See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/239388887
Shear Strength and Stiffness of Silty Sand
Article in Journal of Geotechnical and Geoenvironmental Engineering · May 2000
DOI: 10.1061/(ASCE)1090-0241(2000)126:5(451)
CITATIONS
READS
519
6,194
3 authors:
Rodrigo Salgado
Paola Bandini
Purdue University
New Mexico State University
253 PUBLICATIONS 9,591 CITATIONS
44 PUBLICATIONS 1,421 CITATIONS
SEE PROFILE
SEE PROFILE
Ahmed M Karim
Misr University for Science & Technology
5 PUBLICATIONS 532 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Experimental Study of the Load Response of Large Diameter Closed-ended and Open-ended Pipe Piles Installed in Alluvial Soil View project
Experimental Determination of Displacement and Strain Fields around Piles using Digital Image Correlation (DIC) View project
All content following this page was uploaded by Rodrigo Salgado on 17 September 2014.
The user has requested enhancement of the downloaded file.
SHEAR STRENGTH
AND
STIFFNESS
OF
SILTY SAND
By R. Salgado,1 Member, ASCE, P. Bandini,2 Student Member, ASCE, and A. Karim3
ABSTRACT: The properties of clean sands pertaining to shear strength and stiffness have been studied extensively. However, natural sands generally contain significant amounts of silt and/or clay. The mechanical response
of such soils is different from that of clean sands. This paper addresses the effects of nonplastic fines on the
small-strain stiffness and shear strength of sands. A series of laboratory tests was performed on samples of
Ottawa sand with fines content in the range of 5–20% by weight. The samples were prepared at different relative
densities and were subjected to various levels of mean effective consolidation stress. Most of the triaxial tests
were conducted to axial strains in excess of 30%. The stress-strain responses were recorded, and the shear
strength and dilatancy parameters were obtained for each fines percentage. Bender element tests performed in
triaxial test samples allowed assessment of the effect of fines content on small-strain mechanical stiffness.
INTRODUCTION
It is well established that soils behave as linear elastic materials at shear strains smaller than about 10⫺4 –10⫺3%. For
larger shear strains, the stress-strain relationship is nonlinear.
Peak shear strength develops at relatively large strains (corresponding typically to axial strains in the range of 1–4%),
and critical-state shear strength (corresponding to no volume
change during shearing) develops for axial strains in excess of
25%. As the Mohr-Coulomb failure criterion is commonly
used to describe shear failure in soils, friction angles determined at the peak and critical states can be defined.
Knowledge of the values of small-strain stiffness and critical-state and peak friction angles is very useful for applications
based on constitutive models or analyses that attempt to capture material response from the initial stages of loading up to
shear failure. More immediate application of such knowledge
can be made in analyses that rely predominantly on smallstrain stiffness (such as the design of machine foundations),
or on friction angles only (such as stability analyses of various
forms, where deformations prior to collapse are not considered).
The stress-strain response of sand at small-, intermediate-,
and large-strain levels depends upon soil state variables (the
relative density DR of the sand, the effective stress state, and
fabric) and other factors related to the nature of the sand (particle shape, particle size distribution, particle surface characteristics, and mineralogy). The factors related to the constitution and general nature of the sand particles are referred to as
intrinsic variables (Been et al. 1991; Salgado et al. 1997a,b).
Examples of intrinsic variables are the critical-state friction
angle ␾c, the maximum and minimum void ratios emax and emin,
and the dilatancy parameters Q and R of the peak friction angle
correlation of Bolton (1986).
The properties of clean sands have been extensively studied
under laboratory and field conditions. These include Ottawa,
Ticino, and Monterey #0 sands (Hardin and Richart 1963;
Chung et al. 1984; Bolton 1986; Lo Presti 1987; Lo Presti et
al. 1992). However, in situ soils often contain significant
amounts of fines. If realistic analyses are to be done of soil
1
Assoc. Prof., School of Civ. Engrg., Purdue Univ., West Lafayette, IN
47907. E-mail: rodrigo@ecn.purdue.edu
2
PhD Candidate, School of Civ. Engrg., Purdue Univ., West Lafayette,
IN.
3
Formerly, Postdoctoral Fellow, School of Civ. Engrg., Purdue Univ.,
West Lafayette, IN.
Note. Discussion open until October 1, 2000. To extend the closing
date one month, a written request must be filed with the ASCE Manager
of Journals. The manuscript for this paper was submitted for review and
possible publication on December 10, 1997. This paper is part of the
Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126,
No. 5, May, 2000. 䉷ASCE, ISSN 1090-0241/00/0005-0451–0462/$8.00
⫹ $.50 per page. Paper No. 17169.
mechanics problems involving these materials, information is
needed on their mechanical properties. The same framework
used to describe the small-strain stiffness and the shear
strength of clean sands may be used for silty sands, provided
that the fines content remains below some limit, usually taken
to be in the range of 15–20%. In this paper, we evaluate how
the intrinsic variables that appear in correlations for the smallstrain stiffness and the shear strength of sands vary with the
content of nonplastic fines.
SHEAR STRENGTH
Critical-State Friction Angle
The shear strength of a cohesionless soil can be defined by
the Mohr-Coulomb failure criterion with zero cohesive intercept
s = ␴ tan ␾
(1)
where s = shear strength; ␴ = normal stress on the plane of
shearing; and ␾ = friction angle. For a triaxial test, it is practical to write ␾ in terms of the principal effective stresses
␴ 1⬘
⫺1
␴ 3⬘
sin ␾ =
␴ ⬘1
⫹1
␴ 3⬘
(2)
where ␴ ⬘/␴
⬘3 = effective principal stress ratio or stress obliq1
uity.
In general, a loose sand contracts and a dense sand expands
as it approaches the critical state, usually defined as the state
at which the sand is sheared without changes in either shear
strength or volume. However, whether a sample of sand is
contractive or dilatant depends not only on density but also
on effective confining stress. According to the critical-state
model, when a loose sample is sheared under high effective
confining stress, the shear stress increases monotonically until
it reaches a plateau, after which the sample continues to undergo shear straining without any change in shear stress or
sample volume. The sample is then said to have reached the
critical state, and the corresponding friction angle is known as
the critical-state friction angle ␾c.
During the shearing of a dense sand, the sample contracts
initially and then dilates. The effective principal stress ratio
reaches a peak, associated with a peak friction angle, at which
the dilation rate is maximum. Further loading causes the shear
stress to drop until it reaches the critical state. For practical
purposes, the critical-state friction angle obtained from triaxial
tests is commonly taken as a unique value for a given granular
soil, regardless of the initial relative density and initial confining stress. Such an approach is well justified by results found
in the literature (Rowe 1962, 1971; Bolton 1986; Negussey et
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 / 451
TABLE 1.
