SHEAR MODULUS OF A SATURATED GRANULAR SOIL DERIVED FROM RESONANT-COLUMN TESTS Keywords resonant column; resonant frequency; shear modulus; relative density; effective consolidation pressure; dynamic shear modulus Abstract This paper presents the results of 120 determinations of the shear modulus (G) of a saturated granular soil (20-40 Ottawa sand) in different conditions of relative density (Dr), effective consolidation pressure (o'J and level of torsional excitation (Te). The equipment used was a resonant-column apparatus manufactured by Wykeham Farrance and the tests were performed with relative density values of20, 40, 60 and 80%, effective consolidation pressures of50, 100, 150, 200, 250 and 300 kPa, and torsional excitations of0.025, 0.05, 0.1, 0.2 and 0.4 volts (V), leading to shear strains (y) between 0.002% and 0.023%. The results led to very simple empirical expressions for the shear modulus as a function of the angular strain for different effective consolidation pressures and void-ratio values. 1 INTRODUCTION The dynamic behaviour of granular soils has been intensively studied around the world for several decades now and the results obtained from various research programs are disseminated through the proceedings of international conferences and indexed journals related to geotechnical engineering. Since there is abundant information on the dynamic behaviour of granular soils and many of the topics dealt with are commonplace; this paper will only focus on references directly related to resonant-column tests, either from the point of view of the development of the test itself or from their utilization to obtain shear-wave velocities, shear-stiffness moduli and damping ratios. The resonant column was first used by Ishimato and Iida (1937) [1] and Iida (1938, 1940) [2, 3] to test Japanese soils, and then nearly two decades later by Bishop (1959) [4]. Since the 1960s, this technique has been widely used in many countries and has been subjected to countless modifications in the restraints applied to the specimen ends. Some of the many works on this matter are described below. For the sake of clarity, the references have been grouped by the main objective of the research rather than following a chronological order. Appearing first are the most relevant analyses of the test apparatus itself and of how to use the resonant column. Wilson and Dietrich (1960) [5] used one of the most novel - at that time - resonant columns in the USA to test clay samples. Hall and Richard (1963) [6] designed and developed a "fixed-free" resonant-column apparatus, i.e., the specimen is fixed at the base and free at the upper end, therefore allowing the soil samples to be subjected to torsional and longitudinal vibrations. Drnevich et al. (1966,1967) [7, 8] developed equipment for hollow H. Patiño Universidad Politécnica de Madrid (UPM) Madrid, Spain E. Martínez Universidad Politécnica de Madrid (UPM) Madrid, Spain Jesús González Universidad Politécnica de Madrid (UPM) Madrid, Spain E-mail: jesus.gonzalezg@upm.es A. Soriano Universidad Politécnica de Madrid (UPM) Madrid, Spain Acta Geotechnica Slovenica, 2017/2 19. H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests cylindrical soil specimens, to determine the shear modulus and the damping ratio under large deformations; the reason for using hollow specimens being related to the difficulty in obtaining a representative value of the angular strain in solid samples. In addition, they developed the theory in which the interpretation of the results obtained from the resonant column test is based. The operational principle of resonant-column equipment, the calibration recommendations, the processing of the data and the interpretation of the results were clearly described by Drnevich et al. (1978) [9]. Menq (2003) [10] developed a resonant-column apparatus that allows testing of specimens up to 15 cm in diameter that was used to study the dynamic properties of sand and gravel. Clayton et al. (2009) [11] used aluminium rods of various diameters to evaluate the polar moment of inertia of the excitation system (I0) and found that this value depended on the stiffness of the rod employed in calibrating their apparatus. However, calibrating the resonant column employed for our research led to a constant I0 value. Clayton (2011) [12] refers to some in-situ and laboratory methods to estimate the stiffness and analysed in detail factors influencing the stiffness value obtained from very-small-strain tests, like the range of strains, anisotropy and velocity of loading. Recently, some manufacturers of equipment for obtaining dynamic parameters have marketed relatively sophisticated models for resonant column tests that allow better control and better simulation during execution of the tests; among others Wykeham Farrance in the UK, which made the device used for this investigation. A detailed description