theoretical and experimental investigation of effective
TRANSCRIPT
The Pennsylvania State University
The Graduate School
Department of Civil and Environmental Engineering
THEORETICAL AND EXPERIMENTAL INVESTIGATION OF EFFECTIVE
DENSITY AND PORE FLUID INDUCED DAMPING IN SATURATED
GRANULAR MATERIALS
A Dissertation in
Civil Engineering
by
Yanbo Huang
2014 Yanbo Huang
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
May 2014
ii
The dissertation of Yanbo Huang was reviewed and approved* by the following:
Tong Qiu Assistant Professor of Civil Engineering Dissertation Advisor Chair of Committee
Chun Liu Professor of Mathematics
Farshad Rajabipour Assistant Professor of Civil Engineering
Mansour Solaimanian Senior Research Associate of Geotechnical and Materials Engineering
Peggy A. Johnson Professor of Civil Engineering Head of the Department of Civil and Environmental Engineering
*Signatures are on file in the Graduate School
iii
ABSTRACT
In current geotechnical engineering research and practice, two assumptions are
generally made regarding the dynamics of saturated soil. The first is that pore fluid
induced damping during shear wave excitations is negligible. The second is that
saturated density can be used to calculate shear modulus based on measured shear wave
velocity. The validity of these assumptions depends on the magnitude of fluid motion
relative to solids during shear wave excitations. For soils with low permeability (e.g.,
silts and clays) and under low-frequency excitations (e.g., seismic waves), these
assumptions are generally valid. However, relative fluid motion may be important for
soils with high permeability (e.g., sands and gravels) and under high-frequency
excitations, rendering the above mentioned assumptions questionable.
This study presents an experimental investigation of the concept of effective density
for propagation of small strain shear waves through saturated granular materials. Bender
element tests and resonant column tests were conducted on various granular materials in
dry and saturated conditions. Values of small-strain shear modulus measured for the dry
condition are compared to corresponding values measured for the saturated condition
using saturated density and effective density. Analysis of test results indicates that
effective density instead of saturated density should be used to calculate small-strain
shear modulus. For bender element tests, the use of saturated density produced errors in
shear modulus as high as 28%; whereas the use of effective density resulted in errors
generally less than 5%. For resonant column tests, errors in shear modulus obtained
using saturated density were smaller than those for bender element tests due to the lower
range of excitation frequency.
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This study presents two analytical solutions for Biot flow induced damping in
saturated soil specimens in resonant column tests based on the half-power bandwidth and
free vibration decay methods. These solutions are compared with a closed-form
analytical solution readily available in literature. The solutions indicate that Biot flow
induced damping may provide an important contribution to total soil damping in coarse
sand and gravel, but can be practically neglected for less permeable soils (e.g., fine sand,
silt, and clay). The solutions also indicate that Biot flow induced damping increases as
porosity increases and decreases considerably as the ratio of the mass polar moment of
inertia of the loading system to the specimen increases. It is concluded that Biot flow
induced damping is suppressed by the boundary condition of typical resonant column
apparatuses and is hence difficult to be measured. The solution from the free vibration
decay method is compared to RC test results of various granular materials at dry and
saturated conditions. The comparison suggests that the validity of this analytical solution
is inconclusive, which is largely due to the very small magnitude of Biot flow induced
damping in RC tests.
In addition, a theoretical investigation of energy dissipation in a nearly saturated
poroviscoelastic soil column under quasi-static compressional excitations, which is
applicable to slow phenomena (e.g., consolidation), is also presented in this study.
Different components of the energy dissipation are evaluated and compared. This
investigation indicates that the magnitude of pore fluid induced energy dissipation is
primarily a function of a normalized excitation frequency . For small values of , a
drained soil column is fully relaxed and behaves essentially as a dry column with
negligible pore pressure. In this case, fluid induced energy dissipation is negligible and
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the total damping ratio of the column is essentially the same as that of the solid skeleton.
For very high values of , a drained soil column is fully loaded and the excitation-
generated pore pressure decreases as the fluid becomes more compressible. In this case,
the fluid pressure gradient only exists in a thin boundary layer near the drainage
boundary, where drainage occurs and fluid induces energy dissipation; whereas the rest
of the column is essentially undrained. Significant fluid induced energy dissipation
occurs for moderate values of due to a combination of moderate fluid pressure,
pressure gradient and fluid relative motion throughout the soil column. The effects of
boundary drainage condition, saturation, porosity, and skeleton damping ratio on fluid
induced energy dissipation are discussed.
vi
TABLE OF CONTENTS
List of Figures ............................................................................................................. viii
List of Tables ............................................................................................................... xi
Acknowledgements ...................................................................................................... xii
Chapter 1 Introduction ................................................................................................. 1
1.1 Background ..................................................................................................... 1 1.2 Objectives of Research ................................................................................... 4 1.3 Organization of Dissertation ........................................................................... 5 References ............................................................................................................. 6
Chapter 2 Effective Soil Density for Small Strain Shear Waves in Saturated Granular Materials ................................................................................................ 7
2.1 Introduction ..................................................................................................... 7 2.2 Experimental Program and Results ................................................................ 10
2.2.1 Granular Materials ................................................................................ 10 2.2.2 Bender Element Tests ........................................................................... 11 2.2.3 Resonant Column Tests ........................................................................ 25 2.2.4 Quick Chart .......................................................................................... 32
2.3 Discussion ....................................................................................................... 35 2.4 Conclusions ..................................................................................................... 39 Notations ............................................................................................................... 41 References ............................................................................................................. 42
Chapter 3 Biot Flow Induced Damping in Saturated Poroviscoelastic Soil Specimens in Resonant Column Test ................................................................... 46
3.1 Introduction ..................................................................................................... 46 3.2 Governing Equations ...................................................................................... 49 3.3 Analytical Solutions for Biot Flow Induced Damping ................................... 52
3.3.1 Spectral Response ................................................................................. 53 3.3.2 Damping from Half-Power Bandwidth Method ................................... 57 3.3.3 Damping from Free Vibration Decay Method ..................................... 59
3.4 Resonant Column Test .................................................................................... 63 3.5 Results and Discussion ................................................................................... 64 3.6 Conclusions ..................................................................................................... 70 Notations ............................................................................................................... 71 References ............................................................................................................. 75
vii
Chapter 4 Energy Dissipation in Nearly Saturated Poroviscoelastic Soil Column during Quasi-Static Compressional Excitations ................................................... 80
4.1 Introduction ..................................................................................................... 80 4.2 Governing Equations ...................................................................................... 83 4.3 Analytical Solutions of Steady-State Motion ................................................. 86
4.3.1 General Solution ................................................................................... 86 4.3.2 No-Drainage (ND) Case ....................................................................... 89 4.3.3 Top-Drained (TD) Case ........................................................................ 90 4.3.4 Double-Drained (DD) Case .................................................................. 91 4.3.5 Normalization ....................................................................................... 93
4.4 Energy Dissipation and Damping ................................................................... 94 4.5 Results and Discussions .................................................................................. 100 4.6 Illustrative Example ........................................................................................ 114 4.7 Conclusions ..................................................................................................... 116 Notations ............................................................................................................... 117 References ............................................................................................................. 120
Chapter 5 Conclusions and Recommendations ............................................................ 124
5.1 Conclusions ..................................................................................................... 124 5.2 Recommendations for Future Work ............................................................... 127
Appendix Data from Resonant Column Tests ............................................................ 129
Group 1 ................................................................................................................. 129 Group 2 ................................................................................................................. 140 Group 3 ................................................................................................................. 161
viii
LIST OF FIGURES
Fig. 1-1. Seismic wave propagation in homogeneous soil layer ................................ 2
Fig. 2-1. Particle size distributions for granular materials .......................................... 11
Fig. 2-2. Volumetric strains for saturated granular materials during two loading cycles .................................................................................................................... 14
Fig. 2-3. Small-strain shear modulus for dry specimens: (a) glass beads (6 mm); (b) coarse sand; and (c) fine to medium sand ....................................................... 15
Fig. 2-4. Theoretical values for effective density ratio versus normalized frequency for BE tests on three granular materials .............................................. 17
Fig. 2-5. Comparison of maxG for dry and saturated conditions in BE tests: (a)
glass beads (6 mm); (b) coarse sand; and (c) fine to medium sand ...................... 21
Fig. 2-6. Relative errors of maxG for BE tests on saturated granular materials .......... 21
Fig. 2-7. Comparison of maxG from TS and BE tests on saturated specimens: (a)
silica sand; and (b) Toyoura sand (measured data from Youn et al. 2008) .......... 24
Fig. 2-8. Relative errors of maxG for test data from Youn et al. (2008) ..................... 25
Fig. 2-9. Comparison of maxG for dry and saturated conditions in RC tests: (a)
glass beads (0.4 – 0.6 mm); (b) glass beads (2.9 – 3.5 mm); (c) glass beads (6 mm); and (d) ASTM 20-30 sand .......................................................................... 30
Fig. 2-10. Relative errors of maxG for RC tests on saturated granular materials ........ 30
Fig. 2-11. Influence of effective density ratio on sateff GG max,max, for RC and BE
tests ....................................................................................................................... 32
Fig. 2-12. Test data and chart for quick determination of importance of effective soil density (data for silica sand and Toyoura sand from Youn et al. 2008). Values of 10D are approximate. ............................................................................ 34
Fig. 2-13. Effect of assumptions for mass coupling and non-Poiseuille flow on theoretical effective density ratio: (a) glass beads (6 mm); (b) coarse sand; and (c) fine to medium sand ................................................................................. 39
Fig. 3-1. Boundary conditions of a poroviscoelastic soil specimen in resonant column test ............................................................................................................ 53
ix
Fig. 3-2. Spectral response of A for three specimens with different values (
4.0n 02.0s 0IIt ) ................................................................................. 57
Fig. 3-3. Illustration of half-power bandwidth method for damping determination .. 58
Fig. 3-4. (26) LHS.Eq versus b for three specimens with different values
( 4.0n , 02.0s , 0IIt ) ............................................................................. 62
Fig. 3-5. Comparison of analytical solutions of f based on different methods
( 4.0n , 02.0s , 0IIt ) ............................................................................. 65
Fig. 3-6. f versus for various n ( 02.0s , 0IIt ) ...................................... 66
Fig. 3-7. f versus for various IIt ( 02.0s , 4.0n ) ................................... 67
Fig. 3-8. Comparison of f between solution from FVD and RC test results: (a)
glass beads (0.4 – 0.6 mm); (b) glass beads (1.7 – 2.1 mm); (c) glass beads (3.8 – 4.4 mm); and (d) ASTM 20-30 sand .......................................................... 70
Fig. 4-1. Geometry of poroviscoelastic soil column................................................... 86
Fig. 4-2. G vs. for a ND soil column ( 4.0n , 02.0s ) ............................... 101
Fig. 4-3. G vs. for a TD soil column ( 4.0n , 02.0s ) ................................ 102
Fig. 4-4. Variation of G vs. and for a TD soil column ( 4.0n ,
02.0s ) ............................................................................................................. 102
Fig. 4-5. G vs. for a DD soil column ( 4.0n , 02.0s ) ............................... 103
Fig. 4-6. oP max vs. for a TD soil column ( 4.0n , 02.0s ) ....................... 104
Fig. 4-7. Distribution of fluid pressure and effective stress along a TD soil column for various ( 1.0 , 4.0n , 02.0s ) ......................................... 105
Fig. 4-8. sf EE vs. for a TD soil column ( 4.0n , 02.0s ) ........................ 106
Fig. 4-9. *sDW vs. for a TD soil column ( 4.0n , 02.0s ) ............................. 108
Fig. 4-10. *fDW vs. for a TD soil column ( 4.0n , 02.0s ) .......................... 108
x
Fig. 4-11. vs. for a ND soil column ( 4.0n , 02.0s ) ............................... 110
Fig. 4-12. vs. for a TD soil column ( 4.0n , 02.0s ) ................................ 110
Fig. 4-13. Variation of f vs. and for a TD soil column ( 4.0n ,
02.0s ) ............................................................................................................. 112
Fig. 4-14. Contour plot of 2.0f for a TD soil column with various n (
02.0s ) ............................................................................................................. 113
Fig. 4-15. Contour plot of 2.0f for a TD soil column with various s (
4.0n ) ................................................................................................................ 113
xi
LIST OF TABLES
Table 2-1. Properties of Granular Materials ................................................................ 11
Table 2-2. Theoretical effective density ratios for BE tests on three granular materials. ............................................................................................................... 17
Table 2-3. Analyses of Test Data from Youn et al. (2008). ........................................ 23
Table 2-4. Theoretical effective density ratios for RC tests at effective confining stress of 75 kPa. .................................................................................................... 27
Table 3-1. Typical values of soil parameters ( 2.0L m, 02.0s ) ......................... 56
Table 3-2. Parameters of samples for the analysis of RC tests and FVD. .................. 64
Table 4-1. Parameters Used in Illustrative Example ( 1.0L m, 10 rad/s, 02.0s ) ............................................................................................................. 115
xii
ACKNOWLEDGEMENTS
First and foremost, I wish to express my sincere gratitude to my advisor, Dr. Tong
Qiu, for his patient guidance, valuable suggestions and support through the course of this
study. His passion and enthusiasm in his work always inspired me. Our stimulating
conversations have made this study very interesting. His philosophy and standard of life
and profession motivate me to pay more attention to details. I would also like to offer my
great appreciation to my committee members, Dr. Chun Liu, Dr. Farshad Rajabipour, and
Dr. Mansour Solaimanian for reviewing the manuscript and providing valuable
perspective on my work. I would like to thank Dr. Yaurel Guadalupe-Torres and Dr.
Christopher D.P. Baxter of University of Rhode Island for providing the data of their
bender elements tests. Financial support from the US National Science Foundation under
Grant Nos. CMMI-0826097 and CMMI-1059588 is gratefully acknowledged.
I would like to thank graduate students that used to work or currently still work in
CITEL including Dr. Alireza Akhavan, Chris Cartwright, Cory Kramer, Greg Braun,
Jared Wright, Joe Reiter, Omid Ghasemi, Pezhouhan T.Kheiry, Dr. Wei Chen, and Yin
Gao, for helping me lift the heavy confining chamber of my resonant column device. In
addition, I also enjoyed the nice conversations and fun time with them as well as with
Benjamin T Adams, Chaoyi Wang, Elnaz Kermani, Hamed Maraghechi, Lynsey Reese,
Yao Ling, and Yeh Lin. Special thanks are due to Mr. Daniel Fura for helping me to set
up the resonant column test with efficient saturating system as well as to lift the heavy
confining chamber. You all make my life easier and so enjoyable.
Finally, I want to thank my family for supporting me all the time. Thank my parents
for supporting me to study abroad, it is their hard work and selfless-and-endless love that
xiii
give me the opportunities to pursue my dream. Thank my brothers and sisters-in-law for
their unconditional support.
1
Chapter 1
Introduction
1.1 Background
Shear modulus and damping are two important soil dynamic properties. These two
parameters play crucial roles in ground motion analyses in geotechnical earthquake
engineering. For example, Fig. 1-1 shows a deposit of homogeneous soil layer on top of
bedrock. The fundamental frequency, of , of the soil layer can be estimated as
H
Vf s
o 4 (1-1)
where H is the soil layer thickness and sV is shear wave velocity which can be
calculated as
G
Vs (1-2)
where is the soil density and G is shear modulus. Eq. (1-1) indicates that of depends
on the shear wave velocity and layer thickness. For a typical earthquake ground motion,
the dominant frequency, gf , is generally in the range of 1 – 5 Hz (Kramer 1996). If gf
is close to of , dynamic amplification will occur and large ground motions (e.g.,
acceleration, velocity, and displacement) will be expected. On the other hand, if gf is
significantly different than of and other modes of natural frequencies, deamplification
may occur and the resulted ground motions will be small. As the seismic wave travels
2
between the bedrock and ground surface as shown in Fig. 1-1, wave energy is dissipated
through soil damping. Higher soil damping will result in smaller ground motions as more
energy is dissipated during the wave propagation. Therefore, shear modulus and
damping are considered as input parameters for ground motion analyses.
Fig. 1-1. Seismic wave propagation in homogeneous soil layer
In the current geotechnical engineering research and practice, shear modulus is
generally calculated based on shear wave velocities measured from various field and
laboratory tests involving shear waves, such as the seismic cone penetration tests (SCPT),
bender element (BE) tests, and resonant column (RC) tests, using the following equation
2
sVG (1-3)
This equation is based on the theory of elasticity in a continuum, where is the density
of the continuum (single-phase) under all conditions. Soil is a multi-phase system,
consisting of a solid phase, liquid phase (e.g., water), and gas phase (e.g., air). For dry
G.W.T.
Ground Motion
Bedrock Motion
Soil Layer Thickness H
3
soil, dry density d is used in Equation (1-3); for saturated soil, saturated density sat is
generally used. However, the use of sat for saturated soil assumes no relative motion
between pore fluid and solid skeleton. The validity of this assumption depends on the
magnitude of fluid motion relative to solids during shear wave excitations. For soils with
low permeability (e.g., silts and clays) and under low-frequency excitations (e.g., seismic
waves), this assumption is generally valid. However, relative motion may be important
for soils with high permeability (e.g., sands and gravels) and under high-frequency
excitations based on Biot theory (Biot 1956), rendering this assumption invalid. Qiu and
Fox (2008) proposed the concept of “effective soil density”, eff , that is related to the
fraction of pore fluid that moves with solid skeleton during shear wave propagation. This
effective density is always between d and sat and is the theoretically correct value to
use in Equation (1-3) to calculate shear modulus based on measured shear wave velocity.
Qiu and Fox (2008) provided analytical solutions of eff based on Biot theory (Biot
1956). However, this analytical solution has not been rigorously validated against
laboratory test data for different soils.
Damping is a consequence of energy dissipation due to sliding and rolling at particle
contacts, and the loss and creation of particle contacts when there is particle
rearrangement. This form of energy dissipation is generally considered as “skeleton
damping” (Ellis et al. 2000) and is the only source of material damping in dry soil. For
saturated soil, in addition to skeleton damping, energy is also dissipated due to the
relative motion and viscous drag between pore fluid and solid skeleton (i.e., viscous
coupling). Therefore, saturated soils exhibit higher damping than the same soils in their
4
dry condition. This has been experimentally observed by various researchers (e.g., Hall
and Richart 1963; Bolton and Wilson 1990; Ellis et al. 1998 and 2000). In geotechnical
engineering research and practice, however, pore fluid induced damping is generally
neglected due to the lack of quantitative assessment of its values in various soils. Qiu
and Fox (2006) and Qiu (2010) provided analytical solutions of pore fluid induced
damping in saturated soils during shear wave excitation. These studies suggest that pore
fluid induced damping depends on soil types and may have significant contribution to the
total damping for coarse sands and gravels, in particular at small strain levels. However,
these findings have not been validated by any experimental test data.
1.2 Objectives of Research
The objectives of this research are to quantify effective density and pore fluid induced
damping in granular materials for small strain shear waves using BE and RC tests, and to
conduct additional analytical study on pore fluid induced damping in saturated soils
under quasi-static compressional excitations. This study can potentially improve the
accuracy of how small strain shear modulus and damping are evaluated, especially in
highly permeable granular materials (e.g., coarse sands and gravels) under high-
frequency excitations (e.g., BE tests), which may improve the accuracy of current ground
motion analyses in geotechnical earthquake engineering. The findings of this study will
be of significant value to geotechnical earthquake engineering and soil dynamics.
Ultimately, the benefit will be the reduction of losses to society as a result of earthquakes.
5
1.3 Organization of Dissertation
Chapter 2 presents effective density for small strain shear waves in saturated granular
materials. It presents an experimental investigation consisting of RC and BE tests on
various granular materials in dry and saturated conditions for the concept of effective
density. This chapter is based on a manuscript submitted to the Journal of Geotechnical
and Geoenvironmental Engineering, ASCE.
Chapter 3 presents analytical solutions and their comparison with RC test results for
pore fluid induced damping in saturated granular materials. It presents two analytical
solutions and compares them with a closed-form analytical solution readily available for
RC test in literature. Furthermore, solution of pore fluid induced damping based on the
free vibration decay method is compared with RC test results of various granular
materials in dry and saturated conditions. This chapter is based on a manuscript
submitted to the Soil Dynamics and Earthquake Engineering.
Chapter 4 presents energy dissipation in nearly saturated soil columns during quasi-
static compressional excitations, which is of particular relevance to slow phenomena
(e.g., consolidation). Different components of energy dissipation in a saturated soil
column are derived and compared. This chapter is based on a paper published in the
Journal of Engineering Mechanics, ASCE.
Chapter 5 draws final conclusions of this study and presents suggestions for future
work.
6
References
Biot, M.A. (1956). “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous
Solid. I. Low-Frequency Range. ІІ. Higher Frequency Range.” J. Acoust. Soc. Am.,
28(2), 168-191.
Bolton, M. D., and Wilson, J. N. (1990). “Soil stiffness and damping.” Structural
dynamics, W. B. Kratzig, D. E. Beskos, and I. G. Vardoulakis, eds., Balkema,
Rotterdam, The Netherlands, 209–216.
Ellis, E. A., Soga, K., Bransby, M. F., and Sato, M. (1998). “Effect of pore fluid viscosity
on the cyclic behavior of sands.” Proc., Centrifuge 98, T. Kimura, O. Kusakabe, and
J. Takemura, eds., Balkema, Rotterdam, The Netherlands, 217–222..
Ellis,E.A., Soga, K., Bransby,M.F. and Sato, M. (2000). Resonant Column Testing of
Sands with Different Viscosity Pore Fluids, J. Geotech. Geoenviron. Eng., 126(1),
10-17.
Hall, J.R. and F.E. Richart (1963). “Dissipation of Elastic Wave Energy in Granular
Soils.” J. Soil Mech. and Found. Div., 89(6), 27-56.
Kramer, S.L. (1996). Geotechnical Earthquake Engineering, Prentice Hall, Upper Saddle
River, NJ.
Qiu, T. and Fox, P.J. (2006). “Hydraulic damping of saturated poroelastic soils during
steady-state vibration.” J. Eng. Mech., 132(8), 859-870.
Qiu, T. and Fox, P.J. (2008). “Effective Soil Density for Propagation of Small Strain
Shear Waves in Saturated Soil.” J. Geotech. Geoenviron. Eng., 134(12), 1815-1819.
Qiu, T. (2010). “Analytical Solution for Biot Flow-Induced Damping in Saturated Soils
during Shear Wave Excitations.” J. Geotech. Geoenviron. Eng., 136(11), 1501-1508.
7
Chapter 2
Effective Soil Density for Small Strain Shear Waves in Saturated Granular Materials
2.1 Introduction
The small-strain shear modulus maxG is important for analyses in soil dynamics and
geotechnical earthquake engineering. maxG can be calculated from measured shear wave
velocity sV according to
2
max sVG (2-1)
where soil density. The use of Eq. (2-1) for multiphase soils requires an assumption
regarding material density. For dry soils, is equal to the dry density d because the
density of air in the voids is negligible. For saturated soils, the density of pore water is
not negligible and the value of is generally taken as the saturated density sat , which
implicitly assumes that the solid phase (i.e., soil grains) and fluid phase (i.e., pore water)
move together as a single phase during passage of a shear wave. The validity of this
assumption depends on the magnitude of relative motion between solid and fluid phases.
For seismic waves propagating through saturated soil, this assumption is generally valid
(Zienkiewicz et al. 1999; Santamarina et al. 2001). However, considering Biot (1956)
theory, relative motion may not be negligible and the foregoing assumption may not be
valid for high-hydraulic conductivity soils and/or high-frequency excitations. To account
for such relative motions, Qiu and Fox (2008) introduced the concept of effective soil
8
density, eff , which is the theoretically correct value to be used in Eq. (2-1) for saturated
soils. Values of eff were developed using Biot (1956) theory and the effects of “squirt
flow” at grain contacts (Dvorkin and Nur 1993) were neglected. The motions of solid
and fluid phases are linked through viscous and mass coupling. Values of eff vary from
dry in the absence of coupling forces to sat when strong coupling forces are present.
The effective density ratio, sateff , generally ranges from 0.75 to 1 and is a function of
specific gravity of solids, porosity, hydraulic conductivity, and wave frequency. For
many geotechnical applications, eff is equal to sat for low hydraulic conductivity soils
(e.g., silts and clays) and may be less than sat for high hydraulic conductivity soils (e.g.,
clean sands and gravels). Consequently, the use of sat in Eq. (2-1) may overestimate
maxG , which has been confirmed by analysis of the Hardin and Richart (1963)
experimental data performed by Qiu and Fox (2008) and experimental studies conducted
by Guadalupe-Torres (2013).
In a related study, Youn et al. (2008) measured maxG of dry and saturated sands using
bender element (BE), resonant column (RC), and torsional shear (TS) tests on the same
specimens. In the dry condition, sV values obtained from BE and RC tests were in close
agreement and the resulting values of maxG were close to those obtained from TS tests
and showed no dependence on loading frequency. For saturated conditions, however, sV
values obtained from BE tests were greater than those obtained from RC tests due to the
9
higher wave frequency used in the BE tests. Youn et al. (2008) concluded that a limiting
mass density lim , defined as
fs nn
111lim (2-2)
where n = porosity, s = density of solids, f = density of pore fluid, and = tortuosity
of pore space, should be used to evaluate maxG for saturated clean sands in BE tests. This
conclusion is consistent with the effective density concept and lim is equal to the lower
bound value of eff given by Qiu and Fox (2008), corresponding to conditions of high
frequency and/or high hydraulic conductivity. The investigations of Qiu and Fox (2008)
and Youn et al. (2008) provided experimental data and new insights on the relationship
between material density and shear wave velocity for saturated soils and indicated that
effective density is an important consideration for the evaluation of maxG for saturated
granular materials. However, more work is needed as the existing studies have not
investigated the effects of soil hydraulic conductivity and excitation frequency on
effective density.
This study presents the results of additional research on the effective soil density
concept. Bender element tests and resonant column tests were conducted on dry and
saturated specimens of several granular materials to evaluate effective density for a range
of stress conditions, including unloading and reloading paths. The experimental program
is first described, followed by analysis procedures for effective density and small-strain
shear modulus. Values of small-strain shear modulus obtained for saturated specimens
using both sat and eff are compared with corresponding values obtained for dry
10
specimens. Experimental results are also compared with additional data from literature.
Conclusions are reached on the general importance of effective density for the calculation
of small-strain shear modulus from shear wave velocity measurements in saturated
granular materials.
2.2 Experimental Program and Results
2.2.1 Granular Materials
The experimental program was conducted using several granular materials: coarse
sand, ASTM 20-30 Ottawa sand (ASTM C778), fine to medium quartz sand, and Soda-
lime glass beads of three sizes. Material properties are provided in Table 2-1. Values of
specific gravity sG were provided by the manufacturer or determined according to
ASTM D854. Particle size distribution curves are presented in Fig. 2-1. For the glass
beads, particle sizes are assumed to be uniformly distributed between the minimum and
maximum values. The glass beads and Ottawa sand had rounded particles and the fine to
medium quartz sand had subrounded particles (Guadalupe-Torres 2013).
11
Table 2-1. Properties of Granular Materials
Material Specific Gravity
sG
Particle Size Range
(mm)
Effective Diameter,
10D
(mm)
Glass Beads 2.5 0.40 – 0.60 0.44
2.9 – 3.5 2.98 6.0 6.0
Coarse Sand 2.65 0.85 – 3.35 2.0 ASTM 20-30 Sand 2.65 0.60 – 1.18 0.64
Fine to Medium Sand 2.66 0.075 – 4.75 0.17
Fig. 2-1. Particle size distributions for granular materials
2.2.2 Bender Element Tests
BE tests were performed using a triaxial apparatus to evaluate effective density for
specimens of fine to medium sand, coarse sand, and 6 mm glass beads. Each specimen
was formed in layers by air pluviation, followed by tapping the mold to achieve the
desired dry density. The average diameter and height of the specimens was 70.5 mm and
136.1 mm, respectively. Using bender elements embedded in the triaxial end caps, shear
0
20
40
60
80
100
0.01 0.1 1 10
Glass Beads (0.4 - 0.6 mm)Glass Beads (2.9 - 3.5 mm)Glass Beads (6 mm)Coarse Sand ASTM 20-30 SandFine to Medium Sand
Particle Size (mm)
Perc
ent P
assi
ng (
%)
12
wave velocities were measured as a function of effective stress for dry and saturated
conditions at the same porosity. In BE tests, the difficulty in interpreting the arrival time
is widely recognized (e.g., Lee and Santamarina 2005; Mohsin and Airey 2008). Based
on a detailed comparison of various methods, Youn et al. (2008) concluded that in a
medium with high material damping, the peak–peak method is preferable as reliable
results of sV can be easily obtained. Hence, travel time was determined from the peaks
of transmitted and received signals using the peak-peak method in this study. An input
frequency of 3 kHz produced the clearest signal on average and was used for all BE tests.
