1

1 INTRODUCTION

Laboratory stress-strain test is only one of the important measures of geotechnical engineering research and practice. In addition, as it is usually very difficult to retrieve sufficiently high-quality undisturbed samples of field geomaterial, it is often considered that the laboratory stress-strain test is less direct (so less useful) when compared with the field-loading test. It is particularly the case with ordinary construction projects, in which sophisticated laboratory soil tests may be considered unwarranted, unlike a limited number of huge scale projects. Although it is not essential, another reason why the laboratory stress-strain test has become less popular would be that it is just painstaking and time-consuming, compared with other types of study and investigation, such as those by theoretical and numerical analyses.

It is also true, however, that the proper characterisation of in-situ stress-strain behaviour of geomaterial, which could result into a more rational (so safer and more cost-effective) design, often becomes possible with a help of relevant laboratory stress-strain tests. From a more basic point of view, the proper understanding of the stress-strain-time behaviour of geomaterials is not possible only by field loading tests and back-analysis of full-scale behaviour, but a comprehensive series of relevant laboratory stress-strain tests usually become necessary.

The following four topics, among others, were selected based on a consideration that their vital importance in geotechnical engineering research and practice are not well recognised among practicing engineers and are often ignored even in research: 1) elastic properties at very small strains as well as non-linear pre-failure stress-strain

behaviour of geomaterials; 2) inherent anisotropy in the strength and deformation characteristics of granular materials;

Impacts on Geotechnical Engineering of Several Recent Findings from Laboratory Stress-Strain Tests on Geomaterials

F. Tatsuoka University of Tokyo

ABSTRACT: Significant impacts on the theories and practice of geotechnical engineering of several findings obtained from recent advanced laboratory stress-strain tests on a wide variety of geomaterials that were performed mainly by the author and his colleagues are demonstrated and illustrated. The laboratory stress-strain tests were performed to apply their results to theoretical research as well as practical design. This paper discusses on: 1) elastic properties at very small strains as well as non-linear pre-failure stress-strain behaviour of geomaterials; 2) inherent anisotropy in the strength and deformation characteristics of granular materials; 3) strain localization with shear banding in granular materials; and 4) viscous deformation properties of geomaterials. The importance of knowing the limitations of using over-simplified stress-strain models, such as isotropic linear or perfectly-plastic models, of geomaterials is emphasized. It is attempted to show the important and essential roles of relevant laboratory stress-strain tests of geomaterials in developing the theories and practice of geotechnical engineering. This paper is the lecture note for the 2000 Burmister Lecture, 31st, October 2000, the Columbia University, N.Y., U.S.A.

2

Figure 1.1. Preparation of inclined specimens of sand by pluviation through air and subsequent moistening, followed by a sequence of freezing and thawing (Park andTatsuoka 1994).

0 3 6 9 12 15-2

0

2

4

6

8

a)

εvol - ε1 relations

R - ε1 relations

30o45o

90o

20o

0o

0o 30

o

20o

45o90o Ticino Sand

σ'3=0.8kgf/cm2

OCR=1.0

δ(o) e0.05

90 0.657 45 0.656 30 0.666 20 0.659 0 0.662P

rinci

pal s

tres

s ra

tio,

R=

σ' 1/

σ'3

Axial strain, ε1 (%)

4

0

-4

-8

-12

-16

Vol

umet

ric s

trai

n, ε

vol (

%)

0 3 6 9 12 15-2

0

2

4

6

8

a )

εvol - ε1 relations

R - ε1 relations

3 0o45o

90o

20o

0o

0o 3 0o

20o

45o

90o Ticino Sand

σ'3=0.8kgf/cm

2

OCR=1.0

δ(o) e0.05

90 0.657 45 0.656 30 0.666 20 0.659 0 0.662

Prin

cipa

l stre

ss r

atio

, R

=σ'

1/σ' 3

Axial strain, ε1 (%)

4

0

-4

-8

-12

-16

Vol

umet

ric s

trai

n,

ε vol (

%)

0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0b)

δ(o) e0.05

90 0.657 45 0.656 30 0.666 20 0.659 0 0.662

Dev

iato

r str

ess,

q=σ

' 1-σ

' 3 (kg

f/cm

2 )

Axial strain, ε1 (LDT) (%)

0 3 6 9 12 15-2

0

2

4

6

8

a)

εvol

- ε1 relations

R - ε1 relations

30o45o

90o

20o

0o

0o 30o

20o

4 5o90o Ticino Sand

σ '3=0.8kgf/cm2

OCR=1.0

δ( o) e0.05

90 0.657 45 0.656

30 0.666 20 0.659 0 0.662P

rinci

pal s

tres

s ra

tio,

R=

σ'1/σ

' 3

Axial strain, ε1 (%)

4

0

-4

-8

-12

-16

Vol

umet

ric s

trai

n, ε

vol (

%)0.000 0.001 0.002 0.003 0.004 0.005

0.00

0.02

0.04

0.06

0.08

0.10

0.12c)

Axial strain, ε1 (LDT) (%)

Dev

iato

r st

ress

, q=

σ' 1-

σ' 3 (

kgf/c

m2 )

δ(o) e0.05

90 0.657 45 0.656 30 0.666 20 0.659 0 0.662

Elastic properties

Figure 1.2. Anisotropy in PSC tests on air-dried air-pluviated Ticino sand; δ is the angle of 1σ with respect to

bedding plane: stress-strain relationships a) 1ε ≦11%, b) 1ε≦ 0.5% and c) 1ε ≦ 0.005% (Park and Tatsuoka 1994; Tatsuoka and Kohata 1995).

3) strain localisation with shear banding in granular materials; and 4) time-dependent deformation properties of geomaterials.

Some representative data illustrating these points above are first shown. Fig. 1.1 shows a method to prepare rectangular prismatic specimens having the axis of compression loading in-clined relative to the bedding plane direction, prepared for plane strain compression tests (Park and Tatsuoka 1994). The specimen was first made by pluviating air-dried sand partic les through air, subsequently made moist and then frozen under a restraint against the expansion of specimen upon freezing. The specimen was thawed under partial vacuum after being set in the triaxial cell. The specimen was made fully saturated and isotropically consolidated to 78 kPa. The details of the plane strain apparatus and the plane strain compression procedure are described in Shibuya et al. (1994), Yasin et al. (1999a & b), Masuda et al. (1999) and Yasin and Tatsuoka (2000).

Fig. 1.2 shows a result typical of the tests performed by Park and

3

Figure 1.3. A shear band seen at

aveγ = 11.8 % in a PSC test on Toyoura sand (D50= 0.206 mm;

3σ = 78 kPa) (Yoshida et al. 1995: Yoshida and Tatsuoka 1997).

Shear band in air-dried Toyoura sand (made moist after the test) in a plane strain bearing capacity test of rigid rough strip footing

B= 10 cm

Figure 1.4. A shear band network observed in the sand bed of Toyoura sand in a bearing capacity test of strip footing; the sand was air-dried during loading and made moist after the test to expose this central section (Fig. 4.20) (Tatsuoka et al. 1991).

Tatsuoka (1994). The angle δ means the angle of the direction of the major principal stress 1σ during plane strain compression relative to the bedding plane (see Fig. 1.1). It may be seen that

the stress-strain behaviour of sand is strongly affected by the angle δ (i.e., inherent anisotropy), except for at very small strains and at the residual state. It is likely that the initia l anisotropic structure is substantially damaged until the residual condition in the shear band. At strains less than about 0.005 %, the stress-strain relationships are rather linear (and reversible as shown later); i.e., elastic behaviour. The relationship between the major and minor

principal strains 1ε and 3ε is less sensitive to the angle δ than the stress-strain relation. The reason for such a variance as above is not well understood.

On the other hand, Fig. 1.3 shows a shear band that was observed at the residual conditions on the 2σ plane through the transparent confining platen in a test at 90oδ = (i.e., the conventional plane strain compression test). It may be seen from Fig. 1.3 that the shear band has a noticeable thickness, which means that the deformation characteristics of shear band could have important effects on the kinematics of a failing soil mass in a boundary value problem, such as the bearing capacity problem (as discussed later). Fig. 1.4 shows a shear band network that was observed in a level deposit of Toyoura sand supporting a strip footing (Tatsuoka et al. 1991). These four issues are only a part of a number of important geotechnical engineering issues that are still not well understood, but their proper understanding is essential for the development of geotechnical engineering theories and practice. In the following, these four topics will be explained, trying to show their engineering implications as much as possible. It is the final aim of the paper to show that geotechnical engineering is still young (although a number of myths are already existing in these issues).

2 ELASTIC PROPERTIES AT VERY SMALL STRAINS AND NON-LINEAR PRE-FAILUER STRESS-STRAIN BEHAVIOUR OF GEOMATERIALS

4

To predict ground deformations and structural displacements at working loads;

Why is the elastic property one of the main factors in characterising th pre-failure deformation property ?

- Strains in the ground at working loadare relatively small.

- Deformation properties at these small strainscan be linked to the elastic properties,

although the stress-strain behaviour could be highly non-linear at these small strains.

Figure 2.1. Several reasons why the elastic deformation properties of geomaterials are important in many geotechnical engineering problems.

Figure 2.2. General view of Akashi Kaikyo (Strait) Bridge and geological conditions (Tatsuoka and Kohata 1995).

2.1 Engineering needs

The elastic deformation property is the key reference property for the stress-strain behaviour of a given geomaterial subjected to cyclic loading (e.g. Hardin and Drnevich 1972; Iwasaki and Tatsuoka 1977; Tatsuoka et al. 1978, 1979a & b). It is to be noted however that the elastic deformation property is also an important and essential parameter for so-called static geotechnical loading problems. That is, one of the main features of the recent developments in the charac-terisation of geomaterial pre-failure deformation properties to predict ground deformations and structural displacements at working loads is focusing on the elastic deformation properties of concerned geomaterials (Fig. 2.1) (e.g. Jardine et al. 1985: Burland 1989; Jamiolokowski et al. 1991, Atkinson and Sällfors, 1991, Tatsuoka and Shibuya 1992, Mair 1993; Tatsuoka and Kohata 1995, Jardine 1995; Hight and Higgins 1995; Tatsuoka et al. 1995b, 1999a).

5

a)

180

160

140

120

100

80

6010-6 10-5 10 -4 10-3 10-2

Elevation　

TP-(m)

Vertical strain　εv

Pier 2PPier 3P

b)

Figure 2.3. a) Method used to measure the ground settlement; and b) centre-line vertical strains in the gravel and sedimentary softrock below the piers 2P and 3P for the world’s longest suspension bridge: Akashi Kaikyo Bridge. The average contact pressure and foundation diameters are 5.3 kgf/cm2 and 80 m for Pier 2P; 4.8 kgf/cm2 and 78 m for Pier 3P, respectively (Takeuchi et al. 1997).

