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 exten-
             sively. 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                                                                  mechanics problems involving these materials, information is
                                                                              needed on their mechanical properties. The same framework
   It is well established that soils behave as linear elastic ma-             used to describe the small-strain stiffness and the shear
terials at shear strains smaller than about 10 4 –10 3%. For                  strength of clean sands may be used for silty sands, provided
larger shear strains, the stress-strain relationship is nonlinear.            that the fines content remains below some limit, usually taken
Peak shear strength develops at relatively large strains (cor-                to be in the range of 15–20%. In this paper, we evaluate how
responding typically to axial strains in the range of 1–4%),                  the intrinsic variables that appear in correlations for the small-
and critical-state shear strength (corresponding to no volume                 strain stiffness and the shear strength of sands vary with the
change during shearing) develops for axial strains in excess of               content of nonplastic fines.
25%. As the Mohr-Coulomb failure criterion is commonly
used to describe shear failure in soils, friction angles deter-               SHEAR STRENGTH
mined at the peak and critical states can be defined.
   Knowledge of the values of small-strain stiffness and criti-               Critical-State Friction Angle
cal-state and peak friction angles is very useful for applications               The shear strength of a cohesionless soil can be defined by
based on constitutive models or analyses that attempt to cap-                 the Mohr-Coulomb failure criterion with zero cohesive inter-
ture material response from the initial stages of loading up to               cept
shear failure. More immediate application of such knowledge
can be made in analyses that rely predominantly on small-                                                s=       tan                        (1)
strain stiffness (such as the design of machine foundations),
                                                                              where s = shear strength; = normal stress on the plane of
or on friction angles only (such as stability analyses of various
                                                                              shearing; and = friction angle. For a triaxial test, it is prac-
forms, where deformations prior to collapse are not consid-
                                                                              tical to write in terms of the principal effective stresses
   The stress-strain response of sand at small-, intermediate-,                                                    1
and large-strain levels depends upon soil state variables (the                                                          1
relative density DR of the sand, the effective stress state, and                                       sin    =                              (2)
fabric) and other factors related to the nature of the sand (par-                                                       1
ticle shape, particle size distribution, particle surface charac-                                                  3

teristics, and mineralogy). The factors related to the constitu-              where 1 / 3 = effective principal stress ratio or stress obliq-
tion and general nature of the sand particles are referred to as              uity.
intrinsic variables (Been et al. 1991; Salgado et al. 1997a,b).                  In general, a loose sand contracts and a dense sand expands
Examples of intrinsic variables are the critical-state friction               as it approaches the critical state, usually defined as the state
angle c, the maximum and minimum void ratios emax and emin,                   at which the sand is sheared without changes in either shear
and the dilatancy parameters Q and R of the peak friction angle               strength or volume. However, whether a sample of sand is
correlation of Bolton (1986).                                                 contractive or dilatant depends not only on density but also
   The properties of clean sands have been extensively studied                on effective confining stress. According to the critical-state
under laboratory and field conditions. These include Ottawa,                   model, when a loose sample is sheared under high effective
Ticino, and Monterey #0 sands (Hardin and Richart 1963;                       confining stress, the shear stress increases monotonically until
Chung et al. 1984; Bolton 1986; Lo Presti 1987; Lo Presti et                  it reaches a plateau, after which the sample continues to un-
al. 1992). However, in situ soils often contain significant                    dergo shear straining without any change in shear stress or
amounts of fines. If realistic analyses are to be done of soil                 sample volume. The sample is then said to have reached the
                                                                              critical state, and the corresponding friction angle is known as
    Assoc. Prof., School of Civ. Engrg., Purdue Univ., West Lafayette, IN     the critical-state friction angle c.
47907. E-mail:
    PhD Candidate, School of Civ. Engrg., Purdue Univ., West Lafayette,
                                                                                 During the shearing of a dense sand, the sample contracts
IN.                                                                           initially and then dilates. The effective principal stress ratio
    Formerly, Postdoctoral Fellow, School of Civ. Engrg., Purdue Univ.,       reaches a peak, associated with a peak friction angle, at which
West Lafayette, IN.                                                           the dilation rate is maximum. Further loading causes the shear
   Note. Discussion open until October 1, 2000. To extend the closing         stress to drop until it reaches the critical state. For practical
date one month, a written request must be filed with the ASCE Manager          purposes, the critical-state friction angle obtained from triaxial
of Journals. The manuscript for this paper was submitted for review and
possible publication on December 10, 1997. This paper is part of the
                                                                              tests is commonly taken as a unique value for a given granular
Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126,           soil, regardless of the initial relative density and initial confin-
No. 5, May, 2000. ASCE, ISSN 1090-0241/00/0005-0451–0462/$8.00                ing stress. Such an approach is well justified by results found
   $.50 per page. Paper No. 17169.                                            in the literature (Rowe 1962, 1971; Bolton 1986; Negussey et
                                                    JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 / 451
                                          TABLE 1.     Intrinsic and State Variables of Some Clean Sandsa
           Sand type                        emin           emax             c              Cg                      eg                                 ng            Cu
              (1)                           (2)            (3)            (4)              (5)                     (6)                                (7)           (8)
Monterey #0 sanda,b                        0.57            0.86           37.0             326                    2.97                            0.50              1.6
Ticino sandd,e                             0.57            0.93           34.8             647                    2.27                            0.43              1.5
Toyoura sandd,e                            0.61            0.99           35.1             900                    2.17                            0.40              1.27
Ottawa sand (round)c                       0.48            0.78           29.0             612                    2.17                            0.44              1.48
Sacramento River sanda                     0.61            1.03           33.3             NA                     NA                              NA                1.47
Hokksund sandd,e                           0.55            0.87           36.0             942                    1.96                            0.46              1.91
   Bolton (1986).