Sand type
(1)
Monterey #0 sanda,b
Ticino sandd,e
Toyoura sandd,e
Ottawa sand (round)c
Sacramento River sanda
Hokksund sandd,e
Intrinsic and State Variables of Some Clean Sandsa
emin
(2)
emax
(3)
␾c
(4)
Cg
(5)
eg
(6)
ng
(7)
Cu
(8)
0.57
0.57
0.61
0.48
0.61
0.55
0.86
0.93
0.99
0.78
1.03
0.87
37.0
34.8
35.1
29.0
33.3
36.0
326
647
900
612
NA
942
2.97
2.27
2.17
2.17
NA
1.96
0.50
0.43
0.40
0.44
NA
0.46
1.6
1.5
1.27
1.48
1.47
1.91
a
Bolton (1986).
Chung et al. (1984).
c
Present paper.
d
Lo Presti (1987).
e
Lo Presti et al. (1992).
b
al. 1986; Been et al. 1991; Schanz and Vermeer 1996), which
support the concept that ␾c is unique for silica sands and may
be taken as an intrinsic variable of the sand. The values of the
critical-state friction angles, as well as several other characteristics, of different clean sands are given in Table 1.
Stress-Dilatancy Relation
Stress-dilatancy relations aim to describe the relationship
between the friction and dilatancy angles. The simplest relation can be obtained from a physical analogy—the sawtooth
model—where slippage takes place along a stepped plane separating two blocks of the same material. In this model, slippage takes place when friction between the two blocks on each
side of the plane is overcome and the two blocks move apart,
so that climbing and relative motion between the two blocks
may take place. A simple relation results
␶
= tan ␾ = tan(␾c ⫹ ␺)
␴⬘
(3)
where ␶ = shear stress acting on the plane of shearing; ␴⬘ =
normal effective stress on the plane of shearing; and ␺ = dilatancy angle.
More sophisticated theories have been developed to explain
the relationship between the friction and dilatancy angles. Taylor (1948) suggested an ‘‘energy correction’’ hypothesis to account for the dilation, whereby friction is considered a source
of energy dissipation. The resulting equation for simple shear
is
tan ␾ = tan ␾c ⫹ tan ␺
(4)
Rowe (1962) developed his stress-dilatancy theory based on
the analogy between irregular packings of soil particles and
regular assemblies of spheres or cylinders and on the hypothesis that a minimum energy ratio at failure is achieved. De
Josselin de Jong (1976) questioned the energy minimization
hypothesis made by Rowe, which should not apply to systems
that dissipate energy during loading. He did validate Rowe’s
conclusions through an analysis that does not rely on energy
minimization assumptions. The resulting stress-dilatancy theory, superior to all other attempts to relate shear strength to
dilation, can be best expressed in the form
N = MNc
(5)
where N = flow number = ␴ 1⬘/␴ 3⬘ = stress obliquity; Nc = critical-state flow number = (␴ ⬘/␴
⬘)
1
3 c = stress obliquity at critical
state; M = dilatancy number = 1 ⫺ dεv /dε1; dεv = volumetric
strain increment; and dε1 = major principal strain increment =
axial strain increment in triaxial compression tests. N, M, and
Nc are given in terms of ␾, ␾c, and ␺ by the following expressions:
N=
1 ⫹ sin ␾
= tan2
1 ⫺ sin ␾
Nc =
1 ⫹ sin ␾c
= tan2
1 ⫺ sin ␾c
M=
1 ⫹ sin ␺
= tan2
1 ⫺ sin ␺
冉
冉
冉
冊
冊
冊
45 ⫹
␾
2
(6)
45 ⫹
␾c
2
(7)
45 ⫹
␺
2
(8)
The dilatancy angle, in turn, is defined as
dε1
⫹1
kdε3
sin ␺ = ⫹
dε1
⫺1
kdε3
(9)
where dε1 and dε3 = principal strain increments; k = 1 for plane
strain; and k = 2 for triaxial test conditions.
Bolton (1986) reviewed a large number of triaxial and
plane-strain test results and proposed a much simpler relationship between ␾, ␾c, and ␺, which he found to be operationally
equivalent to (5)
␾ = ␾c ⫹ 0.8␺
(10)
The relationship between the peak friction angle ␾p and the
critical-state friction angle ␾c can be written for both triaxial
and plane-strain tests by modifying (10) so that the dilatancy
angles for both types of test are expressed in terms of the same
quantity IR, referred to as the dilatancy index
␾p = ␾c ⫹ 5IR
(11)
for plane-strain conditions, and
␾p = ␾c ⫹ 3IR
(12)
for triaxial conditions.
The dilatancy index IR is given, for both triaxial and planestrain tests, by
IR = ⫺
10
3
冉 冊
dεv
dε1
(13)
max
and is related to the relative density and effective confining
stress level through
IR = ID
冉
Q ⫺ ln
冊
100p p⬘
PA
⫺R
(14)
where ID = relative density expressed as a number between 0
and 1; p p⬘ = mean effective stress at peak strength; PA = reference stress (=100 kPa = 0.1 MPa ⬇ 1 tsf) in the same units
as p ⬘;
p and Q and R = fitting parameters. Eqs. (11) and (12)
are valid for 0 ⱕ IR ⱕ 4. For higher values of IR the value of
the peak friction angle is taken as the value calculated from
(11) or (12) with IR = 4.
452 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000
Small-Strain Shear Modulus
There are two general forms of empirical equations used to
estimate the shear modulus of sands: one was proposed by
Hardin (1978) and Hardin and Richart (1963) and another by
Roesler (1979). A series of resonant column tests were performed by Hardin and Richart (1963) to obtain the small-strain
shear modulus for round and angular Ottawa sands. They established empirical relations for G0 at a shear strain of ␥ =
10⫺4 or less as
g
G0 = CgP 1⫺n
A
EXPERIMENTAL PROGRAM AND PROCEDURES
(eg ⫺ e)2 ng
␴ ⬘m
1⫹e
(15)
where Cg, eg, and ng = regression constants that depend solely
on the soil (and are therefore intrinsic soil variables); ␴ ⬘m =
mean effective stress; PA = reference stress in the same units
as ␴ m⬘ ; and e = void ratio.
Roesler (1979) developed his correlation from shear-wave
velocity measurements. The equation has the following form:
G0 = CP (1⫺n⫺m)
␴ na␴ mp
A
(16)
where ␴a = normal effective stress acting along the direction
of wave propagation; ␴p = normal effective stress perpendicular to the wave-propagation direction; and m, n, and C =
fitting parameters.
A different form of the Hardin G0 equation, accounting for
the void ratio through a different function, has been proposed
by, among others, Jamiolkowski et al. (1991)
g ag
G0 = CgP 1⫺n
e ␴ ⬘mng
A
(17)
where ag = regression constant (an intrinsic variable of the soil,
if this correlation is used to model soil loading response).
