11th Baltic Sea Geotechnical Conference: ”Geotechnics in Maritime Engineering”,
Gdańsk, Poland, 2008, Vol. 1, pp. 315-322
On the influence of the grain size distribution curve on the secant
stiffness of quartz sand under cyclic loading
Torsten Wichtmann
Rafael Martinez
Francisco Duran Graeff
Emilce Giolo
Miguel Navarette Hernandez
Theodor Triantafyllidis
Institute of Soil Mechanics and Rock Mechanics, University of Karlsruhe
ABSTRACT: The paper presents a study of the influence of the grain size distribution curve on the
small strain shear modulus Gmax of quartz sand. The results of approx. 130 resonant column tests
on 25 different grain size distribution curves are presented. It is demonstrated that while G max is
not influenced by variations in the mean grain size d50 , it significantly decreases with increasing
coefficient of uniformity Cu = d60 /d10 of the grain size distribution curve. The well-known Hardin’s
equation with its commonly used constants may strongly overestimate the stiffness of well-graded
soils. Furthermore it may underestimate the Gmax -values of poorly graded soils. Based on the RC
test results correlations of the constants of Hardin’s equation with Cu have been developed. The
predictions using these correlations are in good accordance with the test data as demonstrated in the
present paper.
1 INTRODUCTION
For studies of the short-time behaviour of structures under cyclic or dynamic loading the secant stiffness of the stress - strain hysteresis is
of interest. The secant stiffness decreases with
increasing strain amplitude if a certain threshold value (γ ampl ≈ 10−5 for sand) is surpassed.
For a given sand, the constant maximum shear
modulus Gmax at very small strain amplitudes
(also denoted as ”dynamic” shear modulus) is
mainly a function of void ratio e and mean pressure p and may be estimated using the equation
of Hardin [4, 3]
(a − e)2
Gmax [MPa] = A
(p[kPa])n
(1)
1+e
It was developed based on tests on Ottawa sand
and on a crushed quartz sand. The constants A
= 6.9, a = 2.17 and n = 0.5 for round grains and
A = 3.2, a = 2.97 and n = 0.5 for angular grains
were recommended by Hardin [3] and are often
used for rough estimations of Gmax -values for
various sands.
However, Eq. (1) with the given constants
does not reflect the strong dependence of the
small strain shear modulus on the grain size
distribution curve, especially on the coefficient
of uniformity Cu = d60 /d10 and on the content of fines. For equal values of e and p the
shear modulus Gmax decreases with increasing
Cu and decreases also with increasing content
Wichtmann et al.
Geotechnics in Maritime Engineering, 2008, Vol. 1, pp. 315-322
a)
of fines. Such dependencies have been reported
also e.g. by Iwasaki & Tatsuoka [5]. Eq. (1) may
strongly overestimate the Gmax -values of wellgraded sands and of sands with a content of
fines. However, no recommendations are given
in the literature how to consider the influence of
the grain size distribution curve when estimating
Gmax .
Finer by weight [%]
100
Silt
coarse
Sand
medium
fine
coarse
fine
Gravel
medium
80
L1
60
L2 L3 L4
L5
L6 L7 L8
40
20
0
0.02
The present paper reports on our effort to extend Eq. (1) by the influence of the grain size distribution curve, in particular by the coefficient
of uniformity Cu . Approx. 130 resonant column
(RC) tests on 25 different grain size distribution
curves have been performed for this purpose.
Gmax was measured at different values of e and
p. A curve-fitting of Eq. (1) to the test data was
performed for each grain size distribution curve.
Correlations of the constants A, a and n with
Cu have been developed. The good prediction
of Hardin’s equation with the new correlations
for the various grain size distribution curves is
demonstrated.
