University of Ontario Institute of Technology Faculty of Engineering and Applied Science MECE 2420U Solid Mechanics Laboratory Manual
Lab 5: The Thin Cylinder
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Lab Objectives
The TQ Thin Cylinder apparatus allows you to set up and perform several experiments related to the stress systems in a thin cylinder. It clearly demonstrates the principles involved and gives practical support to class studies. The use of the Thin Cylinder apparatus gives an appreciation of: A biaxial stress system the use of strain gauges Young’s Modulus Poisson’s ratio Constructing and using Mohr’s Circles of strain.
Background In relation to stress analysis, cylinders are divided into two groups: thick and thin. 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. Thin cylinders or shells are commonplace in engineering. Examples of thin walled cylinders are: pressure pipes, aircraft fuselages and compressed gas containers.
Thick walled cylinders are less common, an example being a gun barrel. For a closed cylinder with an internal pressure there can be three direct stresses acting upon it. Longitudinal stress – the cylinders resistance to stretching along its length (axis). Hoop or Circumferential stress - the cylinders resistance to grow in diameter. Radial stress – gas or fluid compressing the walls of the cylinder. It is equal to the pressure on the inside and zero on the outside
The longitudinal stress and hoop stresses are directly proportional to the pressure and the ratio of diameter to thickness of the cylinder. However the radial stress is related to the pressure alone. Because of their relationship to the geometry, the Longitudinal and Hoop stresses are far greater and more significant than the radial stress in a thin cylinder. It is reasonable and recognized to assume that the radial stress is small enough for it to be ignored for basic calculations.
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Figure 1: Stresses in a thin walled cylinder.
The individual direct stresses are given by: = pd/2t and,
σ H
= pd/4t
σ L
where: -2
= Hoop Stress (Nm ) -2 σ L= Longitudinal Stress (Nm ) σ H
-2
P= Pressure in the cylinder (Nm ) d= Diameter of the cylinder (m) t= Thickness of cylinder walls (m) Nearly all applications of the thin cylinder will have closed ends with the biaxial stress system described previously. However as outlined in the introduction, the equipment allows us to examine the stresses in the cylinder with open ends i.e. with no direct longitudinal stress. Although there is no practical application for a cylinder in this condition, the experiment yields several useful relationships. We can use these relationships in the more complex closed ends condition. Equipment Figure 2 shows the SM1007 Thin Cylinder apparatus. It consists of a thin walled aluminum cylinder of 80 mm inside diameter and 3 mm wall. 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 c ylinder. The cylinder is held in a sturdy frame in which it is free to move along its axis. The strains (and thus the stress) can be measured with the cylinder in two configurations: 1. ”Open” ends – where the axial loads are taken by the frame (not the cylinder), therefore there is no direct axial stress
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2. “Closed” ends – where the axial loads are taken by the cylinder, therefore there must be direct axial stress
Figure 2: Experiment setup
The two configurations are achieved using the large hand wheel at the end of the frame. 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 3. 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 4.
Figure 3: Open Ends Condition
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Figure 4: Closed Ends Condition
Computer Connection To take and record readings and plot graphs, the SM1007 must be connected to a PC. Dedicated software and the connecting lead is supplied with the SM1007. Strain Sensors The sensor used to measure the strains in the walls of the thin cylinder is called a Strain Gauge.
Strain gauges are sensors that experience a change in electrical resistance when they are stretched or compressed, this change in resistance can be shown in terms of displacement (strain). Strain gauges are made from a metal foil cut in a zigzag pattern, they are only a few microns thick so they are mounted on a backing sheet, and this allows them to be handled and electrically insulates the zigzag element. Gauges are bonded to the structural part under examination, thus the strain gauge stretches and compresses the same amount as the surface of the part. To give an direct reading of strain we use a constant called the gauge factor, this is to compensate for the slight differences in manufacture between each batch of gauges, it usually varies between 1.8 and 2.2. It is usual to set this in the strain gauge readout. There are six strain gauges on the cylinder, arranged at various angles to allow the study of how the strain varies at different angles to the axis. Strains are shown on the P.C. screen, directly in micro-strain on a schematic of the thin c ylinder. Note that a negative reading is a compressive strain and a positive reading is a tensile strain the technique of strain gauging is of great importance to structural engineers. This equipment gives you the opportunity to understand their use.
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How to set up the equip ment
Figure 5: Layout of the SM1007
Before using the equipment, always: Visually inspect all parts, including electrical leads, for damage or wear. Check that all electrical connections and other parts are secured correctly and fastenings are sufficiently tight. Position the equipment safely on a solid, level surface, so that it is steady, easily accessible.