Shear wave velocities were calculated using measured travel times and the distance
between bender elements, which varied from specimen to specimen.
Dry specimens of each granular material were subjected to two cycles of isotropic
loading and unloading up to a maximum of confining stress of 400 kPa using
intermediate stress levels of 25, 50, 100, 200, and 300 kPa. BE tests were conducted at
each stress level. These loading/unloading cycles were performed to minimize changes
in void ratio for specimens tested in both the dry and saturated conditions. Each
specimen was then saturated with de-aired water under a back pressure of 500 kPa and
two replicate cycles of loading/unloading were conducted with shear wave velocity
measured at the same stress levels. Shear strains in BE tests are typically less than
0.0001% (Sasanakul 2005) and therefore can be used to obtain maxG .
Volumetric strains for the saturated specimens during loading and unloading, as
measured by water inflow/outflow, are presented in Fig. 2-2. For a given effective
confining stress, volumetric strains are largest for the glass beads, intermediate for the
coarse sand, and smallest for the fine to medium sand, which can be attributed to
13
differences in particle shape and surface roughness for these materials. As such, particle
interlocking is expected to be less for the rounded glass beads and coarse sand than for
the subrounded fine to medium sand. Likewise, interparticle friction is expected to be
less for the smooth glass beads than for the sand specimens. Fig. 2-2 also indicates that,
for the glass beads and coarse sand at a given effective confining stress, the difference in
volumetric strain during the loading or unloading phase for two consecutive cycles is
much less than the difference within each loading cycle. For example, for the glass beads
at a confining stress of 200 kPa, the difference in volumetric strain during loading for the
two cycles is approximately 0.15% while the difference for loading and unloading phases
of the first cycle is approximately 0.41%. Fig. 2-2 indicates that the effect of two loading
cycles on porosity for a saturated specimen at a given confining stress is minor,
particularly when porosity values at consistent points for consecutive cycles are
compared.
Values of maxG for the dry specimens were calculated according to Eq. (2-1) using
measured sV and d and are presented for each loading cycle in Fig. 2-3. Dry densities
were corrected to account for volumetric strain assuming that the dry specimens had the
same strains as corresponding saturated specimens for consistent points of loading. This
assumption is reasonable knowing that the stress paths and strain paths are the same for
these specimens. Fig. 2-3 shows that the difference in maxG between the two loading
cycles is negligible for the coarse sand and fine to medium sand specimens and small, but
non-negligible, for the glass beads specimen. Fig. 2-3(a) also indicates that, for the glass
beads, the difference in maxG between the loading or unloading phase of successive
14
cycles is smaller than the difference within a given loading cycle. These observations are
generally consistent with Fig. 2-2 since maxG is dependent on the porosity at a given
confining stress.
Fig. 2-2. Volumetric strains for saturated granular materials during two loading cycles
0
1
2
3
4
50 100 200 300 400 500
Glass Beads (6 mm)
Coarse Sand
Fine to Medium Sand
Vol
umet
ric
Str
ain
(%)
Effective Confining Stress (kPa)
Second Cycle
First Cycle
0
50
100
150
200
250
0 100 200 300 400 500
First Cycle
Second Cycle
Effective Confining Stress (kPa)
Sm
all-
Str
ain
She
ar M
odul
us (
MPa
)
Glass Beads (6 mm)
(a)
15
Fig. 2-3. Small-strain shear modulus for dry specimens: (a) glass beads (6 mm); (b)
coarse sand; and (c) fine to medium sand
0
50
100
150
200
250
0 100 200 300 400 500
First Cycle
Second Cycle
Sm
all-
Str
ain
She
ar M
odul
us (
MPa
)
Effective Confining Stress (kPa)
(b)
Coarse Sand
0
50
100
150
200
250
0 100 200 300 400 500
First Cycle
Second Cycle
Effective Confining Stress (kPa)
Sm
all-
Str
ain
She
ar M
odul
us (
MPa
)
(c)
Fine to Medium Sand
16
Effective density ratios were calculated using the method of Qiu and Fox (2008),
which is based on matching the shear wave velocity in Eq. (2-1) to that given by Biot
(1956) theory. Specifically, sateff is calculated from sG , n , and f , where f is a
normalized frequency defined as:
ng
fkf
2 (2-3)
and k = hydraulic conductivity, g = gravitational acceleration, and f = wave frequency,
which is taken as the dominant harmonic frequency for a given application. Fig. 2-4
shows theoretical curves for effective density ratio vs. normalized frequency for the 6
mm glass beads, coarse sand, and fine to medium sand, along with corresponding
theoretical effective density ratios for the BE tests on these materials. Properties of the
specimens and parameters needed to calculate f are presented in Table 2-2. The wave
frequency f was taken as the input frequency to the transmitting bender element (i.e., 3
kHz) and the hydraulic conductivity for each material was estimated using the following
empirical relationship for uniform sands and gravels (Chapuis 2004)
7825.032
10
14622.2cm/s
e
eDk (2-4)
where 10D = effective diameter (mm) and e= void ratio. As porosity vary with effective
confining stress (Fig. 2-2), average values of porosity for each BE specimen were used in
the calculations. Fig. 2-4 shows that effective density ratio decreases as f increases,
which results from progressive decoupling of fluid and solid motions during shear wave
propagation. Theoretical values of effective density ratio for the BE tests are 0.897,
17
0.891, and 0.928 for the 6 mm glass beads, coarse sand, and fine to medium sand,
respectively.
Fig. 2-4. Theoretical values for effective density ratio versus normalized frequency for
BE tests on three granular materials
Table 2-2. Theoretical effective density ratios for BE tests on three granular materials.
Specimen n k (cm/s) f
(kHz) f
sat
eff
Glass Beads (6 mm) 0.349 6.7 3.0 369 0.897 Coarse Sand 0.375 1.5 3.0 77 0.891
Fine to Medium Sand 0.345 0.025 3.0 1.4 0.928
For each saturated specimen and effective stress condition, two maxG values were
calculated using measured sV and Eq. (2-1). satGmax, is based on sat and effGmax, is
0.88
0.90
0.92
0.94
0.96
0.98
1.00
0.01 0.1 1 10 100 1000
Glass Beads (6 mm)
Coarse Sand
Fine to Medium Sand
Eff
ecti
ve D
ensi
ty R
atio
Normalized Frequency f
18
based on eff . As such, the ratio of effGmax, to satGmax, is also equal to the effective
density ratio,
sat
eff
sat
eff
G
G
max,
max, (2-5)
Values of sat , eff , and k were calculated using corrected porosity during loading
and unloading cycles. Fig. 2-5 presents values of maxG for the dry specimens versus
satGmax, and effGmax, for the corresponding saturated specimens. Labels 1L and 1U
indicate loading and unloading for the first cycle, and 2L and 2U indicate loading and
unloading for the second cycle, respectively. With no fluid coupling effects, the dry
maxG values should accurately reflect material stiffness. Assuming that no significant
changes in soil structure/fabric occurred during saturation, values of maxG for dry and
saturated conditions should be equal. The data points in each plot generally fall into
clusters with higher G values measured at higher confining stress. Such data clusters for
the 6 mm glass beads, as shown in Fig. 2-5(a) using dashed ovals, are less distinct due to
the effect of loading cycle on specimen porosity. Results for the 6 mm glass beads show
excellent agreement between both satGmax, and effGmax, with actual maxG (i.e., dryGmax, )
at low confining stress levels (low maxG ), where dryGmax, = small-strain shear modulus for
dry conditions. As confining stress increases, values of effGmax, show similar excellent
agreement, whereas values of satGmax, are higher than dryGmax, and show progressively
larger deviation. A similar trend is observed for the coarse sand in Fig. 2-5(b). Data for
19
the fine to medium sand in Fig. 2-5(c) indicates progressive deviation for both satGmax,
and effGmax, with increasing confining stress, with less deviation observed for effGmax, .
Relative errors satE for satGmax, and effE for effGmax, are defined as
%100max,
max,max,
dry
drysatsat G
GGE (2-6a)
%100max,
max,max,
dry
dryeffeff G
GGE (2-6b)
For each data cluster in Fig. 2-5, average values of dryGmax, , satGmax, , and effGmax,
were used to calculate satE and effE . Fig. 2-6 presents satE and effE for each granular
material versus effective confining stress, and indicates that effE is substantially smaller
than corresponding satE for each material. For the 6 mm glass beads and coarse sand,
effE values are generally smaller than 5%, whereas satE values range from 5.5% to 17%.
For the fine to medium sand, Fig. 2-6 shows a gradual increase of satE and effE with
effective confining stress, which is consistent with Fig. 2-5(c). This test was repeated on
an identically prepared specimen of fine to medium sand and similar results were
obtained (Guadalupe-Torres 2013).
20
Fig. 2-5.
Fig.
Compariso
beads (6
2-6. Relati
Rel
ativ
e E
rror
(%
)
on of maxG f
6 mm); (b) c
ive errors of
0
5
10
15
20
25
30
35
40
0 1
satE
Glas(6.0
effE
for dry and
coarse sand;
f maxG for B
100 200
Effective Con
ss Beads 0 mm) Coa
saturated co
; and (c) fin
BE tests on s
300
nfining Stress (
arse SandFine
onditions in
e to medium
saturated gra
400 50
(kPa)
e to Medium Sand
BE tests: (a
m sand
anular mate
00
21
a) glass
erials
22
The above procedure was also used to analyze test data from Youn et al. (2008).
These data are relevant to the current study because measurements of small-strain shear
modulus from TS tests, TSmax,G , can be directly compared to measurements of satGmax,
and effGmax, from BE tests on the same saturated specimens. The test data were obtained
for two saturated silica sand specimens ( sG = 2.63, 10D 0.09 mm) at initial porosity in
of 0.429 and 0.408 and two saturated Toyoura sand specimens ( sG = 2.65, 10D 0.16
mm) at in of 0.448 and 0.412. Table 2-3 presents analysis results, where values of
TSmax,G were digitized from Figs. 13 and 14 of Youn et al. (2008), k and n values were
taken from Tables 4 and 5 of Youn et al. (2008), and results were calculated following
the approach outlined in Table 2-2. Fig. 2-7 compares maxG values from the TS and BE
tests, where the TS values were obtained at low frequencies and should accurately reflect
material stiffness. Similar to Fig. 2-5(c), Fig. 2-7 shows excellent agreement between
effGmax, and TSmax,G at low confining stress levels (low maxG ) and progressive deviation
for both satGmax, and effGmax, with increasing confining stress, with less deviation
observed for effGmax, . Relative errors are presented in Fig. 2-8 and indicate that effE is
substantially smaller than satE for each specimen. Fig. 2-8 also indicates that satE and
effE generally increase with effective confining stress, which is consistent with Fig. 2-
5(c) and Fig. 2-7. For the four specimens, effE values are smaller than 5% except at an
effective confining stress of 400 kPa where the maximum error is 14%, whereas satE
values range from 11% to 28%.
23
Table 2-3. Analyses of Test Data from Youn et al. (2008).
Silica Sand
Effective Confining
Stress (kPa)
TSmax,G
(MPa)
sV
BE Test
(m/s)
k (cm/s)
n sat
(g/cm3)f
sat
eff
satGmax,
(MPa) effGmax,
(MPa)
50 61.8 189.4 0.012 0.427 1.934 1.98 0.891 69.4 61.8 100 85.5 223.5 0.012 0.426 1.936 1.98 0.891 96.7 86.2 200 114.8 264.7 0.011 0.425 1.937 1.82 0.893 135.7 121.2 400 151.0 315.3 0.011 0.424 1.939 1.83 0.894 192.7 172.2
50 76.7 211.2 0.009 0.408 1.965 1.55
0.904 87.7 79.2
100 107.1 248.1 0.009 0.407 1.967 1.560.90
4 121.1 109.4
200 143.3 288.7 0.009 0.407 1.967 1.56 0.904 163.9 148.2 400 191.5 339.2 0.009 0.405 1.970 1.57 0.905 226.6 205.0
Toyoura Sand 50 58.4 184.1 0.028 0.444 1.917 4.45 0.871 65.0 56.6
100 81.6 217.9 0.028 0.442 1.921 4.47 0.872 91.2 79.5 200 111.6 257.5 0.026 0.437 1.929 4.19 0.875 127.9 111.9 400 151.3 306.4 0.026 0.435 1.932 4.21 0.876 181.4 158.9
50 78.6 210.6 0.020 0.410 1.974 3.44 0.891 87.5 77.9
100 106.2 245.9 0.019 0.405 1.982 3.31 0.893 119.8 107.0 200 142.0 285.9 0.019 0.404 1.983 3.31 0.894 162.1 144.9 400 188.1 337.6 0.018 0.403 1.985 3.15 0.895 226.3 202.5
Fig. 2-7.
Comparison
sand; and (b
n of maxG fr
b) Toyoura
rom TS and
sand (measu
BE tests on
ured data fr
n saturated s
om Youn et
specimens: (
t al. 2008)
24
(a) silica
25
Fig. 2-8. Relative errors of maxG for test data from Youn et al. (2008)
2.2.3 Resonant Column Tests
RC tests were performed in general accordance with ASTM D4015 to evaluate the
effective density concept at lower frequencies than for the BE tests. Similarly, resonant
frequencies were measured as a function of effective confining stress for specimens in
both the dry and saturated conditions. RC tests were conducted on solid cylindrical
specimens of ASTM 20-30 sand at initial porosity values of 0.387 and 0.343, and of glass
beads with particle sizes of 0.4 – 0.6 mm, 2.9 – 3.5 mm, and 6 mm. Each specimen had a
diameter of 100 mm and height of 200 mm. The boundary conditions were zero
displacement at the base and harmonic torsional loading at the top. Each dry specimen
was formed in layers by air pluviation, followed by tapping the mold to achieve a desired
dry density. The dry specimens were then subjected to an isotropic confining stress of 25
kPa and 60,000 cycles of torsional excitation at a frequency of 100 Hz using the
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500
satE
effE
Silica Sand
429.0n 408.0 448.0 412.0Toyoura Sand
Effective Confining Stress (kPa)
Rel
ativ
e E
rror
(%
)
26
maximum input voltage (1 V). This initial loading produced shear strains in the range of
0.01 – 0.015% and was conducted to minimize changes in soil porosity and fabric during
subsequent RC testing. Each dry specimen was then subjected to one cycle of isotropic
loading and unloading with confining stress levels ranging from 25 kPa to 150 kPa using
25 kPa increments. To measure maxG and minimize shear modulus degradation during
the RC tests, shear strains were held below 0.0015%. After resonating in the dry
condition, each specimen was saturated by infusing de-aired water from the base and
passing a minimum of three pore volumes through the material. One replicate sequence
of loading (no unloading) was applied to each saturated specimen with resonant
frequencies measured at the same effective stress levels.
From the RC test results, G is calculated using Eq. (2-1) where sV is determined
from
ts
n
s
n
I
LJ
V
Lf
V
Lf
2tan
2 (2-7)
where nf = resonant frequency; L = specimen length; J = polar moment of inertia of
specimen; and tI = mass polar moment of inertia of the loading system. In this study,
values of dryGmax, were obtained using d in Eqs. (2-1) and (2-7), whereas values of
satG max, and effG max, were obtained using sat and eff , respectively. Based on a two-
phase analysis of a saturated poroelastic specimen, Huang and Qiu (2013) proved
theoretically that the use of eff in Eq. (2-7) would result in a value of sV consistent with
Biot (1956) theory. Table 2-4 presents specimen properties, resonant frequencies, and
estimated theoretical effective density ratios for RC tests conducted at an effective
27
confining stress of 75 kPa. Effective density ratios for other stress levels were similarly
estimated and can be found in Huang (2014).
Table 2-4. Theoretical effective density ratios for RC tests at effective confining stress
of 75 kPa.
Specimen n k (cm/s) nf
(Hz) f
sat
eff
Glass Beads (0.4 – 0.6 mm) 0.370 0.14 147.5 0.36 0.975 Glass Beads (2.9 – 3.5 mm) 0.367 2.57 164.1 7.4 0.899
Glass Beads (6 mm) 0.358 7.32 159.6 21 0.898 ASTM 20-30 Sand 0.387 0.28 134.7 0.62 0.944 ASTM 20-30 Sand 0.343 0.19 140.6 0.50 0.961
Fig. 2-9 compares maxG values for dry and saturated RC specimens, with each value
corresponding to a given level of effective confining stress and indicates that the
difference between satGmax, and effGmax, increases with increasing particle size. As shown
in Figs. 2-9(a) and 2-9(d), satGmax, and effGmax, are almost identical for the 0.4 – 0.6 mm
glass beads and ASTM 20-30 sand, which results from the low hydraulic conductivity
and consequently high effective density ratios for these specimens (Table 2-4). For the
2.9 – 3.5 and 6 mm glass beads, differences between satGmax, and effGmax, are larger due
to higher hydraulic conductivity for these materials. Differences between satGmax, and
effGmax, increase with increasing confining stress (i.e., increasing maxG ) because the
resonant frequency also increases, which decreases the effective density ratio. Figure 10
presents relative errors for the same data and shows that most of the satE and effE values
a
v
are smaller
values. Un
effE with ch
than 5% an
like Figs. 2
hanging effe
nd, for each
-6 and 2-8,
ective confin
specimen, E
Fig. 2-10 d
ning stress f
effE values
does not sug
for the RC t
are on aver
ggest any cl
tests.
age smaller
lear trend of
28
r than satE
f satE and
29
Fig. 2-9.
beads (0.4
Fig. 2
Compariso
4 – 0.6 mm)
-10. Relativ
Rel
ativ
e E
rror
(%
)
on of maxG f
); (b) glass b
ve errors of
0
2
4
6
8
10
12
0 25
satE
effE
0.4 –
for dry and
beads (2.9 –
ASTM 20
f maxG for R
50 75
Glass Beads (m
– 0.6 2.9 – 3.5
Effective Con
saturated co
– 3.5 mm); (
-30 sand
RC tests on s
100 125
mm)
6.0
ASTM
3.0n
nfining Stress (
onditions in
(c) glass bea
saturated gra
5 150 17
M 20-30 Sand
387 343.0
(kPa)
RC tests: (
ads (6 mm);
anular mate
75
30
(a) glass
and (d)
erials
31
For the 6 mm glass beads, a comparison between Figs. 2-5(a) and 2-9(c) suggests that
the deviation between satGmax, and effGmax, is higher in BE tests than in RC tests, despite
the effective density ratios in these two tests being practically the same (0.897 for BE and
0.898 for RC). In light of this observation, the effect of sateff on sateff GG max,max, for
RC and BE test conditions is analyzed. In the analysis of RC tests, a smaller produces
a larger sV in Eq. (2-7); therefore, a maxG value calculated based on and sV
according to Eq. (2-1) changes by less than the proportional change in . For BE tests,
however, maxG is proportional to and Eq. (2-5) is valid since sV is measured directly.
To illustrate, Fig. 2-11 presents the influence of effective density ratio on sateff GG max,max,
for RC and BE tests, where satI is the mass polar moment of inertia of saturated
specimen ( LJsat ). Fig. 2-11 indicates that the effect of sateff on sateff GG max,max,
depends on tsat II for RC tests. For small values of tsat II , the effect is negligible
and 1max,max, sateff GG for sateff values in the range of 0.8 – 1.0. As tsat II
increases, the effect increases and the rate of decrease in sateff GG max,max, with decreasing
sateff increases. For a typical RC apparatus and soil specimen, the value of tsat II
is generally less than 1.0. For example, tsat II was approximately 0.8 for the 6 mm
glass beads specimen in the current study. Thus, Figs. 2-9, 2-10 and 2-11 suggest that the
use of sat instead of eff should generally result in smaller errors for RC tests than for
BE tests on saturated soil specimens. Regardless, eff is recommended for analysis of
both tests.
32
Fig. 2-11. Influence of effective density ratio on sateff GG max,max, for RC and BE tests
2.2.4 Quick Chart
Qiu and Fox (2008) presented a chart that allows a user to rapidly determine if
effective density should be considered for a given application. This chart is shown in
Fig. 2-12 along with laboratory data for the sands from the current study. The hatched
zone of the original chart has been replaced with several solid lines corresponding to
specific values of porosity but, otherwise, the chart is unchanged. The solid lines
represent combinations of k and f that yield sateff = 0.95 for sG = 2.7 and three n
values (0.25, 0.3, and 0.6). If a given combination of k and f falls above the line for a
given n value, sateff < 0.95 and consideration of effective density may be important;
otherwise, sateff > 0.95 and saturated density can be used without significant error.
0.80
0.85
0.90
0.95
1.00
0.80 0.85 0.90 0.95 1.00
0.10.20.51.02.0
sat
eff
t
sat
I
Isat
eff
G
G
max,
max,
33
The right-side axis gives approximate values of 10D as calculated from k and the Hazen
(1911) equation. This equation is empirical and was developed for clean loose sands.
Experimental results for sands in the current study are superimposed on the chart based
on f and k values. Fig. 2-12 shows that solid lines for n = 0.3 and 0.6 are closely
spaced, suggesting that combinations of k and f for sateff = 0.95 are relatively
insensitive to porosity in this range. However, the solid line for n = 0.25 is located
significantly higher and indicates that higher k and f values are needed for
consideration of effective density to be important for dense to very dense granular
materials, which occurs because pore fluid accounts for a smaller percentage of total soil
mass. The relative positions of data points in Fig. 2-12 are consistent with their effective
density ratios presented in Tables 2-2, 2-3, and 2-4.
Fig. 2-12 indicates that consideration of effective density will not be important for
clays and is unlikely to be important for silts unless the frequency is very high ( f > 10
kHz). For clean sands, consideration of effective density may be important for
frequencies as low as 10 Hz. Effective density may be important for fine and medium
sands at high-frequency excitations, such as from bender elements, and for coarse clean
sands at lower frequencies, such as for resonant column tests or seismic cone penetration
tests. Consideration of effective density is important for clean gravels at essentially all
frequencies of geotechnical interest. The superimposed experimental data points support
these conclusions. For example, BE data points of the coarse sand, fine to medium sand,
silica sand, and Toyoura sand are above the lines for n = 0.3 and 0.6, indicating that
consideration of effective density is important for BE tests in these materials, which is
34
consistent with Figs. 2-5 and 2-7. Shifting the BE data point of the fine to medium sand
to the left by one order of magnitude (i.e., lowering excitation frequency to RC range)
will result in a new data point below the lines for n = 0.3 and 0.6, suggesting that
consideration of effective density will not be important for RC tests in fine to medium
sand. However, a similar shift for the BE data point of the coarse sand will result in a
new data point remaining above the lines for n = 0.3 and 0.6, suggesting that
consideration of effective density will still be important for RC tests in coarse sands. RC
data points for ASTM 20-30 sand are mostly below the lines for n = 0.3 and 0.6,
suggesting that consideration of effective density is unlikely to be important for RC tests
in medium sand, particularly when the conclusions from Fig. 2-11 are considered.
Fig. 2-12. Test data and chart for quick determination of importance of effective soil
density (data for silica sand and Toyoura sand from Youn et al. 2008). Values of 10D are
approximate.
10-6
10-5
10-4
10-3
10-2
0.1
1
10
0.01
0.1
1
10
0.1 1 10 102 103 104 105
ASTM 20-30 Sand - RCCoarse Sand - BEFine to Medium Sand - BESilica Sand - BEToyoura Sand - BE
D10 (m
m)
f (Hz)
k (m
/s)
Silt
Sand
Gra
vel
%95sat
eff
%95sat
eff
n = 0.25
0.3
0.6
35
2.3 Discussion
Consideration of Biot (1956) theory suggests that effective density ratio for a given
granular material is sensitive to both mass coupling and non-Poiseuille flow effects. For
example, Gajo et al. (1997) found that an erroneous assumption concerning mass and
viscous coupling yielded an error of approximately 15% in values of maxG calculated
using Eq. (2-1). Mass coupling results from tortuosity of the pore space. For a
hypothetical porous medium consisting of straight flow channels, = 1 in Eq. (2-2) and
mass coupling vanishes; however, flow channels are tortuous for granular materials and
>1. Tortuosity is difficult to evaluate and various theoretical and empirical equations
for have been proposed (Ghanbarian et al. 2013). For example, Stoll and Bryan
(1970) proposed 3 for a system of uniform pores, Berryman (1981) proposed
nn 21 for spherical particles, and Sen et al. (1981) suggested 5.0 n for a
random array of spheres. Gajo (1996) compared experimental results from Johnson et al.
(1982) with results from several empirical relationships and found that the Sen et al.
(1981) equation gave the best estimate for . In the current study, this equation was
also used to estimate .
Again using Biot (1956) theory, viscous coupling effects due to non-Poiseuille flow
can be evaluated. At low normalized frequencies ( f < 0.15 for circular pores),
corresponding to low excitation frequency and/or soils with low hydraulic conductivity,
relative fluid motion is of the Poiseuille type (i.e., similar to Darcy’s law) and viscous
coupling forces are calculated using seepage force. At high normalized frequencies ( f >
0.15 for circular pores), relative fluid motion is of the non-Poiseuille type and Biot (1956)
36
introduced a complex function to account for this effect on the viscous coupling force,
which was derived for a series of parallel tubes (Qiu 2010). Considering that non-
Poiseuille flow is highly dependent on local pore structure, viscous coupling forces are
difficult to accurately evaluate in saturated porous media at high normalized frequencies
such as the BE tests conducted in this study.
Fig. 2-13 presents the effect of different assumptions used to characterize mass
coupling and non-Poiseuille flow on the theoretical effective density ratios for the BE
tests with 6 mm glass beads, coarse sand, and fine to medium sand. The solid curves
without symbols are the same as those shown in Fig. 2-4 and represent the full solution,
taking both mass coupling and non-Poiseuille flow into consideration; the dashed curves
without symbols represent a solution that considers mass coupling but neglects non-
Poiseuille flow; the solid curves with circles represent a solution that neglects mass
coupling (i.e., 1 ) but considers non-Poiseuille flow; the dashed curve with circles
represents a solution that neglects both mass coupling and non-Poiseuille flow; and the
vertical dashed lines mark f values for the BE tests. Fig. 2-13 shows that the minimum
value of effective density ratio at high f depends only on mass coupling (i.e., ),
which is consistent with Eq. (2-2). On the other hand, viscous coupling controls the rate
at which effective density ratio reaches the minimum value as f increases.
Figs. 2-13(a) and 2-13(b) indicate that effective density ratios for the 6 mm glass
beads and coarse sand specimens are near-minimum due to the large f values in the BE
tests; therefore, the accuracy of eff is only dependent on the accuracy of for these
specimens (see Eq. (2-2)). These materials had rounded particles; hence, the Sen et al.
37
(1981) equation should provide a reasonable estimation of . This explains the
excellent agreement between effGmax, and dryGmax, in these two specimens as shown in
Figs. 2-5(a) and 5(b). For the fine to medium sand specimen, Fig. 2-13(c) indicates that
the f value falls in a range where the accuracy of eff is dependent on the accuracy of
the chosen model for mass coupling and non-Poiseuille flow. This material had
subrounded particles; hence, the applicability of the Sen et al. (1981) equation is not
strictly satisfied. Inaccurate estimation of the effects of mass coupling and non-Poiseuille
flow may have led to an inaccurate value of eff for the fine to medium sand specimen,
which may have contributed to the lack of good agreement between effGmax, and dryGmax,
as shown in Fig. 2-5(c). Figs. 2-5(c) and 2-7 suggest that the use of eff overpredicts
maxG for the fine to medium sand, silica sand, and Toyoura sand specimens in BE tests
and the overprediction increases with increasing effective confining stress as shown in
Figs. 2-6 and 2-8. This error stems from the measured shear wave velocities at high
effective confining stress levels being greater than those predicted by Biot (1956) theory
(Guadalupe-Torres 2013). Youn et al. (2008) attributed this discrepancy to errors in their
test results. Given these specimens have similar f values, inaccurate values of eff due
to inaccurate estimation of the effects of mass coupling and non-Poiseuille flow may
have also contributed to the discrepancy. The reason for the error increasing with
increasing effective confining stress is currently unknown and warrants additional
investigation.
