Effect of uniformity coefficient on G/G max and damping ratio of uniform to well graded quartz sands

Transcription

1 Journal of Geotechnical and Geoenvironmental Engineering, Vol. 39, No., pp. 59-7, 3 Effect of uniformity coefficient on G/G max and damping ratio of uniform to well graded quartz sands T. Wichtmann i) ; Th. Triantafyllidis ii) Abstract: The modulus degradation curves G(γ)/G max and the damping ratio D(γ) of 7 clean quartz sands with specially mixed grain size distribution curves were measured in approximately 8 resonant column tests. For each material, tests with different pressures and densities were performed. Based on the test data it is demonstrated that the shear modulus degradation is larger for higher values of the uniformity coefficient, C u d 6 /d, while it is rather independent of the mean grain size, d 5. The observed C u -dependence of the curves G(γ)/G max is not adequately described by common empirical equations, because theses equations were developed based on tests on uniform granular materials. In order to consider the influence of the uniformity coefficient, the paper proposes correlations of the parameters of the common empirical equations with C u. Good agreement between the prediction of the extended empirical equations and experimental data collected from the literature is demonstrated. Furthermore, the test data reveal that the curves of damping ratio, D(γ), and the threshold shear strain amplitude indicating the onset of residual deformation accumulation, γ tv, are rather independent of d 5 and C u. The threshold shear strain amplitude at the onset of modulus degradation, γ tl, slightly decreases with increasing values of d 5 and C u. CE Database subject headings: Shear modulus; Damping ratio; Quartz sand; Grain size distribution curve; Uniformity coefficient; Resonant column tests; Introduction The secant shear modulus, G, and the damping ratio, D, are key design parameters for soils subjected to dynamic loading. It is well known that both quantities are strongly straindependent. If a certain threshold value of the shear strain amplitude, γ, is surpassed, the shear modulus, G, decreases with γ while the damping ratio, D, increases. Contrary to the small-strain shear modulus, G max G(γ < 6 ), which can be determined from shear wave velocity measurements, the modulus degradation curves, G(γ)/G max, and the damping ratio, D(γ), cannot be easily quantified in-situ. Therefore, if laboratory test data are not available, the modulus degradation and the damping ratio are often estimated by means of empirical equations or chosen based on typical ranges provided in the literature (see the summary in Section ). The common empirical equations for G(γ)/G max and D(γ) were developed based on tests on granular materials with uniform grain size distribution curves. It has not been proven experimentally yet that these equations are also applicable to well-graded granular materials. Wichtmann & Triantafyllidis [35, 36] have demonstrated that the common empirical equations for the small-strain shear modulus,g max, e.g. the formula proposed by Hardin and Black [4], do not adequately describe the C u -dependence of G max observed in laboratory tests. In those tests, for constant values of void ratio and mean pressure, the small-strain shear modulus, G max, and the P-wave velocity, v P, were found independent of mean grain size, d 5, but strongly decreasing with increasing uniformity coefficient, C u d 6 /d. In order to consider this influence, correlations of the parameters of several empirical equations for G max and v P with C u have i) Research Assistant, Institute of Soil Mechanics and Rock Mechanics, Karlsruhe Institute of Technology (KIT), Germany (corresponding author). torsten.wichtmann@ibf.uka.de ii) Professor and Director of the Institute of Soil Mechanics and Rock Mechanics, Karlsruhe Institute of Technology (KIT), Germany been proposed by Wichtmann & Triantafyllidis [35, 36]. Based on the data from approximately 8 resonant column tests performed on 7 clean quartz sands, the present paper examines the influence of the grain size distribution curve on modulus degradation, G(γ)/G max, on damping ratio, D(γ) and on the threshold shear strain amplitudes, γ tl and γ tv, indicating the onset of modulus degradation or the onset of residual deformation accumulation, respectively. The need for an extension of the common empirical equations describing modulus degradation considering the influence of C u is demonstrated. Empirical equations for G(γ)/G max and D(γ) It is well known that the modulus degradation and the damping ratio are pressure-dependent. For certain shear strain amplitude, the ratio G/G max increases with increasing pressure while the damping ratio, D, decreases (Hardin & Drnevich [6], Kokusho [], Seed et al. [5]). In contrast, the curves G(γ)/G max and D(γ) are rather independent of soil density (e.g. Kokusho []). Several empirical formulas have been proposed in the literature. Those used in this paper are presented in the following. Hardin & Drnevich [5] describe the curves G(γ)/G max by: G G max or by the more flexible function G G max + γ γ r () )] () + γ γ r [ + a exp ( b γγr with two curve-fitting parameters a and b and with the reference shear strain defined as γ r τ max G max (3)

