1

Evaluation of dynamic properties of sandy soil at high cyclic

strains

Shiv Shankar Kumar *

Research Scholar, Department of Civil Engineering, Indian Institute of Technology

Guwahati, Guwahati, Assam – 781039, Email: k.shiv@iitg.ernet.in,

shivshankar.mit@gmail.com

A. Murali Krishna

Associate Professor, Department of Civil Engineering, Indian Institute of Technology

Guwahati, Guwahati, Assam – 781039, Email: amurali@iitg.ernet.in

Arindam Dey

Assistant Professor, Department of Civil Engineering, Indian Institute of Technology

Guwahati, Guwahati, Assam – 781039, Email: arindamdeyiitg16@gmail.com

ABSTRACT

Dynamic loading conditions, such as earthquakes, may result in the generation of high shear

strain (> 5%) in the soil. Conventionally, dynamic properties of soils are estimated from the

tests conducted up to a shear strain of 1% by considering Symmetrical Hysteresis Loop

(SHL). However, it is commonly observed that the hysteresis loops become progressively

asymmetric with increasing shear strain, which leads to the over- or under-estimation of the

conventionally evaluated dynamic properties. Hence, it is necessary to adopt a modified

methodology of evaluating the dynamic properties of saturated sands based on the actual

Asymmetrical Hysteresis Loop (ASHL). Strain-controlled cyclic triaxial tests have been

conducted, for a peak shear strain range of 0.015-4.5% at 1 Hz loading frequency, on test

specimens prepared at different relative density (30-90%) and confining stress (50-150 kPa).

Although, the shear modulus evaluated considering SHL and ASHL are on close agreement,

the damping ratio evaluated considering SHL is approximately 40-70% lesser than that

obtained by considering ASHL. Moreover, in contrast to the classical curves as largely

applied in geotechnical engineering, a noticeable decrement of the damping ratio is observed

beyond 0.75% shear strain.

Keywords: High shear strain, Shear modulus, Damping ratio, Strain-controlled test, Cyclic

triaxial test, Hysteresis loop

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1. Introduction

Past earthquake events have indicated that soils may experience peak shear strain levels

greater than 5% [1-3]. The evaluation of the dynamic characteristics (shear modulus and

damping ratio) of soils at such high strain levels is very essential for design of earthquake

resistant structures. Shear modulus (or, more precisely, secant shear modulus) represents the

stiffness of soil, whereas damping ratio is described as the percentage of energy loss per

cycle of vibration. These dynamic properties are significantly affected by several factors,

namely shear strain amplitude (γ), type and composition of soil, relative density (Dr),

plasticity index, effective confining pressure (σʹc), overconsolidation ratio (OCR), frequency

of loading cycle (f), and number of cycles (N); the details of which are presented in [4-9] and

are not repeated for the sake of brevity.

Several researchers have used different testing methodologies (e.g. resonant column test,

piezoelectric bender element test, cyclic triaxial test and cyclic simple shear test) to determine

the dynamic properties of different soils at varying strain levels [7, 10-21]. It was reported

that the response of soils at high strain levels (> 0.01%) is substantially different than that at

low strain levels (< 0.001%), primarily due to the nonlinear stress-strain behaviour and

damping characteristics at higher strains [6]. In contrary to most of the tests conducted for

low-strain levels [10-21], only limited studies portrayed about the behaviour of soils under

higher strains [2, 22, 23]. This paper presents the dynamic behaviour of sandy soil under high

cyclic strains. Brahmaputra river sand was chosen for the purpose, and strain-controlled

Cyclic Triaxial (CT) tests were performed at 1 Hz loading frequency for a peak shear strain

range of 0.015-4.5%, on the reconstituted specimens prepared at different Dr (30-90%) and

consolidated under different σʹc (50-150 kPa). Although the cyclic tests can be conducted at

various loading frequencies, 1 Hz loading frequency was chosen as recommended by Ishihara

[6], Kramer [8] and commonly adopted by several other researchers [16, 19-21]. The results

obtained were analysed to assess the influence of high shear strain (γ), effective confining

pressure (σʹc) and relative density (Dr) on the evaluated dynamic properties.

