end caps so that there is no contact with the frame. Therefore the axial force is transmitted from the pressurized oil into the cylinder itself. See Figure 5. In relation to stress analysis, cylinders are divided into two groups: thin and thick. The distinction between the two relates to the ratio of internal diameter to wall thickness of a particular cylinder. A cylinder with a diameter to thickness ratio of more than 20 is considered to be thin. A ratio of less than 20 is considered to be thick. This distinction is made as the analysis of a cylinder can be simplified by assuming it is thin. The SM1007 cylinder has a ratio of approximately 27, which is well above the ratio for being considered thin.
PROCEDURES Experiment 1 – Thin Cylinder with Open Ends In this experiment we will pressurize the cylinder in the open ends condition taking readings from all six strain gauges, we will then analyze the results in various ways to establish some important relationships. Examine the cylinder and the diagram on the front panel to understand the notation and placement of the strain gauges in relation to the axis of the 4
cylinder. The experimental method utilizes the SM1007 software to display and take readings. 1. Having set up and familiarized yourself with the equipment open the pump release valve and screw in the hand wheel to set up the open ends condition. 2. In the SM1007 software choose OPEN ENDS CONDITION from the EXPERIMENTS menu option. Then connect the SM1007 unit by selecting CONNECT TO SM1007 from the same menu. The virtual meters on the screen should now display values of pressure and strain. 3. Close the pump release valve and zero the readings by selecting ZERO ALL GAUGES from the EXPERIMENTS menu option. All the virtual strain meters should now read 0±0.3µε, and the pressure meter should read 0±0.01MPa. 4. Take the first set of readings (at zero) into the data table by selecting RECORD GAUGE READINGS from the EXPERIMENTS menu option. Display the data table by selecting DATA TABLE in the RESULTS menu. 5. Pump the handle slowly until a pressure of around 0.5 MPa and record the readings into the data table again by selecting RECORD GAUGE READINGS from the EXPERIMENT menu option. Wait a few seconds between pumps for the gauges to stabilize. 6. Carefully increase the pressure in 0.5 MPa increment, record the readings into the data table until you have reached a value of 3 MPa (Do not exceed a maximum cylinder pressure of 3.5 MPa). 7. You may print the data table if desired by pressing the printer button in the top left corner of the table. 8. Disconnect the communications between the PC and the apparatus by selecting DISCONNECT THE SM1007 from the EXPERIMENT menu option. Experiment 2 – Thin Cylinder with Closed Ends Having completed the analysis of the open ends condition; we will now test the cylinder taking the same readings as in experiment 1 but with the cylinder in the closed ends condition to show the effect of the biaxial stress system. 1. Open the pump release valve and carefully unscrew the hand wheel enough to set up the closed ends condition. To check that the frame is not transmitting any load, close the pump release valve and pump the handle and observe the pressure gauge, you may need to pump a number of times as the oil pushes the pistons outward. 2. Once a pressure of around 3 MPa has been achieved, gently push and pull the cylinder along its axis, the cylinder should move in the frame indicating that the frame is not transmitting any load. If it doesn’t move, wind the hand wheel out some and try again. 3. Release the pressure from cylinder by opening the pump release valve. 4. In the SM1007 software choose CLOSED ENDS CONDITION from the EXPERIMENTS menu option. Then connect the SM1007 unit by selecting CONNECT TO SM1007 from the same menu. The virtual meters on the screen should now display values of pressure and strain. 5. Repeat steps 3 to 8 in Experiment 1.