2 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 where τ max is the shear strength. Hardin & Kalinski [7] demonstrated that a normalization of γ with a simplified factor p/p atm instead of γ r (p effective mean pressure, p atm kpa) is also suitable to purify the modulus degradation curves from the influence of pressure. In that case no information about τ max is necessary. According to Hardin & Kalinski [7], setting b in Eq. () is sufficient in order to describe the modulus degradation curves. A modification of Eq. () was proposed by Stokoe et al. [8]: G G max + (γ/γ r ) α (4) with the curvature parameter α. Unlike Eq. (3), the reference shear strain used in Eq. (4) is defined as γ r γ(g/g max.5). The pressure-dependence of γ r can be described by (Stokoe et al. [8]): γ r γ r (p/p atm ) k (5) with the reference shear strain, γ r, at p p atm kpa and with an exponent k. Hardin & Drnevich [5] proposed the following formulas for the damping ratio, D(γ): D D max D D max γ γ r + γ and (6) γ r )] γ γ r [ + a exp ( b γγr )] (7) + γ γ r [ + a exp ( b γγr Therein D max is an asymptotic value of D at large strain amplitudes and a and b are curve-fitting parameters. Alternatively, damping ratio is often formulated as a function of G/G max (e.g. Hardin & Drnevich [5], Tatsuoka et al. [3], Khouri []). Zhang et al. [4] described this relationship by D D min c (G/G max ) + c (G/G max ) (c + c ) (8) with the minimum damping ratio, D min, at very small shear strain amplitudes and with parameters c.6, c -.36 for torsional shear test data and c.94, c -.65 for resonant column () test data. The equation proposed by Khouri [] based on experimental data collected from the literature is retrieved from Eq. (8) with D min.3, c.95 and c The various empirical equations are inspected in the following sections, based on the data from the present test series. 3 Tested materials, test device and testing procedure The natural quartz sand and gravel used for the present study was obtained from a sand pit near Dorsten, Germany. Its grain shape is sub-angular and the specific gravity is ϱ s.65 g/cm 3. First, the granular material was sieved into 5 gradations with grain sizes between.63 mm and 6 mm. Then, the grain size distribution curves shown in Fig. were mixed from these gradations. They are linear in the semilogarithmic scale. The sands and gravels L to L8 (Fig. a) have a uniformity coefficient of C u.5 and different mean grain sizes in the range. d 5 6 mm. These materials were used to study the influence of d 5. The materials a) Finer by weight [%] b) Finer by weight [%] Silt coarse fine L L L Mean grain size [mm] Silt coarse fine L6 L5 L4 L Sand medium Sand medium L L L coarse L7 L5 coarse L3 L5 L6 L8 fine L7 L9 L8 L7 Gravel medium L8 Gravel fine medium L L Mean grain size [mm] Fig. : Tested grain size distribution curves (adapted from Wichtmann & Triantafyllidis [35]) L4 to L6 (d 5. mm, C u 3), L to L6 (d 5.6 mm, C u 8) and L7 to (d 5 mm, C u 8, Fig. b) were used to study the C u -influence for different values of d 5. The two sand-gravel-mixtures L7 and L8 (Fig. a) have even higher uniformity coefficients (C u.6 and 5.9). The d 5 - and C u -values as well as the minimum and maximum void ratios of the tested materials are summarized in Table. The resonant column () device used for the present study (see a scheme and a foto given by Wichtmann & Triantafyllidis [35]) is a free - free type, meaning both the top and the base mass are freely rotatable. The cuboidal top mass is equipped with two electrodynamic exciters each accelerating a small mass. This acceleration and the resulting acceleration of the top mass are measured with accelerometers. From these signals the torsional moment, M(t), and the angle of twist, θ(t), at the top of the sample can be calculated. The sample is enclosed in a pressure cell that can sustain cell air pressures, σ 3, up to 8 kpa. The state of stress is almost isotropic. A small stress anisotropy results from the weight of the top mass (m 9 kg), such that the vertical stress, σ, is slightly higher than the lateral one, σ 3. However, for higher cell pressures this anisotropy is of secondary importance. Furthermore, test results of Yu & Richart [39] revealed that a stress anisotropy becomes significant only near failure. A sinusoidal electrical signal is generated by a function generator, amplified and applied to the electrodynamic exciters. The frequency of excitation is varied until the resonant frequency, f R, of the system composed of the two end masses and the specimen has been found. By definition, this is the

3 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 Mat. d 5 C u e min e max Range of Mat. d 5 C u e min e max Range of Mat. d 5 C u e min e max Range of [mm] [-] [-] [-] tested D r [mm] [-] [-] [-] tested D r [mm] [-] [-] [-] tested D r L L L L L L L L L L L L L L L L L L L L L L Table : Parameters d 5, C u, e min and e max (determined according to German standard code DIN 86) of the tested grain size distribution curves, range of tested initial relative densities D r case when M(t) and θ(t) have a phase-shift of π/ in time, t. The secant shear modulus ( ) π h fr G ϱ (9) a is calculated from the resonant frequency, the height, h, and the density, ϱ, of the specimen. The parameter a is obtained from: a tan (a) J J J L tan (a) a J J + J J L () In Eq. (), J, J.76 kg m, and J L.663 kg m are the polar mass moments of inertia of the specimen, the base mass and the top mass, respectively. The polar mass moment of inertia of the top mass has been calibrated by means of aluminium rods with different diameters and known stiffnesses. Different shear strain amplitudes can be tested by varying the amplitude of the torsional excitation. All tested specimens had a full cross section and measured d cm in diameter and h cm in height. The variation of the shear strain amplitude with radius, r, and distance, x, from the base of the sample can be calculated from θ(x h) a γ(r, x) r cos (a) J J a sin (a) h [ ( ax ) sin + J ( ax ) ] h J a cos () h with the amplitude of the angle of twist θ(x h) measured at the top of the sample. Evaluating Eq. () for typical a- values reveals only a slight variation of γ with x (about %). The variation with r is considered by calculating a mean value over the sample volume: γ γ(r, x) dv () V V This mean value is simply denoted by γ in the following analysis of the test results. The shear strain amplitudes that can be tested in the device lie in the range 5 7 γ 5 4. Measurements of the shear wave velocity by means of bender elements delivered very similar G max values compared to tests performed with the device [37]. Preliminary tests on hollow cylinder samples (outer diameter d a cm, inner diameter d i 6 cm, h cm), having a more uniform distribution of shear strains over the cross section, showed similar