61 巻 6 号(2009) SEISAN-KENKYU 1069 117 研 究 速 報 1. Introduction Soil modulus and damping ratio are two fundamental parameters in geotechnical engineering problem involving transmission wave through the soil, such as seismic response of soil deposit to the earthquake loads. Soil modulus exhibits quantitatively the tendency of soil movement associated with settlement or deformation during the load application. Meanwhile, damping as the phenomenon of energy dissipation in a vibrating body of the material, exhibits the behavior that causes the strain to lag behind the applied stress. Many studies have been conducted to characterize those two parameters by means of both in situ and laboratory tests, using both static and dynamic measurements. Static measurement, such as plate loading test and cyclic loading in triaxial test, observes those soil parameters resulted from the obtained stress-strain curve that refers to the overall deformation behaviors of the specimen. On the other hand, in the dynamic measurement such as Bender Element, Trigger Accelerometer, or suspension PS logging tests, the parameters reflect the soil response as the elastic wave propagated through the soil. In the field and laboratory tests, Tatsuoka and Kohata (1995) conducted both static and dynamic measurements exploring in detail soil modulus for hard soils and soft rocks. In the laboratory, Hardin (1965) studied on energy dissipation in the soil during cyclic loading at low frequency. However, limited number of studies has been performed to measure damping ratio based on elastic wave propagation. Continuing study on laboratory dynamic measurements using Bender Element and Trigger Accelerometer (Wicaksono et al., 2008), this study focuses on evaluating damping ratio of Toyoura sand. For this purpose, two different methods including Multiple Arrivals method for dynamic measurement and Cyclic Loading method for static measurement were employed. In soil dynamics there are two types of the damping, i.e. soil (or internal or intrinsic) damping and radiation (or geometric) damping. The former is the energy dissipation within a soil element during vibration, while the latter is transmission of energy away from the initial energy by a mechanism of radiation. However, the term damping in this study refers to soil damping. 2. Material, Apparatus, and Test Procedures Air-dried Toyoura sand was used as the test material. The soil particles were poured by a funnel from a certain height to attain relative density of about 90%. Triaxial test apparatus was used in this study. To evaluate dynamic measurement based on elastic wave propagation, as shown in Figure 1, two independent wave measurement methods were employed, i.e. Trigger Accelerometer (TA) and Bender Element (BE). Triggers and accelerometers were combined to observe the wave propagation through the specimens using P and S waves (AnhDan et al. 2002). On the other hand, only S wave was employed during dynamic measurement with BE method. Overall, tests were conducted in the similar manner as the ones explained by Wicaksono et al., (2007). 3. Damping Ratio Based on Wave Propagation The propagation of small-strain elastic waves is a small perturbation phenomenon that assesses the state of the particulate 研 究 速 報 Determination of Damping Ratio Based on Bender Element, Trigger Accelerometer, and Cyclic Loading Measurements Ruta Ireng WICAKSONO* and Reiko KUWANO** * Graduate student, Department of Civil Engineering, The University of Tokyo ** Associate Professor, Institute of Industrial Science, The University of Tokyo 研 究 速 報 Figure 1. Specimen, Trigger, Accelerometer, and Bender Element
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61 巻 6 号(2009) SEISAN-KENKYU 1069
117
研 究 速 報
1. Introduction
Soil modulus and damping ratio are two fundamental
parameters in geotechnical engineering problem involving
transmission wave through the soil, such as seismic response
of soil deposit to the earthquake loads. Soil modulus exhibits
quantitatively the tendency of soil movement associated with
settlement or deformation during the load application. Meanwhile,
damping as the phenomenon of energy dissipation in a vibrating
body of the material, exhibits the behavior that causes the strain
to lag behind the applied stress.
Many studies have been conducted to characterize those two
parameters by means of both in situ and laboratory tests, using
both static and dynamic measurements. Static measurement, such
as plate loading test and cyclic loading in triaxial test, observes
those soil parameters resulted from the obtained stress-strain
curve that refers to the overall deformation behaviors of the
specimen. On the other hand, in the dynamic measurement such
as Bender Element, Trigger Accelerometer, or suspension PS
logging tests, the parameters refl ect the soil response as the elastic
wave propagated through the soil. In the fi eld and laboratory tests,
Tatsuoka and Kohata (1995) conducted both static and dynamic
measurements exploring in detail soil modulus for hard soils and
soft rocks. In the laboratory, Hardin (1965) studied on energy
dissipation in the soil during cyclic loading at low frequency.
