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5th International Conference on Earthquake Geotechnical Engineering, January 2011, 10-13, Santiago, Chile.

ESTIMATION OF THE SMALL-STRAIN STIFFNESS OF GRANULAR SOILS TAKING INTO ACCOUNT THE GRAIN SIZE DISTRIBUTION CURVE

Torsten WICHTMANNi) Miguel NAVARETTE HERNANDEZii) Rafael MARTINEZiii) , , , iv) v) Francisco DURAN GRAEFF, Theodor TRIANTAFYLLIDIS

ABSTRACT Approximately 650 resonant column (RC) tests with additional P-wave measurements have been performed on 64 specially mixed grain size distribution curves of a quartz sand. The tested materials had different mean grain sizes d 50 , coefficients of uniformity Cu = d60 /d10 and fines contents F C. In a first test series, grain size distribution curves with a linear shape in the semi-logarithmic scale were studied. The RC tests revealed that the small-strain shear modulus Gmax and the constrained elastic modulus M max are independent of d50 , but strongly decrease with increasing Cu and F C. Furthermore, the test results showed that Hardin's equation with its commonly used constants can significantly over-estimate the small-strain shear modulus of well-graded granular soils. Therefore, this empirical equation has been extended by the influence of Cu and F C. A similar set of equations has been developed for M max . Correlations for the modulus degradation factor G()/Gmax are also proposed. A second test series was performed with piecewise linear, gap-graded, S-shaped and other smoothly shaped grain size distribution curves. It is demonstrated that the new correlations work well also for these "more complicated" grain size distribution curves. Keywords: small-strain stiffness, granular material, grain size distribution curve, resonant column tests INTRODUCTION For feasibility studies, preliminary design calculations or final design calculations in small projects dynamic soil properties are often estimated by means of empirical formulas. The secant shear modulus G is usually described as a product of its maximum value G max at very small shear strain amplitudes < 10-6 and a modulus degradation factor F (), i.e. G = G max F (). A widely used empirical formula for the small strain shear modulus G max of sand has been proposed by Hardin and Richart [4] and Hardin and Black [1] which is given in its dimensionless form here: Gmax = A

i)

(a - e)2 (patm )1-n pn 1+e

(1)

Research assistant, Institute of Soil Mechanics and Rock Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany, e-mail: [email protected] ii) Project Engineer, Electroandes Geotecnia Minerva, Providencia, Chile, e-mail: [email protected] iii) Master student, Imperial College London, e-mail: [email protected] iv) Project Engineer, RAG Deutsche Steinkohle, Herne, Germany, e-mail:[email protected] v) Professor and Director, Institute of Soil Mechanics and Rock Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany, e-mail: [email protected]

1

2

Wichtmann et al.

with void ratio e, mean pressure p and atmospheric pressure p atm = 100 kPa. The constants A = 690, a = 2.17 and n = 0.5 for round grains, and A = 320, a = 2.97 and n = 0.5 for angular grains were recommended by Hardin and Black [1] and are often used for estimations of Gmax -values for various sands. Hardin and Drnevich [2] proposed the following function for the modulus degradation factor F (): F () = 1

r 1 + a exp -b r

(2)