curves, G(γ) and D(γ), as full cylinder specimens. Therefore, the use of full cylinder specimens was regarded as sufficient for the present study. The comparison was undertaken for a medium coarse uniform sand. However, the conclusion is assumed valid also for more well-graded sands. The settlement of the samples was measured with a noncontact displacement transducer placed approximately mm above the center of the top mass, i.e., in a position where the torsional displacement is very small. This gap sensor measures against an aluminium target rigidly connected to the top mass. All samples were prepared by air pluviation and tested in the air-dry condition. For each material, five to ten tests with different initial relative densities D r (e max e )/(e max e min ) were performed. The tested ranges of relative densities are given in Table. The mean pressure, p, was increased step-wise from p 5 to 4 kpa in order to obtain the G max - and v P -data documented by Wichtmann & Triantafyllidis [35,36]. The curves G(γ) and D(γ) were measured at p 4 kpa. In three additional tests on medium dense specimens, the modulus degradation and the damping ratio were also measured at p 5, and kpa. The shear strength, τ max, is necessary in order to calculate the reference shear strain, γ r, from Eq. (3). For each material, the peak friction angle, ϕ P, was determined from at least three monotonic triaxial tests on isotropically consolidated (p kpa) samples prepared with different relative densities. The measured decrease of the peak-friction angle with increasing void ratio was described by an exponentially decreasing function, ϕ P (e ), where e is the initial void ratio of the sample before shearing. 4 Test results 4. Modulus degradation 4.. Influence of d 5 and C u For a constant value of the strain amplitude, the data from the present test series confirm the well-known increase of G/G max with increasing mean pressure and the independence of density. This becomes obvious from Figure a-d where the modulus degradation curves for different pressures and densities are presented for four selected materials. The first three diagrams show the data for the three sands, L and L6, having a mean grain size of d 5.6 mm and uniformity coefficients between C u.5 and 8. The fourth diagram contains data for the sand-gravel-mixture L8 (C u 5.9). The typical range of G/G max -values specified by 3

4 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 a) Sand b) Sand L c) Sand L6 d) d 5.6 mm d 5.6 mm d. 5.6 mm.. C u.5. C u 3 C u 8 p [kpa] / D r p [kpa] / D r p [kpa] / D r /.7 /.6.8 / p [kpa] / D r /.65 /.69 /.6 /.64 / /.67 5 /.6 5 /.63 5 /.65.6 p 4 kpa, D r p 4 kpa, D r.6 p 4 kpa, D r p 4 kpa, D r e) Sand f) Sand L g) Sand L6 h) Sand L d 5.6 mm C u.5 p [kpa] / D r 5 /.67 /.69 /.7 4 / γ / γ r [-].8.6 d 5.6 mm C u 3 p [kpa] / D r 5 /.6 /.59 /.6 4 / γ / γ r [-] d 5.6 mm C u 8 p [kpa] / D r 5 /.63 /.64 /.65 4 / γ / γ r [-] i) Sand j) Sand L k) Sand L6 l) d 5.6 mm C u.5 p [kpa] / D r 5 /.67 /.69 /.7 4 / γ / (p/p atm ).5 [-].8.6 d 5.6 mm C u 3 p [kpa] / D r 5 /.6 /.59 /.6 4 / γ / (p/p atm ).5 [-] d 5.6 mm C u 8 p [kpa] / D r 5 /.63 /.64 /.65 4 / d 5 mm C u 5.9 p [kpa] / D r 5 /.59 /.6 /.6 4 / γ / γ r [-] γ / (p/p atm ).5 [-] γ / (p/p atm).5 [-] Sand L8 d 5 mm C u 5.9 p [kpa] / D r 5 /.59 /.6 /.6 4 /.67 d 5 mm C u 5.9 Sand L8 Fig. : a-d) Modulus degradation curves G(γ)/G max measured for different materials, pressures and densities (gray shading range specified by Seed et al. [5]), e-h) G/G max versus γ/γ r (solid curves Eq. (), dot-dashed curves Eq. () with b, dashed curves Eq. (4). i-l) G/G max versus γ/ p/p atm (dashed curves Eq. () with b and with p/p atm instead of γ r). Seed et al. [5] (based on Seed & Idriss [4]) is marked by the gray shading in Figure a-d. For poorly graded materials (e.g. ), the curves G(γ)/G max for p 5 and kpa fall into this range while the curves for p and 4 kpa lie slightly above. For intermediate uniformity coefficients (3 C u 5, e.g. L), the data for all tested pressures coincide well with the range specified by Seed et al. [5]. For well-graded granular materials (C u 8, e.g. L6, L8) the data for the lower pressures lie below the lower bound of the range of Seed et al. [5], in particular at higher shear strain amplitudes. The influence of the mean grain size, d 5, on the modulus degradation curves is inspected in Figure 3a which compares the curves G(γ)/G max measured for the sands and gravels L to L8, having the same uniformity coefficient C u.5 but different mean grain sizes in the range. d 5 6 mm. All samples were medium dense and the mean pressure was 4 kpa in all tests. No clear tendency can be observed in Figure 3a. For a closer examination, the G/G max -data for certain values of the shear strain amplitude are plotted versus d 5 in Figure 3b,c. The data are provided for p and 4 kpa. A small tendency for G/G max to decrease with increasing d 5 can be concluded from Figure 3b,c. However, this slight d 5 -dependence can be neglected for practical purpose. The influence of the uniformity coefficient, C u, on the modulus degradation curves is examined in Figure 3d. It compares the G(γ)/G max curves measured for the materials and L7 to, having the same mean grain size d 5 mm but different uniformity coefficients in the range.5 C u 8. The data are provided for a mean pressure p kpa. Obviously, the modulus degradation is larger for higher C u -values. This becomes even clearer from Figure 3e,f where the G/G max -data for certain values of the shear strain amplitude are plotted versus C u. For example, for a pressure p kpa and a shear strain amplitude γ 4, the ratio G/G max drops from about.7 for C u.5 to slightly above.4 for C u 5.9 (Figure 3e). The decrease of G/G max with increasing C u is similar for the three tested mean grain sizes d 5.,.6 and mm (Figure 3e,f). Based on these test results it seems indispensable to consider the influence of the uniformity coefficient in empirical formulas describing modulus degradation. 4.. Inspection of Hardin s equation () The applicability of Eq. () has been inspected in Figure e-h, where the G/G max data measured in the four tests with different pressures are plotted versus a normalized shear strain amplitude γ/γ r. The data are provided for the four materials, L, L6 and L8 but look similar for the other tested materials. The reference shear strain, γ r, was calculated from Eq. (3), with the maximum shear modulus, G max, measured in the test. The shear strength was obtained from τ max p sin ϕ P with the pressure, p, applied in the test and where ϕ P is the peak friction angle. The peak friction angle was calculated from the relationship ϕ P (e ) derived from the monotonic triaxial tests, with e being the 4