Akashi Kaikyo Strait Bridge (Fig. 2.2) is the world longest suspension bridge. This case would be typical of those showing the importance of elastic deformation properties of geomaterials. Fig. 2.3 shows the center-line vertical strains in the gravel and sedimentary softrock below the piers (3P and 2P) (Tatsuoka and Kohata 1995, Tatsuoka et al. 1999a). As may be seen from this figure, the strains in the ground are generally small, lower than 0.5 %, which was due to: 1) the foundations were designed allowing only a limited amount of footing displacement; and 2) the foundations were constructed on relatively stiff ground (although the ground conditions

were worst in the Honshu-Shikoku connection bridge network). Fig. 2.4 shows the relationship between “the secant Young’s modulus values FEME at

different depths that were back-calculated by liner FEM from the measured ground settlement and the known footing load, divided by the respective corresponding elastic Young’s modulus

fE obtained from the field shear wave velocity that was measured before construction” and “the measured ground vertical strain”, for Piers 2P and 3P for Akashi Kaikyo Bridge (Tatsuoka and Kohata 1995). The relationships are compared also with those from the conventional pressure-meter tests. The following trends of behaviour may be noted: 1) the relationships evaluated from the field full-scale behaviour are highly non-linear (n.b.,

the non-linearity of the relationships is not due totally to the strain-non-linearity, but it is also affected by the changes in the pressure level during construction, in particular by those due to ground excavation);

6

0.0001 0.001 0.01 0.1 10.0

0.5

1.0

1.5

Ef: from V

s before construction)

EPMT

/Ef

(strais for E

PMT unreported)

｛

Gravelly soil

range forsoft rock 2

5

1

1

4

( )

3

72

6

3

4

5

6*

987

3P case1 for (p)ave= 0 ∼ 5.2 kgf/cm2

1 ∼ 6 Sedimentary soft rock (P3) 7 Granite * Ef was estimated as 5 x EPMT

2P case1 for (p)ave = 0 ∼ 5.3 kgf/cm2

1 ∼ 5 Gravelly soil (Akashi) 6 ∼ 9 Sedimentary soft rock△

EF

EM/E

f, E

PM

T/E

f

Measured ground vertical strain: ε1 (%)

｛

Ef

EFEM

Figure 2.4. Relationship between “the Young’s modulus back-calculated from the ground settlement divided by the respective elastic Young’s modulus from the field shear wave velocity measured before construction” and “the measured ground strains”, compared with those from the conventional pressure-meter tests, Akashi Kaikyo Bridge Piers 2P and 3P (Tatsuoka and Kohata 1995).

2) as the operating ground strain approaches 0.001 % (i.e., as the concerned soil layer becomes

deeper) , the FEME value approaches the respective corresponding fE value; and

3) the Young’s modulus values PMTE from the conventional pre-bored pressure-meter tests (i.e., linear interpretation of primary loading curves) are noticeably lower than those operating in the ground. This is very likely due to large strains involved in the tests and the effects of wall disturbance and bedding error at the wall face.

At the design stage, there was not a distinct consensus with respect to the stiffness value that should be used to predict the instantaneous settlement of foundation among not only the engineers in charge of this project but also geotechnical engineers in general. In the conventional approach in rock mechanics, the tangent Young’s modulus defined at a deviator stress that is a half of the peak strength ( 50E ) or the PMTE values cited above or the values ( PLTE ) from the conventional plate loading tests (linear interpretation of primary curve or unload/reload cycle curve) have been often used. For such a case of foundation design as this case, the plate loading test results are often considered to be most relevant. In addition, the effects of strain-non-linearity and effects of pressure (or more generally, effects of stress state) on the stiffness were not considered in a systematic way.

Fig. 2.5 compares the Young’s modulus values obtained from a set of field loading tests that were performed at the site of Anchor 1A and laboratory stress-strain tests on samples retrieved from the bottom of the excavation (Tatsuoka and Kohata 1995). The ground, consisting mainly of sedimentary soft rock, was excavated to a depth of about 60 m for constructing the anchorage. In this figure, the values of Young’s modulus ( fE ) from the field shear wave velocities obtained by the down-hole suspension method performed before the excavation are also shown. It may be seen that the statically measured Young’s modulus values (i.e., those from the conventional-type pressure-meter tests, plate loading tests and unconfined compression

7

100 1000 10000 100000-70

-65

-60

-55

-50

Dep

th (m

)

Young's modulus E (kgf/cm2)

Emax from CU and CD TC tests

σc' = σv' (in situ)= 5.2 (kgf/cm2)

Kobe group softrock

1A-11A-21A-31A-41A-51A-6

Ef (from shear wave velocity)

▼＋ 40∼6020∼40,0∼20,▲

range of plate pressure(kgf/cm2)

EPLT

; tangent modulus in primary loading

▲＋▼

EBHLT

; primary loading△

E50

unconfined compression tests(from external axial strains)

□

Emax

Emax

Average (from CD TC tests; axial strains measured with LDTs)

○●

Figure 2.5. Young’s modulus values (in log scale) from field and laboratory tests at Anchor 1A, Akashi Kaikyo Bridge; the diameter of plate in the PLTs= 60 cm (Tatsuoka and Kohata 1995).

tests) are substantially lower than those obtained by the dynamic method (i.e., the field shear wave velocity measurement). It is usual to observe such a large difference as above in such construction projects as this case. This would be one of the origins (perhaps the most important one) for the popular but wrong notion that a given geomaterial mass has static and dynamic Young’s modulus values (or static and dynamic elastic stiffness values) that are different material properties (Tatsuoka and Shibuya 1991; Tatsuoka and Kohata 1995; Tatsuoka et al. 1995a-d, 1999a).

A series of CD and CU triaxial compression tests were performed on undisturbed samples obtained by block sampling at the bottom of excavation performed for the construction of the anchor A1 (Tatsuoka and Kohata 1995; Tatsuoka et al. 1995d; Siddiquee et al. 1994, 1995a). Fig. 2.6 shows the relationship between the deviator stress q and the axial strain 1ε typical of those from CD TC tests on samples of soft sandstone that were isotropically reconsolidated to the respective original field vertical stress. The axial strains were measured locally by means of a pair of LDTs (explained later) as well as externally outside the triaxial cell (i.e., the conventional method). It may be seen from Fig. 2.6 that the axial strains measured externally are utterly unreliable, which is due mostly to significant effects of bedding error at the top and bottom ends of specimen partly to the deformation of the loading piston and specimen cap (i.e., the system compliance). Each drained elastic Young’s modulus at the field stress state was evaluated from the initial linear and reversible part of the respective relationship between q and the locally measured 1ε . By using the relevant Poisson’s ratios for drained and undrained conditions, this drained value of elastic Young’s modulus was converted to the value under the undrained conditions, which is relevant to the field seismic investigation. These undrained values of Young’s modulus ( maxE ) are plotted in Fig. 2.5. It may be seen that these Young’s modulus values ( maxE ) are very similar to those from the field shear wave velocities. This fact indicates that in this case, the dynamically and statically measured elastic Young’s modulus

8

0.00 0.02 0.04 0.06 0.08 0.100.0

0.2

0.4

0.6

0.8

v

External

LDT

Dev

iato

r st

ress

, q (

MP

a)

Axial strain, ε (%)

0 1 2 30

2

4

6

8

10

12

0maxq = 9.39 MPa, E = 1520 MPa

hσ '= 0.51 MPa (CD)Sedimentary soft sandstone (Kobe Formation)

ExternalLDT

v

Dev

iato

r str

ess,

q (M

Pa)

Axial strain, ε (%)

0.0000 0.0005 0.0010 0.0015 0.00200.00

0.01

0.02

0.03

0.04

0E = 1520 MPa

1

LDTv

Dev

iato

r str

ess,

q (M

Pa)

Axial strain, (ε ) (%)

Triaxial compression teston an undisturbed sample

Figure 2.6. Typical relationship between the deviator stress and the axial strain from a CD TC test on a specimen obtained by block sampling at the bottom of excavation for Anchor A1, Akashi Kaikyo Bridge (Tatsuoka and Kohata 1995).

values are very similar to each other, while the effects of anisotropy, discontinuities and inhomogeneity of the ground are not significant, if any.

The direct use of the Young’s modulus values determined by the conventional pressuremeter tests, plate loading tests and unconfined compression tests could largely over-estimate the actual settlement of the plate in the plate loading tests and the full-scale structure in the field as shown below. Fig. 2.7 shows results from three plate loading tests using a 60 cm-diameter rigid plate performed at the bottom of excavation for Anchor A1 for Akashi Kaikyo Bridge (n.b. the data from one test from the four tests performed is excluded, because the result is extraordinary due very likely to the effects of a large joint that was opened by ground excavation,) (Tatsuoka and Kohata 1995; Tatsuoka et al. 1999a). In this figure, the ir simulations by linear FEM using the following Young’s modulus values are also shown; a) the average of the Young’s modulus values from unconfined compression tests (E50),

shown in Fig. 2.5; b) the average of the Young’s modulus values from the pressuremeter tests (EPMT), shown

in Fig. 2.5; and c) the value obtained from the back-analysis of the full-scale behaviour of Pear P3 (shown

in Fig. 2.9). The result from a simulation by a non-linear elasto-plastic FEM using the elastic Young’s

modulus from the field shear wave velocity measurements together with on the pressure-dependency and strain-non-linearity of stiffness evaluated by triaxial compression tests on disturbed samples from the site is also shown in Fig. 2.7 (Siddiquee et al. 1994, 1995a). It may be seen that the only the non-linear elasto-plastic FEM analysis reasonably simulates the results of the plate loading tests. In particular, the concave shape of the relationship between the plate pressure and the plate settlement can be well simulated, which can be attributed to the following:

9

0 2 4 6 8 100

10

20

30

40

50

60

70

1

Es

unloading (S-3)

1

E t

1

D

E=1

0000

kgf/c

m2

(from

full-

scal

e fie

ld b

ehav

iour

of 3

P)

E PMT

=312

5kgf/c

m2

E 50=177

7kgf/cm

2 (from U tes

ts)

linear elastic FEM

elasto-plastic FEM

Plate loadingtest(D=60cm)

S-1 S-3 S-4

(p) av

e.(k

gf/c

m2 )

Settlement, S (mm)

Figure 2.7. Results from plate loading tests performed at the bottom of excavation for Anchor A1 for Akashi Kaikyo Bridge; and their simulations by linear FEM using a) the average Young’s modulus values from unconfined compression tests (E50), pressuremeter tests (EPMT) and the back-analysis of the full-scale behaviour of Pear P3 and by elasto-plastic FEM analysis using the Young’s modulus from the field shear wave velocity and the pressure-dependency and the non-linearity of stiffness from laboratory stress-strain tests (Siddiquee et al. 1994, 1995a; Tatsuoka and Kohata 1995; Tatsuoka et al. 1999a).

a) the elastic deformation occupies a large part of the ground deformation in the plate loading tests; and

b) the pressure-level dependency of elastic deformation characteristic of the ground is properly modelled in the simulation.