    Chung et al. (1984).
   Present paper.
    Lo Presti (1987).
   Lo Presti et al. (1992).

al. 1986; Been et al. 1991; Schanz and Vermeer 1996), which                                             1     sin
support the concept that c is unique for silica sands and may                                    N=                             = tan2           45                         (6)
                                                                                                        1     sin                                           2
be taken as an intrinsic variable of the sand. The values of the
critical-state friction angles, as well as several other charac-                                        1     sin           c                                   c
                                                                                                 Nc =                               = tan2       45                         (7)
teristics, of different clean sands are given in Table 1.                                               1     sin           c                               2
                                                                                                        1         sin
Stress-Dilatancy Relation                                                                        M=                             = tan2           45                         (8)
                                                                                                        1         sin                                       2
   Stress-dilatancy relations aim to describe the relationship                    The dilatancy angle, in turn, is defined as
between the friction and dilatancy angles. The simplest rela-
tion can be obtained from a physical analogy—the sawtooth                                                                               dε1
model—where slippage takes place along a stepped plane sep-                                                                                       1
arating two blocks of the same material. In this model, slip-                                               sin         =                                                   (9)
page takes place when friction between the two blocks on each                                                                                     1
side of the plane is overcome and the two blocks move apart,                                                                            kdε3
so that climbing and relative motion between the two blocks                     where dε1 and dε3 = principal strain increments; k = 1 for plane
may take place. A simple relation results                                       strain; and k = 2 for triaxial test conditions.
                                                                                   Bolton (1986) reviewed a large number of triaxial and
                         = tan     = tan(    c     )                (3)         plane-strain test results and proposed a much simpler relation-
                                                                                ship between , c, and , which he found to be operationally
                                                                                equivalent to (5)
where = shear stress acting on the plane of shearing;          =
normal effective stress on the plane of shearing; and = di-                                                        =            c         0.8                              (10)
latancy angle.
   More sophisticated theories have been developed to explain                      The relationship between the peak friction angle p and the
the relationship between the friction and dilatancy angles. Tay-                critical-state friction angle c can be written for both triaxial
lor (1948) suggested an ‘‘energy correction’’ hypothesis to ac-                 and plane-strain tests by modifying (10) so that the dilatancy
count for the dilation, whereby friction is considered a source                 angles for both types of test are expressed in terms of the same
of energy dissipation. The resulting equation for simple shear                  quantity IR, referred to as the dilatancy index
is                                                                                                                     =                   5IR                             (11)
                                                                                                                   p                c

                       tan    = tan   c      tan                    (4)         for plane-strain conditions, and
   Rowe (1962) developed his stress-dilatancy theory based on                                                      p   =            c      3IR                             (12)
the analogy between irregular packings of soil particles and
regular assemblies of spheres or cylinders and on the hypoth-                   for triaxial conditions.
esis that a minimum energy ratio at failure is achieved. De                        The dilatancy index IR is given, for both triaxial and plane-
Josselin de Jong (1976) questioned the energy minimization                      strain tests, by
hypothesis made by Rowe, which should not apply to systems                                                                 10            dεv
that dissipate energy during loading. He did validate Rowe’s                                                IR =                                                           (13)
                                                                                                                           3             dε1
conclusions through an analysis that does not rely on energy                                                                                    max

minimization assumptions. The resulting stress-dilatancy the-                   and is related to the relative density and effective confining
ory, superior to all other attempts to relate shear strength to                 stress level through
dilation, can be best expressed in the form
                                                                                                                                         100p p
                                 N = MNc                            (5)                            IR = ID        Q                 ln                      R              (14)
where N = flow number = 1 / 3 = stress obliquity; Nc = crit-                     where ID = relative density expressed as a number between 0
ical-state flow number = ( 1 / 3 )c = stress obliquity at critical               and 1; p p = mean effective stress at peak strength; PA = ref-
state; M = dilatancy number = 1       dεv /dε1; dεv = volumetric                erence stress (=100 kPa = 0.1 MPa      1 tsf) in the same units
strain increment; and dε1 = major principal strain increment =                  as p p ; and Q and R = fitting parameters. Eqs. (11) and (12)
axial strain increment in triaxial compression tests. N, M, and                 are valid for 0   IR    4. For higher values of IR the value of
Nc are given in terms of , c, and          by the following ex-                 the peak friction angle is taken as the value calculated from
pressions:                                                                      (11) or (12) with IR = 4.