0.06
0.2
0.6
2
6
20
Mean grain size [mm]
b)
Finer by weight [%]
100
Silt
coarse
Sand
medium
fine
L26
L25
L24
L2
80
60
40
coarse
fine
L12
L11
L10
L4
L13
L14
L15
L16
20
0
0.02
0.06
0.2
Gravel
medium
0.6
2
L20
L21
L22
L23
L19
L18
L17
L6
6
20
Mean grain size [mm]
Figure 1. Tested grain size distribution curves
3 TEST DEVICE AND TESTING PROCEDURE
The resonant column device used for the present
study is explained e.g. by Wichtmann & Triantafyllidis [8]. All specimens were prepared by
pluviating the granular material out of a funnel. They were tested in the dry condition. The
nearly isotropic stress was increased in 7 steps
from p = 50 kPa over p = 75, 100, 150, 200
and 300 kPa to p = 400 kPa. The compaction
was determined by means of non-contact displacement transducers. At each p the small
strain shear modulus Gmax was measured after
a short resting period of 5 minutes. At p = 400
kPa the curves G(γ ampl ) and D(γ ampl ) (damping ratio) were measured. However, the γ ampl dependence as well as additional measurements
of the P -wave velocity will not be discussed in
the present paper. For each sand several such
tests with different initial relative densities were
performed.
2 TESTED MATERIAL
The study was performed with a natural quartz
sand obtained from a sand pit near Dorsten (Germany). The grain shape is sub-angular. The sand
was sieved into 25 gradations with grain sizes
between 0.063 and 16 mm. From these gradations the grain size distribution curves depicted
in Figure 1 have been mixed. They are linear in
the semi-logarithmic scale.
The influence of the mean grain size d50 was
studied in tests on the sands or gravels L1 to
L8 (Figure 1a). L1 to L8 have a coefficient of
uniformity of Cu = 1.5 and different mean grain
sizes in the range 0.1 ≤ d50 ≤ 6 mm. Three test
series differing in the mean grain size have been
performed on the influence of Cu (Figure 1b).
The mean grain size was d50 = 0.2 mm for sands
L2, L24 to L26, d50 = 0.6 mm for sands L4, L10
to L16 and d50 = 2 mm for sands L6, L17 to L23.
The coefficient of uniformity varied in the range
1.5 ≤ Cu ≤ 8. The d50 - and Cu -values of the 25
tested grain size distribution curves are summarized in Table 1.
4 TEST RESULTS
The measured shear moduli Gmax for most of
the 25 grain size distribution curves are given as
a function of void ratio e and for different mean
2
150
100
0.60
0.65
0.70
0.75
0.80
0.85
250
200
150
100
50
0
0.50
0.90
d50 = 0.6 mm
Cu = 1.5
0.55
0.60
Void ratio e [-]
Sand L24
250
p [kPa] =
50
75
100
150
300
200
300
400
Shear modulus Gmax [MPa]
Shear modulus Gmax [MPa]
300
200
150
100
50
0
0.55
d50 = 0.2 mm
Cu = 2
0.60
0.65
0.70
0.75
250
250
0.80
p [kPa] =
50
75
100
150
100
d50 = 0.2 mm
Cu = 2.5
0.55
0.60
0.65
0.70
0.55
0.60
p [kPa] =
50
75
100
150
150
100
d50 = 0.2 mm
Cu = 3
0.55
0.60
0.65
50
d50 = 0.6 mm
Cu = 2.5
0.50
0.55
Prediction of Eq. (2) with the
constants A,a,n in columns
9 to 11 in Table 1 (obtained
with Eqs. (3) to (5))
0.60
0.65
0.70
0.70
250
0.75
150
d50 = 0.6 mm
Cu = 3
0.55
0.60
0.65
0.70
50
d50 = 2 mm
Cu = 2
0.55
100
d50 = 0.6 mm
Cu = 5
0.45
0.50
350
0.55
0.60
250
100
50
d50 = 2 mm
Cu = 2.5