38
f
0.80
0.85
0.90
0.95
1.00
0.01 0.1 1 10 100 1000
No MC, PF Only
No MC, Non-PF
MC, PF Only
Full Solution
Eff
ecti
ve D
ensi
ty R
atio
Normalized Frequency
Glass Beads (6 mm)
(a)
BE
Tes
ts
f
0.80
0.85
0.90
0.95
1.00
0.01 0.1 1 10 100 1000
No MC, PF Only
No MC, Non-PF
MC, PF Only
Full Solution
Normalized Frequency
Eff
ecti
ve D
ensi
ty R
atio
Coarse Sand
(b)
BE
Tes
ts
39
Fig. 2-13. Effect of assumptions for mass coupling and non-Poiseuille flow on
theoretical effective density ratio: (a) glass beads (6 mm); (b) coarse sand; and (c) fine to
medium sand
2.4 Conclusions
This study presents an experimental investigation of the concept of effective density
for propagation of small strain shear waves through saturated granular materials. Bender
element tests and resonant column tests were conducted on various granular materials in
the dry and saturated condition. Values of small-strain shear modulus for the dry
condition, which accurately reflect material stiffness, are compared to those for the
saturated condition calculated using both saturated density and effective density.
Analyses were also conducted on similar data taken from the literature. The following
conclusions are reached as a result of this investigation:
0.80
0.85
0.90
0.95
1.00
0.01 0.1 1 10 100 1000
No MC, PF Only
No MC, Non-PF
MC, PF Only
Full Solution
Normalized Frequency
Eff
ecti
ve D
ensi
ty R
atio
Fine to Medium Sand
(c)
f
BE
Tes
ts
40
Effective density ratio (i.e., effective soil density/saturated soil density) is a function
of specific gravity of solids, porosity, hydraulic conductivity, and shear wave
frequency. For a given specific gravity of solids and porosity, the value of effective
density ratio decreases with increasing normalized frequency. Viscous coupling
controls the rate at which effective density ratio decreases, whereas mass coupling
determines the minimum value of effective density at high frequency.
Effective density should be used instead of saturated density to calculate small-strain
shear modulus from measured shear wave velocity in bender element tests on
saturated granular materials. The use of saturated density will generally overestimate
these modulus values. In the current study, the use of saturated density yielded errors
up to 28%; whereas the use of effective density resulted in substantially smaller
errors, which were generally less than 5%.
To calculate small-strain shear modulus from measured resonant frequency in
resonant column tests on saturated granular materials, the errors for using saturated
density are smaller than those for bender element tests. The error increases with
increasing ratio of mass polar moment of inertia of saturated specimen to that of the
loading system. In the current study, the use of saturated density and effective
density yielded errors generally less than 5%, although the former yielded slighter
higher errors on average.
The quick chart is a useful tool for a user to rapidly determine if effective density
should be considered for a given application. Effective density may be important for
fine and medium sands at high-frequency excitations, such as from bender elements,
41
for coarse clean sands at lower frequencies, such as for resonant column tests, and for
clean gravels at essentially all frequencies of geotechnical interest.
Notations
The following symbols are used in this chapter:
10D = effective diameter;
e = void ratio;
effE = relative error for effGmax, ;
satE = relative error for satGmax, ;
f = wave frequency;
f = normalized wave frequency;
nf = resonant frequency;
g = acceleration of gravity;
G = shear modulus;
maxG = small-strain shear modulus;
effGmax, = small-strain shear modulus of saturated specimen based on
effective density;
dryGmax, = small-strain shear modulus of dry specimen;
satGmax, = small-strain shear modulus of saturated specimen based on
saturated density;
42
TSmax,G = small-strain shear modulus from torsional shear test;
sG = specific gravity of solids;
tI = mass polar moment of inertia of loading system
satI = mass polar moment of inertia of saturated specimen
J = polar moment of inertia of specimen;
k = hydraulic conductivity;
L = specimen length;
n = porosity;
in = initial porosity;
sV = shear wave velocity;
= tortuosity of pore space;
= soil density;
d = dry density;
eff = effective density;
lim = limiting mass density;
sat = saturated density
43
References
Berryman, J.G. (1981). “Elastic wave propagation in fluid-saturated porous media.” J.
Acoust. Soc. Am., 69(2), 416–424.
Biot, M.A. (1956). “Theory of propagation of elastic waves in a fluid-saturated porous
solid. II. Low-frequency range. ІІ. Higher frequency range.” J. Acoust. Soc. Am.,
28(2), 168-191.
Chapuis, R.P. (2004). “Predicting the saturated hydraulic conductivity of sand and gravel
using effective diameter and void ratio.” Can. Geotech. J., 41(5), 787-795.
Dvorkin, J. and Nur, A. (1993). “Dynamic poroelasticity: A unified model with the squirt
and the Biot mechanisms.” Geophysics, 58(4), 524-533.
Gajo, A. (1996). “The effects of inertial coupling in the interpretation of dynamic soil
tests.” Geotechnique, 46(2), 245-257.
Gajo, A., Fedel, A., and Mongiovia, L. (1997). “Experimental analysis of the effects of
fluid-solid coupling on the velocity of elastic waves in saturated porous media.”
Geotechnique, 47(5), 993-1008.
Ghanbarian, B., Huntb, A.G., Ewingc, R.P., and Sahimid, M. (2013). “Tortuosity in
porous media: a critical review.” Soil Sci. Soc. Am. J., 77(5), 1461–1477.
Guadalupe-Torres, Y. (2013). “Unique relationship between small strain shear modulus
and effective stresses at failure.” Ph.D. Dissertation, University of Rhode Island,
Kingston, RI.
Hardin, B.O. and Richart, F.E. (1963). “Elastic wave velocities in granular soils.” J. Soil
Mech. and Found. Div., 89(1), 33–65.
44
Hazen, A. (1911). “Discussion of ‘Dams on sand foundations’ by A. C. Koenig.” Trans.
Am. Soc. Civ. Eng., 73, 199–203.
Huang, Y.B. (2014). “Theoretical and experimental investigation of effective density and
pore fluid induced damping in saturated granular materials.” Ph.D. Dissertation, The
Pennsylvania State University, University Park, PA.
Huang, Y.B. and Qiu, T. (2013). “Evaluation of effective soil density in resonant column
tests.” IACGE 2013: Challenges and Recent Advances in Geotechnical and Seismic
Research and Practices, GSP 232, J.P. Hu, J.L. Ma, J. Meneses, T. Qiu, X. Yu, and
X.W. Zeng, eds., Reston, VA, 685-693.
Johnson, D.L., Plona, T.J., Scala, C., Pasierb, F., and Kojima, H. (1982). “Tortuosity and
acoustic slow waves.” Phys. Rev. Lett., 49(25), 1840-1844.
Lee, J.S. and Santamarina, J.C. (2005). “Bender elements: performance and signal
interpretation.” J. Geotech. Geoenviron. Eng., 131(9), 1063-1070.
Mohsin, A.K.M. and Airey, D.W. (2008). “Using maxG measurements to monitor
degradation of an artificially cemented sand.” Deformational Characteristics of
Geomaterials, IOS Press, Amsterdam, Netherlands, 1, 305–310.
Qiu, T. (2010). “Analytical solution for Biot flow-induced damping in saturated soils
during shear wave excitations.” J. Geotech. Geoenviron. Eng., 136(11), 1501-1508.
Qiu, T. and Fox. P.J. (2008). “Effective soil density for propagation of small strain shear
waves in saturated soil.” J. Geotech. Geoenviron. Eng., 134(12), 1815-1819.
Santamarina, J.C., Klein, K.A., and Fam, M.A. (2001). Soils and Waves: Particulate
Materials Behavior, Characterization, and Process Monitoring, Wiley, New York.
45
Sasanakul, I. (2005). “Development of an electromagnetic and mechanical model for a
resonant column and torsional testing device for soils.” Ph.D. Dissertation, The Utah
State University, Logan, Utah.
Sen, P.N., Scala, C., and Cohen, J.H. (1981). “A self-similar model for sedimentary rocks
with application to the dielectric constant of fused glass beads.” Geophysics, 46(5),
781-795.
Stoll, R.D. and Bryan, G.M. (1970). “Wave attenuation in saturated sediments.” J.
Acoust. Soc. Am., 47(5), 1440–1447.
Youn, J.U., Choo, Y.W., and Kim, D.S. (2008). “Measurement of small-strain shear
modulus Gmax of dry and saturated sands by bender element, resonant column, and
torsional shear tests.” Can. Geotech. J., 45(10), 1426-1438.
Zienkiewicz, O.C., Chan, A.H.C., Pastor, M., Schrefler, B.A., and Shiomi, T. (1999).
Computational Geomechanics with Special Reference to Earthquake Engineering,
Wiley, Chichester, England.
46
Chapter 3
Biot Flow Induced Damping in Saturated Poroviscoelastic Soil Specimens in Resonant Column Test
3.1 Introduction
It has long been recognized that pore fluid induces additional damping in saturated
soils (Hall and Richart 1963, Bolton and Wilson 1990, Ellis et al. 1998, 2000); however,
past research has almost exclusively focused on the characterization of solid skeleton
damping (Hardin 1965, Vucetic and Dobry 1991, Darendeli 2001, Phillips and Hashash
2009) and little attention has been paid to pore fluid induced damping. In addition to
solid skeleton damping, energy dissipation is induced due to the relative motion between
viscous pore fluid and solid particles in saturated soils. Conceptually, pore fluid induced
damping can occur from two mechanisms: Biot flow and squirt flow. Biot flow is
relative fluid motion that, on average, occurs parallel to the direction of solid motion and
results from pore fluid moving past soil particles. Squirt flow is relative fluid motion that
occurs as fluid is forced out of small cracks and voids at contacts between solid particles
due to solid skeleton deformation (Dvorkin and Nur 1993). The effect of pore fluid
induced damping was observed in silts by Stoll (1989) and in sands by Ellis et al. (1998,
2000). Stoll showed that at low frequencies and for materials with low permeability or
saturated with fluids with high viscosity (e.g., silicone oil), squirt flow is a more
important energy dissipation mechanism than Biot flow. Though they occur
simultaneously and are coupled, Biot flow and squirt flow are often modeled separately.
47
Biot flow is often treated in a macroscopic sense (e.g., Biot 1956), whereas squirt flow is
typically treated in a microscopic sense based on microfabric properties such as geometry
of individual pores (Mavko and Nur 1979, Miksis 1988). Pore fluid induced damping
due to squirt flow is therefore difficult to assess theoretically and is generally evaluated
using laboratory tests. Pore fluid induced damping due to Biot flow, on the other hand,
can be evaluated analytically using macroscopic parameters (e.g., porosity and hydraulic
conductivity).
A theoretical study of Biot flow induced damping was conducted by Bardet (1995).
In his work, the spectral response of a two-phase poroelastic soil column subjected to
harmonic compression waves was obtained analytically. Biot flow induced damping was
quantified using the theory of a damped single-degree-of-freedom (SDOF) system based
on the following equation
2max12
1
A (3-1)
where max
A maximum amplification and damping. Bardet showed that for
saturated gravels and dense sands, the equivalent damping ratio contributed by Biot flow
varies from 0 to 0.7; while the skeleton damping ratio for such soils typically varies from
0.01 to 0.07 (Richard et al. 1979). It was therefore concluded that Biot flow induced
damping in saturated gravels and sands is not negligible and may be large compared to
soil skeleton damping under compression vibrations. Although Bardet’s finding is
insightful, his approach has challenges in interpreting the unusually high damping values
obtained for frequencies near higher modes of resonances, which is due to the limitation
of applying the theory of a damped SDOF system to wave propagation along a
48
continuous soil column. Following Bardet’s work, Qiu and Huang (2012) provided the
analytical solution for Biot flow induced damping in a nearly saturated poroviscoelastic
soil column during quasi-static compressional excitations, which is relevant to slow
phenomena discussed by Zienkiewicz et al. (1999). In this work, total damping in the
soil column is evaluated based on
s
D
W
W
4 (3-2)
where DW and sW energy dissipated in one loading cycle and the peak strain energy
stored in the system at the maximum displacement, respectively. Using Eq. (3-2), Qiu
(2008 and 2010) provided an analytical solution for Biot flow induced damping in
saturated poroviscoelastic soil during shear wave excitations for the cases of resonant
column (RC) tests and 1-D site response analysis. Qiu (2010) showed that Biot flow
induced damping is mainly dependent on two parameters: hydraulic conductivity and
excitation frequency, and for coarse sands and clean gravels, Biot flow-induced damping
may be significant. Using a different approach, Michaels (2006 and 2008) proposed a
Kelvin-Voigt-Maxwell-Biot (KVMB) model to connect pore fluid induced damping to
permeability and inertial coupling of saturated soil. In the proposed KVMB model, fluid
and soil skeleton masses are permitted to have separate motions; while in the traditional
Kelvin-Voigt (KV) model, the masses are lumped together as a single unit.
Although Qiu’s (2010) solution is of interest to soil dynamics and geotechnical
earthquake engineering, Eq. (3-2) cannot be readily implemented in resonant column
tests, in which the energy dissipation and strain energy cannot be measured directly or
determined indirectly through the stress-strain hysteresis loops. In resonant column tests,
49
damping can be obtained from the half-power bandwidth (HPB) method which is
applicable to small strain levels, or the free vibration decay (FVD) method which is
applicable to a wide range of strain levels (Kim 1991 and Hwang 1997).
In this study, analytical solutions for Biot flow induced damping in a saturated
poroviscoelastic soil specimen in resonant column tests are derived based on the HPB
method and the FVD method. For the analytical study, the solid skeleton is treated as
linearly viscoelastic thus limiting the application of this study to small strain levels. The
solution from the FVD method is compared to RC test results of various granular
materials at dry and saturated conditions. In the following sections, the governing
equations and their stead-state solutions are first reviewed, followed by derivations of the
analytical solutions. These solutions are compared to the solution provided by Qiu
(2010) based on Eq. (3-2). Effects of porosity and the ratio of mass polar moment inertia
between the driving system and soil specimen on Biot flow induced damping are
discussed. Finally, the difficulty in measuring Biot flow induced damping in typical
resonant column apparatuses is discussed.
3.2 Governing Equations
Following Biot theory (1956) and the recent work of Qiu (2010), governing equations
for shear wave propagation along x direction in saturated poroviscoelastic soil can be
expressed as
)()1(2
2
3
2
2
2
2
2
2
fsfssf
as
as tF
k
gn
txxG
ttn
(3-3a)
50
)(2
2
2
2
2
fsff
afs
a tF
k
gn
tn
t
(3-3b)
where n , , , t , G , , g and k denote porosity, density, angular displacement,
time, shear modulus, shear coefficient of viscosity, acceleration of gravity and hydraulic
conductivity, respectively. Subscripts s and f denote solid phase and fluid phase,
respectively. Coefficient a represents the mass coupling between the solid and fluid
phases. Biot (1962) proposed the following relationship:
11 fa n (3-4)
where 1 is a structural factor representing the “added mass” caused by the tortuosity of
the pore space (Stoll and Bryan 1970). Sen et al. (1981) proposed 5.01
n for a random
array of spheres. Gajo (1996) compared results from the experiments of Johnson et al.
(1982) with results from various theoretical models and found that Sen’s equation gave
the best estimate overall. In this study, Sen’s equation is used to estimate 1 . In Eq. (3-
3), F is a complex function that accounts for the effect of non-Poiseuille flow at high
frequencies and is given by (Biot 1956)
Qi
QF
84
2
(3-5)
ibeiber
ibeiberQ
''
(3-6)
f2 (3-7)
ng
fk
f
ff
c
2 (3-8)
51
where f is the wave frequency, cf is the characteristic frequency ( kng 2/ ), f is a
normalized frequency, 2 is a factor dependent on the geometry of the pores and is
approximately 8 for circular and slit-like pores (Biot 1956), ber and bei are Kelvin
functions of the first kind and zero order, the prime operator denotes differentiation, and
1i . For detailed discussions on F , Poiseuille flow, and non-Poiseuille flow, please
refer to Biot (1956) and Qiu (2010). For most geotechnical applications, the Darcy
assumption (i.e., Poiseuille flow) is valid.
Eq. (3-3a) and Eq. (3-3b) are obtained by considering the motions of the solid phase
and fluid phase separately. To consider the two phases as a whole system, the two
equations can be combined to yield
txx
Gtt
n ssff
ss
2
3
2
2
2
2
2
2
)1(
(3-9)
where the viscous coupling term vanishes as it becomes an internal force. It is evident
that Eq. (3-9) reduces to the classic governing equation for torsional wave propagation in
a single-phase soil by Bishop (1959) and Hardin (1965) for the case of dry soil by setting
0f and for the case of saturated soil by setting fs (i.e., ignoring the relative
motion between the solid and fluid phases). Eq. (3-3) also reduces to the governing
equation provided by Qiu and Fox (2006) by setting 11 (i.e., ignoring mass coupling)
and 1F (i.e., ignoring non-Poiseuille flow).
52
3.3 Analytical Solutions for Biot Flow Induced Damping
Eq. (3-3) is applied to a saturated poroviscoelastic soil specimen in resonant column
test in Hall’s fixed base model (Hall 1962). As shown in Fig. 3-1, the specimen with
length L is fixed at its base ( 0x ) and subject to a torsional excitation at its top ( Lx
). Energy dissipation in viscoelastic materials and during wave propagation in soil can be
evaluated using various parameters, including the coefficient of attenuation, loss angle,
specific damping capacity, quality factor, logarithmic decrement, and damping ratio and
these parameters are interrelated mathematically (Kramer 1996; Santamarina et al. 2001).
Within the context of resonant column testing, damping ratio of the specimen can be
obtained from the HPB method or the FVD method. In the latter, resonant frequency of
the specimen is first identified through a frequency sweep, the specimen is then excited at
the resonant frequency until a steady state is reached. After shutting off power supply to
the electromagnetic loading system, the specimen undergoes damped free vibration.
Damping ratio can then be obtained from logarithmic decay response of the free vibration
based on the theory of a damped SDOF system. In this section, spectral response of the
saturated poroviscoelastic soil specimen is first derived. Based on the spectral response,
resonant frequency of the specimen is obtained and damping ratio of the specimen is
evaluated based on the HPB method. Decay response of the soil specimen under damped
free vibration after resonance is subsequently derived. Damping ratio of the specimen is
then evaluated based on the FVD method.
53
Fig. 3-1. Boundary conditions of a poroviscoelastic soil specimen in resonant column
test
3.3.1 Spectral Response
General solution for the steady-state response of the poroviscoelastic soil specimen
shown in Fig. 3-1 under harmonic torsional excitation tioeTT at its top, where T is
the applied torque and oT is the torque amplitude, can be expressed as
tiss extx , (3-10a)
tiff extx , (3-10b)
where circular frequency; xs and xf amplitudes of the angular displacement
of the solid and fluid phases, respectively. Based on the work of Qiu (2008 and 2010),
the solution for xs can be expressed in a normalized form by using a normalized
frequency and a normalized hydraulic conductivity as
Lx
tLTT ,
tioeT
tI
I
Fixed
54
1
2
11
1
sincos21
sin
I
Ii
L
x
GJ
LTx
ts
os (3-11)
where J polar moment of inertia of the soil specimen; tI mass polar moment of
inertia of the loading system connected to the specimen at the top; I mass polar
moment of inertia of the saturated specimen (i.e., LJI sat ). The solid skeleton is
considered as equivalent linear (i.e., viscoelastic) and the skeleton damping ratio s is
defined as
Gs 2
(3-12)
This treatment of solid skeleton damping is widely used in geotechnical earthquake
engineering and soil dynamics (Ishihara 1995 and Kramer 1996). Dimensionless
complex parameter 1 is given by
ninF
inFi s
1
11 21 (3-13)
sGnn
1
1
(3-14)
where sG specific gravity of solids. The normalized frequency and hydraulic
conductivity are defined as
sV
L (3-15a)
gL
Vk s (3-15b)
55
where sats GV and sat saturated soil density.
Values of soil parameters used in this study are presented in Table 3-1. The range of
k for different soil types is based on Das (2010). In order to emphasize the effect of
hydraulic conductivity on Biot flow induced damping in different soils, constant values
of n , sG , and sV are used. This treatment is justifiable because the variation of these
parameters among different soils is significantly less than the variation of hydraulic
conductivity. The height of the test specimen is assumed to be 0.2 m (i.e., 2.0L m) in
the calculation of . The pore fluid is assumed to be water as in most cases in
geotechnical engineering; however, other types of pore fluid can be considered with the
corresponding values of k and (e.g., silicone-saturated soil in centrifuge testing). The
skeleton damping ratio s increases with shear strain. At small strain levels, s is
generally less than 0.02; therefore, s is considered to be 0.02 for simplicity in this study.
The dynamic amplification factor at the top of the specimen can be expressed as
A where the transfer function A is
1
2
11
1
sincos21
sin
I
Ii
At
s
(3-16)
Fig. 3-2 presents the spectral response of A for three soil specimens with 4.0n ,
02.0s and 0IIt , but different values. Values of 310 and 0.1 correspond
to fine sand and coarse sand, respectively (see Table 3-1). Value of 310 corresponds
to a material with hydraulic conductivity much higher than gravels as suggested by Table
56
3-1. Fig. 3-2 demonstrates that the three specimens have different resonant frequencies.
This can be explained using two extreme cases: an impermeable soil specimen (i.e.,
0 ) and an infinitely permeable soil specimen (i.e., ) (Qiu 2010, Qiu and Fox
2008, Youn et al. 2008). For the impermeable specimen, there is no relative motion
between the solid and fluid phases and the shear wave velocity is sV . This velocity
produces the first resonance at 2o . Conversely, there is no viscous coupling
between the two phases for the infinitely permeable specimen and shear wave propagates
at a higher speed nVs 11 , yielding a higher resonant frequency
68.111 no . For the soil specimen with 1.0 , its resonant
frequency is between o and . The dependence of the first resonant frequency on
is discussed in detail by Qiu (2010) and will not be further discussed herein.
Table 3-1. Typical values of soil parameters ( 2.0L m, 02.0s )
Soil Type n sG
(m/s) k
(m/s)
Gravel 0.4 2.66 200 0.01 – 1 1 – 100Coarse Sand 0.4 2.66 200 10-4 – 10-2 0.01 – 1Fine Sand 0.4 2.66 200 10-5 – 10-4 10-3 – 10-2
Silt 0.4 2.66 200 10-7 – 10-5 10-5 – 10-3
sV
57
Fig. 3-2. Spectral response of A for three specimens with different values ( 4.0n
02.0s 0IIt )
3.3.2 Damping from Half-Power Bandwidth Method
Fig. 3-2 shows that the soil specimens with 310 and 310 yield significantly
higher peaks in the spectral response than does the specimen with 1.0 , which
suggests that the total damping of the specimen with 1.0 is significantly higher.
Given that the skeleton damping of these specimens is assumed to have a constant value
of 0.02, the difference in total damping among these specimens is attributed to Biot flow
induced damping. Based on the spectral response, the total damping in the specimens can
be evaluated using the HPB method as demonstrated in Fig. 3-3, using the soil specimen
with 1.0 as an example. For the spectral response of a given soil specimen, the
resonant frequency r that produces the maximum amplification max
A and the
0.01
0.1
1
10
100
0 1 2 3 4 5 6
A
3101.0310
o
58
frequencies 1 and 2 that produce the amplitude 2max
A can be analytically obtained
from Eq. (3-16); the total damping of the soil specimen can be calculated as
r
2
12 (3-17)
It is widely recognized that r primarily depends on IIt and r decreases as IIt
increases. It can also be observed from Eqs. (3-13), (3-15), and (3-16) that parameters
such as n , s , and k have secondary effects on r (Qiu 2010). Both Biot flow induced
damping and solid skeleton damping contribute to the total damping in the specimen;
therefore, an equivalent damping ratio for Biot flow induced damping can be calculated
as
sf (3-18)
Fig. 3-3. Illustration of half-power bandwidth method for damping determination
1
10
1 1.2 1.4 1.6 1.8 2
A
r
maxA
2max
A
1 21.0
59
3.3.3 Damping from Free Vibration Decay Method
Analytical solutions of transient phenomena in saturated poroelastic media based on
Biot theory have been provided by many researchers (Garg et al. 1974, Simon 1984, Gajo
and Mongiovi 1995, Kamero et al. 2008). A comprehensive literature review on this
subject is provided by Schanz (2009). The solution procedures typically utilize Laplace
transform or Fourier transform, which are cumbersome, particularly when complex
parameters are involved. In this study, an approximate solution is provided by assuming
that the free vibration follows a logarithmic decay. Thus, the solution for decay response
of the soil specimen following a harmonic steady-state vibration at its resonant frequency
r can be expressed as
tibss
rextx , (3-19a)
tibff
rextx ,
(3-19b)
where r angular velocity at the resonance; b is a real number and is related to the
logarithmic decrement as
2
b (3-20)
Substituting Eq. (3-19) into Eq. (3-3) and solving for xs gives
2
2
1
G
ibV
ib
x
x
rs
r
s
s
(3-21)
where 2 is a dimensionless complex parameter
60
Fn
ib
Fn
ibn
r
r
1
1
2
(3-22)
The general solution for xs can be expressed as
x
G
ibV
ibAx
G
ibV
ibAx
rs
r
rs
rs
1
exp
1
exp 22
21
(3-23)
where constants 1A and 2A can be determined by the boundary conditions provided
below (Qiu 2010)
0,0 ts (3-24a)
0,,, 2
2
2
tx
tLJ
x
tLGJ
t
tLI sss
t
(3-24b)
Eqs. (3-24a) and (3-24b) correspond to the fixed boundary condition at the bottom and
free vibration condition at the top of the soil specimen, respectively. Substituting Eq. (3-
23) into Eq. (3-24) yields 21 AA and
01
1
tanh
1
1 2222
G
ibV
Lib
G
ibV
Lib
I
I
rs
r
rs
rt
(3-25)
Eq. (3-25) can be presented in a dimensionless form as
01
21tanh
21
1 2222
ib
ib
ib
ib
I
I
s
r
s
rt
(3-26)
For a soil specimen in resonant column test with known r , , and tI , Eq. (3-26) is a
function of only one unknown variable b . For a single-phase viscoelastic soil specimen
61
(e.g., dry soil by setting 0f ), a closed-form solution of b can be obtained from Eq.
(3-26) and this solution is consistent with the classical solution provided by Hardin
(1965). For a two-phase poroviscoelastic soil specimen, however, a closed-form solution
of b is not available to the authors’ knowledge due to the complex nature of Eq. (3-26).
Instead, a numerical solution can be obtained by searching for a b value that satisfies Eq.
(3-26). Fig. 3-4 presents the absolute value of the left hand side (LHS) of Eq. (3-26) as a
function of b for the three soil specimens considered in Fig. 3-2. Fig. 3-4 demonstrates
that the numerical solution of b can be obtained by searching for a b value to yield
0 Eq.(26)LHS. (i.e., satisfies Eq. (3-26)) for the cases corresponding to 310 and
310 . However, a b value that strictly satisfies Eq. (3-26) doesn’t exist for the case
corresponding to 1.0 and the b value that yields the minimum absolute value of
LHS of Eq. (3-26) can be considered as an approximate solution. Once the solution for
b is obtained, the total damping of the specimen can be calculated as
21 b
b
(3-27)
For small values of b , b . Biot flow induced damping ratio f can be obtained by
substituting Eq. (3-27) into Eq. (3-18).
62
Fig. 3-4. (26) LHS.Eq versus b for three specimens with different values ( 4.0n ,
02.0s , 0IIt )
Fig. 3-4 indicates that the total damping ( b ) is essentially the same as the
skeleton damping s (i.e., 0.02) for the cases of 310 and 310 . In these two cases,
Biot flow induced damping is negligible. However, the total damping is much higher
than the skeleton damping for the case of 1.0 (coarse sand), in which Biot flow
induces significant damping. The nonexistence of a b value that strictly satisfies Eq. (3-
26) suggests that, when Biot flow induced damping is significant, the decay response of a
specimen under free vibration after resonance may not strictly follow logarithmic decay.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.02 0.04 0.06 0.08
b
3101.0310
63
3.4 Resonant Column Test
RC tests were performed on granular materials to quantify Biot flow induced
damping and compare with the analytical solution based on the FVD method. The
strategy is to measure damping ratio as a function of effective confining stress for
specimens in both the dry and saturated conditions. The difference in damping values
between the dry and saturated conditions is considered to be Biot flow induced damping
(i.e., f ). These tests were performed following the same procedure as presented in
Chapter 2 and two granular materials: ASTM 20-30 Ottawa sand (ASTM C778) and
Soda-lime glass beads of three sizes, were utilized. Both glass beads and Ottawa sand
had rounded particles. Material and specimen properties are presented in Table 3-2.