Small-Strain Shear Modulus                                                        where = total mass density of the soil, and Vs = shear-wave
   There are two general forms of empirical equations used to                        Because a triaxial test can be performed on the same sample
estimate the shear modulus of sands: one was proposed by                          where Vs was measured using bender elements, bender element
Hardin (1978) and Hardin and Richart (1963) and another by                        testing has been increasingly used for measuring G0 (Shirley
Roesler (1979). A series of resonant column tests were per-                       and Hampton 1977; Dyvik and Madshus 1985; Viggiani and
formed by Hardin and Richart (1963) to obtain the small-strain                    Atkinson 1995a,b). It is used in this paper to study the small-
shear modulus for round and angular Ottawa sands. They es-                        strain stiffness of clean Ottawa sand and Ottawa sand with 5,
tablished empirical relations for G0 at a shear strain of =                       10, 15, and 20% added silt.
10 4 or less as
                                                                                  EXPERIMENTAL PROGRAM AND PROCEDURES
                                     (eg       e)2
                   G0 = CgP 1
                                                                          (15)       As discussed in the preceding sections, the small-strain stiff-
                                       1       e
                                                                                  ness and the shear strength of sand may be expressed in terms
where Cg, eg, and ng = regression constants that depend solely                    of a number of intrinsic variables ( c, Q, R, Cg, eg, ng). The
on the soil (and are therefore intrinsic soil variables); m =                     intrinsic variables are a function of the nature of the sand and
mean effective stress; PA = reference stress in the same units                    thus change with fines content. For a given soil density, this
as m; and e = void ratio.                                                         is a valid approach as long as the fines content remains below
  Roesler (1979) developed his correlation from shear-wave                        a limit beyond which the fines may dominate over the sand
velocity measurements. The equation has the following form:                       matrix, changing the mechanical response of the soil in a more
                                                                                  fundamental way. This paper focuses on sand and silty sand
                      G0 = CP (1
                                     n m)      n
                                                     p                    (16)    where sand-to-sand particle contact prevails; the ranges of den-
where a = normal effective stress acting along the direction                      sity and fines content for which this assumption holds will be
of wave propagation; p = normal effective stress perpendic-                       discussed later.
ular to the wave-propagation direction; and m, n, and C =                            A series of triaxial and bender element tests was performed
fitting parameters.                                                                to assess how the shear strength and small-strain stiffness of
   A different form of the Hardin G0 equation, accounting for                     Ottawa sand change as an increasing percentage of nonplastic
the void ratio through a different function, has been proposed                    fines is added to it. Ottawa sand, designated as ASTM C 778,
by, among others, Jamiolkowski et al. (1991)                                      is a standard, clean quartz sand with the grain size distribution
                                                                                  shown in Fig. 1. The diameters of the sand particles range
                       G0 = CgP 1
                                       ng ag
                                           e    m
                                                                          (17)    from 0.1 to 0.6 mm. Ottawa sand is defined as SP according
                                                                                  to the Unified Soil Classification System. The coefficient of
where ag = regression constant (an intrinsic variable of the soil,                uniformity Cu is 1.48, and the mean grain size D50 is 0.39 mm.
if this correlation is used to model soil loading response).                      The maximum and minimum void ratios emax and emin are 0.78
   Most recent correlations [e.g., Iwasaki and Tatsuoka (1977)                    and 0.48, respectively. Its specific gravity GS is 2.65. Ottawa
and Yu and Richart (1984)] were developed based on the form                       sand particles are round to subround.
proposed by Hardin (1978) and Hardin and Richart (1963).                             The nonplastic fines are #106 Sil-Co-Sil ground silica from
Values of the curve-fitting parameters required to calculate G0                    U.S. Silica Co., Ottawa, Ill., which passes the #200 sieve and
for different sands from previous studies are listed in Table 1.                  is composed of SiO2 (99.8%), with Al2O3( 0.05%) and Fe2O3
Iwasaki and Tatsuoka (1977) studied G0 of clean sands, natural                    (0.035%) as secondary components. Its specific gravity is 2.65,
sands with fines, and artificially graded sand with fines. They                      with the grain size distribution shown, together with the grain
proposed a correlation for G0 with the form                                       size distribution of pure Ottawa sand, in Fig. 1.
                                                                                     Static, drained triaxial compression tests were conducted on
                          (2.17  e)2 1
             G0 = C( )B             PA               m( )
                                                         (     )m(
                                                                          (18)    isotropically consolidated sand samples with 0, 5, 10, 15, and
                             1 e                                                  20% nonplastic fines. To obtain homogeneous samples, the
where C( ) and m( ) = fitting parameters depending on the                          slurry deposition method of Kuerbis and Vaid (1988) was
strain level of the test; and B = fitting parameter that is inde-                  used. According to Kuerbis and Vaid (1988), the slurry-dep-
pendent of shear strain , void ratio e, and confining stress                       osition method has the following advantages:
  m. The results of Iwasaki and Tatsuoka (1977) indicate that
                                                                                    1. The method produces loose to dense samples in the com-
G0 decreases with increasing fines content. Results of resonant                         monly observed density range of in situ soils.
