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ENGINEERING
MMAN3400 Mechanics of Solids 2
7KLQ &\OLQGHU Pressure Vessel
Name: …………………………….. Student No. …………………... Alfred
2
SECTION 1: INTRODUCTION
Introduction
This lab guide describes how to set up and perform experiments related to thin walled
pressure vessels and determine the associated stresses, modulus of elasticity and Poisson’s
ratio.
Description
Figure 1: Pressure Vessel Apparatus
Figure 1 shows a thin-walled aluminium alloy cylinder (pressure vessel). Inside each end of
the cylinder is a free moving piston. The cylinder sits inside a sturdy frame, on top of a steel
box. Fixed to the surface of the cylinder is a set of electrical strain gauges. A digital display
on the front of the apparatus shows the strain measured by each gauge.
A hydraulic hand pump is used to force oil into a cylinder to apply internal pressure. A
mechanical Bourdon type pressure gauge shows the oil pressure in the cylinder. Fitted as
standard to the pressure line is an electronic pressure transducer. The hand pump includes a
Pressure Control for the operator to control the pressure in the cylinder and a built-in pressure
relief valve to help prevent damage to the equipment. The body of the hand pump is the oil
reservoir.
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Open and Closed Ends
A Hand Wheel at the end of the frame sets the cylinder for the open and closed ends
experiments.
When the user screws in the Hand Wheel, it clamps the free-moving pistons in the
cylinder. The frame then takes the axial (longitudinal) stress and not the cylinder wall,
as if the cylinder has no ends. This allows ‘Open Ends’ experiments (Figure 2).
When the user unscrews the Hand Wheel, the pistons push against caps at the end of
the cylinder and become ‘Closed Ends’ of the cylinder. The cylinder wall then takes
the axial (longitudinal) stress (Figure 3).
Figure 2 Open Ends Conditions
Figure 3 Closed Ends Conditions
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Strain Gauges
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 as shown in Figure 4.
Figure 4 Strain Gauge Positions
Cylinder Dimensions
Outer diameter = 86.04 mm; Length = 359 mm, Wall thickness = 2.9 mm.
Experiment 1 – Open Ends
Aims
To show the linearity of the strain gauges in the Open Ends condition
To find the hoop stress and strain relationship (Modulus of Elasticity) for the cylinder
material
To find the longitudinal and hoop strain relationship (Poisson’s Ratio) for the cylinder
material
Construction of Mohr’s Circle
Procedure:
1. Create a blank table of results, similar to Table 1.
2. Switch on power to the cylinder and leave it for at least five minutes before you do
the experiment.
3. Open (turn anticlockwise) the Pressure Control and screw in the Hand Wheel to set up
the Open Ends condition.
4. Shut (turn clockwise) the Pressure Control and use the ‘Press and & hold to zero’
button to zero the strain gauge display readings. All the strain gauge readings should
now read 0 (∓5) and the pressure meter should read 0 MNm-2 (∓0.05 MNm-2).
5. Enter first set of readings (at zero pressure) into your blank results table (Table 1).
6. Pump the Hand Pump until the pressure is approximately 0.5 MNm-2. Wait a few
seconds for the readings to stabilize and record the readings into the results table.
inthelab:86.04 mm
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7. Carefully increase the pressure in 0.5 MNm-2 up to 3 MNm-2. At each increment, wait
for the readings to stabilize and record the readings into your results table.
8. Open (turn anticlockwise) the Pressure Control to reduce the indicated pressure back
to 0 MNm-2.
Table 1 Blank Results Table
Results Analysis: Open Ends
(a) Strain Gauge Linearity
On one chart, plot strain (vertical axis) against pressure (horizontal axis) for all six strain
gauges. You should note three things:
The results are linear
The readings for strain gauge 3 (30o to the axis) remain at approximately zero
whatever the pressure.
The readings for strain gauge 2 (longitudinal strain) are negative.
ME
0.5 90 -32 0 30 59 91
I 181 - 64 -2 54 118 182
1,5 274 - 98 -3 84 778 273
2 369 -129 - 4 120 240 370
2.5 461 - 162 - 4 150 302 465
3 554 - 193 - 5 182 361 556
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E The Hoop stress and Strain Relationship
Calculate the direct Hoop Stress at each pressure using the theory.
Use the results of gauges 1 and 6 to plot a graph of the Hoop Stress (vertical axis) vs
Hoop Strain (horizontal axis) and find the Modulus of Elasticity of the cylinder
material.
F The Longitudinal and Hoop Strain Relationship
Plot a graph of the Longitudinal Strain (from gauge 2) vs Hoop Strain (from gauges 1
or 6). Find the Poisson’s ratio for the cylinder material from the gradient of the graph.
(d) Principal Strains
The principal strains are only in the longitudinal and hoop directions (two angles). Construct
a Mohr’s Circle to find the strains at other angles.
