ORIGINAL PAPER

Steady State Strength Behavior of Yamuna Sand

Sourav De Æ Prabir Kumar Basudhar

Received: 6 November 2006 / Accepted: 12 November 2007 / Published online: 30 November 2007

� Springer Science+Business Media B.V. 2007

Abstract The paper pertains to the study of steady

state or residual strength of sandy soils (Yamuna sand

lying in the Indo-Gangetic alluvial plains) by con-

solidated rebounded drained triaxial test with volume

change measurements and strain-controlled consoli-

dated undrained test as well. The observed behavior

obtained from these two tests is then compared to

check their comparative merit. The same was also

compared with those of Ganga and Toyoura sand,

and with the predicted behavior obtained by using a

semi empirical model. The results obtained from

rebounded drained and undrained tests are found to

be in good agreement. The curvature of ultimate

steady state line of Yamuna sand is similar in trend to

Ganga and Toyoura sand in the initial mean effective

principle stress range; but the experimental observa-

tions with reference to Yamuna sand is not in good

agreement with the model predictions in the region of

higher mean normal stress. A semi empirical general

model has been developed fitting the data for better

prediction of the steady state behavior.

Keywords Liquefaction � Rebounded drained

triaxial test � Steady state strength

Nomenclature

(Dr)c Relative Density after consolidation

(dimensionless)

Cc Coefficient of curvature (dimensionless)

Cu Uniformity coefficient (dimensionless)

D50 Mean grain size (mm)

dI Intermediate particle dimension (mm)

dL Longest particle dimension (mm)

Dro Relative density of the steady state line,

in percentage (dimensionless)

Drs Relative density at the steady state line

expressed as a ratio (dimensionless)

dS Shortest particle dimension (mm)

e Void ratio (dimensionless)

emax Maximum void ratio (dimensionless)

emin Minimum void ratio (dimensionless)

G Specific gravity (dimensionless)

p0 Effective mean stress (kPa)

k Slope of the steady state line (dimensionless)

r03 Effective confining pressure after rebound

(kPa)

r3c Consolidation pressure (kPa)

1 Introduction

Evaluation of steady state strength of sands from

static drained and undrained tests and interpretation

of the results has drawn some attention in recent

years. Castro (1969) observed that cohesionless

soils of different densities exhibit three basic

Sourav De � P. K. Basudhar (&)

Department of Civil Engineering, Indian Institute of

Technology Kanpur, Kanpur 208016, Uttar Pradesh, India

e-mail: pkbd@iitk.ac.in

Sourav De

e-mail: sourav_de_net@yahoo.co.in

123

Geotech Geol Eng (2008) 26:237–250

DOI 10.1007/s10706-007-9160-5

phenomenological behaviors termed liquefaction,

limited liquefaction and dilation that are different

over a broad range of strain levels. This state, along

with the requirement of constant velocity, has been

defined as the steady state of deformation (Castro

and Poulos 1977; Poulos 1981). Poulos et al. (1985)

made systematic measurement of steady state

strength through stress-controlled CU triaxial tests.

Poulos et al. (1985) and Fear and Robertson (1995)

studied the effect of grain characteristics and grain

size compositions of soils and Been and Jefferies

(1985), Sladen et al. (1985) and Pitman et al.

(1994) studied the influence of fines on the position

and slope of the steady state line. Norris et al.

(1997) presented an alternative effective stress

approach for determining the undrained stress–

strain curve and the undrained effective stress path

of a monotonic test, up to large strain level from

consolidated-rebounded drained triaxial tests with

accompanying volume change measurements.

Based on experimental studies on Toyoura sand,

Cubrinovski and Ishihara (1999, 2000) developed a

semi empirical model to predict the steady state

response of sands.

In the present paper details of a study on steady

state strength behavior of Yamuna sand (belong-

ing to the Indo-Gangetic Alluvial plains) obtained

from consolidated rebounded drained triaxial test

with volume change measurements and strain-

controlled consolidated undrained test as well. The

observed behaviors obtained from these two tests

were then compared to check the comparative

merit of the two approaches. A comparative study

of the steady state response of Yamuna sand, as

observed, with those of Ganga and Toyoura sand

and with their respective predicted behavior

obtained by using a semi empirical model has

also been made with a view to supplement to the

present state of knowledge with data on regional

soil deposits.

2 Experimental Investigation

In this study the steady state strength behavior of

Yamuna sand has been assessed by using the method

proposed by Norris et al. (1997). The details of

experimental method can be obtained from the

original paper.

2.1 Sample Preparation

Moist Tamping and Dry Pluviation (Mulilis et al.