0.50
150
d50 = 0.6 mm
Cu = 8
0.40
0.45
0.50
Void ratio e [-]
0.55
0.60
0.65
p [kPa] =
50
75
100
150
Sand L19
250
200
300
400
150
100
50
d50 = 2 mm
Cu = 3
0.50
0.55
0.60
p [kPa] =
50
75
100
150
Sand L21
300
0.65
250
200
300
400
200
150
100
50
d50 = 2 mm
Cu = 5
0.40
0.45
0.50
0.55
0.60
Sand L23
300
250
p [kPa] =
50
75
100
150
0.55
200
300
400
200
150
100
50
0
0.35
d50 = 2 mm
Cu = 8
0.40
0.45
Void ratio e [-]
Figure 2. Comparison of measured shear moduli with the G max -values predicted by Eq. (2)
3
0.70
200
350
200
300
400
100
0
0.35
200
300
400
200
0
0.35
0.65
200
50
0.75
Void ratio e [-]
p [kPa] =
50
75
100
150
Sand L16
0.70
p [kPa] =
50
75
100
150
Sand L18
Void ratio e [-]
300
0.65
150
350
200
300
400
150
0
0.40
0.60
250
0
0.45
0.75
200
50
200
300
400
Void ratio e [-]
p [kPa] =
50
75
100
150
250
p [kPa] =
50
75
100
150
Sand L17
Void ratio e [-]
Sand L14
0.80
100
300
200
300
400
100
0.50
0.75
200
0
0.45
0.75
200
50
0.70
Void ratio e [-]
p [kPa] =
50
75
100
150
Sand L12
0.65
150
300
200
300
400
100
300
Prediction of Eq. (2) with the
constants A,a,n in columns
6 to 8 in Table 1
0.60
Void ratio e [-]
p [kPa] =
50
75
100
150
Sand L11
Void ratio e [-]
Curve-fitting of Eq. (2) to the
experimental data of the
individual pressure steps
d50 = 2 mm
Cu = 1.5
250
0
0.50
0.75
200
0
0.45
Shear modulus Gmax [MPa]
0
0.50
0.70
150
300
200
300
400
200
50
50
Void ratio e [-]
Shear modulus Gmax [MPa]
250
0.65
250
0
0.45
0.75
Shear modulus Gmax [MPa]
Shear modulus Gmax [MPa]
Sand L26
100
300
200
300
400
d50 = 0.6 mm
Cu = 2
Void ratio e [-]
300
200
Void ratio e [-]
100
50
200
300
400
150
0
0.55
0.80
200
300
200
300
400
200
0
0.50
250
Void ratio e [-]
150
50
0.75
150
0
0.50
Shear modulus Gmax [MPa]
Shear modulus Gmax [MPa]
Sand L25
0.70
p [kPa] =
50
75
100
150
Sand L10
Void ratio e [-]
300
0.65
p [kPa] =
50
75
100
150
Sand L6
Void ratio e [-]
Shear modulus Gmax [MPa]
0
0.55
d50 = 0.2 mm
Cu = 1.5
300
300
200
300
400
Shear modulus Gmax [MPa]
50
p [kPa] =
50
75
100
150
Shear modulus Gmax [MPa]
200
Sand L4
Shear modulus Gmax [MPa]
250
350
200
300
400
Shear modulus Gmax [MPa]
Shear modulus Gmax [MPa]
p [kPa] =
50
75
100
150
Shear modulus Gmax [MPa]
Sand L2
300
Geotechnics in Maritime Engineering, 2008, Vol. 1, pp. 315-322
Shear modulus Gmax [MPa]
Wichtmann et al.
0.50
Wichtmann et al.
Geotechnics in Maritime Engineering, 2008, Vol. 1, pp. 315-322
Shear modulus Gmax [MPa]
200
two or three sands (e.g. L24, L10 and L17)
with equal Cu but different d50 fall together confirming the d50 -independence of Gmax . For e =
constant the decrease of Gmax with increasing
Cu is obvious in Figures 2 and 4. It becomes
even more evident from the diagram in Figure
5 where the Gmax -values at e = 0.55 are plotted
versus Cu (the Gmax -values were obtained with
Eq. (2) and the constants in columns 6 to 8 of
Table 1, see remarks below). For p = 50 kPa the
shear modulus (mean value of the three test series) is Gmax = 115 MPa for the sand with Cu
= 1.5 but only Gmax = 53 MPa for the sand with
Cu = 8 (54 % decrease). For p = 400 kPa the values are Gmax = 277 MPa for Cu = 1.5 and Gmax
= 169 MPa for Cu = 8 (39 % decrease).