5 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 a). p 4 kpa b) c) γ. p kpa D -4 r Sand / d 5 [mm] L /. L /. L3 /.35 /.6 L5 /. / L7 / 3.5 C u L8 / 6 D r Mean grain size d 5 [mm] Shear strain amplitude d 5 [mm] L7/ d)..6 L8 γ e) 5 p kpa f). p kpa p 4 kpa D r γ Mean grain size d 5 [mm] p 4 kpa d 5 [mm]..6 L7/ L8 γ Sand / C u /.5 L7 / L8 /.5 L9 / 3 L / 4 L / 5 / 6 / 8 d 5 mm D r D r D r Shear strain amplitude 5 5 Uniformity coefficient C u d 6 / d [-] Uniformity coefficient C u d 6 / d [-] Fig. 3: Dependence of modulus degradation on a)-c) mean grain size d 5 and d)-f) uniformity coefficient C u actual void ratio of the test sample at pressure p. As obvious from Figure e-h, the curves G(γ/γ r )/G max for different pressures fall together. The prediction by Eq. () has been added as solid curves in Figure e-h. While Eq. () fits well for poorly graded sands (e.g. ), it significantly underestimates the modulus degradation for well-graded granular materials (e.g. L6 and L8) Extension of Hardin s equation () Eq. () with b could be fitted well to the data of all tested materials (dot-dashed curves in Figure e-h). The a parameter of Eq. () is plotted versus the uniformity coefficient in Figure 4a. The nearly linear increase of a with ln(c u ) can be approximated by (solid line in Figure 4a): a.7 ln(c u ) (3) 4..4 Modification of Hardin s equation () Based on the experimental results in this study, a simple modification of Eq. () was found suitable as well (dashed curves in Figure e-h): G G max + d γ/γ r (4) The curve-fitting parameter d of Eq. (4) is shown as a function of C u in Figure 4b. The following correlation between d and C u could be established (solid line in Figure 4b): d ln(c u ) (5) 4..5 Normalization with p/p atm instead of γ r The diagrams in Figure i-l demonstrate that the normalization of the shear strain amplitude with a factor p/p atm as proposed by Hardin & Kalinski [7] is also suitable. For each tested material, the data G(γ/ p/p atm )/G max for different pressures fall together into a narrow band which can be approximated by Eq. () with b and with p/p atm instead of γ r (dashed curves in Figure i-l). The resulting a parameter of Eq. () is plotted versus C u in Figure 4c. The increase of a with C u can be described by (solid line in Figure 4c): a ln(c u ) (6) The curves obtained from a curve-fitting of Eq. (4) with p/patm instead of γ r are very similar to the dashed curves in Figure i-l. The correlation between the parameter d of Eq. (4) and the uniformity coefficient can be sufficiently well described by Eq. (6) with d instead of a Extension of Stokoe et al. [8] equations (4) and (5) A curve-fitting of Eq. (4) to the G(γ)/G max -curves of all tested materials delivered the reference shear strain, γ r γ(g/g max.5), and the curvature parameter, α. The parameter α neither depends on density nor on pressure. The latter is in accordance with Zhang et al. [4]. Subsequently γ r was plotted versus mean pressure p and Eq. (5) was fitted to these data, delivering the parameters γ r and k. In Figure 4d-f the parameters α, γ r and k obtained for the various materials are plotted as a function of the uniformity coefficient. The slight increase of α with increasing C u (Figure 4d) can be neglected for practical purpose. It is recommended to use the mean value α.3. The parameter γ r strongly decreases with increasing C u (Figure 4e) which can be described by (solid curve in Figure 4e): γ r exp[.59 ln(c u )] (7) 5

6 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 a) 3.5 b) 4 c) 8 Parameter a of Eq. () [-] d 5. mm d 5.6 mm d 5. mm L7 / L8 Eq. (3) Parameter d of Eq. (4) [-] Uniformity coefficient C u [-] Uniformity coefficient C u [-] Uniformity coefficient C u [-] d) e) 8 f) Parameter of Eq. (4) [-] r of Eq. (5) [ -4 ] d 5. mm d 5.6 mm. k.4 d 5. mm.6 L7 / L8 Eq. (7) Uniformity coefficient C u [-] Uniformity coefficient C u [-] Uniformity coefficient C u [-] Parameter g) h) i) Peak friction angle ϕ P [ ] 5 L L L3 45 L5 L7 4 L8 35 L L L L3 L5 L6 L7 L8 L9 L L Eq. (9) L4 L5 L6 L7 L Initial relative density D r [-] Parameter c of Eq. (8) [-] Eq. (5) Eq. (3) d 5. mm d 5.6 mm d 5. mm L7 / L8 d 5. mm d 5.6 mm d 5. mm L7 / L8 d 5. mm d 5.6 mm d 5. mm L7 / L8 Parameter a of Eq. () [-] Parameter k of Eq. (5) [-] Parameter c of Eq. (8) [-] Uniformity coefficient C u [-] Uniformity coefficient C u [-] Eq. (6) Eq. (4) d 5. mm d 5.6 mm d 5. mm L7 / L8 d 5. mm d 5.6 mm d 5. mm L7 / L8 d 5. mm d 5.6 mm d 5. mm L7 / L8 Fig. 4: Correlation of the parameters of various empirical equations with the uniformity coefficient C u: a) Parameter a in Eq. (), b) Parameter d in Eq. (4), c) Parameter a in Eq. () with p/p atm instead of γ r, d) Parameter α in Eq. (4), e,f) Parameters γ r and k in Eq. (5), g) Peak friction angle ϕ P versus initial relative density D r, h),i) Parameters c and c in Eq. (8) The parameter k shows a significant scatter when plotted versus C u (Figure 4f). For a practical application it is recommended to use the mean value k.4 (solid line in Figure 4f). The error made by this assumption increases with increasing pressure. It has been inspected for the largest tested pressure p 4 kpa and a shear strain amplitude γ 5 4, using α.3 and γ r according to Eq. (7). The maximum or minimum k values.6 and. were obtained for sands with C u or C u 6, respectively (see Figure 4f). For C u, the predicted G/G max values are.67 for k.6 and.6 