1977; DeGregorio 1990) have been used for sample

preparation. For preparing the sample of relative

density 20%, 124.5 g of dry sand was thoroughly

mixed with 8% of distilled water and was divided in

three equal parts. Each part was then carefully poured

into the circular split-mould by a spoon and leveled

with a spatula. The encapsulating rubber membrane

was kept in position by applying a suction corre-

sponding to 330 mm of mercury. Each soil layer was

gently tamped 10 times by a tamping rod, 10 mm in

diameter. The top of the compacted layer was then

scratched to promote bonding of the next layer. The

procedure was repeated for each of the three layers.

For preparing the sample of relative density 35%,

128 g of dry sand was weighed and placed inside an

8 cm diameter funnel keeping the opening covered

with a finger. Positioning the funnel over the split

mould, the finger was removed and the sand was

allowed to rain freely from the funnel into the split

mould from a height of 20 mm over the sand level. No

tamping was done here, but the suction as specified

was provided. Some alteration in the above procedure

was made to prepare samples of higher relative

density values. For preparing a sample of 55% relative

density, 133.1 g of dry sand was taken into a bowl.

Then it was placed inside the funnel and rain freely

from approximately 40 mm height into the circular

split mould. After dropping 1/3rd of 133.1 g, 5 blows

were provided by a 6 mm diameter tamper. All the

while, suction was provided as stated earlier. But due

to the alteration of the sample volume during spec-

imen preparation, the initial relative density values

that can be achieved for three types range between 24

to 28%, 38 to 42% and 58 to 62% respectively.

2.2 Description of Material Used

The sand that is used for experiments is natural river

sand from the Yamuna river system from Kalpi

region in Yamuna basin (Fig. 1). The basic properties

of this sand as determined in the laboratory are given

in the Table 1.

From the grain size distribution curve for the soil

shown in Fig. 2 the soil can be classified as per

Unified Soil Classification System (USCS) as poorly

238 Geotech Geol Eng (2008) 26:237–250

123

graded sand with little gravel and fine content with a

group symbol SP. A microscopic view of the grains

(Fig. 3) at around 50 times zoom shows the grains to

be angular.

Tables 2 and 3 show respectively the mineralog-

ical composition and morphological characteristics of

Kalpi sand (Rahim 1989). Some of the terminologies

used here are defined in the literature as follows.

Wadell (1932) defined sphericity as the ratio of the

surface area of a sphere with the same volume as that of

the particle and the actual surface area of the particle;

roundness is a measure of the sharpness of the corners

and edges of the grains and is defined as the ratio of

the average radius of curvature of the corners to the

radius of the largest inscribed circle. The shape factor

(SF) is an index to define the aspects of particle shape.

SF ¼ dSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dL=dIð Þp ð1Þ

where, dL, dI, dS represent the longest, intermediate

and the shortest particle dimensions respectively.

Fig. 1 Yamuna basin

showing the Kalpi area in

the circle

Table 1 Properties of Kalpi sand

Sand emax emin G D50 (mm) Cu Cc

Kalpi 0.91 0.57 2.66 0.48 2.60 0.72

Geotech Geol Eng (2008) 26:237–250 239

123

2.3 Planning of Experiment

After preparing a sample and de-aired water was

flushed through the specimen to accelerate the

saturation process. When the B-value was found to

be greater than 0.95 the specimen was kept under

the process of consolidation pressure of 300 kPa.

The volume change was recorded (using a U-tube

burette with the least count of 0.00188 ml/mm)

during isotropic consolidation so that the relative

density at the end of consolidation process could be

determined. It was found that the relative density

after consolidation process ranges for three types of

samples between 34 ± 1%, 49 ± 1% and 69 ± 1%.

After that the specimen was rebounded to different

cell pressures of 280, 260, 240, 220 and 200 kPa

and the rebounded volume change was noted. Then

the strain-controlled drained triaxial tests were

conducted with the strain rate of 0.24 mm/min.

For each rebounded cell pressure three tests were

carried out with utmost care to avoid the inconsis-

tency. Also the results were checked with the

undrained triaxial tests carried out with the samples

prepared in the same fashion. Fifty samples were

tested with rebounded drained and undrained triax-

ial tests with three types of relative densities.

Corresponding to each set, several tests were

performed and average values of the test results

that did not vary significantly have been reported.

Thus, the sample behavior may be construed to be

representative. The tests were conducted up to 20%

strain level beyond which it was difficult to perform

the tests due to excessive bulging. The experiments

showed that as long as the drained and undrained

test samples were formed in the same fashion, the

undrained test response could be assessed from

drained triaxial test.