Sand L1
Sand L2
Sand L3
Sand L4
Sand L5
Sand L6
Sand L7
Sand L8
p = 100 kPa
150
100
p = 100 kPa
150
100
50
0.40
0.50
0.60
0.60
0.70
0.80
300
p = 400 kPa
0.90
Sand L1
Sand L2
Sand L3
Sand L4
Sand L5
Sand L6
Sand L7
Sand L8
250
200
0.70
0.80
p = 400 kPa
300
50
0.50
Void ratio e [-]
Shear modulus Gmax [MPa]
200
L2, L4, L6
(Cu = 1.5)
L24, L10, L17 (Cu = 2)
L26, L12, L19 (Cu = 3)
L14, L21 (Cu = 5)
L16, L23 (Cu = 8)
Hardin eq., round grains
Hardin eq., angular grains
Void ratio e [-]
Shear modulus Gmax [MPa]
Shear modulus Gmax [MPa]
pressures p in Figure 2. Other presentations of
the test results were given by by Martinez [6],
Duran Graeff [1], Giolo [2] and Hernandez [7].
Figure 3 compares the Gmax -values for the
eight materials L1 to L8 with the same Cu = 1.5
but with different mean grain sizes 0.1 ≤ d50 ≤ 6
mm. Independent of the pressure p, the data
points for the seven sands L1 to L7 lay on a
unique curve, i.e. the variation in d50 does not
influence Gmax . The values of the gravel L8 lay
slightly below those of L1 to L7. Martinez [6]
demonstrated, that these lower values are due to
an insufficient interlocking between the tested
material and the end plates which were glued
with coarse sand. If end plates with small wings
penetrating into the specimen were used, the values of L8 coincided with those of the sands L1
to L7. Thus, from this test series it may be concluded that d50 does not influence Gmax . The
mean grain size has not to be considered in an
Equation of the type (1).
250
200
150
100
0.40
0,50
0,60
0,70
0,80
Void ratio e [-]
150
100
0.50
0.60
0.70
0.80
Figure 4. Comparison of Gmax (e) of the grain size
distribution curves with different C u -values at p =
100 kPa and p = 400 kPa
0.90
Void ratio e [-]
In Figure 4 the curves predicted by Hardin’s
Eq. (1) for round and for angular grains have
been supplemented. In general, in our RC tests
the void ratio-dependence was found slightly
larger than predicted by Eq. (1). From Figure 4
it is obvious that Eq. (1) overestimates the Gmax -
Figure 3. Comparison of Gmax (e) of the eight sands
L1 - L8 (Cu = 1.5, 0.1 ≤ d50 ≤ 6 mm) at p = 100 kPa
and p = 400 kPa
In Figure 4 the data of the test series on the
influence of Cu are combined. The data of the
4
Wichtmann et al.
Geotechnics in Maritime Engineering, 2008, Vol. 1, pp. 315-322
Shear modulus Gmax [MPa]
300
d50 = 0,2 mm
d50 = 0,6 mm
d50 = 2 mm
Appr. by Eqs. (2)-(5)
250
200
curve (solid line, resulting from the fitting of
f = k F (e) to the data for each pressure) is
small.
The aim of the current study was to develop
a unique formula for the prediction of Gmax for
different grain size distribution curves. For this
purpose correlations of the constants A, a and n
in Eq. (2) with the coefficient of uniformity were
developed. In Figure 6a the constant a is plotted
versus Cu . For each sand the seven values for the
seven tested pressures are given. The decrease of
a with Cu may be described by the exponential
function
p [kPa] =
400
150
200
100
100
50
0
50
1
2
3
4
5
6
7
8
9
Coefficient of uniformity Cu [-]
Figure 5. Shear modulus Gmax at e = 0.55 as a function of Cu for different pressures p
a = c1 exp(−c2 Cu )
values for well-graded soils, especially at large
void ratios, i.e small relative densities. However,
Eq. (1) may underestimate the shear moduli for
poorly graded soils at small void ratios.