Never apply excessive loads to any part of the equipment. 1. The main switch is at the rear of the unit. Move it to the on position, the green LED on the front should illuminate. 2. Start the SM1007 software on the PC. 3. Leave the SM1007 for 5 minutes to allow the gauges to warm up and reach a steady state. 4. To check that the Hydraulic circuit is functioning, and familiarize yourself with the controls: a). Open the release valve on the pump (turn anticlockwise) b). Wind the hand wheel fully in c). Fully close the release valve on the pump d). Operate the pump steadily observing the pressure on the gauge. Stop at a 2 pressure of 3 MN/m . e). Leave the unit for 1 minute, observing the pressure gauge, the pressure should not vary in this time f). Open the release valve on the pump, the pressure should fall to zero Finally, to set up the unit with the computer, select ‘Options’ from the menu bar and set the Gauge factor to that given on the SM1007 front panel. Press ‘Test SM1007’ to ensure that the SM1007 is communicating with the PC. The PC should give the message: ‘Commun icatio ns wi th SM1007 OK’ .
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Safety Instructions There is a risk of electric shock. Always unplug first. During test do not touch any parts of the test stand. Do not exceed a cylinder pressure of 3.5 MNm-2 Follow lab dress codes Use assistance and follow the correct handling procedures when moving this apparatus. Always disconnect after each experiment. IMPORTANT! Never attempt any form of machine maintenance. Should anyone have any Safety Concerns or Questions, Please Ask!
Experimentation 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 cylinder. The experimental method utilizes the SM1007 software to display and take readings.
1. Familiarize yourself with the equipment by following the instructions in section 1.3. 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 to 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.
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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
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should now read 0±0.3 με, and the pressure meter should read 0±0.01 MNm . 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. -2
5. Pump the handle slowly until a pressure of around 0.5 MNm and record the readings into the data table again by selecting ‘Record Gauge Readings’ from the ‘Experiments’ menu option. Wait a few seconds between pumps for the gauges to stabilize. -2
6. Carefully increase the pressure in 0.5 MNm increments, recording the readings -2
into the data table until you have reached a value of 3 MNm . Try to get as close -2
as possible to 3 MNm as it will allow you to make direct comparisons with established theoretical values at this pressure. Do not exceed a cylinder pressure -2
of 3.5 MNm 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 ‘Experiments’ menu option. Experiment 2 - Thin Cylinder with Closed Ends For experiment 2 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 MNm-2 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 -2
some more and try again. Do not exceed a cylinder pressure of 3.5 MNm 3. In the SM1007 software choose ‘Closed Ends Condition’ from the ‘Experiments’ menu option. Then connect to 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. 4. 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 -2
now read 0±0.3 με, and the pressure meter should read 0±0.01 MNm .
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5. 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. -2
6. Pump the handle slowly until a pressure of around 0.5 MNm and record the readings into the data table again by selecting ‘Record Gauge Readings’ from the ‘Experiments’ menu option. Wait a few seconds between pumps for the gauges to stabilize. -2
7. Carefully increase the pressure in 0.5 MNm increments, recording the readings -2
into the data table until you have reached a value of 3 MNm . Do not exceed a -2
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cylinder pressure of 3.5 MNm . Try to get as close as possible to 3 MNm as it will allow you to make direct comparisons with established theoretical values at this pressure. 8. You may print the data table if desired by pressing the printer button in the top left corner of the table. 9. Disconnect the communications between the PC and the apparatus by selecting ‘Discon nect the SM1007’ from the ‘Experiments’ menu option. Analysis For Experiment 1
The Stress Strain Relationship
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? You should conclude that the hoop strain remains constant along the length of the cylinder. Plot a graph of Average Hoop Stress versus Hoop Strain either by hand (from the data table), or by using the graphing facility in the SM1007. To plot the graph using the SM1007 software, select ‘Hoop Stress v Hoop Strain’ from the ‘Results’ menu option. A chart will appear with crosses marking the results. What is the relationship between stress and strain? Your graph should reveal a linear relationship between stress and strain, the gradient of which is a measure of stiffness called the Young’s Modulus. i.e
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E = σ/ε Where: E = Young’s Modulus (Nm-2) σ = Stress (Nm-2) ε = Strain (με) Find a value of the Young’s Modulus for the cylinder material from your graph. To find the value of the gradient and therefore the Young’s Modulus from the graph plotted by the software, a line needs to be drawn through the results. The slope of the line is indicated when you release the mouse button. The Young’s Modulus varies from material to material, but is a constant for each material, so long as it has uniform properties (homogenous and isotropic). For the aluminum alloy used for the SM1007 cylinder the Young’s modulus is nominally 70 -2
GNm Does the value of Young’s modulus from your graph agree with the theoretical value stated? If there is a discrepancy between the values then name any sources of error that may be present. Steel is approximately three times stiffer than aluminum having a Young ’s Modulus of -2 210 GNm . If the cylinder had been made of steel would the measured strain be higher or lower for the same stress? Conclusion The use of the apparatus has established the stress strain relationship and experimental value of Young’s Modulus. Note that since the Young’s Modulus remains constant for any given homogenous isotropic material, strain gauges are a reliable means of measuring stress on the surface of a structural part. The Ratio of Hoop Strain to Longitudinal Strain in an Open Cylinder In the introduction to the experiments it was stated that there is no Direct longitudinal strain in the open ends conditions. With reference to the small diagram on the front of the SM1007, identify the gauge which measures the longitudinal strain. Does this gauge register zero?
