Student: ALIOUNE BADARA SAMBSupervisor: HIROFUMI TOYOTA
Date: 2020/03/10

Nagaoka University of Technology
Geotechnical Laboratory

SMALL STRAIN DEFORMATION CHARACTERISTICS IN ANISOTROPICALLYCONSOLIDATED COHESIVE SOIL
1. Introduction
This study investigated the deformation characteristics at small strain (about 0.001%) oflateritic soil which is classified as cohesive soil. According to Mitchell and Soga (2005), a soil
is classified as cohesive if the amount of fines (silt and clay-sized material) exceeds 50% byweight. This type of soil is widely distributed in Africa and used for multi earthworks. Heavy
damages have been reported during and after civil engineering constructions. These damages
are often caused by a lack of understanding of the ground behavior. In order to accurately
predict the soil response underneath geotechnical structures, it is important to take account thedeformation at very small strain (<0.001%) and the anisotropic effect. The deformation
characteristics at very small strain is mainly quantified by the initial shear modulus G0. Thisparameter is evaluated by triaxial apparatus equipped with bender elements (BEs) and localsmall strain (LSS) measurement for dynamic and static method, respectively. The soil’sanisotropy, which has paid a little attention, refers to the directional property, in which
properties are different in different directions. In order to simulate the anisotropy in laboratory,
the specimen is consolidated at different stress ratio K ('3/'1). The factors influencing G0 suchas the anisotropy and the overconsolidation ratio OCR have been elucidated by applying thewell-known empirical equation of Hardin and Blandford (1989). The parameters of thisequation applicable to lateritic soil is also assessed in this research.
The purpose of this study is to provide deep understanding of fundamental mechanicalproperties of lateritic soil for construction and consulting companies in Africa and to assess theparameters of the Hardin’s equation for application to lateritic soil. This equation is in the formof:
G0= Sij F(e) (OCR)k Pa(1-ni-nj) (σ’i)ni (σ’j)nj

1.1

where Sij, n and k are material contant depending on the soil fabric, F(e) is the void ratio function,
OCR is the overconsolidation ratio which is defined as the ratio between the maximum effective
stress and the current stress, Pa is the atmospheric pressure, σ’i and σ’j are the effective stressin the i and j planes
2. Littérature review
The initial shear modulus G0 is the basic parameter for evaluating the soil response induced byseismic or dynamic loading and for description of the non-linear behaviour of soil in a wide
range of strain. Atkinson (1993) demonstrated that the strain range in the ground of many
structures (retaining wall, foundation…) is less than 0.1%. Soils have highly non-linear
mechanical behavior. Hence, he postulated three regions: very small strain, small strain, andlarge strain. The stiffness of geomaterial is higher at small strains than at the large strains.
Although the anisotropy has important effects on G0 for non-cohesive soil (Oda et al., 1985;
Tatsuoka et al., 1979; Yu and Richart, 1984; Belloti et al., 1996), the stress induced anisotropy
1has minor influence on G0 for cohesive soils (Sagae et al., 2006; Jovcic, 1997;
Teachavorasinkun and Lukkanaprasit, 2008). Hardin and Blandford (1989) demonstrated thatthe initial shear modulus G0 is influenced many factors such as current stress state,overconsolidation ratio, density, void ratio, microstructure, and so on. Several equations havebeen proposed by researchers to evaluate G0. The maximum stiffness of geomaterial can bemeasured by dynamic and static methods. Laboratory measurements includes resonant column,
torsional shear, bender elements, and triaxial shear with local strain measurement (Schneider etal., 1999).
In the case of BE test, the corresponding shear stiffness (G0) is calculated by measuring theshear wave velocity (s) and bulk density () of geomaterial by the formula as:
G0 = ρ × vs 2

2.1

Shear wave velocity is determined by ratio between distance of bender elements (d) and thetravel time of shear wave (Δt).
d2.2

vs = Δt

The start to start method illustrated in Fig. 1 is adopted to calculate the shear wave travel time(Δt) corresponding to the time difference between the starting of the input signal and the firstarrivals of the received signals through visual inspection (Yang and Gu, 2013).
The local small strain (LSS) measurement is nowadays widely used by researchers to assess
more accurately soil response and minimize errors such as bedding error, sample tilting,apparatus compliance. G0 is calculated by the following formula:q

G0 = 3Δε2.3s

where q is the deviatoric stress and 𝜀𝑠 corresponds to the shear strain
The comparison of G0 obtained from BE and LSS test indicates a good agreement with eachother (Jovcic, 1997; Kiku and Yoshida, 2000; Tatsuoka et al., 1995). G0 from BE is sometimefound to be higher than that from LSS (Yamashita et al., 2003; Kung, 2007).
Output wave(mV)

Input wave(V)1050-50.8
0.4
0.0-0.4
-0.8

-10
0.4

0.5Time(ms)

0.6

0.4

0.6

0.8

Time(ms)

