7 liquefaction
TRANSCRIPT
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Final Report
Vibration Induced Liquefaction of a Granular Material
Group #7 2 May 2007
ME 241
Gerald AbtGlenn DillerHeather Howard
Abstract
We investigated properties of a liquefied granular material subjected to horizontal
shaking. The density of the glass beads we used was = 2523 kg/m3; upon liquefaction, we
found an average density of = 2021 kg/m3. We found that viscosity of the liquefied beads
varies linearly with angular frequency, with the relation (f) = -349.7f + 6722.76 where
frequency f is in rad/s and viscosity is in kg/ms
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Introduction
Liquefaction is the process by which a granular material transforms from a solid state to a
liquid state. Because of its connection with seismic activity, the liquefaction of soils has long
been an area of interest. Many of the catastrophic effects of earthquakes are related to the
behavior of liquefied sand. Several reports have been published describing the onset of
liquefaction; the causes of the phenomenon are generally understood. However, to our
knowledge very little research has been done on the properties of fluidized soil. Our goal was to
determine the effective density of liquefied sand when exposed to various frequencies of
vibration, using a system of small glass beads subjected to horizontal shaking as a model for
earthquake-induced vibrations.
Last year, a group of students in ME 241 investigated the conditions required for the
onset of liquefaction [1]. They used a dimensionless parameter, , to measure this critical point.
is the dimensionless acceleration and is defined by
g
2= (1)
where A is the amplitude, is the angular frequency, and g is the gravitational acceleration.
Their determination of the critical value for liquefaction corroborated the work done by
Metcalfe et al [2]. We built on the work of these groups to test the properties of a liquefied
medium.
Formulation & Procedure
We had originally planned to use a peanut can filled partially with lead shot to determine
the density of the liquefied beads. However, initial experimentation showed that when the
shaker table was turned on the can would sink into the beads at an angle. This would have made
calculations of the submerged volume extremely difficult so we decided to change the object
used to a ping-pong ball (Figure 1). This allowed us to determine the volume of the submerged
portion of the ball, no matter how it sank into the beads, using the equation for the volume of a
spherical cap
=
3
2 hrhVsubmerged (2)
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where Vsubmergedis the submerged volume, h is the height of the submerged portion of the ball,
and ris the balls radius. We filled the ping-pong ball with lead shot to vary the effective density
of the ball.
After we found the volume of the ball that was submerged under the surface, we
determined the density of the liquefied beads using Archimedes Principle (3),
SubmergedB gVF = (3)
whereFb is the buoyant force, is the effective density of the fluid,gis gravitational
acceleration and VSubmergedis the volume submerged.
Our shaker table was initially set up to simply run from an input voltage. To allow us to
directly control frequency from LabVIEW, we determined the voltage-frequency relationship
using a stopwatch to time ten rotations at each voltage (Figure 2). This allowed us to modify
our LabView virtual instrument for frequency input, rather than voltage input (Figures 3, 4). We
then calibrated a Linear Variable Displacement Transducer (LVDT) to determine the relationship
between LVDT voltage and the displacement of the shaker tables moving platform (Figure 5).
The LVDT output a plot of displacement versus time to the computer, and enabled us to
determine the amplitude of shaking. Using frequency and the amplitude calculated with the
LVDT, we were able to quantitatively determine when we reached cr.
To measure viscosity, we attached the LVDT vertically to the fish tank using C Clamps
(Figure 6). The LVDT was then screwed into a nut that had been glued to a ping-pong ball. We
modified our LabView setup to output displacement and time as the ball sank, allowing us to
obtain data for the rate at which the ball sank into the beads. Initially we performed the test by
setting the ball on top of the sand and using the flattest portion of the displacement curve to
calculate velocity. We then used Stokes Law (4) for a sphere sinking in an infinite fluid [3]
gagVVa d )3
4(6 3 =+ (4)
where is the viscosity, is the density of the material in which the object is sinking, Vis the
terminal velocity of the sphere, a is the radius of the sphere, andgis the acceleration of gravity.
However, the viscosities we obtained from this method were extremely high, causing us to
question the accuracy of our results.