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 stiff-
ness 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 shear-
wave 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
                            G0 = V 2
                                   s                                      (19)     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.                              sipation of pore-water pressures during loading, and the tests
  3. The samples have a homogeneous fabric and fairly uni-          were discontinued at 20–33% axial strain.
     form void ratio throughout.                                       The bender element tests were performed after consolidation
  4. There is no particle segregation, regardless of gradation      to a given confining stress was completed. Two piezoceramic
     or fines content.                                               plates or bender elements are used in the test: one embedded
  5. The method simulates the natural soil deposition mode          in the base pedestal and the other at the top platen of the
     and is easy to duplicate.                                      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
   With respect to Item 5, it is important to point out, as does    the transmitter element, causing it to produce a shear wave.
Vaid (1994), that a triaxial sample is intended to represent an     This rectangular pulse is typically on the order of 10 V in
element within a soil profile and, as such, must be homoge-          order to generate a shear disturbance that is sufficiently large
neous. Particle segregation is possible during deposition of soil   to reach the other end of the sample and distort the receiver
through water in natural settings or in the construction of hy-     element, producing another voltage pulse on the order of
draulic fills, which may cause the fines contents to vary with        about 1–2 mV. The signal is recorded and analyzed using an
depth within the soil deposits resulting from such processes.       HP3566A/67A digital analyzer (Hewlett-Packard Co., Palo
However, the fines content at any given point within such de-        Alto, Calif.). The velocity of the shear wave transmitted
posits is unique, and it is the element or ‘‘point’’ within the     through the soil is calculated as the ratio of the effective length
soil profile (not the entire deposit) that the laboratory sample     of the test sample to the shear-wave travel time. The effective
is intended to represent. Thus the slurry deposition method         length of the sample is taken as the length between the tips of
simulates the process of soil formation while preserving one        the bender elements, and this value is used along with travel
of the requirements of a laboratory sample: that it be reason-      time to calculate the velocity of the shear wave. Following
ably uniform.                                                       Viggiani and Atkinson (1995a), the arrival of the shear wave
   Samples were prepared by first estimating the weights of          corresponds to the first significant inversion of the received
sand and silt needed for a desired fines content. These amounts      signal, determining the travel time of the shear wave. Fig. 2
of silt and sand were then mixed in a cylindrical plexiglass        illustrates a bender element test on a sample of sand with 5%
tube completely filled with deaired water. A vacuum is applied       silt, void ratio of 0.577, and effective confining stress of 80
for at least 6 h to the mix of sand, silt, and water through a      kPa. The arrival signal was magnified 5,000 times so that it
valve contained in the rubber cap used to seal the tube to          could be plotted on the same graph as the originating pulse.
minimize entrapped air bubbles. The silt and sand are thor-         Points A and B illustrate the starting and ending points for the
oughly mixed by vigorous shaking of the plexiglass tube for         calculation of the travel time.
approximately 20 min to achieve sample uniformity. After-              The small-strain shear modulus G0 is computed for a bender
ward, the rubber cap is removed, a very small amount of de-         element test using (19). It is important to stress that the ac-
aired water is added to raise the water level back to the top       curacy of G0 values obtained using bender element tests is not
of the tube, and the tube is topped with a 0.12         0.50 m2     perfect; the errors in G0 values may, in extreme cases, be on
piece of 0.43-mil high-density polyethylene film. The tube           the order of 15%, as discussed by Viggiani and Atkinson
containing the slurry is quickly inverted and positioned inside     (1995a,b). Arulnathan et al. (1998) reached similar conclu-
the triaxial sample split mold, where a stretched, thin mem-        sions, although they focused on originating pulses of a sinu-
brane, completely filled with deaired water, is already in place.    soidal shape. According to Arulnathan et al. (1998), errors in
The contents of the tube are released into the membrane by          G0 values exist mainly due to (1) deviations from 1D wave
raising the tube. Densification of the sample is accomplished        propagation, which is assumed in the calculations; (2) wave
by carefully and symmetrically tapping the sides of the sample      interference at the caps; (3) the different time delays between
mold immediately after slurry deposition. Because the mass of       the generation of the electrical signal and its transformation
sand and silt used in sample preparation can be accurately          into a mechanical impulse at the source bender element and
estimated, it is possible to obtain a relative density that is      the reverse process at the receiving bender element; and (4)
reasonably close to a target value by measuring the height of       near field effects. Near field effects may be significant only
the sample as it densifies. Samples had heights of the order of      when shear-wave arrival is identified with first motion at the
165 mm and diameters of the order of 70 mm. Backpressures           receiving bender element, which is not how wave arrival is
up to 500 kPa were applied to the samples to ensure that B          defined in this paper; however, the other factors are reflected
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
   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 consolida-
tion 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 some-
times 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 appa-
ratus 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     FIG. 2. Bender Element Test on Dense (e = 0.577) Ottawa Sand
at axial strain rates that were slow enough to allow full dis-      Sample with 5% Silt under Effective Confining Stress of 80 kPa

in the results. As pointed out by Arulnathan et al. (1998), such      content is well explained by Lade and Yamamuro (1997). As
factors sometimes balance each other out, but sometimes do            fines are added to either a dense or loose sand matrix, most
not. Thus, in the context of the current state of knowledge of        particles initially occupy the voids between sand particles.