of the equipment is presented later on. In general, the resonant column test is the most commonly used laboratory technique to measure the dynamic properties of soils subjected to a low level of deformation. The various designs developed so far imply the application of axial or torsional harmonic loads to solid or hollow specimens by means of electro-magnetic systems capable of accurately controlling the frequency and amplitude of the different types of waves that can be generated. On the other hand, Al-Sanad and Aggour (1984) [13] applied random loads and Tawfig et al. (1988) [14], impulsive loads. Resonant-column tests also make it possible to determine the velocity of shear waves and to analyse their influence on other test parameters. Some researches on this point are presented below. Hardin and Richart (1963) [15] measured the shear-wave propagation velocity in samples prepared with Ottawa sand, with crushed quartz sand and with crushed quartz silts, subjected to small strains, and they proposed empirical correlations to calculate the shear modulus as a function of the void ratio and the effective consolidation pressure. Hardin (1965) [16], based on the theory of linear vibrations of a cylindrical rod, presents an expression to calculate the shear wave propagation velocity (Vs) as a function of the resonant frequency, the polar moment of inertia, the height of the specimen and the polar moment of inertia of the system. Richart et al. (1970) [17] proved mathematically that proportionality exists between the resonant frequency of the specimen and the corresponding shear-wave propagation velocity. Santamarina and Cascante (1996) [18] used a resonant column apparatus capable of applying both compressive and tensile devia-toric stresses to measure the velocity of shear and damping waves under small strains. These velocities turned out to depend mainly on the isotropic stress, while the deviatoric stress played a lesser role. Probably, the factor most often analysed with this equipment has been the shear modulus, obtained in cyclic shear tests. Kuribayashi et al. (1975) [19] found that the shear modulus of several materials is not a function of the relative density, but rather of the void ratio. Iwasaki et al. (1978) [20] present the average variation trend of the shear modulus in eight different types of sand as a function of the angular strain. In addition, they found that in the case of Toyura sand, within a wide range of deformations, a linear relationship exists between the shear modulus and the effective consolidation pressure. Tatsuoka et al. (1979) [21] determined that the shear modulus, within a wide range of deformations, is not affected by the initial structure of the tested specimens. Alarcon-Guzman et al. (1989) [22] investigated the effect of the principal stress ratio on the shear modulus, concluding that this factor has a less important effect on the determination of the maximum shear modulus, but drastically affects the secant shear modulus. Saxena et al. (1989) [23] extensively reviewed empirical relations for obtaining Gmax and the damping (D) under small strain and conducted resonant-column tests on Monterey No. 0 Sand and showed that published relations overestimated Gmax ar,d underestimated D for this sand. Lo Presti et al. (1997) [24] evaluated the influence of the strain rate in the determination of the shear modulus of granular soils, and found that this factor has a very small effect on the maximum shear modulus. Diaz-Rodriguez and Lopez-Flores (1999) [25] proposed an empirical function (a potential expression) between the shear modulus and the isotropic consolidation stresses (o'c). Wichtmann and Triantafyllidis (2004) [26] analysed the influence of the history of dynamic loading on the properties of dry sands; the results thus obtained indicated that a dynamic pre-stressing moderately affects the shear modulus under small deformations. Gu et al. (2013) [27] used bender 34. Acta Geotechnica Slovenica, 2017/2 H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests elements to test three different sands subjected to small strains and found that both G0 (shear modulus) and M0 (constraint modulus) increase with the density and the confining pressure. They found G0 to be more sensitive to E0 (Youngs modulus) and proposed empirical relations between the Poisson ratio and G0 and M0. Finally, some works are presented that analyse how the soil identification properties (grading, particle shape, etc.) influence results. Chang and Ko (1982) [28] tested 23 samples of Denver sand and found that the maximum shear modulus is - to a large extent - a function of the coefficient of uniformity, whereas the effect of the mean size of the particles is minimal. Koono et al. (1993) [29] executed what can be regarded as a field resonant-column test in a gravel deposit. Wichtmann and Triantafyllidis (2009, 2013 and 2014) [30, 