Values of sG were provided by the manufacturer. Effective diameter 10D values for the
glass beads are obtained assuming that particle sizes are uniformly distributed within the
given range; for ASTM 20-30 Ottawa sand, 10D value is from the grain size distribution
curve provided by the manufacturer (see Fig. 2-1). Value of tI is obtained from
calibration tests and satI is calculated based on the geometry of specimens (i.e., solid
cylindrical specimens) and density of tested materials. Value of k is estimated based on
empirical relationship for uniform sands and gravels (Chapuis 2004), which assumes it is
a function of effective diameter and void ratio/porosity (see Eq. (2-4)).
64
Table 3-2. Parameters of samples for the analysis of RC tests and FVD.
Specimen sG Particle Size Range (mm)
10D
(mm) n
sat
t
I
I k
(cm/s)
Glass Beads (mm) 2.5 0.4 – 0.6 0.436 0.370 1.273 0.14 1.7 – 2.1 1.754 0.376 1.272 1.25 3.8 – 4.4 3.874 0.370 1.260 4.11
ASTM 20-30 sand 2.65 0.6 – 1.18 0.641 0.387 1.274 0.28 0.6 – 1.18 0.641 0.343 1.239 0.19
3.5 Results and Discussion
Fig. 3-5 presents the comparison of the analytical solutions of f versus based on
Eq. (3-2) as presented in (Qiu 2010), HPB method, and FVD method for the case of
4.0n , 02.0s , 0IIt . Fig. 3-5 indicates that f maybe significant for values
within the range of coarse sand and gravel but is negligible for values outside of this
range. For soil specimens with low values (e.g., medium sand and less permeable
soils), f is negligible because the relative motion between the solid and fluid phases is
negligible; for soil specimens with high values (e.g., more permeable than gravel), f
is negligible because the viscous coupling force between the two phases is negligible
despite a potentially large relative motion. Energy dissipation is a product of the
dissipative viscous coupling force and relative motion between the two phases. In coarse
sand and gravel, the combination of a moderate viscous coupling force and a moderate
relative motion may produce significant damping. Fig. 3-5 demonstrates that the three
analytical solutions are generally consistent with only minor differences around their
peak values. For a damped SDOF system where its damping is purely viscous, the three
65
methods would yield identical damping values for the system. This suggests that Biot
flow induced damping can be described as viscous with little penalty.
Fig. 3-5. Comparison of analytical solutions of f based on different methods ( 4.0n ,
02.0s , 0IIt )
Given that the FVD method is the most commonly used method to evaluate damping
in resonant column tests, subsequent discussions in this study will be based on the
solutions of f from this method. Fig. 3-6 presents f versus for 3.0n , 0.4 and
0.5 for the case of 02.0s and 0IIt . All three curves reach their maximum values
of f for in the range of approximately 0.1 to 0.2, corresponding to specimens of
coarse sand. Fig. 3-6 demonstrates that f increases as porosity increases due to the fact
0
0.01
0.02
0.03
10-3 10-2 10-1 1 101 102 103
f
Eq. (2)
Coarse Sand Gravel
HPB Method
FVD Method
Eq.(3-2)
66
that a larger fluid mass is capable of producing more energy loss through its viscous
interaction with the solid skeleton.
Fig. 3-6. f versus for various n ( 02.0s , 0IIt )
Fig. 3-7 presents the effect of IIt on f , which indicates that f decreases
considerably as IIt increases. The decrease of f is due to the decrease of the resonant
frequency as IIt increases and the resulted different flow and displacement patterns
(i.e., mode shapes) along the specimens as discussed in detail by Qiu (2010). For a
typical resonant column apparatus and soil specimen, the value of IIt is generally
larger than 1. For example, the resonant column apparatus at The Pennsylvania State
University has 25.1IIt for a soil specimen with 100 mm in diameter and 200 mm in
height; the resonant column apparatus at the Utah State University has 10IIt for a
f
Coarse Sand Gravel
0
0.01
0.02
0.03
0.04
0.05
10-3 10-2 10-1 1 101 102 103
5.0n
4.0n
3.0n
67
soil specimen with 35 mm in diameter and 70 mm in height (Sasanakul 2005). Hence,
f is suppressed by the boundary condition and difficult to be measured in typical
resonant column apparatuses. On the other hand, the boundary condition of 1-D site
response is similar to the case of 0tI and f can be significant (2010).
Fig. 3-7. f versus for various IIt ( 02.0s , 4.0n )
Fig. 3-8 presents a comparison of Biot flow induced damping between RC test data
and corresponding analytical solution from the FVD method. For the 1.7 – 2.1 mm glass
beads, there is a good agreement between the RC tests data and the analytical solution as
shown in Fig. 3-8(b). However, Biot flow induced damping is overestimated by the
analytical solution for the 0.4 – 0.6 mm glass beads and ASTM 20-30 sand as shown in
Figs. 3-8(a) and 3-8(d) and underestimated by the analytical solution for the 3.8 – 4.4 mm
0
0.01
0.02
0.03
10-3 10-2 10-1 1 101 102 103
f
Coarse Sand Gravel
IIt
0
5.01
2
4
68
glass beads as shown in Fig. 3-8(c). Fig. 3-8(d) indicates that Biot flow induced damping
is dependent on porosity based on the analytical solution; however, this dependence is not
reflected in the RC test results. Fig. 3-8 indicates that the RC test data scattered around
the FVD solution in a non-predictable pattern, thus the validity of FVD solution is
considered as inconclusive based on these data. This is largely due to the very small
magnitude of Biot flow induced damping in RC tests based on the conclusions from Fig.
3-7.
0
0.001
0.002
0.003
0.004
0.005
0.006
10-3 10-2 10-1 100 101 102 103
RC test
FVD
f
K
Glass beads (0.4 - 0.6 mm)
(a)
69
0
0.001
0.002
0.003
0.004
0.005
0.006
10-3 10-2 10-1 100 101 102 103
RC test
FVD
K
f
Glass beads (1.7 - 2.1 mm)
(b)
0
0.001
0.002
0.003
0.004
0.005
0.006
10-3 10-2 10-1 100 101 102 103
RC test
FVD
Glass beads (3.8 - 4.4 mm)
f
K
(c)
70
Fig. 3-8. Comparison of f between solution from FVD and RC test results: (a) glass
beads (0.4 – 0.6 mm); (b) glass beads (1.7 – 2.1 mm); (c) glass beads (3.8 – 4.4 mm); and
(d) ASTM 20-30 sand
3.6 Conclusions
Biot flow induced damping is due to the viscous coupling between solid and fluid
phases in saturated porous media. This study presents two analytical solutions of Biot
flow induced damping in saturated poroviscoelastic soil specimens in resonant column
tests using the half-power bandwidth and free vibration decay methods. Resonant
column tests were conducted to validate the analytical solution from the free vibration
decay method. The following conclusions are reached as a result of this investigation:
The two methods yield consistent solutions of Biot flow induced damping, which are
also consistent with a previous solution based on the dissipated energy and maximum
0
0.001
0.002
0.003
0.004
0.005
0.006
10-3 10-2 10-1 100 101 102 103
f
K
n = 0.387 0.343
RC test
FVD
(d)
ASTM 20-30 Sand
71
strain energy during one cycle of vibration. The consistent solutions from these three
methods suggest that Biot flow induced damping can be described as viscous.
The analytical solutions indicate that Biot flow induced damping may provide an
important contribution to total soil damping in coarse sand and gravel, but can be
practically neglected for less permeable soils (e.g., fine sand, silt, and clay). The
analytical solutions indicate that Biot flow induced damping increases as porosity
increases and decreases considerably as IIt increases. Given the range of IIt
values for typical resonant column apparatuses, Biot flow induced damping is
suppressed by the boundary condition and hence difficult to be measured.
The comparison between the solution from the FVD method and RC test data
suggests that the validity of this analytical solution is inconclusive. This is largely
due to the very small magnitude of Biot flow induced damping in RC tests.
Notations
10D = effective diameter;
A = transfer function at top of soil specimen;
max
A = maximum amplification;
1A , 2A = constants determined by boundary conditions;
b = real number related to logarithmic decrement;
F = complex function accounting for non-Poiseuille flow;
f = excitation frequency;
72
f = normalized frequency;
cf = characteristic frequency;
G = shear modulus;
sG = specific gravity of solids;
g = acceleration of gravity;
I = mass polar moment of inertia of soil specimen;
tI = mass polar moment of inertia of loading system;
i = 1 ;
J = polar moment of inertia of soil specimen;
k = hydraulic conductivity;
L = length of soil specimen;
n = porosity;
Q = complex function involved in the calculation of F ;
T = applied torque;
oT = amplitude of torque;
t = time;
sV = reference shear wave velocity in soil;
DW = energy dissipated in one loading cycle;
sW = peak strain energy stored in one loading cycle;
x = spatial coordinate;
1 , 1 = dimensionless complex parameter;
73
= dimensionless parameter;
= logarithmic decrement;
= amplitude of angular displacement;
= function involved in the calculation of F ;
= shear coefficient of viscosity;
= angular displacement;
= density;
a = mass coupling coefficient;
sat = saturated density;
1 = structural factor representing “added mass”;
2 = factor dependent on pore geometry;
= circular frequency;
r = circular frequency at resonance;
= damping;
= normalized hydraulic conductivity;
= normalized frequency;
1 , 2 = normalized frequencies corresponding to 2
maxA
;
r = normalized resonant frequency;
o = normalized frequency at first resonance for impermeable soil;
= normalized frequency at first resonance for infinitely permeable soil;
74
ber , bei = Kevin functions of the first kind and zero order;
Subscripts:
f = fluid phase;
s = solid phase;
75
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Zienkiewicz, O.C., Chan, A.H.C., Pastor, M., Schrefler, B.A. and Shiomi, T. (1999).
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Qiu, T. (2010). “Analytical Solution for Biot Flow-Induced Damping in Saturated Soils
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Shear Waves in Saturated Soil.” J. Geotech. and Geoenviron. Eng., 134(12), 1815-
1819.
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modulus Gmax of dry and saturated sands by bender element, resonant column, and
torsional shear tests.” Can. Geotech. J., 45(10), 1426-1438.
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Simon, B.R., Zienkiewicz,C. and Paul, D.K. (1984). “An Analytical Solution for the
Transient Response of Saturated Porous Elastic Solids.” Int. J. Numer. Anal. Methods
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80
Chapter 4
Energy Dissipation in Nearly Saturated Poroviscoelastic Soil Column during Quasi-Static Compressional Excitations
4.1 Introduction
The dynamic response of a saturated porous column under compression wave
excitations has been investigated by numerous researchers based on Biot’s theory (Biot
1956). Zienkiewicz et al. (1980) discovered that three zones of problems exist depending
on the speed of the fluid and solid motions. Zone I corresponds to slow phenomena that
inertial effects can be reasonably neglected and consolidation equations provide
satisfactory solutions. Zone II corresponds to problems involving moderate speeds where
the inertial terms cannot be neglected, however, the relative acceleration between the
pore fluid and solid phases can be neglected. Zone III corresponds to fast phenomena
where the full solution of Biot’s theory has to be sought. Zienkiewicz et al. (1980) and
(1999) concluded that most problems related to geotechnical earthquake engineering fall
into Zone II, where the solutions can be obtained using the u-p dynamic formulation. A
comprehensive literature review on the dynamic response of a saturated poroelastic
column under compression wave excitations is provided by Schanz (2009) and this
review is not presented herein. Most of the work conducted to date focuses on the wave
propagation (e.g., Simon et al. 1984, Hiremath et al. 1988, Gajo and Mongiovi 1995),
spectrum response (e.g., Bardet 1995, Yang and Sato 2001) and pore pressure generation
(e.g., Madsen 1978, Yamamoto 1981, Mei and Foda 1981) aspects of the dynamic
81
behavior. Few investigations have explored the energy dissipation in the poroelastic
column resulted from the viscous interaction between the solid and fluid phases.
Gent and Rusch (1966) investigated the viscoelastic behavior of open-cell foams to
explain the large mechanical damping observed due to the energy dissipation in forcing
air in and out of the foam during a deformation cycle. The mechanical damping in open
foams has also been investigated more recently by Jaouen et al. (2005) and Lind-
Nordgren and Goransson (2010). Wijesinghe and Kingsbury (1979) investigated the
dynamic behavior of a saturated 1-D poroelastic column with the inertial terms neglected.
The storage modulus and loss tangent were used to infer the energy dissipation inside the
poroelastic column. They observed that the dissipative effect manifests itself at relatively
low frequencies compared to the frequencies at which the inertia effects begin to
influence the behavior. Bardet (1995) presented a theoretical investigation on pore fluid
induced energy dissipation in saturated poroelastic soil during compression wave
excitations. In his work, pore fluid induced damping was quantified using the theory of a
viscously damped single-degree-of-freedom (SDOF) system based on the vibration
amplitude of the soil column. Bardet showed that for saturated gravels and dense sands,
the equivalent damping ratio contributed by pore fluid varies from 0 to 0.7; while the
skeleton damping ratio for such soils typically varies from 0.01 to 0.07 (Richart et al.
1970). It was therefore concluded that pore fluid induced damping in saturated gravels
and sands is not negligible and may be large compared to soil skeleton damping under
compression vibrations. Bardet’s work highlighted the importance of various factors
including the stiffness of solid skeleton, bulk modulus of pore fluid (i.e., degree of
saturation), hydraulic conductivity, and excitation frequency on the dynamic response of
82
saturated poroelastic soil and pore fluid induced damping. In Bardet’s work, however, the
evaluation of damping in the column based on the vibration amplitude is unable to
account for deamplifications of the column. Following Bardet’s work, Qiu and Fox
(2006) investigated the effect of boundary drainage conditions on pore fluid induced
damping in a saturated poroelastic soil column. Recently, the dynamic response of
poroviscoelastic column has been investigated (e.g., Abousleiman et al. 1996,
Abousleiman and Kanj 2004, Schanz and Cheng 2001). In these investigations, energy
dissipation in the solid skeleton has been incorporated by considering the skeleton
viscoelastic. Grasley and Leung (2011) studied the quasi-static axial damping of
poroviscoelastic cylinders, with special reference to cementitious materials.
This study presents a theoretical investigation on the energy dissipation in a 1-D
poroviscoelastic soil column under quasi-static compressional excitations, where the
inertial effects are neglected. Therefore, this study is applicable to slow phenomena in
zone I (Zienkiewicz et al. 1980). Neglect of the inertial terms is also justifiable for two
other reasons. First, without the interference from phenomena due to inertia (e.g.,
resonance, slow and fast waves), this study can focus on the energy dissipation due to the
fluid diffusion alone (Wijesinghe and Kingsbury 1979). Second, the error due to this
simplification is considered to be small compared to the errors due to the uncertainty of
geotechnical engineering properties of soils in the field (Yamamoto 1981). In this study,
the solid skeleton is treated as linearly viscoelastic thus limiting the application of this
study to small strain levels. In this study, the governing equations and their solutions are
first presented, followed by the development of closed-form analytical solutions of the
steady-state response, fluid induced energy dissipation, solid skeleton induced energy
83
dissipation, and total damping ratio of the soil column for two boundary drainage
conditions. An illustrative example is also presented to demonstrate the application of
this study to a practical situation.
4.2 Governing Equations
Following Biot theory (1956), governing equations in various forms have been
proposed for the propagation of 1-D compression waves in saturated poroelastic soil.
Equations originally proposed by Zienkiewicz and Shiomi (1984) based on the general
theory of mixtures and recently modified by Qiu and Fox (2008) are adopted herein due
to their simplicity. These equations are modified to incorporate the viscosity of the solid
skeleton. The modified equations for 1-D compression waves in nearly saturated
viscoelastic soil are
)()()1(2
2
3
2
2
2
2
t
u
t
u
k
gn
x
pn
tx
u
x
uD
t
un sffss
ss
s
(4-1a)
)(2
2
2
t
u
t
u
k
gn
x
pn
t
un sfff
f
(4-1b)
where , u , sD , , p , n , k , g , t , and x denote density, displacement, constrained
modulus of the soil skeleton, coefficient of viscosity of the soil skeleton, fluid pressure,
porosity, hydraulic conductivity, acceleration of gravity, time, and spatial coordinate
along axis of wave propagation, respectively, and is an effective stress parameter.
Subscripts s and f indicate solid and fluid phases, respectively. Eqs. (4-1a) and (4-1b)
are obtained from the application of Newton’s second law to the solid phase and fluid
phase in a control volume, respectively. The last term on the right-hand side (RHS) of
84
each equation represents the seepage force between the two phases. If the inertial terms
are ignored, Eq. (4-1) becomes
0)()(2
2
3
2
2
t
u
t
u
k
gn
x
pn
tx
u
x
uD sffss
s
(4-2a)
0)(2
t
u
t
u
k
gn
x
pn sff
(4-2b)
Zienkiewicz and Shiomi (1984) developed a storage equation that accounts for fluid
mass conservation during fluid flow through a deformable porous medium. The volume
of pore space inside a given section of the solid skeleton changes as the section deforms.
Depending on the compressibility of the solid and fluid phases, fluid must flow in or out
of this section to maintain saturation. The storage equation for isothermal conditions is
expressed as
sf
ssf
K
n
K
n
t
p
tx
u
tx
u
tx
un
222
(4-3)
where fK and sK are the bulk modulii of the fluid and solid grains, respectively. The
first term of the RHS represents storage due to expansion of the solid skeleton (i.e.,
overall increase in volume) and the second term represents storage due to compressibility
(i.e., compression) of the fluid and solid particles.
The effective stress parameter can be expressed as
sKK1 (4-4)
where K is the bulk modulus of the soil skeleton. For most soils, sKK and 1 .
More discussion on the physical meaning of is provided by Nur and Byerlee (1971),
Garg and Nur (1973), and Oka (1996). In this study, sK is considered for simplicity;
85
therefore, 01 sK in Eqs. (4-3) and (4-4). For most soils under full saturation, the
compressibility of pore fluid (e.g., pure water in most cases) is negligible as compared to
that of the soil skeleton (i.e., KK f ). However, as the degree of saturation S
decreases, fK decreases drastically (Verruijt 1969). For example, for the fully saturated
case (i.e., %100S ), 9102.2 fK Pa (i.e., pure water); whereas for %95S ,
6100.2 fK Pa under one atmosphere pressure (Qiu and Fox 2008). It is assumed that
the mixture of pore water and pore air can be simplified as a single fluid phase with a
bulk modulus decreasing with decreasing S . This assumption is generally valid for
%90S .
Eqs. (4-2a), (4-2b), and (4-3) have three unknown variables: su , fu , and p and can
be simplified by eliminating fu . Combining Eqs. (4-2a) and (4-2b) and considering
1 yield
02
3
2
2
x
p
tx
u
x
uD ss
s (4-5a)
Differentiating Eq. (4-2b) over x and substituting the resultant equation to Eq. (4-3)
yield
pK
n
x
u
tk
g
x
p
f
sf2
2
(4-5b)
Eq. (4-5) only has two unknown variables: su and p . Eq. (4-5) is consistent with the
governing equations proposed by Wijesinghe and Kingsbury (1979) for poroelastic
materials (i.e., 0 ). Due to its simplicity, Eq. (4-5) is used as the governing equation
for 1-D low-frequency compression excitations in nearly saturated poroviscoelastic soil.
86
4.3 Analytical Solutions of Steady-State Motion
Fig. 4-1 presents the geometry and boundary conditions for a 1-D poroviscoelastic
soil layer/column constrained from lateral movement. The layer is nearly saturated with a
length L , a free top boundary ( 0x ) and a fixed bottom boundary ( Lx ). Both the top
and bottom boundaries can be undrained or drained with zero fluid pressure; whereas the
side boundary is always undrained (i.e., 1-D). The soil layer is subject to a harmonic low-
frequency compressional excitation at the top: tioe , where total stress, o
amplitude of total stress, circular frequency, and 1i .
Fig. 4-1. Geometry of poroviscoelastic soil column
4.3.1 General Solution
The general solution for the steady-state response of the soil column can be expressed
as
tioet ),0(
0),( tLus
x
L
Side: undrained
Top and Bottom: drained/undrained
87
tis exUtxu , (4-6a)
tiexPtxp , (4-6b)
xeBxU 1 (4-6c)
xeBxP 2 (4-6d)
where xU and xP are the amplitudes of the solid displacement and fluid pressure,
respectively. Parameters 1B , 2B , and are determined in this section. Substituting Eq.
(4-6) into Eq. (4-5) yields two linear equations with two unknowns 1B and 2B
021 212 BBiD ss (4-7a)
022
1
Bi
k
g
K
nBi
k
g f
f
f
(4-7b)
where
s
s D2
(4-8)
The skeleton damping ratio s is assumed to be of Kelvin-Voigt type as commonly
defined in soil dynamics for the mode of shear excitations, in which case the shear
modulus is used in Eq. (4-8) instead of sD . It is assumed that this definition can be
extended to the mode of compressional excitations. It should be noted that s is
independent of frequency for the frequencies of geotechnical interests (Kramer 1996). To
ensure this frequency independency, Eq. (4-8) implies that is inversely proportional to
.
88
To have a nontrivial solution, the determinant of Eq. (4-7) must be equal to zero,
which gives the following characteristic equation
0
21121 22
f
ssfss K
inDi
k
giD
(4-9)
Eq. (4-9) has three roots and they are
01 (4-10a)
fss
f
K
n
iDk
ig
21
13,2 (4-10b)
For 1 (i.e., zero root), Eq. (4-7b) yields
02 B (4-11a)
For 3,2 (i.e., non-zero roots), Eq. (4-7a) yields
ss iDB
B
21
1
2
1
(4-11b)
Therefore, the general solution for xP and xU can be expressed as
sxsx eCeCxP 21 (4-12a)
sxsx
ss
eCeCsiD
CxU
213
1
21
1
(4-12b)
where
fss
f
K
n
iDk
igs
21
1 (4-13)
and 1C , 2C , and 3C are constants that are determined by the boundary conditions. By
observing the characteristics of Eq. (4-5), a particular solution can be identified
89
oCxP (4-14a)
xCK
nxU o
f
(4-14b)
where oC is a constant that is determined by the boundary conditions. The solution for
xP and xU can be obtained by combining the general solution in Eq. (4-12) and the
particular solution in Eq. (4-14) as
sxsxo eCeCCxP 21 (4-15a)
sxsx
sso
f
eCeCisD
xCK
nCxU
213 21
1
(4-15b)
The effective stress amplitude x can be expressed as
x
xUiDx ss
21 (4-15c)
4.3.2 No-Drainage (ND) Case
For the soil column with impermeable top and bottom boundaries, the boundary
conditions are
00
xx
P (4-16a)
ox
P 0
(4-16b)
0
Lxx
P (4-16c)
0LU
(4-16d)
90
Substituting Eq. (4-15) into Eq. (4-16) gives four linear equations with four unknown
coefficients jC ( j = 0, 1, 2, 3) and the solution for jC is
s
o
inC
2110 (4-17a)
021 CC
(4-17b)
os
CD
LnC
3 (4-17c)
where
f
s
K
D (4-18)
For saturated soil, 1 but as the degree of saturation decreases, increases
drastically. Substituting Eq. (4-17) to Eq. (4-15) gives
s
o
inxP
211 (4-19a)
L
x
in
n
D
LxU
ss
o 1211
(4-19b)
s
so in
inx
211
21
(4-19c)
4.3.3 Top-Drained (TD) Case
For the soil column drained only at the top, the boundary conditions are
00 P (4-20a)
ox
0
(4-20b)
91
0
Lxx
P (4-20c)
0LU
(4-20d)
Similarly, the following solutions can be obtained
s
o
inC
2110 (4-21a)
sL
o
e
CC
21 1
(4-21b)
sL
o
e
CC
22 1 (4-21c)
os
CD
LnC
3 (4-21d)
sL
Lx
sL
inxP
s
o
cosh
1cosh
1211
(4-22a)
sLsLi
Lx
sL
L
xn
inD
LxU
sss
o
cosh21
1sinh
1211
1
(4-22b)
sLi
Lx
sL
nin
ix
ss
so cosh21
1cosh
211
21
(4-22c)
4.3.4 Double-Drained (DD) Case
For the soil column drained at both the top and bottom, the boundary conditions are
92
00 P (4-23a)
ox
0
(4-23b)
0LP (4-23c)
0LU (4-23d)
Similarly, the following solutions can be obtained
s
o
inC
2110 (4-24a)
sLsL
sL
o ee
eCC
1
1
(4-24b)
sLsL
sL
o ee
eCC
12 (4-24c)
sL
sL
isDL
K
nCC
ssfo sinh
cosh1
21
13
(4-24d)
sL
sxLx
sL
inxP
s
o
sinh
sinh1sinh
1211
(4-25a)
sLsLi
sxLx
sLsL
L
xn
inD
LxU
sss
o
sinh21
cosh1coshcosh1
1211
1
(4-25b)
sLi
sxLx
sL
nin
ix
ss
so sinh21
sinh1sinh
211
21
(4-25c)
93
Eq. (4-25a) implies that the fluid pressure gradient (i.e, xP ) is zero at the midheight
of the soil column (i.e., 2Lx ), which is consistent with Terzaghi’s 1-D consolidation
theory (Terzaghi 1943) corresponding to the case of 0 and 0 .
The solutions given by Eqs. (4-19), (4-22), and (4-25) are complex functions. The
real parts of these functions generally represent the corresponding physical quantities, the
imaginary parts are related to the phase angles, and the absolute values represent the
amplitude or maximum values. For example, the fluid pressure distribution along the soil
column can be represented as xPRe , where function Re returns the real part of its
argument; whereas the maximum fluid pressure distribution along the soil column can be
represented as xP .
4.3.5 Normalization
It can be observed that the solutions discussed above involve a dimensionless term
sL . Eq. (4-13) gives
nikD
iLgsL
ss
f
21
12
(4-26)
The coefficient of consolidation vc of the soil column is
g
kDc
f
sv (4-27)
Substituting Eq. (4-27) to Eq. (4-26) yields
n
iisL
s21
1 (4-28)
94
where
vc
L2 (4-29)
Dimensionless parameter is a normalized excitation frequency, which reflects the
relative pace between two concurrent underlying processes: excitation and relaxation
(i.e., consolidation). For small , corresponding to low-frequency excitations and/or
small column thickness and/or large coefficients of consolidation, the soil column is
relaxed as the consolidation (i.e., fluid pressure dissipation) outpaces the loading. As
increases, the effect of loading gradually increases and the soil column becomes less
relaxed or more loaded.
4.4 Energy Dissipation and Damping
The amplitude of solid displacement at 0x can be expressed as
G
D
LU
s
o0 (4-30)
where G is the amplification factor and the transfer function G can be expressed as
sin
nG
211 (4-31a)
for the ND case and
sLsLi
sLn
inG
ss cosh21
sinh
211
1
(4-31b)
for the TD case and
95
sLsLi
sLn
inG
ss sinh21
cosh22
211
1
(4-31c)
for the DD case. It should be noted that G can also be interpreted as the amplitude of
solid displacement normalized by the amplitude corresponding to the fully relaxed state
of the column: dry column under a static load o .
Energy dissipation in viscoelastic materials and during wave propagation in saturated
poroelastic media has been evaluated using various parameters, including the coefficient
of attenuation, loss angle, specific damping capacity, quality factor, logarithmic
decrement, and damping ratio. These parameters are interrelated mathematically
(Cascante 1996). In soil dynamics and geotechnical earthquake engineering, the energy
dissipation in soil is often evaluated using the damping ratio defined as
W
WD
4 (4-32)
where DW and W are the energy dissipated in one loading cycle and the peak strain
energy stored in the system at the maximum displacement, respectively. Eq. (4-32) is
widely used to evaluate soil skeleton damping in low-frequency tests for the mode of
shear excitations, such as cyclic simple shear tests (e.g., Vucetic et al. 1998) and cyclic
torsional shear tests (e.g., Drnevich 1972). Based on this equation, Qiu (2010) provided
analytical solutions of pore fluid induced damping in saturated soils during shear wave
excitations for the cases of resonant column tests and 1-D site response. Eq. (4-32) is
adopted herein to evaluate the total damping in the soil column under compressional
excitations.
