d
3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
Maximum and minimum void ratios and median grain size of
granular soils: their importance and correlations with material
properties
B.M. Das
California State University, Sacramento, USA,brajamdas@gmail.com
N. Sivakugan
James Cook University, Australia, siva.sivakugan@jcu.edu.au
C. Atalar
Near East University, North Cyprus, catalar@neu.edu.tr
KEYWORDS: Maximum void ratio, minimum void ratio, median grain size, idealized spheres,
angularity, volumetric strain potential, compaction, cone penetration test
ABSTRACT: The relationships between maximum and minimum void ratios of granular soil with
various percentages of fine contents have been presented. The maximum and minimum void ratios
are functions of soil properties such as grain size distribution, uniformity coefficient, angularity, and
percentage of fine contents. Based on the existing results, it appears that the difference between the
maximum and minimum void ratios, not maximum void ratio or minimum void ratio alone, is the
controlling parameter for compressibility, relative density, and strength of granular soils. In spite of
some scatter, the difference between the maximum and minimum void ratio bears a unique
relationship to the median grain size. Several correlations relating the median grain size with the
strength and compressibility are presented.
1 INTRODUCTION
According to the Classification System of Soils and Soil-Aggregate Mixtures for Highway
Construction Purposes provided in Test Designation D-3282 [(ASTM 2010); also referred to as
AASHTO Classification System], sand is defined as particles passing 2-mm sieve (US No. 10) and
retained on 0.075-mm sieve (US No. 200). Particles that pass through 2-mm sieve (US No. 10) and
are retained on 0.425-mm sieve (US No. 40) are defined as coarse sand. Similarly, particles passing
0.425-mm sieve and retained on 0.075-mm sieve are fine sand. However, according to the Unified
Soil Classification System (ASTM D-2487), sand particles can be divided into the following three
major categories:
 Coarse sand: Particles passing through 4.75-mm sieve (US No. 4) and retained on 2-mm sieve
(US No. 10);
 Medium sand: Particles passing through 2-mm sieve (US No. 10) and retained on 0.425-mm
sieve (US No. 40); and
 Fine sand: Particles passing through 0.425-mm sieve (US No. 40) and retained on 0.075-mm
sieve (US No. 200).
From the viewpoint of soil-separate size limits, the US No. 10 sieve (2-mm opening) is the moreaccepted upper limit for sand.
In nature, sand is generally a combination of particles of variation sizes and shapes and may
contain some plastic and/or non-plastic fines. The stress-strain behavior of sand is primarily a
function of the following: grain-size distribution, fine content, shape of grain angularity,
mineralogy, relative density (Dr), and state of effective stress.
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
In many cases the behavior of sand is represented by one or two parameters such as fine content (Fc)
and the uniformity coefficient (Cu) without consideration to other material properties. It has been
clearly pointed out by Cubrinovski and Ishihara (1999, 2002) that two sand samples with identical
fine contents (Fc) can show remarkably different stress-strain characteristics. For that reason they
suggested that emax – emin (emax = maximum void ratio and emin = minimum void ratio) and, hence, the
relative density may be a more appropriate parameter to describe the behavior of sand. In addition,
relative density indicates the relative position of field void ratio between emax and emin and is defined
as
Dr 
emax  e
emax  emin
(1)
where e = in situ void ratio
The minimum and maximum void ratios can be determined respectively according to the
procedures given by ASTM test designations D-4253 and D-4254. These test methods are applicable