for k.4. For C u 6, the values are.37 for k. and.44 for k.4. These differences are considered small enough to justify the assumption of a constant k Estimation of reference shear strain, γ r For an application of Eq. (), one also needs an estimate of the reference shear strain, γ r τ max /G max. The smallstrain shear modulus can be obtained from the correlations proposed by Wichtmann and Triantafyllidis [35]. For cohesionless soils, the maximum shear stress, τ max, can be calculated from: ( ) ( ) + K K τ max σ v sin ϕ P (8) with the vertical effective stress, σ v, and the lateral stress coefficient, K σ h /σ v. Figure 4g collects the data of the peak friction angle, ϕ P, as a function of initial relative density D r for all tested materials. The ϕ P -values at D r were obtained from the inclination of a loosely pluviated cone of sand while all other ϕ P -values were obtained from drained monotonic triaxial tests. No clear dependence of ϕ P on the mean grain size, d 5, or on the uniformity coefficient, C u, can be detected from the data in Figure 4g. The formula ϕ P 34. exp(.7 D r.8 ) (9) (solid curve in Figure 4g) can be used for an estimation of ϕ P if no triaxial test data are available. Alternatively, Eq. (5) can be applied to estimate γ r used in Eq. (). Based on the data of the present study, the parameter γ r of Eq. (5) is almost independent of C u. The relationship between γ r and initial relative density can be 6

7 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 Predicted Eq. () with b and a from Eq. (3) L L L Predicted - Measured. -. L3 L5 L6 Predicted Eq. (4) with d from Eq. (5) L L L L3 L5 L6 Predicted Eq. () with b, (p/p atm ).5 instead γ r and a from Eq. (6) L L L L3 L5 L6 Predicted Eq. (4) with (p/p atm ).5 instead γ r and d from Eq. (6) L L L L3 L5 L6 Predicted Eqs. (4) and (5) with α.3, k.4 and γ r from Eq. (7) L L L L3 L5 L Measured Measured Measured Measured Measured Predicted L7 L8 L9 L L Measured Predicted L7 L8 L9 L L Measured Predicted..8 L7 L8 L9.6 L.4 L Measured Predicted..8 L7 L8 L9.6 L.4 L Measured Predicted L7 L8 L9 L L Measured Fig. 5: Predicted versus measured values of G/G max for materials with d 5.6 mm (upper row) or d 5 mm (lower row) and various C u-values. For each material the data from the four tests with different pressures and medium-dense specimens are shown. described by γ r (D r ) 6. 4 [.4(D r.6)] () The decrease of the exponent k of Eq. (5) with increasing uniformity coefficient can be approximated by: k.6.9 ln(c u ) () 4..8 Summary of equations and correlations for G/G max The C u -dependent modulus degradation can be estimated from five different sets of equations:. Eq. () with b and a from Eq. (3). Eq. (4) with d from Eq. (5) 3. Eq. () with b, with p/p atm instead of γ r and with a from Eq. (6) 4. Eq. (4) with p/p atm instead of γ r and with d a from Eq. (6) 5. Eqs. (4) and (5) with α.3, k.4 and γ r from Eq. (7) The quality of prediction is inspected in Figure 5, where the predicted G/G max -values are plotted versus the measured ones. Most data points plot close to the bisecting line. Using the first set of equations in the list above, the differences between predicted and measured G/G max data (analyzed for the experimental data with G/G max <.9) are less than.5 in 86% of cases, while they lie between.5 and. in 3% of cases. For the second, third, fourth and fifth sets of equations these values are 85/4%, 86/4%, 86/4% and 87/%, respectively. Therefore, the prediction quality of the different sets of equations is quite similar. These observations are further inspected in Section 5. by means of literature data. 4. Damping ratio Curves of damping ratio, D, versus shear strain amplitude, γ, are given exemplary for sand L in Figure 6. For a given shear strain amplitude, D decreases with increasing pressure while it is nearly independent of density. The data of the present study agrees well with the D-values reported by Kokusho [] for Toyoura sand and similar pressures (gray shading in Figure 6). The data lie at the lower bound of the range of typical damping ratios reported by Seed et al. [5]. Damping ratio [-] Sand L d 5 mm, C u 4 Seed et al. (986) (typical range for sands) Kokusho et al. (98) (p 5-3 kpa) Shear strain amplitude [-] p [kpa] / D r 4 /.65 4 /.7 4 /.8 4 /.9 4 /. 5 /.63 /.65 /.66 Fig. 6: Damping ratio D versus shear strain amplitude γ for different mean pressures and relative densities 4.. Influence of d 5 and C u The curves D(γ) are not significantly affected by the mean grain size. This is evident in Figure 7a, where the curves D(γ) of the sands and gravels L to L8 are compared. All curves were measured in tests on medium-dense samples at p kpa. A closer inspection of the d 5 -influence is undertaken in Figure 7b,c, where the damping ratio, D, is plotted as a function of d 5. The data are provided for medium 7

8 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 a) Damping ratio d) Damping ratio p kpa Sand / d 5 [mm] L /. L /. L3 /.35 /.6 L5 /. / L7 / 3.5 L8 / Shear strain amplitude [-] p 4 kpa Sand / C u /.5 L7 / L8 /.5 L9 / 3 L / 4 L / 5 / 6 / 8 C u.5 D r d 5 mm, D r Shear strain amplitude [-] b) Damping ratio.6.4. e).8 Damping ratio.6.4. p kpa D r γ Mean grain size d 5 [mm] p 5 kpa d 5 [mm]..6 5 Uniformity coefficient C u d 6 / d [-] c) Damping ratio Damping ratio.4.. p 4 kpa.79/ f) γ 5-5 p 4 kpa -4.4 D r Mean grain size d 5 [mm] D r D r Uniformity coefficient C u d 6 / d [-] Fig. 7: Dependence of damping ratio D on a)-c) mean grain size d 5 and d)-f) uniformity coefficient C u dense samples and for different pressures and shear strain amplitudes. For small pressures, D slightly decreases with d 5 (Figure 7b) while it is nearly independent of d 5 at larger pressures (Figure 7c). For practical purposes, this slight variation of D with d 5 can be neglected in empirical formulas. Figure 7d demonstrates that, in contrast to the modulus degradation curves, the curves D(γ) are hardly influenced by the uniformity coefficient, C u. The curves D(γ) measured for materials with different C u -values lie close to each other. The diagrams in Figure 7e,f allow a closer inspection and reveal that the C u -influence on D is somewhat ambiguous. D is almost independent of C u for small pressures and small shear strain amplitudes (Figure 7e). For small pressures in combination with larger shear strain amplitudes, D decreases with C u (Figure 7e). An increase of D with increasing C u was observed for large pressures, independent of the shear strain amplitude (Figure 7f). However, with reference to Figure 7d, it seems justified to neglect the C u -influence in empirical equations for D(γ). 