31, 32] evaluated the influence of the coefficient of uniformity and of the grain size distribution for 27 types of clean sand in the determination of the maximum shear modulus: the results obtained indicate that for equal values of the void ratio and of the effective consolidation pressure, the maximum shear modulus decreases as the coefficient of uniformity increases, whereas it does not change with the mean particle size. Martinez (2012) [33] studied the influence of the soil index properties on the determination of the dynamic parameters of a saturated granular soil. Senetakis et al. (2012) [34] tested sands with different grading curves, particle origin and shape under very small strains. Volcanic sands showed significantly lower G0 values than those of quartz sands, whereas their D0 were only slightly lower compared to quartz sands. Yang and Gu (2013) [35] found that, in the range of small strains, the shear modulus varies very little in terms of particle size. Senetakis and Madhusudhan (2015) [36] tested quartz sands and angular-grained gravels and they proposed potential functions to relate G0 with p'. The exponent nG was shown to be dependent on the specimen preparation procedure. Finally, Payan et al. (2016) [37, 38] observed that the published formulae cannot accurately relate the shear modulus under small strain with the void ratio and confining pressure, probably because the particle shape was not taken into account. Based on critical-state theories, they propose a new expression, including the effect of grading curves and particle shapes. Taking into account the background information presented above, the objective of this investigation is an in-depth study of the influence of the relative density, effective consolidation pressure and torsional excitation values on the shear modulus of a saturated granular sand and to develop simple empirical functions to correlate these parameters. 2 MATERIAL USED The tests were performed on 20-40 Ottawa sand (maximum, minimum and average particle sizes are 0.85, 0.43 and 0.64 mm, respectively). It is a standard material employed in many other investigations into the behaviour of granular soil. Its main characteristics are: very hard, uniform particles (the coefficient of uniformity turned out to be Cu = 1.35), fine and rounded grains and nearly pure quartz in composition. The index properties of particles passing mesh 20 and retained in mesh 40 are as follows: specific gravity Gs = 2.669, maximum void ratio emax= 0.754, and minimum void ratio emin = 0.554. The initial properties of the specimens tested are presented in Table 1. Table 1. Properties of specimens tested. Relative Height, Diameter, Mass, Dry density Void density, H, D, g Pd> ratio, e Dr, % mm mm g/cm3 20 105 49.5 314.59 1.557 0.714 40 105 49.5 322.13 1.594 0.674 60 105 49.5 330.05 1.633 0.634 80 105 49.5 338.36 1.675 0.594 2.2 Description of the equipment used The resonant-column apparatus consists of a forced oscillation system with a single torsional degree of freedom that makes the specimen vibrate within a range of frequencies in which its first natural mode can be found. In this particular investigation, the specimen remained fixed at its base and was free to vibrate at its upper end. Testing was performed with the resonant-column device manufactured by Wykeham Farrance, Fig. 1. The frequency of the resonant-column tests is higher than 10 Hz, while in cyclic torsional shear mode the equipment typically operates at frequencies below 2 Hz. In this research, the frequency range was between 74 Hz and 140 Hz. This instrumented and automated equipment provides a series of advantages, among which mention can be made of the following. It combines resonant-column and simple torsional shear functions. It determines automatically the resonant frequency, the shear-wave velocity, the shear modulus, the angular strain and the damping ratio, this latter parameter by using the Half-Power 34. Acta Geotechnica Slovenica, 2017/2 H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests Figure 1. Resonant-column apparatus manufactured by Wykeham Farrance. method or the Free Vibration Decay method. There is no need to externally use either an oscilloscope or a function generator. The internal floating structure for the excitation system allows the execution of tests in which the specimens can experience large axial deformations during consolidation. It makes it possible to visualize, in real time, the response of the sensors during the test. The equipment is basically constituted by two polycarbonate hollow cylindrical cells allowing, by means of the internal cell, the application of the consolidation pressure to the specimen through a fluid, without the electronic components being submerged, and - through the external cell - the application to the fluid of a confining pressure provided by a pneumatically operated system; a