We then tried the same test on a substance for which we knew the viscosity, 10,000
centistokes silicone oil, to estimate the error involved in our procedure. Unfortunately, the tests
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with the oil failed to give us the expected viscosity. To improve our method, we modified our
starting position so that the entire ping-pong ball began fully submerged in the liquid. This was
done to better simulate an infinite fluid, bringing the conditions of the test closer to those that can
be modeled by the Stokes flow equation. This test produced a result of 4227.31 centistokes for
the oil. After comparing the 4227.31 cS found to the correct 10,000 cS, our tests appeared to be
off by a factor of 2.37. We then tested the glass beads with an initially submerged ball and found
the viscosities of the beads at different frequencies. Then we adjusted these values with the error
factor of 2.37 to predict the actual viscosity of the beads.
Results
Using the angular frequency of the table and the amplitude of shaking, we were able to
calculate the critical value of our system. Based on a qualitative observation of when the top
layer of beads appeared to liquefy, we found crto be 0.57. This compares favorably with the
value of cr= 0.52 found by Metcalfe et al [2], as liquefaction could possibly have occurred at a
lower frequency then we observed, which would have brought our value closer to 0.52.
To evaluate whether our values for density of the liquefied beads made sense, we first
found the density of the stationary beads. We used an optical microscope to measure the
diameters of ten individual beads and found an average diameter of 1.133 mm. We weighed
beads individually and used our data to calculate a density of = 2523 kg/m3
for the beads
themselves.
With the density of the stationary beads determined, we began to measure the density of
the liquefied beads. Using balls of mass 43.43 g (Ball G) and 19.36 g (Ball B), we found the
densities to be = 2103 kg/m3
and = 1950 kg/m3, respectively (Figures 7, 8). This resulted in
an average density of = 2021 kg/m3. The standard deviation was 165 kg/m
3, which is small in
comparison to the average value (about 8%).
We also showed that a low-density object experienced upward motion when placed
beneath the surface of the beads. To demonstrate this, we placed an empty ping-pong ball on the
bottom of the tank, underneath the beads. When the beads were fully liquefied, the ball rose
quickly to the top and remained on the surface.
Our final viscosities ranged from 465.95 to 1517.40, varying linearly with frequencies
ranging from 15 Hz to 18 Hz. We used Microsoft Excels trendline option to find a relationship
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(f) = -349.7f + 6722.76 (Figure 9) between the angular frequency f and the viscosity . The
RMS error value was approximately 56.96.
Conclusions/Discussion
The average density of the liquefied beads we measured is significantly less than that of
the stationary beads. This makes sense as, with the notable exception of water, the density of
materials in their liquid state is less than the density of the solid state. As shown in Figures 7
and 8, there was a significant variation in the measured density of the liquefied beads across the
range of frequencies tested. This was mostly due to the human error in our measurement method
of using a ruler to determine how much of the ball was initially and finally submerged in the
beads. The smallest scale on the ruler was in millimeters, and even a small error in the
measurement would have had a significant effect on our calculations when the ball itself was
only 38 mm in diameter. However, we discovered that when we input a frequency into the
control and turned the table on, the initial sudden movement caused the ball to sink in further
than if a gradual frequency increase was used. To avoid this error, we modified our procedure
and used a slow increase in the frequency to obtain our data.
At high frequencies the rapid movement of the tank caused waves to form in the beads.
This appeared to be amplified by the small size (two gallons) of our tank. The high vibration
frequencies also caused the ball to travel towards the ends of the tanks where it encounters the
most severe wave action. As a result, the ball may have sunk deeper into the beads than if it had
not been affected by waves. To test whether the waves had a significant impact on the ball we
submitted a request for more beads to Flex-o-Lite, the manufacturer. This would have allowed
us to utilize a larger tank where the ball could have been placed further from the tank edges and
presumably be less affected by the wave action. Unfortunately, despite the company
representatives assurance that the new beads were on their way, the package did not arrive in
time for us to perform this additional testing.
We encountered some confusion while evaluating the density of our liquefied material.