Fig. 3 Microscopic view

of Yamuna (Kalpi) sand

(509)

0

10

20

30

40

50

60

70

80

90

100

0.01 0.10 1.00 10.00

Sieve size (mm)

Per

cen

t fi

ner

Fig. 2 Particle size distribution curve

Table 2 Mineralogical composition of Kalpi sand (Rahim

1989)

Sand Percentage mineral content

Quartz Feldspar Mica Carbonate Chlorite

Kalpi 40 40 1–2 18 0

Table 3 Morphological characteristics of Kalpi sand (Rahim

1989)

Sand Average shape

factor

Average

sphericity

Average

roundness

Kalpi 0.596 0.698 0.10–0.20

240 Geotech Geol Eng (2008) 26:237–250

123

3 Results and Discussions

3.1 Stress–Strain and Volumetric Strain-Axial

Strain Behavior in Strain-controlled Drained

Triaxial Tests

Figure 4 depicts the stress–strain characteristics of

five samples of relative density, (Dr)c of 34 ± 1%,

which were all initially consolidated to a cell pressure

of 300 kPa, then rebounded to 280, 260, 240, 220 and

200 kPa separately and subsequently subjected to

deviator stress under drained condition. Figure 5

depicts the volume change characteristics of the same

samples. From Figs. 4 and 5 and from the isotropic

volumetric strain data during rebound as presented in

Table 4, the whole undrained response was evaluated

by the Norris et al. (1997) method and is shown in the

bold line. The steady state is obtained where the

undrained as well as corresponding rebounded

drained stress–strain response become horizontal. If

this does not happen within the strain levels up to

which testing were done, then the stress–strain curves

for the drained tests may have to be extrapolated to

obtain the steady state. Extrapolation of data may not

be very correct if the region is far off from the

measured domain. However, due to practical diffi-

culties as the samples undergone large amount of

bulging while the deviator stress was being applied,

some tests could not be performed till the steady state

is reached. In such an eventuality extrapolation had to

be adopted to find the value of the steady state

strength. It may not be out of place to mention here

that similar extrapolation procedure had been adopted

earlier in presenting the stress–strain behavior of sand

and predicting its residual shear strength and critical

void ratio and were reported in standard text books

(Lambe and Whitman 1969). This is particularly true

when the density becomes larger. Also a sample with

rebounded cell pressure lying between two actually

conducted rebounded tests may have to be interpo-

lated to obtain the steady state (Tables 5, 6).

Similarly Figs. 6 and 7 represent stress–strain and

volume change data from strain-controlled rebounded

CD tests for (Dr)c of 49 ± 1%. Figures 8 and 9

0

100

200

300

400

500

600

700

800

900

0 4 8 12 16 20

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

280 kPa

260 kPa

240 kPa220 kPa

200 kPa

(Dr)c = 34±1%,σ3c = 300 kPa

3σ ′

Fig. 4 Stress–Strain behavior for (Dr)c = 34 ± 1%

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

0 4 8 12 16 20

Axial Strain (%)

Vo

lum

etri

c S

trai

n (

%)

280 kPa

260 kPa

240 kPa

220 kPa

200 kPa

(Dr)c = 34±1%,σ3c = 300 kPa

3σ ′

Fig. 5 Volume change behavior for (Dr)c = 34 ± 1%

Table 5 Volumetric strain during rebound to lower cell

pressures for (Dr)c = 49 ± 1%

Rebounded cell

pressure (kPa)

Volumetric strain

during rebound (%)

280 0.1010

260 0.1525

240 0.2122

220 0.2520

200 0.2829

Table 4 Volumetric strain during rebound to lower cell

pressures for (Dr)c = 34 ± 1%

Rebounded cell

pressure (kPa)

Volumetric strain

during rebound (%)

280 0.3227

260 1.1176

240 1.4698

220 1.5383

200 1.6201

Geotech Geol Eng (2008) 26:237–250 241

123

represent the same respectively for (Dr)c of 69 ± 1%.

It can be noted from the volumetric strain-axial strain

curves (Figs. 5, 7 and 9) that volume change behav-

ior depends greatly on the relative density of the soil

sample. While loose sand shows completely contrac-

tive behavior, medium or dense sand tend to expand

beyond a certain axial strain. From Fig. 4 we can say

that it shows strain-softening behavior as the sand is

in a loose state with the final relative density of

34 ± 1%. Figure 6 shows quasi-steady state behavior

for relative density 49 ± 1%. This dilative tendency

increases with the increase in relative density. Also

this method of determination of steady state strength

amply depicts the importance of this volume change

characteristics. It can be seen that the undrained

response can rise after formation of a quasi-steady

state to even beyond the initial peak stress for

medium or dense samples. This is possible only if the

drained response of a particular rebounded sample is

intersected twice by the vertical lines generated from

corresponding volumetric strain curve, which in turn,

is possible only if this corresponding volume change

Table 6 Volumetric strain during rebound to lower cell

pressures for (Dr)c = 69 ± 1%

Rebounded cell

pressure (kPa)