96
The peak energy stored in the system at the maximum displacement consists of two
components: the strain energy in the solid skeleton sE and the strain energy in the
compressible pore fluid fE
fs EEW (4-33)
The parameter sE can be calculated as
dxADE
L
xss 0
2
2
1 (4-34)
Where A is the cross-section area of the soil column; dxxdUx is the
compressive strain in the solid skeleton. Substituting Eqs. (4-19b), (4-22b), and (4-25b)
into Eq. (4-34) yields
2
222
2112 ss
os
in
n
D
ALE
(4-35a)
2
2
211
1
2 ss
os
inD
ALE
2222
cosh21
2sinh41
21
cosh21
sinh2
sLsLi
sLsL
sLsLi
sLnn
ss
(4-35b)
2
2
211
1
2 ss
os
inD
ALE
22
22
sinh21
sinhcosh2sinh21
sinh21
1cosh4
sLsLi
sLsLsLsLsL
sLsLi
sLnn
ss
(4-35c)
97
for the ND, TD, and DD cases, respectively. The first term on the RHS of Eq. (4-35)
represents the strain energy stored in the soil skeleton under a static load o for the case
of dry soil and the rest of the terms accounts for the interaction between pore fluid and
solid skeleton during the excitation.
To derive the solution for fE , let’s first consider some compressible fluid with a
volume of oV under a reference pressure of zero. At this state the strain energy in the
fluid is zero. As the fluid pressure changes to P due to an external perturbation, the
volume changes to V and the strain energy fE in the fluid can be expressed as
V
Vfo
dVVPE (4-36)
The fluid volume V is related to oV and P through
f
o
KP
VV
exp (4-37)
Substituting Eq. (4-37) into (36) gives
fffo
P
ff
of K
PKPKVdP
K
PP
K
VE expexp
0 (4-38)
For most applications, fKP and Eq. (4-38) can be approximated as
22
2
2
1
2
11 P
K
V
K
P
K
PKPKVE
f
o
ffffof
(4-39)
For the pore fluid in the soil layer, fE can be calculated as
dxxP
K
AndVP
KE
L
fff
0
22
22
1 (4-40)
Substituting Eqs. (4-19a), (4-22a), and (4-25a) into Eq. (4-40) gives
98
22
2112 ss
of
in
n
D
ALE
(4-41a)
sL
sL
sLsL
sL
in
n
D
ALE
ss
of
tanh2
cosh4
2sinh2
12112 22
2
(4-41b)
2
2
2112 ss
of
in
n
D
ALE
sLsL
sLsLsLsLsL
sLsL
sL2sinh
sinhcosh2sinh21
sinh
cosh141
(4-41c)
for the ND, TD, and DD cases, respectively.
The total energy dissipated in the soil column in one loading cycle is equal to the
work done by the externally applied load on the top boundary in one loading cycle (Qiu
2010)
GD
ALdt
dt
deUeAW
s
otioD
ti
Im0ReRe2
2
0
(4-42)
where function Im returns the imaginary part of its argument. Substituting Eq. (4-31)
into (42) yields DW for the ND, TD, and DD cases. Based on Eq. (4-32), the energy
dissipated in the solid skeleton due to the skeleton damping sDW can be expressed as
sssD EW 4 (4-43)
and sDW can be normalized based on the strain energy stored in the soil skeleton under a
static load o as
99
s
o
sDsD
D
ALWW
2
2*
(4-44)
Substituting Eq. (4-35) into Eqs. (4-43) and (4-44) gives sDW and *sDW for the ND, TD,
and DD cases. Both the solid skeleton and pore fluid contribute to the total energy
dissipation. Therefore, the energy dissipated due to pore fluid fDW can be expressed as
sDDfD WWW (4-45)
Similarly fDW can be normalized as
s
o
fDfD
D
ALW
W
2
2*
(4-46)
Substituting Eqs. (4-42) and (4-43) into Eqs. (4-45) and (4-46) yields fDW and *fDW
for
the ND, TD, and DD cases.
A closed-form solution of can be obtained by substituting Eqs. (4-33), (4-35), (4-
41), and (4-42) into Eq. (4-32). Both pore fluid induced damping and solid skeleton
damping contribute to the total energy dissipation in the soil column. Therefore, an
equivalent damping ratio for pore fluid induced damping f can be expressed as
sf (4-47)
TD and DD Equivalency
Based on the formulations presented above, it can be mathematically proven that a
TD column subjected to a normalized excitation of has the same amplification factor
and total damping as a DD column subjected to a normalized excitation of 4 . For
example, 4G in Eq. (4-31c) is mathematically identical as G in Eq. (4-31b).
100
This implies that if vc is the same, a DD column would have the same amplification
factor and total damping ratio as a TD column if the DD column has four times the
excitation frequency or twice the length as the TD column.
4.5 Results and Discussions
Figs. 4-2 and 4-3 present G vs. for a soil column with 4.0n and 02.0s for
the ND case and DD case, respectively. For the ND case, deformation of the soil column
is contributed solely by the instant compression of the solid skeleton and pore fluid;
therefore, G is frequency independent. Fig. 4-2 shows that G monotonically increases
as the pore fluid becomes more compressible (i.e., increases). For a soil column with
incompressible pore fluid (i.e., 0 ), 0G and 1G as . Fig. 4-3 shows
that G is frequency dependent for the TD case. For small values of , corresponding to
low-frequency excitations and/or small column thickness and/or large coefficient of
consolidation, 1G because the pore pressure dissipation (i.e., consolidation) outpaces
the loading and the soil column is fully relaxed. As increases, the loading gradually
outpaces consolidation and the soil column is increasingly loaded (i.e., less relaxed)
resulting in smaller G values. Fig. 4-4 presents the contour plot of G vs. and for
the TD case. It can be observed that for 1.0 , the column is essentially fully relaxed
regardless of fluid compressibility. Fig. 4-5 presents G vs. for the DD case. All the
curves in Fig. 4-5 can be obtained by shifting the corresponding curves in Fig. 4-3
horizontally to the right by a distance of 4log (i.e., 4loglog4log ) due to the
101
equivalency between TD and DD cases as discussed earlier. In contrary to the
amplifications observed in Bardet (1995) and Qiu and Fox (2006), no amplification (i.e.,
1G ) is observed in Figs. 4-2 through 4-5. This is due to the soil column being in
quasi-static condition.
Fig. 4-2. G vs. for a ND soil column ( 4.0n , 02.0s )
0
0.2
0.4
0.6
0.8
1
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.01
= 0.1
= 1
= 10
= 100
G
102
Fig. 4-3. G vs. for a TD soil column ( 4.0n , 02.0s )
Fig. 4-4. Variation of G vs. and for a TD soil column ( 4.0n , 02.0s )
0
0.2
0.4
0.6
0.8
1
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.01
= 0.1
= 1
= 10
= 100
G
0.01
0.1
1
10
100
10-410-3 10-2 0.1 1 10 102 103 104
|G| = 0.99
0.90
0.75
0.50
0.20
0.10
0.05
103
Fig. 4-5. G vs. for a DD soil column ( 4.0n , 02.0s )
Fig. 4-6 presents the maximum value of normalized fluid pressure in the column
oP max vs. for a TD soil column with 4.0n and 02.0s . It can be observed that
for small values of , the soil column is relaxed with negligible oP max . As
increases, the maximum fluid pressure increases. For large values of , the soil column
is fully loaded and oP max becomes constant. This constant value decreases as the fluid
compressibility increases, as more load is transferred to the solid skeleton.
0
0.2
0.4
0.6
0.8
1
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.01
= 0.1
= 1
= 10
= 100
G
104
Fig. 4-6. oP max vs. for a TD soil column ( 4.0n , 02.0s )
Fig. 4-7 presents the distribution of normalized fluid pressure oxP /Re and
effective stress ox /Re along a TD soil column with 1.0 , 4.0n , and
02.0s for different values of . For all values of , the sum of oxP /Re and
ox /Re is one, which is consistent with the effective stress principle and force
equilibrium. For the case of 210 , 0/Re oxP and 1/Re ox as the soil
column is fully relaxed and the solid skeleton takes all the load, which is consistent with
Fig. 4-6. As increases, oxP /Re increases and ox /Re decreases, indicating
that the applied load is gradually shifted to the pore fluid as the loading gradually
outpaces pore pressure dissipation. Fig. 4-7 demonstrates that for the case of 410 ,
fluid pressure gradient only exists in a region near the drainage boundary. The rest of the
column has a constant fluid pressure, the value of which corresponds to the ND case.
10-5
10-4
10-3
10-2
0.1
1
10
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.01
= 0.1
= 1
= 10
= 100
o
P
max
105
This implies that for the cases of large , drainage only “penetrates” into a thin layer of
soil adjacent to the drainage boundary; whereas the rest of the column essentially remains
undrained (i.e., no relative fluid motion). This observation is consistent with the boundary
layer theory of Mei and Foda (1981).
Fig. 4-7. Distribution of fluid pressure and effective stress along a TD soil column for
various ( 1.0 , 4.0n , 02.0s )
Fig. 4-8 presents sf EE vs. for a TD soil column with 4.0n and 02.0s .
Fig. 4-8 shows that sf EE monotonically increases with as more load is transferred
to pore fluid, which is consistent with Figs. 4-6 and 4-7. Fig. 4-8 also shows that for a
0
0.2
0.4
0.6
0.8
1-0.2 0 0.2 0.4 0.6 0.8 1 1.2
Re[P(x)]/o
Re['(x)]/o
= 10-2 1 102 104
oo
xxP
]Re[
or ]Re[
L
x
106
given , sf EE first increases with , but beyond certain values of , sf EE
decreases as increases further. This can be explained by two extreme cases:
incompressible pore fluid (i.e., 0 ) and infinitely compressible pore fluid (i.e.,
). For both cases, pore fluid does not contain strain energy and 0sf EE . Therefore,
maximum values of sf EE correspond to some intermediate values of for a given
. Fig. 4-8 also indicates that generally more strain energy is stored in the solid skeleton
than in the pore fluid (i.e., 1sf EE ) unless for the cases of large values of
combined with certain values of .
Fig. 4-8. sf EE vs. for a TD soil column ( 4.0n , 02.0s )
Figs. 4-9 and 4-10 present *sDW and *
fDW vs. for a TD soil column with 4.0n
and 02.0s , respectively. Fig. 4-9 indicates that for small values of (i.e., up to
10-6
10-5
10-4
10-3
10-2
0.1
1
10
0.1 1 10 102 103 104
= 0.01
= 0.1
= 1
= 10
= 100
s
f
E
E
107
0.02), *sDW has a constant value of s4 (see Eq. (4-43)) as the solid skeleton is fully
relaxed and the soil column behaves essentially as a dry column. As increases, the
solid skeleton induced energy dissipation decreases due to the load transfer from the solid
skeleton to pore fluid, which is consistent with the previous discussions. For large
values as the column is fully loaded, the skeleton induced energy dissipation increases as
the pore fluid becomes more compressible (i.e., increases). Fig. 4-10 indicates that
fluid induced energy dissipation is frequency dependent due to its viscous nature, and
significant energy dissipation occurs for values within the range of 0.1 to 310 . For
small values of , the soil column is fully relaxed and behaves essentially as a dry
column. The excitation-generated pore pressure is negligible (see Fig. 4-6) resulting in
negligible *fDW . For very large values of , the excitation only induces relative fluid
motion within a thin boundary layer (Mei and Foda 1981) where pore fluid induces
energy dissipation; whereas the rest of the column essentially remains undrained without
fluid induced energy dissipation, as shown in Fig. 4-7. Significant fluid induced energy
dissipation occurs for moderate values of due to a combination of moderate fluid
pressure, pressure gradient and fluid relative motion (i.e., drainage) throughout the soil
column (see Fig. 4-7). Fig. 4-10 also indicates that fluid induced energy dissipation
decreases as fluid becomes more compressible.
108
Fig. 4-9. *sDW vs. for a TD soil column ( 4.0n , 02.0s )
Fig. 4-10. *fDW vs. for a TD soil column ( 4.0n , 02.0s )
0
0.05
0.10
0.15
0.20
0.25
0.30
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.1
= 1
= 10
= 100
*sDW
0
0.5
1.0
1.5
2.0
2.5
3.0
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.1
= 1
= 10
= 100
*fDW
109
Figs. 4-11 and 4-12 present vs. for a soil column with 4.0n and 02.0s
for the ND case and TD case, respectively. All the mechanisms discussed above are
manifested in these two figures. Fig. 4-11 indicates that for the ND case, is
independent of and increases as the pore fluid becomes more compressible (i.e.,
increases). For a soil column with incompressible pore fluid (i.e., 0 ), 0 due to
the lack of any motion (i.e., 0G ) and as , 02.0 s . Eq. (4-47) seems to
suggest negative values of f (i.e., s ). The explanation is that for the ND case, the
viscous coupling between pore fluid and solid does not produce any energy dissipation
(i.e., 0 fDW and sDD WW ) due to the absence of relative fluid motion (i.e., ND
boundary condition). However, pore fluid contributes to the stiffness of the soil column
and results in more strain energy in the soil column. Therefore, the total damping ratio of
the soil column is smaller than that of the solid skeleton (see Eq. (4-32)). Qiu (2010)
presented an analytical solution of pore fluid induced damping in saturated soils during
shear wave excitations. For shear wave excitations, pore fluid does not contribute to the
skeleton stiffness and nor to the total strain energy, but always produces energy
dissipation due to the viscous coupling force. Therefore, Qiu (2010) demonstrated that a
saturated soil column/layer always has higher damping ratio than the skeleton damping
(i.e., s ) under shear wave excitations. Fig. 4-11 indicates that pore fluid induced
damping under compressional excitations is more complex than under shear wave
excitations, and pore fluid may increase or decrease the total damping ratio of the soil
column during compressional excitations.
110
Fig. 4-11. vs. for a ND soil column ( 4.0n , 02.0s )
Fig. 4-12. vs. for a TD soil column ( 4.0n , 02.0s )
0
0.01
0.02
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.01
= 0.1
= 1
= 10
= 100
0.01
0.1
1
10-4 10-3 10-2 0.1 1 10 102 103 104
= 0
= 0.01
= 0.1
= 1
= 10
= 100
111
Fig. 4-12 indicates that pore fluid may significantly contribute to the total damping in
a TD soil column and the total damping ratio is frequency dependent. This dependency is
contributed solely by the fluid motion (i.e., f ) since s is frequency independent. Fig.
4-12 shows that for small values of (up to 210 ), 02.0 s as the column is fully
relaxed and it behaves essentially as a dry column. As increases, increases due to
increases in pore fluid induced energy dissipation (see Fig. 4-10). After reaches its
peak values, decreases as further increases due to a combination of decreases in
skeleton induced energy dissipation (see Fig. 4-9) and pore fluid induced energy
dissipation (see Fig. 4-10). Fig. 4-13 shows the contour plot of f vs. and for a
TD soil column with 4.0n and 02.0s . It can be observed that for 10 and
2.0 , f can be significant as compared to the solid skeleton damping ratio (i.e.,
0.02).
112
Fig. 4-13. Variation of f vs. and for a TD soil column ( 4.0n , 02.0s )
Fig. 4-14 presents the contour plots of 2.0f for a TD soil column with 02.0s
and 3.0n , 0.4, and 0.5 to show the effect of n on f . It is observed that as n
increases, the contour line for 2.0f shrinks inward; similar trend is observed for the
contour lines of different f values. This indicates that f decreases as n increases for
a given combination of and . Fig. 4-15 presents the contour plots of 2.0f for a
TD soil column with 4.0n and 02.0s , 0.05, and 0.1 to demonstrate the effect of
s on f . It is observed that as s increases, the contour line for 2.0f shrinks
inward suggesting that f decreases as s increases for a given combination of and
. By comparing Figs. 4-14 and 4-15, it can be concluded that between n and s , the
former has a greater influence on f .
0.01
0.1
1
10
100
10-4 10-3 10-2 0.1 1 10 102 103 104
f = 0
0.05
0.1
0.2
0.5
f = 0
0.85
113
Fig. 4-14. Contour plot of 2.0f for a TD soil column with various n ( 02.0s )
Fig. 4-15. Contour plot of 2.0f for a TD soil column with various s ( 4.0n )
0.01
0.1
1
10
0.1 1 10 102 103 104
n = 0.3
n = 0.4
n = 0.5
0.01
0.1
1
10
0.1 1 10 102103 104
s = 0.02
s = 0.05
s = 0.10
114
4.6 Illustrative Example
The solutions, results, and discussions presented above are based on dimensionless
parameters and . This section presents an illustrative example to show values of
and for a potential practical situation. The soil column presented in Fig. 4-1 may be
considered as a nearly saturated soil specimen fixed at its base, constrained from lateral
movement, and subjected to low-frequency compressional excitations from the top in an
experimental setup similar to consolidation tests. It is assumed that the height of the soil
column is 0.1 m (i.e., 1.0L m) and the column is in TD condition for illustration
purpose.
Values of soil parameters and the corresponding values of and for the soil
column are presented in Table 4-1. The range of k for different soil types is based on
Das (2010). In order to emphasize the effect of hydraulic conductivity on various
phenomena discussed in this study, constant values of n and sD are used. This treatment
is justifiable because the variation of these parameters among different soils is
significantly less than the variation of hydraulic conductivity. For example, k decreases
by more than eight magnitudes as the soil changes from gravel to clay. At small strain
levels, the constrained modulus sD is highly dependent on the soil type and effective
stress. For simplicity, 100sD MPa is considered, which is based on the shear wave
velocities reported by Hardin and Richart (1963) for well-graded quartz sands (No 20 –
No 140) with 4.0n and confining pressure of 50 kPa. To satisfy the assumption of no-
inertial effects, the excitation frequency should be much smaller than the fundamental
frequency of the soil column and please refer to Zienkiewicz et al. (1980) for the exact
115
conditions needed to satisfy this assumption. For simplicity and illustration purpose,
10 rad/s is assumed, corresponding to a frequency of 1.6 Hz. The pore fluid is
assumed to be water as in most cases in geotechnical engineering, however, other types
of pore fluid can be considered with the corresponding values of k and fK (e.g.,
silicone-saturated soil in geotechnical centrifuge testing).
Different behaviors are expected for the soil column consisting of different soils. For
example, Fig. 4-6 indicates that the excitation-generated pore pressure is negligible for a
clean gravel or coarse sand column as the column is fully relaxed regardless of saturation;
whereas the excitation generates maximum pore pressure in a clay column as the column
is fully loaded. A fine sand or silty clay column has intermediate behavior between the
fully relaxed and fully loaded conditions. Figs. 4-10 and 4-13 indicate that pore fluid
induced energy dissipation and damping ratio are negligible for a clean gravel or coarse
sand column regardless of saturation; whereas they are significant in a fine sand and silty
clay column when the saturation is larger than 99%.
Table 4-1. Parameters Used in Illustrative Example ( 1.0L m, 10 rad/s, 02.0s )
Soil Type n sD (MPa)
k (m/s)
%90S %99S %100S
Clean Gravel 0.4 100 0.01 – 1 10-5 – 10-3 100 10 0.05 Coarse Sand 0.4 100 10-4 – 10-2 10-3 – 0.1 100 10 0.05
Fine Sand 0.4 100 10-5 – 10-4 0.1 – 1 100 10 0.05 Silty Clay 0.4 100 10-7 – 10-5 1 – 100 100 10 0.05
Clay 0.4 100 < 10-8 >103 100 10 0.05
116
4.7 Conclusions
Pore fluid induced damping is a manifestation of the viscous coupling force and
relative motion between solid and fluid phases in a saturated porous medium. This paper
presents a theoretical investigation on the energy dissipation in a nearly saturated
poroviscoelastic soil column under quasi-static compressional excitations. Different
components of the energy dissipation are evaluated and compared. Damping ratio of the
column is evaluated based on the dissipated energy and peak strain energy stored in the
column in one loading cycle. The effects of boundary drainage condition, saturation,
excitation frequency (normalized), porosity, and skeleton damping ratio on fluid induced
energy dissipation are discussed. The following conclusions are reached as a result of
this investigation:
The magnitude of fluid induced energy dissipation is primarily a function of a
normalized excitation frequency , which reflects the relative pace between two
concurrent underlying processes: excitation and relaxation (i.e., consolidation). For
small values of , a drained soil column is fully relaxed and behaves essentially as a
dry column with negligible pore pressure. For such soil column, fluid induced energy
dissipation is negligible and the total damping ratio of the column is essentially the
same as that of the solid skeleton. For very high values of , a drained soil column is
fully loaded and the excitation-generated fluid pressure decreases as the fluid
becomes more compressible. For such soil column, the fluid pressure gradient only
exists in a thin boundary layer near the drainage boundary, where drainage occurs and
fluid induces energy dissipation; whereas the rest of the column is essentially
undrained and fluid induces no energy dissipation. Significant fluid induced energy
117
dissipation occurs for moderate values of due to a combination of moderate fluid
pressure, pressure gradient and fluid relative motion throughout the soil column.
Unlike the case of a soil column under shear wave excitations where the total
damping ratio of the column is always higher than that of the solid skeleton, a
saturated soil column under compressional excitations may have a smaller damping
ratio than that of the solid skeleton due to the additional strain energy stored in the
pore fluid.
Pore fluid induced energy dissipation generally decreases as the fluid becomes more
compressible. Between the porosity and solid skeleton damping ratio, the former has
a greater effect on fluid induced energy dissipation.
Notations
The following symbols are used in this chapter:
A = cross-section area of soil column;
1B , 2B , = parameters in steady-state solution;
oC , 1C , 2C , 3C = constants determined by boundary conditions;
vc = coefficient of consolidation;
sD = constrained modulus of solid skeleton;
E = strain energy;
G = transfer function at top of soil column;
g = acceleration of gravity;
118
i = 1 ;
K = bulk modulus;
k = hydraulic conductivity;
L = length of soil column;
n = porosity;
P = amplitude of fluid pressure;
maxP = maximum fluid pressure along soil column;
p = fluid pressure;
t = time;
S = degree of saturation;
s = complex function in steady-state solution;
U = amplitude of displacement;
u = displacement;
V = fluid volume;
oV = initial fluid volume at reference state;
DW = energy dissipated in one loading cycle;
fDW = energy dissipated due to pore fluid in one loading cycle;
sDW = energy dissipated due to skeleton damping in one loading cycle;
*fDW , *
sDW = normalized dissipated energy;
W = peak strain energy stored in one loading cycle;
x = spatial coordinate;
= effective stress parameter;
119
= ratio of bulk modulus of solid skeleton over that of pore fluid;
x = compressive axial strain;
= shear coefficient of viscosity;
= density;
= total stress;
o = amplitude of total stress;
= effective stress;
= circular frequency;
= damping ratio;
= normalized circular frequency;
Im = function that returns the imaginary part of complex number;
Re = function that returns the real part of complex number;
Subscripts:
f = fluid phase;
s = solid phase;
120
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Mech. and Found. Div., 89(1), 33-66.
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124
Chapter 5
Conclusions and Recommendations
5.1 Conclusions
Shear modulus and damping ratio are two important dynamic soil properties that
govern the behavior of soil under dynamic loading. In saturated soil, shear modulus is
typically evaluated using saturated density and Eq. (1-3) and pore fluid induced damping
is assumed to be negligible due to the lack of quantitative assessment of its values in
various soils. This study presents: 1) an experimental investigation of the concept of
effective density for propagation of small strain shear waves through saturated granular
materials; 2) an experimental and analytical investigation that aims to quantify pore fluid
induced damping in saturated granular materials under shear wave excitations; and 3) an
analytical investigation that aims to quantify the energy dissipation in nearly saturated
soil columns under quasi-static compressional excitations. Based on this study, the
following conclusions can be made.
Effective Density
Effective density ratio (i.e., effective soil density/saturated soil density) is a
function of specific gravity of solids, porosity, hydraulic conductivity, and shear
wave frequency. For a given specific gravity of solids and porosity, the value of
effective density ratio decreases with increasing normalized frequency. Viscous
125
coupling controls the rate at which effective density ratio decreases, whereas mass
coupling determines the minimum value of effective density at high frequency.
Effective density should be used to calculate small-strain shear modulus from
measured shear wave velocity in bender element tests on saturated granular
materials because the use of saturated density will generally overestimate these
modulus values. For the experimental study conducted, the use of saturated
density yielded errors up to 28%; whereas the use of effective density resulted in
substantially smaller errors, which were generally less than 5%.
For the calculation of small-strain shear modulus in resonant column tests on
saturated granular materials, the errors for using saturated density are smaller than
those from bender element tests. The error increases with increasing ratio of mass
polar moment of inertia of saturated specimen to that of the loading system. In
this study, the use of saturated density and effective density yielded errors
generally less than 5%, although the former yielded slighter higher errors on
average.
The quick chart is a useful tool for a user to determine whether effective density
should be considered for a given application. Effective density may be important
for fine and medium sands at high-frequency excitations, such as from bender
elements, for coarse clean sands at lower frequencies, such as for resonant column
tests, and for clean gravels at essentially all frequencies of geotechnical interest.
Biot Flow Induced Damping
The results of Biot flow induced damping from half-power bandwidth and free
vibration decay methods are consistent and they are also consistent with the
126
closed-form solutions based on dissipated energy and maximum strain energy
during one cycle of vibration. The consistency among these solutions indicates
that Biot flow induced damping is of viscous nature.
Analytical studies indicate that Biot flow induced damping may provide an
important contribution to total soil damping in coarse sand and gravel, but can be
practical neglected for less permeable soils ( e.g., fine sand, silt, and clay).
Biot flow induced damping increases as porosity increases and decreases
significantly as the ratio of mass polar moment of inertia of the saturated sample
to that of the loading system, IIt , increases. For typical values of IIt , Biot
flow induced damping is suppressed by the boundary condition and hence
difficult to be measured.
The comparison between the solution from the free vibration decay method and
RC test data suggests that the validity of this analytical solution is inconclusive.
This is largely due to the very small magnitude of Biot flow induced damping in
RC tests.
Energy Dissipation in Nearly Saturated Soil during Quasi-Static Compressional
Excitations
Pore fluid induced energy dissipation is governed by the relative pace between
two concurrent underlying processes: excitation and relaxation, which is
represented by the normalized excitation frequency . For low values of , a
drained soil column is fully relaxed and behaves essentially as a dry column with
negligible pore pressure. In this case, pore fluid induced energy dissipation is
127
negligible and the total damping ratio is essentially the same as that of the solid
skeleton in the soil column. For very high values of , a drained soil column is
fully loaded and the fluid pressure decreases as the compressibility of fluid
increases. In this case, the fluid pressure gradient only exists in a thin boundary
layer near the drainage boundary, where drainage occurs and fluid induces
energy dissipation; whereas the rest of the column is essentially undrained and
fluid induces no energy dissipation. Large values of fluid induced energy
dissipation occur at moderate values of due to a combination of moderate
fluid pressure, pressure gradient and fluid relative motion throughout the soil
column.
Unlike the case of a soil column under shear wave excitations where the total
damping ratio of the column is always higher than that of the solid skeleton, a
saturated soil column under compressional excitations may have a smaller
damping ratio than that of the solid skeleton because of the additional strain
energy stored in the pore fluid.
Pore fluid induced energy dissipation generally decreases as the fluid becomes
more compressible. Between the porosity and solid skeleton damping ratio, the
former has a greater effect on fluid induced energy dissipation.
5.2 Recommendations for Future Work
Further experimental studies are recommended to investigate the effects of following
parameters on effective density and pore fluid induced damping:
128
Wave frequency – the current study utilized resonant column tests with wave
frequencies in the range of sub-200 Hz and bender element tests with wave
frequencies of 3 kHz and 11 kHz. Additional tests with wave frequencies
between these values and beyond 11 kHz (e.g., ultrasonic waves) may elucidate
the applicability of effective density concept over a wide range of wave
frequencies.
Soil type – the current study focused on clean granular materials with negligible
fine contents. Additional tests for granular material with varying percentages of
fine contents (e.g., 5%, 10%, and 15%) will address the applicability of effective
density concept and quantify pore fluid induced damping over a wide range of
soil types.
Pore fluid – the current study utilized water as pore fluid, which is most common
in geotechnical engineering. However, other pore fluids (e.g., silicon oil) have
been utilized in geotechnical research (e.g., geotechnical centrifuge tests).