Eq. (1) in its dimensionless form
Gmax = A
(a − e)2
patm 1−n pn
|
{z
}
1
+
e
| {z }
F (e)
(3)
with the constants c1 = 1.94 and c2 = 0.066 (solid
line in Figure 6a). Figure 6b shows the constant
n as a function of Cu . The exponent n increases
with increasing Cu which can be expressed by
the potential function
(2)
n = c 3 Cu c4
(4)
F (p)
with the constants c3 = 0.40 and c4 = 0.18. The
constants a and n were calculated from Eqs. (3)
and (4) for each tested grain size distribution
curve and are summarized in columns 10 and 11
of Table 1. Using these constants the functions
F (e) and F (p) were re-calculated and from the
data Gmax /F (e)/F (p) the constant A was determined. In Figure 6c it is plotted versus Cu . The
relationship A(Cu ) may be approximated by a
function consisting of a constant and a potential
portion:
with patm = 100 kPa has been fitted separately
to the test data for each grain size distribution
curve. First, the constant a was determined for a
certain pressure p by fitting the function f (e) =
k F (e) to the data Gmax (e) (the constant k is not
further used). The a-value in column 7 of Table 1 is the mean value of the seven values obtained for the different pressures p. Afterwards
the shear moduli were divided by the void ratio
function F (e) and the data Gmax /F (e) was plotted versus p. The function f (p) = k F (p) was
fitted to the data of each test resulting in the exponent n. The n-value in column 8 of Table 1
is the mean value of the values for the different
tests. The data Gmax /F (e)/F (p) is equal to the
constant A. First, a mean value of A was determined for each test. Afterwards the values from
the different tests were averaged resulting in the
A-values given in column 6 of Table 1.
The good approximation of Eq. (2) with the
constants A, a and n given in columns 6 to 8
of Table 1 is demonstrated in Figure 2 where
the prediction is shown as the dashed curves.
In most cases the deviation from the best-fit
A = c 5 + c 6 Cu c7
(5)
with the constants c5 = 1563, c6 = 3.13 and c7 =
2.98. The A-values calculated from Eq. (5) are
collected in column 9 of Table 1.
The prediction of Eq. (2) with the constants
A, a and n obtained from Eqs. (3), (4) and
(5) is compared to the test data in Figure 2
(dot-dashed lines). An alternative presentation
is given in Figure 7 where the predicted Gmax values (for the same e and p) are plotted versus
the measured ones. Since the deviations of the
meas.
data from the line Gpred.
max = Gmax are small, the
good prediction of the proposed correlations is
5
Wichtmann et al.
Geotechnics in Maritime Engineering, 2008, Vol. 1, pp. 315-322
3.0
2.5
Constant a [-]
the mean grain size d50 but that it significantly
decreases with increasing coefficient of uniformity Cu = d60 /d10 . The constants A, a and
n of the well-known Hardin’s equation have
been correlated with Cu . Using Equation (2) and
the proposed correlations A, a, n(Cu ), the Gmax values of the tested grain size distribution curves
are well predicted for different void ratios and
pressures.
S-shaped and step-shaped grain size distribution curves (in the semi-logarithmic scale) will
be tested in future. The applicability of the proposed correlations will be studied. The influence of the content of fines is also being studied
and will be integrated into the proposed correlations. The small-strain constrained elastic modulus Mmax , Poisson’s ratio ν and the curves
G(γ ampl ) and D(γ ampl ) will be discussed in separate papers in future.
Sands with d50 =
0.2 mm
0.6 mm
2 mm
2.0
1.5
1.0
0.5
0
1
2
3
4
5
6
7
8
9
Coefficient of uniformity Cu [-]
0.70
0.65
Exponent n [-]
0.60
0.55
0.50
0.45
Sands with d50 =
0.2 mm
0.6 mm
2 mm
0.40
0.35
0.30
1
2
3
4
5
6
7
8
ACKNOWLEDGEMENTS
9
Coefficient of uniformity Cu [-]
The presented study has been performed within
the framework of the project ”Influence of the
coefficient of uniformity of the grain size distribution curve and of the content of fines
on the dynamic properties of non-cohesive
soils” founded by the German Research Council (DFG, project No. TR218/11-1). The authors are grateful to DFG for the financial support. The RC tests have been performed at RuhrUniversity Bochum.