Most recent correlations [e.g., Iwasaki and Tatsuoka (1977)
and Yu and Richart (1984)] were developed based on the form
proposed by Hardin (1978) and Hardin and Richart (1963).
Values of the curve-fitting parameters required to calculate G0
for different sands from previous studies are listed in Table 1.
Iwasaki and Tatsuoka (1977) studied G0 of clean sands, natural
sands with fines, and artificially graded sand with fines. They
proposed a correlation for G0 with the form
G0 = C(␥)B
(2.17 ⫺ e)2 1⫺m(␥)
PA
(␴ ⬘m)m(␥)
1⫹ e
(18)
where C(␥) and m(␥) = fitting parameters depending on the
strain level of the test; and B = fitting parameter that is independent of shear strain ␥, void ratio e, and confining stress
␴ m⬘ . The results of Iwasaki and Tatsuoka (1977) indicate that
G0 decreases with increasing fines content. Results of resonant
column tests on Ticino sand by D. C. F. Lo Presti (personal
communication, 1996) showed that the coefficient Cg of (15)
is reduced by about 50% when the fines content increases from
0 to 25%, while ng increases slightly. Randolph et al. (1994)
also recognized a significant reduction in the small-strain stiffness of sand with addition of silt. According to these authors,
the small-strain stiffness of silty sand with 5–10, 10–15, and
15–20% silt content ranges might be reduced to about 50, 25,
and 19% of the G0 value of clean sand, respectively.
Another nondestructive laboratory testing method used to
measure the small-strain stiffness of soils is the bender element
test. In this test, a shear wave is generated at one end of the
sample and its arrival detected at the other end. The shearwave velocity is calculated from the sample length and travel
time. The small-strain shear modulus G0 of sands is calculated
from the velocity of the shear wave as it travels through the
sample
G0 = ␳V 2s
where ␳ = total mass density of the soil, and Vs = shear-wave
velocity.
Because a triaxial test can be performed on the same sample
where Vs was measured using bender elements, bender element
testing has been increasingly used for measuring G0 (Shirley
and Hampton 1977; Dyvik and Madshus 1985; Viggiani and
Atkinson 1995a,b). It is used in this paper to study the smallstrain stiffness of clean Ottawa sand and Ottawa sand with 5,
10, 15, and 20% added silt.
(19)
As discussed in the preceding sections, the small-strain stiffness and the shear strength of sand may be expressed in terms
of a number of intrinsic variables (␾c, Q, R, Cg, eg, ng). The
intrinsic variables are a function of the nature of the sand and
thus change with fines content. For a given soil density, this
is a valid approach as long as the fines content remains below
a limit beyond which the fines may dominate over the sand
matrix, changing the mechanical response of the soil in a more
fundamental way. This paper focuses on sand and silty sand
where sand-to-sand particle contact prevails; the ranges of density and fines content for which this assumption holds will be
discussed later.
A series of triaxial and bender element tests was performed
to assess how the shear strength and small-strain stiffness of
Ottawa sand change as an increasing percentage of nonplastic
fines is added to it. Ottawa sand, designated as ASTM C 778,
is a standard, clean quartz sand with the grain size distribution
shown in Fig. 1. The diameters of the sand particles range
from 0.1 to 0.6 mm. Ottawa sand is defined as SP according
to the Unified Soil Classification System. The coefficient of
uniformity Cu is 1.48, and the mean grain size D50 is 0.39 mm.
The maximum and minimum void ratios emax and emin are 0.78
and 0.48, respectively. Its specific gravity GS is 2.65. Ottawa
sand particles are round to subround.
The nonplastic fines are #106 Sil-Co-Sil ground silica from
U.S. Silica Co., Ottawa, Ill., which passes the #200 sieve and
is composed of SiO2 (99.8%), with Al2O3( 0.05%) and Fe2O3
(0.035%) as secondary components. Its specific gravity is 2.65,
with the grain size distribution shown, together with the grain
size distribution of pure Ottawa sand, in Fig. 1.
Static, drained triaxial compression tests were conducted on
isotropically consolidated sand samples with 0, 5, 10, 15, and
20% nonplastic fines. To obtain homogeneous samples, the
slurry deposition method of Kuerbis and Vaid (1988) was
used. According to Kuerbis and Vaid (1988), the slurry-deposition method has the following advantages:
1. The method produces loose to dense samples in the commonly observed density range of in situ soils.
FIG. 1.
Grain Size Distribution of Ottawa Sand and Sil-Co-Sil
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 / 453
2. The samples are easy to saturate.
3. The samples have a homogeneous fabric and fairly uniform void ratio throughout.
4. There is no particle segregation, regardless of gradation
or fines content.
5. The method simulates the natural soil deposition mode
and is easy to duplicate.
With respect to Item 5, it is important to point out, as does
Vaid (1994), that a triaxial sample is intended to represent an
element within a soil profile and, as such, must be homogeneous. Particle segregation is possible during deposition of soil
through water in natural settings or in the construction of hydraulic fills, which may cause the fines contents to vary with
depth within the soil deposits resulting from such processes.
However, the fines content at any given point within such deposits is unique, and it is the element or ‘‘point’’ within the
soil profile (not the entire deposit) that the laboratory sample
is intended to represent. Thus the slurry deposition method
simulates the process of soil formation while preserving one
of the requirements of a laboratory sample: that it be reasonably uniform.
Samples were prepared by first estimating the weights of
sand and silt needed for a desired fines content. These amounts
of silt and sand were then mixed in a cylindrical plexiglass
tube completely filled with deaired water. A vacuum is applied
for at least 6 h to the mix of sand, silt, and water through a
valve contained in the rubber cap used to seal the tube to
minimize entrapped air bubbles. The silt and sand are thoroughly mixed by vigorous shaking of the plexiglass tube for
approximately 20 min to achieve sample uniformity. Afterward, the rubber cap is removed, a very small amount of deaired water is added to raise the water level back to the top
of the tube, and the tube is topped with a 0.12 ⫻ 0.50 m2
piece of 0.43-mil high-density polyethylene film. The tube
containing the slurry is quickly inverted and positioned inside
the triaxial sample split mold, where a stretched, thin membrane, completely filled with deaired water, is already in place.
The contents of the tube are released into the membrane by
raising the tube. Densification of the sample is accomplished
by carefully and symmetrically tapping the sides of the sample
mold immediately after slurry deposition. Because the mass of
sand and silt used in sample preparation can be accurately
estimated, it is possible to obtain a relative density that is
reasonably close to a target value by measuring the height of
the sample as it densifies. Samples had heights of the order of
165 mm and diameters of the order of 70 mm. Backpressures
up to 500 kPa were applied to the samples to ensure that B
values in excess of 0.96 were obtained for all samples (most
B values were in excess of 0.98). More details on the sample
preparation and testing procedures can be found in Bandini
(1999).