2. Material characteristics

Brahmaputra sand (BS) obtained from Guwahati region (Assam, India) has been used for the

study. The FESEM (Field Emission Scanning Electron Microscope) image of BS (Fig. 1)

exhibits the particles to be profoundly angular and possessing noticeable surficial roughness.

Particle size distribution of BS (Fig. 2), determined by conducting dry sieve analysis [24],

classifies the soil to be poorly graded as per the relevant standards [25-26]. It can be observed

3

that the soil belongs to the category of severely liquefiable soils zone [27]. Index properties

of the soil (specific gravity, minimum and maximum dry unit weight) were determined as per

relevant standards [28-29] and are presented in Table 1.

3. Test apparatus and testing programme

The experimental investigations were conducted using an automated pneumatic controlled

CT testing apparatus (Fig. 3). The apparatus consists of a 100 kN capacity loading frame

fitted with a pneumatic dynamic actuator having a double amplitude displacement of 30 mm

(±15 mm) and operational frequency range of 0.01-10 Hz. The system comprises of a triaxial

cell having a maximum confining capacity of 2000 kPa, and an air compressor with a

maximum capacity of 800 kPa. The instrumentations available with the apparatus are: two

linear variable differential transducers, one each for cyclic and static measurements, having a

measuring range of ±15 mm and up to 50 mm respectively; one submersible load cell of

capacity 25 kN; three pressure transducers of capacity 1000 kPa to measure cell pressure,

back pressure and pore-water pressure; and, one automatic volume change measuring device.

The testing was controlled by a compact dynamic controller unit, which conveys the

instructions provided by the DYNATRIAX software and also records the data with the help

of a data logger.

All the tests were conducted on the remoulded cylindrical soil specimens of dimensions 70

mm diameter and 140 mm height [30]. Different specimen preparation techniques were

reported in literature, namely dry pluviation, moist tamping and water sedimentation [30, 31].

For the present investigation, dry pluviation technique was adopted. Each of the specimens

were prepared in three layers. Dry sand was poured through a cone shaped funnel into the

specimen-forming mould. A vacuum pressure of 15-20 kPa was applied before removing the

mould to maintain the verticality of the specimen. Subsequently, the triaxial cell was

mounted on the base plate and then filled with water, followed by simultaneous application of

cell pressure (15-20 kPa) and release of vacuum pressure [31].

The specimen preparation was followed by subsequent saturation and consolidation stages.

In order to expedite the saturation process, the specimen was flushed with CO2 for 10-15

minutes at a pressure lower than 15-20 kPa, the initial cell pressure [31]. Subsequently, de-

aired water was passed through the CO2 flushed specimen. To attain the saturation, the cell

pressure (CP) and back pressure (BP) were then gradually increased in stages while

maintaining an almost constant differential pressure of 10 kPa. After each increment of CP,

the Skempton pore-pressure parameter (B) was estimated to check the saturation status. The

4

specimen was considered to be completely saturated when the B-value was obtained to be

greater than 0.95. After attaining the saturation, the specimen was isotropically consolidated

to a targeted σʹc by increasing the CP, while maintaining a constant BP. The consolidated

specimens were then subjected to monotonic and cyclic loading.

Monotonic and cyclic tests were performed on the soil specimens prepared at different Dr

(30, 60 and 90%) and consolidated to varying σʹc (50, 100 and 150 kPa). Undrained

monotonic tests were conducted at a constant axial displacement rate of 1.2 mm/min, the

details of which are enlisted in Table 2. Strain-controlled CT tests were conducted on

isotropically consolidated reconstituted specimens subjected to sinusoidal waveform (f = 1

Hz) and various testing conditions, as presented in Table 3.