with a reference shear strain r = max /Gmax and two constants a and b. max is the shear strength. Eq. (1) with its commonly used constants does not consider the strong dependence of the small strain shear modulus of granular soils on the grain size distribution curve. A respective literature review has been given by Wichtmann and Triantafyllidis [8]. For example, Iwasaki and Tatsuoka [5] demonstrated that Gmax does not depend on the mean grain size d 50 but strongly decreases with increasing coefficient of uniformity C u = d60 /d10 and with increasing fines content F C. Iwasaki and Tatsuoka [5] performed a single test on each sand. They did not propose an extension of Eq. (1) by the influence of C u and F C. However, their experiments demonstrated that Hardins equation (1) with its commonly used constants can significantly overestimate the small-strain stiffness of well-graded granular materials. An extension of Eq. (1) considering the influence of the grain size distribution curve is the purpose of the present study. TESTED MATERIAL, TEST DEVICE, SAMPLE PREPARATION, TESTING PROCEDURE A natural quartz sand was sieved into 25 single gradations with grain sizes between 0.063 mm and 16 mm. The grains have a subangular shape and the specific weight is s = 2.65 g/cm3 . From these gradations the grain size distribution curves shown in Figure 1 and in the first column of Figures 14 and 15 were mixed. The grain size distribution curves of the materials L1 to L28 (Figure 1) are linear in the semi-logarithmic scale and contain no fines. The eight sands or gravels L1 to L8 (Figure 1a) were used to study the d 50 -influence. These materials had different mean grain sizes in the range 0.1 mm d 50 6 mm and the same coefficient of uniformity C u = 1.5. The Cu -dependence was examined in tests on the materials shown in Figure 1b. The mean grain sizes of the sands L10 to L26 were d 50 = 0.2, 0.6 or 2 mm, respectively, while the coefficients of uniformity varied in the range 2 C u 8. Two sand-gravel mixtures (L27 and L28, Figure 1c) with larger coefficients of uniformity (C u = 12.6 or 15.9) were also tested. The influence of the fines content (defined according to German standard as the percentage of grains with diameters d < 0.063 mm) was tested by means of the six grain size distribution curves F1 to F6 shown in Figure 1c. These materials have fines contents in the range 0 % F C 20 %. A quartz meal was used for the fines. In the range d > 0.063 mm, the grain size distribution curves of the sands F1 to F6 are parallel to those of the materials L1 to L8 (C u = 1.5). The resonant column (RC) device used for the present study has been explained in detail by Wichtmann and Triantafyllidis [8]. The device belongs to the "free-free" type, that means both, the top and the base mass are freely rotatable. Cylindrical specimens with a diameter of 10 cm and a height of 20 cm were tested. In order to measure the P-wave velocity the end plates of the RC device have been additionally equipped with a pair of piezoelectric elements. The measuring technique and the analysis of the signals have been presented by Wichtmann and Triantafyllidis [9]. The samples were prepared by air pluviation and tested in the dry condition. For each grain size distribution curve several samples with different initial relative densities I D0 = (emax -

Small-strain stiffness of granular soils taking into account the grain size distribution curve

a)

100

fine Sand med. coarse Gravel fine med.

3

b)

fine

Sand med.

coarse

Gravel fine med.

c)

Silt coarse

fine

Sand med.

coarse

Gravel fine med.

Finer by weight [%]

80 60 40 20 0 0.06 0.2 0.6 2 6 20 0.06

L1 L2

L3 L4

L5 L6

L7 L8

L26 L25 L24 L2

L19 L18 L17 L6 6 20

Finer b.w. [%]

L12 L11 L10 L4 L13 L14 L15 L16

L20 L21 L22 L23

F6 F5 F4 F3 F2 F1 0.02 0.06

L27 L1

L28

For Eqs. (6)-(8) + (14)-(16): FC > 10 % 1

Cu

d>0.063

0.063 Grain size [mm]

Grain size [mm]

0.2

Grain size [mm]

0.6

2

0.2

Grain size [mm]

0.6

2

6

20

Fig. 1: Tested grain size distribution curves with linear shape

e)/(emax - emin ) were tested. In each test the isotropic stress was increased in seven steps from p = 50 to 400 kPa. At each pressure the small-strain shear modulus G max and the P-wave velocity vP were measured. At p = 400 kPa the curves of shear modulus G and damping ratio D versus shear strain amplitude were determined. In three additional tests on medium dense samples, the curves G() and D() were also measured at smaller pressures p = 50, 100 and 200 kPa.

TEST RESULTS AND CORRELATIONS FOR LINEAR GRAIN SIZE DISTRIBUTION CURVES Influence of d50 and Cu on Gmax Figure 2 presents results of the RC tests performed on the materials L1 to L8 with C u = 1.5 and with different mean grain sizes in the range 0.1 d 50 6 mm. Similar as in the tests of Iwasaki and Tatsuoka [5], for constant values of void ratio and mean pressure, no dependence of Gmax on d50 could be found. The gravel L8 showed slightly lower G max -values which can be explained with an insufficient interlocking between the tested material and the end plates which were glued with coarse sand (Martinez [6]).