You will find that the gauge does not register zero, in fact a significant compressive strain is measured. This is because there is an Indirect Longitudinal strain which is a result of the hoop stress. This indirect strain is generated by the fact that as the cylinder increases diameter the length decreases (the opposite can be seen to occur if you stretch elastic band). To explore this relationship, plot a graph of the Longitudinal Strain Versus Average Hoop Strain, either by hand (from the data table), or by using the graphing facility in the
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SM1007. To plot the graph using the SM1007 software, select longitudinal v Hoop Strain from the Results menu. A chart will appear with crosses marking the results. Find the gradient of your plot. To find the value of the gradient from the graph plotted by the software, a line needs to be drawn through the results. The slope of the line (gradient) is indicated when you release the mouse button. The magnitude of the gradient of your plot is a called Poisson’s ratio it can be defined as the ratio of indirect strain to direct strain. In mathematical terms it is given as: v = -εLo/εHo Where: v = Poisson’s Ratio εLo = Longitudinal Strain (in the open ends condition) εHo = Hoop Strain (in the open ends condition) The Poisson’s ratio changes from material to material but is constant for any given homogenous isotropic material. Most metals have a value of Poisson’s ratio of around 0.3, although the range of values is 0.1 to 0.5 the extremes being concrete and rubber respectively. The cylinder is manufactured from an aluminum alloy which has a Poisson’s ratio of 0.33. Compare this to the gradient of your graph. Principle Strains and the Mohr’s Circle In the case of a cylinder the maximum and minimum strains are always at right a ngles to each other. These strains are called the principle strains. As discovered previously, in the open ends condition, the thin cylinder has principle strains of:
εLo = Longitudinal Strain (in the open ends condition) εHo = Hoop Strain (in the open ends condition)
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v= -ε Lo / ε Ho where:
v= Poission’s Ratio ε Lo= Longitudinal ε Ho= Hoop
Strain (in the open ends condition)
Strain (in the open condition)
The Poisson’s ratio changes from material to material but is constant for any given homogenous isotropic material. Most metals have a value of Poisson’s ratio of around 0.3, although the range of values is 0.1 to 0.5 the extremes being concrete and rubber respectively. The cylinder is manufactured from an aluminum alloy which has a Poisson’s ratio of 0.33. Compare this to the gradient of your graph. Principle Strains and the Mohr’s Circle In the case of a cylinder the maximum and minimum strains are always at right a ngles to each other. These strains are called the principle strains. As discovered previously, in the open ends condition, the thin cylinder has principle strains of Ε Ho= σ Ho /E (direct Ε Lo=
from the hoop stress)
-vσ Ho /E (indirectly due to the Poisson effect)
But how to we quantify strains at other than right angles to the axis of the cylinder? The answer lies in a simple yet effective graphical method known as a Mohr’s Circle. Mohr’s circles can be used to solve a variety of strain, stress and deflection problems. First, construct a Mohr’s Circle by hand so that you understand how the method works. The SM1007 software will also construct a Mohr’s circle from your results. You can use this facility to check your work. We have already established that the strain gauges give a very linear response to pressure so we will use only the values for the maximum test -2 pressure (3 MNm ). 1. On graph paper construct an axis which allows for the minimum principle strain (gauge 2) and the maximum principle strain (the average of gauges 1 and 6) on the x axis and a strain of around –400 to +400 on the y axis. The x and y axes must be to the same scale . 2. As can be seen from the equations in the theory section, on a Mohr’s Circle the strains are related to 2 θ where θ = the angle from the axis. This is an important thing to remember.
