Apparatus SM1011– Thick Cylinder (TQ Education and Training Ltd)
Thick Cylinder Experiment
Dr P Wardle
p.wardle@staffs.ac.uk
1. Description of the Cylinder
This laboratory exercise enables the student to investigate the distribution of radial and hoop stresses and strains throughout the walls of a thick- walled cylinder under internal pressure and to compare experimental results with the theoretical Lamé predictions. The experiment aims to teach students about:
Stress and Strain distributions in the walls of a thick cylinder under internal pressure.
How to predict the stress and strain in thick cylinder.
The use of strain gauges in mechanical design.
Shear stress in thick cylinders.
The cylinder is made from aluminium alloy in two halves cemented together. One face of the joint has an eccentric shallow groove containing ten strain gauges at carefully determined radii and orientation. These measure radial and hoop strains from which the corresponding stresses are calculated. The groove is completely filled with jointing cement.
Additional strain gauges on the inner and outer walls enable the measurement of longitudinal and circumferential strains. A digital display on the front of the apparatus shows the strains measured at each gauge.
The cylinder is mounted in a sturdy frame and the whole unit complete with a hydraulic hand pump for applying pressure is fitted to a modular steel base. A mechanical Bourdon pressure gauge shows oil pressure in the cylinder and an electronic pressure transducer is fitted to the pressure line to allow connection to TQ Versatile Data Acquisition System VDAS
(Optional).
All strain gauges are temperature compensated forming a full bridge high stability circuit for each channel.
2. General Information
Material – Aluminium Alloy type HE15
Apparatus Nett weight – 30kg
Young’s Modulus (E) - 73GN/m2
Poisson’s Ratio (ν) – 0.33
Max. Test Pressure – 7MN/m2
Strain Gauges – Electrical Resistance
Five Hoop strain
Five Radial strain
Two Circumferential
One Longitudinal
Thick Cylinder Experiment
Dr P Wardle
p.wardle@staffs.ac.uk
3. Experimental Procedure and Data Aquisition
i.)
Allow current to flow through the gauges for 30 minutes at least and with zero gauge pressure obtain a balance reading for each gauge. N.B The more time you allow for the system to stabilise, the more repeatable and accurate your results will be.
ii.)
Record the position and orientation of each strain gauge. See Figure 1 below.
Figure 1. Distribution of strain gauges through the cylinder wall. iii.) Increase the internal pressure by increments of 1MN/m2 up to 7MN/m2 and for each increment take a reading from each of the 13 strain gauges. (Use the data sheet provided). At each increment, wait for the readings to stabilise and record the readings in you results table. WARNING - DO NOT EXCEED A
CYLINDER PRESSURE OF 7MN/m2.
iv.)
Reduce the pressure back to 0.MN/m2 and repeat iii.) to verify your strain gauge data.
Thick Cylinder Experiment
Dr P Wardle
p.wardle@staffs.ac.uk
4. Theory
Recall the elementary Lamé equations for thick cylinders:-
r A
B r2 [1]
H A
B r2 [2]
and
H
1
H r
E
r
1
r H
E
L
E
[3]
[4]
H r
[5]
For a thick walled cylinder of internal radius ri and external radius ro acting under an internal pressure P, the general expressions for radial and hoop stresses may be given by:
[6]
ri 2 ro2
1
P 2 ro ri 2 r 2
[7]
ri 2
r P 2 2 ro ri
H
ro2
1 2
r
From Eqn`s 3 & 4 it can be shown that,
H
E
H r
1 2
[8]
and
r
E
r H
1 2
[9]
Hence Eqn`s 8 & 9 can be used to derive stresses from experimentally measured strains.
Thick Cylinder Experiment
Dr P Wardle
p.wardle@staffs.ac.uk
According to Eqn`s 6 & 7 the variation of the two principal stresses
r
and
H is shown
plotted through the cylinder wall in Figure 2 below.
Figure 2. Variation of r and H through the cylinder wall.
5. Results
i.)
ii.) iii.) iv.)
v.)
vi.) vii.) viii.)
Plot experimental values of ε against radial position for all gauges.
Plot theoretical values of ε against radial position for all gauges.
Compare theoretical and experimental data from i.) & ii.)
Plot experimental values of ε against pressure for all gauges.
For a pressure of 4.5MPa, obtain “faired” values of strain from iv.)
Compare experimental “faired” values of strain with the theoretical predictions. Plot and compare experimental and theoretical stress distributions throughout the cylinder wall.
Plot a Lamé line for each pressure increment. Show both theoretical and experimental results on each graph.
Thick Cylinder Experiment
Dr P Wardle
p.wardle@staffs.ac.uk
6. Discussion and Conclusions
i.)
Do the Lamé equations predict the stress and strain in thick cylinders?
ii.)
Do the results prove the linearity of the strain/pressure response?
iii.)
What do you notice about the measured hoop and radial strains as they get nearer to the cylinder bore?
iv.)
What do you notice about the longitudinal strain at gauge 12?
v.)
If there are any discrepancies between measured and calculated stresses and strains, give the percentage errors and signify at what radius these occur.
vi.)
Does the technique of placing strain gauges throughout a component seem a useful one for three dimensional investigations? Comment on the practical difficulties of making the cylinder in two halves and how this “imperfection” might affect the results.