The testing apparatus used to perform the tests is a CKC
automatic triaxial testing system (Soil Engineering Equipment
Co., San Francisco) (Chan 1981). Consolidation of the sample
is accomplished by applying the desired effective consolidation stress to the sample in the course of a time ranging from
30 min (for dense silty sand and low confining stress) to 180
min (for loose silty sand and moderately high confining stress).
Consolidation to moderately high confining stresses was sometimes done in stages to allow bender element testing between
stages. The volume change of the sample was measured using
a sensitive differential pressure transducer. The testing apparatus uses a pneumatic pressure loading system, and the axial
loading is applied through a double-acting oil piston. The test
is computer-controlled, and the stress-strain data are recorded
automatically. All triaxial tests for this study were performed
at axial strain rates that were slow enough to allow full dis-
sipation of pore-water pressures during loading, and the tests
were discontinued at 20–33% axial strain.
The bender element tests were performed after consolidation
to a given confining stress was completed. Two piezoceramic
plates or bender elements are used in the test: one embedded
in the base pedestal and the other at the top platen of the
triaxial apparatus. These elements have the property that they
bend when subjected to a voltage, and, in turn, produce a
voltage when bent. A rectangular voltage pulse is applied to
the transmitter element, causing it to produce a shear wave.
This rectangular pulse is typically on the order of 10 V in
order to generate a shear disturbance that is sufficiently large
to reach the other end of the sample and distort the receiver
element, producing another voltage pulse on the order of
about 1–2 mV. The signal is recorded and analyzed using an
HP3566A/67A digital analyzer (Hewlett-Packard Co., Palo
Alto, Calif.). The velocity of the shear wave transmitted
through the soil is calculated as the ratio of the effective length
of the test sample to the shear-wave travel time. The effective
length of the sample is taken as the length between the tips of
the bender elements, and this value is used along with travel
time to calculate the velocity of the shear wave. Following
Viggiani and Atkinson (1995a), the arrival of the shear wave
corresponds to the first significant inversion of the received
signal, determining the travel time of the shear wave. Fig. 2
illustrates a bender element test on a sample of sand with 5%
silt, void ratio of 0.577, and effective confining stress of 80
kPa. The arrival signal was magnified 5,000 times so that it
could be plotted on the same graph as the originating pulse.
Points A and B illustrate the starting and ending points for the
calculation of the travel time.
The small-strain shear modulus G0 is computed for a bender
element test using (19). It is important to stress that the accuracy of G0 values obtained using bender element tests is not
perfect; the errors in G0 values may, in extreme cases, be on
the order of 15%, as discussed by Viggiani and Atkinson
(1995a,b). Arulnathan et al. (1998) reached similar conclusions, although they focused on originating pulses of a sinusoidal shape. According to Arulnathan et al. (1998), errors in
G0 values exist mainly due to (1) deviations from 1D wave
propagation, which is assumed in the calculations; (2) wave
interference at the caps; (3) the different time delays between
the generation of the electrical signal and its transformation
into a mechanical impulse at the source bender element and
the reverse process at the receiving bender element; and (4)
near field effects. Near field effects may be significant only
when shear-wave arrival is identified with first motion at the
receiving bender element, which is not how wave arrival is
defined in this paper; however, the other factors are reflected
FIG. 2. Bender Element Test on Dense (e = 0.577) Ottawa Sand
Sample with 5% Silt under Effective Confining Stress of 80 kPa
454 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000
in the results. As pointed out by Arulnathan et al. (1998), such
factors sometimes balance each other out, but sometimes do
not. Thus, in the context of the current state of knowledge of
bender element testing, it may be stated that the actual G0
values may differ from those measured in the present testing
program by as much as 15%, although the actual difference is
probably smaller due to self-compensating effects. It is important to stress, however, that one of the main goals of the
testing program is to assess the extent to which the presence
of nonplastic fines changes the stiffness of silica sand. For
such comparisons, where the focus is on the ratios of stiffness
of silty and clean sands measured in the same way, errors are
expected to be quite small.
content is well explained by Lade and Yamamuro (1997). As
fines are added to either a dense or loose sand matrix, most
particles initially occupy the voids between sand particles.
This represents a reduction in void ratio with increasing the
amount of fines. Some particles, however, end up between the
surface of adjacent sand particles. Such particles would tend
to cause an increase in void ratio, as they do not occupy the
ANALYSIS OF RESULTS
Minimum and Maximum Void Ratios
The concept of relative density is used in this paper despite
having been subjected to some criticism. The criticism has
focused on difficulties in obtaining emax and emin, particularly
for sands with more than 15% fines content (Burmister 1948;
Tavenas and La Rochelle 1972; Selig and Ladd 1973). However, careful execution of a specific procedure to determine
emax and emin does lead to reasonably reproducible numbers
(and a relative density reproducible to ⫾5%). Additionally,
important advantages are offered by the use of relative density,
notably that relative density allows unification of the description of the density or degree of compaction of granular soils
with fines content ranging from 0 to 20% with respect to the
densest and loosest possible states for these soils.
In their study of the undrained properties of Brenda tailings
sand, Kuerbis et al. (1988) found that the maximum and minimum void ratios of silty sand decreased as silt content increased from 0 to 20%. Similar results were observed by Lade
and Yamamuro (1997) for Nevada and Ottawa sands mixed
with nonplastic fines and, in the present study, for Ottawa sand
mixed with Sil-Co-Sil. Minimum and maximum void ratios
were determined in this study according to ASTM D 4253 and
ASTM D 4254. Minimum density was obtained by pouring
sand into a standard compaction mold with a volume of 2,830
cm3 using a thin-wall cylindrical tube. Maximum density was
achieved by densifying dry sand in a compaction mold of
2,830 cm3 using an electromagnetic, vertically vibrating table
with a frequency of 60 Hz. A double amplitude of vertical
vibration of 0.379 mm was found to be optimum for all gradations. Even though the ASTM recommended procedure is
applicable for fines contents up to 15%, no difficulties were
found when using it for 20% silt content. Table 2 gives maximum and minimum void ratios of clean and silty Ottawa
sands as a function of silt content; it is clear that emax and emin
of silty sands decrease as the fines content increases from 0
to 20%. The rate of decrease drops as the fines content approaches 20%, and Kuerbis et al. (1988) and Lade and Yamamuro (1997) observed in their studies that emax and emin increase after the fines content exceeds about 25%.
This pattern of decreasing emax and emin with increasing fines
FIG. 3.