lower base through which the back pressure is applied and drainage of the specimen is allowed during the consolidation stage; a corrugated head piece with no possibility of drainage, to transmit the torsional forces to the specimen; a driving mechanism constituted by eight coils and four magnets to apply the torsional load to the specimen; an accelerometer attached to the mechanism to generate the torsional action and to provide the information necessary to calculate the shear-wave propagation velocity (Vs); an LVDT to measure axial deformations (with a stroke of +/- 12.5 mm and an accuracy of 0.2%), two proximity transducers to measure angular deformations in case the data supplied by the accelerometer is not used to calculate them; three pressure transducers to measure the chamber pressure (ctc), the back pressure (Bp) and the pore water pressure (u); a transducer to register volume changes during the consolidation stage; a compact unit fitted with a power source, a manual pressure regulator, two electric pressure regulators, eight electronic components for signal conditioning and a control and data-acquisition module; and a computer for equipment control and data acquisition. 3 THEORETICAL BACKGROUND According to the theory of torsional vibrations in a cylindrical rod, expression (1) relates the shear-wave propagation velocity (vs) to the shear modulus (G) and to the unit mass density (p). ys=M (i) v P Expression (2), obtained by Hardin (1965) [13], calculates the shear-wave propagation velocity (vs) as a function of the resonant frequency (Fr), the polar moment of inertia of the excitation mechanism about its symmetry axis (I0), the polar moment of inertia of the specimen about its symmetry axis (I) and the height of the sample (h). T „ 2n-¥h /? ■ tan = —; where /3 =-1— (2) I() Vs Implicit equation (2) can be represented graphically as a function of j3, as depicted in Fig. 2. Li- ter ids t infiiiitv svttcn |J = X pta n|i= l/l V 0.001 0.01 0.1 I 10 100 1000 10000 t/lo Figure 2. Graphical representation of the implicit equation (2). Equation (3) is obtained from (1) and (2). G = 4*Vp 2 (3) P2 For this particular equipment the height of the specimen and the polar moment of the excitation mechanism are fixed constants. Their values are: h = 10.5 cm I0 = 13.1kg • cm2. I0 was obtained by calibration with two rods of the same dimensions and made up of different materials and turned out to be independent of the rod's stiffness. However, Clayton et al. (2009) [11] found that, in their equipment, I0 was dependent on the rods stiffness. Our I0 value lies outside the range reported by them (2.99 to 4.32 kg-cm2) and it seems as though low values of I0 will depend on the rods stiffness, while high I0 values will not. The specimen diameter is D = 4.95 cm. 34. Acta Geotechnica Slovenica, 2017/2 H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests The densities of the samples used for this experiment range from psat(min)=1.974 gr/cm3 to psai(max)=2.048 gr/cm3 or psat = 2.011 ±0.037 gr/cm3 For the central value, psat = 2 gr/cm3, the value of the polar moment of inertia of the specimen is: 1= J-7rD4hp= 1.24 kg • cm2 (4) 32 The corresponding value of /3; obtained from equation (2) P- tg/J= —=0.0946 is p= 0.303 rd (p= 17.4°) 13.1 The shear-wave velocity can be obtained from the resonant frequency, Fr, measured during the test Vs = 2-18 Fr — (Fr en Hz) (5) P s and the corresponding value of the shear modulus G = vs2 ■ p = 9.51 Fr2 kN/m2 (Fr en Hz). 3.1 Experimental program This investigation was aimed at determining the effect of the relative density, the effective consolidation pressure and the magnitude of the torsional excitation on the shear modulus. A total of 120 determinations of the resonant frequency for saturated specimens were made in specimens measuring 49.5 millimetres in diameter and 105 millimetres in height. They all had a height-to-diameter ratio equal to 2.12, thus eliminating the uncertainty related to the slenderness of the specimens; the ratio specimen diameter to particle diameter was of about 120, therefore eliminating the scale effect. The total number of tests is a result of the combination of relative densities equal to 20, 40, 60 and 80%, effective consolidation pressures equal to 50,100,150, 200, 250 and 300 kPa and amplitudes of sinusoidal waves equal to 0.025, 0.05, 0.1, 0.2 and 0.4 volts. The frequency varied between 74 Hz and 140 Hz, which corresponds to angular deformations between 0.002% and 0.023%. The backpressure was equal to 400 kPa for all the tests. Figure 3. Basic elements for the specimen preparation: 1) lifting device of the three-part mould; 2) fixed lower base; 3) porous stone; 4) three-part split mould; 5) latex membrane; 6) O-Ring; 7) O-Ring stretcher; 8) extension of three-part mould; 9) 500-cm3 beaker with de-aired water; and 10) loading head. The sample-preparation procedure was similar to other laboratory tests using sand. The need to reproduce specimens complying with a certain relative density led to a setting process that was very careful and repetitive. 