1.0

1.2

Fig.1. Determination method of shear wave travel time by start to start method
3. Soil characteristics
The lateritic soil is named as Yoneyama silt and was sampled in costal area in Niigata. Table 1
shows some physical characteristics of the soil. From Fig. 2, silt and clay contents are dominant
in the soil. The sand content is less than 10%. The soil is then termed as cohesive soil.
2

Table 1: physical characteristics of Yoneyama siltParameters

Data
3Specific gravity (g/cm )2.746

Liquid limit (%)

52.4

Plastic limit (%)

29.9

Plasticity index (%)

22.5

Sand content (%)

8.5

Silt content (%)

65.5

Clay content (%)

26
Fig.2. Grain size distribution of Yoneyama silt

4. Experiment procedures and methodologies

Deviator stress, q:kPa

The triaxial apparatus is represented schematically in Fig. 3. The BEs are installed on top cap(transmitter function) and pedestal (receiver function) and the travel time is determined by startto-start method. The LSS technique was incorporated into the triaxial apparatus. The verticalaxial displacement is measured using 2 separated targets directly glued on the membrane of thespecimen. The change in the diameter is calculated through the lateral proximity transducer
installed on the middle part of the specimen.

600

K Consolidation andK Swelling

400
Shearing

200

Drained q-loading

00Fig.3. triaxial apparatus
methodology

0.350.43
1.0100 200 300 400 500 600
Effective pressure, p' :kPaFig.4. Representative stress path of the testing

Two testing methodologies are applied in this study. In the first methodology, shearing isapplied at effective stress p’ of 300 kPa. This methodology investigated anisotropy of G0 at thesame p’ and the aging effect, which is skipped in this resume. Figure 4 illustrated the second
methodology in which the overconsolidation ratio OCR effect was investigated. During the3

specimen preparation, soil and de-aired water are mixed and stirred to produce slurry. Onedimensionally pre-consolidation pressure of 70 kPa was applied on this slurry in a cylindricalmould. The specimen set in the triaxial cell was saturated using the vacuum saturation
procedure and consolidated isotropically under p’= 50 kPa with back-pressure of 200 kPa.
Drained q-loading was completed under constant p’ of 50 kPa up to a certain K. Five cases ofK are defined in this study: 0.35, 0.43, 0.6, 0.8, and 1.0. K consolidation was performed withmaintaining K constant and p’ increases gradually (50, 100, 200, 300, 400, 500, and 600 kPa).
K swelling was conducted by gradual decrease of p’ up to 50 kPa. BE test is performed 24h
after p’ reached a target value. LSS test was performed with the monotonic loading at shear
strain rate of 0.005 %/min.
5. Experimental results and discussion
5. 1 Stress ratio effect on G0Figure 5 shows linear behaviour of G0 at very small strain and decreasing in stiffness with theincrease of strain for all cases of stress ratio K. Highest value of G0 is observed at the stressratio K=0.43. Furthermore, the increase of the stress ratio K lead to a slight decrease in theinitial stiffness. Similar results have been observed in BE test (Fig. 6). G0 decays by about 10%
and 17% in BE and LSS tests, respectively (Fig. 7). Therefore, greater degree of anisotropy wasobserved in G0 obtained from LSS test than that from BE test. Results also elucidated theincrease of G0 with p’ during K consolidation and K swelling stage (Fig. 8). This may be related
to the decrease of the void ratio. Furthermore, at a constant value of p’, there is small variationof G0 with K as shown in the previous results.

Secant shear modulus, Gsec (MPa)

150
10050

K = 0.35
K = 0.43
K=0.6
K=0.8
K=1.0

00.0001

0.001
0.01
Shear strain, εs (%)

0.1

Fig.5. Relation between K and G0 (LSS test)

160
Initial shear modulus, Go (MPa)

Saturated Yonenama sandy silt
Triaxial compression
εs = 0.005 %/min p'=300 kPa
Local small strain test

200

150
140
130

120
110
100
0.3

130120110

G0 (BE)G0 (LSS)

100
0.35

0.45

0.55

0.65

K

0.75

0.85

0.95

0.5

0.6

K

0.7

0.8

0.91Shear modulus vs effective stresses

220

Shear modulus, G0 (MPa)

Initial shear modulus.Go(MPa)

Comparison LSS and BE
140

0.4

Fig.6. Relation between K and G0 (BE test)

160
150

K=0.35
K=0.43
K=0.6
K=0.8
K=1.0

BE

K Swelling
170
120
K Consolidation

70
20

0200

K=0.35
K=0.43
K=0.6
K=1.0400
600
Effective stresses,p' (kPa)800

Fig.7. Comparison of G0 from BE and LSS test Fig.8. Relation between K, p’ and G04

5. 2 Assessment of the parameters of Hardin’s equation in lateritic soil
The factors affecting G0 is considered by applying the proposed equation by Hardin andBlandford (1989), which is illustrated in equation 1.1. Different parameters of this equation
have been defined in the literature for different type of soil.
Tables 2 and 3 indicate the variation of the parameters Sij, n and K summarized from theliterature reviews. These parameters are found to be dependent on the soil fabric; k, theexponent of the OCR is mostly controlled by the soil plasticity.
Table 2. Values of n and Sij summarized from the literature
Type of soilnSij