Volumetric strain

during rebound (%)

280 0.0774

260 0.1273

240 0.1591

220 0.2122

200 0.2481

0

200

400

600

800

1000

1200

1400

0 4 8 12 16 20

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

280 kPa

260 kPa

240 kPa

220 kPa

200 kPa

(Dr)c = 49±1%, σ3c = 300 kPa

3σ ′

Fig. 6 Stress–Strain behavior for (Dr)c = 49 ± 1%

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

0 4 8 12 16 20

Axial Strain (%)

Vo

lum

etri

c S

trai

n (

%)

280 kPa

260 kPa

240 kPa

220 kPa

200 kPa

(Dr)c = 49±1%, σ3c = 300 kPa

3σ ′

Fig. 7 Volume change behavior for (Dr)c = 49 ± 1%

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 4 8 12 16 20

Axial Strain (%)

Vo

lum

etri

c S

trai

n (

%)

280 kPa

260 kPa

240 kPa

220 kPa

200 kPa

(Dr)c = 69±1%, σ3c = 300 kPa

3σ ′

Fig. 9 Volume change behavior for (Dr)c = 69 ± 1%

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 4 8 12 16 20

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

280 kPa

260 kPa

240 kPa

220 kPa

200 kPa

(Dr)c = 69±1%,σ3c = 300 kPa

3σ ′

Fig. 8 Stress–Strain behavior for (Dr)c = 69 ± 1%

242 Geotech Geol Eng (2008) 26:237–250

123

curve of the rebounded sample shows dilative

tendency. Figure 8 shows almost a complete dilative

behavior of no-flow condition.

3.2 Comparison of Stress–Strain Curves

Determined by Rebounded CD and CU

Triaxial Tests

Figures 10–12 show the comparison of stress–strain

behavior (variation of deviator stress with axial

strain) as evaluated from the series of rebounded

strain-controlled consolidated drained triaxial tests

and strain-controlled consolidated undrained triaxial

test, at the final relative density of 34 ± 1%,

49 ± 1% and 69 ± 1% respectively. The dotted

lines indicate the extrapolated values.

It is seen that the behavior evaluated from the

series of strain-controlled consolidated drained tests

on rebounded samples with volume change measure-

ments, is in excellent agreement with the result

obtained from the strain-controlled consolidated

undrained tests. The small deviation of the predicted

points from the observed undrained behavior may be

due to the variation in the density of the drained test

samples. It was very difficult to measure the volume

of the formed test samples accurately especially the

bulked samples.

As the behavior observed from the two different

types of tests that were performed are in very close

agreement with each other, it can be concluded that

the sample preparation and measurements made are

reliable.

3.3 Stress–Strain Behavior in Strain-controlled

Consolidated Undrained Triaxial Tests

Strain-controlled consolidated undrained (CU) tests

were conducted on samples with three relative

densities, subjected to different consolidation pres-

sure. The stress–strain behavior is presented in

Fig. 13.

Figure 13 depicts that samples with same initial

relative density, when tested under different consol-

idation stresses, tend to converge to same stress at

large strain levels. Thus the position of the ultimate

0

100

200

300

400

500

600

700

800

900

0 4 8 12 16 20

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

From CU test

From rebounded

CD test

(Dr)c = 34±1%,σ3c = 300 kPa

Fig. 10 Comparison of stress–strain behavior at 34 ± 1%

relative density

0

200

400

600

800

1000

1200

1400

0 4 8 12 16 20

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

From CU test

From rebounded

CD test

(Dr)c = 49±1%, σ3c = 300 kPa

Fig. 11 Stress–strain behavior at 49 ± 1% relative density

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 4 8 12 16 20

Axial Strain (%)

Dev

iato

r S

trss

e (k

Pa)

From CU test

From rebounded

CD test

(Dr)c = 69±1%, σ3c = 300 kPa

Fig. 12 Stress–strain behavior at 69 ± 1% relative density

Geotech Geol Eng (2008) 26:237–250 243

123

steady state line is unique with respect to the initial

mean stresses but the position of the quasi-steady

state lines for different initial mean stresses are

different. They deviate more from the ultimate steady

state line for lower initial mean stresses. This

indicates that the behavior is highly sensitivity to

the density. This observation supports the existence

of a unique fabric developed during the steady state.

3.4 Comparison of Behavior at Different Relative

Densities

Figure 14 shows stress–strain behavior of samples

prepared at three different relative densities that were

initially subjected to the same consolidation pressure

of 300 kPa and Fig. 15 shows the corresponding

effective stress path.