3.3 Effect of the sample density on the ratio Fr The value of the ratio between the shear modulus and the square of the resonant frequency, G/F2, turns out to be only slightly affected by the sample density. In fact, the theoretical value of that ratio is: .21,2 (6) p G, 4/r h R= —r= p2 P When the density, p, increases, the value of P also increases and the result is that the value of R is almost unchanged. In fact, taking the derivative of R with respect to p, we obtain: dR = R _ 2R dfi (7) dp p P dp From equation (2), and taking the previously indicated value of I, we obtain: a «. a 1 ^D4h (8) 3.2 Preparation and setting of specimens The accessories depicted in Fig. 3 that are necessary for making specimens with the sedimentation method were used to carry out the tests reported herein. Differentiating with respect to p gives: dp I cos P P^P (9) to obtain: 34. Acta Geotechnica Slovenica, 2017/2 H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests — = — —-—; being a = ——— dp p l + a sen 2/? (10) and, with the help of equation (7) dR = dp(a-\\ R p W + lJ With a > 1, any increase in the density always produces an increase in the value of R. (11) For the particular case of psat = 2 gr/cm3 equations (10) and (11) are: Figure 5. Typical variation trends of the shear modulus G, and its inverse 1/G as a function of the angular strain for different effective consolidation pressures and relative densities of 20%. The degradation of the shear modulus (or the inverse of G) as a function of the angular strain has low values and reaches a maximum of 24% when the relative density is equal to 20%, the effective consolidation pressure amounts to 50 kPa and the angular strain increases from 0.0035% to 0.019% (large dots 1 and 2, Fig. 5). The simplest mathematical model used to simulate the degradation of the shear modulus G as the strain, y, increases is the one suggested by Hardin and Drnevich (1972) [39]. x G J_ G„ 1 + - Y ref (12) where G0 and yrej are the two parameters of the model. In order to find values of G0 and yrey of the Hardin and Drnevich model that can contribute to better analyse the results of the investigation, a diagram of 1/G versus y has been plotted and the best fit for straight lines was obtained. The resulting G0 and yrej values are given in Table 4. Fig. 6 represents the relationship between shear modulus (G0) and the void ratio (e); in this case the void ratio is denoted by e to distinguish it from number e, the base of natural logarithms. 34. Acta Geotechnica Slovenica, 2017/2 H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests Table 4. Values of G0 and yref that best fit the test results. Value of G0 (MPa) Dr Consolidation pressure a'c (kPa) 50 100 150 200 250 300 20 75.8 107.5 133.3 151.5 166.7 185.2 40 78.1 116.3 140.9 161.3 178.6 192.3 60 78.7 109.9 137.0 158.7 175.4 192.3 80 80.0 119.1 144.9 166.7 185.2 204.1 Value of yref (%) Dr Consolidation pressure a'c (kPa) 50 100 150 200 250 300 20 0.045 0.092 0.100 0.124 0.122 0.118 40 0.068 0.103 0.114 0.124 0.114 0.147 60 0.062 0.104 0.124 0.141 0.118 0.122 80 0.060 0.085 0.106 0.124 0.130 0.107 Dr = 20% iv) 200 S i«> 2 O 100 Ml •..... ------ G,*2E+37e1:4«« R1 » 0.973 ......... ...........« • am o to* 97» oras an om o?w Void ratio, c Figure 6. Value of shear modulus G0 vs void ratio (e). As we can see in Table 4, the fact that G0 is sometimes even larger in samples with Dr = 40% than in samples with Dr = 60% is due to the narrow range of variation coupled with the unavoidable experimental scatter of results. The low sensitivity of G0 can be attributed to the nature of the sand, made of hard quartz grains (rounded and very uniform in size), which implies only little variation between the maximum and minimum void ratios. 4.2 Increase of G0 with a'c The values of the shear modulus for small strains, G0, obtained as indicated in the previous paragraph, are given in Table 4. It is clear that for each value of the relative density, the value of G0 increases as the consolidation pressure increases. See Fig. 7. Usually, the relation among these values (G0 and a'c) is thought to be of the type. G = K o", nN Po ■Po (13) where K is a "modulus number", N is an "exponent number" and p0 is the value of a standard reference pressure. For this particular investigation a value of Po = 98.1 kPa is used. 0l>»% MO 190 < & 1,- 0 0W&? '0*77**. . H*-»» rt «« a t» 0 100 1» 300 Figure 7. Values of G0 and (fc for Dr = 20%. In order to investigate whether the expression (13) is applicable to this particular case, values of G were plotted versus the corresponding values of a'c on a log-log diagram. Fig. 8 is the plot that corresponds to Dr = 20%. S Figure 8. Double log plot of G0 and a'c data for Dr = 20%. From this