1. Effect of stress state on shear wave-velocity of Ganga sand using Bender elements
Presented By :
DebayanBhattacharya
Department of Civil Engineering
IIT Gandhinagar
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Project Advisor :
Dr.AmitPrashant
Department of Civil Engineering
IIT Gandhinagar
Main Author : SARASWATHI GUNDLAPALLI
Department of Civil Engineering-IIT K (Sept.2007)

2. 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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3. Past work done by Gundlapalli.S (former Research fellow at IITK)
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A series of undrainedtriaxial compression tests on
i.2% FC sand by isotropicallyconsolidating to 100 kPa,200 kPa,300 kPaand 500 kPa.
ii.Natural composition of Ganga river sand(10% FC) by isotropicallyconsolidating to pi‟ = 200 kPa, 300 kPaand 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 isotropicallyconsolidating to pi‟ =300 kPawith 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-correlationhas been used to estimate the time-lag (phase-difference).

4. 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')
Shear wave velocity(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)

5. 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 undrainedconditions
•Use of Isotropic Elasticity model using J2‟ effect (Lade Nelson‟s Model-Modelling the Elastic Behavior of Granular Materials-1987)

6. Comparison/ Applicability of other Models to capture Elastic behaviour
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Models
Limitations/Whythe fail
Pa =atm. Pressure ;n = exponent =rate of variation of Eiwith
σ3„ ; Ei= initial tangent modulus –power function of initially isotropic
effective confining pressure σ3„ .Ki= modulus number.
Widelyused as it captured elastic behavioursobserved in triaxial & isotropic compression & material parameters can be determined from conventional triaxial tests
Zytynskiet.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 ofteninfluenced 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 soilswhich were tried to express through B (related to hydrostatic stress component) & G (related to deviatoric stress component )-led to similar forms
Hardin & Black(empiricalexpression ) based on wave propagation velocities & from small amplitude cyclic-simple shear tests.(K=exponent may be related to the P.I. of soil)
Limitationas these were case specific & many such general equations of the form as given below are there.
(it includes the effect of void ratio onshear modulus)
' 3()niiaaEKpp   2' max(2.973) 1230(1) koocreGe     max'()noGAFe

7. 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
{ } 0
2
T
Elastic work     (for any change in stress)
1 2 (I , ) n E  f J

8. 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 stresshas been used.
E varies with stress-state,
σm‘ was replaced I1while σd was replaced with J2’
Pa = atmospheric pressure and M and λ are constants (M = modulus number)
'(',)(,)nnmdEfpqf ' 1,2(I)nEfJ 2' 122aaaIJEMPRPP     1612R    

9. 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 3 I      
     2 
3 1
2
2 3
2
2 1 2 6
1
J                
1 1 3 I    2 
 2
2 1 3
1
3
J     
E = 2Gmax(1+ υ)
E = Young‟s modulus
Gmax = Small strain shear modulus
where σ1‟, σ2‟, σ3‟ = Major,
intermediate, and minor principal
stresses respectively.

10. 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.
' max12(,)nGfIJ 2max.sGv

11. 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.

12. 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-PoulV.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”, 16thASCE 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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