4.. Inspection of Hardin s equations (6) and (7) If damping ratio, D, is plotted versus the normalized shear strain amplitude, γ/γ r, with γ r τ max /G max, the curves D(γ/γ r ) for different pressures nearly coincide (Figure 8a). Eq. (6) overestimates the damping ratios measured in the present study (solid curve in Figure 8a), independently of the grain size distribution curve of the tested material. Figure 8b collects the curves D(γ/γ r ) for most of the tested materials. These curves were measured at p 4 kpa in tests on medium dense samples. Obviously, there is hardly any influence of the grain size distribution curve on the D(γ/γ r )- data. The data in Figure 8b can be described by Eq. (7) with b, modified by a minimum damping ratio, D min, at small shear strain amplitudes: D D min D max D min )] γ γ r [ + a exp ( γγr )] () ( γγr + γ γ r [ + a exp with D min.6, D max.3 and a -.64 (solid curve in Figure 8b). No significant influence of pressure and density on D min could be found in the present study Normalization with p/p atm instead of γ r The normalization of the shear strain amplitude with p/patm instead of γ r works well also for the damping ratio. Figure 8c demonstrates that the curves D(γ/ p/p atm ) measured for the various granular materials fall into a narrow band, which can be approximated by Eq. () with p/p atm instead of γ r and with a constant a 843 (solid curve in Figure 8c) Correlation of D D min with G/G max Figure 9 presents plots of D D min versus G/G max, exemplary for the materials, L and having different C u -values. Since the modulus degradation depends on the uniformity coefficient, the correlation between D D min and G/G max is also C u -dependent. For each tested material, the data could be approximated by Eq. (8) (solid curves in Figure 9). In Figure 4h-i the parameters c and c of Eq. (8) are presented as functions of the uniformity coefficient. The decrease of c and the increase of c with C u can be approximated by (solid lines in Figure 4h-i): c.6.74 ln(c u ) (3) c ln(c u ) (4) Eq. (8) with the parameters derived from Khouri [] (dotdashed curve in Figure 9) overestimates the data of the 8

9 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 a). b) Sand. L c). L4 Eq. (6) L5.8.8 L6.8 Damping ratio.6.4. p [kpa] / D r 5 /.67 /.69 /.7 4 /.67 d 5.6 mm C u.5 Damping ratio.6.4. L7 L8 L L L L3 L5 L6 L7 L8 L9 L L p 4 kpa, D r Eq. () Damping ratio.6.4. L L4 L5 L6 L7 L8 L L L L3 L5 L6 L7 L8 L9 L L p 4 kpa, D r Eq. () with (p/p atm ).5 instead of γ r γ / γ r [-] γ / γ r [-] γ / (p/p atm ).5 [-] Fig. 8: a) Damping ratio D as a function of γ/γ r for sand ; Comparison of curves b) D(γ/γ r) and c) D(γ/ p/p atm ) for different materials D - D min [%] d 5 mm C u.5 Sand p [kpa] / D r 4 /.48 4 /.55 4 /.64 4 /.69 4 /.78 4 /.86 4 /.95 5 /.63 /.63 /.66 D - D min [%] 8 p [kpa] / D r 4 /.6 4 /.7 4 / /.9 4 /.99 5 /.64 /.6 4 /.65 d 5 mm C u 5 Sand L D - D min [%] d 5 mm C u 8 Sand p [kpa] / D r 4 /.45 4 /.5 4 /.6 4 /.73 4 /.83 4 /.86 4 /.97 4 /.4 5 /.57 /.63 / best-fit of Eq. (8) Khouri (984) Zhang et al. (5), torsional shear Zhang et al. (5), Fig. 9: Damping ratio D D min as a function of G/G max present study, in particular for the well-graded materials. Eq. (8) with the constants proposed by Zhang et al. [4], either for torsional shear or test data (dashed curves in Figure 9), agrees well with the experimental data of the present study for intermediate uniformity coefficients C u 3. However, the D D min -values for poorly graded sands (C u.5) are slightly underestimated while the values for C u 4 are overestimated Summary of equations and correlations for D The damping ratio can be estimated from three different sets of equations: Eq. () with D min.6, D max.3 and a -.64 Eq. () with p/p atm instead of γ r and with a 843 Eq. (8) with Eqs. (3) and (4) Using Eq. (), the differences between predicted and measured damping ratio data (analyzed for the experimental data with G/G max <.9) are less than. in 8% of cases, while they lie between. and. in 4% of cases. For the second and third set of equations in the list above, these values are 83/5% and 93/6%, respectively. Therefore, the correlation between D D min and G/G max delivers the best approximation of the measured D data. This is also confirmed by Figure where the D values calculated from Eq. (8) are plotted versus the measured ones. a) Predicted b). - Measured L. L L.8 -. Predicted.4 L3 L5 L Measured Predicted.8 L7 L8 L9.4 L L.4.8 Measured Fig. : Damping ratios D predicted by Eq. (8) with Eqs. (3) and (4) versus measured D-values for materials with a) d 5.6 mm and b) d 5 mm and various C u-values. For each material the data from the four tests with different pressures and medium-dense specimens are shown. 4.3 Threshold amplitudes The threshold shear strain amplitude indicating the transition from the linear elastic to the nonlinear elastic behaviour was defined as γ tl γ(g/g max.99) (Vucetic [3]). An increase of γ tl with increasing mean pressure was observed (compare the curves G(γ)/G max in Figure a-d). In Figure b-c, γ tl is plotted versus d 5 or C u, respectively. The given γ tl -values are mean values from the four tests with different pressures. A slight tendency for a decrease of γ tl with in- 9