bender element testing, it may be stated that the actual G0           This represents a reduction in void ratio with increasing the
values may differ from those measured in the present testing          amount of fines. Some particles, however, end up between the
program by as much as 15%, although the actual difference is          surface of adjacent sand particles. Such particles would tend
probably smaller due to self-compensating effects. It is im-          to cause an increase in void ratio, as they do not occupy the
portant 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.

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). How-
ever, 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 descrip-
tion 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       FIG. 3.   Limit Void Ratio for 5, 10, 15, and 20% Silt Content
sand, Kuerbis et al. (1988) found that the maximum and min-
imum void ratios of silty sand decreased as silt content in-
creased 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 gra-
dations. 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 max-
imum 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 ap-
proaches 20%, and Kuerbis et al. (1988) and Lade and Ya-
mamuro (1997) observed in their studies that emax and emin in-
crease after the fines content exceeds about 25%.
   This pattern of decreasing emax and emin with increasing fines

TABLE 2. Minimum and Maximum Void Ratios for Clean and
Silty Ottawa Sands
         (%)                    emin                  emax
          (1)                   (2)                   (3)
           0                   0.48                   0.78
           5                   0.42                   0.70
          10                   0.36                   0.65
          15                   0.32                   0.63            FIG. 4. Determination of Critical-State Friction Angle c from
          20                   0.29                   0.62            Drained Triaxial Compression Test on Loose Clean Sand Sam-
                                                                      ple (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         For each gradation, a limit void ratio (and a corresponding
particles apart. Due to the methods of preparation of loose and      limit relative density) can be defined. For Ottawa sand, with
dense samples, more particles are found between the surfaces         emax = 0.78, these relative densities are 3% for 5% fines, 17%
of adjacent sand particles in loose than in dense sands, hence       for 10% fines, 38% for 15% fines, and 59% for 20% fines.
the larger drops in emin than in emax for a given increase in fines   For relative densities lower than the limit relative density, the
content observed in Table 2.                                         fines control and the behavior becomes that of a sandy silt or
   For a given overall void ratio, there is a fines content for       sandy clay, depending on the nature of the fines. For soils
which the fines completely (or almost completely) separate            denser than the limit relative density, the behavior is that of
adjacent sand particles. An easy way to determine the fines           sand, modified by the presence of 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     Peak and Critical-State Shear Strength
ratio of the silty sand calculated as if the fines were voids
                                                                        The peak and critical-state friction angles are obtained ac-
                              1    e                                 cording to (1) at the points of peak strength and critical state,
                        esk =          1                     (20)
                              1    f                                 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          FIG. 6. Drained Triaxial Compression Tests on Dense Sam-
of Ottawa Sand with Various Silt Contents under Moderately           ples of Ottawa Sand with Various Silt Contents under Low Effec-
High Effective Confining Stress                                       tive Confining Stress

           TABLE 3.       Static Triaxial Test Results             and volume with increasing shear strain. The critical-state fric-
                                                                   tion angle was obtained from those tests that, for practical
          Fines                                                    purposes, did reach critical state. The volumetric strain versus
         content                 DR        3                pp     axial strain plot was the main check of whether critical state
  Test     (%)       e          (%)     (kPa)        p    (kPa)    was approached. The critical-state friction angle was deter-
   (1)      (2)     (3)          (4)     (5)       (6)     (7)
                                                                   mined at the axial strain at which the volumetric strain versus
A1          0      0.633        49.1     200       32.4   359      axial strain plot becomes horizontal (i.e., the dilatancy angle
A2          0      0.590        63.3     400       34.7   756      becomes zero). As a secondary check, the value of friction
A3          0      0.643        45.8     100       31.0   173      angle corresponding to the first peak in the volumetric strain
A4          0      0.674        35.3     100       30.1   167
A5          0      0.635        48.4     200       32.0   352
                                                                   versus axial strain plot, which also corresponds to a horizontal
A6          0      0.632        49.3     200       31.1   346      tangent and thus to zero dilatancy angle, was determined as
A7          0      0.678        33.9     100       31.4   177      well. These two estimates of the critical-state friction angle c
A8          0      0.662        39.3     200       30.9   342      were practically the same in the tests where the volume change
A9          0      0.674        35.2     300       31.2   523      curve became horizontal at large axial strains. Fig. 4 illustrates
A10         0      0.659        40.2     200       32.4   357      one instance of critical-state friction angle determination.