300

Shear modulus Gmax [MPa]

Shear modulus Gmax [MPa]

250 200 150 100 50 0 0.50

Sand / d50 = L1 / 0.1 mm L2 / 0.2 mm L3 / 0.35 mm L4 / 0.6 mm

L5 / 1.1 mm L6 / 2 mm L7 / 3.5 mm L8 / 6 mm

300

e = 0.55

250 200 150 100 50 0

d50 = 0.2 mm d50 = 0.6 mm d50 = 2 mm

p [kPa] = 400 200 100 50

p=

400 kPa

p = 10 0 kPa

0.60

0.70

0.80

0.90

1

2

3

4

5

6

7

8

9

Void ratio e [-]

Coefficient of uniformity Cu [-]

Fig. 2: No dependence of Gmax on mean grain size d50 , Wichtmann and Triantafyllidis [8]

Fig. 3: Decrease of Gmax with increasing coefficient of uniformity Cu , data for a constant void ratio e = 0.55, Wichtmann and Triantafyllidis [8]

Figure 3 demonstrates the significant decrease of the small-strain shear modulus G max with increasing coefficient of uniformity. It was measured in the RC tests on the sands L24 to L26 (d50 = 0.2 mm and 2 Cu 3), L10 to L16 (d50 = 0.6 mm and 2 Cu 8) and L17 to L23 (d50 = 2 mm and 2 Cu 8). For same values of void ratio and pressure, G max at Cu = 1.5 is approximately twice larger than at C u = 8. Hardin's equation (1) with its commonly

4

Wichtmann et al.

used constants overestimates the G max -values of well-graded granular materials while the shear modulus of uniform sands can be underestimated (Wichtmann and Triantafyllidis [8]). The parameters A, a and n of Eq. (1) have been obtained from a curve-fitting of Eq. (1) to the Gmax -data of each sand. Figure 4 shows A, a and n as functions of the coefficient of uniformity Cu . The following correlations could be established (solid lines in Figure 4): a = 1.94 exp(-0.066Cu ) n = 0.40 (Cu )

0.18 2.98

(3) (4) (5)

c) 4

Parameter A [103]

3 2 1

Sands with d50 = 0.2 mm 0.6 mm 2 mm

A = 1563 + 3.13 (Cu )

a) 3

Parameter a [-]

2 1 0

Parameter n [-]

Sands with d50 = 0.2 mm 0.6 mm 2 mm

b) 0.7

0.6 0.5 0.4 0.3 1 2 3 4 5

Sands with d50 = 0.2 mm 0.6 mm 2 mm

1

2

3

4

5

6

7

8

9

6

7

8

9

1

2

3

4

5

6

7

8

9

Coefficient of uniformity Cu [-]

Coefficient of uniformity Cu [-]

Coefficient of uniformity Cu [-]

Fig. 4: Correlations of the parameters A, a and n of Eq. (1) with the coefficient of uniformity C u , Wichtmann and Triantafyllidis [8]

In Figure 5 the shear moduli predicted by Eq. (1) with the correlations (3) to (5) are plotted versus the measured values. All data plot close to the bisecting line (also for the sand-gravel mixtures L27 and L28 with Cu -values up to 16), confirming the good approximation of the experimental data by the extended Hardin's equation. Wichtmann and Triantafyllidis [8] have demonstrated that Eq. (1) with (3) to (5) also predicts well the shear moduli for various sands documented in the literature. Based on the RC test results a correlation of G max with relative density Dr could be also established (Wichtmann and Triantafyllidis [8]). It is less accurate than Eq. (1) with (3) to (5) but may suffice for practical purposes.

300 200 100

d50 = 0.2 mm

Sand L2 L24 L25 L26

Predicted Gmax [MPa]

300 200 100

Sand L4 L10 L11 L12

L13 L14 L15 L16

300 200 100

Sand L6 L17 L18 L19

L20 L21 L22 L23

400 300 200

Sand L27 L28

d50 = 0.6 mm

d50 = 2 mm

100 300 0 0 100 200 300 400

0 0

100

200

300

0

0

100

200

300

0

0

100

200

Measured Gmax [MPa]

Measured Gmax [MPa]

Measured Gmax [MPa]

Measured Gmax [MPa]

Fig. 5: Comparison of the shear moduli Gmax predicted by Eqs. (1) and (3) to (5) with the experimental data, Wichtmann and Triantafyllidis [8]

Influence of fines content on Gmax Figure 6 demonstrates the strong decrease of G max with increasing fines content for F C 10 %. This figure shows the data for a constant void ratio e = 0.825 obtained from RC tests performed on sands F1 to F6. For the same void ratio and the same pressure, the G max -values for a clean

Small-strain stiffness of granular soils taking into account the grain size distribution curve

160

5

Shear modulus Gmax [MPa]

Approx. by Eqs. (1), (6) to (8) (1), (3) to (5), (9)

120

p [kPa] = 150 100 75 50

10

400 300 200

8

Parameter a [-]