The points a & b are also relevant to the field full-scale behaviour. Fig. 2.8, for example, shows the relationships between the average contact pressure and the total, elastic and irreversible settlements (S , eS and ir eS S S= − ) of Pier 3P, for which the elastic part was obtained by the FEM analysis based on the elasticity model (Tatsuoka et al. 1999a). A similar result has also been obtained from Pier 2P (Siddiquee et al. 1994, 1995a). For the decomposition of the footing settlement, FEM analysis based on a hypo-cross elasticity model, which is explained later in this paper, was performed. The parameters of the model were determined based on the initial elastic stiffness values from the field shear wave velocities measured at the sites of Piers 2P & 3P and the pressure-dependency of the elastic deformation properties that were obtained from laboratory cyclic and monotonic triaxial tests on undisturbed samples retrieved from the site. The separation of the total settlement into two components (elastic and irreversible), not into three components (elastic, plastic and viscous), is not arbitrary. This issue is discussed in detail by Tatsuoka et al (2000).

It is to be noted from Fig. 2.8 that the tangential slope of the relationship between the average contact pressure at the footing base and the elastic component of the settlement of the pier

10

increases with the increase in the contact pressure. This behaviour can be attributed to the fact that the elastic Young’s modulus of the sedimentary soft sandstone increases with the increase in the pressure, as observed in the triaxial compression tests. This issue is discussed in detail in Tatsuoka et al. (1995a-d, 1999a) and Kohata et al. (1997).

A fluctuation seen in the relationships between the footing contact pressure and the total or irreversible footing settlement is not due to simple measurement errors, but it is very likely that this fluctuation was basically due to the viscous deformation property of the ground (as discussed later). That is, the tangential slope is small when the construction rate was slow and vice versa, and the stiffness becomes very large immediately after construction was restarted at a normal construction speed following a long period of construction stop. It will be shown later that a relevant elasto- “viscoplastic” model could explain this behaviour.

Fig. 2.9 shows the full-scale behaviour of Pear P3 until the average footing pressure became about 95 kPa (refer to the full-range behaviour show in Fig. 2.8). The simulation of this behaviour by the following methods are also shown: a) linear FEM using:

i) the average Young’s modulus value from unconfined compression tests (E50); and ii) that from the pressuremeter tests (EPMT) obtained at the site of A1; and;

b) the non-linear elasto-plastic FEM analysis using the elastic Young’s modulus from the field shear wave velocity and the pressure-dependency and non-linearity of stiffness from laboratory stress-strain tests on undisturbed samples obtained at the site of 3P (Tatsuoka and Kohata 1995; Tatsuoka et al. 1999a; Siddiquee et al. 1994, 1995a).

It may be seen again that the field behaviour can be reasonably simulated only by the relevant FEM analysis b). By comparing Figs. 2.7 and 2.9, it may be seen that for the result of a linear isotropic analysis to fit the full-scale behaviour of Pier 3P, a Young’s modulus value that is much larger than the value to be used to fit the plate loading test result should be used. This difference is due to the so-called scale effect on the stiffness of ground. This scale effect in this respect can be attributed to an initially inhomogeneous distribution of elastic stiffness and the pressure- and strain-non-linearity of stiffness. The latte factor can be evaluated by relevant laboratory stress-strain tests.

-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

0

2

4

6

8

10

.Sir = 0.05 (mm/day)

.Sir (mm/day)

b)

Fitted to

3P

St

Sir

Se

0.05

0.00-0.05 0.05-0.10 0.10-0.15 0.15-0.20 0.20-0.25 >0.25A

vera

ge c

onta

ct p

ress

ure,

(p) av

e (k

gf/c

m2 )

Settlement, S (mm)

Figure 2.8. Decomposition of the measured settlement of Pier 3P into elastic and irreversible components, Akashi Kaikyo Bridge (Tatsuoka et al. 2001a).

11

0

2

4

6

8

10

12

0 20 40 60 80 100Settlement, S (mm)

(p)a

ve.

(kgf

/cm2)

measured

E=10000kgf/cm2

E50(from 1A site)

=1777kgf/cm2

FEM

EPMT(from 3P site)

=2890kgf/cm2

Figure 2.9. Full-scale behaviour of Pear P3 for Akashi Kaikyo Bridge; and its simulations by linear FEM using a) the average Young’s modulus values from unconfined compression tests (E50), and pressuremeter tests (EPMT) obtained at the site of A1; and b) elasto-plastic FEM using the Young’s modulus from the field shear wave velocities measured at the site of 3P and the pressure-dependency and non-linearity of stiffness from laboratory stress-strain tests (Sddiquee et al. 1994, 1995a; Tatsuoka and Kohata 1995; Tatsuoka et al. 1999a).

Another typical example is the behaviour of the foundations for another suspension bridge, Rainbow Bridge, located in the Tokyo area (Fig. 2.10). This bridge was completed in 1993. The foundations were caissons that were constructed on a thick layer of sedimentary soft mudstone of a geological age of about 1.5 million years having a compressive strength of about 2.0 - 3.0 MPa (from CD triaxial compression tests). It was considered at the initial design stage that the foundation should be supported by pile foundations, anticipating excessive settlements of foundations. The four foundations were constructed directly on the mudstone layers based on the judgement made referring to the results of analysis using stiffness values from conventional pressure-meter tests (Tatsuoka and Kohata 1995: Izumi et al. 1997). Fig. 2.11 shows the comparison between the measured instantaneous settlement of anchorage A4 that took place when constructing the top anchorage block having a weight of 140,000 tonf and its predictions. As seen from Fig. 2.11, the actually observed settlement was substantially smaller than the value predicted before the start of construction based on the stiffness values obtained from the primary loading curves of oedometer tests on undisturbed samples. This substantially large over-estimation of the observed value is most likely due to very large effects of bedding error at the top and bottom surfaces of a thin specimen with a thickness of as small as 2 cm. The settlement predicted based on the unload/reload stiffness from the oedometer tests is better, but it still largely overestimates the observed value. In addition, there is no sound reason for the use of the stiffness values of unload/reload cycles, unlike the field stress history, in the oedometer tests. The prediction based on the pressuremeter stiffness is also noticeably larger than the observed value.

12

Figure 2.10. Rainbow Bridge

- Over-estimation of the footing settlement by the conventional methods.- Accurate estimation when based on .

140,000 tonf

AnchorageObserved

Non-liner FEMusing the shearmodulus from Vsand the non-linearityfrom CD TC tests

Observed

Based on mv fromprimary loadingcurves of oedometertests

Based on PMTs

Based on mv fromunload/reload curvesof oedometer tests

Figure 2.11. Comparison between the observed behaviour of Anchor A4 and the predictions by the conventional method and the simulation by non-linear 3D FEM using stiffness values obtained from SOA geotechnical investigation, Rainbow Bridge (Izumi et al. 1997).

To find a reason(s) for this inconsistency, after the completion of the bridge, a SOA geotechnical investigation was carried out, consisting of a suspension-type downhole seismic survey, undisturbed sampling by rotary core tube sampling, triaxial compression tests measuring axial strains locally using LDTs and non-linear 3D FEM analysis based on the Young’s modulus (or shear modulus) from the field shear wave velocity and strain-non-linearity from the triaxial compression tests (Izumi et al. 1997). In this FEM analysis, the effects of changes in the effective pressure on the stress-strain behaviour of the sedimentary soft mudstone were not considered based on the triaxial test results. The result from this SOA investigation is also shown in Fig. 2.11. It may be seen that the observed behaviour is best simulated by this non-linear FEM analysis using the stiffness obtained from the SOA geotechnical investigation. Fig. 2.12 compares the shear modulus values, as a function of shear strain, obtained from the CD triaxial compression tests, the conventional pressuremeter tests and the field full-scale behaviour for some range of depth below the anchorage, which is typical of similar comparisons in this case (Izumi et al. 1977). The following trends of behaviour may be seen from Fig. 2.12, which were basically the same as those with the foundations for Akashi Kaikyo Bridge: that is: 1) the conventional pressuremeter stiffness values are too small, representing the stiffness

2

f sG Vρ= ⋅

13

Full-scale behaviour CD Triaxial compressiontests using LDTs

Pressure-meter tests(Linear interpretation of primary loading curve)

Shear wave velocities(Suspension method)

1 (%)Strain ε0.1 1.0 100.010.0010.0001

Shea

r m

odul

us, G

(M

Pa)

Figure2.12. Typical comparison between the stiffness values as a function of strain from CD triaxial compression tests (GL –54 – 64 m and 64 – 74 m), conventional pressuremeter tests and field full-scale behaviour, Anchor A4, Rainbow Bridge (Izumi et al. 1997).

values at strains that are considerably larger than those operated in the ground; and 2) the stiffness values from the CD TC tests are consistent with:

a) the elastic shear modulus from field shear wave velocities (defined at strains less than about 0.0001 %);

b) those from the conventional pressuremeter tests measured at strains from a range from about 0.5 % to about 1.0 %, and

c) those from the field full-scale behaviour at strains of about 0.01 %.

Summary: The elastic deformation characteristics that are to be referred to when predicting the ground deformation and structural displacements in field static loading cases are often considered to have no link to the dynamically measured deformation properties. In addition, linear deformation analysis of filed loading tests and full-scale behaviour is stiff popular. In many cases, the strains operated in the ground are relatively small. In such cases, prediction of ground deformation based on the elastic stiffness obtained from field shear wave velocities, while considering non-linearity and pressure level-dependency of stiffness at relatively small strains, could be relevant. Laboratory stress-strain tests can contribute in many important aspects to such geotechnical engineering issues as described above. 2.2 Modelling of small strain stiffness

A brief overview: Previously, the most common practice to obtain the elastic deformation properties was dynamic tests such as the resonant-column (RC) tests and the wave propagation tests (e.g. Hardin and Richart 1963; Hardin and Back 1968). Later, Hardin and Drnevich (1972), Iwasaki et al (1978), Teachavorasinskun et and (1991a & b), Shibuya et al. (1992) and others showed that the strain rate -dependency of the stiffness at small strains of geomaterials in cyclic torsional shear is very low. Based on such experimental results as above, Woods (1991) and Tatsuoka and Shibuya (1992) pointed out that it is not necessary to distinguish between dynamically and statically measured elastic stiffness values when they are measured under otherwise the same conditions.

In the meantime, we have become rather confident with reliable measurements of strains less than about 0.001 % and associated small stresses in both triaxial and torsional shear tests. It has therefore become rather popular to obtain the stiffness and damping values under cyclic loading

14

Figure 2.14. Relationships between the peak-to-peak secant shear modulus and the single amplitude shear strain from a torsional RC test (Lo Presti 1989, Lo Presti et al. 1993), a cyclic torsional shear test and a cyclic triaxial test; and between the secant shear modulus and the shear strain from a monotonic torsional shear test and a monotonic triaxial compression test, all on an isotropically consolidated specimen of air-pluviated Ticino sand (Teachavorasinskun et al. 1991a & b: Tatsuoka et al. 1995b, 1999a).

1) Elastic deformation characteristics :can be obtained only by dynamic tests ?

2) Statically and dynamically determined elastic deformation properties are different ?