The undrained behavior corresponding to the three

relative density values, as shown in Fig. 14, are

distinct from each other. At comparatively loose state

i.e. (Dr)c = 34 ± 1%, after the peak stress is attained,

the shear strength drops until it reaches the ultimate

steady state or in other word residual state. For sands

with medium relative density (i.e. 49 ± 1%), there is

a drop after the peak shear stress is attained, till a

quasi steady state is reached, after which the sample

regains strength and takes more stress till the ultimate

steady state is obtained at large deformation level. At

dense state i.e. (Dr)c = 69 ± 1%, there is hardly any

drop in shear stress till the ultimate steady state is

obtained at very large strain level.

Similar behavior is depicted also in the effective

stress paths shown in Fig. 15. The effective stress

path at 34 ± 1% relative density shows continuous

loss in shear strength as it traces towards the origin

but for 49 ± 1% relative density it shows some loss

in q as it starts moving towards origin during the

quasi steady state stage before it shoots back again

showing increasing trend in strength. But the effec-

tive stress path at 69 ± 1% relative density shows a

dilative behavior all through.

It is seen that as density increases the true or

ultimate steady state is obtained at a larger strain

level. Figure 16 shows, for same initial consolidation

stress of 300 kPa, ultimate steady state was obtained

after 40%, 30% and 26% axial strain respectively for

final relative densities of around 69 ± 1%, 49 ± 1%

and 34 ± 1% with deviator stress of around 1400,

0

200

400

600

800

1000

1200

1400

1600

1800

0 4 8 12 16 20

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

(Dr)c = 69±1%, σ3c = 400kPa, 300 kPa and 200 kPa

Fig. 13 Stress–strain behavior at different consolidation

stresses

0

200

400

600

800

1000

1200

1400

0 5 10 15 20Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

(Dr)c = 34±1%

(Dr)c = 49±1%

(Dr)c = 69±1%

σ3c = 300 kPa

Fig. 14 Stress–strain behavior at different relative densities

0

200

400

600

800

1000

1200

1400

1600

0 500 1000

p'=(σ '1+2σ '3)/3 (kPa)

= q

σ1-

σ3

k( P

a)

(Dr)c = 34±1%

(Dr)c = 49±1%

(Dr)c = 69±1%

σ3c = 300 kPa

Fig. 15 Effective stress paths at different relative densities

244 Geotech Geol Eng (2008) 26:237–250

123

920 and 270 kPa, respectively. Figure 17 shows that

for different relative densities the effective stress path

comes over a common steady state envelope.

In Figs. 15–17 the dotted lines indicate the

extrapolated values to obtain the ultimate steady

state strength as the axial strain level of experiment

ended much before that. This is because triaxial

specimens tend to bulge at large strain. So the steady

state strength becomes inaccurate due to uncertainty

in the actual shape of the specimen. It also shows that

the line of extrapolation is much more at higher

relative density as the steady state is obtained at

higher deformation level. This may be considered a

limitation of the present study.

3.5 Ultimate Steady State Line (USSL) of

Yamuna Sand

Under undrained loading condition, irrespective of

the initial state of the specimens, when the strain

level is large enough, the soil mass tends to be in a

state of continuous deformation under constant shear

stress and constant mean stress, there exists a

correlation between the void ratio and the mean

effective principle stress. This line is referred to as

the Ultimate Steady State Line (USSL). Figure 18

shows the ultimate steady state line for Yamuna sand.

In triaxial compression tests, in the range of initial

effective mean stresses up to 300 kPa, the rate of

decrease in void ratio with mean stress is lesser in

comparison to the same there after. This implies that

during undrained loading, the initial effective mean

stresses in this range have lesser effect on the

compressibility behavior of sand than in the higher

range. As the samples were prepared in different

ways to attain different relative densities, it is

assumed here that the position of the ultimate steady

state line is independent of the method of sample

preparation.

Figure 19 shows the shear stress, q/2, and the

mean effective principle stress, p0, computed for each

relative density. It is seen that all the data points fall

in a straight line passing through the origin indicating

that a unique friction angle is mobilized at the

ultimate steady state, a fact earlier reported by

Negussey et al. (1988) and Verdugo and Ishihara

(1996). For Yamuna sand, the experimental results

indicated that the slope of the line is close to 0.824,

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30 35 40 45 50

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

(Dr)c = 34±1%

(Dr)c = 49±1%

(Dr)c = 69±1%

σ3c = 300 kPa

Fig. 16 Ultimate steady strength at different relative densities

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800 1000

p'=(σ '1+2σ '3)/3 (kPa)

= q

σ1-

σ3

k( P

a)

(Dr)c = 34±1%

(Dr)c = 49±1%

(Dr)c = 69±1%

σ3c = 300 kPa

steady state envelope

Fig. 17 Effective stress paths showing the steady state

envelope

0.65

0.7

0.75

0.8

0.85

100 1000

Mean effective principal stress,p' (kPa)

Vo

id r

atio

,e

Fig. 18 Ultimate steady state line (USSL) of Yamuna Sand

Geotech Geol Eng (2008) 26:237–250 245

123

confirming a friction angle in the steady state of

deformation of around 39.5�.