References

Six undisturbed Italian clay

0.2-0.29

520-810

Jamiolkowski et al., 1995

Ottawa Sand

0.56900

Hardin and black (1968)

Table 3. Values of k, exponent of OCR summarized from the literature
Type of soil

PI (PlasticityindexkReferencesSand<40>4000.5

Hardin and black (1966)

London clay

10-40

0.2-0.25

Viggiani and Atkinson(1995)

Boston blue clay

22.70.15

Santagata (2005)

In this study, ni and nj are assumed to be equal to n = 0.25. From the experimental results, theparameters of the Hardin’s equation are assessed in the case of lateritic soil as follow: Sij =730,
k=0.24. This result is similar to the finding of other researchers (Jamiolkowski et al., 1995;
Viggiani and Atkinson, 1995). Figure 9 shows that the predicted value of G0 (pred G0)
calculated from the Hardin’s equation agrees well with the true value of G0 (true G0) from BE
test. Several void ratio functions have been found in the literature. Hardin and Richart (1963)
proposed the most frequently applied void ratio function to estimate the initial shear modulus:f(e) =f(e) =(2.17−e)21+e(2.97−e)21+e

5.1
5.2

Lo Presti (1989) and Jamiolkowski et al. (1995) also proposed an empirical void ratio function
on the following form.

f(e) = e−x

5.3

5Therefore, one of the mostly applied function corresponding to the equation 5.1 was used inthis study. In the case of lateritic soil, the equation of Hardin can be rewritten as the followingform:
G0 = 730(2.17−e)1+e20.24(OCR)

Pa(1−0.5) σi 0.25 σj 0.25
K = 0.43

200
150100true G050

Pred G000200

400600800

Initial shear modulus, G0(MPa)

Initial shear modulus, G0(MPa)K = 0.35
250250200150100
true G050

Pred G000

Effective pressure, p' (KPa)

150100true G0
Pred G000200
400600Effective pressure, p' (KPa)

800

Initial shear modulus, G0(MPa)

Initial shear modulus, G0(MPa)20050800

K = 1.0

K = 0.6250200400600Effective pressure, p' (KPa)

250200150
10050

true G0

Pred G000200
400
600
800
Effective pressure, p' (KPa)

Fig.9. True and predicted G06. Conclusion
The conclusions of this study can be summarized as follows: G0 decreases with the increase of K under constant p’. Thus, there is effect of K on G0.
 The degradation of G0 with increased K is about 10% and 17% in BE test and LSS test,
respectively. Greater degree of anisotropy is observed in G0 obtained from LSS than thatfrom BE.
 G0 can be predicted for the lateritic soil using the following Hardin’s equation:
G0= 730 F(e) (OCR)0.24 Pa(1-0.5) (σ’i)0.25 (σ’j)0.25
7. References
1) Atkinson, J.H (1993): An introduction to the mechanics of soils and foundations.
Mcgraw-hill international (uk) limited. Publication.
2) Bellotti, R., Jamiolkowski, M., Lo Presti, D. C. F., and O’Neil, D. A. (1996): Anisotropyof small strain stiffness in Ticino sand. Geotechnique, 46(1), 115–131.
3) Hardin, B.O., and Black, W.L. (1966): Sand stiffness under various triaxial stresses.
Journal of Soil Mechanics and Foundation Division, ASCE, 92(2), 667–692.
64) Hardin, B. 0, and Black, W.L. (1968): Vibration modulus of normally consolidated clay.
J. Soil Mech. Fdns Div. Am. Soc. Civ. Engrs 95, SM6, 1531-1537.
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ASCE, Vol. 115, No. 6.
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7) Jamiolskowki, M., Lancellota, R., and Lo Presti, D.C.F. (1995). Remarks on the stiffness
at small strains of six Italian clays. Proc. 1st International conference on ‘Pre-failure
Deformation Characteristics of Geomaterials’, Sapporo2, 817-836.
8) Jovcic, V. (1997): The measurement and interpretation of small strain stiffness of soils.
(Unpublished Doctoral thesis, City University London).
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J Chin Instit Eng, 30 (4), pp. 579-591.
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978-0-471-46302-3.
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shear strength to vertical effective stress. Proc. Schl. Eng. Tokai Univ., Ser. E 31 21-25.
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pp. 430-441.
15) Schneider, J.A., Hoyos, L., P.W. Mayne, E.J. Macari, G.J. Rix (1999): Field andlaboratory measurements of dynamic shear modulus of piedmont residual soils, in:Behavioral Characteristics of Residual Soils, GSP 92, ASCE, Reston, pp. 12–25.
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anisotropy on elastic stiffness of soft clays. Soils Found 48(1):127–132.
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Géotechnique, vol. 45(2), pp. 249-265.
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1, 187–194. Lisse: Swets & Zeitlinger.
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it depend on grain size? Géotechnique 63 (2), 165–179.
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