To draw the Mohr’s Circle,
You would normally draw a chart of each strain gauge’s response and find its gradient to give
an average value. However, your results should be very linear, so you could use any row of
your results. For comparison with theoretical values you must use the values for the
maximum test pressure (3 MNm-2). Calculate the average maximum value for gauge 1 and 6.
From your results, find the maximum positive value of principal strain (the average
maximum of gauges 1 and 6) Find the maximum negative value of principal strain
(from gauge 2).
Half-way down a piece of graph paper, construct a horizontal axis that allows for the
maximum and minimum principal strain values. Construct a vertical axis as shown in
Figure 5. Plot these two points on the horizontal axis.
The two points are the extremes (diameter) of the Mohr’s Circle. Find the centre of
the circle (half of the diameter) and draw a circle.
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Figure 5
Take the Modulus of Elasticity of the cylinder material as 69 GN/m2 and its
Poisson’s ratio as 0.33. Use them with the Principal strain (Open Ends) equations
(σH/E, - µ σH/E) to calculate theoretical principal strains with your calculated
Hoop Stress at 3 MNm-2 pressure.
Enter your results into the bottom lines of your results table. Use these values to
construct a theoretical Mohr’s Circle on the same axis as your actual Mohr’s
Circle. Enter your theoretical values into your results table. Are the results
similar? If there are any differences, what do you think are the causes?
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Experiment 2 – Closed Ends
Aim
To use the experience gained from the Open Ends experiment to analyse the more ‘real
world’ application of a ‘Closed Ends’ (biaxial stressed) cylinder.
Procedure:
Table 2 Blank Results Table
1. Create a blank table of results, similar to Table 2.
2. Switch on power to the cylinder and leave it for at least five minutes before you do
the experiment.
3. Open (turn anticlockwise) the Pressure Control and screw in the Hand Wheel to set up
the Closed Ends condition.
To check that the frame is not taking any load:
Shut (turn clockwise) the Pressure Control valve and use the Hand Pump until the
pressure gauge reaches 3 MNm-2 (you may need to pump many times).
NE
0.5 76 18 35 49 62 77
I 154 32 66 95 124 156
1.5 233 49 98 142 186 232
2 309 64 129 188 247 311
2.5 383 79 158233307 386
3 462 95 190 280 368 463
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Gently push and pull the cylinder along its axis, the cylinder should move in the
frame. This indicates that the frame is not taking any load. If the cylinder does not
move, wind the Hand Wheel out some more and try again.
Open the Pressure Control to release the pressure.
4. Shut (turn clockwise) the Pressure Control and use the ‘Press and & hold to zero’
button to zero the strain gauge display readings. All the strain gauge readings should
now read 0 (∓5) and the pressure meter should read 0 MNm-2 (∓0.05 MNm-2).
5. Enter first set of readings (at zero pressure) into your blank results table (Table 2).
6. Pump the Hand Pump until the pressure is approximately 0.5 MNm-2. Wait a few
seconds for the readings to stabilize and record the readings into the results table.
7. Carefully increase the pressure in 0.5 MNm-2 up to 3 MNm-2. At each increment, wait
for the readings to stabilize and record the readings into your results table.
8. Try to get as close as possible to 3 MNm-2 so that you can compare your results with
theoretical values (later).
9. Open (turn anticlockwise) the Pressure Control to reduce the indicated pressure back
to 0 MNm-2.
Results Analysis_Closed Ends
(a) Strain Gauge Linearity
On one chart, plot strain (vertical axis) against pressure (horizontal axis) for all six strain
gauges. You should note three things:
The results are linear
The readings for strain gauge 3 (30o to the axis) do not remain at approximately as in
the Open Ends experiment.
The readings for strain gauge 2 (longitudinal strain) are positive.
E Mohr’s Circle and Shear Strain
Use the relevant equation equations from the theory, and calculate the direct Hoop
Stress at each pressure.
Use your actual results for the Principal Strains (Gauges 1, 6 and 2) to construct a
Mohr’s Circle as in the Open Ends experiment. Does this Mohr’s Circle accurately
predict the values of gauges 3, 4 and 5?
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What do you notice about the diameter of this circle compared to the circle from the
Open Ends experiment, and how does this affect the Shear Strain?
F Theoretical Principal Strains from Superposition
Take the given values of Modulus of Elasticity (E= 69 GN/m2) of the cylinder material
and Poisson’s ratio (µ = 0.33). Use them with the Principal Strain (Closed Ends) super
position equations to calculate theoretical principal strains with your calculated direct
Hoop Stress at 3 MNm-2 pressure.
Enter your results into the bottom lines of your results table. Use these values to
construct a theoretical Mohr’s Circle on the same axis as your actual Mohr’s Circle.
Enter your theoretical values into your results table. Are the results similar? If there
are any differences, what do you think are the causes?
Writing a laboratory report
In writing a technical report, you are expected to provide information which is specific and
precise. Please adopt the following points in writing your report:
Type your report (including equations) in the main body of the report.