6

Sand F1 Sand F2 Sand F3 Sand F4 Sand F5 Sand F6

80

4

40

2

a = 1.76 exp(0.065 FC) a = 1.76 for clean sands with Cu = 1.5

clean sands

0

0

0 5 10 15 20 25

0

5

10

15

20

25

Fines content FC [%]

Fines content FC [%]

Fig. 6: Decrease of small-strain shear modulus Gmax with increasing fines content, data for a constant void ratio e = 0.825

Fig. 7: Parameter a of Eq. (2) as a function of fines content

sand (F C = 0) are about twice larger than the shear moduli for a sand with a fines content of 10 %. The following extension of the correlations (3) to (5) by the influence of the fines content is proposed (see the exemplary plot of a versus F C in Figure 7): a = 1.94 exp(-0.066Cu ) exp(0.065F C) n = 0.40 (Cu )

0.18

(6) (7)

1.10

[1 + 0.116 ln (1 + F C)]

2.98

A = 0.5[1563 + 3.13 (Cu )

][exp(-0.30F C

) + exp(-0.28F C

0.85

)]

(8)

For the parameter A, a very flexible function is necessary. For fines contents F C > 10 %, the d>0.063 of the grain size distribution curve in the range of grain sizes d > 0.063 mm inclination Cu (see the scheme in Figure 1c) should be set into Eqs. (6) to (8) for C u . The solid curves in Figure 6 were generated with Eqs. (1) and (6) to (8). They confirm the good approximation of the test data by the new correlations. As an alternative, the G max -values obtained for clean sand from Eqs. (1) and (3) to (5) can be reduced by a factor f r depending on fines content (see the less accurate prediction given as dashed curve in Figure 6): fr (F C) = 1 - 0.043F C for F C 10% 0.57 for F C > 10% (9)

Based on the RC test results, for the sands containing fines, G max could not be correlated with relative density. Influence of d50 and Cu on Mmax The constrained elastic modulus Mmax = (vP )2 was calculated from the P-wave velocity. Similar to Gmax , for constant values of void ratio and pressure, M max does not depend on the mean grain size (Figure 8) but decreases with increasing coefficient of uniformity (Figure 9). For each tested material, Eq. (1) with M max instead of Gmax has been fitted to the experimental data: Mmax = A (a - e)2 (patm )1-n pn 1+e (10)

6

1000

Wichtmann et al.

p 40 = 0k Pa

Sand / d50 [mm] = L5 / 1.1 L1 / 0.1 L6 / 2 L2 / 0.2 L7 / 3.5 L3 / 0.35 L4 / 0.6

1000 d50 = 0.2 mm d50 = 0.6 mm d50 = 2 mm

Constrained modulus Mmax [MPa]

Constrained modulus Mmax [MPa]

800 600 100 400

800

kPa

600

p [kPa] = 400

400

100

200

200

0 0.50

0

0.60

0.70

0.80

0.90

0

1

2

3

4

5

6

7

8

9

Void ratio e [-]

Coefficient of uniformity Cu [-]

Fig. 8: No influence of mean grain size d50 on constrained elastic modulus Mmax , Wichtmann and Triantafyllidis [9]

Fig. 9: Decrease of constrained elastic modulus Mmax with increasing coefficient of uniformity, Wichtmann and Triantafyllidis [9]

The following correlations between the parameters A, a and n of Eq. (10) and C u could be derived: a = 2.16 exp(-0.055 Cu ) n = 0.344(Cu )

0.126 2.42

(11) (12) (13)

A = 3655 + 26.7(Cu )

The Mmax -values calculated from Eqs. (10) to (13) agree well with the experimental data (Wichtmann and Triantafyllidis [9]). The correlation between M max and relative density is rather rough. With Gmax from Eqs. (1) and (3) to (5) and Mmax from Eqs. (10) to (13), Poisson's ratio can be calculated. For constant values of void ratio and pressure, increases with increasing Cu (Wichtmann and Triantafyllidis [9]).