4. Test results and discussions

4.1. Undrained monotonic shear

Stress-strain responses of saturated BS specimens obtained from monotonic tests at

different σʹc and Dr are presented in Fig. 4. It is observed that the peak deviator stresses and

the associated strain levels are significantly affected by the variation of σʹc and Dr. At any

given σʹc, the increase in peak deviator stress is substantial (~ 60%) for the increase in Dr

from 30% to 60%, whereas it is marginal (~ 8%) for an increase in Dr from 60% to 90%.

Hence, it can be stated that the effect of variation of σʹc is more prominent for sands with low

Dr, i.e. in the range of loose-to-medium dense sands.

4.2. Undrained cyclic shear

Strain-controlled CT tests were performed on the consolidated specimens. Each test

specimen was subjected to 40 cycles of sinusoidal strain-controlled loading [30].

Corresponding to peak axial strain of 0.20% applied at f = 1 Hz (Fig. 5a) on a soil specimen

prepared at σʹc = 100 kPa and Dr = 30%, Fig. 5(b-d) exhibits the response obtained. Figure 5b

represents the exponential decay of deviator stress with increasing N which can be attributed

to the deformation of soil specimen. The increase in excess pore water pressure (PWP)

generation is portrayed in Fig. 5c, which indicates the gradual increase of excess PWP ratio

(ru) to 1 (i.e. excess PWP = σʹc) during the cyclic loading. As presented in Fig. 5d, the

variation of deviator stress with axial strain (hysteresis loops) depicts the degradation of

damping ratio and shear stiffness of soil with increasing number of loading cycles (N).

During an undrained cyclic loading, the rise of PWP in saturated sand causes the reduction of

inter-granular forces resulting in the decrease of effective stress and soil stiffness [9, 32].

5

Similar results have been obtained from different cyclic tests conducted at varying levels of

peak axial strain (), which have been subsequently used to evaluate the dynamic properties

of the soils.

4.3. Evaluation of dynamic properties

Out of all the loading cycles applied during the experimental investigation, dynamic

properties of soil can be mathematically evaluated by considering any one particular

hysteresis loop obtained from a specific loading cycle. Literature indicate the use of different

loading cycles to compute the dynamic soil properties, e.g. 10th

cycle [12], 5th

cycle [14], 3rd

cycle [33] and 1st cycle [19, 34-37]. Matasovic and Vucetic [32] have used separately the 1

st

cycle to calculate the shear modulus and 2nd

cycle to compute damping ratio. The selection of

the hysteresis loop was guided by the assumption that up to the nth

cycle, the hysteresis loop

remained symmetrical.

Figure 6 presents the hysteresis loops for initial two cycles, from the strain-controlled CT

tests conducted at different peak shear strain levels. It is seen that the hysteresis loops

becomes gradually asymmetric with increasing peak shear strain. Beyond γ = 0.15%, even the

1st cycle hysteresis loop is observed to be asymmetric (Fig. 6d-h). For the present study, the

1st cycle of loading and the corresponding hysteresis loop has been chosen for the evaluation

of dynamic properties. Figure 7a represents a typical Symmetrical Hysteresis Loop (SHL),

and along with portrays the suggested conventional method of calculation of dynamic

properties [30]. In this method, the dynamic shear modulus (G) is evaluated from secant

Young’s modulus (Esec), obtained by the slope of the line joining the points of peak

compressive and tensile stress-strain points; the damping ratio is evaluated from the stored

strain energy by the triangle in the first quadrant (considering symmetricity for the entire

loading cycle). However, in the case of Asymmetrical Hysteresis Loop (ASHL) as shown in

Fig. 6(d-h), this method will not yield appropriate values as it will lead to the underestimation

of damping ratio due to the consideration of higher strain energy for the entire loop. To take

into the effect of actual strain energy and moduli values in compression and tension, true

representation of asymmetric nature of hysteresis loop should be considered. In that case,

dynamic soil properties are to be evaluated as described in Fig. 7b, which is a regenerated

version of the original methodology stated by Kokusho [12]. Therefore, for an ASHL, a

modified method has been adopted to evaluate the shear modulus (Ga) and damping ratio

(D#). Nevertheless, the conventional method, as stated for a SHL, has also been used to

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evaluate the shear modulus (G) and damping ratio (D), and the same are compared against

those obtained from ASHL for the sake of understanding the necessity of adopting the

modified methodology.