A11         0      0.610        56.7     100       33.3   181         Figs. 5 and 6 show results of tests on samples with 0, 5,
A12         0      0.586        64.6     100       34.0   185      10, 15, and 20% silt at 3 = 400 kPa and DR approximately
A13         0      0.537        80.9     100       36.5   198
A14         0      0.558        74.1     100       35.9   195
                                                                   equal to 30% and 3 = 100 kPa and DR approximately equal
A15         0      0.645        44.9     400       31.2   687      to 80%, respectively. Of the tests on the loose samples in Fig.
A16         0      0.665        38.3     400       31.5   692      5, the sample with 20% silt content is clearly below the limit
A17         0      0.699        27.1     400       30.2   669      relative density and thus has a floating fabric. With 20% silt,
B1          5      0.660        14.4     150       33.8   275      in most situations of relevance in geotechnical practice the soil
B2          5      0.581        42.3     200       36.8   402      will have a floating fabric. It is clear from Figs. 5 and 6 that
B3          5      0.661        14.0     250       33.2   456      the critical-state friction angle increases with fines content. The
B4          5      0.495        73.4     200       38.7   426
                                                                   values of c are 29 for clean Ottawa sand, 30.5 for 5%
B5          5      0.630        24.9     200       34.5   379
B6          5      0.587        40.4     250       36.8   501      fines content, 32 for 10%, 32.5 for 15%, and 33 for
B7          5      0.657        15.3     200       33.2   366      20%. An increase of dilatancy with fines content is also ob-
B8          5      0.634        23.7     300       33.6   549      served. Because of the increase in both c and dilatancy, the
B9          5      0.609        32.5     200       35.6   390      peak friction angle p increases with increasing fines content.
B10         5      0.475        80.3     100       40.4   133      Table 3 contains the essential information for all triaxial tests
B11         5      0.502        70.8     100       40.8   225      performed as part of the current testing program. The Q and
B12         5      0.612        31.4     300       33.7   554      R values of (14) are obtained for each silt-sand gradation using
B13         5      0.632        24.3     400       32.5   709
                                                                   the p from each test performed on samples with that gradation
C1         10      0.583        23.1     250       35.9   489
C2         10      0.564        29.6     100       37.0   201      and the c corresponding to the gradation. Substituting (14)
C3         10      0.569        28.0     250       37.0   504      into (12) and rearranging, we obtain the following linear equa-
C4         10      0.581        23.9     350       35.8   685      tion:
C5         10      0.571        27.2     300       37.0   607
C6         10      0.447        69.9     150       39.0   317                          p          c              100p p
C7         10      0.567        28.8     200       35.6   393
                                                                                                        ID ln             = IDQ    R           (21)
                                                                                           3                      PA
C8         10      0.500        51.7     200       37.3   408
C9         10      0.447        70.0     100       40.5   224         Bolton (1986) found that R = 1 and Q = 10 provided a
C10        10      0.420        79.3     100       41.3   230      good fit for several different clean silica sands. Table 4
C11        10      0.563        30.2     400       33.7   733      shows the results of linear regression following (21) on the
C12        10      0.560        31.0     400       34.1   741
D1         15      0.500        41.9     100       35.8   199
                                                                   data for Ottawa sand with 0, 5, 10, 15, and 20% silt contents.
D2         15      0.512        37.9     200       34.9   382      Only data corresponding to relative densities higher than the
D3         15      0.363        86.1     100       44.7   258      limit relative density, below which the sand particles are
D4         15      0.410        70.9     100       39.5   217      completely or nearly completely floated by the fines, were
D5         15      0.390        77.5     100       42.4   238      considered for 5 and 10% silt contents. The limit relative den-
D6         15      0.366        85.1     100       43.1   244      sities were found earlier to be 3, 17, 38, and 59% for fines
D7         15      0.412        70.4     100       41.9   235      contents equal to 5, 10, 15, and 20%, respectively. The best
D8         15      0.375        82.4     100       42.1   235
                                                                   fit for clean Ottawa sand gives Q = 9.0 and R = 0.49, with an
D9         15      0.392        76.8     100       44.4   256
D10        15      0.320       100.0     100       45.5   265      excellent coefficient of correlation (r = 0.96).