Limit Void Ratio for 5, 10, 15, and 20% Silt Content
TABLE 2. Minimum and Maximum Void Ratios for Clean and
Silty Ottawa Sands
Silt
(%)
(1)
emin
(2)
emax
(3)
0
5
10
15
20
0.48
0.42
0.36
0.32
0.29
0.78
0.70
0.65
0.63
0.62
FIG. 4. Determination of Critical-State Friction Angle ␾c from
Drained Triaxial Compression Test on Loose Clean Sand Sample (DR = 27.1%) under Effective Confining Stress of 400 kPa
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 / 455
natural void space left by the sand matrix, and to push sand
particles apart. Due to the methods of preparation of loose and
dense samples, more particles are found between the surfaces
of adjacent sand particles in loose than in dense sands, hence
the larger drops in emin than in emax for a given increase in fines
content observed in Table 2.
For a given overall void ratio, there is a fines content for
which the fines completely (or almost completely) separate
adjacent sand particles. An easy way to determine the fines
content for which this happens is based on the concept of the
skeleton void ratio esk (Kuerbis et al. 1988), which is the void
ratio of the silty sand calculated as if the fines were voids
1⫹e
esk =
⫺1
1⫺f
(20)
For each gradation, a limit void ratio (and a corresponding
limit relative density) can be defined. For Ottawa sand, with
emax = 0.78, these relative densities are 3% for 5% fines, 17%
for 10% fines, 38% for 15% fines, and 59% for 20% fines.
For relative densities lower than the limit relative density, the
fines control and the behavior becomes that of a sandy silt or
sandy clay, depending on the nature of the fines. For soils
denser than the limit relative density, the behavior is that of
sand, modified by the presence of fines.
Peak and Critical-State Shear Strength
The peak and critical-state friction angles are obtained according to (1) at the points of peak strength and critical state,
respectively. Critical state is identified as constant shear stress
where e = overall void ratio of soil; and f = ratio of weight of
fines to total weight of solids. Whenever esk is greater than the
maximum void ratio (emax)f=0 of clean sand, the sand matrix
exists with a void ratio higher than it could achieve in the
absence of fines, which means that the sand particles are, on
average, not in contact, and mechanical behavior is no longer
controlled by the sand matrix. Fig. 3 shows the skeleton void
ratio as a function of void ratio for 5, 10, 15, and 20% fines.
FIG. 5. Drained Triaxial Compression Tests on Loose Samples
of Ottawa Sand with Various Silt Contents under Moderately
High Effective Confining Stress
FIG. 6. Drained Triaxial Compression Tests on Dense Samples of Ottawa Sand with Various Silt Contents under Low Effective Confining Stress
456 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000
TABLE 3.
Test
(1)
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A12
A13
A14
A15
A16
A17
B1
B2
B3
B4
B5
B6
B7
B8
B9
B10
B11
B12
B13
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11
D12
D13
D14
D15
D16
D17
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
E11
Static Triaxial Test Results
Fines
content
(%)
(2)
e
(3)
DR
(%)
(4)
␴ ⬘3
(kPa)
(5)
␴p
(6)
p ⬘p
(kPa)
(7)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
10
10
10
10
10
10
10
10
10
10
10
10
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
20
20
20
20
20
20
20
20
20
20
20
0.633
0.590
0.643
0.674
0.635
0.632
0.678
0.662
0.674
0.659
0.610
0.586
0.537
0.558
0.645
0.665
0.699
0.660
0.581
0.661
0.495
0.630
0.587
0.657
0.634
0.609
0.475
0.502
0.612
0.632
0.583
0.564
0.569
0.581
0.571
0.447
0.567
0.500
0.447
0.420
0.563
0.560
0.500
0.512
0.363
0.410
0.390
0.366
0.412
0.375
0.392
0.320
0.607
0.587
0.588
0.551
0.533
0.530
0.522
0.423
0.384
0.402
0.470
0.494
0.535
0.448
0.531
0.484
0.476
0.487
49.1
63.3
45.8
35.3
48.4
49.3
33.9
39.3
35.2
40.2
56.7
64.6
80.9
74.1
44.9
38.3
27.1
14.4
42.3
14.0
73.4
24.9
40.4
15.3
23.7
32.5
80.3
70.8
31.4
24.3
23.1
29.6
28.0
23.9
27.2
69.9
28.8
51.7
70.0
79.3
30.2
31.0
41.9
37.9
86.1
70.9
77.5
85.1
70.4
82.4
76.8
100.0
7.4
13.7
13.5
25.6
31.2
32.1
34.8
59.8
71.5
66.0
45.4
38.3
25.9
52.2
27.0
41.2
43.5
34.4
200
400
100
100
200
200
100
200
300
200
100
100
100
100
400
400
400
150
200
250
200
200
250
200
300
200
100
100
300
400
250
100
250
350
300
150
200
200
100
100
400
400
100
200
100
100
100
100
100
100
100
100
100
200
200
100
100
400
400
500
350
450
200
100
300
450
500
400
400
400
32.4
34.7
31.0
30.1
32.0
31.1
31.4
30.9
31.2
32.4
33.3
34.0
36.5
35.9
31.2
31.5
30.2
33.8
36.8
33.2
38.7
34.5
36.8
33.2
33.6
35.6
40.4
40.8
33.7
32.5
35.9
37.0
37.0
35.8
37.0
39.0
35.6
37.3
40.5
41.3
33.7
34.1
35.8
34.9
44.7
39.5
42.4
43.1
41.9
42.1
44.4
45.5
32.4
33.9
33.2
35.0
33.0
34.9
33.8
37.5
38.4
38.8
34.5
35.2
35.0
37.4
34.7
34.5
34.7
34.3
359
756
173
167
352
346
177
342
523
357
181
185
198
195
687
692
669
275
402
456
426
379
501
366
549
390
133
225
554
709
489
201
504
685
607
317
393
408
224
230
733
741
199
382
258
217
238
244
235
235
256
265
179
376
364
191
182
754
738
408
524
428
380
198
572
305
280
747
754
744
and volume with increasing shear strain. The critical-state friction angle was obtained from those tests that, for practical
purposes, did reach critical state. The volumetric strain versus
axial strain plot was the main check of whether critical state
was approached. The critical-state friction angle was determined at the axial strain at which the volumetric strain versus
axial strain plot becomes horizontal (i.e., the dilatancy angle
becomes zero). As a secondary check, the value of friction
angle corresponding to the first peak in the volumetric strain
versus axial strain plot, which also corresponds to a horizontal
tangent and thus to zero dilatancy angle, was determined as
well. These two estimates of the critical-state friction angle ␾c
were practically the same in the tests where the volume change
curve became horizontal at large axial strains. Fig. 4 illustrates
one instance of critical-state friction angle determination.
Figs. 5 and 6 show results of tests on samples with 0, 5,
10, 15, and 20% silt at ␴ 3⬘ = 400 kPa and DR approximately
equal to 30% and ␴ 3⬘ = 100 kPa and DR approximately equal
to 80%, respectively. Of the tests on the loose samples in Fig.