to soils that may contain up to 15%, by dry mass, of soil particles passing 0.075-mm sieve (fines).
However, there are other methods in use to obtain emax and emin. The Japanese Geotechnical Society
(2000) has a test method to obtain emax and emin with less than 5% fines. These methods may provide
slightly different values for the extreme void ratios.
The purpose of this paper is to review the general nature of variation of emax, emin, and emax – emin
of sand and the factors controlling them. The importance of emax, emin and the median grain size (D50)
of several material properties will be discussed.
2 MAXIMUM AND MINIMUM VOID RATIOS OF IDEALIZED SPHERES
In order to understand the factors on which the values of emax, emin, and emax – emin of soil particles
depend, it is desirable to initially evaluate the void ratio variation of idealized spheres. Graton and
Fraser (1935), White and Walton (1937), and McGeary (1961) have studied this in detail. It has
been recently well summarized by Lade et al. (1998).
Figure 1 shows the five possible modes of packing of single-sized spherical particles along with
the void ratio for each type of packing. They are
1. Simple cubic (e = 0.91)—This is the loosest possible form of packing.
2. Single stagger (e = 0.65)—In this packing, each sphere in its own layer touches six other
spheres. The spheres in consecutive layers are stacked directly over each other.
3. Double stagger (e = 0.43)—It is similar to single stagger. However, each sphere in a layer
slides over and down to contact two spheres in the second layer.
4. Pyramidal (e = 0.35)—In this packing, each sphere in a layer lies over the depression between
and is in contact with four spheres in the layer below.
5. Tetrahedral (e = 0.35)—This is similar to pyramidal packing. However, each sphere in one
layer lies in the depression between and is in contact with three spheres.
.
Figure 1. Possible modes of packing of single-size spherical particles (adapted from Lade et al. 1998).
McGeary (1961) performed tests of single-sized spheres to determine its minimum void ratio [also
see Lade et al. (1998)], the average of which was about 0.6—with about 80% of spheres in single
stagger packing, and the remaining 20% in double stagger packing.
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Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.
Next we consider the packing of two types of spheres (binary packing) with two different
diameters (D = diameter of larger spheres and d = diameter of smaller spheres). Figure 2 shows a
schematic diagram of the theoretical variation of the minimum void ratio in a binary packing
arrangement (for small D/d ratios). It has primarily two stages:
1. Stage AB—When the volume of small spheres is zero, the minimum void ratio of the primary
fabric (larger spheres) is e1. With the increase in the percent of small spheres by volume, the
void space of the primary fabric is filled with smaller spheres reaching the lowest point B.
This is the filling of the void stage. At point B, the void ratio of smaller spheres is e2, and the
minimum void ratio of the larger and small spheres combined is emin(T).
2. Stage BC—Beyond point B, if additional smaller spheres are added, they will replace the
larger spheres. At point C, when the percent of smaller spheres is 100% by volume, the
minimum void ratio is e2. This phase is called replacement of smaller spheres.
At point B, assuming that the specific gravity of solids of large and small spheres is the same, it can
be shown that
emin(T ) 
n1n2
(2)
n1
n
 n1 2
e1
e2
where e1, n1 = respectively, minimum void ratio and porosity of larger spheres (point A)
e2, n2 = minimum void ratio and porosity, respectively, of smaller spheres (point C)
Figure 2. Variation of minimum void ratio emin
in binary packing.
Thus, percent of smaller spheres at B is given by
n 
(n1 ) 2 
weight of smaller spheres
 e2  (100)  emin(T ) (100)
VT 