D11        15      0.607         7.4     100       32.4   179         Referring to (21), dilatancy increases with increasing Q and
D12        15      0.587        13.7     200       33.9   376
D13        15      0.588        13.5     200       33.2   364
D14        15      0.551        25.6     100       35.0   191      TABLE 4. Values of Dilatancy Parameters Q and R for Clean
D15        15      0.533        31.2     100       33.0   182      and Silty Ottawa Sands
D16        15      0.530        32.1     400       34.9   754                                                             Trendline with
D17        15      0.522        34.8     400       33.8   738
                                                                                                      Best Fit               R = 0.5
E1         20      0.423        59.8     500       37.5   408            Silt                                                              Number
E2         20      0.384        71.5     350       38.4   524            (%)                Q            R        r2        Q      r2      of tests
E3         20      0.402        66.0     450       38.8   428             (1)              (2)          (3)       (4)      (5)     (6)        (7)
E4         20      0.470        45.4     200       34.5   380
                                                                   0                        9.0          0.49     0.93     9.0     0.93      17
E5         20      0.494        38.3     100       35.2   198
E6         20      0.535        25.9     300       35.0   572      5                        9.0          0.50     0.98    11.0     0.92      13
E7         20      0.448        52.2     450       37.4   305      10                       8.3          0.69     0.97    10.6     0.87      12
E8         20      0.531        27.0     500       34.7   280      15   (DR   > 38%)       11.4          1.29     0.97    10.3     0.96      10
E9         20      0.484        41.2     400       34.5   747      15   (DR   < 38%)        7.9          0.04     0.86     9.6     0.82       7
E10        20      0.476        43.5     400       34.7   754      20   (DR   > 59%)       10.1          0.85     0.95     9.5     0.95       3
E11        20      0.487        34.4     400       34.3   744      20   (DR   < 59%)        7.3          0.08     0.82     8.7     0.79       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 5% Nonplastic
Silt; (c) Ottawa Sand    10% Nonplastic Silt; (d) Ottawa Sand     15% Nonplastic Silt and Nonfloating Fabric; (e) Ottawa Sand 15%
Nonplastic Silt and Floating Fabric; (f) Ottawa Sand 20% Nonplastic Silt and Nonfloating Fabric; (g) Ottawa Sand 20% Nonplastic
Silt and Floating Fabric
decreases with increasing R. Bolton (1986) discussed the case       cable only within the range of realtive densities for which test
in which the calculated peak friction angle results less than       data are available, and use of (21) outside such ranges should
the critical-state friction angle. This would be seen when the      be made with caution.
strains necessary to reach critical-state shear strength are so        A direct comparison of the dilatancy of the sand-silt
large that p is selected at an earlier, lower value of shear        mixtures is possible through examination of the values of both
strength. A positive value of R would suggest this type of          Q and R for each silt content. It is easier, although not entirely
scenario for very low relative densities. A negative value of       correct from a fundamental point of view, to compare the di-
R, on the other hand, would imply that the p of very loose          latancies of sand with 0, 5, 10, 15, and 20% silt by comparing
sand would still be higher than c . An implication of the role      the values of Q obtained if a single R value is used, for which
of Q and R in (21) is that, for a set of triaxial tests performed   the coefficient of correlation is satisfactory for all gradations.
on a given material, the value of R would be affected by the        A value of R = 0.5 works relatively well for all gradations (r
selected value of c , but Q would remain unchanged. Ulti-           = 0.96, 0.96, 0.93, 0.98, and 0.97), producing Q values equal
mately, however, both Q and R are fitting parameters, and            to 9, 11, 10.6, 10.3, and 9.5 for 0, 5, 10, 15, and 20% silt
interpretations of their physical meaning should be used within     contents, respectively, and relative densities higher than the
bounds. The values of Q and R in Table 4 are strictly appli-        limit relative density. Fig. 7 illustrates graphically the best fit
                                                                    for all gradations. It is observed that Q for samples with non-
                                                                    floating 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 in-
                                                                    creases throughout the range from 0 to 20% silt content, di-
                                                                    latancy increases initially, as the fines content is raised to 5%,
                                                                    and then drops with further addition of fines, remaining how-
                                                                    ever 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 neighbor-
                                                                    ing sand particles, increasing particle interlocking and causing
                                                                    the soil to become more dilative.

                                                                    FIG. 9. Results of Bender Element Tests for Samples of Sand
FIG. 8. Results of Bender Element Tests for Samples of Clean        with 5% Nonplastic Silt at Various Initial Void Ratios: (a) Low Ef-
Sand at Various Initial Void Ratios: (a) and (b) Low Effective      fective Confining Stresses; (b) Higher Effective Confining
Confining Stresses; (c) Higher Effective Confining Stresses           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 con-
tent is increased from 0 to 15% silt but then appears to sta-
bilize as the silt content is increased further (Fig. 5). The deg-
radation 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 den-
sity (DR = 25–40%), but different fines content. If the ap-
proximate 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 progres-
sively 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 con-
tracts and the sand particles come in closer contact, with the

                                                                            FIG. 11. Results of Bender Element Tests for Samples of
FIG. 10. Results of Bender Element Tests for Samples of Sand                Sand with: (a) 15% Nonplastic Silt; (b) 20% Nonplastic Silt
with 10% Nonplastic Silt at Various Initial Void Ratios: (a) Low            at Low Effective Confining Stresses; (c) 20% Nonplastic Silt
Effective Confining Stresses; (b) Higher Effective Confining                  at Higher Effective Confining Stresses, at Various Initial Void
Stresses                                                                    Ratios

                    TABLE 5.     Regression Parameters Cg , eg , ag , and ng for Calculation of G0 Using (15) and (17)

                                       Using Eq. (15)                                                 Using Eq. (17)
     (%)             Cg              eg             ng                r               Cg             ag            ng             r2
      (1)            (2)             (3)            (4)              (5)              (6)            (7)           (8)           (9)
0                    612            2.17           0.439             0.96             547            1.051        0.443          0.97
5                    454            2.17           0.459             0.94             410            1.044        0.458          0.95
10                   357            2.17           0.592             0.91             135            2.376        0.557          0.96
15                   238            2.17           0.745             0.85             101            2.069        0.715          0.94
20 (DR > 59%)        270            2.17           0.686             1.00              —             —             —              —
20 (DR < 59%)        207            2.17           0.809             0.98              —             —             —              —

fines actually helping to enhance dilatancy and shear strength.         with a prescribed function of the void ratio, either (1) a power
At 15% silt content, the fines start controlling, and at 20% silt       ag of the void ratio; or (2) according to (eg         e)2/(1   e).