Additional tests for granular materials saturated with different pore fluids (e.g., 30
cS and 100 cS silicone oil) may further address the applicability of effective
density concept and quantify pore fluid induced damping in more general
conditions.
129
Appendix
Data from Resonant Column Tests
All specimens are cylindrical solid specimens with a diameter of 0.1 m. Hydraulic
conductivity, k , is estimated based on the Chapuis (2004) empirical equation. All
measurements are repeated three times and the average value is reported herein.
Notations: n =porosity, 10D =effective diameter, sG =specific gravity, c =effective
confining pressure, =maximum shear strain, rf =resonant frequency, and =total
damping ratio. The mass polar moment of inertial of the driving system is 59.47396
g·cm2.
Group 1
Note: Five rounds of RC tests were conducted for 75c kPa.
Glass beads: 3.10.1 mm (particle diameter)
Height of specimen: 191.0 m
376.0n , 04.110 D mm, 551.0k cm/s, 5.2sG
Test Date: May 2011 Dry condition
c (kPa) Input voltage (V) rf (Hz)
75 Round 1 0.05 1.082E-05 157.1 7.655E-03 0.10 1.985E-05 155.2 8.796E-03 0.15 2.934E-05 153.8 9.485E-03 0.20 3.898E-05 152.5 9.289E-03 0.25 4.897E-05 151.5 9.607E-03
130
0.50 8.718E-05 148.0 1.135E-02 0.75 1.194E-04 145.7 1.368E-02 0.90 1.294E-04 144.8 1.488E-02
Round 2 0.05 1.072E-05 155.8 8.085E-03 0.10 2.105E-05 154.3 8.514E-03 0.15 3.074E-05 152.8 8.578E-03 0.20 4.051E-05 151.6 8.944E-03 0.25 5.046E-05 150.4 9.284E-03 0.50 8.928E-05 147.2 1.137E-02 0.75 1.210E-04 144.9 1.310E-02 0.90 5.263E-04 144.0 1.511E-02
Round 3 0.05 1.075E-05 155.2 8.111E-03 0.10 2.047E-05 153.5 8.631E-03 0.15 3.052E-05 152.0 9.103E-03 0.20 4.066E-05 150.8 9.167E-03 0.25 5.090E-05 149.6 9.618E-03 0.50 9.012E-05 146.5 1.121E-02 0.75 1.222E-04 144.3 1.342E-02 0.90 1.330E-04 143.4 1.534E-02
Round 4 0.05 1.077E-05 154.7 8.049E-03 0.10 2.066E-05 152.8 8.557E-03 0.15 3.159E-05 151.7 7.947E-03 0.20 4.167E-05 150.2 8.727E-03 0.25 5.196E-05 149.1 8.939E-03
0.50 9.156E-05 145.8 1.080E-02 0.75 1.237E-04 143.6 1.350E-02 0.90 1.342E-04 142.8 1.523E-02
Round 5 0.05 1.074E-05 154.3 7.931E-03 0.10 2.102E-05 152.3 8.371E-03 0.15 3.118E-05 150.7 8.679E-03 0.20 4.186E-05 149.5 8.971E-03 0.25 5.205E-05 148.3 9.421E-03 0.50 9.195E-05 145.2 1.110E-02 0.75 1.247E-04 143.0 1.352E-02 0.90 1.356E-04 142.2 1.521E-02
131
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
75 0.05 9.743E-06 146.3 1.028E-02 0.10 1.747E-05 144.8 1.231E-02 0.15 2.466E-05 143.7 1.337E-02 0.20 3.164E-05 143.0 1.423E-02 0.25 3.872E-05 142.4 1.442E-02 0.50 7.076E-05 139.7 1.694E-02 0.75 1.023E-04 137.6 1.840E-02 0.90 1.190E-04 136.3 2.025E-02
132
Glass beads: 1.27.1 mm (particle diameter)
Height of specimen: 193.0 m
380.0n , 75.110 D mm, 297.1k cm/s, 5.2sG
Test Date: May 2011 Dry condition
c (kPa) Input voltage (V) rf (Hz)
75 Round 1 0.05 1.235E-05 147.7 7.750E-03 0.10 2.181E-05 146.1 9.262E-03 0.15 3.050E-05 144.8 1.031E-02 0.20 3.853E-05 143.7 1.150E-02 0.25 4.577E-05 142.8 1.257E-02 0.50 7.279E-05 139.9 1.633E-02 0.75 9.546E-05 138.2 1.964E-02 0.90 1.072E-04 137.3 2.031E-02
Round 2 0.05 1.190E-05 147.6 8.430E-03 0.10 2.137E-05 146.0 9.467E-03 0.15 3.003E-05 144.7 1.049E-02 0.20 3.804E-05 143.6 1.181E-02 0.25 4.552E-05 142.6 1.243E-02 0.50 7.379E-05 139.7 1.639E-02 0.75 9.486E-05 138.1 1.902E-02 0.90 1.070E-04 137.2 2.047E-02
Round 3 0.05 1.171E-05 147.6 8.339E-03 0.10 2.110E-05 145.9 9.772E-03 0.15 2.966E-05 144.7 1.087E-02 0.20 3.765E-05 143.6 1.200E-02 0.25 4.526E-05 142.5 1.268E-02 0.50 7.332E-05 139.5 1.642E-02 0.75 9.530E-05 92.1 1.941E-02 0.90 1.064E-04 137.1 2.110E-02
Round 4 0.05 1.163E-05 147.6 7.899E-03 0.10 2.093E-05 145.9 9.766E-03 0.15 2.964E-05 144.6 1.107E-02 0.20 3.790E-05 143.3 1.184E-02 0.25 4.508E-05 142.5 1.317E-02 0.50 7.278E-05 139.6 1.632E-02 0.75 9.524E-05 137.7 1.922E-02
133
0.90 1.059E-04 137.0 2.087E-02 Round 5
0.05 1.145E-05 147.4 8.759E-03 0.10 2.072E-05 145.9 9.713E-03 0.15 2.942E-05 144.4 1.106E-02 0.20 3.780E-05 143.2 1.185E-02 0.25 4.479E-05 142.4 1.285E-02 0.50 7.240E-05 139.5 1.682E-02 0.75 9.479E-05 137.7 1.935E-02 0.90 1.063E-04 137.1 2.084E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
75 0.05 1.697E-05 120.9 8.923E-03 0.10 3.015E-05 119.9 1.057E-02 0.15 4.235E-05 119.5 1.182E-02 0.20 5.308E-05 119.1 1.261E-02 0.25 6.308E-05 118.9 1.346E-02 0.50 9.976E-05 117.1 1.758E-02 0.75 1.272E-04 115.7 2.095E-02 0.90 1.380E-04 115.3 2.286E-02
134
Glass beads: 4.20.2 mm (particle diameter)
Height of specimen: 194.0 m
374.0n , 05.210 D mm, 575.1k cm/s, 5.2sG
Test Date: April 2011 Dry condition
c (kPa) Input voltage (V) rf (Hz)
75 Round 1 0.05 1.265E-05 142.7 8.026E-03 0.10 2.257E-05 141.0 9.814E-03 0.15 3.219E-05 139.9 1.056E-02 0.20 4.261E-05 138.7 1.076E-02 0.25 5.349E-05 137.7 1.093E-02 0.50 1.009E-04 134.7 1.241E-02 0.75 1.372E-04 133.0 1.426E-02 0.90 1.492E-04 132.4 1.633E-02
Round 2 0.05 1.259E-05 141.5 8.610E-03 0.10 2.213E-05 140.2 1.008E-02 0.15 3.207E-05 139.0 1.098E-02 0.20 4.219E-05 138.0 1.126E-02 0.25 5.293E-05 137.0 1.138E-02 0.50 9.828E-05 134.1 1.303E-02 0.75 1.355E-04 132.4 1.493E-02 0.90 1.483E-04 132.1 1.630E-02
Round 3 0.05 1.240E-05 141.8 8.440E-03 0.10 2.226E-05 139.9 1.016E-02 0.15 3.172E-05 138.7 1.110E-02 0.20 4.147E-05 137.7 1.172E-02 0.25 5.083E-05 137.3 1.196E-02 0.50 9.559E-05 134.3 1.371E-02 0.75 1.343E-04 132.3 1.494E-02 0.90 1.481E-04 131.8 1.643E-02
Round 4 0.05 1.222E-05 141.3 8.297E-03 0.10 2.204E-05 139.9 1.036E-02 0.15 3.093E-05 139.0 1.148E-02 0.20 4.066E-05 137.8 1.178E-02 0.25 5.074E-05 136.9 1.203E-02 0.50 9.320E-05 134.3 1.374E-02 0.75 1.318E-04 132.3 1.547E-02
135
0.90 1.477E-04 131.6 1.689E-02 Round 5
0.05 1.211E-05 141.2 8.557E-03 0.10 1.856E-05 140.0 1.053E-02 0.15 3.084E-05 138.6 1.200E-02 0.20 3.986E-05 137.8 1.204E-02 0.25 4.998E-05 136.9 1.228E-02 0.50 9.307E-05 133.9 1.400E-02 0.75 1.316E-04 132.0 1.549E-02 0.90 1.465E-04 131.5 1.746E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
75 0.05 1.029E-05 151.7 8.769E-03 0.10 1.760E-05 150.1 1.087E-02 0.15 2.432E-05 149.3 1.220E-02 0.20 3.074E-05 148.6 1.318E-02 0.25 3.705E-05 148.2 1.439E-02 0.50 6.513E-05 145.4 1.719E-02 0.75 9.150E-05 143.2 1.926E-02 0.90 1.053E-04 142.0 2.006E-02
136
Glass beads: 4.48.3 mm (particle diameter)
Height of specimen: 194.0 m
365.0n , 87.310 D mm, 91.3k cm/s, 5.2sG
Test Date: April 2011 Dry condition
c (kPa) Input voltage (V) rf (Hz)
75 Round 1 0.05 1.362E-05 155.1 5.751E-03 0.10 2.457E-05 153.7 6.801E-03 0.15 3.469E-05 152.6 7.714E-03 0.20 4.356E-05 151.8 8.212E-03 0.25 5.234E-05 151.0 8.955E-03 0.50 8.852E-05 148.6 1.144E-02 0.75 1.147E-04 147.0 1.370E-02 0.90 1.248E-04 146.2 1.472E-02
Round 2 0.05 1.267E-04 145.8 1.486E-02 0.10 1.149E-04 146.0 1.388E-02 0.15 8.758E-05 147.0 1.186E-02 0.20 4.942E-05 149.2 9.766E-03 0.25 4.172E-05 150.0 8.880E-03 0.50 3.263E-05 150.7 8.626E-03 0.75 2.259E-05 151.8 7.793E-03 0.90 1.176E-05 153.6 7.470E-03
Round 3 0.05 1.197E-05 153.9 7.029E-03 0.10 2.330E-05 152.2 7.655E-03 0.15 3.363E-05 151.1 8.440E-03 0.20 4.263E-05 150.2 8.673E-03 0.25 5.084E-05 149.4 9.665E-03 0.50 8.926E-05 146.7 1.201E-02 0.75 1.167E-04 145.2 1.364E-02 0.90 5.117E-04 144.7 1.532E-02
Round 4 0.05 1.231E-04 144.7 1.499E-02 0.10 1.050E-04 145.4 1.347E-02 0.15 7.295E-05 146.8 1.164E-02 0.20 4.494E-05 148.5 1.055E-02 0.25 3.649E-05 149.3 9.772E-03 0.50 2.709E-05 150.4 9.592E-03 0.75 1.767E-05 151.9 8.817E-03
137
0.90 1.118E-05 153.3 7.767E-03 Round 5
0.05 1.849E-05 152.3 8.026E-03 0.10 2.831E-05 150.9 8.451E-03 0.15 3.750E-05 149.8 9.496E-03 0.20 4.664E-05 148.9 9.867E-03 0.25 7.475E-05 146.8 1.118E-02 0.50 1.073E-04 144.7 1.345E-02 0.75 1.241E-04 143.9 1.501E-02 0.90 1.288E-04 143.7 1.534E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
75 0.05 1.315E-05 156.0 5.984E-03 0.10 2.363E-05 154.7 7.194E-03 0.15 3.265E-05 153.5 8.037E-03 0.20 4.069E-05 152.5 9.008E-03 0.25 4.813E-05 151.4 9.527E-03 0.50 7.666E-05 148.9 1.310E-02 0.75 1.001E-04 146.8 1.491E-02 0.90 1.164E-04 145.0 1.663E-02
138
Glass beads: 0.6 mm (particle diameter)
Height of specimen: 194.0 m
367.0n , 0.610 D mm, 91.7k cm/s, 5.2sG
Test Date: April 2011 Dry condition
c (kPa) Input voltage (V) rf (Hz)
75 Round 1 0.05 9.553E-06 151.4 9.533E-03 0.10 1.723E-05 149.8 1.107E-02 0.15 2.468E-05 148.6 1.202E-02 0.20 3.180E-05 147.6 1.276E-02 0.25 3.702E-05 146.3 1.388E-02 0.50 7.224E-05 143.0 1.591E-02 0.75 1.002E-04 140.7 1.735E-02 0.90 1.111E-04 139.6 1.904E-02
Round 2 0.05 9.803E-06 149.6 9.278E-03 0.10 1.753E-05 148.1 1.105E-02 0.15 2.445E-05 146.8 1.261E-02 0.20 3.146E-05 145.6 1.344E-02 0.25 3.878E-05 144.7 1.418E-02 0.50 7.409E-05 141.3 1.582E-02 0.75 1.024E-04 139.2 1.696E-02 0.90 1.123E-04 138.2 1.918E-02
Round 3 0.05 9.840E-06 148.3 1.032E-02 0.10 1.739E-05 146.6 1.167E-02 0.15 2.476E-05 145.4 1.300E-02 0.20 3.250E-05 144.1 1.337E-02 0.25 4.005E-05 143.2 1.382E-02 0.50 7.419E-05 140.2 1.570E-02 0.75 1.028E-04 137.9 1.765E-02 0.90 1.130E-04 137.0 1.908E-02
Round 4 0.05 9.870E-06 147.1 9.904E-03 0.10 1.770E-05 145.4 1.154E-02 0.15 2.537E-05 144.1 1.154E-02 0.20 3.306E-05 143.0 1.348E-02 0.25 3.928E-05 144.2 1.372E-02 0.50 7.009E-05 142.1 1.599E-02 0.75 9.848E-05 140.3 1.815E-02
139
0.90 1.126E-04 139.3 1.953E-02 Round 5
0.05 9.687E-06 149.8 9.374E-03 0.10 1.703E-05 148.2 1.128E-02 0.15 2.428E-05 147.0 1.273E-02 0.20 3.183E-05 146.1 1.301E-02 0.25 3.956E-05 145.1 1.312E-02 0.50 7.101E-05 142.2 1.615E-02 0.75 1.010E-04 140.2 1.740E-02 0.90 1.138E-04 139.1 1.879E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
75 0.05 1.014E-05 149.8 9.496E-03 0.10 1.937E-05 148.5 9.973E-03 0.15 2.741E-05 147.4 1.101E-02 0.20 3.509E-05 146.8 1.169E-02 0.25 4.175E-05 146.5 1.207E-02 0.50 7.114E-05 143.0 1.540E-02 0.75 9.557E-05 140.8 1.773E-02 0.90 1.064E-04 139.5 1.945E-02
140
Group 2
Notes: One cycle of loading and unloading of RC tests were conducted for confining
pressures of 25, 50, 75, 100, 125, and 150 kPa with a maximum input voltage of 0.5 V
under dry condition.
Glass beads: 6.04.0 mm (particle diameter)
Height of specimen: 197.0 m
4.0n , 44.010 D mm, 173.0k cm/s, 5.2sG
Test Date: April 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 1.945E-05 95.6 1.397E-02 0.10 3.349E-05 93.2 1.766E-02 0.25 6.631E-05 89.2 2.528E-02 0.50 9.806E-05 86.5 3.341E-02
50
0.05 1.893E-05 121.0 7.883E-03 0.10 3.120E-05 118.9 1.024E-02 0.25 6.104E-05 114.1 1.532E-02 0.50 1.002E-04 109.0 2.117E-02
75
0.05 1.792E-05 134.2 6.228E-03 0.10 3.006E-05 132.5 8.010E-03 0.25 5.831E-05 128.6 1.167E-02 0.50 9.407E-05 124.3 1.625E-02
100
0.05 1.718E-05 143.8 5.443E-03 0.10 2.926E-05 142.2 6.965E-03 0.25 5.843E-05 138.5 9.873E-03 0.50 9.532E-05 134.6 1.341E-02
125
0.05 1.623E-05 151.7 5.029E-03 0.10 2.830E-05 150.2 6.170E-03 0.25 5.659E-05 147.0 8.551E-03 0.50 9.480E-05 143.0 1.167E-02
150
0.05 1.567E-05 158.1 4.610E-03 0.10 2.689E-05 156.8 5.867E-03 0.25 5.438E-05 153.8 8.255E-03 0.50 9.150E-05 150.4 1.066E-02
141
Unloading
150
0.05 1.382E-05 155.8 5.915E-03 0.10 2.456E-05 154.6 6.939E-03 0.25 5.373E-05 152.4 8.424E-03 0.50 9.475E-05 150.0 1.027E-02
125
0.05 1.757E-05 146.4 4.790E-03 0.10 3.204E-05 145.2 5.729E-03 0.25 6.589E-05 143.1 7.438E-03 0.50 1.034E-04 141.2 1.101E-02
100
0.05 1.919E-05 137.0 5.178E-03 0.10 3.461E-05 135.7 6.196E-03 0.25 6.878E-05 133.7 8.854E-03 0.50 1.052E-04 131.9 1.277E-02
75
0.05 2.040E-05 126.2 6.042E-03 0.10 3.619E-05 124.7 7.464E-03 0.25 7.014E-05 122.3 1.090E-02 0.50 1.058E-04 120.2 1.549E-02
50
0.05 2.224E-05 111.7 7.798E-03 0.10 3.844E-05 110.0 9.724E-03 0.25 7.161E-05 107.4 1.490E-02 0.50 1.060E-04 105.2 2.088E-02
25
0.05 2.382E-05 90.9 1.213E-02 0.10 4.012E-05 88.8 1.543E-02 0.25 7.291E-05 86.1 2.411E-02 0.50 1.042E-04 85.7 3.234E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 1.950E-05 91.1 1.518E-02 0.10 3.885E-05 89.4 1.605E-02 0.25 8.384E-05 85.4 2.167E-02 0.50 1.225E-04 82.3 3.175E-02
50
0.05 1.733E-05 120.8 8.525E-03 0.10 3.153E-05 118.2 1.011E-02 0.25 7.093E-05 111.5 1.376E-02 0.50 1.213E-04 105.4 1.873E-02
75
0.05 1.523E-05 134.3 7.464E-03 0.10 2.785E-05 132.2 8.668E-03 0.25 6.178E-05 126.7 1.146E-02 0.50 1.113E-04 120.5 1.481E-02
100 0.05 1.507E-05 143.9 6.133E-03 0.10 2.801E-05 142.1 6.944E-03
142
0.25 6.057E-05 138.7 9.194E-03 0.50 1.093E-04 133.1 1.169E-02
125
0.05 1.276E-05 152.8 6.488E-03 0.10 2.389E-05 151.1 7.443E-03 0.25 5.467E-05 146.3 9.077E-03 0.50 1.007E-04 141.3 1.108E-02
150
0.05 1.149E-05 159.8 6.286E-03 0.10 2.181E-05 158.2 7.411E-03 0.25 4.860E-05 154.4 8.986E-03 0.50 9.321E-05 148.7 1.080E-02
143
Glass beads: 3.10.1 mm (particle diameter)
Height of specimen: 193.0 m
372.0n , 04.110 D mm, 536.0k cm/s, 5.2sG
Test Date: April 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 5.237E-06 132.6 2.675E-02 0.10 1.014E-05 130.3 2.855E-02 0.25 2.244E-05 125.0 3.713E-02 0.50 3.810E-05 119.5 4.785E-02
50
0.05 5.693E-06 157.5 1.662E-02 0.10 1.089E-05 155.4 1.830E-02 0.25 2.339E-05 150.8 2.307E-02 0.50 4.020E-05 145.3 2.851E-02
75
0.05 6.043E-06 171.2 1.241E-02 0.10 1.144E-05 169.5 1.357E-02 0.25 2.438E-05 165.5 1.738E-02 0.50 4.148E-05 160.6 2.209E-02
100
0.05 6.273E-06 180.9 1.028E-02 0.10 1.186E-05 179.4 1.118E-02 0.25 2.580E-05 175.8 1.368E-02 0.50 4.356E-05 171.2 1.792E-02
125
0.05 6.250E-06 188.6 9.093E-03 0.10 1.199E-05 187.4 9.814E-03 0.25 2.666E-05 183.8 1.186E-02 0.50 4.542E-05 179.8 1.492E-02
150
0.05 5.947E-06 195.0 8.854E-03 0.10 1.143E-05 193.7 9.353E-03 0.25 2.568E-05 190.5 1.130E-02 0.50 4.488E-05 186.9 1.366E-02
Unloading
150
0.05 5.763E-06 193.1 9.315E-03 0.10 1.117E-05 192.1 9.761E-03 0.25 2.567E-05 189.5 1.136E-02 0.50 4.590E-05 187.1 1.340E-02
125
0.05 6.067E-06 184.4 1.002E-02 0.10 1.185E-05 183.2 1.047E-02 0.25 2.730E-05 180.7 1.205E-02 0.50 4.658E-05 178.2 1.506E-02
100 0.05 5.907E-06 174.8 1.205E-02
144
0.10 1.152E-05 173.8 1.263E-02 0.25 2.603E-05 171.1 1.501E-02 0.50 4.438E-05 168.2 1.791E-02
75
0.05 5.430E-06 164.1 1.542E-02 0.10 1.075E-05 162.7 1.582E-02 0.25 2.476E-05 159.5 1.826E-02 0.50 4.150E-05 156.0 2.295E-02
50
0.05 5.273E-06 149.4 2.036E-02 0.10 1.036E-05 148.1 2.057E-02 0.25 2.380E-05 144.4 2.408E-02 0.50 3.951E-05 140.6 3.067E-02
25
0.05 5.123E-06 129.5 2.827E-02 0.10 1.007E-05 127.7 3.000E-02 0.25 2.288E-05 122.8 3.793E-02 0.50 3.883E-05 118.6 4.776E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 4.560E-06 133.0 2.939E-02 0.10 8.843E-06 131.7 3.231E-02 0.25 2.050E-05 127.6 3.724E-02 0.50 3.769E-05 122.5 4.687E-02
50
0.05 4.720E-06 156.2 2.006E-02 0.10 9.127E-06 154.5 2.040E-02 0.25 2.091E-05 150.6 2.448E-02 0.50 3.808E-05 146.1 3.005E-02
75
0.05 4.957E-06 168.8 1.557E-02 0.10 9.590E-06 167.5 1.628E-02 0.25 2.209E-05 164.4 1.875E-02 0.50 3.887E-05 161.0 2.266E-02
100
0.05 4.870E-06 179.6 1.340E-02 0.10 9.510E-06 178.6 1.386E-02 0.25 2.226E-05 175.8 1.572E-02 0.50 3.973E-05 172.7 1.866E-02
125
0.05 4.957E-06 187.7 1.190E-02 0.10 9.737E-06 186.6 1.219E-02 0.25 2.290E-05 184.1 1.350E-02 0.50 4.121E-05 181.0 1.592E-02
150
0.05 4.720E-06 194.0 1.118E-02 0.10 9.247E-06 193.1 1.188E-02 0.25 2.171E-05 190.7 1.311E-02 0.50 3.953E-05 187.7 1.523E-02
145
Glass beads: 1.27.1 mm (particle diameter)
Height of specimen: 194.0 m
376.0n , 75.110 D mm, 244.1k cm/s, 5.2sG
Test Date: April 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 1.486E-05 110.6 1.400E-02 0.10 2.395E-05 108.7 1.773E-02 0.25 4.716E-05 105.3 2.496E-02 0.50 7.467E-05 104.3 3.119E-02
50
0.05 1.482E-05 142.4 7.035E-03 0.10 2.516E-05 140.5 8.599E-03 0.25 4.907E-05 136.5 1.283E-02 0.50 7.674E-05 132.1 1.771E-02
75
0.05 1.374E-05 156.6 6.101E-03 0.10 2.380E-05 154.8 7.207E-03 0.25 4.951E-05 151.0 9.655E-03 0.50 7.813E-05 147.5 1.341E-02
100
0.05 1.343E-05 166.5 4.918E-03 0.10 2.377E-05 164.9 6.414E-03 0.25 4.684E-05 161.7 8.631E-03 0.50 7.649E-05 158.0 1.203E-02
125
0.05 1.343E-05 174.4 4.249E-03 0.10 2.426E-05 172.7 5.120E-03 0.25 4.854E-05 169.7 7.300E-03 0.50 7.649E-05 165.8 1.068E-02
150
0.05 1.312E-05 180.5 3.788E-03 0.10 2.387E-05 179.1 4.716E-03 0.25 4.972E-05 176.2 6.244E-03 0.50 7.971E-05 173.6 8.742E-03
Unloading
150
0.05 1.195E-05 177.5 4.690E-03 0.10 2.237E-05 176.3 5.268E-03 0.25 5.001E-05 174.6 6.398E-03 0.50 8.231E-05 173.1 8.504E-03
125
0.05 1.245E-05 168.7 5.066E-03 0.10 2.373E-05 167.8 5.560E-03 0.25 5.133E-05 166.2 7.416E-03 0.50 8.412E-05 164.6 9.862E-03
0.05 1.326E-05 160.6 5.629E-03
146
100 0.10 2.529E-05 159.6 5.915E-03 0.25 5.278E-05 157.7 8.170E-03 0.50 8.375E-05 156.0 1.095E-02
75
0.05 1.481E-05 150.2 5.867E-03 0.10 2.758E-05 149.1 6.806E-03 0.25 5.494E-05 147.0 9.501E-03 0.50 8.359E-05 144.9 1.316E-02
50
0.05 1.532E-05 136.3 7.475E-03 0.10 2.822E-05 134.7 8.875E-03 0.25 5.346E-05 132.2 1.258E-02 0.50 7.920E-05 129.9 1.762E-02
25
0.05 1.560E-05 113.2 1.231E-02 0.10 2.805E-05 110.8 1.483E-02 0.25 5.073E-05 107.4 2.223E-02 0.50 7.463E-05 105.1 3.043E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 1.198E-05 116.9 1.489E-02 0.10 2.236E-05 114.7 1.673E-02 0.25 4.885E-05 110.2 2.147E-02 0.50 7.962E-05 106.6 2.813E-02
50
0.05 1.207E-05 139.4 9.506E-03 0.10 2.188E-05 137.8 1.070E-02 0.25 4.690E-05 134.1 1.365E-02 0.50 7.686E-05 130.3 1.773E-02
75
0.05 1.119E-05 154.6 7.756E-03 0.10 2.069E-05 153.1 8.971E-03 0.25 4.544E-05 151.5 1.085E-02 0.50 7.693E-05 148.2 1.321E-02
100
0.05 1.040E-05 164.0 7.125E-03 0.10 1.998E-05 162.6 7.592E-03 0.25 4.679E-05 160.7 8.679E-03 0.50 7.842E-05 158.6 1.098E-02
125
0.05 1.101E-05 172.9 5.613E-03 0.10 2.045E-05 171.4 6.350E-03 0.25 4.451E-05 168.7 8.297E-03 0.50 7.587E-05 166.2 1.022E-02
150
0.05 1.071E-05 181.5 5.034E-03 0.10 1.993E-05 180.1 5.804E-03 0.25 4.440E-05 177.4 7.008E-03 0.50 7.734E-05 174.5 8.621E-03
147
Glass beads: 4.20.2 mm (particle diameter)
Height of specimen: 194.0 m
372.0n , 05.210 D mm, 546.1k cm/s, 5.2sG
Test Date: May 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 1.167E-05 117.7 1.518E-02 0.10 2.113E-05 115.2 1.794E-02 0.25 4.081E-05 110.8 2.465E-02 0.50 6.594E-05 105.2 3.497E-02