4.4. Evaluation of dynamic shear modulus

Based on the obtained experimental data, G for SHL and Ga for ASHL are evaluated based

on the descriptions provided in Figs. 7(a, b). As the saturated sandy specimen being sheared

under undrained conditions, the Poisson’s ratio (υ) was suitably adopted to be 0.5 as per the

recommendations in [12, 37].

For BS specimens prepared at Dr = 30% and σʹc = 100 kPa, Fig. 8a depicts the variation of

G with N for different strain levels (γ = 0.045% to 0.60%). For γ ≥ 0.075% a noticeable

reduction in G is observed with increasing N. However, after a few loading cycles, depending

on the strain level, the reduction becomes marginal. For BS specimens at Dr = 30% and σʹc =

50, 100 and 150 kPa, Fig. 8b depicts the variation of G with γ for first loading cycle (N = 1).

It is observed that for a given shear strain amplitude, the shear modulus increases with the

increase of σʹc; similar observations were also reported by other researchers [7, 10, 15, 19-

21]. Figure 8c represents the variation of shear modulus (G and Ga) with shear strain (BS

specimens at Dr = 30% and σʹc = 100 kPa). The results indicate that estimations of shear

modulus from SHL and ASHL are practically same when evaluated from the 1st cycle due to

the similar mathematical averaging technique followed in both the approaches.

A modulus reduction curve is used to represent the degradation of shear modulus with

shear strain in terms of reduction in modulus ratio (G/Gmax) [8]. Gmax is the maximum shear

modulus of the soil that is generally defined at a very low shear strain (γ ≤ 5×10-6

) [8, 33].

For the present study, in the absence of availability of such experimental data at sufficiently

low strain (γ ≈ 10-6

), empirical correlation proposed by Chung et al. [38] has been used for

evaluating Gmax (Eqn. 1).

0.48

0.52

max 2

( )kPa = 523 '

(0.3 0.7 )

k

a o

OCRG p

e

(1)

where, OCR is the overconsolidation ratio, k is an index parameter (0 for sands), e is the void

ratio, ap is the atmospheric pressure and σʹo is the mean principal effective stress in kPa.

Estimated Gmax values for BS soil specimens at different Dr and σʹc are presented in Table 4.

The variation of G/Gmax for BS with , at different Dr and σʹc, are presented in Fig. 9. The

scatter of the obtained data can be confined by upper- and lower-bounds, as represented by

7

dotted lines in the figure. It can be observed that, at higher strain levels, the scatter of G/Gmax

gets significantly narrowed. For all practical purposes, the obtained estimates can be

represented by an average G/Gmax curve (Fig. 9). The obtained average G/Gmax curve for BS

has been compared with the G/Gmax curves, for different sandy soils, available in the literature

(Fig. 10). In spite of a similar trend, the G/Gmax estimates of BS show lower values in

comparison to the classical curves [10, 12, 16]; although, it is in close agreement to those

reported for Indian soils [19-21]. As each of the established modulus reduction models are

developed based on the dynamic testing of a specific soil [10, 12, 16], adoption of any

particular model to determine the dynamic response of different other soils might be

incorrect. It should be customary that the dynamic properties of each soil be judiciously

determined before its application for any practical geotechnical engineering problems.