The density that we found for the liquefied beads was 2092 kg/m3, compared to 2523 kg/m
3for
the material of just the beads themselves. Initially this seemed impossible because the liquefied
density of 2092 kg/m3
is greater than the 1867 kg/m3
obtained from Gauss 74% maximum close
packing factor for spheres. When we examined the beads under an optical microscope, we found
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that the diameter of the beads was not uniform at all. We took measurements of 10 individual
beads and found an average diameter of 1.133 mm (Figure 10). This is only a rough average,
however, as there were all sorts of imperfections in our beads. Diameters in our sample of beads
ranged from 1.05 mm to 1.26 mm, and in addition there were some beads attached to other
beads. Since the beads were non-uniform, we hypothesized that the 74% maximum close
packing factor does not apply. We also discussed the matter with Professor Lambropoulos, and
he agreed with our assessment that with the uneven bead size, the liquefied density could be
greater than 74% of the density of the glass material [4]. The Gauss close-packing model only
applies to uniform beads, and in the non-uniform case the smaller beads can fit interstitially
between larger ones, packing closer and increasing the density.
Our viscosity calculation was based on a scaling factor that we assumed to be constant.
However, we did not prove that this scaling factor is accurate. The most appropriate way to
confirm our error factor of 2.37 would be to test another control fluid with a high, known
viscosity. If the viscosity calculated for this fluid could be scaled to the correct value using our
error factor, our result would be much more convincing. A fluid that is more viscous than
10,000 cS would be useful, as a slower sinking rate could allow us to take more data and confirm
the validity of our experimentation. The error factor may be a result from a problem with our
Stokes flow assumption. The Stokes model assumes an infinite fluid, but we had only a limited
supply of beads and control fluid. More beads would have made Stokes flow a better
approximation. Also, as mentioned before, a wave effect began at high frequencies, disrupting
the falling motion and giving inaccurate velocity values. Even with these inaccuracies, the RMS
error value of 56.96 is small compared to our viscosity values (at most 7%).
Also, Stokes flow only applies for low Reynolds numbers. We initially assumed low
Reynolds number and after calculating the viscosity with Stokes law, we solved for the Re for
the system. We found Re to be less than 0.001 using our experimentally determined viscosity
and density, confirming that we satisfied that condition for Stokes law.
References
[1] Dhillon, Tejpal. Greenwald, Jesse. Shay, Tom. Granular Flow as a Model for Earthquake
Induced Liquefaction of Soils, May 2006.
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[2] G. Metcalfe, S.G.K. Tennakoon, L. Kondic, D.G. Schaeffer, and R.P. Behringer,
Granular friction, coulomb failure, and the fluid-solid transition for horizontally shaken
granular materials, inPhysical Review E, vol. 65, pp 1-15, February 2002.
[3]Batchelor, G. K. An Introduction to Fluid Dynamics. Chapter 6. Cambridge University
Press (1967).
[4] Lambropoulos, J. C. Personal Communication.
Figures
Figure 1. Shaker Table.
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Voltage vs. Frequency
y = 53.872x + 0.1162
R2 = 0.9999
0
1
2
3
4
5
6
0 0.02 0.04 0.06 0.08 0.1 0.12
Shaker Table Voltage (V)
Frequency(Hz)
Figure 2. Frequency Calibration.
Figure 3. Block Diagram.
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Displacement vs. LVDT Voltage
y = 2.5732x - 7.9116
R2 = 1
0
2
4
6
8
10
12
0 1 2 3 4 5 6 7 8
LVDT voltage (V)
Displacement(in)
Figure 4. Front Panel.
Figure 5. LVDT Calibration.
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Ball G
1500
1700
1900
2100
2300
2500
11 12 13 14 15 16 17 18
Angular Frequency (rad/s)
Density(kg/m3)
Figure 6. Viscosity Measuring Configuration.
Figure 7. Liquefied bead density measured with Ball G of mass 43.43 g.
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Ball B
1500
1700
1900
2100
2300
2500
10 11 12 13 14 15 16 17 18
Angular Frequency (rad/s)
Density(kg/m
3)
Viscosity vs. Frequencyy = -3.49699E+02x + 6.72276E+03
R2 = 9.74152E-01
400
600
800
1000
1200
1400
1600
14.5 15 15.5 16 16.5 17 17.5 18 18.5
Frequency
Viscosity(k
g/m*s)
Figure 8. Liquefied bead density measured with Ball B of mass 19.364 g.
Figure 9. Predicted Viscosity of Liquefied Beads vs. Frequency.
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Figure 10. Beads under an optical microscope.