50
0.05 1.205E-05 144.3 8.774E-03 0.10 2.054E-05 142.2 1.042E-02 0.25 4.085E-05 137.7 1.464E-02 0.50 6.648E-05 132.8 1.978E-02
75
0.05 1.178E-05 157.6 7.257E-03 0.10 2.035E-05 155.9 8.530E-03 0.25 4.083E-05 152.2 1.154E-02 0.50 6.656E-05 148.4 1.466E-02
100
0.05 1.168E-05 167.7 6.037E-03 0.10 2.016E-05 166.2 7.300E-03 0.25 3.961E-05 163.0 1.052E-02 0.50 6.752E-05 159.2 1.277E-02
125
0.05 1.164E-05 175.3 5.215E-03 0.10 2.073E-05 173.8 6.356E-03 0.25 4.151E-05 170.9 8.663E-03 0.50 6.912E-05 167.5 1.141E-02
150
0.05 1.138E-05 181.8 4.865E-03 0.10 2.046E-05 180.4 5.650E-03 0.25 4.276E-05 177.6 7.363E-03 0.50 6.976E-05 175.1 9.634E-03
Unloading
150
0.05 1.023E-05 179.3 5.676E-03 0.10 1.977E-05 178.2 6.143E-03 0.25 4.282E-05 176.4 7.379E-03 0.50 7.225E-05 174.6 9.592E-03
125
0.05 1.035E-05 171.3 6.541E-03 0.10 1.929E-05 170.2 7.416E-03 0.25 4.292E-05 168.3 8.822E-03 0.50 7.466E-05 166.3 1.052E-02
100 0.05 1.035E-05 162.4 7.188E-03
148
0.10 2.079E-05 161.1 7.809E-03 0.25 4.605E-05 158.9 9.655E-03 0.50 7.408E-05 157.0 1.166E-02
75
0.05 1.137E-05 151.8 8.026E-03 0.10 2.162E-05 150.4 8.621E-03 0.25 4.434E-05 147.9 1.148E-02 0.50 7.043E-05 145.7 1.503E-02
50
0.05 1.147E-05 137.3 1.034E-02 0.10 2.128E-05 135.5 1.184E-02 0.25 4.253E-05 132.4 1.595E-02 0.50 6.618E-05 129.8 2.078E-02
25
0.05 1.114E-05 114.4 1.711E-02 0.10 2.012E-05 112.0 2.001E-02 0.25 3.977E-05 107.4 2.817E-02 0.50 6.467E-05 104.1 3.611E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 8.857E-06 125.6 1.752E-02 0.10 1.612E-05 124.3 1.967E-02 0.25 3.614E-05 121.0 2.385E-02 0.50 6.196E-05 117.8 2.883E-02
50
0.05 9.104E-06 148.0 1.138E-02 0.10 1.646E-05 146.3 1.258E-02 0.25 3.530E-05 142.7 1.572E-02 0.50 6.254E-05 138.6 1.946E-02
75
0.05 8.750E-06 160.3 9.665E-03 0.10 1.626E-05 158.7 1.094E-02 0.25 3.640E-05 155.3 1.291E-02 0.50 6.420E-05 151.6 1.564E-02
100
0.05 8.857E-06 167.4 8.090E-03 0.10 1.706E-05 166.5 8.568E-03 0.25 3.875E-05 164.7 1.041E-02 0.50 6.424E-05 162.8 1.299E-02
125
0.05 9.110E-06 177.5 6.843E-03 0.10 1.694E-05 176.3 7.666E-03 0.25 3.798E-05 173.8 9.209E-03 0.50 6.511E-05 171.2 1.114E-02
150
0.05 9.277E-06 184.3 6.010E-03 0.10 1.738E-05 183.0 6.716E-03 0.25 3.876E-05 180.3 7.978E-03 0.50 6.742E-05 177.6 9.681E-03
149
Glass beads: 45.385.2 mm (particle diameter)
Height of specimen: 193.0 m
372.0n , 92.210 D mm, 685.2k cm/s, 5.2sG
Test Date: May 2012 Dry condition
c (kPa) Input voltage (V) rf
Loading
25
0.05 1.068E-05 123.4 1.520E-02 0.10 2.033E-05 121.6 1.671E-02 0.25 3.960E-05 118.2 2.259E-02 0.50 6.298E-05 114.6 2.960E-02
50
0.05 1.149E-05 154.7 8.000E-03 0.10 2.027E-05 153.2 9.326E-03 0.25 4.021E-05 149.6 1.290E-02 0.50 6.475E-05 145.1 1.633E-02
75
0.05 1.167E-05 169.5 6.000E-03 0.10 2.083E-05 168.0 6.865E-03 0.25 4.082E-05 164.6 9.560E-03 0.50 6.641E-05 160.5 1.330E-02
100
0.05 1.156E-05 179.6 4.801E-03 0.10 2.090E-05 178.1 5.841E-03 0.25 4.340E-05 175.1 7.724E-03 0.50 6.886E-05 172.3 1.023E-02
125
0.05 1.079E-05 187.6 4.605E-03 0.10 1.984E-05 186.3 5.316E-03 0.25 4.267E-05 183.3 6.859E-03 0.50 7.225E-05 180.4 8.647E-03
150
0.05 1.056E-05 194.1 4.244E-03 0.10 1.953E-05 192.9 4.881E-03 0.25 4.224E-05 190.2 6.270E-03 0.50 7.201E-05 187.6 7.857E-03
Unloading
150
0.05 9.610E-06 191.1 5.024E-03 0.10 1.828E-05 190.2 5.523E-03 0.25 4.231E-05 188.6 6.308E-03 0.50 7.507E-05 187.1 7.825E-03
125
0.05 1.051E-05 183.1 5.167E-03 0.10 2.047E-05 182.2 5.380E-03 0.25 4.687E-05 180.7 6.292E-03 0.50 7.840E-05 179.1 8.085E-03
100 0.05 1.033E-05 174.9 6.149E-03
150
0.10 1.979E-05 174.0 6.584E-03 0.25 4.488E-05 172.3 7.804E-03 0.50 7.379E-05 170.6 9.740E-03
75
0.05 1.070E-05 164.3 6.902E-03 0.10 2.075E-05 163.4 7.401E-03 0.25 4.495E-05 161.3 9.241E-03 0.50 7.128E-05 159.0 1.277E-02
50
0.05 1.115E-05 150.4 8.451E-03 0.10 2.104E-05 149.0 9.607E-03 0.25 4.353E-05 146.2 1.208E-02 0.50 6.714E-05 143.5 1.664E-02
25
0.05 1.082E-05 126.2 1.408E-02 0.10 2.029E-05 123.9 1.596E-02 0.25 4.103E-05 119.8 2.139E-02 0.50 6.283E-05 116.6 2.837E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 8.710E-06 132.7 1.536E-02 0.10 1.766E-05 129.9 1.608E-02 0.25 3.893E-05 126.0 2.016E-02 0.50 6.481E-05 123.3 2.496E-02
50
0.05 8.917E-06 157.6 9.692E-03 0.10 1.659E-05 156.0 1.112E-02 0.25 3.709E-05 150.5 1.319E-02 0.50 6.184E-05 146.6 1.708E-02
75
0.05 9.303E-06 168.1 7.602E-03 0.10 1.712E-05 165.9 9.241E-03 0.25 3.762E-05 162.6 1.104E-02 0.50 6.414E-05 158.5 1.402E-02
100
0.05 9.197E-06 176.2 6.790E-03 0.10 1.718E-05 175.0 7.501E-03 0.25 3.835E-05 174.2 8.790E-03 0.50 6.574E-05 171.9 1.100E-02
125
0.05 8.697E-06 185.6 6.536E-03 0.10 1.621E-05 184.1 7.024E-03 0.25 3.853E-05 182.8 7.724E-03 0.50 6.718E-05 180.6 9.390E-03
150
0.05 8.530E-06 192.0 5.883E-03 0.10 1.599E-05 190.2 6.546E-03 0.25 3.654E-05 189.5 7.406E-03 0.50 6.607E-05 187.2 8.774E-03
151
Glass beads: 4.48.3 mm (particle diameter)
Height of specimen: 192.0 m
369.0n , 87.310 D mm, 068.4k cm/s, 5.2sG
Test Date: May 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 9.737E-06 128.7 1.509E-02 0.10 1.783E-05 126.2 1.757E-02 0.25 3.638E-05 121.9 2.256E-02 0.50 5.926E-05 116.6 2.864E-02
50
0.05 1.054E-05 152.9 9.103E-03 0.10 1.785E-05 151.1 1.070E-02 0.25 3.609E-05 146.9 1.407E-02 0.50 5.923E-05 142.9 1.852E-02
75
0.05 1.081E-05 166.9 6.811E-03 0.10 1.867E-05 165.3 8.212E-03 0.25 3.475E-05 162.3 1.194E-02 0.50 5.905E-05 157.9 1.536E-02
100
0.05 1.110E-05 176.5 5.660E-03 0.10 1.955E-05 175.0 6.536E-03 0.25 4.069E-05 171.6 8.758E-03 0.50 6.196E-05 169.0 1.199E-02
125
0.05 1.043E-05 184.3 5.268E-03 0.10 1.905E-05 182.8 5.910E-03 0.25 4.161E-05 179.5 7.496E-03 0.50 6.879E-05 176.7 9.446E-03
150
0.05 9.510E-06 191.0 5.268E-03 0.10 1.730E-05 189.5 6.080E-03 0.25 3.725E-05 186.8 7.581E-03 0.50 6.536E-05 184.0 9.273E-03
Unloading
150
0.05 8.740E-06 188.4 6.016E-03 0.10 1.664E-05 187.4 6.557E-03 0.25 3.876E-05 185.5 7.395E-03 0.50 6.810E-05 183.8 9.016E-03
125
0.05 1.074E-05 180.2 5.146E-03 0.10 2.085E-05 179.2 5.507E-03 0.25 4.601E-05 177.4 7.035E-03 0.50 7.236E-05 175.8 9.289E-03
100 0.05 1.061E-05 172.1 6.090E-03
152
0.10 2.009E-05 170.9 6.780E-03 0.25 4.258E-05 169.0 8.663E-03 0.50 6.486E-05 167.4 1.167E-02
75
0.05 9.660E-06 161.5 9.008E-03 0.10 1.905E-05 160.1 8.864E-03 0.25 3.920E-05 157.8 1.160E-02 0.50 6.245E-05 155.8 1.442E-02
50
0.05 1.029E-05 147.4 1.015E-02 0.10 1.888E-05 145.4 1.136E-02 0.25 3.792E-05 142.6 1.486E-02 0.50 6.026E-05 140.3 1.845E-02
25
0.05 1.015E-05 124.3 1.564E-02 0.10 1.869E-05 121.8 1.857E-02 0.25 3.610E-05 118.4 2.423E-02 0.50 5.891E-05 115.8 2.996E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 8.910E-06 128.8 1.643E-02 0.10 1.773E-05 126.2 1.770E-02 0.25 3.728E-05 122.3 2.297E-02 0.50 5.717E-05 119.5 2.924E-02
50
0.05 8.917E-06 149.6 1.143E-02 0.10 1.571E-05 147.9 1.343E-02 0.25 3.405E-05 144.1 1.630E-02 0.50 5.737E-05 140.7 1.961E-02
75
0.05 8.360E-06 162.3 9.819E-03 0.10 1.670E-05 161.1 1.036E-02 0.25 3.759E-05 158.8 1.206E-02 0.50 6.116E-05 156.4 1.496E-02
100
0.05 9.337E-06 175.1 6.976E-03 0.10 1.706E-05 173.8 7.984E-03 0.25 3.759E-05 170.9 9.771E-03 0.50 6.372E-05 168.5 1.173E-02
125
0.05 9.213E-06 179.9 6.334E-03 0.10 1.762E-05 178.7 6.881E-03 0.25 3.914E-05 176.7 8.589E-03 0.50 6.598E-05 174.6 1.046E-02
150
0.05 8.357E-06 186.4 6.849E-03 0.10 1.601E-05 185.3 7.215E-03 0.25 3.753E-05 182.6 7.947E-03 0.50 6.714E-05 180.1 9.390E-03
153
F50 Ottawa Sand: 6.01.0 mm (particle diameter)
Height of specimen: 190.0 m
43.0n , 18.010 D mm, 055.0k cm/s, 65.2sG
Test Date: May 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 3.923E-05 82.8 9.230E-03 0.10 6.243E-05 81.9 1.228E-02 0.25 1.028E-04 80.0 1.965E-02 0.50 1.446E-04 77.9 2.737E-02
50
0.05 3.321E-05 108.3 5.194E-03 0.10 5.481E-05 106.2 7.056E-03 0.25 1.033E-04 100.8 1.148E-02 0.50 1.446E-04 95.3 1.924E-02
75
0.05 2.856E-05 121.9 4.419E-03 0.10 4.724E-05 119.8 6.859E-03 0.25 9.388E-05 115.3 9.098E-03 0.50 1.422E-04 109.5 1.403E-02
100
0.05 2.716E-05 131.4 3.746E-03 0.10 4.549E-05 129.7 4.971E-03 0.25 8.343E-05 124.5 9.633E-03 0.50 1.317E-04 120.3 1.178E-02
125
0.05 2.587E-05 139.3 3.226E-03 0.10 4.455E-05 137.6 4.340E-03 0.25 8.010E-05 134.5 6.987E-03 0.50 1.174E-04 130.2 1.312E-02
150
0.05 2.437E-05 145.3 3.050E-03 0.10 4.287E-05 143.6 4.005E-03 0.25 8.264E-05 140.4 6.032E-03 0.50 1.180E-04 136.9 9.236E-03
Unloading
150
0.05 2.134E-05 139.4 4.483E-03 0.10 3.686E-05 138.3 5.523E-03 0.25 7.236E-05 136.2 8.398E-03 0.50 1.223E-04 133.3 1.152E-02
125
0.05 2.801E-05 129.5 3.671E-03 0.10 4.998E-05 128.7 4.594E-03 0.25 9.631E-05 127.0 7.162E-03 0.50 1.434E-04 125.0 1.058E-02
100 0.05 3.004E-05 120.6 4.197E-03
154
0.10 5.298E-05 119.5 5.252E-03 0.25 1.026E-04 117.5 7.947E-03 0.50 1.508E-04 115.7 1.179E-02
75
0.05 3.322E-05 109.0 5.029E-03 0.10 5.824E-05 107.8 6.339E-03 0.25 1.075E-04 106.3 9.878E-03 0.50 1.531E-04 104.9 1.486E-02
50
0.05 3.766E-05 95.3 6.456E-03 0.10 6.303E-05 94.3 8.610E-03 0.25 1.105E-04 92.8 1.337E-02 0.50 1.532E-04 91.9 2.001E-02
25
0.05 4.195E-05 77.8 9.512E-03 0.10 6.832E-05 76.1 1.350E-02 0.25 1.135E-04 74.7 2.118E-02 0.50 1.507E-04 74.7 2.946E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 3.043E-05 79.5 1.249E-02 0.10 5.407E-05 77.8 1.489E-02 0.25 1.056E-04 74.7 2.138E-02 0.50 1.451E-04 73.8 2.850E-02
50
0.05 3.048E-05 102.2 6.509E-03 0.10 5.276E-05 100.0 8.207E-03 0.25 1.035E-04 94.0 1.275E-02 0.50 1.452E-04 88.5 2.064E-02
75
0.05 2.671E-05 113.2 6.021E-03 0.10 4.716E-05 111.2 7.347E-03 0.25 9.102E-05 108.0 1.017E-02 0.50 1.454E-04 99.9 1.642E-02
100
0.05 2.359E-05 127.4 4.865E-03 0.10 4.262E-05 125.8 5.915E-03 0.25 8.411E-05 122.6 8.318E-03 0.50 1.379E-04 115.1 1.272E-02
125
0.05 2.143E-05 133.7 4.966E-03 0.10 3.888E-05 132.2 5.820E-03 0.25 7.767E-05 129.0 7.910E-03 0.50 1.253E-04 122.9 1.198E-02
150
0.05 1.996E-05 139.2 4.758E-03 0.10 3.645E-05 137.6 5.613E-03 0.25 7.434E-05 134.5 7.761E-03 0.50 1.200E-04 128.5 1.133E-02
155
F50 Ottawa Sand: 6.01.0 mm (particle diameter)
Height of specimen: 195.0 m
359.0n , 18.010 D mm, 03.0k cm/s, 65.2sG
Test Date: May 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 2.452E-05 109.9 7.342E-03 0.10 4.025E-05 108.3 9.597E-03 0.25 7.206E-05 105.4 1.460E-02 0.50 1.047E-04 102.8 2.036E-02
50
0.05 2.442E-05 129.0 4.584E-03 0.10 3.917E-05 128.1 6.366E-03 0.25 6.691E-05 125.2 1.130E-02 0.50 1.029E-04 123.0 1.459E-02
75
0.05 2.200E-05 142.4 3.995E-03 0.10 3.744E-05 141.4 4.998E-03 0.25 6.981E-05 139.3 7.554E-03 0.50 9.918E-05 137.2 1.152E-02
100
0.05 1.994E-05 152.3 3.666E-03 0.10 3.510E-05 151.3 4.531E-03 0.25 6.880E-05 149.1 6.531E-03 0.50 1.051E-04 146.6 9.082E-03
125
0.05 1.855E-05 160.3 3.273E-03 0.10 3.291E-05 159.4 4.244E-03 0.25 6.555E-05 157.2 6.080E-03 0.50 1.040E-04 154.7 8.424E-03
150
0.05 1.793E-05 167.0 3.061E-03 0.10 3.176E-05 166.2 3.766E-03 0.25 6.451E-05 164.1 5.353E-03 0.50 1.045E-04 161.5 7.284E-03
Unloading
150
0.05 1.580E-05 165.9 3.995E-03 0.10 2.824E-05 164.9 4.759E-03 0.25 6.155E-05 162.9 5.836E-03 0.50 1.078E-04 161.0 7.263E-03
125
0.05 1.904E-05 157.3 3.347E-03 0.10 3.512E-05 156.3 3.809E-03 0.25 7.322E-05 154.8 5.491E-03 0.50 1.146E-04 153.5 7.814E-03
100 0.05 2.147E-05 148.9 3.236E-03
156
0.10 4.005E-05 148.1 3.851E-03 0.25 7.895E-05 146.7 5.857E-03 0.50 1.151E-04 145.7 8.610E-03
75
0.05 2.473E-05 139.2 3.295E-03 0.10 4.268E-05 138.4 4.382E-03 0.25 7.572E-05 137.1 7.141E-03 0.50 1.070E-04 134.6 1.181E-02
50
0.05 2.723E-05 126.4 4.133E-03 0.10 4.565E-05 125.3 5.623E-03 0.25 8.087E-05 123.6 8.944E-03 0.50 1.165E-04 121.9 1.293E-02
25
0.05 3.021E-05 107.3 5.862E-03 0.10 4.850E-05 106.0 8.196E-03 0.25 8.252E-05 103.9 1.287E-02 0.50 1.176E-04 102.0 1.868E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 2.743E-05 110.2 5.814E-03 0.10 4.670E-05 107.4 7.809E-03 0.25 9.490E-05 100.2 1.243E-02 0.50 1.231E-04 95.3 2.242E-02
50
0.05 2.357E-05 130.4 4.313E-03 0.10 4.082E-05 128.7 5.777E-03 0.25 7.641E-05 124.8 8.796E-03 0.50 1.258E-04 115.1 1.351E-02
75
0.05 2.400E-05 142.3 4.069E-03 0.10 3.677E-05 140.8 4.950E-03 0.25 7.065E-05 137.6 7.141E-03 0.50 1.160E-04 127.4 1.200E-02
100
0.05 1.738E-05 150.8 4.048E-03 0.10 3.362E-05 149.8 4.345E-03 0.25 7.363E-05 148.0 6.011E-03 0.50 1.136E-04 143.1 8.726E-03
125
0.05 1.471E-05 160.9 4.578E-03 0.10 2.800E-05 159.5 5.485E-03 0.25 6.026E-05 156.3 6.456E-03 0.50 1.111E-04 149.4 8.127E-03
150
0.05 1.538E-05 167.3 3.650E-03 0.10 2.828E-05 167.1 4.297E-03 0.25 6.029E-05 165.2 5.342E-03 0.50 1.051E-04 160.1 7.570E-03
157
ASTM 20-30 Ottawa Sand: 85.06.0 mm (particle diameter)
Height of specimen: 197.0 m
394.0n , 46.010 D mm, 157.0k cm/s, 65.2sG
Test Date: June 2012 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25
0.05 4.176E-05 86.4 7.358E-03 0.10 6.138E-05 84.9 1.047E-02 0.25 9.769E-05 82.5 1.812E-02 0.50 1.303E-04 81.9 2.596E-02
50
0.05 2.892E-05 115.4 5.517E-03 0.10 4.915E-05 113.1 6.563E-03 0.25 9.449E-05 107.2 1.041E-02 0.50 1.306E-04 102.6 1.696E-02
75
0.05 2.600E-05 131.4 3.729E-03 0.10 4.413E-05 129.5 4.934E-03 0.25 7.393E-05 125.5 8.547E-03 0.50 1.168E-04 118.3 1.456E-02
100
0.05 2.325E-05 142.6 3.475E-03 0.10 4.105E-05 140.7 4.281E-03 0.25 7.903E-05 135.9 6.822E-03 0.50 1.074E-04 132.7 1.040E-02
125
0.05 2.116E-05 151.3 3.157E-03 0.10 3.783E-05 149.6 3.963E-03 0.25 7.873E-05 144.7 6.048E-03 0.50 1.137E-04 140.1 9.135E-03
150
0.05 1.962E-05 158.8 2.982E-03 0.10 3.518E-05 157.2 3.862E-03 0.25 7.519E-05 152.3 5.400E-03 0.50 1.159E-04 147.5 8.016E-03
Unloading
150
0.05 1.785E-05 150.3 4.202E-03 0.10 3.379E-05 149.1 4.838E-03 0.25 7.469E-05 147.4 5.878E-03 0.50 1.179E-04 146.1 8.329E-03
125
0.05 2.604E-05 139.1 2.902E-03 0.10 4.576E-05 138.2 4.005E-03 0.25 8.049E-05 137.1 6.541E-03 0.50 1.141E-04 136.6 9.787E-03
100 0.05 2.939E-05 128.8 3.379E-03
158
0.10 5.148E-05 127.8 4.737E-03 0.25 9.085E-05 126.2 7.751E-03 0.50 1.249E-04 124.7 1.220E-02
75
0.05 3.331E-05 117.3 3.666E-03 0.10 5.810E-05 116.1 4.950E-03 0.25 1.010E-04 114.7 8.345E-03 0.50 1.348E-04 113.5 1.336E-02
50
0.05 3.753E-05 102.9 4.695E-03 0.10 6.480E-05 101.4 6.414E-03 0.25 1.073E-04 100.5 1.067E-02 0.50 1.403E-04 100.0 1.675E-02
25
0.05 4.308E-05 83.9 7.013E-03 0.10 6.948E-05 82.5 9.894E-03 0.25 1.118E-04 81.7 1.694E-02 0.50 1.411E-04 81.9 2.601E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 2.407E-05 85.7 1.386E-02 0.10 4.399E-05 83.6 1.571E-02 0.25 9.498E-05 80.9 2.020E-02 0.50 1.275E-04 80.6 3.019E-02
50
0.05 2.078E-05 117.1 7.353E-03 0.10 3.674E-05 114.6 9.024E-03 0.25 7.937E-05 107.1 1.280E-02 0.50 1.226E-04 99.4 2.001E-02
75
0.05 1.772E-05 134.6 6.058E-03 0.10 3.278E-05 132.2 7.125E-03 0.25 6.643E-05 125.9 1.074E-02 0.50 1.129E-04 116.5 1.515E-02
100
0.05 1.603E-05 143.9 5.809E-03 0.10 2.966E-05 141.7 6.669E-03 0.25 6.387E-05 134.9 9.305E-03 0.50 1.004E-04 125.7 1.467E-02
125
0.05 1.394E-05 153.7 5.819E-03 0.10 2.662E-05 151.7 6.398E-03 0.25 6.031E-05 146.3 8.021E-03 0.50 9.724E-05 139.2 1.143E-02
150
0.05 1.329E-05 158.7 5.724E-03 0.10 2.495E-05 157.1 6.207E-03 0.25 5.711E-05 152.1 7.777E-03 0.50 1.003E-04 144.2 1.018E-02
159
ASTM 20-30 Ottawa Sand: 85.06.0 mm (particle diameter)
Height of specimen: 197.0 m
347.0n , 46.010 D mm, 052.0k cm/s, 65.2sG
Test Date: June 2012 Dry condition
c (kPa) Input voltage (V) rf
Loading
25
0.05 3.332E-05 91.3 9.247E-03 0.10 5.526E-05 90.0 1.077E-02 0.25 1.035E-04 88.0 1.624E-02 0.50 1.397E-04 86.0 2.501E-02
50
0.05 2.765E-05 125.3 4.106E-03 0.10 4.605E-05 123.4 5.400E-03 0.25 9.207E-05 117.3 8.562E-03 0.50 1.431E-04 109.5 1.415E-02
75
0.05 2.302E-05 142.7 3.475E-03 0.10 3.938E-05 140.9 4.355E-03 0.25 7.895E-05 135.6 6.860E-03 0.50 1.326E-04 127.3 1.041E-02
100
0.05 1.903E-05 155.5 3.565E-03 0.10 3.258E-05 153.7 5.342E-03 0.25 7.243E-05 148.0 5.926E-03 0.50 1.227E-04 140.7 8.318E-03
125
0.05 1.749E-05 165.3 3.172E-03 0.10 3.080E-05 163.7 3.984E-03 0.25 6.081E-05 157.2 7.125E-03 0.50 1.109E-04 151.6 7.600E-03
150
0.05 1.741E-05 172.7 2.690E-03 0.10 3.103E-05 171.1 3.385E-03 0.25 5.959E-05 167.2 5.278E-03 0.50 1.050E-04 158.4 8.011E-03
Unloading
150
0.05 1.383E-05 162.3 4.849E-03 0.10 2.735E-05 160.7 4.939E-03 0.25 6.506E-05 158.6 5.522E-03 0.50 1.129E-04 156.9 7.475E-03
125
0.05 2.267E-05 151.0 2.674E-03 0.10 4.304E-05 150.0 3.088E-03 0.25 8.977E-05 148.2 4.546E-03 0.50 1.356E-04 146.3 6.716E-03
100 0.05 2.658E-05 139.7 2.801E-03
160
0.10 4.929E-05 138.5 3.294E-03 0.25 9.822E-05 136.7 5.379E-03 0.50 1.442E-04 134.9 8.175E-03
75
0.05 3.079E-05 126.5 3.199E-03 0.10 5.535E-05 125.4 4.143E-03 0.25 1.042E-04 123.7 6.573E-03 0.50 1.490E-04 121.8 1.029E-02
50
0.05 3.632E-05 110.4 4.064E-03 0.10 6.223E-05 109.2 5.560E-03 0.25 1.101E-04 107.4 9.034E-03 0.50 1.525E-04 105.6 1.429E-02
25
0.05 4.292E-05 88.0 6.467E-03 0.10 6.851E-05 86.6 9.140E-03 0.25 1.136E-04 85.0 1.534E-02 0.50 1.533E-04 84.0 2.410E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25
0.05 2.104E-05 93.7 1.268E-02 0.10 4.365E-05 91.9 1.285E-02 0.25 9.547E-05 89.2 1.609E-02 0.50 1.409E-04 82.8 2.900E-02
50
0.05 1.836E-05 126.9 6.711E-03 0.10 3.335E-05 124.5 8.026E-03 0.25 7.570E-05 116.6 1.095E-02 0.50 1.301E-04 106.2 1.665E-02
75
0.05 1.602E-05 142.6 5.836E-03 0.10 2.881E-05 140.7 6.897E-03 0.25 6.376E-05 134.7 9.422E-03 0.50 1.154E-04 124.9 1.274E-02
100
0.05 1.353E-05 157.3 5.528E-03 0.10 2.488E-05 155.5 6.504E-03 0.25 5.509E-05 149.4 8.573E-03 0.50 1.008E-04 141.2 1.065E-02
125
0.05 1.205E-05 165.0 5.438E-03 0.10 2.195E-05 163.2 6.647E-03 0.25 4.764E-05 158.1 8.732E-03 0.50 9.276E-05 149.6 1.039E-02
150
0.05 1.204E-05 174.2 4.621E-03 0.10 2.243E-05 172.7 5.390E-03 0.25 4.922E-05 168.2 7.013E-03 0.50 8.075E-05 162.4 9.957E-03
161
Group 3
Notes: One cycle of loading and unloading of RC tests were conducted for confining
pressures of 25, 50, 75, 100, 125, and 150 kPa with a maximum input voltage of 0.02,
0.04 or 0.05 V under dry condition.