Static tests (monotonic or cyclic); stress-strain properties from stresses and strains !

Dynamic tests (RC tests & wave-propagation tests); stress-strain properties from dynamic responses !

Figure 2.13. Some relevant questions with respect to the elastic deformation characteristics of geomaterials .

conditions for a full range of concerned strain (usually from lower than 0.001 % to around 1 %) by cyclic static loading tests using a single specimen. It is only recent however that it becomes possible to evaluate confidently the elastic deformation properties, as well as the whole pre-peak stress-strain behaviour and peak strength, by monotonic loading tests using a single specimen (e.g. Tatsuoka and Shibuya 1992; Tatsuoka 1994, Tatsuoka et al. 1990c, 1994a & b, 1995a & b).

It seems however that it is still popular in ordinary engineering practice to define and treat separately dynamic and static stiffness values of a given mass of geomaterial. Therefore, the questions listed in Fig. 2.13 are still relevant. In Fig. 2.14, the shear modulus values of Ticino sand that was isotropically consolidated to 49 kPa obtained from dynamic and static (cyclic and monotonic) tests are compared. It may be seen that the shear modulus at shear strains less than about 0.001 % is essentially the same among the dynamic tests (i.e., resonant-column tests) and the static tests (i.e., monotonic and cyclic triaxial and torsional tests). It has also been shown that for various types of fine-graded sands, the elastic stiffness from RC tests and static tests is essentially the same with the values from the bender element tests, as summarised in Tatsuoka et al. (1999a). The differences seen among the shear modulus values from the different testing methods at strains exceeding the elastic limit strain, equal to about 0.001 %, is due mostly to different stress paths and different strain histories (i.e., with and without cyclic loading) and due only partly to different strain rates.

Fig. 2.15 shows a summary of data showing the effects of strain rate on the very small strain Young’s modulus Ev defined for (∆εv)SA of 0.001 % or less (Tatsuoka et al., 1999a &b). These data were obtained from the following series of tests, which are mostly static tests; cyclic tests and monotonic tests (only for kaolin), except for dynamic tests on hard rock cores, concrete and mortor: 1) cyclic triaxial tests (U; undrained and D; drained):

a) Sagamihara soft mudstone ( ' 'v hσ σ= = 4.8 kgf/cm2; Tatsuoka et al. 1995a & b); b) OAP clay ( 'vσ =6.9 kgf/cm2 and 'hσ =3.4 kgf/cm2; Tatsuoka et al. 1995a);

15

Sandy gravel (D)

sand (U)

Hostun sand (D)

Resonant-column

Hard rock core

Ultrasonic waveConcreteMortar

Sagamihara soft rock (U)

OAP clay (U)

Air-dried

Wet Chiba gravel (D)

Metramo silty sand (U)

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104

103

104

105

106

Vallericca clay

.

E v (kg

f/cm

2 )

Axial strain rate, dεv/dt (%/min)

N.C. Kaolin (CU TC)

Saturated Toyoura

Figure 2.15. Summary of the effects of strain rate on the very small strain Young’s modulus Ev

defined for (∆εv)SA of 0.001 % or less (Tatsuoka et al., 1999a,b).

c) Chiba gravel (e= 0.247, w0= 3.7 % and ' 'v hσ σ= = 0.2 kgf/cm2; Jiang et al. 1999;

Tatsuoka et al. 1999b); d) air-pluviated Toyoura sand (e= 0.658 and ' 'v hσ σ= = 1.0 kgf/cm2; Tatsuoka et al. 1995a);

16

Figure 2.16. Large triaxial specimen (30 cm in dia) with local strain measurements at the University of Tokyo (Tatsuoka et al., 1994a).

Figure 2.17. Triaxial testing systems using a small cylindrical specimen at the University of Tokyo (Tatsuoka et al., 1995a, 1999a) .

Phosphor bronzestrain-gaged strip

LDT

Pseudo-hinge

Membrane

Heart of LDT(includes electric resistance strain gages, terminals, wiring, sealant)

Scotch tape used to fix wireon the specimen surface

Instrument Leadwire

Membrane Surface

Figure 2.18. Local deformation transducer (Goto et al. 1991; Hoque et al. 1997).

e) air-pluviated Hostun sand (e= 0.72, 'vσ = 0.8 - 2.5 kgf/cm2 and 'hσ = 0.8 kgf/cm2; Hoque 1996, 1997; Hoque and Tatsuoka 1998; Tatsuoka et al., 1999b); and

f) compacted Metramo silty sand ( ' 'v hσ σ= = 4.0 kgf/cm2; Santucci de Magistris et al. 1998, 1999, Santucci de Magistris and Tatsuoka 1999).

2) CU TC tests on NC kaolin (pc= 3.0 kgf/cm2 and Kc=0.6 - 1.0) (Tatsuoka et al. 1995a). 3) unconfined cyclic tests and

ultrasonic tests on hard rocks, concrete and mortar (Sato et al. 1997a, b); the strain rates in the ultra-sonic tests were evaluated from the wave frequency and particle velocity. The following trends of behaviour

could be seen from Fig. 2.15: a) With hard rock cores, concrete

and mortar specimens, effects of strain rate on the elastic Young’s modulus are very small in the static tests. These test results indicate that the effects of strain rate are very small commonly with these materials.

b) With concrete and mortar speci-mens, a good agreement can be seen between the elastic stiffness values from the resonant column tests and static tests at the

17

-0.0010 -0.0005 0.0000 0.0005 0.0010-4

-3

-2

-1

0

1

2

3

4

f (Hz) Ev(s)(MPa) 10 477.9 5 479.0 1 484.8 0.2 476.0 0.1 469.0 0.02 470.3 0.01 458.3 0.002 455.3

Chiba gravel

5 th cycleσ

h=19.6 kPa

Dev

iato

r stre

ss, q

(kPa

)Axial strain, ε

v (%)

a)

-0.0010 -0.0005 0.0000 0.0005 0.0010-4

-3

-2

-1

0

1

2

3

4

Chiba gravel

5th cycleσh=19.6 kPa

f(Hz) d εv/dt (%/min)

10 3.6x10 -1

5 1.8x10 -1

1 3.6x10 -2

0.2 7.2x10-3

0.1 3.6x10-3

0.02 7.2x10 -4

0.01 3.6x10 -4

0.002 7.2x10 -5

Dev

iato

r stre

ss, q

(kPa

)

Axial strain, εv (%)

b) Figure 2.19. Relationships between the deviator stress q and the axial strain εv (measured with a pair of LDTs) at the fifth cycle at different strain rates from drained cyclic triaxial tests performed at a constant confining pressure σ h= 19.6 kPa on an isotropically consolidated very dense specimen of mo ist well-compacted well-graded gravel of crushed sand stone (Chiba gravel) (Jiang et a1. 1999; Tatsuoka et al. 1999b); and b) overlapped relationships of the stress-strain curves in Fig. A).

same strain rate. Considering that the average material property of a given specimen is measured in both types of test, these test results indicate that with these materials, the measuring methods, dynamic or static, have no effects on the measured stiffness value.

c) With hard rocks and mortar, a good agreement can be seen also between the elastic stiffness values from the-propagation tests and the static tests. Considering that if the specimen is not homogeneous, the body wave velocity reflects the property of stiffer zones to a larger extent than that of softer zones, these tests results indicate that these materials are essentially homogeneous in terms of the wave length in the wave propagation tests. A similar result has also been obtained with cement-mixed sand related to the ground improvement work for the Trans-Tokyo Bay High project (Tatsuoka and Shibuya 1991; Tatsuoka et al. 1997b) and sedimentary softrock (Tatsuoka et al. 1997a).

d) With concrete, on the other hand, the stiffness from the wave propagation tests is noticeably larger than that from the static tests even when compared at the same strain rate. These test results indicate that concrete is noticeably heterogeneous in terms of the wave length in the wave propagation tests. A similar result has been obtained with a gravel (Modoni et al. 1999, 2000). Tanaka et al. (1995) showed that the ratio of the elastic stiffness evaluated by the wave propagation test to that by the static test increased with the increase in the particle size when the particle size exceeded some limit. Souto et al. (1994) showed that the ratio of the elastic stiffness of gravels, as used for road pavement, evaluated by the bender element test to that by the resonant-column test increased with the particle diameter. Sato et al. (1977a & b) showed that with hard rock cores, the ratio of the elastic stiffness evaluated by the wave propagation test to that by the static test became a minimum, much less than unity, when the density of crack was a certain intermediate value (as quoted in Fig. 7.5 of

18

-0.0008 -0.0004 0.0000 0.0004 0.0008-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

5th cycleσh= 19.6 kPa

f (Hz) 10 5 1 0.2 0.1 0.02 0.01 0.002

Rad

ial s

train

, ε h (

%)

Axial strain, εv (%)

a)

-0.0008 -0.0004 0.0000 0.0004 0.0008-0.0003

-0.0002

-0.0001

0.0000

0.0001

0.0002

0.0003

Chiba gravel

5th cycleσ

h= 19.6 kPa

f (Hz) 10 5 1 0.2 0.1 0.02 0.01 0.002

Rad

ial s

train

, ε h (

%)

Axial strain, εv (%)

b) Figure 2.20. a) Relationships between the radial strain εh (measured with four pairs of proximity transducers) and εv (measured with a pair of LDTs) at the fifth cycle for different strain rates (the εh values have been shifted arbitrarily so that the loops do not overlap) from cyclic triaxial tests on an isotropically consolidated very dense specimen of Chiba gravel (e0= 0.247 and w0= 3.73 %) (Jiang et a1., 1999; Tatsuoka et al. 1999b); and b) overlapped hysteresis loops.

Tatsuoka et al. 1999a). Sato et al. (1997a & b) also showed that the ratio was nearly one when the amount of crack was negligible (as shown in Fig. 2.15), while the ratio tended to approach the unity as the crack density increased exceeding the above-mentioned intermediate value (as it is with fine-grained granular materials). All these results indicate that the wave propagation test becomes not relevant for the purpose of evaluating the average elastic de-formation property when the material becomes discontinuous when compared with the wave length in the wave propagation test.

e) Also with the other types of geomaterials , the effects of strain rates on the elastic stiffness evaluated by triaxial tests are generally not significant (as discussed more in detail below).