5, the sample with 20% silt content is clearly below the limit
relative density and thus has a floating fabric. With 20% silt,
in most situations of relevance in geotechnical practice the soil
will have a floating fabric. It is clear from Figs. 5 and 6 that
the critical-state friction angle increases with fines content. The
values of ␾c are 29⬚ for clean Ottawa sand, 30.5⬚ for 5%
fines content, 32⬚ for 10%, 32.5⬚ for 15%, and 33⬚ for
20%. An increase of dilatancy with fines content is also observed. Because of the increase in both ␾c and dilatancy, the
peak friction angle ␾p increases with increasing fines content.
Table 3 contains the essential information for all triaxial tests
performed as part of the current testing program. The Q and
R values of (14) are obtained for each silt-sand gradation using
the ␾p from each test performed on samples with that gradation
and the ␾c corresponding to the gradation. Substituting (14)
into (12) and rearranging, we obtain the following linear equation:
␾p ⫺ ␾c
⫹ ID ln
3
冉 冊
100p ⬘p
PA
= IDQ ⫺ R
(21)
Bolton (1986) found that R = 1 and Q = 10 provided a
good fit for several different clean silica sands. Table 4
shows the results of linear regression following (21) on the
data for Ottawa sand with 0, 5, 10, 15, and 20% silt contents.
Only data corresponding to relative densities higher than the
limit relative density, below which the sand particles are
completely or nearly completely floated by the fines, were
considered for 5 and 10% silt contents. The limit relative densities were found earlier to be 3, 17, 38, and 59% for fines
contents equal to 5, 10, 15, and 20%, respectively. The best
fit for clean Ottawa sand gives Q = 9.0 and R = 0.49, with an
excellent coefficient of correlation (r = 0.96).
Referring to (21), dilatancy increases with increasing Q and
TABLE 4. Values of Dilatancy Parameters Q and R for Clean
and Silty Ottawa Sands
Silt
(%)
(1)
0
5
10
15
15
20
20
(DR
(DR
(DR
(DR
> 38%)
< 38%)
> 59%)
< 59%)
Trendline with
R = 0.5
Best Fit
Q
(2)
R
(3)
r2
(4)
Q
(5)
r2
(6)
Number
of tests
(7)
9.0
9.0
8.3
11.4
7.9
10.1
7.3
0.49
⫺0.50
⫺0.69
1.29
0.04
0.85
0.08
0.93
0.98
0.97
0.97
0.86
0.95
0.82
9.0
11.0
10.6
10.3
9.6
9.5
8.7
0.93
0.92
0.87
0.96
0.82
0.95
0.79
17
13
12
10
7
3
8
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 / 457
FIG. 7. Visual Illustration of Best-Fit Q and R Values for Triaxial Tests on: (a) Clean Ottawa Sand; (b) Ottawa Sand ⴙ 5% Nonplastic
Silt; (c) Ottawa Sand ⴙ 10% Nonplastic Silt; (d) Ottawa Sand ⴙ 15% Nonplastic Silt and Nonfloating Fabric; (e) Ottawa Sand ⴙ 15%
Nonplastic Silt and Floating Fabric; (f) Ottawa Sand ⴙ 20% Nonplastic Silt and Nonfloating Fabric; (g) Ottawa Sand ⴙ 20% Nonplastic
Silt and Floating Fabric
458 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000
decreases with increasing R. Bolton (1986) discussed the case
in which the calculated peak friction angle results less than
the critical-state friction angle. This would be seen when the
strains necessary to reach critical-state shear strength are so
large that ␾p is selected at an earlier, lower value of shear
strength. A positive value of R would suggest this type of
scenario for very low relative densities. A negative value of
R, on the other hand, would imply that the ␾p of very loose
sand would still be higher than ␾c . An implication of the role
of Q and R in (21) is that, for a set of triaxial tests performed
on a given material, the value of R would be affected by the
selected value of ␾c , but Q would remain unchanged. Ultimately, however, both Q and R are fitting parameters, and
interpretations of their physical meaning should be used within
bounds. The values of Q and R in Table 4 are strictly appli-
FIG. 8. Results of Bender Element Tests for Samples of Clean
Sand at Various Initial Void Ratios: (a) and (b) Low Effective
Confining Stresses; (c) Higher Effective Confining Stresses
cable only within the range of realtive densities for which test
data are available, and use of (21) outside such ranges should
be made with caution.
A direct comparison of the dilatancy of the sand-silt
mixtures is possible through examination of the values of both
Q and R for each silt content. It is easier, although not entirely
correct from a fundamental point of view, to compare the dilatancies of sand with 0, 5, 10, 15, and 20% silt by comparing
the values of Q obtained if a single R value is used, for which
the coefficient of correlation is satisfactory for all gradations.
A value of R = 0.5 works relatively well for all gradations (r
= 0.96, 0.96, 0.93, 0.98, and 0.97), producing Q values equal
to 9, 11, 10.6, 10.3, and 9.5 for 0, 5, 10, 15, and 20% silt
contents, respectively, and relative densities higher than the
limit relative density. Fig. 7 illustrates graphically the best fit
for all gradations. It is observed that Q for samples with nonfloating fabric increases with the addition of 5% silt and then
drops as the silt content is increased further, but never returns
to the value for clean sand. These results indicate that the peak
friction angle of sands increases with fines content not only
because the critical-state friction angle ␾c increases, but also
because dilatancy increases. In contrast with ␾c, which increases throughout the range from 0 to 20% silt content, dilatancy increases initially, as the fines content is raised to 5%,
and then drops with further addition of fines, remaining however higher than that of clean sand. The test results suggest
that, for low silt contents (about 20% or less) and a fabric
mostly or completely associated with sand-to-sand particle
contact, the silt particles occupy spaces adjacent to neighboring sand particles, increasing particle interlocking and causing
the soil to become more dilative.
FIG. 9. Results of Bender Element Tests for Samples of Sand
with 5% Nonplastic Silt at Various Initial Void Ratios: (a) Low Effective Confining Stresses; (b) Higher Effective Confining
Stresses
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 / 459
While not apparent in dense samples under low confining
stresses, the stiffness of loose samples at 400-kPa confinement
at moderately small strains decreases noticeably as the silt content is increased from 0 to 15% silt but then appears to stabilize as the silt content is increased further (Fig. 5). The degradation and then recovery of shear stiffness can be understood
by referring again to the concept of the limit relative density.
All of the samples have approximately the same relative density (DR = 25–40%), but different fines content. If the approximate relative density of the samples is compared with the
limit relative density for each fines content, it is observed that
it nearly matches the limit relative density for 15% fines, but
then falls definitely below the limit relative density for 20%
fines. From 0 to 15% fines content, the fabric gets progressively weaker, as the fines separate the sand particles more
and more. It is noted that the lower stiffness resulting from
the addition of fines to the sand is a phenomenon observed at
relatively small strain; as shearing takes place, the sample contracts and the sand particles come in closer contact, with the
FIG. 11. Results of Bender Element Tests for Samples of
Sand with: (a) 15% Nonplastic Silt; (b) 20% Nonplastic Silt
at Low Effective Confining Stresses; (c) 20% Nonplastic Silt
at Higher Effective Confining Stresses, at Various Initial Void
Ratios
FIG. 10. Results of Bender Element Tests for Samples of Sand
with 10% Nonplastic Silt at Various Initial Void Ratios: (a) Low
Effective Confining Stresses; (b) Higher Effective Confining
Stresses
TABLE 5.