4.5. Evaluation of damping ratio

Figure 11 illustrates the variations of damping ratio with N, computed using both

conventional (D) and the modified method (D#) (as shown in Figs. 7 a, b). For the evaluation

of D#, the stored strain energy in one complete cycle has been considered (represented by the

hatched polygon in Fig. 7b), which comprises of two triangles and a rectangle having the area

AΔ1, AΔ2 and A□ respectively. AL is the area enclosed by the hysteresis loop and is expressed in

Eqn. (2) [39], where i and σi are respectively the axial strain and deviator stress (i = 1 to n,

where n is number of points used to construct the hysteresis loop).

1 2 2 1 2 3 3 2 1 1

1( ) ( ) .... ( )

2L n nA (2)

It is observed that for γ ≤ 0.075%, the differences between D and D# are insignificant,

which is primarily attributed to the nearly symmetrical loop (SHL) at these low shear strains

for all loading cycles. Significant difference is noted between D and D#, at γ ≥ 0.15%, where

D# is found to be noticeably higher than D, due to enhanced asymmetricity of the hysteresis

loops at these strain levels (Fig. 6). This observation inadvertently indicates that the

estimation of damping ratio following conventional method may underestimate the actual

value of damping, and therefore, the modified method should be adopted. For a typical

variation of damping ratio (γ = 0.60%, Fig. 11e), it can be observed that the estimated values

of D and D# become closer, thus suggesting formation of near-symmetrical loops

approximately beyond 30th

cycle. Similar observations are made for the damping ratios

obtained for other peak shear strains (γ > 0.60%, Fig. 11f).

8

Figures 12- 14 depict the variations of damping ratio (D and D#) with shear strain for test

specimens prepared at different Dr and σʹc. Figure 12a shows the effect of σʹc on D and D#

for

BS specimens at Dr = 30%. It is observed that specimens subjected to lower σʹc (50 kPa)

depicts higher damping ratio in comparison to that obtained at higher σʹc (100 and 150 kPa),

which is attributed to the relatively higher stiffness imparted in the specimens due to higher

confinement. The damping ratio is not found to vary noticeably for σʹc greater than 100 kPa.

It can also be noted that D attains a peak magnitude at a γ = 0.5%, while D# attains the peak

magnitude at γ ≈ 1% (Fig. 12a). Beyond the stated peak values, a significant reduction in

damping ratio is observed at higher strain levels which are noticeably different from that

observed in the damping ratio curves from earlier studies [10-21]. It should be noted that, for

the earlier studies, tests were conducted up to strain levels of about 1%. Very few researchers

have provided the experimental evidence of estimated damping ratio beyond 1% shear strain

[2, 22-23, 32] which followed a similar trend as obtained in the present study. Figure 12b

portrays that up to γ ≈ 0.5%, Dr has an insignificant effect on D and D#

estimated for BS

specimens subjected to σʹc = 100 kPa.

For all practical purposes, Fig. 13 presents the average trends of the estimated D and D#

for BS specimens prepared at different Dr and σʹc. It is observed that, the average trend of D#

exceeds D by 40-70%, with higher deviation at higher strain levels beyond 0.50%. From the

overall observations of Figs. 12-13, it can be stated that the modified method to estimate

damping ratio is significantly important and that conventional method largely underestimates

the actual damping developed in the specimen, especially at higher strain levels.

Most of the existing literature reported the estimated damping ratio for γ ≤ 1%, and the

same is extrapolated to estimate the damping ratio for beyond 1% (such as in the existing

commercial software for ground response analyses or any other seismic studies). However,

the present study unambiguously reflects that the damping ratio exhibits non-conventional

behavior γ ≥ 1%, where a noticeably decreasing trend is observed rather than the

conventionally assumed increasing or asymptotic variation. Hence, the current observation

provides a better phenomenological understanding and a scope of improvement of the

existing methods utilizing damping ratio curves beyond γ = 1% by defining a curvilinear

variation for the entire strain range used for the present study. The obtained variation of

damping ratio from the present study has been compared with several data available in

literature as shown in Fig. 14. It can be observed that most of the earlier researchers have

restricted the data up to γ = 1%. The results from the present study also follows the observed

trend (within 1%). Based on the present observation related to the variation of damping ratio

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over the entire strain range, it can be stated that it is not judicious to extrapolate the trend of

damping ratios for γ ≤ 1% to obtain the same for γ >1%. Thus, it is suggested that a new

functional variation of damping ratio should be developed in order to use the same for

various geotechnical engineering purposes involving wider range of shear strain.