To evaluate the stress-strain behaviour under both monotonic and cyclic loading conditions of undisturbed samples of geomaterial, the triaxial testing method is the most practical and relevant one. Note however that to obtain reliable stress-strain behaviour from strains less than 0.0001 % to that at peak by this testing method, strains and stresses should be measured accurately for this full range of strain. The data from static loading tests presented in Fig. 2.15 were obtained by local axial strain measurements (except for the data of reconstituted kaolin). Indeed, the effects of bedding error in triaxial compression tests on specimens that do not exhibit noticeable compression during consolidation could be significant (e.g. Kim et al. 1994; Tatsuoka et al., 1995a; see also Fig. 2.6). Different methods have been developed to locally measure axial strains (as summarised by Tatsuoka et al. 1999a,b); such as the inclinometer at the Imperial College (UK), LVDTs at several laboratories; and LDTs (local deformation transducers at the University of Tokyo (Tatsuoka 1988; Tatsuoka et al. 1991; Goto et al. 1991; Hoque et al. 1997; Santucci di Magistris et al., 1999). Figs. 2.16 and 2.17 show how LDTs are used in triaxial compression tests and Fig. 2.18 shows the details of the arrangement of a LDT. The LDT is a sort of clip gauge made of a thin narrow strip of phosphor bronze, which is pinched, after slightly bent, between two small metal pieces glued on the lateral surface of the latex membrane of specimen. Electric -resistant strain gauges attached to the central part of the strip form a full bridge, which detects very sensitively the compression of the strip in the axial direction. The details of manufacturing, calibrating and setting of LDTs are described by Goto et al. (1991), Hoque et al. (1996, 1997) and Santucci de Magistris et al. (1999).

19

0.00001 0.00010 0.00100 0.01000 0.100000

200

400

600

800

1000

1200

1400

σ'c = 98.1 kPa σ'c =196.2 kPa σ'c =392.4 kPa

.

(a)

Initi

al Y

oung

's m

odul

us, E

0 (MPa

)

Axial strain rate, Év (%/min)0.00001 0.00010 0.00100 0.01000 0.10000

0

2

4

6

8

10

.

Metramo silty sandtest MO03undrainedε

a,SA = 0.00075 %

e0 = 0.307

σ'c = 98.1 kPa

σ'c =196.2 kPa σ'

c =392.4 kPa

(b)

Initi

al d

ampi

ng ra

tio, h

0 (%

)

Axial strain rate, Év (%/min)

Figure 2.21a. Relationships between; a) the peak-peak secant Young’s modulus; and b) damping ratio and the strain rate from cyclic undrained triaxial tests on an isotropically consolidated saturated specimen of compacted Metramo silty sand (Santucci de Magistris et al. 1999).

Strain rate-dependency of small strain stiffness: Fig. 2.19 shows the relationships between the deviator stress and the axial strain at very small strains obtained from a series of cyclic triaxial tests at different loading frequencies on Chiba gravel (shown in Fig. 2.15; Jiang et al. 1999, Tatsuoka et al. 1999b). The confining pressure was kept constant during each of the tests. It may be seen that for the examined wide range of strain rate, the relationship between the deviator stress and the axial is rather independent of loading frequency (i.e., independent of strain rate). The associated relationships between the axial and lateral strains are presented in Fig. 2.20. A similar insensitivity of stress-strain behaviour to changes in the strain rate may be seen in these relationships.

It may be seen by carefully examining Fig. 2.15, however, that the dependency of the peak-to-peak secant Young’s modulus of the gravel is not perfectly negligible, in particular at smaller strain rates, while the dependency becomes smaller as the strain rate becomes larger. It is also the case in a more clear manner with a high-compacted silty sand, which was used as the core material for a rockfill dam (Metramo silty sand from Italy; Santucci de Magistris et al. 1999, Santucci de Magistris and Tatsuoka 1999). The detailed result for this silty sand is presented in Figs. 2.21a. The behaviour presented in Fig. 2.21a was simulated based on a linear three-component rheology model by Di Benedetto and Tatsuoka (1997).

0.0000 0.0005 0.0010 0.00150

2

4

6

8

10

12

14

16

18

20

22

larger strain rate

start of loading

Metramo silty sandMO03UT

3rd

cycleσ'

c= 392.4 kPa

(a)

Axial strain rate εv

3.52x10-5 %/min

1.55x10-4 %/min

4.11x10-4 %/min

8.08x10-4 %/min

2.44x10-3

%/min

8.00x10-3

%/min

2.44x10-2 %/min

Dev

iato

r stre

ss in

crem

ent,

∆q (k

Pa)

Axial strain increment, ∆εv (%)-0.0015 -0.0010 -0.0005 0.0000-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

larger strain rate

start of unloading(b)

Axial strain rate εv

3.52x10 -5 %/min

1.55x10-4

%/min

4.11x10-4

%/min

8.08x10 -4 %/min

2.44x10 -3 %/min

8.00x10 -3 %/min

2.44x10 -2 %/min

Dev

iato

r stre

ss in

crem

ent,

∆q (k

Pa)

Axial strain increment, ∆εv (%)

Figure 2.21b. Relationships between the deviator stress q and the axial strain vε (measured with a pair of LDTs) at strains less than about 0.0015 % during a) loading ( q∆ > 0) and b) unloading ( q∆ <0) from cyclic undrained triaxial tests on isotropically consolidated Metramo silty sand (Santucci de Magistris et al. 1999); in each figure, the origins have been reset to the common starting point of loading and unloading.

vε&

vε&

20

0.00001 0.00010 0.00100 0.01000 0.10000800

900

1000

1100

1200

1300

1400

Axial strain, 2(∆εv)sa

1.05x10-6

2.02x10-6

5.00x10-6

1.47x10-5

Metramo silty sandMO03UT

3rd cycleσ'

c= 392.4 kPa

.

Seca

nt Y

oung

's m

odul

us, E

sec (

MPa

)Axial strain rate, Év (%/min)

Elastic property

Quasi-elasticproperty

Figure 2.22. Relationships between the secant Young’s modulus defined at different small strains obtained from the data presented in Fig. 2.21 (Santucci de Magistris et al. 1999).

　　 q Elastic limiting line Slope; Eo at Increasing strain rate 1 constant strain rate 　Strain rate 1 　　　　　　　　　　　　　　　 　　　　　　　　　　　 ; Limit of elastic behaviour at each strain rate

0 0.001 % ε

Figure 2.23. Framework of stress-strain relationship at small strains (Tatsuoka and Shibuya 1992: Tatsuoka et al. 1999a).

To examine the dependency of the stiffness and damping ratio on the strain rate seen in Fig. 2.21a, the respective stress and strain relationships during loading and unloading in cyclic loading tests were plotted separately (Fig. 2.21b). It may be seen that the stress-strain relation becomes more non-linear and its strain rate-dependency becomes more obvious with the increase in the strain level (defined from the moment of reversing the loading direction): in other words, the stress-strain relationship becomes more

linear with the increase in the strain rate, while the initial slope is rather insensitive to the strain rate. Fig. 2.22 shows the secant modulus values that are defined at different strain levels from the moment of reversing the loading direction, plotted against the average strain rate in each loading cycle. It may be seen that the secant Young’s modulus increases with the decrease in the strain level even at strains less than 0.001 %, while the dependency of Young’s modulus on the strain rate decreases with the decrease in the strain level. At a strain of 0.0001 %, the Young’s modulus is essentially independent of strain rate, which could therefore be defined as the elastic Young’s modulus. The Young’ modulus values defined at a strain of 0.001 % exhibits strain rate-dependency with a noticeable damping value (Fig.2.21), which should therefore be called the quasi-elastic Young’s modulus (Tatsuoka et al. 1999a).

The test results shown above indicate that the framework shown in Fig. 2.23 is relevant to the stress-strain relationships at small strains of geomaterial (Tatsuoka and Shibuya 1991; Tatsuoka et al. 1999a). That is, as the strain rate becomes larger, the stress-strain relation approaches an upper-bound relation. So, the length of elastic (i.e., reversible and rate -independent) stress-strain relation becomes larger as the strain rate becomes larger, while, if the strain rate is very low, the initial stress-strain relation could be located below the upper-bound relation from the start of loading. This framework is consistent with the linear three-component

vε&

21

10 100 1000 500010

100

1000

5000

RCTBS+DC

(1:2)(1:

1)

Range for Soft rocks andCement-treated soils(BS+DC) and clays

RCTBS+DC

Kazusa

Sedimentary soft rock

Local axial strain measurements

G0=

E0/{

2(1+

ν)} (

MP

a)(ν

=0.5

for

clay

s an

d 0.

42 fo

r so

ftroc

ks)

Gf=ρ(Vs)vh2 (MPa)

Kobe

Sagara

Miura

Uraga-A

Slurry

Dry

Pleistocene clay site

Cement-treated soilDMM

TSBS

Tokyo bay

Osaka bay

OAP

RCT=rotary coringDC=direct coringBS=block samplingTS=fixed-piston thin-wall sampling

Suginami

Uraga-B

Tokoname

Figure 2.24. Comparison of the shear modulus 2

f sG Vρ= ⋅ from the field downhole seismic survey

with the respective corresponding value 00

02(1 )E

Gν

=+

from triaxial tests on undisturbed specimens

of stiff clays, sedimentary softrocks and cement-mixed sand and clay obtained either by “block or direct sampling” or by rotary core tube sampling (Tatsuoka et al. 1995a & c, 1999a).

rheology model (Di Benedetto and Tatsuoka 1997) and the non-linear three-component rheology models described later.

Today, the pre-peak stress-strain relationships for a strain range from less than 0.001 % to that at the peak stress state can be evaluated by means of a relevant static stress-strain test using a single specimen. The relationships between static and dynamic experiments, between laboratory and field techniques, and between testing and field full-scale behaviour, which have been rather understood in a separated manner, can be better described based on the elastic deformation characteristics, while referring to other important features; including a) effects of strain and pressure non-linearity; b) kinematic yielding; c) effects of recent stress-time history; d) anisotropy; e) structuration and destructuration; and e) effects of cyclic loading (Tatsuoka et al., 1999a). This methodology is considered more relevant when the elastic modulus values

22

　　　　　　　 ; independent of

; independent of

00( )

m

vvv EE

σσ

=

⋅ hσ

vσ

1) Ev/Eh is proportional to the inherent anisotropy.2) Ev/Eh increases in a non-linear fashion with .

0

0

( )( )

v v

m

h

v

hh

EE

EE

σσ

⋅

=

00( )

m

hhh EE

σσ

=

⋅

Figure 2.26. Cross-anisotropic elasticity model (Tatsuoka et al., 1999a).