n1
n
totalweight of spheres
e2
 n1 2
e1
e2
(3)
In obtaining Eqs. (2) and (3), it is assumed that small spheres can pass through the openings in the
void between the larger spheres. As mentioned before, McGeary (1961) observed e1 ≈ e2 ≈ 0.6 (for
single spheres). Thus, n1 = n2 = 0.375. Using these values in Eqs. (2) and (3),
39
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
emin(T ) 
n1n2
n1
n
 n1 2
e1
e2

(0.375)2
 0.164
0.375 (0.375)2

0.6
0.6
(4)
and percent of smaller spheres of fines,
n2
e2
(0.375) 2
0.6
VT 
(100) 
(100)  27.2%
n1
n2
0.375 (0.375) 2
 n1

e1
e2
0.6
0.6
n1
(5)
McGeary (1961) also conducted some laboratory tests with two different sizes of steel spheres. The
larger spheres had a diameter of 3.15 mm (D). The diameters of the small spheres (d) varied from
0.91 mm to 0.16 mm. This provided D/d ratios varying from 3.46 to 19.69. The minimum void
ratios of binary packing thus obtained for D/d = 3.46 and 4.77 is shown in Figure 3. The
approximate values of emin(T) and VT as defined in Figure 2 are given in Table 1 and are also plotted in
Figure 4. It can be seen that, for D/d ≥ 7, the magnitudes of emin(T) and VT are approximately constant
with emin(T) ≈ 0.2 and VT ≈ 27%. These values are not too far off from those calculated in Eqs. (4)
and (5).
Figure 3. Variation of emin vs. percent of smaller
steel spheres (based on McGeary 1961).
Figure 4. Variation of emin(T) and VT with D/d
(based on McGeary, 1961).
Table 1. Interpolated values of emin(T) and VT from
binary packing obtained from McGeary’s tests (1961)
D/d
3.46
4.77
6.56
11.25
16.58
19.69
emin(T)
0.425
0.344
0.256
0.216
0.213
0.192
VT (%)
41.3
26.2
25.0
27.5
26.3
27.5
3 BEHAVIOR OF TWO NATURAL SOIL MIXTURES
The concept presented above for the variation of void ratio of binary mixtures of steel spheres has
been compared with the behavior of mixtures of two soils (i.e., mixture of poorly graded sand, and
non-plastic fines) by Lade et al. (1998). Two types of poorly graded sand (Nevada sand 50/80 and
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Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.
80/200) were used. The median grain size (D50) of Nevada sand 50/80 was 0.211 mm and that of
Nevada sand 80/200 was 0.120. The median grain size (d50) of Nevada non-plastic fines was 0.050.
Figure 5 shows the variation of emax and emin with percent of fines for the two binary mixtures. The
general trend is similar to that shown in Figure 3. It is important to note that, in Nevada 50/80 sand
and fine mixture, the zone of filling the void is clearly seen, and the replacement of larger particles
starts when the fine percent reaches about 30% (D50/d50 = 4.2). These two phases are not clear for
Nevada 80/200 sand and fine mixtures (D50/d50 = 2.4) since the fine grains do not fit as well between
the void spaces present in the sand (Nevada 80/200) compared to those with higher D50/d50 (i.e.,
Nevada 50/80 and fines).
The change in the packing mode in granular soils (i.e. from filling the void to replacement of
larger particles) can also be seen from a plot of emax – emin vs. Fc (% of fines). Figure 6 shows a plot
for several granular soils (Cubrinovski and Ishihara 2002). The average plot of Fc ≤ 30% is given by
the relationship
emax  emin  0.43  0.0087Fc (%)
(6)
Equation (6) is generally related to filling the void zone. If Fc > 30% it becomes the zone of
replacement of larger particles and thus the slope of the average plot changes. Or,
emax  emin  0.57  0.004Fc (%)
Figure 5. Variation of emax and emin with
percent of fines (Lade et al. 1998).
(7)
Figure 6. Variation of emax – emin of sand with fine contents
(based on Cubrinovski and Ishihara, 2002).
4 EFFECT OF GRAIN SHAPE ON emax AND emin
Grain shape of natural clean sand has an effect on the maximum and minimum void ratios. The
shape of particles can be expressed by a term called roundness (R) which can be defined as
R
average radius of cornersand edges
radius of the maximum inscribed sphere
(8)
Figure 7 shows the approximate values of R for various particle shapes. Youd (1973) provided
relationships of emax and emin as functions of R and uniformity coefficient Cu (=D60/D10; where D60
and D10 are the diameters through which 60% and 10% of soil passes through, respectively). These
are shown in Figures 8 and 9 and are suggested to be valid with normal to moderately skewed grainsize distribution.
41
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
Figure 7. Definition of roundness R (Youd, 1973).
Figure 8. Variation of emax with Cu and R
(Youd, 1973).
Figure 9. Variation of emin with Cu and R
(Youd, 1973).
