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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 ered). The stress-strain response of sand at small-, intermediate-, 1 and large-strain levels depends upon soil state variables (the 1 3 relative density DR of the sand, the effective stress state, and sin = (2) 1 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 1 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: rodrigo@ecn.purdue.edu 2 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 3 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 a Bolton (1986). b Chung et al. (1984). c Present paper. d Lo Presti (1987). e 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 kdε3 arating two blocks of the same material. In this model, slip- sin = (9) dε1 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) PA 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. 452 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 Small-Strain Shear Modulus where = total mass density of the soil, and Vs = shear-wave velocity. 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 A ng m ng (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 A n m) n a m 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 A ng ag e m ng (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( 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 sample 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 (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 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 454 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 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. 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). 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 Silt (%) 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 456 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 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 458 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 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) Silt 2 (%) 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 — — — — 460 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000 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 element testing. APPENDIX I. REFERENCES These observations that lower stiffness results from the ad- Arulnathan, R., Boulanger, R. W., and Riemer, M. F. (1998). ‘‘Analysis dition of ﬁnes indicate that analyses of granular soil masses of bender element tests.’’ Geotech. Testing J., 21(2), 120–131. where clean sand G0 values are used can be in signiﬁcant error Bandini, P. (1999). ‘‘Static response and liquefaction of silty sands,’’ Mas- if the soil has even a small amount of ﬁnes. ter thesis, Purdue University, West Lafayette, Ind. Been, K., Jefferies, M. G., and Hachey, J. (1991). ‘‘The critical state of ´ sands.’’ Geotechnique, London, 41(3), 365–381. SUMMARY AND CONCLUSIONS ´ Bolton, M. D. (1986). ‘‘The strength and dilatancy of sands.’’ Geotech- nique, London, 36(1), 65–78. At small shear strains (typically <10 4 –10 3%) the shear Burmister, D. (1948). The importance and practical use of relative density stress versus shear strain relationship of sand is linear, but for in soil mechanics, Spec. Publ. No. 48, ASTM, West Conshohocken, larger shear strains it becomes strongly nonlinear. If the sand Pa., 1–20. is dilative, a peak shear strength is reached at axial strains of Chan, C. K. (1981). ‘‘An electropneumatic cyclic loading system.’’ Geo- the order of 2–3%. At large strains (of the order of 25–40%), tech. Testing J., 4(4), 183–187. the sand reaches its critical state. In the analysis of soil prob- Chung, R. M., Yo Kel, F. Y., and Drenevich, V. P. (1984). ‘‘Evaluation of dynamic properties of sands by resonant column testing.’’ Geotech. lems, it becomes important to describe the loading response Testing J., 7(2), 60–69. of sands. In this paper, we studied the effects of various levels De Josselin de Jong, G. (1976). ‘‘Rowe’s stress-dilatancy relation based of silt content on the stress-strain properties of sand at small ´ on friction.’’ Geotechnique, London, 26(3), 527–534. and large shear strains. Dyvik, R., and Madshus, C. (1985). ‘‘Laboratory measurement of Gmax The small-strain shear modulus G0 describes soil response using bender elements.’’ Advances in the art of testing soils under in the initial, elastic stress-strain range and is a function of the cyclic conditions, ASCE, New York, 186–196. Hardin, B. O. (1978). ‘‘The nature of stress-strain behavior of soils.’’ stress state and degree of compactness of the sand. The small- Proc., ASCE Geotech. Engrg. Div.., Spec. Conf., Vol. 1, ASCE, New strain shear modulus G0 increases with a power ng (usually in York, 3–90. the range of 0.4–0.8, depending on the silt content) of the Hardin, B. O., and Richart, F. E., Jr. (1963). ‘‘Elastic wave velocities in mean effective stress. The stiffness is further assumed to vary 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 formation parameters for soil.’’ Proc., Roscoe Memorial Symp., Stress- on dynamic shear moduli of sands.’’ Soils and Found., Tokyo, 17(3), Strain Behavior of Soils, R. H. G. Parry, ed., Foulis, Henley on Thames, 19–35. U.K., 143–194. Jamiolkowski, M., Leroueil, S., and Lo Presti, D. C. F. (1991). ‘‘Theme Salgado, R., Boulanger, R. W., and Mitchell, J. K. (1997a). ‘‘Lateral stress lecture: Design parameters from theory to practice.’’ Proc., Geo-Coast effects on CPT liquefaction resistance correlations.’’ J. Geotech. and ’91, 1–41. Geoenvir. Engrg., ASCE, 123(8), 726–735. Kuerbis, R., Negussey, D., and Vaid, Y. P. (1988). ‘‘Effect of gradation Salgado, R., Mitchell, J. K., and Jamiolkowski, M. (1997b). ‘‘Cavity ex- and ﬁnes content on the undrained response of sand.’’ Hydraulic ﬁll pansion and penetration resistance in sand.’’ J. Geotech. and Geoenvir. structures, Geotech. Spec. Publ. No. 21, ASCE, New York, 330–345. Engrg., ASCE, 123(4), 344–354. Kuerbis, R., and Vaid, Y. P. (1988). ‘‘Sand sample preparation—The Schanz, T., and Vermeer, P. A. (1996). ‘‘Angles of friction and dilatancy slurry deposition method.’’ Soil and Found., Tokyo, 28(4), 107–118. ´ of sand.’’ Geotechnique, London, 46(1), 145–151. Lade, P., and Yamamuro, J. (1997). ‘‘Effects of non-plastic ﬁnes on static Selig, E. T., and Ladd, R. S. (1973). ‘‘Evaluation of relative density mea- liquefaction of sands.’’ Can. Geotech. J., Ottawa, 34(6), 918–928. surement and applications.’’ Evaluation of relative density and its role Lo Presti, D. C. F. (1987). ‘‘Mechanical behavior of Ticino sand from in geotechnical projects involving cohesionless soils, ASTM STP 523, resonant column tests,’’ PhD thesis, Politecnico di Torino, Turin, Italy. ASTM, West Conshohocken, Pa., 487–504. Lo Presti, D. C. F., Pedroni, S., and Crippa, V. (1992). ‘‘Maximum dry Shirley, D. J., and Hampton, L. D. (1977). ‘‘Shear-wave measure- ments in laboratory sediments.’’ J. Acoust. Soc. Am., 63(2), 607– density of cohesionless soil by pluviation and by ASTM D 4253-83: a 613. comparative study.’’ Geotech. Testing J., 15(2), 180–189. Tavenas, F., and La Rochelle, P. (1972). ‘‘Accuracy of relative density Negussey, D., Wijewickreme, W. K. D., and Vaid, Y. P. (1986). Constant ´ measurements.’’ Geotechnique, London, 22(4), 549–562. volume friction angle of granular materials, Soil Mech. Ser. No. 94, Taylor, D. W. (1948). Fundamentals of soil mechanics. Wiley, New York. Dept. of Civ. Engrg., University of British Columbia, Vancouver. Vaid, Y. P. (1994). ‘‘Liquefaction of silty soils.’’ Ground failures under Randolph, M. F., Dolwin, J., and Beck, R. (1994). ‘‘Design of driven seismic conditions, Geotech. Spec. Publ. No. 44, Shamsher Prakash and ´ piles in sand.’’ Geotechnique, London, 44(3), 427–448. Panos Dakoulas, eds., ASCE, New York, 1–16. Roesler, S. K. (1979). ‘‘Anisotropic shear modulus due to stress anisot- Viggiani, G., and Atkinson, J. H. (1995a). ‘‘Interpretation of bender ele- ropy.’’ J. Geotech. Engrg. Div., ASCE, 105(7), 871–880. ´ ment tests.’’ Geotechnique, London, 45(1), 149–154. Rowe, P. W. (1962). ‘‘The stress-dilatancy relation for static equilibrium Viggiani, G., and Atkinson, J. H. (1995b). ‘‘Stiffness of ﬁne-grained soil of an assembly of particles in contact.’’ Proc., Royal Soc., London, ´ at very small strains.’’ Geotechnique, London, 42(2), 249–265. A269, 500–527. Yu, P., and Richart, F. E., Jr. (1984). ‘‘Stress ratio effect on shear modulus Rowe, P. W. (1971). ‘‘Theoretical meaning and observed values of de- of dry sands.’’ J. Geotech. Engrg., ASCE, 110(3), 331–345. 462 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / MAY 2000