10 0 0

10 0 0

10 0 0 11

22 10 0 0 0 0 11

22 10 0 0 0 01

21

0 0 0 0 0

hv hv

v h hev v

vh hhe

hh v h he

hh vh hh

e v h hvhvh

e vhhv

ehvhh

hh

E E Ed d

dd E E Edd

E E E dd

G dd

Gd

G

ν ν

ε σν νσεσε ν ν

τγ

γ

γ

− − − − − − = ⋅

12

vh

hhd

τ

τ

Figure 2.25. Compliance matrix for cross-anisotropic materials (Tatsuoka and Kohata, 1995).

from the corresponding field shear wave velocity measurements and laboratory stress-strain tests are consistent to each other. Fig. 2.24 compares the respective pair of the shear modulus

2f sG Vρ= ⋅ obtained from

the field downhole seismic survey with the corresponding value obtained from triaxial tests on undisturbed specimens reconsolidated to the field pressure level (Tatsuoka et al. 1995a & c, 1999a). The respective value of 0G shown in this figure is the averaged of several data corresponding to a range of depth for which the value of

fG was measured. Each

value of 0G was obtained from a Young’s modulus value 0E measured at axial strains less than about 0.001 % using a relevant Poisson’s ratio

0ν . In this global comparison, the possible effects of inherent and stress system-induced anisotropy and discontinuities are considered to be secondary. It may be seen that with stiff clays of Pleistocene Era, these two types of shear modulus are consistent to each other. With sedimentary soft rocks, the agreement is generally

satisfactory when the values of 00

02(1 )E

Gν

=+

were evaluated by using undisturbed samples

obtained by block sampling or direct coring at the site. It can be seen on the other hand that for

many data points, the values of 0G are noticeably lower than the corresponding 2f sG Vρ= ⋅

value. With most of these data points, samples that were obtained by the rotary core tube sampling operated from the ground surface were used for the triaxial tests. It is very likely that these samples were more-or-less disturbed. This issue is discussed in detail by Tatsuoka et al. (1995c, 1999a). Taking into account the effects of sample disturbance, it could be concluded based on the data presented in Fig. 2.24 that when the ground consists of fine-grained soils or rocks without a noticeable amount of discontinuity, the elastic deformation property of geomaterial in the field could be evaluated by the field shear wave velocity measurement.

/v hσ σ

23

Proximeter for εh

seeFig. 2b

Lateral LDT

Vertical LDT

W=23 cm

W=23 cm

σv

σh

H=57 cm

Proximeter for εv

Lubricated 9.5 cm

19 cm

9.5 cm

3.5 cm

Figure 2.27. Rectangular prismatic specimen with local strain measurements of axial and lateral strains for triaxial tests (Jiang et al. 1997, 1999: Hoque et al., 1996).

0.01 0.1 150

100

1000

2000

hh

vv

h

h

v

σ ' (kPa)

σ ' (kPa)

49 98 147 196 245 294 343 392 441

hv

E

or E

(

MP

a)

hvσ ' or σ ' (MPa)

25~108 49~216 74~323 98~431 123~539 147~647 172~755 196~862 221~970

Nerima gravelTC test (σ '=49kPa)

E ~ σ '

E ~ σ '

stress statesIsotropic

Figure 2.28. Relationships between: a) vertical Young’s modulus and vertical stress; and b) lateral Young’s modulus and lateral Young’s modulus, at isotropic stress states and triaxial stress states (Kohata et al. 1997).

Modelling of elastic stress-strain behaviour (hypo-elastic models): The definition of elasticity for geomaterials is not simple, because the elastic deformation properties of geomaterial are usually not constant with respect to changes in the stress state even for a given element of geo-material. Moreover, for a given type of geomaterial at a certain stress state, they are also a function of density, stress and strain history and so on. So, only hypo-elasticity models, for which elastic strain increments are related to stress increments through a stiffness or compliance matrix that is a function of instantaneous stress state (and density, stress and strain history and others) are relevant. For example, Fig. 2.25 shows the compliance matrix for a cross-anisotropic material having the axis of symmetry in the vertical direction, which is relevant to, for example, a mass of geomaterial that has horizontal bedding planes and the principal stresses working in the vertical and horizontal directions. In addition, careful distinctions should be made between elastic, plastic and viscous properties as follows;

24

0.0 0.5 1.0 1.5 2.0 2.50.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

hv

hv

anisotropyStress system-induced

Inherent anisotropy

Perfectly isotropic material

Toyoura sandSLB sandTicino sandHime gravelChiba gravelNerima gravel

E

/E

σ '/σ '

0

0

( )( )

v v

m

h

v

hh

EE

EE

σσ

⋅

=

Figure 2.30. Relationships between the ratio of vertical and horizontal elastic Young’s modulus values and the principal stress ratio for cross-anisotropic sands and gravels (Kohata et al. 1997, Tatsuoka et al. 1999a).

0.01 0.1 1

100

1000m

h

0e

v

Toyoura sand 0.49 SLB sand 0.47 Ticino sand 0.53 Hostun sand 0.47 Hime gravel 0.51 Nerima gravel 0.52 Chiba gravel 0.52 Nagoya gravel 0.57

E

/f(e)

(=E

/f(

e))

(MP

a)

σ ' =σ ' (MPa)

Power m≒0.5 for of uncemented materials,

( )mv vE σ∝

Figure 2.29. Relationships between the elastic Young’s modulus and the confining pressure at isotropic stress states of uncemented sands and gravels (Kohata et al. 1997).

- elasticity; reversible and time-independent deformation properties for a given stress history;

- plasticity; irreversible (with energy dissipation) and time-independent deformation properties for a given stress history; and

- viscosity; irreversible (with energy dissipation) and time-dependent deformation properties for a given stress-history.

One of the most primitive hypo-elastic models for uncemented geomaterial is the one in which the elastic deformation property is isotropic and the elastic Young’s modulus E is a function of instantaneous

minor principal stress 3σ . It is readily seen that this model is too simple when compared with the actual elastic deformation properties of geomaterials, which is inherently anisotropic and can also become more anisotropic at more anisotropic stress states. Hardin (1978) proposed on

the other hand that, for sands, the elastic Young’s modulus xx

x

Eσε

∂=∂

in any particular direction

25

X is a unique function of the normal stress xσ working in direction X, independent of the normal stresses acting in the other orthogonal directions.

The deformation characteristics developed at very small to intermediate strains of a variety of geomaterials are now often evaluated by testing using modern laboratory stress-strain tests. A great amount of such data as above supports the proposal of Hardin (1978) described above (e.g. Kohata et al. 1994, 1997, Tatsuoka et al. 1999a, Jiang et al. 1997, Hoque and Tatsuoka 1998). Based on results from such tests, a hypo-elasticity model with inherent and stress system-induced anisotropy has been developed by extending the above proposal by Hardin (Tatsuoka et al., 1999a, b & c). The major feature of this model is summarised in Fig. 2.26.

Fig. 2.27 shows a rectangular prismatic large triaxial specimen with local strain measurements by means of a pair of vertical LDTs for axial strains and four pairs of lateral LDTs for lateral strains. In this way of testing, both axial and lateral strains that are free from effects of bedding error at the top, bottom and lateral surfaces of specimen can be evaluated. Very small unload/reload cycles of vertical stress, with strain amplitudes of the order of 0.001 %, were applied at a constant confining pressure, while very small unload/reload cycles of lateral stress were applied at a constant vertical stress. Such cyclic tests were performed at various isotropic and anisotropic stress states. As the two lateral orthogonal principal stresses are always the same, the lateral Young’s modulus hE was obtained as

( ) .(1 ) /

vh hh h h const

Eσ

ν σ ε=

= − ⋅ ∂ ∂ , while assuming that the Poisson’ ratio hhν in this case is equal

to the value of vhν at the stress ratio where v hE E= (Jiang et al. 1997). Fig. 2.28 shows typical results for a very dense well-graded gravel consisting of crushed sandstone (Kohata et al. 1997). It may be seen that the vertical Young’s modulus vE measured at isotropic and anisotropic stress states is essentially a unique function of the vertical stress vσ , while the lateral Young’s modulus hE is essentially unique function of the lateral stress hσ . Similar results have been obtained for poor-graded sands (Hoque and Tatsuoka, 1998), for a reconstituted gravel (Jiang et al. 1997, Balakrishnaiyer et al. 1998) and for a undisturbed gravel (Koseki et al., 1999). This result shows that the model illustrated in Fig. 2.26 is relevant. Some data points of vE and hE obtained near the failure state in triaxial compression deviate to values lower than the respective value that is obtained at isotropic stress states. This is due likely to effects of damage by shearing to the elastic deformation properties (Flora et al. 1994; Kohata et al. 1997; Tatsuoka et al. 1999a,b; Koseki et al. 1998). Fig. 2.29 summarises the power law at isotropic stress states for inherently cross-anisotropic geomaterials (Kohata et al. 1997). It may be seen that the power m is around 0.5.

Referring to Fig. 2.26, we have the following relationship between the vertical and horizontal elastic modulus values:

0

0

( )( )

m

v v v

h h h

E EE E

σσ

= ⋅

(2.1)

This equation means that the ratio v

h

EE

increases in a non-linear fashion with v

h

Rσσ

= . This

prediction is supported by the data (Fig. 2.30). It is reasonable to assume that the compliance matrix for quasi-elastic strain increments for

sands and gravels is symmetric as:

vh hv

v hE Eν ν= (2.2)

26

With m= 0.5, we can then assume the following equations for the Poisson’s ratios vhν and h vν using the value for the isotropic behaviour 0ν :

26

Figure 3.1. Typical results from plane strain compression tests of air-pluviated specimens of SLB sand (Park and Tatsuoka 1994).

0.50 ( )m

vh a Rν ν= ⋅ ⋅ and 0.5

0 /( )mhv a Rν ν= ⋅ (2.3)

where a is “ /v hE E at

1R = ”; and 0ν is vh hvν ν=

when 1/ nR a−= (Tatsuoka and Kohata 1995). Eq. 2.3 means that the elastic Poisson’s ratio is not constant either, but it changes with the stress ratio (as supported by the data presented in Fig. 2.31). Similar data are reported also in Tatsuoka et al. (1999b). In this respect, this model is somewhat different from the one proposed by Hardin and Bradford (1989). Summary: The elastic Young’s modulus of uncemented sands and gravels can be modelled by a hypo-elastic model representing both inherent and stress system-induced anisotropy. A model for a cross-anistropic case with the vertical symmetric axis is shown.

Another important topic that could not be touched upon in this report is the non-linearity of stress-strain relation due to changes in strain and stress sta te under cyclic loading conditions (e.g., Iwasaki and Tatsuoka 1977, Tatsuoka et al., 1978, 1979a & b) and that under monotonic loading conditions (e.g., Shibuya et al., 1991, Tatsuoka et al., 1999a). 3 INHERENT ANISOTROPY IN THE STRENGTH AND DEFORMATION CHARACTERISTICS OF GRANULAR MATERIALS Arthur,R. (UK) and Oda,M. (Japan) are two among the pioneers who disclosed systematically inherently anisotropic deformation and strength characteristics of granular materials (Tatsuoka 1987). Tatsuoka et al. (1986a), Lam and Tatsuoka (1988a & b) and Park and Tatsuoka (1994) performed a systematic study on this subject. Fig. 3.1 shows another example showing the inherently anisotropic stress-strain behaviour of sand (Park and Tatsuoka 1994). Fig. 3.2 shows grain size distribution curves of poorly graded sands

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

0.15

0.20

.

Toyoura Sand

Expt. data

νvh = (νvh)R=1 (σv/σh)

nv/2

ν vh

σv' / σh' Figure 2.31. Relationship between the elastic Poisson’s ratio vhν

and v

h

Rσσ

= at triaxial extension and compression stress states

and the isotropic stress state and the fitting of the data by Eq. 2.3 (Hoque and Tatsuoka 1998).

27

Figure 3.3. Summary of the effects of inherent anisotropy on the angle of internal friction for poorly graded granular materials obtained from plane strain compression tests (see Fig. 3.2 for the gradings of the sands) (Park and Tatsuoka 1994).