Silt
(%)
(1)
0
5
10
15
20 (DR > 59%)
20 (DR < 59%)
Regression Parameters Cg , eg , ag , and ng for Calculation of G0 Using (15) and (17)
Using Eq. (15)
Using Eq. (17)
2
Cg
(2)
eg
(3)
ng
(4)
r
(5)
Cg
(6)
ag
(7)
ng
(8)
r2
(9)
612
454
357
238
270
207
2.17
2.17
2.17
2.17
2.17
2.17
0.439
0.459
0.592
0.745
0.686
0.809
0.96
0.94
0.91
0.85
1.00
0.98
547
410
135
101
—
—
⫺1.051
⫺1.044
⫺2.376
⫺2.069
—
—
0.443
0.458
0.557
0.715
—
—
0.97
0.95
0.96
0.94
—
—
460 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000
fines actually helping to enhance dilatancy and shear strength.
At 15% silt content, the fines start controlling, and at 20% silt
content, they fully control soil response, with the soil fabric
becoming more stable and the stiffness stabilizing with increasing fines content.
Small-Strain Shear Modulus G0
Figs. 8–11 contain the essential information from the
bender element tests performed for this study. Eqs. (15) and
(17) were used to fit the data from the bender element tests
on Ottawa sand with 0, 5, 10, 15, and 20% silt. Research by
Iwasaki and Tatsuoka (1977) indicated that eg = 2.17 can be
used in (15) with satisfactory results for sand particles ranging
from round to angular in shape. Thus eg was assumed equal
to 2.17, and the values of Cg and ng in (15), found through
regression analysis, are listed in Table 5. It was found that (15)
works very well for both clean and silty sand (with coefficients
of correlation r = 0.98, 0.97, 0.95, 0.92, and 0.99 for 0, 5, 10,
15, and 20% silt content, respectively).
The values of Cg, ag, and ng in (17) are found by finding
the best fit through the data points. The correlation in (17)
worked better than (15) for all stress levels and silt contents
up to 15%. For the available data, a reasonable correlation was
not found in (17) for 20% silt content. The correlation parameters are listed in Table 5 for different fines contents. Based
on the values of Cg, ag, and ng from Table 5, it is clear that
the shear modulus of sand decreases dramatically with fines
content. For instance, at a confining pressure of 100 kPa and
DR = 50%, the value of G0 is 89 MPa for clean sands, but it
drops to 75, 66, 46, and 42 MPa for sands with 5, 10, 15, and
20% fines, respectively. The stiffness reduction with fines content may be partially explained by the way in which the fines
interact with the sand matrix. If the fines are positioned within
the sand matrix in such a way that they do not have welldeveloped contacts with the sand particles, shear waves (or
static stresses) are not effectively transferred through the fine
particles. Thus lower void ratios due to the addition of fines
do not lead to increases in G0; accordingly, a silty sand at the
same void ratio as a clean sand has a lower G0. Even when
silt particles have better developed contacts with the sand particles, the silt particles may more easily move sideways under
shear stress application or shear-wave propagation, leading to
lower shear stiffness. These two effects, related to the fabric
of silty sands, lead to the lower Cg values measured in bender
element testing.
These observations that lower stiffness results from the addition of fines indicate that analyses of granular soil masses
where clean sand G0 values are used can be in significant error
if the soil has even a small amount of fines.
SUMMARY AND CONCLUSIONS
At small shear strains (typically <10⫺4 –10⫺3%) the shear
stress versus shear strain relationship of sand is linear, but for
larger shear strains it becomes strongly nonlinear. If the sand
is dilative, a peak shear strength is reached at axial strains of
the order of 2–3%. At large strains (of the order of 25–40%),
the sand reaches its critical state. In the analysis of soil problems, it becomes important to describe the loading response
of sands. In this paper, we studied the effects of various levels
of silt content on the stress-strain properties of sand at small
and large shear strains.
The small-strain shear modulus G0 describes soil response
in the initial, elastic stress-strain range and is a function of the
stress state and degree of compactness of the sand. The smallstrain shear modulus G0 increases with a power ng (usually in
the range of 0.4–0.8, depending on the silt content) of the
mean effective stress. The stiffness is further assumed to vary
with a prescribed function of the void ratio, either (1) a power
ag of the void ratio; or (2) according to (eg ⫺ e)2/(1 ⫹ e).
Another constant, Cg, also appears in the correlations. These
two sets of constants—(Cg, eg, ng) and (Cg, ag, ng)—were determined for clean Ottawa sand and Ottawa sand with 5, 10,
15, and 20% silt contents. It was observed that the small-strain
stiffness at a given relative density and confining stress level
decreases dramatically with the addition of even small percentages of silt. This is an important result, as analyses of
problems involving silty sand using stiffness properties of
clean sand can be in serious error.
Results of triaxial tests were analyzed to assess both the
peak and the critical-state friction angles of clean and silty
Ottawa sands. It was observed that the addition of even small
percentages of silt to clean sand considerably increases both
the peak friction angle at a given initial relative density and
the critical-state friction angle. This study suggests that silty
sands with nonfloating fabric in the 5–20% silt content range
are more dilatant than clean sands; dilatancy appears to peak
at around 5% silt content, but even at 20% silt content it remains above that of clean sand.
It is interesting to note that, although small-strain stiffness
drops, peak and critical-state strengths increase with increasing
fines content. This may be interpreted as follows: initially the
fine particles are not positioned in a way to provide optimum
interlocking and small shear strains are imposed on the soil
with greater ease than if the fines were not present. As shearing
progresses, the fines reach more stable arrangements and ultimately increase interlocking, dilatancy, and shear strength.
The soil response observed in this study is strictly applicable
only to the silt and sand gradations used in the testing. Further
study is needed to assess the effects of different gradations on
the behavior of silty sand.
ACKNOWLEDGMENTS
This material is based upon work supported by the National Science
Foundation, Washington, D.C., Earthquake Hazards Mitigation Program,
under grant No. CMS-9410361. However, any opinions, findings, and
conclusions or recommendations expressed in this material are those of
the writers and do not necessarily reflect the views of the National Science Foundation. The support of Dr. Michael F. Riemer and his useful
insights into bender element testing are greatly appreciated. The assistance of Bryan Scott and Yongdong Zeng with some of the tests is also
appreciated. Dr. Vincent P. Drnevich assisted with the data acquisition
software for the bender element tests.