Influence of fines content on Mmax Similar to Gmax , the constrained elastic modulus M max decreases with increasing fines content, at least in the range F C 10 % (Figure 10). Therefore, the correlations (11) to (13) have been extended by the influence of F C: a = 2.16 exp(-0.055 Cu )(1 + 0.116F C) n = 0.344(Cu )

0.126

(14) (15)

1.10

[1 + 0.125 ln (1 + F C)]

2.42

A = 0.5[3655 + 26.7(Cu )

][exp(-0.42F C

) + exp(-0.52F C

0.60

)]

(16)

The solid curves in Figure 10 were generated with Eqs. (10) and (14) to (16). They approximate the experimental data well. For a simplified procedure, the M max -value obtained for clean sand from Eqs. (10) to (13) can be reduced by a factor f r (see the prediction given as dashed curve in Figure 10): fr (F C) = 1 - 0.041F C for F C 10% 0.59 for F C > 10% (17)

The small dependence of Poisson's ratio on fines content can be neglected for practical purposes.

Small-strain stiffness of granular soils taking into account the grain size distribution curve

500

7

Constrained modulus Mmax [MPa]

Approx. by Eqs. (10), (14) to (16) (10) to (13), (17)

400

p [kPa] = 150 100 75 50

400 300 200

1.1 1.0 0.9

Sand L11

G/Gmax [-]

300

0.8 0.7 0.6

200

p [kPa] / ID0 = 50 / 0.60 100 / 0.61 200 / 0.61 400 / 0.60

100

clean sands

0.5 0.4 -7 10

typical range of curves according to Seed et al. (1986)

0

0

5

10

15

20

25

10

-6

10

-5

10

-4

10

-3

Fines content FC [%]

Shear strain amplitude [-]

Fig. 10: Decrease of small-strain constrained elastic modulus Mmax with increasing fines content, data for a constant void ratio e = 0.825

Fig. 11: Typical curves G()/Gmax for four different pressures, shown exemplary for sand L11

Influence of d50 and Cu on the curves G()/Gmax and D() Figure 11 presents typical curves G()/G max measured in four tests with different pressures. In accordance with the literature, the modulus degradation is larger for smaller pressures. The curves G()/Gmax measured in the present study fall in the range of typical values specified by Seed et al. [7] (gray region in Figure 11). An influence of density on the curves G()/G max could not be detected. The modulus degradation becomes larger with increasing coefficient of uniformity. This is evident in Figure 12: For a certain value of shear strain amplitude , the ratio G/G max decreases with increasing Cu . No significant influence of the mean grain size on the curves G()/G max could be found. For each material, G/G max was plotted versus the normalized shear strain amplitude /r and Eq. (2) was fitted to the data. The reference shear strain r was determined from monotonic triaxial tests. The parameter b in Eq. (2) was set to 1 which is sufficient in order to describe the modulus degradation curves (see also Hardin and Kalinski [3]). The relationship between the parameter a in Eq. (2) and the coefficient of uniformity C u (Figure 13) can be described by: a = 1.070 ln (Cu ) (18)

Damping ratio D increases with decreasing pressure, but does not depend on density. A comparison of the damping ratios measured for the sands L1 to L8 revealed that D does not significantly depend on mean grain size. The influence of the coefficient of uniformity on D depends on shear strain amplitude and pressure. For larger pressures, D increases with C u , independently of the shear strain amplitude. For smaller pressures, D is almost independent of C u if the shear strain amplitudes is small, while a decrease of D with C u was observed at larger -values. During the increase of shear strain amplitude the axial deformation of the samples was measured. From this data the threshold shear strain amplitude tv at the onset of settlement could be determined. The threshold shear strain amplitude tl at the transition from the linear to the nonlinear elastic behavior was defined as the amplitude for which the shear modulus has decreased to 99 % of its initial value, i.e. G = 0.99G max . The threshold amplitudes tl and tv neither depend on d50 nor on Cu .

8

1.0 0.9

Wichtmann et al.

p = 400 kPa

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

1 2 5 10 20

L1 - L26 with d50 [mm] = 0.2 0.6 2 L27/L28

0.7 0.6 0.5 0.4 0.3

d50 [mm] = 0.79/ 0.2 0.6 2 1

5 1 2 4 = 10-5 10-4 10-4 10-4

Parameter a [-]

0.8

G/Gmax [-]

1

2

3

4

5

6 7 8 9 10

20

Coefficient of uniformity Cu = d60 / d10 [-]

Coefficient of uniformity Cu [-]

Fig. 12: Factor G/Gmax as a function of Cu , plotted for different shear strain amplitudes

Fig. 13: Correlation of the parameter a in Eq. (2) with the coefficient of uniformity