However, limited number of studies has been performed to
measure damping ratio based on elastic wave propagation.
Continuing study on laboratory dynamic measurements using
Bender Element and Trigger Accelerometer (Wicaksono et al.,
2008), this study focuses on evaluating damping ratio of Toyoura
sand. For this purpose, two different methods including Multiple
Arrivals method for dynamic measurement and Cyclic Loading
method for static measurement were employed.
In soil dynamics there are two types of the damping, i.e. soil
(or internal or intrinsic) damping and radiation (or geometric)
damping. The former is the energy dissipation within a soil
element during vibration, while the latter is transmission of
energy away from the initial energy by a mechanism of radiation.
However, the term damping in this study refers to soil damping.
2. Material, Apparatus, and Test Procedures
Air-dried Toyoura sand was used as the test material. The soil
particles were poured by a funnel from a certain height to attain
relative density of about 90%.
Triaxial test apparatus was used in this study. To evaluate
dynamic measurement based on elastic wave propagation, as
shown in Figure 1, two independent wave measurement methods
were employed, i.e. Trigger Accelerometer (TA) and Bender
Element (BE). Triggers and accelerometers were combined to
observe the wave propagation through the specimens using P and
S waves (AnhDan et al. 2002). On the other hand, only S wave
was employed during dynamic measurement with BE method.
Overall, tests were conducted in the similar manner as the ones
explained by Wicaksono et al., (2007).
3. Damping Ratio Based on Wave Propagation
The propagation of small-strain elastic waves is a small
perturbation phenomenon that assesses the state of the particulate
研 究 速 報
Determination of Damping Ratio Based on Bender Element, Trigger
Accelerometer, and Cyclic Loading Measurements
Ruta Ireng WICAKSONO* and Reiko KUWANO**
* Graduate student, Department of Civil Engineering, The University of Tokyo
** Associate Professor, Institute of Industrial Science, The University of Tokyo
研 究 速 報
Figure 1. Specimen, Trigger, Accelerometer, and Bender Element
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medium without altering the fabric or causing permanent effects.
In term of media in the context of wave propagation,
several models have been suggested to represent geomaterial
(Kjartansson, 1979), such as viscoelastic medium (Kelvin-Voigt
model), constant or nearly-constant energy loss per cycle, and
frictional models. As the concept of equivalent viscous damping
fi rstly proposed by Jacobsen (1930), this study adopted analytical
solution employing the single-degree-of-freedom system of
Kelvin-Voigt (KV) model having a mass (m) that corresponds to a
purely elastic spring with spring stiffness (k) and a purely viscous
dashpot with damping coeffi cient (c) in parallel combination.
Additionally, Richart et al. (1970) described in detail the formula
of this system according to the model as shown in Figure 2.
Free vibration system as shown in Figure 2 are described
by differential equation of motion employing Newton’s second
law and measuring displacement (z) from the rest position as in
Equation (1).
…………………………………… (1)
By letting z = e (βt), the equation of
…………………………………… (2a)
will have the following solution for β, as follows:
………………………… (2b)
By considering Equation (2b), there are three possible cases
depending upon whether the roots are real (for over damped case),
complex (for under damped case), or equal (for critically damped
case). However, the case related to this study is that under damped
which has the fi nal equation, as follows:
…………………………………… (2c)
where
……………… (2d)
……………………………………… (2e)
…………………………………………… (2f)
…………………………………………… (2g)
………………………… (2h)
where ωd is damped natural angular frequency, ωn is undamped
natural angular frequency, and D is damping ratio. Additionally,
z = z0 and dz/dt = v0 at t = 0.
Multiple Arrivals Method was developed to determine the
material damping ratio by BE method considering multiple
arrivals of a signal caused by refl ections on the pedestal and
top caps. Those refl ections are observed in the time histories as
repetitions of the fi rst arrival with decreased amplitude and shifted
time. As shown in Figure 3b, Arrival I and Arrival II describe the
fi rst and refl ected arrivals, respectively.