4 Comparison

4.1 Comparison of the Behavior of Yamuna Sand

with Ganga and Toyoura Sand

The test data of Yamuna sand is compared with the

same on Ganga sand (Datta 2005) and Toyoura sand

(Yoshimine and Ishihara 1998). A comparison of the

grain size distributions of the three sands as presented

in Fig. 20 shows that except for Yamuna sand the

other two sands (Ganga sand and Toyoura sand) show

very less variation in this respect. A comparative

study of the three sands is presented in Table 7; it

shows that even though the gradation curves of the

Ganga and Toyoura sands are similar and very close

to each other, their respective uniformity coefficient

values, 2.30 and 1.7 differ significantly. Toyoura sand

is poorly graded sand. But, the values of the

coefficient of curvature (Cc) of Yamuna and Ganga

sand being 0.72 and 0.90 respectively are much close

to each other than Toyoura sand. According to

Unified Soil Classification System (USCS) both the

Yamuna as well as Ganga are poorly graded sand

with the group symbol SP. The particles of Toyoura

sand are composed of 75% quartz, 22% feldspar and

3% magnetite (Oda et al. 1978). The mineralogical

composition and morphological characteristics of

Yamuna (Kalpi) and Ganga sand as shown in

Tables 8 and 9 indicate significant difference in these

respects.

Stress–strain behavior at different void ratio for a

particular confining stress presented in Fig. 21 indi-

cates that Ganga and Toyoura sands show close

resemblance with respect to strength and stress–strain

characteristics. This may be due to similarity of their

grain size distribution curve and particle shape. In

contrast, Yamuna sand being very angular and larger

in size shows more strength compared to the others.

Figure 22 shows the comparison of the variation of

effective stress path with different relative densities

for the above sands. The qualitative trend of p0-q

diagram of Yamuna sand matches reasonably well

with that of Ganga and Toyoura sand. But, it is

interesting to note from the p0-q diagram of Ganga

sand that its strength is very close to that of Yamuna

sand in spite of the fact that they are so much

different from each other with respect to their

mineralogical composition and morphological

characteristics.

Finally Fig. 23 compares the ultimate steady state

lines for the three sands. As for Yamuna and Ganga

sand the experiment is done with three different final

relative densities, and a trend line is drawn with these

three points. As the grain size and shape are similar

for Ganga and Toyoura sand, their ultimate steady

state line (USSL) overlaps for low ranges of mean

effective principle stress beyond which there is a

deviation. These studies have shown the variation in

the steady state properties is to a great extent is due to

differences in the grain characteristics and grain size

compositions of soils. Poulos et al. (1985) pointed

out that the vertical position of the steady state line is

chiefly influenced by the grain distribution, whereas

the grain shape affects the slope of the line. The fines

0

200

400

600

800

0 200 400 600 800 1000

Mean effective principal stress, p' (kPa)

Sh

ear

stre

ss, q

/2 (

kPa)

Yamuna sand (at steady state)

Fig. 19 Strength envelope at the steady state

0

10

20

30

40

50

60

70

80

90

100

0.01 0.10 1.00 10.00

Sieve size (mm)

Per

cen

t fi

ner

Ganga sand

Yamuna sand

Toyoura sand

Fig. 20 Comparison of grain size distribution

246 Geotech Geol Eng (2008) 26:237–250

123

also to some extent influence both the position and

the slope of the steady state line (Been and Jefferies

1985; Sladen et al. 1985; Pitman et al. 1994). The

curves are also observed to be in good agreement

with the fact that sands with rounded grains have

fairly flat steady state lines whereas angular sands

have much steeper steady state lines (Poulos et al.

1985; Vaid and Chern 1985; Konrad 1990a, b). It can

be seen from the Fig. 23 that the steady state lines of

Ganga sand and Toyoura sand are very close to each

other over a larger portion up to the mean effective

principal stress value equal to 650 kPa depicting a

flatter slope. Also at the same time the shape and

curvature of the steady state line of Ganga sand

closely follows that of Yamuna sand especially up to

the mean effective principal stress of 700 kPa. Even

after 700 kPa there is not much difference in the

slope. It may be due to the fact that even though

Ganga sand is similar to that of Toyoura sand in

terms of gradation curve, maximum and minimum

values of void ratio, its uniformity coefficient, shape

factor, sphericity and degree of roundness values are

closer to that of Yamuna sand. Thus, Ganga sand

showed a mixed behavior being closer to both the

sands depending on the value of the mean effective

principal stress to which the sample was subjected.