Figure 3.2. Grain size distribution curves of sands for which inherent anisotropy was evaluated by plane strain compression tests (see Fig. 3.3) (Park and Tatsuoka 1994).

for which the effects of inherent anisotropy on the stress-strain properties were evaluated by plane strain compression tests (as shown in Fig. 3.1). Fig. 3.3 summarises the effects of inherent anisotropy on the angle of internal friction in plane strain compression of air-pluviated samples of these sands. Each angle of internal friction has been divided by the respective value at

90oδ = for the same void ratio. It may be seen that all these sands, having different origins in the world, have a very similar and marked trend of inherently anisotropic strength. Figs. 3.4 and 3.5 show a similar, but more marked, trend of inherent anisotropy for undisturbed samples of sand (Tatsuoka et al. 1990a). Fig. 3.6 compares the angles of internal friction, as a function of the angle δ , obtained from a series of triaxial compression tests, plane strain compression tests and torsional simple shear tests performed on air-pluviated specimens of Toyoura sand (Tatsuoka et al. 1988, 1990a). Each angle of internal friction has been divided by the respective value from the plane strain compression test at 90oδ = for the same void ratio. Note that the data presented in Fig. 3.6 include those from triaxial compression tests, plane strain compression test and torsional simple shear tests in which the

3σ value at failure was very low (such as 10 kPa or less). In such tests, very precise measurement of the effective confining pressure and relevant stress correction for the effect of membrane force is essential to obtain accurate results (Tatsuoka et al. 1986a & b, 1988). It can be seen from Fig. 3.6 that the strength (and also deformation) properties obtained from different testing methods can be linked to each other only when taking into account the effects of inherent anisotropy, among other parameters (Tatsuoka et al. 1996c; Tatsuoka 1988; Tatsuoka et al. 1988; Pradhan et al., 1988a & b). A more comprehensive analysis in this respect is made in Lam and Tatsuoka (1988a & b). Summary: It has been shown above that commonly with different types of granular materials, the pre-peak deformation properties and peak strength could be markedly anisotropic. Then, for numerical analysis and design of boundary value stability problems in geotechnical engineering: e.g., earth pressure, slope stability, bearing capacity of footing and so on, the following important questions arise: 1) What is the meaning of using an isotropic value of φ in analysis and design related to the

28

Figure 3.5. Effects of inherent anisotropy of undisturbedsamples of resedimented Shirasu (volcanic pumice) (see Fig. 3.4) (Tatsuoka et al. 1990a).

Figure 3.6. Comparison of φ as a function of δ , from a series of triaxial compression (TC) tests, plane strain compression (PSC) tests and torsional simple shear (TSS) tests of air-pluviated specimens of Toyoura sand (Tatsuoka et al. 1988, 1990a); the details of the torsional shear testing method are described in Tatsuoka et al. (1986c) and Pradhan et al. (1988a).

Figure 3.4. Effects of inherent anisotropy of undisturbedsamples of resedimented Shirasu (volcanic pumice) (Tatsuoka et al. 1990a).

failure of sand and gravel ? 2) What is the meaning of the classical limit equilibrium stability analysis assuming

isotropic perfectly-plastic properties of soil and gravel ?

In fact, the stress-strain behaviour of soil is over-simplified in the classical soil mechanics. The factor of anisotropic strength and deformation characteristics is only one of several essential factors that are ignored in such classical theories, which includes: a) effects of pressure level on

φ b) different definitions of φ

between compression tests and simple and direct shear tests; and

c) progressive failure associated with shear banding, thereby associated with effects of particle sizes.

These factors, or most of them, are often unduly ignored even in recent research, as discussed by Tatsuoka et al. (1989a & b, 1992, 1994c; Tatsuoka and Huang 1991) and even in some modern numerical analysis by FEM. The consideration of the effects of these factors is equally important when analysing the

29

Figure 4.1. Relationships between the stress and the strain (averaged for the whole specimen size) from a series of special plane strain compression tests on different granular materials having a wide range of particle size (see Fig. 4.2: Yoshida et al. 1995: Yoshida and Tatsuoka 1997).

failure of reinforced soil (Huang and Tatsuoka 1990; Huang et al. 1994; Huang and Tatsuoka 1994). 4 STRAIN LOCALISATION WITH SHEAR BANDING IN GRANULAR MATERIALS 4.1 Strain localisation in plane strain compression tests Fig. 4.1 shows results from a series of special plane strain compression tests performed at

3σ = 78 kPa and 392 kPa, with lubricated top and bottom ends, on a number of different types of granular material having a wide range of particle size (Fig. 4.2) (Yoshida et al. 1995; Yoshida and Tatsuoka 1997). A deformable grid, which was made of latex rubber, had been printed on one 2σ plane of the specimen membrane (Fig. 1.3). Pictures of deformed grid-printed 2σ plane were taken through the transparent confining platen and the triaxial cell at a number of loading stages in each plane strain compression test. Figs. 4.3 and 4.4 shows three typical shear strain fields immediately before and after the peak stress state and immediately after the start of residual state. These figures were constructed from observed displacement fields on the 2σ plane, as shown in Fig. 1.3. It may be seen from Figs. 4.3 and 4.4 that the strain has already been noticeably localised into some zones before the peak stress state. In the post-peak resume, only one of these shear zones seen at the peak stress state developed into a distinct shear band while the remaining part of specimen was unloaded with negative local shear strain increments. From such displacement field seen on the 2σ plane as seen in Fig. 1.3, shear deformation and volume change of a shear band at each loading stage, defined as shown in Fig. 4.5, were obtained. Fig. 4.6 shows typical relationships between the shear stress level, defined by Eq. 4.1, and the shear deformation of shear band in the post-peak regime for the different types of

30

Figure 4.2. Grain size distributions of granular materials used in the plane strain compression tests to observe strain localisation (see Fig. 4.1) (Yoshida et al. 1995: Yoshida and Tatsuoka 1997).

Figure 4.3. Shear strain fields constructed from the observed displacement field on the 2σ plane:

A) immediately before the peak stress state; B) immediately before the peak stress state; and C) immediately after the start of residual condition (SLB sand; 3σ = 78.4 kPa; see Fig. 4.1a for the locations of A, B and C) (Yoshida et al. 1995).

granular material (i.e., the shear defor-mation is defined as zero at the peak stress state).

1 3 1 3

1 3 1 3

( / ) ( / )

( / ) ( / )peak

npeak residual

Rσ σ σ σ

σ σ σ σ

−=

− (4.1)

Rn= 1.0 means the peak stress state and Rn= 0.0 means the residual stress state. The corresponding relationships between the shear stress level and the volume change of shear band are reported in Yoshida et al. (1995). It may be seen from Fig. 4.6 that the post-peak stress-shear deformation relationship is markedly different for different grain sizes: i.e., the shear displacement increment *su that takes place as the stress state changes from the peak state to the residual state increases with the increase in the particle size. This feature is summarised in Fig. 4.7 (Yoshida and Tatsuoka 1997). The value of *su is slightly smaller when 3σ = 392 kPa than when 3σ = 78 kPa. The reason for the above is not known. It is also to be noted that the value of *su is not proportional to the mean diameter 50D , but the ratio 50* /su D decreases noticeably with the increase in 50D . It seems that this is due to that particle properties other than 50D change with the changes in 50D .

A similar result has been obtained from triaxial compression tests and plane strain compression tests on sedimentary soft rock (Tatsuoka and Kim 1995; Hayano et al. 1999). These results indicate that the post-peak deformation properties of geomaterials are controlled by a characteristic scale that is specific to each geomaterial type under each specific stress

D (mm)

31

Figure 4.5. Definition of shear deformation and volume change of shear band.

Figure 4.6. Relationships between the shear stress level and the shear displacement increment from the peak stress state for differentparticle sizes at 3σ = 78 kPa (Yoshida and Tatsuoka 1997).

Figure 4.4. Shear strain fields constructed from the observed displacement field on the 2σ plane: A) immediately before the peak stress state; B) immediately before the peak stress state; and C) immediately after the start of residual condition (Karlsrule sand; 3σ = 392 kPa; see Fig. 4.1b for the locations of A, B and C) (Yoshida et al. 1995).

conditions (and others). It seems that for granular materials, the mean diameter 50D is the most important parameter representing this characteristic scale. This fact indicates that the prototype soil mass (in a large scale) cannot be scaled down into a smaller model by using the same type of soil as the prototype. This point is discussed in the next section.

32

Figure 4.8. Bearing capacity on sand of a strip footing in plane strain.

4.2 Implications of shear banding in the issue of the bearing capacity of strip footing on sand and particle size effects Most of the classical theories for the bearing capacity of strip footing on sand assume that the sand is an isotropic perfectly plastic material having a constant angle of internal friction. For the upper bound analysis and the limit equilibrium analysis, it is further assumed that the thickness of failure planes is zero, or independent of particle size at best. It has already been shown in the above however that these assumptions overly simplify the actual behaviour of real soils. In this section, the bearing capacity of strip footing on sand, as defined in Fig. 4.8, will be discussed to demonstrate some essential limitations of the classical stability analysis and the importance of particle size effects. In the following, only the mechanism of the bearing capacity of a rigid strip footing with a rough and smooth footing base placed on the surface of a homogeneous level sand layer subjected to vertical central load (i.e., the basic case) will be

Figure 4.7. Relationship between the shear displacement increment that is needed for the stress state changes from the peak to residual states and the mean particle size (Yoshida and Tatsuoka 1997).

33

Figure 4.9. Summary of strength anisotropy in plane strain compression tests of air-pluviated Toyoura sand (see Fi. 3.6) (Tatsuoka et al. 1986a, 1991).

Figure 4.10. Summary of pressure level-dependency in plane strain compression tests of air-pluviated Toyoura sand (Tatsuoka et al. 1986a, 1991).

examined. Toyoura sand was used in the

model tests that are explained below. The strength characteristics of Toyoura sand are first summarised. 1) Fig. 4.9 summarises the

inherent anisotropy of strength (i.e. the angle of internal friction) obtained from a series of plane strain compression tests (Tatsuoka et al. 1986a; see Fig. 3.6).

2) Fig. 4.10 shows the pressure level-dependency of

1 3

1 3 max

arcsinσ σ

φσ σ

−= +

at

90oδ = from the plane strain compression tests (Tatsuoka et al. 1986a).

3) Fig. 4.11a compares the relationships between the friction angles φ and the void ratio obtained from the following different tests: a) φ at 90oδ = from plane

strain compression tests, which is the largest value with respect to the angle δ (i.e. the conventional plane strain compression tests);

b) φ at 23oδ = from plane strain compression tests, which is the smallest value with respect to the angle δ ;

c) φ at 90oδ = from triaxial compression tests, which is noticeably smaller than the

corresponding value from the plane strain compression test at 90oδ = (i.e. the conventional triaxial compression tests); and

d) the simple shear angle of friction defined as max

arctan hvSS

v

τφ

σ

=

from torsional simple

shear tests using hollow cylindrical specimens in which the bedding plane is horizontal (Pradhan et al. 1988a & b).