A time history captured by receiver BE is decomposed into
different parts bi(t), where the script i denotes the order of the
wave arrival. A rectangular window having length of LW (Figure
3b) is used to separate the arrivals, with the center corresponding
to the maximum response of the arrival under consideration. The
time histories bi(t) are transformed to the frequency domain. The
spectral amplitude are denoted as , where is
the Fourier transform of bi(t). The attenuation coeffi cient (αS)
is defi ned as the natural logarithm of the spectral ratio of two
amplitudes , divided by the travel path length 2 (j ‒ i) L
Figure 2. Kelvin-Voigt model Figure 3. S wave time history by Bender Element method
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of the wave, as shown in Equation (3) as follows:
…………………………………… (3)
where 2 (j ‒ i) is the number of refl ection between the i-th and
the j-th arrivals, while L is the distance between two corresponding
sensors.
Furthermore, through a point of the mean dominant frequency
of the fi rst and the second (refl ected) arrivals, a line with slope
S is fi tted to the curve of the attenuation coeffi cient αS within a
bandwidth of frequency as shown in Figure 4. By considering the
values of shear wave velocity (VS), the damping ratio is evaluated
using Equation (4), as follows:
……………………………………………… (4)
In this study, beside the waves captured by BE method, those by
TA method were also applied using the Multiple Arrivals method.
In TA method cases the waves captured by accelerometers 1 and
2 consecutively were considered as multiple arrivals without any
refl ection. As shown in Figures 5 and 6, in each TA method the
peak point of the fi rst half-wave captured by Accelerometer 1 was
considered as the center of Arrival I, while those by Accelerometer
2 was that of Arrival II. Consequently, the term of 2 (j ‒ i) L of
Equation (3) is simply replaced with that of L.
4. Damping Ratio Based on Cyclic Loading
By performing cyclic-loading as static measurement, hysteretic
loop is exhibited in the plot of stress-strain curve, as shown in
Figure 7. The area enclosed by the ellipse (ΔW) is related to the
amount of energy (per unit volume) dissipated by the material
during one cycle of loading at a certain circular frequency (ω).
From a thermodynamic point of view, ΔW is equal to the amount
of entropy produced in one cycle of harmonic loading and due to
unrecoverable mechanical work (Lai and Rix, 1998). Meanwhile,
the triangle area (W) is related to the maximum strain energy
stored during that cycle.
In this system, by considering Equations (1) and (2c) with
α = 0, the rate of dissipating energy with time for one cycle can
be solved as follows:
…………………… (5a)
Figure 4. Curve of attenuation coeffi cient in the Multiple Arrivals method
Figure 5. S wave time history by TA with S wave method
Figure 6. P wave time history by TA with P wave method
Figure 7. Damping ratio obtained from cyclic loading
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……………………………………… (5b)
The negative sign in equation above denotes the energy
dissipated with time.
The total energy of the system (W) to mobilize one cyclic
loading can be expressed either as the maximum potential energy
or as the maximum kinetic energy, as follows:
………………………… (6)
Thus, the ratio of equations (5b) and (6) yield in damping ratio
as follows:
……………… (7a)
…………………………………………… (7b)
5. Test Results and Discussions
5.1 Damping Ratios in Different Frequencies
Figures 8 to 10 show graphs plotting damping ratio with
frequency of the excitation for TA with S wave, P wave, and BE
methods, respectively. Additionally, in the case of Multiple Arrival
method, in order to neglect the effects of performance due to
fabrication inherent between 2 accelerometers, sensitivity (mV/g)
of each accelerometer was considered to measure gravitational
amplitude of signal, instead of voltage amplitude.
To observe the effects of the length of rectangular window (LW)
in Multiple Arrivals method (Figure 3b), two different number of
length in time domain were evaluated, i.e. 500 μs and 1000 μs,
which the results were plotted in the graphs as Multiple Arrivals1
and Multiple Arrivals2, respectively.
As shown in Figures 8 to 10, the damping ratio value was
relatively sensitive with the frequency. However, the values of
damping ratio having the smallest deviation among those methods
were observed when those values were evaluated at dominant
frequency, especially for TA-S wave (2 kHz) and BE methods
(7.8 kHz).
The facts occurred due to possibly that signal excited in
dominant frequency yielded in more stable wave form. The more
stable wave form eventually results in relatively similar values of
damping ratio regardless the length values of rectangular window
(LW). Hence, in each method in dynamic measurements, dominant
frequency was employed in evaluation for further analyses and
evaluation.
5.2 Damping Ratios under Isotropic Stress States
Figures 11 to 14 that correspond to the data obtained from TA-S
wave, TA-P wave, BE, and cyclic loading methods respectively,
show graphs plotting damping ratio versus isotropic effective
confi ning stress at 50, 100, 200, and 400 kPa.