The grains of Yamuna sand being very angular, its

steady state line is expected to have higher slope

indicating greater compressibility in comparison

to Ganga and Toyoura sand whose particles are

Table 7 Comparison of

emax, emin, G, D50, Cu and

grain shape for three sands

Sand emax emin Specific gravity, G D50 (mm) Cu Grain shape

Yamuna 0.91 0.57 2.66 0.48 2.60 Angular

Ganga 0.97 0.64 2.67 0.16 2.30 Sub-angular

Toyoura 0.977 0.597 2.65 0.17 1.7 Sub-rounded

Table 8 Mineralogical compositions of some sands (Rahim

1989)

Sand Percentage mineral content

Quartz Feldspar Mica Carbonate Chlorite

Kalpi 40 40 1–2 18 0

Ganga 60–65 20–25 8–10 2–3 2–3

Table 9 Morphological characteristics of some sands (Rahim

1989)

Sand Average

shape factor

Average

sphericity

Average

roundness

Kalpi 0.596 0.698 0.10–0.20

Ganga 0.539 0.678 0.15–0.25

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

Ganga sand [(Dr)c =20%, 55%

and 75%,σ3c = 350 kPa]

Toyour a sand [(Dr)c=20%,

31% and 57%,σ3c = 490 kPa]

Yamuna sand [(Dr)c=34±1%, 49±1%

and 69±1%,σ3c = 300 kPa]

Fig. 21 Comparison of

stress–strain behavior at

different relative densities

Geotech Geol Eng (2008) 26:237–250 247

123

sub-angular to angular and sub-rounded respectively.

But according to Fig. 23 Ganga sand is showing

higher slope after 700 kPa. This may be due to the

fact that as the fine content increases the steady state

line moves downwards in the e-p0 diagram. Further

more, Ganga sand with large mica content is more

compressible at higher effective stresses. Had the

relative densities and consolidation stresses for

the sands been very close better agreement among

the stress–strain diagrams and effective stress paths

showing the complete loading history up to the steady

state point would perhaps have been attained.

4.2 Comparison with Model Behavior

As per the guidelines provided by Cubrinovski and

Ishihara (2000), Yamuna sand is clean sand (fine

content \ 5%) with 1.91% fines content. In a previ-

ous study (Cubrinovski and Ishihara 1999) suggested

that the void ratio range is a useful parameter in

quantifying the grain-size characteristics of sandy

soils, and that it is possible to classify soils according

to (emax - emin). For clean sand having fines content

less than 5%, the void ratio range is expected to

lie within 0.25–0.50. So a void ratio range of 0.34

for Yamuna sand is fairly consistent with this

observation.

Cubrinovski and Ishihara (2000) suggested the

following expressions to estimate the slope of the

steady state line.

k ¼ �0:01þ 0:125 emax � eminð Þfor round�grained sands ð2Þ

k ¼ �0:02þ 0:25 emax � eminð Þ for angular sands

ð3Þ

This is valid in the range of p0 = 10 to 200 kPa.

As the angular sandy soils have much steeper

steady state lines than round-grained soils, it was

found that for a given void ratio range, the average

slope k of angular sands is twice as large as that of

round-grained sands. The Yamuna sand particles as

shown in Fig. 3 illustrates that the grains are

predominantly very angular in nature. So, in our

case for Yamuna sand, (emax - emin) = 0.34 and

putting this value in the above equation for angular

sands, we get, k = 0.065. For Ganga sand, putting

(emax - emin) = 0.33, we get k = 0.0625 as it

0

200

400

600

800

1000

1200

1400

0 100 200 300 400 500 600 700 800

p'=(σ '1+2σ '3)/3 (kPa)

= q

σ1-

σ3

k( P

a)

Ganga sand [ (Dr)c =20%, 55%

and 75%, σ3c = 350 kPa]

Toyoura sand [ (Dr) c =20%,

31% and 57%, σ3c = 490

kPa]

Yamuna sand [ (Dr)c=34±1%,

49±1% and 69±1%, σ3c = 300kPa]

Fig. 22 Comparison of

effective stress paths at

different relative densities

0.65

0.7

0.75

0.8

0.85

0.9

0.95

100 1000 10000

Mean effective principal stress, p' (kPa)

Vo

id r

atio

,e

USSL, Toyoura sand

USSL, Ganga sand

USSL, Yamuna sand

Fig. 23 Comparison of steady state lines

248 Geotech Geol Eng (2008) 26:237–250

123

contains sub-angular grains while for Toyoura sand,

putting (emax - emin) = 0.38, we get k = 0.0375 as

it posses sub-rounded grains.