Influence of fines content on the curves G()/G max and D() Hardly no influence of the fines content on the curves G()/G max and D() could be found in the RC tests on sands F1 to F6. However, due to the decrease of G max with increasing F C, the reference shear strain r = max /Gmax significantly increases with increasing fines content, resulting in an increase of the parameter a in Eq. (2). Therefore, the following extension of Eq. (18) is proposed: a = 1.070 ln (Cu ) exp(0.053F C) (19)

For small pressures (p = 50 kPa) the damping ratio D decreases by almost a factor 4 when the fines content is increased from 0 to 10 %. For larger fines contents the damping ratio stays almost constant. For larger pressures (p = 400 kPa) the decrease of D with F C is less pronounced. While the linear elastic threshold shear strain amplitude tl does hardly depend on fines content, the cumulative threshold shear strain amplitude tv increases with increasing F C. APPLICABILITY OF THE CORRELATIONS FOR PIECEWISE LINEAR, GAPGRADED, S-SHAPED AND OTHER SMOOTHLY SHAPED GRAIN SIZE DISTRIBUTION CURVES All correlations presented above have been developed based on experimental data for grain size distribution curves with a linear shape in the semi-logarithmic scale. In the meantime, the new correlations have been inspected for piecewise linear, gap-graded, S-shaped and other smoothly shaped grain size distribution curves. All these materials did not contain fines. Figures 14 and 15 show some of the tested grain size distribution curves, together with the measured G max (e)- and Mmax (e)-data for pressures p = 100 and 400 kPa. In Figures 14 and 15, the shear moduli G max (e) predicted by Eq. (1) with (3) to (5) and the constrained elastic moduli M max (e) predicted by Eqs. (10) to (13) have been added as thick solid curves. These curves were generated using Cu = d60 /d10 as input for the correlations. The equivalent linear grain size distribution curves, which have the same d10 - and Cu -values, are shown as thick solid lines in the first column of diagrams in Figures 14 and 15. For most of the "more complicated" grain size distribution curves, the experimental data is well approximated by the new correlations. However, for a few materials (see e.g. PL7 and GG2 in Figures 14 and 15) too low G max - and Mmax -values are predicted. Therefore, a possible improvement of the prediction by using the equivalent linear grain size distribution curves shown as dashed thick lines in the first column of diagrams in Figures 14 and 15 has been checked. They have the same d 10 but the inclination Cu,A is chosen

Small-strain stiffness of granular soils taking into account the grain size distribution curve

Silt coarse fine Sand med. coarse Gravel fine med.

9

300

1000

Finer by weight [%]

Gmax [MPa]

100 80 60 40

Sand PL1

200 100 0 0.50 250

800

Sand PL1

Mmax [MPa]

0.55 0.60 0.65 0.70 0.75

20 0 0.02 0.06

Sand PL1 d50 = 0.35 mm Cu = 1.50 (Cu,A = 1.95)

600 400 200 0 0.50 800 0.55 0.60 0.65 0.70 0.75

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

Void ratio e [-]

Void ratio e [-]

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

A1 A1 = A2

Gmax [MPa]

20 0 0.02 0.06

Sand PL2 d50 = 0.20 mm Cu = 1.50 (Cu,A = 2.16)

150 100 50 0 0.60 300 0.65 0.70 0.75 0.80

Mmax [MPa]

A2

200

Sand PL2

Sand PL2

600 400 200 0 0.60 1000 0.65 0.70 0.75 0.80

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

Void ratio e [-]

Void ratio e [-]

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand PL3 Gmax [MPa]

200 100 0 0.50 300

Mmax [MPa]

800 600 400 200 0 0.50 0.55 0.60

Sand PL3

20 0 0.02 0.06

Sand PL3 d50 = 0.20 mm Cu = 2.5 (Cu,A = 3.2)

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

0.55

0.60

0.65

0.70

0.65

0.70

Void ratio e [-]

1200 1000

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand PL4 Gmax [MPa]

200 100 0 0.35

Sand PL4

Mmax [MPa]

Sand PL4 d50 = 2.0 mm Cu = 10.0 (Cu,A = 7.7)

800 600 400 200 0 0.35 0.40

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

0.40

0.45

0.50

0.55

0.60

Void ratio e [-]

300

Void ratio e [-]

0.45

0.50

0.55

0.60

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand PL5

1000

Sand PL5

Gmax [MPa]

Sand PL5 d50 = 1.1 mm Cu = 4.5 (Cu,A = 4.2)

200 100 0 0.40 300

Mmax [MPa]

800 600 400 200

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

0.45

Void ratio e [-]