Figure 10. Comparison of emax vs. R.
Shimobe and Moreto (1995) have determined the variation of emax with R for 40 uniform clean sand
samples having a uniformity coefficient Cu ≤ 2. The experimental range of emax with R is shown in
Figure 10. The average plot can be expressed as
emax  0.642R 0.354
(9)
Also shown in Figure 10 is the variation predicted by Figure 8 for Cu = 2 which falls below the
average line [i.e. Eq. (9)] and close to the lower limit of the test results by Shimobe and Moreto
42
Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.
(1995). It is obvious from the test results that the predicted values of emax can vary over a wide range
since roundness is not an exact quantity.
Miura et al. (1997) conducted an extensive study of the physical characteristics of about 200
samples of granular material which included mostly clean sand, some glass beads, and lightweight
aggregates (L.W.A.). The grain shape was represented by a quantity called two-dimensional
angularity (A2D) Lees (1964a, 1964b) which is an alternative to roundness. Referring to Figure 11,
 x
Angularity  (180  ) 
r
(10)
A2 D   angularity
(11)
and
To determine A2D of a soil sample, the following procedure was used by Miura et al. (1997).
1. Consider about 20 sand grains having the size of about D50 (medium grain size) from each
sample.
2. Obtain enlarged photographic images of the grains taken in the vertical direction.
3. Determine A2D for each projection with the aid of Figure 12.
4. Calculate the average value of A2D for the soil.
Figure 11. Definition of angularity.
Figure 12. Angularity, A2D, estimation chart (Lees, 1964a and 1964b).
Figure 13 shows the results of the study of Miura et al. (1997) in the form of a plot of emax – emin
versus A2D that shows three representative linear relationships: (a) for D50 < 0.3 mm, (b) 0.3 mm ≤
D50 < 0.6 mm, and (c) D50 ≥ 0.6 mm. The slope of these lines increases with the decrease in D50
indicating that, for similar values of A2D, the range of emax – emin is higher for fine sand. This also
confirms the fact that, for a given value of D50, a decrease in angularity is also accompanied by a
decrease in emax – emin.
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
Figure 13. Variation of emax – emin with A2D and D50
(after Miura et al. 1997).
5 CORRELATIONS BETWEEN emax AND emin
Figure 14 shows a plot of emax and emin with median grain size (D50) for clean sand as reported by
Miura et al. (1997). The minimum and maximum void ratios show a general tendency to decrease
with the increase of median grain size. Based on the details shown in Figure 14, Figure 15 shows a
plot of emax versus emin. A regression analysis gives
emax  1.62emin
(12)
Figure 14. Variation of emax and emin with D50 for
clean sand (Miura et al. 1997).
Figure 15. Plot of emax and emin from Figure 14
(Miura et al. 1997).
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Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.
Cubrinovski and Ishihara (2002) analyzed a large number of clean sand samples (fine fraction
with grain size less than 0.075 mm, Fc ≤ 5%). Based on this analysis they suggested the following
relationship
emax  0.072  1.53emin
(13)
The data points upon which Eq. (13) is based and an additional 55 data points for clean sand given
by Patra et al. (2010) are shown in Figure 16. From this figure it appears that Eq. (12) may be taken
as a good average approximation. The difference in the angularity or roundness of the particles of
different soils is another major factor causing the scatter.
Figure 16. Plot of emax vs. emin for clean sand.
Based on best-fit linear regression lines, Cubrinovski and Ishihara (2002) also provided the
following relationships for other soils.
 Sand with fines (5 < Fc ≤ 15%)
emax  0.25  1.37emin
(14)
 Sand with fines and clay (15 < Fc ≤ 30%; Pc = 5% to 20%)
emax  0.44  1.21emin
(15)
 Silty soils (30 < Fc ≤ 70%; Pc = 5% to 20%)
emax  0.44  1.32emin
(16)
where Fc = fine fraction for which grain size is smaller than 0.075 mm
Pc = clay-size fraction (< 0.005 mm)
6 CORRELATIONS BETWEEN emax – emin AND MEDIAN GRAIN SIZE, D50
Based on a very large database, Cubrinovski and Ishihara (1999, 2002) developed a unique
relationship. The database included results from clean sand, sand with fines, and sand with clay,
silty soil, gravelly sand, and gravel. This relationship is shown in Figure 17. In spite of some scatter,
the average line can be given by the relation,
45
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
emax  emin  0.23 
0.06
D50 (mm)
(17)
It appears that the upper and lower limits of emax – emin vs. D50 as shown in Figure 17 can be