4000
Sands with d50 =
0.2 mm
0.6 mm
2 mm
Constant A [-]
3500
3000
2500
2000
1500
1000
1
2
3
4
5
6
7
8
9
Coefficient of uniformity Cu [-]
REFERENCES
Figure 6. Constants a, n and A in dependence of C u
[1] F. Duran Graeff. Influence of the coefficient of
uniformity of the grain size distribution curve
on the stiffness and the damping ratio of noncohesive soils at small strains (in German).
Diploma thesis, Institute for Soil Mechanics
and Foundation Engineering, Ruhr-University
Bochum, 2008.
confirmed. Another proof for the proposed correlations are the predicted Gmax -values for e =
0.55 in Figure 5 (solid lines). The prediction by
Eqs. (2) to (5) describes well the decrease of
Gmax with Cu .
[2] E. Giolo. Influence of the coefficient of uniformity of the grain size distribution curve on the
stiffness and the damping ratio of non-cohesive
soils at small strains (in German). Project thesis, Institute for Soil Mechanics and Foundation
Engineering, Ruhr-University Bochum, 2008.
5 SUMMARY AND OUTLOOK
Approximately 130 resonant column tests have
been performed on 25 different grain size distribution curves of a quartz sand. It has been
demonstrated that for e, p = constant the smallstrain shear modulus Gmax does not depend on
[3] B.O. Hardin and W.L. Black. Sand stiffness un-
6
300
Sand L2
250
d50 = 0.2 mm
Cu = 1.5
200
p [kPa] =
50
75
100
150
200
300
400
150
100
50
0
0
50
100
150
200
250
300
Sand L4
250
d50 = 0.6 mm
Cu = 1.5
200
p [kPa] =
50
75
100
150
200
300
400
150
100
50
0
300
0
50
100
150
200
250
300
Sand L6
250
d50 = 2 mm
Cu = 1.5
200
p [kPa] =
50
75
100
150
200
300
400
150
100
50
0
300
0
50
Measured Gmax [MPa]
100
150
300
Sand L10
300
Sand L17
250
d50 = 0.2 mm
Cu = 2
250
d50 = 0.6 mm
Cu = 2
250
d50 = 2 mm
Cu = 2
p [kPa] =
50
75
100
150
200
300
400
150
100
50
0
0
50
100
150
200
250
200
p [kPa] =
50
75
100
150
200
300
400
150
100
50
0
300
Predicted Gmax [MPa]
Sand L24
200
0
50
Measured Gmax [MPa]
100
150
200
250
100
50
0
300
0
50
100
150
300
Sand L19
250
d50 = 0.2 mm
Cu = 3
250
d50 = 0.6 mm
Cu = 3
250
d50 = 2 mm
Cu = 3
50
50
100
150
200
250
p [kPa] =
50
75
100
150
200
300
400
100
50
0
300
0
Measured Gmax [MPa]
50
100
150
200
250
100
50
0
300
0
50
100
150
300
Sand L21
250
d50 = 0.6 mm
Cu = 5
250
d50 = 2 mm
Cu = 5
200
p [kPa] =
50
75
100
150
200
300
400
50
0
0
50
100
150
200
250
100
50
0
300
0
50
100
150
300
Sand L23
250
250
d50 = 2 mm
Cu = 8
Predicted Gmax [MPa]
Sand L16
200
0
p [kPa] =
50
75
100
150
200
300
400
0
50
100
150
200
250
Measured Gmax [MPa]
200
250
200
p [kPa] =
50
75
100
150
200
300
400
150
100
50
300
0
0
50
100
150
200
250
Measured Gmax [MPa]
Figure 7. Shear moduli predicted by Eqs. (2) to (5) versus measured G max -values
7
300
Measured Gmax [MPa]
d50 = 0.6 mm
Cu = 8
50
300
p [kPa] =
50
75
100
150
200
300
400
150
300
100
250
200
Measured Gmax [MPa]
150
200
Measured Gmax [MPa]
Sand L14
100
300
p [kPa] =
50
75
100
150
200
300
400
150
300
150
250
200
Measured Gmax [MPa]
Predicted Gmax [MPa]
0
Predicted Gmax [MPa]
0
200
Predicted Gmax [MPa]
100
Predicted Gmax [MPa]
Sand L12
Predicted Gmax [MPa]
300
150
200
Measured Gmax [MPa]
Sand L26
150
300
p [kPa] =
50
75
100
150
200
300
400
150
300
p [kPa] =
50
75
100
150
200
300
400
250
200
Measured Gmax [MPa]
200
200
Measured Gmax [MPa]
300
Predicted Gmax [MPa]
Predicted Gmax [MPa]
Measured Gmax [MPa]
Predicted Gmax [MPa]
Predicted Gmax [MPa]
Geotechnics in Maritime Engineering, 2008, Vol. 1, pp. 315-322
Predicted Gmax [MPa]
Predicted Gmax [MPa]
Wichtmann et al.