5. Conclusions

Strain controlled cyclic triaxial tests have been conducted on saturated Brahmaputra sand

to evaluate the dynamic properties over a wide range of shear strain. Hysteresis loops at

different loading cycles manifested the dynamic behavior of the soil and exhibited noticeable

asymmetric nature, especially at higher strain levels. Since conventional methods do not

account for such asymmetry in the estimation of the dynamic properties of the soil, a

modified method of the evaluation of the same was adopted and has been reported in the

article.

Shear modulus (G) of BS soil is observed to be significantly affected by the variations in

σʹc and Dr. However, the scatter of the estimate becomes lower when expressed in terms of

the modulus reduction (G/Gmax) curve. In comparison to the classical curves, G/Gmax curve of

BS specimens depicted lower range of modulus ratio; however, the trend was well-matching

with those reported for Indian soils. It can be concluded that the direct application of the

existing modulus reduction models may lead to improper estimation of the dynamic response

of any other soil, and hence, a proper characterization of the soil is extremely necessary.

In comparison to higher σʹc, it was observed that BS specimens subjected to lower σʹc

revealed higher damping ratio due to the higher shear stiffness imparted in the specimens by

higher confinement. It is observed that, on an average D# exceeds D by 40-70%, with higher

deviation exhibited at higher strain levels beyond 0.50%. The obtained variation of damping

ratio, from the present study, has been compared with several data available in literature. A

prominent decrease in damping ratio is noted beyond γ = 1%, portraying a marked difference

from the conventional extrapolated estimates. This decrement clearly manifests that the

conventional trends of damping ratio at higher shear strains will result in the adoption of

significantly different magnitudes leading to an improper geotechnical analysis. Hence, it is

imperative to develop and adopt a judicious functional variation of the damping ratio over a

wider strain range for practical purposes.

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Table 1 Physical properties of collected Brahmaputra sand

Unit Weight

(kN/m3)

Specific

Gravity

D10

(mm)

Uniformity

Coefficient

Cu

Coefficient

of Curvature

Cc γmax γmin

16.84 13.85 2.7 0.13 1.47 1.09

Table 2 Testing program for monotonic triaxial tests

Sand

Specimen

Relative Density

Dr (%)

Confining Pressure σʹc

(kPa)

Displacement

rate (mm/min)

BS

30 50, 100, 150

1.2 60 50, 100, 150

90 50, 100, 150

Table 3 Investigation parameters of the strain controlled cyclic triaxial tests

Sand

Specimen

Relative

density

Dr (%)

Confining

pressure

σʹc (kPa)

Frequency

f (Hz)

Shear strain

γ (%)

BS

30

50

1

0.015, 0.045, 0.075, 0.15, 0.30,0.45,0.60,

0.75, 1.0, 1.5, 3.0

100 0.045, 0.075, 0.15, 0.30,0.45,0.60, 0.75, 1.5

150 0.045, 0.075, 0.15, 0.30, 0.45, 0.60, 0.75

60

50

0.15, 0.60, 1.0, 1.5, 3.0, 4.5 100

150

90

50 0.045, 0.075, 0.15, 0.30, 0.45, 0.60, 0.75,

1.5

100 0.045, 0.075, 0.15, 0.30, 0.45, 0.60, 1.0,

1.5, 2.0

150 0.045, 0.075, 0.15, 0.30, 0.45, 0.60, 0.75,

1.0, 1.5, 2.0

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Table 4 Estimated Gmax for BS