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Since gauge 2 is on the axis of the cylinder, then the angle is 0° on the Mohr’s circle Since gauges 1 and 6 are 90° from the axis, then the angle is 180° on the Mohr’s circle 3. Plot these 2 points on the axis as shown in sketch 1.
Sketch 1 4. The two points are the extremes of the Mohr’s circle. On this basis find the centre either by calculation or by construction, then draw a circle as shown in sketch 2. This is the Mohr’s circle based on your results for the principle strains. You can now use this to find the direct strain at any angle from the axis. 5. To find the direct strain at 60° from the axis, draw a line from the centre of the circle at 120° clockwise from 0° (remember, twice the angle! ) until it intersects the circle, then draw a line vertically down to the x axis. 6. Read off the value of strain and compare this to the reading for the strain gauge which is placed at 60° from the axis (gauge 5) you should find that the two readings are in close agreement. 7. Repeat the process for 30° and 45° and compare the readings from your Mohr’s Circle to those measured on the equipment. Does the Mohr’s circle accurately predict the direct strain at any angle? As you know, the x axis on the Mohr’s circle plot is the direct strain, but what is the y axis? The answer is the shear strain , from which we can quantify the shear stress.
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Sketch 2
Sketch 3
To plot the Mohr’s circle using the SM1007 software, select Plot Mohr’s Circle from the Results menu. A chart will appear with crosses marking the results. Readings of strain may be taken from the graph by moving the mouse cursor around the circle. Find the values of shear strain for each of the angles on the cylinder in the same way you found the direct strain. At what angle is the maximum shear strain? Will it always be this angle for a thin cylinder? Calculate theoretical principle strains (using the equations given in the theory section) -2
with a pressure of 3 MNm , a Poisson’s ratio of 0.33 and a Young’s Modulus of 69 -2
MNm . Construct a Mohr’s circle using these values and compare it to your experimental one. Do the theoretical and experimental Mohr’s circles agree? What are the sources of error? For Experiment 2
Having completed the analysis of the open ends condition we will now analyze the cylinder in the closed ends condition to show the effect of the biaxial stress system. Examine the data table. With reference to the stress equations, confirm that longitudinal stress is half the value of the hoop stress. Check, using a method of differences, or b y
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sketching graphs, that the readings from the gauges are still linear in the closed ends condition.
Using the Method o f Superposition to Find the Principle Strains In the previous experiment, we established that for the direct hoop stress we created and indirect longitudinal strain, due to the Poisson effect. It follows therefore, that if we created a purely longitudinal stress that we wou ld have an indirect hoop strain. If the cylinder were stretched it will have a tendency to reduce in diameter (which is exactly what happens if you stretch an elastic band.) To find the principle strains we ca n use the idea of superposition. This means that we can simply work out the principle strains for each stress case (hoop and longitudinal) in isolation and then sum them to give the principle strains for the biaxial system. i.e.: “The principle strain in the longitudinal direction is the sum of the direct longitudinal strain and the indirect longitudinal strain ” and “The principle strain in the circumferential direction is the sum of the direct hoop strain and the indirect hoop strain .” In mathematical terms, the hoop stress will cause strains of: ε H = σ H /
E
ε L=-v σ H /E
(due to the Poisson effect)
Conversely, the longitudinal stress will cause strains of: ε L= σ L /
E
ε H =-v σ L /
E (due to the Poisson effect)
The algebric sum of these strains is ε H =( σ H -
v σ L )/E
ε L=( σ L -
v σ H )/E
Construct a Mohr’s circle from your results at 3 MNm-2 either by hand or by using the SM1007 Software using the same method as used in experiment 1. Read off values for the direct strains and compare them to values obtained from the equipment. Your predicted and experimental value should be in close agreement. What do you notice about
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the overall diameter of the Mohr’s circle compared to the open ends condition? How does this affect the shear strain? Calculate theoretical principle strains using the equations given in the theory section with a pressure of 3 MNm-2, a Poisson’s ratio of 0.33 and a Young’s Modulus of 70 MNm-2. Construct a Mohr’s circle using these values and compare it to your experimental one. Do the theoretical and experimental Mohr’s circles agree? References:
Callister, William D. Materials Science and Engineering: An Introduction (4 Wiley & Sons, Inc.: New York, 1997.
th
Ed.). John
TQ Education and Training Ltd 2000 “SM1007 Thin Cylinder STUDENT’S GUIDE”
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