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MMAN3400 Mechanics of Solids
Creation date：2024-05-23 16:00:09
Belonging category：工程力学engineering mechanics
Mechanics of Solids

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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.
3
 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
4
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
5
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
6
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.
7
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?
8
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
9
 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?
10
 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.
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COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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