σʹc (kPa) 50 100 150

Dr (%) Gmax (MPa)

30 49.64 69.24 84.12

60 57.73 80.52 97.82

90 67.25 84.12 113.95

15

Fig. 1. FESEM image of Brahmaputra Sand (BS)

1E-4 1E-3 0.01 0.1 1 10

0

20

40

60

80

100

Percen

tag

e f

iner

Praticle size (mm)

Boundaries for

partial liquefiable zone

Boundaries for

severely liquefiable zone

Brahmaputra sand (BS)

Fig. 2. Particle size distribution of BS

16

Fig. 3. Cyclic triaxial setup and components

Fig. 4. Response of monotonic compression shear on BS at different Dr and σʹc

17

Fig. 5. Typical test result plots at ε = 0.20%, f = 1 Hz, σʹc = 100 kPa and Dr = 30% (a) Axial

strain vs N (b) Deviator stress vs N (c) PWP ratio vs N (d) Stress vs strain

a

c d

b

18

Fig. 6. (a-h) Typical shear stress-shear strain plot for initial two cycles at Dr = 30%, σʹc = 100

kPa and f = 1 Hz for different γ

a b c

d e f

g h

Dr = 30%, σ'c =100 kPa, f = 1 Hz, ── N = 1, ˗ ˗ ˗ N = 2

19

Fig. 7. A typical (a) SHL (Redrawn after Kramer, 1996) and (b) ASHL

1.

1

4LA

AD

AΔ

AL

Axial

strain (ε)

Deviator

stress (σd)

εmax

εmin

σd,max

σd,min

Emax 1

Esec 1

(a)

1.

Axial

strain (ε)

Deviator

stress (σd)

εmax

εmin

σd,max

σd,min

Emax

1 Esec1

1

Esec2 1

Loading Curve

Unloading Curve

a

c

d

ef

g

AΔ1

bAL

o

AΔ2 A

(b)

sec

sec

,max ,min

max min

/

/[2(1 )]

(1 )

1

4

d d

d

L

E

G E

AD

A

sec1 sec2sec,

sec,

( )#

1 2

2

/[2(1 )]

(1 )

1

a

a a

L o a b c d

E EE

G E

AD

A A A

20

Fig. 8. Variation of shear modulus with (a) N (b) σʹc (c) SHL and ASHL

a b

c

21

Fig. 9. Variation of G/Gmax curve with γ at different σʹc and Dr

Fig. 10. Comparison of G/Gmax curves for different soils

22

0 10 20 30 400

5

10

15

20

25

30

35

40 D

# from ASHL approach, D from SHL approach

D

am

pin

g r

ati

o (

%)

Number of cycles (N)

= 0.045%

0 10 20 30 40

5

10

15

20

25

30

35

40

Number of cycles (N)

D

am

pin

g r

ati

o (

%)

D# from ASHL approach, D from SHL approach

0.075%

0 10 20 30 400

5

10

15

20

25

30

35

40 = 0.15%

Da

mp

ing

ra

tio

(%

)

Number of cycles (N)0 10 20 30 40

0

5

10

15

20

25

30

35

40

D

am

pin

g r

ati

o (

%)

0.30%

Number of cycles (N)

0 10 20 30 400

5

10

15

20

25

30

35

40

0.60%

Da

mp

ing

ra

tio

(%

)

Number of cycles (N) 0 10 20 30 40

0

10

20

30

40

0.75%

Da

mp

ing

ra

tio

(%

)

Number of cycles (N)

Fig. 11. (a-f) Variation of D# and D with N for γ = 0.045-0.75%

a b

c d

e f

23

Fig. 12. Variation of damping ratio (D and D#) with (a) σʹc (b) Dr

Fig. 13. Comparison of D and D# of BS

a b

24

Fig. 14. Comparison of damping ratio from available data in literature