10 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 a) p [kpa] / D r Sand L b) c) 4 / tv 4 /.5 d 5. mm tl 4 /.66 C u 4 / /.8 4 /.9 4 /. h [mm] Settlement. /.6 /.6 5 / Shear strain amplitude [-] [-] Shear strain amplitude Mean grain size d 5 [mm] [-] Shear strain amplitude -4-5 D r D r d 5 [mm]..6 L7/ L8 5 Uniformity coefficient C u d 6 / d [-] tv tl Fig. : a) Settlement h(γ) for different pressures and densities, shown exemplary for sand L; Threshold shear strain amplitudes γ tl and γ tv in dependence of b) mean grain size d 5 and c) uniformity coefficient C u creasing values of d 5 and C u can be concluded from these diagrams. The mean values from all tested materials were γ tl 4. 6, 6. 6, 8. 6 and. 5 for p 5,, and 4 kpa, respectively. These values lie slightly above the threshold amplitude γ tl 3. 6 proposed by Vucetic [3] for granular materials. Figure a shows typical curves of the settlement of the test specimens as a function of shear strain amplitude. In accordance with cyclic triaxial test data [34], the settlement increased with decreasing pressure, decreasing initial density and with increasing uniformity coefficient, C u, of the tested material. The threshold shear strain amplitude at the onset of settlement, γ tv, does not significantly depend on pressure and density. For each material mean values of γ tv are plotted versus d 5 or C u, respectively, in Figure bc. These diagrams demonstrate that γ tv is almost independent of the grain size distribution curve. The mean value γ tv obtained in the present study (upper solid line in Figure b-c) lies at the lower bound of the range 3 5 γ tv 4. 4 reported by Vucetic [3] for granular materials. 5 Comparison with data from the literature 5. Modulus degradation Several G(γ)/G max curves for clean sands were collected from the literature, together with the tested pressures and void ratios or relative densities. These G(γ)/G max data are shown by the filled symbols or solid lines, respectively, in Figure a-u. Each diagram in Figure belongs to a different sand or test series. In some studies (e.g. Ray & Woods [], Yu [38]) the data are given in terms of a normalized shear strain γ/γ r only. Such data are shown in the last three diagrams (v-x) in Figure. With the given values of p, e or D r and C u, the G(γ)/G max data predicted by the five different sets of equations summarized in Section 4..8 have been calculated and compared with the experimental data. The prediction by Eqs. (4) and (5) with α.3, k.4 and γ r from Eq. (7) is shown as dashed curves in Figure a-u. A good agreement between experimental and predicted data can be concluded for most of the data from the literature. Slightly different curves G(γ)/G max were reported in the literature for the same sand. For example (Figure a-d), for a given shear strain, the modulus degradation measured by Iwasaki et al. [8] ( / torsional shear tests) and Lo Presti et al. [5] () for Toyoura sand is larger than that reported by Kokusho [] (undrained cyclic triaxial tests) and Wang & Tsui [33] (). This may be due to the different test devices, drainage conditions or batches of the tested sand. The prediction of the extended empirical equations derived from the present study is better for the data of Iwasaki et al. [8] and Lo Presti et al. [5] while the modulus degradation in the tests of Kokusho [] and Wang & Tsui [33] is overestimated. For Monterey No. sand (Figure e-g), the data of Iwasaki et al. [8] and Chung et al. [] is predicted well, while the curves G(γ)/G max measured for small pressures by Saxena & Reddy [3] cannot be reproduced. For Ottawa sand (Figure h-i), a good agreement of the prediction with the data of Shen et al. [6] and Hardin & Kalinski [7] (not shown in Figure ) was found, while the modulus degradation observed by Alarcon-Guzman et al. [] is slightly underestimated. The new correlations predict well the G(γ)/G max curves measured for Ticino sand by Lo Presti et al. [6] (Figure j), for clean dry sand by Hardin & Drnevich [6] (Figure l), for several more well-graded sands by Iwasaki et al. [8] (Figure m-p), for Santa Monica beach sand by Lanzo et al. [] (Figure q), for uniform HongKong beach sand by Li et al. [4] (Figure r) and for mortar sand tested by Menq & Stokoe [7]. The differences between measured and predicted data are larger for Leighton Buzzard sand tested by Li & Cai [3] (Figure k), sand studied by Wang & Tsui [33] (not shown in Figure ) and parts of the crushed limestones studied by Hardin & Kalinski [7] (Figure s). In contrast to the extended empirical equations for G max discussed by Wichtmann & Triantafyllidis [35], there is no clear correlation between the quality of prediction and grain shape in Figure. For example, for Ottawa sand and Monterey No. sand with subrounded to rounded grain shape, the modulus degradation is sometimes overestimated (Saxena & Reddy [3], Figure g), but in some other cases underestimated (Alarcon-Guzman et al. [], Figure i). Zhang et al. [4] proposed to apply Eqs. (4) and (5) with parameters α, k and γ r depending on the plasticity index PI and the geological age of the soil. The G(γ)/G max curves of Zhang et al. [4] for PI and quaternary soil can be reproduced approximately with the author s equations for C u. (Figure t). The curves of Zhang et al. [4] for PI and tertiary or residual/saprolite soils are equivalent to the author s equations with C u 4 or.5, respectively. The G(γ)/G max curve proposed for PI by Vucetic [3] can be reproduced well if the new correlations are applied with C u 4 (Figure u). The prediction by the equations using p/p atm as a reference quantity is similar to the dashed curves in Figure a-u. The prediction by the correlations necessitating an estima-