Glass beads: 6.04.0 mm (particle diameter)
Height of specimen: 195.0 m
37.0n , 44.010 D mm, 135.0k cm/s, 5.2sG
Test Date: March 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 3.460E-06 113.3 1.067E-02 0.02 5.840E-06 111.7 1.361E-02 0.05 1.206E-05 108.5 1.818E-02
50 0.01 3.950E-06 140.6 4.907E-03 0.02 6.923E-06 139.5 6.058E-03 0.04 1.184E-05 137.8 7.634E-03
75 0.01 3.513E-06 154.0 4.366E-03 0.02 6.430E-06 153.1 4.987E-03 0.04 1.145E-05 151.7 5.750E-03
100 0.01 3.280E-06 164.0 3.767E-03 0.02 6.030E-06 163.4 4.494E-03 0.04 1.067E-05 162.1 5.496E-03
125 0.01 3.177E-06 172.0 3.087E-03 0.02 5.910E-06 171.6 3.708E-03 0.04 1.056E-05 170.5 4.514E-03
150 0.01 3.077E-06 178.6 2.870E-03 0.02 5.753E-06 178.1 3.257E-03 0.04 1.048E-05 177.2 3.921E-03
Unloading
150 0.01 3.000E-06 178.8 1.182E-02 0.02 5.647E-06 178.3 3.289E-03 0.04 1.045E-05 177.5 4.000E-03
125 0.01 3.113E-06 171.3 3.225E-03 0.02 5.773E-06 170.7 3.698E-03 0.04 1.043E-05 169.6 4.600E-03
162
100 0.01 3.170E-06 160.5 4.334E-03 0.02 5.817E-06 159.8 4.769E-03 0.04 1.041E-05 158.7 6.132E-03
75 0.01 3.683E-06 147.7 4.589E-03 0.02 6.653E-06 146.8 5.342E-03 0.04 1.158E-05 145.3 6.504E-03
50 0.01 4.083E-06 131.1 5.719E-030.02 7.230E-06 130.1 6.759E-03 0.04 1.259E-05 128.6 8.488E-03
25 0.01 4.557E-06 106.4 9.029E-03 0.02 7.923E-06 104.9 1.097E-02 0.04 1.384E-05 102.9 1.337E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 3.470E-06 105.7 1.142E-02 0.02 6.340E-06 104.3 1.356E-02 0.04 1.173E-05 101.9 1.617E-02
50 0.01 2.857E-06 131.0 8.530E-03 0.02 5.330E-06 130.1 9.597E-03 0.04 9.793E-06 128.3 1.108E-02
75 0.01 2.553E-06 148.2 7.008E-03 0.02 4.833E-06 147.5 7.819E-03 0.04 8.947E-06 146.2 8.727E-03
100 0.01 2.187E-06 160.3 7.273E-03 0.02 4.237E-06 159.8 7.199E-03 0.04 8.020E-06 158.7 8.165E-03
125 0.01 2.130E-06 169.3 6.297E-03 0.02 4.103E-06 169.0 6.488E-03 0.04 7.780E-06 167.9 7.135E-03
150 0.01 2.100E-06 175.9 5.406E-03 0.02 4.050E-06 175.7 5.613E-03 0.04 7.720E-06 174.9 6.276E-03
163
Glass beads: 3.10.1 mm (particle diameter)
Height of specimen: 196.0 m
371.0n , 04.110 D mm, 532.0k cm/s, 5.2sG
Test Date: March 2013 Dry condition
c (kPa) Input voltage (V) rf
Loading
25 0.01 1.253E-06 130.5 2.020E-02 0.02 2.387E-06 129.6 2.292E-02 0.05 5.610E-06 127.5 2.601E-02
50 0.01 1.170E-06 154.5 1.614E-02 0.02 2.270E-06 154.1 1.714E-02 0.05 5.500E-06 152.7 1.761E-02
75 0.01 1.203E-06 169.1 1.322E-02 0.02 2.383E-06 168.9 1.324E-020.05 5.800E-06 167.9 1.315E-02
100 0.01 1.310E-06 178.3 1.020E-02 0.02 2.267E-06 178.0 1.002E-02 0.05 6.310E-06 177.0 1.088E-02
125 0.01 1.327E-06 185.5 8.339E-03 0.02 2.630E-06 185.3 8.669E-03 0.05 6.467E-06 184.5 9.029E-03
150 0.01 1.290E-06 191.7 7.908E-03 0.02 2.560E-06 191.5 8.053E-03 0.05 6.400E-06 190.6 8.268E-03
Unloading
150 0.01 1.330E-06 191.9 7.607E-03 0.02 2.647E-06 191.6 7.915E-03 0.05 6.440E-06 190.9 8.254E-03
125 0.01 1.287E-06 184.4 9.315E-03 0.02 2.540E-06 184.1 9.543E-03 0.05 6.160E-06 183.3 9.825E-03
100 0.01 1.140E-06 174.5 1.220E-02 0.02 2.247E-06 174.3 1.248E-02 0.05 5.460E-06 173.4 1.302E-02
75 0.01 8.667E-07 163.4 1.993E-02 0.02 1.720E-06 163.4 2.036E-02 0.05 4.350E-06 162.3 1.993E-02
50 0.01 1.050E-06 146.5 2.002E-02 0.02 2.050E-06 146.2 1.899E-02 0.05 4.977E-06 145.1 2.116E-02
164
25 0.01 1.120E-06 125.3 2.661E-02 0.02 2.163E-06 124.8 2.760E-02 0.05 5.190E-06 123.2 2.961E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 8.967E-07 132.3 2.726E-02 0.02 1.740E-06 132.0 2.906E-02 0.05 4.247E-06 130.0 3.258E-02
50 0.01 9.567E-07 151.7 2.035E-02 0.02 1.860E-06 151.4 2.042E-02 0.05 4.480E-06 149.9 2.197E-02
75 0.01 9.200E-07 166.5 1.581E-02 0.02 1.807E-06 166.2 1.681E-02 0.05 4.417E-06 165.0 1.810E-02
100 0.01 8.833E-07 176.8 1.551E-02 0.02 1.740E-06 176.6 1.608E-02 0.05 4.300E-06 175.6 1.673E-02
125 0.01 8.900E-07 184.8 1.441E-02 0.02 1.780E-06 184.7 1.405E-02 0.05 4.403E-06 184.1 1.398E-02
150 0.01 9.200E-07 191.2 1.151E-02 0.02 1.870E-06 191.1 1.160E-02 0.05 4.810E-06 190.0 1.149E-02
165
Glass beads: 1.27.1 mm (particle diameter)
Height of specimen: 196.0 m
376.0n , 75.110 D mm, 248.1k cm/s, 5.2sG
Test Date: March 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 2.777E-06 124.6 1.067E-02 0.02 5.060E-06 123.1 1.214E-02 0.05 1.077E-05 119.5 1.586E-02
50 0.01 2.817E-06 145.3 7.157E-03 0.02 5.103E-06 144.5 7.851E-03 0.05 1.086E-05 142.5 1.029E-02
75 0.01 2.613E-06 159.7 5.745E-03 0.02 4.827E-06 159.1 6.350E-030.05 1.030E-05 157.4 8.085E-03
100 0.01 2.687E-06 169.7 4.902E-03 0.02 4.997E-06 169.1 4.955E-03 0.05 1.083E-05 167.6 6.546E-03
125 0.01 2.677E-06 176.8 3.830E-03 0.02 5.017E-06 176.4 4.403E-03 0.05 1.100E-05 175.1 5.522E-03
150 0.01 2.607E-06 182.8 3.703E-03 0.02 4.923E-06 182.4 4.085E-03 0.05 1.095E-05 181.1 4.981E-03
Unloading
150 0.01 2.583E-06 182.8 3.846E-03 0.02 4.887E-06 182.4 3.952E-03 0.05 1.104E-05 181.4 4.923E-03
125 0.01 2.493E-06 174.6 4.589E-03 0.02 4.717E-06 174.2 4.987E-03 0.05 1.055E-05 173.1 5.942E-03
100 0.01 2.380E-06 164.4 5.830E-03 0.02 4.490E-06 163.9 6.398E-03 0.05 1.008E-05 162.8 7.793E-03
75 0.01 2.460E-06 152.3 7.544E-03 0.02 4.627E-06 151.7 7.793E-03 0.05 1.027E-05 150.2 8.875E-03
50 0.01 2.640E-06 137.0 9.300E-03 0.02 4.877E-06 136.2 9.793E-03 0.05 1.054E-05 134.4 1.216E-02
166
25 0.01 2.933E-06 116.4 1.157E-02 0.02 5.320E-06 115.3 1.345E-02 0.05 1.113E-05 112.7 1.761E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 2.340E-06 120.5 1.326E-02 0.02 4.330E-06 119.4 1.514E-02 0.05 9.710E-06 116.8 1.847E-02
50 0.01 2.263E-06 141.0 9.512E-03 0.02 4.220E-06 140.2 1.045E-02 0.05 9.340E-06 138.3 1.277E-02
75 0.01 2.140E-06 155.9 7.724E-03 0.02 4.030E-06 155.4 8.223E-03 0.05 9.013E-06 153.8 9.978E-03
100 0.01 2.050E-06 166.9 6.981E-03 0.02 3.907E-06 166.5 7.369E-03 0.05 8.823E-06 165.2 8.514E-03
125 0.01 2.060E-06 175.0 5.931E-03 0.02 3.930E-06 174.6 6.403E-03 0.05 9.007E-06 173.4 7.353E-03
150 0.01 2.010E-06 181.2 5.453E-03 0.02 3.880E-06 180.9 5.629E-03 0.05 8.983E-06 179.9 6.456E-03
167
Glass beads: 4.20.2 mm (particle diameter)
Height of specimen: 197.0 m
369.0n , 05.210 D mm, 504.1k cm/s, 5.2sG
Test Date: February 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 2.637E-06 115.4 1.315E-02 0.02 4.850E-06 114.0 1.504E-02 0.05 1.071E-05 111.7 1.872E-02
50 0.01 2.707E-06 141.9 7.517E-03 0.02 4.997E-06 141.0 8.669E-03 0.05 1.128E-05 138.9 1.019E-02
75 0.01 2.523E-06 157.3 6.313E-03 0.02 4.753E-06 156.6 6.748E-030.05 1.054E-05 154.9 8.414E-03
100 0.01 2.670E-06 168.6 4.812E-03 0.02 4.967E-06 167.9 5.172E-03 0.05 1.101E-05 166.3 6.292E-03
125 0.01 2.620E-06 176.5 3.963E-03 0.02 4.927E-06 175.9 4.408E-03 0.05 1.108E-05 174.5 5.289E-03
150 0.01 2.447E-06 183.0 4.196E-03 0.02 4.527E-06 182.4 4.568E-03 0.05 1.009E-05 181.2 5.549E-03
Unloading
150 0.01 2.423E-06 182.9 3.926E-03 0.02 4.590E-06 182.3 4.138E-03 0.05 1.036E-05 181.4 5.321E-03
125 0.01 2.377E-06 174.3 4.870E-03 0.02 4.543E-06 173.9 5.188E-03 0.05 1.051E-05 172.9 5.963E-03
100 0.01 2.280E-06 164.9 6.085E-03 0.02 4.363E-06 164.4 6.446E-03 0.05 1.010E-05 163.4 7.629E-03
75 0.01 2.257E-06 153.4 8.058E-03 0.02 4.320E-06 152.8 8.085E-03 0.05 1.007E-05 151.7 8.774E-03
50 0.01 2.370E-06 138.4 9.385E-03 0.02 4.483E-06 137.7 1.067E-02 0.05 1.021E-05 136.2 1.205E-02
168
25 0.01 2.577E-06 116.6 1.362E-02 0.02 4.757E-06 115.5 1.467E-02 0.05 1.071E-05 113.2 1.775E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 1.863E-06 121.5 1.597E-02 0.02 3.547E-06 120.6 1.794E-02 0.05 8.443E-06 118.6 1.990E-02
50 0.01 1.797E-06 140.3 1.263E-02 0.02 3.430E-06 139.6 1.335E-02 0.05 8.217E-06 138.1 1.430E-02
75 0.01 1.810E-06 156.0 1.023E-02 0.02 3.493E-06 155.5 1.018E-02 0.05 8.247E-06 154.2 1.098E-02
100 0.01 1.933E-06 167.1 7.029E-03 0.02 3.690E-06 166.6 7.745E-03 0.05 8.590E-06 165.4 8.615E-03
125 0.01 1.963E-06 175.6 5.799E-03 0.02 3.767E-06 175.1 6.414E-03 0.05 8.787E-06 173.9 7.125E-03
150 0.01 1.940E-06 182.2 5.146E-03 0.02 3.747E-06 181.9 5.629E-03 0.05 8.807E-06 180.7 6.276E-03
169
Glass beads: 45.385.2 mm (particle diameter)
Height of specimen: 198.0 m
367.0n , 92.210 D mm, 561.2k cm/s, 5.2sG
Test Date: February 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 2.230E-06 130.6 1.171E-02 0.02 4.123E-06 129.4 1.351E-02 0.05 9.243E-06 126.7 1.584E-02
50 0.01 2.300E-06 152.4 8.127E-03 0.02 4.280E-06 151.7 8.690E-03 0.05 9.440E-06 149.8 1.043E-02
75 0.01 2.407E-06 167.5 5.995E-03 0.02 4.440E-06 166.8 6.228E-030.05 9.653E-06 165.0 7.793E-03
100 0.01 2.507E-06 177.9 4.218E-03 0.02 4.663E-06 177.3 4.770E-03 0.05 1.022E-05 175.8 5.894E-03
125 0.01 2.483E-06 185.5 3.868E-03 0.02 4.670E-06 185.0 4.162E-03 0.05 1.038E-05 183.7 5.125E-03
150 0.01 2.407E-06 191.8 3.369E-03 0.02 4.520E-06 191.3 3.730E-03 0.05 9.520E-06 190.3 5.088E-03
Unloading
150 0.01 2.330E-06 191.5 3.751E-03 0.02 4.293E-06 191.2 4.260E-03 0.05 9.620E-06 190.4 4.928E-03
125 0.01 2.220E-06 182.2 4.531E-03 0.02 4.243E-06 181.9 4.865E-03 0.05 9.823E-06 181.0 5.676E-03
100 0.01 2.127E-06 172.4 5.146E-03 0.02 4.067E-06 171.9 6.382E-03 0.05 9.383E-06 171.0 7.284E-03
75 0.01 1.927E-06 160.7 8.101E-03 0.02 3.760E-06 160.2 8.615E-03 0.05 8.900E-06 159.2 9.342E-03
50 0.01 2.080E-06 146.1 9.618E-03 0.02 3.920E-06 145.5 1.062E-02 0.05 8.810E-06 144.0 1.255E-02
170
25 0.01 2.170E-06 125.4 1.322E-02 0.02 4.023E-06 124.4 1.501E-02 0.05 8.940E-06 122.3 1.793E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 1.680E-06 129.2 1.607E-02 0.02 3.170E-06 128.3 1.745E-02 0.05 7.347E-06 126.3 2.028E-02
50 0.01 1.763E-06 149.8 1.100E-02 0.02 3.327E-06 149.0 1.202E-02 0.05 7.620E-06 147.2 1.404E-02
75 0.01 1.780E-06 164.6 8.551E-03 0.02 3.337E-06 164.1 8.865E-03 0.05 7.580E-06 162.6 1.081E-02
100 0.01 1.880E-06 175.2 6.616E-03 0.02 3.573E-06 174.7 6.854E-03 0.05 8.183E-06 173.3 8.164E-03
125 0.01 1.903E-06 183.8 5.613E-03 0.02 3.633E-06 183.3 5.862E-03 0.05 8.337E-06 182.0 6.844E-03
150 0.01 1.867E-06 190.6 5.114E-03 0.02 3.577E-06 190.2 5.268E-03 0.05 8.273E-06 188.9 6.133E-03
171
Glass beads: 4.48.3 mm (particle diameter)
Height of specimen: 197.0 m
370.0n , 87.310 D mm, 11.4k cm/s, 5.2sG
Test Date: February 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 1.917E-06 123.8 1.564E-02 0.02 3.610E-06 122.6 1.736E-02 0.05 8.247E-06 120.4 1.982E-02
50 0.01 2.103E-06 148.1 9.045E-03 0.02 3.940E-06 147.3 9.973E-03 0.05 8.997E-06 145.3 1.171E-02
75 0.01 2.137E-06 163.8 6.881E-03 0.02 4.003E-06 163.1 7.565E-030.05 9.107E-06 161.6 8.711E-03
100 0.01 2.273E-06 175.1 5.289E-03 0.02 4.267E-06 174.9 5.666E-03 0.05 9.497E-06 173.2 7.034E-03
125 0.01 2.283E-06 184.0 4.005E-03 0.02 4.310E-06 183.4 4.727E-03 0.05 9.630E-06 182.0 5.735E-03
150 0.01 2.250E-06 190.5 3.852E-03 0.02 4.260E-06 190.1 4.334E-03 0.05 9.687E-06 188.8 5.072E-03
Unloading
150 0.01 2.210E-06 190.3 4.085E-03 0.02 4.217E-06 189.9 4.313E-03 0.05 9.747E-06 189.0 5.029E-03
125 0.01 1.980E-06 180.9 5.592E-03 0.02 3.823E-06 180.5 5.767E-03 0.05 8.940E-06 179.7 6.546E-03
100 0.01 1.880E-06 171.3 6.844E-03 0.02 3.587E-06 170.9 7.682E-03 0.05 8.377E-06 170.0 8.329E-03
75 0.01 1.743E-06 159.7 8.785E-03 0.02 3.360E-06 159.1 9.925E-03 0.05 7.807E-06 158.2 1.110E-02
50 0.01 1.817E-06 144.4 1.089E-02 0.02 3.457E-06 143.7 1.234E-02 0.05 8.080E-06 142.2 1.385E-02
172
25 0.01 1.950E-06 123.5 1.573E-02 0.02 3.643E-06 122.5 1.661E-02 0.05 8.267E-06 120.6 1.995E-02
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 1.597E-06 127.3 1.580E-02 0.02 3.033E-06 126.5 1.960E-02 0.05 7.110E-06 124.7 2.149E-02
50 0.01 1.547E-06 148.3 1.214E-02 0.02 2.933E-06 147.6 1.379E-02 0.05 6.820E-06 146.1 1.491E-02
75 0.01 1.607E-06 163.3 1.006E-02 0.02 3.043E-06 162.8 1.046E-02 0.05 7.080E-06 161.4 1.134E-02
100 0.01 1.730E-06 173.7 7.432E-03 0.02 3.297E-06 173.3 8.106E-03 0.05 7.620E-06 172.0 8.902E-03
125 0.01 1.780E-06 182.1 6.032E-03 0.02 3.393E-06 181.6 6.637E-03 0.05 7.843E-06 180.3 7.570E-03
150 0.01 1.813E-06 189.1 5.209E-03 0.02 3.470E-06 188.5 5.751E-03 0.05 8.033E-06 187.4 6.658E-03
173
Glass beads: 6 mm (particle diameter)
Height of specimen: 198.0 m
358.0n , 610 D mm, 319.7k cm/s, 5.2sG
Test Date: February 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 1.660E-06 126.2 1.642E-02 0.02 3.253E-06 124.7 1.790E-02 0.05 7.700E-06 122.0 2.023E-02
50 0.01 1.710E-06 147.7 1.128E-02 0.02 3.363E-06 146.6 1.151E-02 0.05 8.033E-06 144.4 1.253E-02
75 0.01 1.760E-06 161.2 8.875E-03 0.02 3.477E-06 160.3 9.305E-030.05 8.293E-06 158.2 9.766E-03
100 0.01 1.790E-06 171.2 7.358E-03 0.02 3.513E-06 170.3 7.703E-03 0.05 8.323E-06 168.3 8.472E-03
125 0.01 1.797E-06 178.9 6.615E-03 0.02 3.553E-06 177.9 6.663E-03 0.05 8.483E-06 176.1 7.257E-03
Unloading
125 0.01 1.883E-06 177.9 6.181E-03 0.02 3.670E-06 177.4 6.271E-03 0.05 8.620E-06 176.2 6.971E-03
100 0.01 1.667E-06 166.6 8.281E-03 0.02 3.297E-06 166.0 8.403E-03 0.05 7.940E-06 164.8 9.305E-03
75 0.01 1.557E-06 155.2 1.086E-02 0.02 3.053E-06 154.5 1.163E-02 0.05 7.317E-06 153.3 1.236E-02
50 0.01 1.547E-06 141.2 1.408E-02 0.02 3.017E-06 140.5 1.459E-02 0.05 7.057E-06 139.1 1.649E-02
25 0.01 1.603E-06 121.9 1.878E-02 0.02 3.090E-06 120.8 1.982E-02 0.05 7.180E-06 119.0 2.281E-02
174
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 1.190E-06 126.5 1.925E-02 0.02 2.333E-06 125.5 2.463E-02 0.05 5.723E-06 123.2 2.719E-02
50 0.01 1.310E-06 145.5 1.552E-02 0.02 2.537E-06 144.5 1.648E-02 0.05 6.060E-06 142.5 1.808E-02
75 0.01 1.303E-06 160.6 1.293E-02 0.02 2.607E-06 159.6 1.217E-02 0.05 6.353E-06 157.6 1.332E-02
100 0.01 1.457E-06 171.6 9.199E-03 0.02 2.823E-06 170.5 9.761E-03 0.05 6.643E-06 168.5 1.077E-02
125 0.01 1.520E-06 179.2 7.363E-03 0.02 2.940E-06 178.4 8.047E-03 0.05 6.930E-06 176.3 8.955E-03
175
ASTM 20-30 Ottawa Sand: 85.06.0 mm (particle diameter)
Height of specimen: 197.0 m
387.0n , 46.010 D mm, 284.0k cm/s, 65.2sG
Test Date: December 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 8.813E-06 101.4 3.831E-03 0.02 1.548E-05 100.7 5.109E-03
50 0.01 6.937E-06 126.4 2.764E-03 0.02 1.237E-05 126.0 3.369E-03
75 0.01 5.957E-06 139.9 2.446E-03 0.02 1.082E-05 139.5 2.817E-03
100 0.01 5.273E-06 150.1 2.133E-03 0.02 9.630E-06 149.8 2.552E-03
125 0.01 4.370E-06 158.7 2.584E-03 0.02 8.213E-06 158.4 2.945E-03
150 0.01 4.320E-06 166.1 1.995E-03 0.02 8.063E-06 165.8 2.371E-03
Unloading
150 0.01 4.267E-06 166.3 2.011E-03 0.02 8.077E-06 166.0 2.318E-03
125 0.01 4.357E-06 159.0 2.440E-03 0.02 8.187E-06 158.6 2.817E-03
100 0.01 5.247E-06 149.5 2.111E-03 0.02 9.737E-06 149.1 2.557E-03
75 0.01 6.167E-06 138.6 2.196E-03 0.02 1.116E-05 138.1 2.690E-03
50 0.01 7.377E-06 124.3 2.398E-03 0.02 1.319E-05 123.7 3.167E-03
25 0.01 9.387E-06 102.0 3.703E-03 0.02 1.572E-05 101.3 4.849E-03
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 6.660E-06 105.1 5.528E-03 0.02 1.296E-05 103.1 6.170E-03
50 0.01 5.207E-06 121.8 5.045E-03 0.02 1.023E-05 120.2 5.369E-03
176
75 0.01 4.243E-06 135.3 4.499E-03 0.02 8.203E-06 134.7 5.029E-03
100 0.01 3.710E-06 146.5 4.573E-03 0.02 7.253E-06 145.7 4.769E-03
125 0.01 3.303E-06 154.0 4.425E-03 0.02 6.430E-06 153.7 4.567E-03
150 0.01 2.927E-06 161.6 4.674E-03 0.02 5.714E-06 161.3 4.801E-03
177
ASTM 20-30 Ottawa Sand: 85.06.0 mm (particle diameter)
Height of specimen: 195.0 m
343.0n , 46.010 D mm, 192.0k cm/s, 65.2sG
Test Date: December 2013 Dry condition
c (kPa) Input voltage (V) rf (Hz)
Loading
25 0.01 7.823E-06 102.9 4.382E-03 0.02 1.428E-05 101.8 5.326E-03
50 0.01 6.623E-06 131.8 2.303E-03 0.02 8.387E-06 131.3 2.966E-03
75 0.01 5.840E-06 144.7 1.772E-03 0.02 1.056E-05 144.4 2.377E-03
100 0.01 4.900E-06 155.1 1.974E-03 0.02 9.263E-06 154.6 2.319E-03
125 0.01 4.207E-06 164.0 2.164E-03 0.02 7.973E-06 163.7 2.494E-03
150 0.01 4.247E-06 171.7 1.512E-03 0.02 8.020E-06 171.3 1.873E-03
Unloading
150 0.01 4.217E-06 171.7 1.628E-03 0.02 7.993E-06 171.4 1.862E-03
125 0.01 4.263E-06 164.5 2.026E-03 0.02 8.040E-06 164.3 2.440E-03
100 0.01 4.973E-06 154.8 1.862E-03 0.02 9.407E-06 154.5 2.186E-03
75 0.01 5.907E-06 143.9 1.889E-03 0.02 1.088E-05 143.5 2.398E-03
50 0.01 6.770E-06 130.0 2.318E-03 0.02 1.245E-05 129.2 2.934E-03
25 0.01 8.967E-06 103.5 3.337E-03 0.02 1.598E-05 101.9 4.636E-03
Saturated condition
c (kPa) Input voltage (V) rf (Hz)
25 0.01 6.753E-06 103.1 5.708E-03 0.02 1.322E-05 99.5 6.234E-03
50 0.01 4.947E-06 125.2 4.652E-03 0.02 9.827E-06 123.5 4.982E-03
178
75 0.01 4.017E-06 141.3 4.361E-03 0.02 7.787E-06 140.6 4.478E-03
100 0.01 3.370E-06 152.6 4.398E-03 0.02 6.613E-06 151.9 4.695E-03
125 0.01 2.983E-06 160.1 4.748E-03 0.02 5.817E-06 159.8 4.880E-03
150 0.01 2.880E-06 167.7 4.021E-03 0.02 5.603E-06 167.4 4.106E-03
ACADEMIC VITA Yanbo Huang
302 Vairo Boulevard, Apartment D, State College, PA, 16803 e-mail: [email protected]
EDUCATION Ph.D. in Civil Engineering, 2014,The Pennsylvania State University, University Park, PA M.S. in Civil Engineering, 2009, Chengdu University of Technology, Chengdu, China B.S. in Civil Engineering, 2006, Chengdu University of Technology, Chengdu, China ASSOCIATION MEMBERSHIPS Certified Engineer-in-Training (EIT) in Civil Engineering, MI, 2013 Student Member of ASCE (American Society of Civil Engineers) PROFESSIONAL EXPERIENCE 2010 – Present Research Assistant The Pennsylvania State University, University Park, PA 2009 – 2010 Research Assistant Clarkson University, Potsdam, NY SELECTED PUBLICATIONS Qiu, T., Huang, Y.B., Guadalupe-Torres, Y., Baxter, C.D.P., and Fox, P.J. (2014). “Effective Soil Density for Small Strain Shear Waves in Saturated Granular Materials,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, under review.
Huang, Y.B. and Qiu, T. (2014). “Analytical Solutions for Biot Flow Induced Damping in Saturated Poroviscoelastic Soil Specimen in Resonant Column Test,” Soil Dynamics and Earthquake Engineering, under review.
Huang, Y.B. and Qiu, T. (2013). “Evaluation of Effective Soil Density in Resonant Column Tests,” Proceedings of 2nd IACGE International Conference on Geotechnical and Earthquake Engineering, Chengdu, Sichuan, China, October 25 – 27, pp. 1-9.
Qiu, T. and Huang, Y.B. (2012). “Energy Dissipation in Nearly Saturated Poroviscoelastic Soil Column during Quasi-Static Compressional Excitations,” Journal of Engineering Mechanics, ASCE, 138(10), 1263–1274
Huang, Y.B. and Li Jing (2008). “Quality Evaluation of Rock Mass Based on Indicators Measured in Situ,” Shanxi Water Resources, Issue Number 5, (in Chinese)