Effect of stress state on shear wave-velocity of Ganga sand

using Bender elements

Presented By :

Debayan Bhattacharya

Department of Civil Engineering

IIT Gandhinagar

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Project Advisor :

Dr. Amit Prashant

Department of Civil Engineering

IIT Gandhinagar

Main Author : SARASWATHI GUNDLAPALLI

Department of Civil Engineering-IIT K (Sept.2007)

Outline

Overview of the past work done

Important Conclusions of the study

Comparisons of Models/Justification of the model used

Isotropic Elasticity Model-Lade & Nelson

Future Work

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Past work done by Gundlapalli.S (former Research fellow at IITK)

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A series of undrained triaxial compression tests on

i. 2% FC sand by isotropically consolidating to 100 kPa,200 kPa,300 kPa and 500

kPa.

ii. Natural composition of Ganga river sand(10% FC) by isotropically consolidating

to pi‟ = 200 kPa, 300 kPa and 500 kPa. The series was then repeated to the same

values of pi‟ but q/ pi‟ was kept at 0.85(anisotropic consolidation).

iii. Ganga sand – silt mix by isotropically consolidating to pi‟ =300 kPa with varying

fine content in the range of 2%,10%,31%,70% and 100%.

iv. Bender element readings were taken at every 2% of axial strain & it was

primarily used to measure the shear wave velocity(vs )-Cross-correlation has

been used to estimate the time-lag (phase-difference) .

Important Conclusions made from the study as conducted

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• Pore Pressure response at small strains was largely governed by pi’ –positive pore pressure

increased with pi’ ; at phase transformation (contractive to dilative response 1-2 % axial strains

); decrease in pore pressure was independent of pi’ .

• Bender-element tests data showed that vs increased with increase in pi‟ -found true for all the

tests carried out.

140

160

180

200

220

240

260

280

300

320

0 100 200 300 400 500 600 700

Effective mean stress(p')

Sh

ear

wav

e v

elo

cit

y(m

/sec)

pi΄=200kPa

pi΄=300kPa

pi΄=500kPa

(Shear wave velocity vs. Mean

effective stress of isotropic &

anisotropically consolidated

specimens of Ganga sand)

(*Source :Effect of Stress State & Silt-Content on

Shear Wave Velocity of Ganga Sand using Bender

Elements)

Important Conclusions made from the study as conducted

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• The failure point (often defined as either peak shear stress location or the location of

peak shear stress ratio is same as the angle of maximum obliquity (βmax).

• Peak shear stress occurred close to 24 % of axial strain (εa) & the peak of βmax

observed ≈ 10-15 % of axial strain.

• Shear wave velocity increased with the increase in initial mean effective stress and

deviatoric stress. Post peak shear stress ratio, the shear wave velocity decreased

although the mean effective stress and deviatoric stress both increased continuously

up to ultimate state.

• After anisotropic consolidation, the shear wave velocity showed sudden decrease in its

value when the specimen was sheared under undrained conditions

• Use of Isotropic Elasticity model using J2‟ effect (Lade Nelson‟s Model-Modelling the Elastic

Behavior of Granular Materials-1987)

Comparison/ Applicability of other Models to capture Elastic behaviour

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Models Limitations/Why the fail

Pa =atm. Pressure ;n = exponent =rate of variation of Ei with

σ3„ ; Ei = initial tangent modulus –power function of initially

isotropic

effective confining pressure σ3„ .Ki = modulus number.

Widely used as it captured elastic behaviours observed

in triaxial & isotropic compression & material

parameters can be determined from conventional

triaxial tests

Zytynski et.al pointed out this model for the

variation of E results in violation of principle

of conservation of energy.

As initial slope of stress-strain curve often influenced by the non-recoverable plastic deformation

(εp )- Duncan & Chang proposed to use slope of an unloading-reloading cycle from a triaxial

compression test as Elastic Modulus

Deformation characteristics of soils which were tried to express through B (related to hydrostatic

stress component) & G (related to deviatoric stress component )-led to similar forms

Hardin & Black (empirical expression ) based on wave propagation

velocities & from small amplitude cyclic-simple shear

tests.(K=exponent may be related to the P.I. of soil)

Limitation as these were case

specific & many such general

equations of the form as given

below are there.

(it includes the effect of void ratio on shear modulus)

'

3( )n

i i a

a

E K pp

2'

max

(2.973 )1230

(1 )

k

o

ocr eG

e

max

'( ) n

oG AF e

Comparison/ Applicability of other Models to capture Elastic behavior contd…

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• Rowe –also had proposed some relationship (Theoretical meaning & observed values of

deformation parameters of soil)

• Coon & Evans –Recoverable deformation of cohesion less soils.

• Limitations of the above two models: Considerable number of material constants required & in

part due to the requirement of results from complicated or advanced tests.

• Theoretical Limits of Poisson‟s ratio are -1<ν<0.5 . These limits are obtained on the basis of

• Experimental evidences indicates that Poisson’s ratio is isotropic & practically constant for a solid

at a given void ratio.