approximated as
 Lower limit
emax  emin  0.16 
0.045
D50 (mm)
(18)
0.079
D50 (mm)
(19)
 Upper limit
emax  emin  0.29 
Figure 17. Correlation between emax – emin and D50
(after Cubrinovski and Ishihara, 1999, 2002)
The approximate ranges of emax – emin for clean sand and silty sands are given in Table 2.
Table 2. Range of emax – emin for sandy soils
Soil
Clean sand
Silty sand
Silty sand
Silty sand
Fines (%)
<5
5 – 10
10 – 20
20 – 30
Gravel (%)
<5
<5
<5
<5
emax – emin
0.25 – 0.45
0.45 – 0.55
0.5 – 0.6
0.6 – 0.7
Based on the preceding review it appears that there are several factors such as uniformity
coefficient, angularity, grain size distribution, and fine contents that affect the maximum and
minimum void ratios of granular soils. However, the stress-strain relationships, compressibility, and
other practical geotechnical parameters of granular soils can be reasonably predicted from emax – emin,
but not separately from emax and/or emin. In spite of some scatter, which is to be expected, emax – emin
bears a unique relationship with the median grain size (D50). Hence D50 can be used as a parameter
to approximately predict correlation of volumetric strain, relative density, compaction characteristics,
strength, and other geotechnical properties of granular soils. Some examples of this are given in the
following sections.
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Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.
7 VOLUMETRIC STRAIN POTENTIAL
The volumetric strain potential (v) is the volumetric strain that a granular soil will undergo when it
is densified and the void ratio changes from emax to emin, or
v 
emax  emin
1  emin
(20)
A similar parameter like volumetric strain potential was also proposed by Terzaghi [in
Erdbaumechanik (Terzaghi, 1925); also see Bjerrum et al. (1960)] which was referred to as
compactibility (C), or
C  compactibility 
emax  emin
emin
(21)
Volumetric strain potential will have significant influence on liquefaction of granular soils.
Based on the results of Cubrinovski and Ishihara (2002) and Patra et al. (2010), the volumetric strain
potential of several sands is shown in Figure 18. The average plot is practically linear. With a
variation of about ±10 to 15%, the volumetric strain potential can be expressed as
v (%)  22(emax  emin )  11
(22)
Combining Eqs. (17) and (22), we obtain
v (%)  5.06 
1.32
 11
D50 (mm)
(23)
Figure 18. Plot of v vs. emax – emin.
47
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
8 CORRELATION BETWEEN RELATIVE DENSITY OF CLEAN SAND WITH D50 AND
COMPACTION ENERGY
Patra et al. (2010) conducted Proctor compaction tests on 55 clean sand samples—most of which
were poorly graded. They also determined emax and emin based on ASTM test procedures given in D4253 and D-4254. The compaction tests included standard Proctor tests (compaction energy, E =
600 kN-m/m3), reduced standard Proctor tests with 15 standard Proctor hammer blows per layer (E
= 360 kN-m/m3), modified Proctor tests (E = 2700 kN-m/m3), and reduced modified Proctor tests
with 12 Proctor hammer blows per layer (E = 1300 kN-m/m3).
Based on the laboratory tests, it has been shown that emax, emin, and void ratio at maximum density
of compaction are as follows:
emax  0.6042D500.304
( r 2  0.7623)
(24)
emin  0.3346D500.491 (r 2  0.8516)
(25)
es  0.4484D500.356
( r 2  0.8040)
(26)
em  0.3825D500.04
( r 2  0.8095)
(27)
ers  0.5039D500.327
(r 2  0.7809)
(28)
erm  0.4087D500.389
( r 2  0.8076)
(29)
where D50 is in mm, and es = void ratio from standard Proctor tests, ers= void ratio from reduced
standard Proctor tests, em = void ratio from modified Proctor tests, and erm = void ratio from reduced
modified Proctor tests
The maximum relative density of compaction can then be correlated to the energy of compaction
(Patra et al. 2010) as,
Dr  AD50 B
(30)
A  0.216ln E  0.850
(31)
B  0.03ln E  0.306
(32)
where
In Eq. (30), Dr is in fraction, and D50 is in mm.
Figure 19 shows a comparison of the relative densities predicted by Eqs. (30)–(32) and those
obtained from laboratory tests.
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Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.
Figure 19. Comparison of Dr predicted from Eqs. (30)–(32) with the
experimental results (Patra et al. 2010).
9 CORRELATIONS BETWEEN RELATIVE DENSITY (Dr), D50, AND STANDARD
PENETRATION NUMBER (N)
The field standard penetration number in granular soils, N, varies with the effective overburden
pressure,  o . It can be normalized to a standard effective overburden pressure of 98 kN/m2 (≈
atmospheric pressure) as (Liao and Whitman, 1986)