As shown in graphs for TA-S wave, TA-P wave, and BE
methods, by increasing the isotropic stresses the values of
damping ratio were plotted in relatively small scattered values
having standard deviation of 4.0%, 1.0%, and 2.5%, respectively.
Figure 8. Damping ratio values with different frequency evaluated from TA-S wave method
Figure 9. Damping ratio values with different frequency evaluated from TA-P wave method
Figure 10. Damping ratio values with different frequency evaluated from BE wave method
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This indicates that each method yields in damping ratio value that
is insensitive to the isotropic confi ning stress.
Furthermore, the values of damping ratio obtained from static
measurement at strain level of 0.001% were in the range of
0.3 %–2% as shown in Figure 14, which were about 10% lower
than the highest of those from dynamic measurements.
5.3 Damping Ratios Obtained from Static & Dynamic
Measurements
Figure 15 shows degradation curve that plot the values of
Young’s modulus, damping ratio, and strain level in a graph. The
values of Young’s modulus and damping ratio obtained from TA-S
wave and BE methods were plotted at the strain level of 10–4% as
experimentally estimated by Wicaksono et al. (2008).
As suggested by JGS (2000), cyclic loading as static
measurement with 11 cycles was performed. The Young’s modulus
and damping ratio values of the 5th and 10th cycle were evaluated
as shown in Figure 15. Furthermore, shear modulus value of
dynamic measurement obtained in this study was converted into
the Young’s modulus value, and then was plotted in the same
graph.
Similarly to that reported in Wicaksono et al. (2009), stiffness
modulus value obtained from static measurement was smaller
Figure 11. Damping ratio values with different isotropic stress evaluated from TA-S wave method
Figure 12. Damping ratio values with different isotropic stress evaluated from TA-P wave method
Figure 13. Damping ratio values with different isotropic stress evaluated from BE method
Figure 14. Damping ratio values with different isotropic stress evaluated from Cyclic Loading method
Figure 15. Comparison of the values of damping ratio obtained from static and dynamic measurements
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than that from dynamic measurement. Additionally, damping ratio
values obtained from dynamic measurement was about 5% higher
than those obtained from static measurement.
In accordance to these facts, relevant study by Dobry and
Vucetic (1987), Stoll (1979, 1985), and Rix and Meng (2005)
mention that the higher material damping ratio revealed by
dynamic measurement is believed to be related to the higher
frequency range during testing.
6. Summary
1. Wave form of the signal captured by the sensor plays important
role in evaluating damping ratio based on elastic wave. In each
case of TA-S wave and BE methods, the wave form obtained
from excitation that was at dominant frequency yielded in
relatively similar value of damping ratio while employing the
rectangular window length of 500 μs and 1000 μs.
2. At very small strain level, the values of damping ratio obtained
from static measurement were in the range of 0.3%–2%,
which were about 10% lower than the highest of those from
dynamic measurements.
(Manuscript received. November 10, 2009)
References
1) AnhDan, L.Q., Koseki, J. and Sato, T. (2002). “Comparison of
Young’s Moduli of Dense Sand and Gravel Measured by Dynamic
and Static Methods,” Geotechnical Testing Journal, ASTM, Vol. 25
(4), pp. 349–368.
2) Hardin, B. (1965). “The Nature of Damping in Sands, “J. Soil
Mechanics and Foundation, Div. 911, 63–97.
3) Jacobsen, L. S., (1930), “Steady Forced Vibration as Infl uenced by
Damping,” ASME 52, Part I, 169–181.
4) JGS 0542 (2000), “Method for Cyclic Triaxial Test to Determine
Deformation Properties of Geomaterials.
5) Kjartansson, E. (1979). “Constant Q-wave Propagation and
Attenuation,” Journal of Geophysical Research, 84, 4737–4748.
6) Lai, C. G. and Rix, G. J., (1998), “Simultaneous Inversion of
Rayleigh Phase Velocity and Attenuation for Near-surface Site
Characterization,” GIT-CEE/GEO-98-2, Georgia Institute of
technology, School of Civil and Environmental Engineering.
7) Richart Jr., F. E., Hall Jr., J. R., and Woods, R. D. (1970),
“Vibrations of Soils and Foundations,” Prentice Hall Inc.,