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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.

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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.,

Englewood Cliffs, New Jersey.

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Hime Gravel,” Bulletin of Earthquake Resistant Structure Research

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Stiffness of Toyoura Sand Obtained from Various Measurement

Techniques,” Bulletin Earthquake Resistant Structure, The

University of Tokyo, 41.

10) Tatsuoka, F. and Kohata, Y. (1995), “Stiffness of hard soils and soft

rocks in engineering applications,” Keynote Lecture, Proc. of Int.

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