98
N1  N 
2 
 o (kN/m ) 
0.5
(33)
where N1 = normalized standard penetration number
Figure 20 shows a correlation between N1 / Dr2 and emax – emin provided by Cubrinovski and
Ishihara (1999) which is based on high quality undisturbed samples of silty sand, clean sand, and
gravels recovered from natural soil deposits. The average plot shown in Figure 20 can be expressed
as
N
9
(34)
C D  12 
Dr (emax  emin )1.7
Hence
N

Dr   1 (emax  emin )1.7 
9

0.5
(35)
Now, combining Eqs. (17), (33) and (35),
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
1.7
 

0.06 

 N  0.23 

D50   98 


 
Dr  

9
 o 



0.5
(36)
Figure 20. Development of Eq. (34) (Cubrinovski and Ishihara, 1999).
It is important to note that the N value referred to in Eqs. (34)‒(36) approximately relates to an
average energy ratio of 78% for the SPT tests.
Figures 21 and 22 show comparisons between Dr predicted using Eq. (34) with that measured,
respectively, for gravelly and sandy soils. Kulhawy and Mayne (1990) also provided the following
relationship to estimate Dr in which D50 is a factor, or
Figure 21. Measured Dr vs. Eq.(34) for gravel
deposits (Cubrinovski and
Ishihara, 1999)
Figure 22. Measured Dr vs. Eq.(34) for sandy
soil deposits (Cubrinovski and
Ishihara, 1999)
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Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.


( N1 )60
Dr  

 (60  25log D50 )CA COCR 
where
0.5
(37)
D50 = median grain size, in mm
(N1)60 = normalized standard penetration number for an energy ratio of 60%
 t years
CA  factorfor aging effect  1.2  0.05log

 100 
(38)
COCR  factor for overconsol idation  OCR1.8
(39)
where OCR = overconsolidation ratio
10 CORRELATIONS OF CONE PENETRATION TEST RESULTS WITH D50
Based on the results of a large number of cone penetration tests, Anagnostopoulos et al. (2003) have
provided a correlation of friction ratio, Rf, with the median grain size, or
R f (%)  1.45  1.36log(D50 )
(electriccone)
(40)
R f (%)  0.7811 1.6111log(D50 )
(mechanical cone)
(41)
where D50 is in mm, and
R f (%) 
fc
(100)
qc
(42)
where fc = frictional resistance of sleeve located above the cone
qc = cone penetration resistance
The plot of Rf versus D50 from which Eqs. (40) and (41) were developed is shown in Figure 23.
Figure 23. Plot of Rf vs. D50 (after Anagnostopoulos, 2003).
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3r International Conference on New Developments in Soil Mechanics and Geotechnical Engineering,
28-30 June 2012, Near East University, Nicosia, North Cyprus
11 CORRELATION BETWEEN CONE PENETRATION RESISTANCE (qc), STANDARD
PENETRATION RESISTANCE (N60) AND D50
The cone penetration resistance (qc), field standard penetration resistance (N60), and the median grain
size (D50) have been correlated by several investigators in the past. These correlations can be
expressed in a general form as
 qc 
 
 pa   cD a
50
N 60
(43)
where D50 is in mm and pa = atmospheric pressure with the same units as qc
Results of some of these studies are summarized below.
 Burland and Burbidge et al. (1985)
 qc 
 
 pa   8D 0.305
50
N 60
(see Figure 24a)
(44)
 Based on the data of Robertson and Campanella (1983) and Seed and de Alba (1986)
 qc 
 
 pa   6D 0.228
50
N 60
(see Figure 24a)
(45)
Figure 24. Continued.
52
Maximum and minimum void ratios and median grain size of granular soils: their importance and correlations with material properties
Das, B.M., Sivakugan, N., Atalar, C.
Figure 24. Variation of (qc /N60) with D50. (a) Adapted from Terzaghi et al.
(1996); (b) Adapted from Anagnostopoulos (2003); (c) Adapted from
Kulhawy and Mayne (1990).
 Based on the data of Anagnostopoulos et al. (2003)
 qc 
 
 pa   7.6429 D 0.26
50
N 60
(see Figure 24b)
(46)
 Based on Kulhawy and Mayne (1990)
 qc 
 
 pa   5.44 D 0.288
50
N 60
(see Figure 24c)
(47)
12 CONCLUSIONS
A review of the various parameters controlling the magnitudes of maximum and minimum void
ratios of sandy soils is presented. The magnitudes of emax and emin are primarily functions of
uniformity coefficient, angularity, and fine contents. Based on the available results in the literature,
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28-30 June 2012, Near East University, Nicosia, North Cyprus
the relationships between emax and emin for soils with various percentages of fine contents are
included. Also the experimental results suggest that emax – emin (but not emax and emin separately)
control the primary geotechnical properties of granular soil, such as volumetric strain potential (v),
relative density (Dr), cone penetration resistance (qc), and standard penetration number (N). In spite
of some scatter, the median grain size (D50) is probably the best parameter for correlation with emax –
emin and, hence, the geotechnical properties related to strength and compression. Also provided in
this paper are some correlations presently available in literature between D50 and Dr, v, and qc /N60.
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