0.50

0.55

0.60

0.65

0 0.40 1000

0.45

Void ratio e [-]

0.50

0.55

0.60

0.65

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand PL6 Gmax [MPa]

200 100 0 0.40

Sand PL6 d50 = 2.0 mm Cu = 5.6 (Cu,A = 5.6)

Mmax [MPa]

800 600 400 200 0 0.40 0.45 0.50

Sand PL6

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

0.45

Void ratio e [-]

0.50

0.55

0.60

0.65

Void ratio e [-]

0.55

0.60

0.65

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

1200 300

Sand PL7

1000

Sand PL7

Gmax [MPa]

Mmax [MPa]

Sand PL7 d50 = 2.0 mm Cu = 15.6 (Cu,A = 9.8)

200 100 0 0.35

800 600 400 200 0 0.35 0.40 0.45 0.50 0.55

20 0 0.02 0.06

Grain size [mm]

0.2

0.6

2

6

20

0.40

0.45

0.50

0.55

Void ratio e [-]

Void ratio e [-]

Fig. 14: Results from tests on piecewise linear grain size distribution curves

10

Silt coarse fine Sand med. coarse Gravel fine med.

Wichtmann et al.

1000 300

Finer by weight [%]

100 80 60 40

Sand GG1 Mmax [MPa]

Gmax [MPa]

Sand GG1 d50 = 1.1 mm Cu = 4.7 (Cu,A = 3.7)

800 600 400 200

Sand GG1

200 100 0 0.40 0.45 0.50 0.55 0.60 0.65

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

0 0.40 1200

0.45

Void ratio e [-]

300

Void ratio e [-]

0.50

0.55

0.60

0.65

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand GG2

1000

Sand GG2

Mmax [MPa]

0.45 0.50 0.55

Gmax [MPa]

Sand GG2 d50 = 2.0 mm Cu = 15.6 (Cu,A = 6.6)

200 100 0 0.40 300

800 600 400 200 0 0.40

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

Void ratio e [-]

0.45

0.50

0.55

Void ratio e [-]

1200 1000

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand GG3 Gmax [MPa]

200 100 0 0.35

Sand GG3

Mmax [MPa]

0.40 0.45 0.50 0.55

Sand GG3 d50 = 1.0 mm Cu = 7.8 (Cu,A = 7.1)

800 600 400 200 0 0.35

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

Void ratio e [-]

0.40

Void ratio e [-]

0.45

0.50

0.55

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

300

Sand S1

1000

Sand S1

Gmax [MPa]

200 100 0

Mmax [MPa]

Sand S1 d50 = 0.80 mm Cu = 6.1 (Cu,A = 5.4)

800 600 400 200 0 0.35 0.40 0.45 0.50 0.55 0.60

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

0.40

Void ratio e [-] Sand S2

0.50

0.60

Void ratio e [-]

Finer by weight [%]

80 60 40 20 0 0.02 0.06

Sand S2 d50 = 0.37 mm Cu = 2.9 (Cu,A = 3.5)

Gmax [MPa]

200 100 0 0.45 400

Mmax [MPa]

100

Silt coarse

fine

Gravel fine med.

300

1000 800 600 400 200

Sand S2

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

0.50

Void ratio e [-]

0.55

0.60

0.65

0 0.45

0.50

Void ratio e [-]

0.55

0.60

0.65

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand S3 Gmax [MPa]

300 200 100 0 0.30 400 0.35 0.40 0.45 0.50

1200

Sand S3

Mmax [MPa]

Sand S3 d50 = 0.89 mm Cu = 7.4 (Cu,A = 8.7)

800 400 0 0.30

20 0 0.02 0.06

Grain size [mm]

Sand med. coarse

0.2

0.6

2

6

20

Void ratio e [-]

0.35

0.40

0.45

0.50

Void ratio e [-]

1200

Finer by weight [%]

100 80 60 40

Silt coarse

fine

Gravel fine med.