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Experiment 3 – Thick Cylinder Strain distribution Select this option to create the Strain Table and determine the difference between measured and theoretical strain values through the cylinder wall. 1. Ensure the cylinder is at zero pressure by checking that the hand wheel turns freely and the pressure gauge reads zero. 2. Select ZERO READINGS to zero the pressure and strain signals. 3. Increase the pressure to about 6.5 MPa, allowing about 5 seconds for the pressure and strain readings to stabilize and then select TAKE READING to copy the current readings to the data table. 4. Select PRINT to print the data table if desired, ensure a printer is connected, on-line and correctly set up. 5. Select GRAPH to draw a graph of strain distribution through the cylinder wall. Experimental hoop strains are shown by green circles, experimental radial strains by blue circles and theoretical measurements by white circles. 6. Select PRINT to print the graph if desired. Stress distribution Select this option to create the Stress Table and display and print graphs showing stress distribution through the cylinder wall and calculated stresses against a Lame line. The following steps should be taken: 1. Select PRINT to print the data table if desired, ensure a printer is connected, on-line and correctly set-up. 2. Select GRAPH to show a graph of stress distribution through the cylinder wall. Experimental hoop stresses are shown by green circles, experimental radial stresses by blue circles and theoretical stresses by white circles. 3. Select LAME GRAPH to show a graph of calculated stresses against a Lame line for the thick cylinder at the pressure used. The Lame line is shown as a black line and the experimental data is shown as blue and green circles. Note that, in order to obtain a straight-line relationship, the X-axis of the graph is 1 r . 2
4. Select PRINT to print the graph that is currently displayed if desired. 5. Finally select EXIT to return to the main screen. 6. Once the strain table has been created, the results from it may be used to calculate the principal stresses on the thick cylinder. The calculation is shown on screen and select PRINT to print out the result.
EXPERIMENTAL RESULTS 1. Thin Cylinder with Open Ends a. The stress relationship
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• The data table calculates the hoop stress for each pressure reading.
Select one pressure reading (other than zero) and check the calculation of stress using the equations given in the previous section and the data on the front panel of the SM1007. • From your examination of the positioning of the strain gauges you
will have noticed that gauges 1 and 6 have been placed so that they are measuring the hoop strain in the cylinder. Examine the results for gauges 1 and 6, what can you say about the magnitude of the hoop strain as you move along the axis of the cylinder? • Plot a graph of Average Hoop Stress versus Hoop Strain and find a
value of the Young’s Modulus for the cylinder material from the graph. b. The Ratio of Hoop Strain to Longitudinal Strain in an Open Cylinder Plot a graph of the Longitudinal Strain Versus Average Hoop Strain and find gradient of the graph (magnitude of the gradient/slope is called Poisson’s ratio, ν).
2. Thin Cylinder with Closed Ends Calculate theoretical principle strains with a pressure 3 MPa, a Poisson’s ratio, ν = 0.33 and a Young’s Modulu,s E = 70 MPa.
3. Thick cylinder In all calculations the following values for Young’s Modulus and Poisson’s ratio are used: ν = 0.33 E = 73.1 GPa a. Outlines the method for calculating the theoretical strain values from the theory outlined earlier. Calculate the values for εH and εR and tabulate them along with the measured values in table below. b. Plot the two (experimental & theoretical) strain distributions. c. Outlines the method for calculating the theoretical stress values and also the method of calculating the derived stress values from the measured strains. Tabulate the two set of value for σH and σR in a table below. d.
Plot the two (experimental & theoretical) stress distributions.
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DISCUSSION (N.B. This part of the report must at least describe or discuss the following, but not necessarily limited to those ideas. This must be done in the paragraphs format rather than the points form format)
• •
•
Explain and discuss the main results and observations obtained in this work and explain any discrepancies observed. In experiment 1, the Young’s Modulus varies from material to material but is a constant for each material, so as long as it has uniform properties (homogenous and isotropic). For the aluminum alloy used for the thin cylinder, the Young’s Modulus is nominally 70 GPa. Does the value of Young’s Modulus from your graph agree with the theoretical value stated? If there is discrepancy between the values then name any sources of error that may be present.
In thin cylinder analysis (open ends), the thin cylinder is manufactured from an aluminum alloy that has a Poisson’s ratio of 0.33. Compare this to the gradient of your graph and give your comment about the differences.
CONCLUSION Give your conclusion and summary of this experimental work. State whether its main objectives have been achieved or not.
QUESTIONS (For FORMAL report only) 1. Steel is approximately three times stiffer than aluminum having a Young’s Modulus of 210 GPa. If the cylinder had been made of steel would the measured strain be higher or lower for the same stress? Justify your answer. 2. In experiment 1, there is no direct longitudinal strain (εL) in the open
ends conditions. However, the gauge, which measures the longitudinal strain, does not register zero reading. Explain this phenomenon and give your reason why it happened.
3. Give two examples for each category of pressure vessels in industry that you could consider as the thin and thick cylinders.
REFERENCES (For FORMAL Report Only) List at least 3 main references that have been referred to write the formal report of this laboratory exercise. 8
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