• Constant ν , but E changing with stress state is the very basis of formulation of Lade-Nelson’s

model which represents

1

{ } 02

TElastic work (for any change in stress)

1 2(I , )nE f J

Isotropic elasticity model considering J2’effect-Lade & Nelson (1987)

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Isotropic elastic model developed by Lade and Nelson (1987), which assumes ν = constant and the

elastic moduli E and thus shear moduli G as a function of the mean effective stress and the

deviatoric stress has been used.

E varies with stress-state,

σm‘ was replaced I1 while σd was replaced with J2’

Pa = atmospheric pressure and M and λ are constants (M = modulus number)

'( ', ) ( , )n n

m dE f p q f

'

1, 2(I )nE f J

2'

1 2

2a

a a

I JE MP R

P P

16

1 2R

Isotropic elasticity model considering J2’effect-Lade & Nelson (1987) contd...

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For generalized stress-state & it’s degenerated form for triaxial test:

For Triaxial Test (degenerated forms of I1 & J2’)

σ2‟ = σ3‟

1 1 2 3I

2

13

2

32

2

2126

1 J

1 1 32I

2

2 1 3

1

3J

E = 2Gmax(1+ υ)

E = Young‟s modulus

Gmax = Small strain shear modulus

where σ1‟, σ2‟, σ3‟ = Major,

intermediate, and minor principal

stresses respectively.

Isotropic elasticity model considering J2’effect-relating it to G & vs

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From theoretical considerations (as like that derived for expression for E- considering the closed

loop stress path-similar approach of Lade & Nelson);

Now also,

Thus, in turn we have shear wave velocity (vs )as a function of the first stress invariant (I1) second

invariant of the deviatoric stress-tensor (J2’).

Shear-wave velocity (vs ) -important parameter in many geo-physical investigations & in turn

shear-modulus is essential for vibration isolation measures, & analysis of soil-structure

interaction problems as well as design of deep & shallow foundations.

'

max 1 2( , )nG f I J

2

max. sG v

Future work

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To estimate the material parameters of the model of Lade & Nelson for 100% fine –content

(almost pure silt) & 0% fine content (almost pure sand) for both isotropic ally consolidated as

well as anisotropically consolidated specimens.

To estimate the model‟s fit by regression from the existing data as available from the previous

work of Gundlapalli S.(former IITK research fellow)

To explore if any other model can capture the elastic behavior within the framework of elastic

considerations.

Finally, to consolidate the data & results & its interpretation & ultimately publish the data &

discussions & justifications.

References:

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1. Effect of Stress State & Silt-Content on Shear wave velocity of Ganga Sand using Bender Elements-A Thesis Submitted In the Partial

Fulfillment of the Requirements for the degree of Master of Technology (IIT Kanpur)-Saraswathi Gundlapalli.

2. Modelling the Elastic behavior of Granular Materials-Poul V.Lade & Richard B. Nelson-International Journal for Numerical &

Analytical Methods in Geomechanics-vol.11-pp 521-542 (1987)

3. Dano,C.,and Hitcher,P.Y.(2002). “Evolution of elastic shear modulus in granular materials along isotropic and deviatoric stress

paths,”15th ASCE Engineering Mechanics Conference,Columbia University, New York, NY

4. Dano,C., and Hicher,P.Y.(2003), “Characterization of Loire river sand in the small strain domain using new bender-extender

elements”, 16th ASCE Engineering Mechanics Conference, University of Washington, Seattle

5. Kuwano, R., and Jardine, R. J. (2002). “On the applicability of cross anisotropic elasticity to granular materials at very small

strains,” Geotechnique, 52(10), pp. 727-749.

6. N. Janbu, „Soil compressibility as determined by odometer and triaxial tests‟, Proc. European Conf. Soil Mech.

7. H.-Y. KO and R. F. Scott, „Deformation of sand in hydrostatic compression‟, J. Soil Mech. Found. Div., A.S.C.E.,

8. H.-Y. KO and R. F. Scott, ‟Deformation of sand in shear‟, J. Soil Mech. Found. Div., A.S.C.E., 93(SM5), 283-310 (1967).

9. J. M. Duncan and C.-Y. Chang, „Nonlinear analysis of stress and strain in soils‟, J. Soil Mech. Found. Diu., A.S.C.E.,

10. M. Zytynski, M. F. Randolph, R. Nova and C. P. Wroth, „On modelling the unloading-reloading behavior of soils‟,Int. J . numer. anal.

methods geomech., 2, 87-94 (1978).

11. B. O. Hardin and W. L. Black, Closure to „Vibration modulus of normally consolidated clay‟, J. Soil Mech. Found.

12. B. O. Hardin, „The nature of stress-strain behavior for soils‟, Proc. Spec. Cod. Earthquake Eng. Soil Dyn., A.S.C.E.,

13. P. W. Rowe, „Theoretical meaning and observed values of deformation parameters for soil‟, Proc.‟ Roscoe Memorial

14. M. D. Coon and R. J. Evans, „Recoverable deformation of cohesionless soils‟, J. Soil Mech. Found. Div., A.S.C.E., pp. 1-4. 7(SM2),

375-391 (1971).

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