Sand S4 Gmax [MPa]

300 200 100 0 0.35 0.40 0.45 0.50 0.55

Sand S4

Mmax [MPa]

800 400 0 0.35

20 0 0.02 0.06

Sand S4 d50 = 0.37 mm Cu = 3.2 (Cu,A = 6.8)

Grain size [mm]

0.2

0.6

2

6

20

Void ratio e [-]

0.40

Void ratio e [-]

0.45

0.50

0.55

Fig. 15: Results from tests on gap-graded and smoothly shaped grain size distribution curves

Small-strain stiffness of granular soils taking into account the grain size distribution curve

11

such way that the areas enclosed between the original and the equivalent linear curve, above and below the original curve, are equal (see the scheme in the diagram for sand PL2, Figure 14). For most tested materials, in particular for PL7 and GG2, the difference between the measured and the predicted Gmax - and Mmax -values is less if Cu,A is used instead of Cu = d60 /d10 . It can be concluded that the new correlations work well also for "more complicated" grain size distribution curves. It is recommended to apply the correlations with C u,A instead of Cu . SUMMARY AND CONCLUSIONS Approx. 650 resonant column tests with additional P-wave measurements have been performed on 64 quartz sands with different grain size distribution curves. First, grain size distribution curves with a linear shape in the semi-logarithmic scale were tested. These tests showed that the small-strain shear modulus Gmax and the small-strain constrained elastic modulus M max are independent of mean grain size d50 but strongly decrease with increasing coefficient of uniformity Cu = d60 /d10 . A fines content leads to a further reduction of G max and Mmax . The well-known Hardin's equation for Gmax has been extended by the influence of the grain size distribution curve. For that purpose the parameters have been correlated with C u and F C. A similar set of equations has been developed for Mmax . For a certain shear strain amplitude, the modulus degradation factor G()/G max decreases with increasing coefficient of uniformity, but hardly depends on fines content. An empirical formula for the modulus degradation factor has been extended by the influence of the grain size distribution curve. Damping ratio D decreases or increases with C u , depending on pressure and shear strain amplitude. A fines content reduces the damping ratio. The decrease is more pronounced at low pressures. The linear elastic threshold shear strain amplitude tl depends neither on Cu nor on F C. The cumulative threshold shear strain amplitude tv is not affected by the coefficient of uniformity, but increases with increasing fines content. In a second test series, piecewise linear, gap-graded, S-shaped and other smoothly shaped grain size distribution curves have been tested. For most of these materials, the G max and Mmax -values predicted by Hardin's equation with the new correlations agree well with the measurements. For a few materials, a better congruence between the predicted and the experimental data is achieved when the new correlations are applied with an inclination factor C u,A instead of Cu = d60 /d10 . Cu,A is defined as the uniformity coefficient of an equivalent linear grain size distribution curve, for which the areas enclosed between the original and the equivalent linear curve, above and below the original curve, are equal. AKNOWLEDGEMENTS The presented study has been funded by the German Research Council (DFG, projects No. TR218/11-1 and TR218/17-1). The authors are grateful to DFG for the financial support. References [1] B.O. Hardin and W.L. Black. Sand stiffness under various triaxial stresses. Journal of the Soil Mechanics and Foundations Division, ASCE, 92(SM2):2742, 1966. [2] B.O. Hardin and V.P. Drnevich. Shear modulus and damping in soils: design equations and curves. Journal of the Soil Mechanics and Foundations Division, ASCE, 98(SM7):667692, 1972. [3] B.O. Hardin and M.E. Kalinski. Estimating the shear modulus of gravelly soils. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 131(7):867875, 2005.

12

Wichtmann et al.

[4] B.O. Hardin and F.E. Richart Jr. Elastic wave velocities in granular soils. Journal of the Soil Mechanics and Foundations Division, ASCE, 89(SM1):3365, 1963. [5] T. Iwasaki and F. Tatsuoka. Effects of grain size and grading on dynamic shear moduli of sands. Soils and Foundations, 17(3):1935, 1977. [6] R. Martinez. Influence of the grain size distribution curve on the stiffness and the damping ratio of non-cohesive soils at small strains (in German). Diploma thesis, Institute of Soil Mechanics and Foundation Engineering, Ruhr-University Bochum, 2007. [7] H.B. Seed, R.T. Wong, I.M. Idriss, and K. Tokimatsu. Moduli and damping factors for dynamic analyses of cohesionless soil. Journal of Geotechnical Engineering, ASCE, 112(11):10161032, 1986. [8] T. Wichtmann and T. Triantafyllidis. On the influence of the grain size distribution curve of quartz sand on the small strain shear modulus G max . Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 135(10):14041418, 2009. [9] T. Wichtmann and T. Triantafyllidis. On the influence of the grain size distribution curve on P-wave velocity, constrained elastic modulus M max and Poisson's ratio of quartz sands. Soil Dynamics and Earthquake Engineering, 30(8):757766, 2010.

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