type of plot the model parameters can be automatically obtained from the data fit, made by minimizing the sum of the squares of the deviations of the test results that correspond to each relative density. If the results of this research are compared to those recently obtained by Senetakis and Madhusudhan (2015) [36], Fig. 9 shows that even though the particle 34. Acta Geotechnica Slovenica, 2017/2 H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests Table 5. Automatically adjusted values of the dimensionless model parameters K and N. g _ /max r. (14) Relative density K N 20% 1084 0.495 40% 1143 0.501 60% 1122 0.502 80% 1175 0.517 size differs roughly over an order of magnitude, the trend of the variation of G0 with a'c for the Ottawa sand coincides with that of specimen 6-2 tested by Senetakis. Dr = 20% ■ This research • Senetakis S6-1 A Senetakis S6-2 i G„ = 1! l,308o,cH 394 < r* H = 0,999 i' 1 1 —rji . M [ . - -_'! ( ! r a'c kPa Figure 9. Values of G0 and a'c for Dr = N tends to decrease slightly when the void ratio increases; it could even be suggested that N remains at a constant value of 0.50. This behaviour is different to the results obtained by Gu et al (2013) [27] using Toyoura sand, for which N tends to increase slightly with the void ratio. In reality, when the variation range is very narrow, some sands will show a tendency to increase N with an increase of the void ratio and in some others N will decrease slightly, as in Ottawa Sand. 4.3 Increase of Yref with o'c From the values of yreyin Table 4 it seems that this parameter could be considered to be a constant but only for consolidation pressures above 200 kPa; for lower values of the consolidation pressure, the value of yrey decreases and it can no longer be considered as a constant in an hypothetical mathematical model. A better option, which would account for the effect of large G degradation rates for lower values of the consolidation pressure, could be based on considering Yrefas a variable that depends on the consolidation pressure. It seems appropriate to assume that the degradation of the modulus G, when the angular deformation increases, should be mainly conditioned by the ratio Being iy max the 'maximum shear stress applied' and if the shear strength of the sand. The value of xy max can be approximated by ^y max — G Ymax and if can be estimated by T/= cr'c-tg(|) With these considerations, the following degradation equation can be proposed: G = Go (1 - A (15) where A = dimensionless constant that would mainly depend on the shear strength of the tested sand, Xj. The value of yrejwould then be given by the following expression: Yref= (16) G0A Furthermore, it is known that G0 increases with the square root of a'c and, as a consequence, a value of yref increasing with the square root of a'c should be expected. For this reason, a value of yrefcan be found to reasonably fit the data with an expression involving (a'c/p0)0'5. The best fit is given in Fig. 10. As we can see, the relative density also has an effect on yrep but it is not easy to draw a clear figure showing the effect of the relative density at present. veo 1.40 120 1 00 oeo 060 040 020 0.00 • 20 >40 «60 c«0 ■ ^S0^ p^T J > l/*^- 1 so 300 100 ISO 200 2S0 Consolidation pressure, a\ (KPa) Figure 10. Reference values of the shear deformation. This should be valid for values of y with the interval 2 x 10"5 < y < 23 x 10"5 and for the range of densities of this particular investigation and for consolidation pressures lower than 200 kPa. 34. Acta Geotechnica Slovenica, 2017/2 H. Patino et al.: Shear modulus of a saturated granular soil derived from resonant-column tests 4.4 Influence of void ratio on the shear modulus Four nominal values of the relative densities are used to prepare the samples for testing. These are relatively precise data, since the volume of the sample and the associated mass are known with a margin of error of about 0.1%. During the process of consolidation some reduction in the volume takes place that increases the relative density This change of relative density has been investigated by running an oedometer test on a sample prepared with an initial relative density of 20%. The results are given in Fig. 11. Since the void ratio is a better parameter, the values of the relative density have been translated into void-ratio values. For this particular sand, with values of emax= 0.754 and emin. = 0.554, the following relation exists between the relative densities and the void ratios: D = emax-e _ 0.754-e _ 0.754-e r emax - emjn 0.754-0.554 0.20 Based on this expression and taking into consideration the small increase of the relative densities during the consolidation process of the samples, the values of the void ratios to represent the expected value of each test are given in Table 6. r-l-1—- O. ■ Xrv MOT«* .UM | kr,» J«»M Or. 0 007 o uo no loo <00 soo Eflvctiv* cor^vo diiwi p«*»«ur».