11 Wichtmann & Triantafyllidis b) f) Monterey No..4 d5.44 mm Cu.38, SR-R. p 98 kpa p 5 / 3 kpa p 68 / kpa / torsional shear Ban-nosu sand B.4 d5.9 mm Cu p 98 kpa -4.8 Ban-nosu sand 5.4 d5.88 mm Cu p 98 kpa r) Santa Monica beach sand.4 d5.3 mm Cu.4. p / kpa (K.5 assumed).6.4 Uniform HongKong beach sand Cu.5 (assumed). p 96 kpa.6 s) u).4. PI curves plotted for Cu test data p 5 / 88 kpa torsional shear Ohgi-Shima sand.4 d5.37 mm Cu.5. p 98 kpa / torsional shear Crushed limestone d5 3.3 mm Cu.4, A Crushed limestone d5.9 mm Cu.9, A p 53 kpa p 53 kpa t).4. Quaternary soil.6 PI curves plotted.4 for C. u / torsional shear Mambucaba s. d5.55 mm Cu 3., SA Yu (988) p kpa / torsional shear.. Lubuga sand d5.65 mm Cu 6.7 p kpa. For diagrams v)-x):. γ / γr [-] γ / γr [-] Eqs. (4) + (5) with α.3, k.4, γr from (7) - Zhang et al. (5) -6-4 x). -3. p 5 / 3 kpa Ray & Woods (988) Iwasaki et al. (978).8 Hardin & Kalinski (5).. p kpa / torsional shear -4. Ottawa sand d5.7 mm Cu., SR-R. Clean dry sand Cu. (assumed) Iruma sand.4 d5.5 mm Cu.98. p 98 kpa -6 - Ray & Woods (988) γ / γr [-] For diagrams a)-u): -3 w).. p kpa simple shear -6 v) Vucetic (994) p) Hardin & Drnevich (97). Iwasaki et al. (978) simple shear p / 3 kpa -6 - Li et al. (998). p 78 / 568 kpa free torsional vibration.8 Lanzo et al. (997) Ottawa C-9 Cu. (assumed) SR-R / torsional shear / torsional shear -3 - Leighton Buzzard sand Cu Iwasaki et al. (978) -5-3 o) / torsional shear.4-6 Li & Cai (999) l). n) -6-4 Iwasaki et al. (978) Shen et al. (985) Dr 9 %...8 p 8 kpa Monterey No..4 d5.36 mm Cu.5, SR-R. p 49 / 588 kpa. Ticino sand d5.54 mm Cu.5, SA -6 - m) - Lo Presti et al. (993) h) Saxena & Reddy (989) Dr 5 % k) q) j) Alarcon-Guzman et al. (989) p kpa. Ottawa -3 d5.7 mm Cu., SR-R -3 g) Monterey No. d5.36 mm Cu.5, SR-R -6 - i) Toyoura sand d5.3 mm Cu.4, SA.6. p / 3 kpa -6 - Chung et al. (984) / torsional shear Iwasaki et al. (978) -6.4 Toyoura sand d5. mm Cu.35, SA / torsional shear e).6 Toyoura sand.4 d5.6 mm Cu.46, SA. p 49 / 96 kpa Wang & Tsui (9) p 5 / 3 kpa Cyclic triaxial Lo Presti et al. (997). d)..4 Toyoura sand d5.9 mm Cu.3, SA Iwasaki et al. (978) c). Kokusho (98). a) J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 Eqs. () + (3) Eqs. (4) + (5) Fig. : Comparison of G(γ)/Gmax data for various sands from the literature with the prediction by the extended empirical equations (grain shape: R round, SR subrounded, SA subangular, A angular)

12 Wichtmann & Triantafyllidis J. Geotech. Geoenviron. Eng., ASCE, Vol. 39, No., pp. 59-7, 3 a).5 b).4 c) d). Kokusho (98) Tatsuoka et al. (978). Lo Presti et al. (997). Toyoura sand.3 Toyoura sand Toyoura sand.5 d 5.9 mm d 5.6 mm d.8 5. mm C u.3, SA C. u.46, SA.8 C u.35, SA. p 5 / 3 kpa p 49 / 96 kpa p / 3 kpa cyclic triaxial.4.5 N. torsional.4 shear, N Chung et al. (984) Monterey No. d 5.36 mm C u.5, SR-R p / 3 kpa e). f). g).3 h). Saxena & Reddy (989) Shen et al. (985) Silver & Seed (97) Hardin & D r 5 % Drnevich (97) Monterey No. d 5.36 mm C u.5, SR-R p kpa Ottawa C-9 C u. (assumed), SR-R p kpa free torsional vibration.. Crystal silica sand d 5.66 mm C u.4 p 6-8 kpa (K.5 assum.) simple shear.5..5 Clean dry sand C u. (assumed) p 5-88 kpa torsional shear N i).5 j). k).3 l) Tatsuoka et al. (978) Li et al. (998) Vucetic (994). Ban-nosu sand B.5 Uniform d. PI mm HongKong C curves plotted u beach sand for C u 4. p 98 kpa torsional C u.5 (assumed) shear, p 96 kpa. p kpa N.5 simple shear.5 N Yu (988).. Lubuga sand d 5.65 mm C u 6.7 p kpa... γ/γ r [-] For diagrams a)-k): test data Eq. () with (p/patm).5 instead γ r via G/G max with Eqs. (8), (3), (4) For diagram l): Eq. () with γ r Fig. 3: Comparison of D(γ) data for various sands from the literature with the prediction by the extended empirical equations (grain shape: R round, SR subrounded, SA subangular, A angular) tion of the reference shear strain, γ r, was found slightly more inaccurate, probably due to uncertainties in the γ r -values calculated with estimations of G max and τ max. The diagrams in Figure v-x demonstrate a good congruence between the G(γ/γ r )/G max data of Ray & Woods [] and Yu [38] and the curves calculated from Eqs. () and (4) with the parameters from Eqs. (3) or (5), respectively. However, due to the uncertainties in the estimated γ r -values, for practical purposes it is recommended to apply the extended Stokoe s equation (4) or the formulas using normalization with p/p atm. 5. Damping ratio A similar comparison with the literature has been undertaken for the damping ratio. Some of the curves D(γ) collected from the literature are given as solid lines or filled symbols in Figure 3. The prediction by Eq. () with p/p atm instead of γ r and with a 843 has been added as dashed curves in Figure 3a-k. For some of the test series in the literature (e.g. test data for several sands measured by Tatsuoka et al. [3], Figure 3b,i, Shen et al. [6], Figure 3f, Silver & Seed [7], Figure 3g, Hardin & Drnevich [6], Figure 3h, Li et al. [4], Figure 3j), the prediction by Eq. () with p/p atm is satisfying, while in some other cases the experimental data are underestimated (torsional shear test data of Tatsuoka et al. [3], Figure 3b,i, simple shear tests of Vucetic et al. [3]) or overestimated (Chung et al. [], Figure 3d, Saxena & Reddy [3], Figure 3e, Wang & Tsui [33], Khan et al. [9]). There is no clear tendency with grain shape or type of test. The differences could be due to equipmentgenerated damping (see Stokoe et al. [9], not determined in the present test series) or due to a different number of applied cycles, N. A significant decrease of D with N was found by several researchers (e.g. Hardin & Drnevich [5], Li & Cai [3]). The curve D(γ) proposed by Vucetic [3] for PI (Figure 3k) is approximately obtained with the same C u 4 that was necessary to reproduce the G(γ)/G max data in Figure u. Similarly, the damping data of Zhang et al. [4] for PI is well reproduced with the same C u -values applied for the shear modulus degradation. Although the data D(γ/γ r ) shown by Ray & Woods [] and Yu [38] is well approximated by Eq. () with D min.6, D max.3 and a -.64 (Figure 3l), for most of the test series from the literature the prediction of Eq. () with γ r as a reference quantity was found less reliable, probably due to an inaccurate estimation of the reference shear strain, γ r. If G/G max measurements were available in the literature, these values were used to calculate the prediction of Eq. (8) with (3) and (4). As shown by the dot-dashed curves in Figure a-k, in most cases this prediction is closer to the experimental data than that of Eq. () with p/p atm. Therefore, Eq. (8) with the correlations (3) and (4) is recommended for a practical application.