content, they fully control soil response, with the soil fabric        Another constant, Cg, also appears in the correlations. These
becoming more stable and the stiffness stabilizing with in-            two sets of constants—(Cg, eg, ng) and (Cg, ag, ng)—were de-
creasing fines content.                                                 termined for clean Ottawa sand and Ottawa sand with 5, 10,
                                                                       15, and 20% silt contents. It was observed that the small-strain
Small-Strain Shear Modulus G0                                          stiffness at a given relative density and confining stress level
                                                                       decreases dramatically with the addition of even small per-
   Figs. 8–11 contain the essential information from the               centages of silt. This is an important result, as analyses of
bender element tests performed for this study. Eqs. (15) and           problems involving silty sand using stiffness properties of
(17) were used to fit the data from the bender element tests            clean sand can be in serious error.
on Ottawa sand with 0, 5, 10, 15, and 20% silt. Research by               Results of triaxial tests were analyzed to assess both the
Iwasaki and Tatsuoka (1977) indicated that eg = 2.17 can be            peak and the critical-state friction angles of clean and silty
used in (15) with satisfactory results for sand particles ranging      Ottawa sands. It was observed that the addition of even small
from round to angular in shape. Thus eg was assumed equal              percentages of silt to clean sand considerably increases both
to 2.17, and the values of Cg and ng in (15), found through            the peak friction angle at a given initial relative density and
regression analysis, are listed in Table 5. It was found that (15)     the critical-state friction angle. This study suggests that silty
works very well for both clean and silty sand (with coefficients        sands with nonfloating fabric in the 5–20% silt content range
of correlation r = 0.98, 0.97, 0.95, 0.92, and 0.99 for 0, 5, 10,      are more dilatant than clean sands; dilatancy appears to peak
15, and 20% silt content, respectively).                               at around 5% silt content, but even at 20% silt content it re-
   The values of Cg, ag, and ng in (17) are found by finding            mains above that of clean sand.
the best fit through the data points. The correlation in (17)              It is interesting to note that, although small-strain stiffness
worked better than (15) for all stress levels and silt contents        drops, peak and critical-state strengths increase with increasing
up to 15%. For the available data, a reasonable correlation was        fines content. This may be interpreted as follows: initially the
not found in (17) for 20% silt content. The correlation param-         fine particles are not positioned in a way to provide optimum
eters are listed in Table 5 for different fines contents. Based         interlocking and small shear strains are imposed on the soil
on the values of Cg, ag, and ng from Table 5, it is clear that         with greater ease than if the fines were not present. As shearing
the shear modulus of sand decreases dramatically with fines             progresses, the fines reach more stable arrangements and ul-
content. For instance, at a confining pressure of 100 kPa and           timately increase interlocking, dilatancy, and shear strength.
DR = 50%, the value of G0 is 89 MPa for clean sands, but it               The soil response observed in this study is strictly applicable
drops to 75, 66, 46, and 42 MPa for sands with 5, 10, 15, and          only to the silt and sand gradations used in the testing. Further
20% fines, respectively. The stiffness reduction with fines con-         study is needed to assess the effects of different gradations on
tent may be partially explained by the way in which the fines           the behavior of silty sand.
interact with the sand matrix. If the fines are positioned within
the sand matrix in such a way that they do not have well-              ACKNOWLEDGMENTS
developed contacts with the sand particles, shear waves (or               This material is based upon work supported by the National Science
static stresses) are not effectively transferred through the fine       Foundation, Washington, D.C., Earthquake Hazards Mitigation Program,
particles. Thus lower void ratios due to the addition of fines          under grant No. CMS-9410361. However, any opinions, findings, and
do not lead to increases in G0; accordingly, a silty sand at the       conclusions or recommendations expressed in this material are those of
same void ratio as a clean sand has a lower G0. Even when              the writers and do not necessarily reflect the views of the National Sci-
silt particles have better developed contacts with the sand par-       ence Foundation. The support of Dr. Michael F. Riemer and his useful
                                                                       insights into bender element testing are greatly appreciated. The assis-
ticles, the silt particles may more easily move sideways under         tance of Bryan Scott and Yongdong Zeng with some of the tests is also
shear stress application or shear-wave propagation, leading to         appreciated. Dr. Vincent P. Drnevich assisted with the data acquisition
lower shear stiffness. These two effects, related to the fabric        software for the bender element tests.
of silty sands, lead to the lower Cg values measured in bender
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