APPENDIX I.
REFERENCES
Arulnathan, R., Boulanger, R. W., and Riemer, M. F. (1998). ‘‘Analysis
of bender element tests.’’ Geotech. Testing J., 21(2), 120–131.
Bandini, P. (1999). ‘‘Static response and liquefaction of silty sands,’’ Master thesis, Purdue University, West Lafayette, Ind.
Been, K., Jefferies, M. G., and Hachey, J. (1991). ‘‘The critical state of
sands.’’ Géotechnique, London, 41(3), 365–381.
Bolton, M. D. (1986). ‘‘The strength and dilatancy of sands.’’ Géotechnique, London, 36(1), 65–78.
Burmister, D. (1948). The importance and practical use of relative density
in soil mechanics, Spec. Publ. No. 48, ASTM, West Conshohocken,
Pa., 1–20.
Chan, C. K. (1981). ‘‘An electropneumatic cyclic loading system.’’ Geotech. Testing J., 4(4), 183–187.
Chung, R. M., Yo Kel, F. Y., and Drenevich, V. P. (1984). ‘‘Evaluation
of dynamic properties of sands by resonant column testing.’’ Geotech.
Testing J., 7(2), 60–69.
De Josselin de Jong, G. (1976). ‘‘Rowe’s stress-dilatancy relation based
on friction.’’ Géotechnique, London, 26(3), 527–534.
Dyvik, R., and Madshus, C. (1985). ‘‘Laboratory measurement of Gmax
using bender elements.’’ Advances in the art of testing soils under
cyclic conditions, ASCE, New York, 186–196.
Hardin, B. O. (1978). ‘‘The nature of stress-strain behavior of soils.’’
Proc., ASCE Geotech. Engrg. Div.., Spec. Conf., Vol. 1, ASCE, New
York, 3–90.
Hardin, B. O., and Richart, F. E., Jr. (1963). ‘‘Elastic wave velocities in
granular soils.’’ J. Soil Mech. Found. Div., ASCE, 89(1), 33–65.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 / 461
Iwasaki, T., and Tatsuoka, F. (1977). ‘‘Effect of grain size and grading
on dynamic shear moduli of sands.’’ Soils and Found., Tokyo, 17(3),
19–35.
Jamiolkowski, M., Leroueil, S., and Lo Presti, D. C. F. (1991). ‘‘Theme
lecture: Design parameters from theory to practice.’’ Proc., Geo-Coast
’91, 1–41.
Kuerbis, R., Negussey, D., and Vaid, Y. P. (1988). ‘‘Effect of gradation
and fines content on the undrained response of sand.’’ Hydraulic fill
structures, Geotech. Spec. Publ. No. 21, ASCE, New York, 330–345.
Kuerbis, R., and Vaid, Y. P. (1988). ‘‘Sand sample preparation—The
slurry deposition method.’’ Soil and Found., Tokyo, 28(4), 107–118.
Lade, P., and Yamamuro, J. (1997). ‘‘Effects of non-plastic fines on static
liquefaction of sands.’’ Can. Geotech. J., Ottawa, 34(6), 918–928.
Lo Presti, D. C. F. (1987). ‘‘Mechanical behavior of Ticino sand from
resonant column tests,’’ PhD thesis, Politecnico di Torino, Turin, Italy.
Lo Presti, D. C. F., Pedroni, S., and Crippa, V. (1992). ‘‘Maximum dry
density of cohesionless soil by pluviation and by ASTM D 4253-83: a
comparative study.’’ Geotech. Testing J., 15(2), 180–189.
Negussey, D., Wijewickreme, W. K. D., and Vaid, Y. P. (1986). Constant
volume friction angle of granular materials, Soil Mech. Ser. No. 94,
Dept. of Civ. Engrg., University of British Columbia, Vancouver.
Randolph, M. F., Dolwin, J., and Beck, R. (1994). ‘‘Design of driven
piles in sand.’’ Géotechnique, London, 44(3), 427–448.
Roesler, S. K. (1979). ‘‘Anisotropic shear modulus due to stress anisotropy.’’ J. Geotech. Engrg. Div., ASCE, 105(7), 871–880.
Rowe, P. W. (1962). ‘‘The stress-dilatancy relation for static equilibrium
of an assembly of particles in contact.’’ Proc., Royal Soc., London,
A269, 500–527.
Rowe, P. W. (1971). ‘‘Theoretical meaning and observed values of de-
formation parameters for soil.’’ Proc., Roscoe Memorial Symp., StressStrain Behavior of Soils, R. H. G. Parry, ed., Foulis, Henley on Thames,
U.K., 143–194.
Salgado, R., Boulanger, R. W., and Mitchell, J. K. (1997a). ‘‘Lateral stress
effects on CPT liquefaction resistance correlations.’’ J. Geotech. and
Geoenvir. Engrg., ASCE, 123(8), 726–735.
Salgado, R., Mitchell, J. K., and Jamiolkowski, M. (1997b). ‘‘Cavity expansion and penetration resistance in sand.’’ J. Geotech. and Geoenvir.
Engrg., ASCE, 123(4), 344–354.
Schanz, T., and Vermeer, P. A. (1996). ‘‘Angles of friction and dilatancy
of sand.’’ Géotechnique, London, 46(1), 145–151.
Selig, E. T., and Ladd, R. S. (1973). ‘‘Evaluation of relative density measurement and applications.’’ Evaluation of relative density and its role
in geotechnical projects involving cohesionless soils, ASTM STP 523,
ASTM, West Conshohocken, Pa., 487–504.
Shirley, D. J., and Hampton, L. D. (1977). ‘‘Shear-wave measurements in laboratory sediments.’’ J. Acoust. Soc. Am., 63(2), 607–
613.
Tavenas, F., and La Rochelle, P. (1972). ‘‘Accuracy of relative density
measurements.’’ Géotechnique, London, 22(4), 549–562.
Taylor, D. W. (1948). Fundamentals of soil mechanics. Wiley, New York.
Vaid, Y. P. (1994). ‘‘Liquefaction of silty soils.’’ Ground failures under
seismic conditions, Geotech. Spec. Publ. No. 44, Shamsher Prakash and
Panos Dakoulas, eds., ASCE, New York, 1–16.
Viggiani, G., and Atkinson, J. H. (1995a). ‘‘Interpretation of bender element tests.’’ Géotechnique, London, 45(1), 149–154.
Viggiani, G., and Atkinson, J. H. (1995b). ‘‘Stiffness of fine-grained soil
at very small strains.’’ Géotechnique, London, 42(2), 249–265.
Yu, P., and Richart, F. E., Jr. (1984). ‘‘Stress ratio effect on shear modulus
of dry sands.’’ J. Geotech. Engrg., ASCE, 110(3), 331–345.
462 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000
View publication stats