By introducing this relationship, it is possible to

establish generalized expressions for the steady state

line where the void ratio range would only be the

soil-property variable. Steady state line in the relative

density-p0 diagram for the range of p0 = 10 to

200 kPa was given as

Drs¼�0:4þ1:4 emax�eminð Þ

þ0:01� 1� logp0ð Þ �0:02þ0:25 emax�eminð Þf gemax�eminð Þ

ð4Þ

Here, Drs is the relative density at the steady state line

expressed as a ratio, not as a percentage. Dro, the

relative density of the steady state line, in percentage,

at p0 = 0 is given as

Dro ¼ �40þ 140ðemax � eminÞ ð5Þ

The comparison between steady state lines as

obtained from the model and experiments, in the

relative density-p0 plot is shown in Fig. 24.

It is to be noted that Cubrinovski and Ishihara

(2000) proposed their model for the mean effective

principle stress range of 10–200 kPa. They also

stated that the slope of the steady state line, as given

by k is not valid for inconsequential values of

consolidating pressure (p0\ 10 kPa) and expressed

the steady state line as combination of two straight

lines within and beyond 10 kPa in the e-log p0 curve.

However, the same expression was used here to

check for the deviation, if there be any, from the

Cubrinovski and Ishihara (2000) predictions. It can

be seen from the Fig. 24 that the experimental points

are in good agreement with the model behavior up to

500 kPa and after that there is a considerable

difference between the two. The slope of the steady

state line of Yamuna sand is observed to be steeper

than the predicted values when p0[ 500 kPa. The

slope of the experimental steady state line in e-log p0

curve is constantly changing with change in p0 and it

is observed that the steady state line in e-log p0 curve

can be approximated in a much better way by a

quadratic form which can be obtained from the trend

lines of Fig. 23. So either separate equations for

slopes are needed to be developed for different p0

ranges or a quadratic equation can be fitted. Therefore

a new relationship for predicting the steady state lines

for Yamuna, Ganga and Toyoura sand is proposed as

follows, for p0[ 200 kPa, respectively.

e ¼ �2� 10�07 p02 þ 1� 10�05p0 þ 0:7963 ð6Þ

e ¼ �7� 10�08 p02 � 3 � 10�05p0 þ 0:9073 ð7Þ

e ¼ 5� 10�09 p02 � 8� 10�05p0 þ 0:9102 ð8Þ

By using the above equations, the ultimate steady

state lines of Yamuna, Ganga and Toyoura sand are

presented in Fig. 25 that fit the experimental data

better.

5 Conclusions

Based on the results, discussions and comparisons

presented in the above sections the following con-

clusions are drawn.

0

10

20

30

40

50

60

70

80

100 1000Mean effective principal stress, p' (kPa)

Rel

ativ

e d

ensi

ty ,

Dr (%

) From Cubrinovski and Ishihara (2000) Model

From Experiment

Fig. 24 Comparison of steady state line obtained from model

and experiment

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

100 1000 10000

Mean effective principal stress, p' (kPa)

Vo

id r

atio

, e

Yamuna sand

Ganga sand

Toyoura sand

Fig. 25 Ultimate steady state lines obtained from the sug-

gested quadratic model

Geotech Geol Eng (2008) 26:237–250 249

123

The results obtained by an effective stress

approach involving traditional consolidated drained

triaxial tests with volume change measurements on

rebounded samples are in good agreement with that

found by strain-controlled consolidated undrained

tests with pore pressure measurements. The study

also corroborated the method proposed by Norris

et al. (1997) to find the different stress–strain

diagrams using interpolation and extrapolation tech-

niques from limited experiments.

As observed earlier, the ultimate steady state is

generally achieved irrespective of the initial state of

the soils sample and whether the response is

contractive or dilative. But due to errors arising from

bulging, we may not observe such a trend.

The steady state behavior of the Yamuna (Kalpi)

sand has been found to be of similar trend with those

of Ganga and Toyoura sands with some variations

when the mean effective principal stress is less than

650 kPa even though its grain characteristics and

grain size compositions are different from those of

Ganga and Toyoura sand. Though the initial portion

of the ultimate steady state line of Yamuna sand is

similar to that of Toyoura sand up to the mean

effective principle stress of 650 kPa, the behavior of

Yamuna sand is in close analogy with Ganga sand

over the whole range of mean principal stress.

The observed steady state line for Yamuna sand

matches well with the predicted values (Cubrinovski

and Ishihara 2000) only up to the mean effective

principle stress of 500 kPa; appreciable deviation

between the observed and predicted behavior was

noted beyond this value. The proposed semi-empir-

ical models representing the steady state behavior for

Yamuna, Ganga and Toyoura sand have been found

to provide better predictions.

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