300
Wichtmann et al.
Sand
L1
L2
L3
L4
L5
L6
L7
L8
L10
L11
L12
L13
L14
L15
L16
L17
L18
L19
L20
L21
L22
L23
L24
L25
L26
d50
[mm]
0.1
0.2
0.35
0.6
1.1
2
3.5
6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
2
2
2
2
2
2
2
0.2
0.2
0.2
Geotechnics in Maritime Engineering, 2008, Vol. 1, pp. 315-322
Cu
[-]
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
2
2.5
3
4
5
6
8
2
2.5
3
4
5
6
8
2
2.5
3
emin
[-]
0.674
0.596
0.591
0.571
0.580
0.591
0.626
0.634
0.541
0.495
0.474
0.414
0.394
0.387
0.356
0.555
0.513
0.491
0.439
0.401
0.401
0.398
0.559
0.545
0.540
emax
[-]
1.122
0.994
0.931
0.891
0.879
0.877
0.817
0.799
0.864
0.856
0.829
0.791
0.749
0.719
0.673
0.827
0.810
0.783
0.728
0.703
0.553
0.521
0.958
0.937
0.920
Fitting of (2) for each sand
A
a
n
636
2.34
0.44
1521
1.79
0.43
1620
1.77
0.42
2023
1.67
0.41
1570
1.77
0.43
1035
2.04
0.43
852
2.13
0.45
734
2.16
0.45
1207
1.85
0.46
2240
1.47
0.48
2489
1.39
0.50
2969
1.27
0.51
2771
1.26
0.54
4489
1.08
0.53
2388
1.27
0.54
1325
1.78
0.47
1194
1.79
0.48
3018
1.30
0.49
1197
1.67
0.51
1402
1.54
0.54
3345
1.15
0.55
1382
1.47
0.58
1446
1.77
0.43
1434
1.72
0.44
2451
1.42
0.46
From correlations (3) to (5)
A
a
n
1573
1.76
0.43
1573
1.76
0.43
1573
1.76
0.43
1573
1.76
0.43
1573
1.76
0.43
1573
1.76
0.43
1573
1.76
0.43
1573
1.76
0.43
1588
1.70
0.45
1611
1.64
0.47
1646
1.59
0.49
1758
1.49
0.51
1942
1.39
0.53
2215
1.31
0.55
3100
1.14
0.58
1588
1.70
0.45
1611
1.64
0.47
1646
1.59
0.49
1758
1.49
0.51
1942
1.39
0.53
2215
1.31
0.55
3100
1.14
0.58
1588
1.70
0.45
1611
1.64
0.47
1646
1.59
0.49
Table 1. Parameters d50 , Cu , emin and emax of the tested grain size distribution curves; Summary of the constants A, a and n of Eq. (2)
Mechanics and Foundation Engineering, RuhrUniversity Bochum, 2007.
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Mechanics and Foundations Division, ASCE,
92(SM2):27–42, 1966.
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8