In the torsional simple shear tests, as the magnitudes and directions of the three principal stresses were continuously measured. In these tests, the directions of the principal stresses rotate before reaching the residual state. Fig. 4.11b compares the friction angles,

1 3

1 3 max

arcsinσ σ

φσ σ

−= +

and max

arctan hvSS

v

τφ

σ

=

, from the torsional simple shear tests. It

may be seen that the friction angle 1 3

1 3 max

arcsinσ σ

φσ σ

−= +

is consistently larger than the

34

Figure 4.11a. Comparison of the relationships between the friction angles φ and the void ratio from different tests for air-pluviated Toyoura sand (Tatsuoka et al. 1991; Siddiquee et al. 1999).

Figure 4.11b. Comparison of the values of φ and SSφ for the same torsional simple shear test data for air-pluviated Toyoura sand (Tatsuoka et al. 1991; Siddiquee et al. 1999).

simple shear angle max

arctan hvSS

v

τφ

σ

=

(Pradhan et al. 1989, Pradhan and Tatsuoka 1989,

Tatsuoka et al. 1991). This is due to the fact that the horizontal plane, along which the normal strain is always zero, is not the plane of maximum stress obliquity due to the fact that the dilatancy angle is significantly smaller that the angle of internal friction (e.g. Pradhan et al. 1989a & b: Tatsuoka et al. 1988). It is to be noted that SSφ values obtained by conventional direct shear tests could be subjected to some large experimental errors, which could mask the relationship shown in Figs. 4.11a & b (Qui et al. 2000).

For these data, all the specimens were prepared by the same method (pluviation of air-dried particles through air). The same preparation method was also used to prepare the sand bed for the model tests that are described below. It may be seen from Fig. 4.9 – 4.11 that the range of strength for the same void ratio is very large among these different test methods performed under otherwise the same testing conditions.

Fig. 4.12 summarises the representative classical bearing capacity theories for the coefficient for sand weight Nγ (Tani 1986; Tatsuoka et al. 1991). Then, only from the fact that the friction angle is not unique for the same mass of sand, it is understood that it is very hard to predict the realistic bearing capacity of a footing in sand based on any of these cla ssical bearing capacity theories. Even based on the same assumptions (i.e., isotropic perfectly plasticity with a constant φ and zero shear band thickness), the Nγ values by these classical theories differ from each other due to different assumptions with respect to the distribution of the friction angle on the footing base or the failure mechanism of the active zone immediately below the footing. Although the variation in the Nγ value among the classical theories is not small with a range of about two times, the difference between the classical theories and the actual value is much larger, as shown below.

35

Figure 4.12. Summary of relationships between the factor of soil weight Nγ and φ by the representative classical bearing

capacity theories (Tani 1986; Tatsuoka et al. 1991).

Fig. 4.13 compares the relationships between the value of Nγ value and the friction angle φ according to the classical bearing capacity theories, which are denoted by a band in the figure, with experimental results for air-pluviated Toyoura sand (Tatsuoka et al. 1991). The experimental results were obtained from a comprehensive series of element tests, as described above, and plane strain model tests using different sizes of strip footing under gravitational acceleration (1g). All the experimental results were obtained with air-pluviated Toyoura sand and corrected to the same void ratio (e= 0.66). Therefore, these data points should collapse into a single point if the sand were an isotropic perfectly–plastic material having a single value of φ . The data scatter very largely, however, due to the following reasons: 1) In the model tests (1g), the Nγ value decreases with the increase in the physical footing size

( 0B ). This behaviour has been called the scale effect.

2) For a single value of Nγ for the respective model test condition with a single value of 0B ,

different values of φ and SSϕ from different types of shear tests are plotted. These friction angles were obtained at the respective 3σ value equal to one tenth of the evaluated average footing pressure. So, these values decrease with the increase in 0B .

It is seen from Fig. 4.13 that if we substitute the value of φ from plane strain compression

tests at 90oδ = into a classical bearing capacity theory, the actual Nγ value is largely over-estimated to a degree that cannot be covered by the global safety factor that is used in usual practice, such two to three. For example, ( ; 90 ) 49o oPSCφ δ = = for 0B = 50 cm is substituted into the classical bearing capacity theories, we obtain Nγ ≈ 800, which is about eight times as large as the measured value (about 100). This extremely large overestimation is due to the fact that not only the effects of strength anisotropy but also the effects of progressive failure of sand bed, which is explained in detail below, are not considered in the above-mentioned procedure to obtain the Nγ value. That is, the consideration of the pressure level-dependency of φ is not sufficient to explain the discrepancy between the classical theories and the real behaviour of sand. To validate the above-mentioned notion, results obtained from a comprehensive series of model bearing capacity tests of a strip footing on sand are shown below.

36

Figure 4.13. Comparison between the classical bearing capacity theories and the experimental results for air-pluviated Toyoura sand with a void ratio of 0.66 with Nγ values from FEM analysis plotted

against ( ; 90 )oPSCφ δ = (Tatsuoka et al. 1991).

Small- scale 1g model tests (Tani 1986): Fig. 4.14 shows the sand box in which a sand bed of Toyoura sand was made by the air-pluviation technique. The side walls were lubricated by using a thin latex membrane smeared with a controlled thickness (0.05 mm) of a selected silicone grease. A very low friction angle with this configuration has been confirmed (Tatsuoka et al. 1984; Goto et al. 1993). The footing load was measured by using eleven load cells, each measuring normal and shear loads. These local load cells were arranged at the central third of footing (Fig. 4.15); so the footing load measured in this way is therefore essentially free from the effects of side wall friction, if any. A grid had been printed on the surface of the membrane, and displacements at the nodes of the grid were obtained from pictures of the grid that were taken at several loading stages during each test. Strain fields in the sand bed were then obtained from these observations. In some tests, the sand bed, which was air-dried during each loading test, was made wet after each loading test had been over so that stable vertical faces of the sand bed could be excavated and exposed without a support (as shown in Fig. 4.17). It was confirmed that the deformation of the membrane in contact with the side wall, which was seen through the transparent side wall, be essentially the same as that seen in the exposed central section of the sand bed (Tani 1986). The sand box was made very stiff so that the plane strain conditions could be satisfied. The bottom face of footing was made rough in most series of tests, while it was lubricated in one series of tests. The loading of footing was made by displacement control. Similar configurations were taken in the other series of model tests (i.e. large scale 1g tests and centrifuge tests that are described later in this paper).

The relationships between the normalised footing pressure and the normalised footing settlement obtained from two typical 1g tests using a rough footing with a width 0B of 10 cm are shown in Fig. 4.16. Fig. 4.17 shows the central section of the sand bed in the test in which

37

Plane strain bearing capacity testsair-dried Toyoura sand (B0= 10 cm in 1 g)

Sand box: 40 cm wide, 183 cm long and 49 cm (sand depth )

Lubricated

Figure 4.14. Small scale plane strain sand box (Tani 1986).

B0= 10 cm

1/3 of footing length = 40/3 cm

Eleven two-component load cells (normal and shear stresses) Figure 4.15. View from the bottom of the strip footing having a width of 10 cm (Tani 1986).

the loading was terminated when the footing load became nearly the peak state (see Fig. 4.16). Thin layers of black-dyed Toyoura sand seen in the picture had been placed around the central section of the sand bed when it was prepared. It may be seen from Figs. 4.16 and 4.17 that the footing settlement was about 5 % of 0B at the peak footing load, and by this moment, shear bands had developed for some length from the edges of the footing. It may also be seen that the developed shear bands were only a part of the full potential shear bands which are assumed to have developed at the moment of peak footing load in the classical bearing capacity theories. Fig. 4.18 shows the shear strain field corresponding to Fig. 4.17. It can also be seen from this figure that the strain in the sand bed are extremely non-uniform, so it is the case along the potential failure planes. This fact indicates a highly progressive nature of the failure of the ground.

Fig. 4.19 shows the central section of sand bed that was exposed in another test in which loading was continued until the footing settlement became as large as about 70 % of 0B . It may be seen that the full potential shear bands, reaching the ground surface, had developed only far

38

Figure 4.17. Picture of the exposed central section of the sand bed in the test in which loading wasstopped around the peak footing load (the footing width= 10 cm; the horizontal back colour strips are black-dyed Toyoura sand placed only around the central section of the sand bed) (see Fig. 4.16) (Tatsuoka et al. 1991).

after the peak footing load had been attained. Fig. 4.20 is the zoom-up of a local zone below the footing. It can be seen from Fig. 4.20 that the failure planes are not ‘planes without a thickness’, but they are bands having an intrinsic thickness, and the shear deformation is not uniform along the shear bands.

It is practically impossible to evaluate the local stress states inside the model sand bed in such model tests as those in this series of tests. Therefore, the stress field was estimated as follows. 1) The relationship between the

mobilised angle of friction

1 3

1 3

arcsinmob

σ σφ

σ σ −

= + and

the shear strain 1 3γ ε ε= − for different angles of δ were constructed from the results of the plane strain compression tests of Toyoura sand explained before (Fig. 4.21) (Tatsuoka et al. 1991). The shear strains shown in this figure are local values that were averaged for a 1 cm-wide band including a shear band. A value of 1 cm was selected to be equal to the spacing between the lines of the grid printed on the

Figure 4.16. Typical relationships between normalised footing pressure and normalised footing settlement from two typical 1g model tests with 0 10B cm= (Tatsuoka et al. 1991) .

Fig. 4.17

39

membrane in the model tests. Note that the decreasing rate of stress level in the post-peak regime is very low in these stress-strain curves when compared with those seen in the usual relationships between the stress and the strain that is averaged for the whole of a specimen. In this sense, relationships between the stress and the strain that is averaged for the whole of a specimen are not objective in the sense that the post-peak stress-strain curves are controlled by the ratio of the particle size to the specimen size.

2) It was assumed that the direction of 1σ be the same as the direction of

1ε measured at each

point in the sand bed. This was based on the consideration that the most part of the strains that takes place by the moment when the footing load becomes the peak value are inelastic and the principal directions of inelastic strain increment are close to the instantaneous principal directions of stress, as validated with Toyoura sand by Pradhan et al. (1988a & b, 1989) and Pradhan and Tatsuoka (1989). .

Figure 4.18. Local shear strain (%) contours in the zone below the footing around the peak footing load state (the grid spacing is 1 cm) (Tatsuoka et al. 1991).

Figure 4.19. Picture of the exposed central section of the sand bed in which loading was continued to S/B0 of about 0.7, far after the moment when the peak footing load was attained; a shear band has developed up to the surface of the sand bed in each side of the footing (Tatsuoka et al. 1991).

40

- The shear displacement along shear band is not uniform,indicating the progressive failure of the ground !

- The shear band has an intrinsic thickness !

Figure 4.20. Zoom up of the central zone of Fig. 4.19.

Figure 4.21. Relationships between the mobilised angle of friction 1 3

1 3

arcsinmob

σ σφσ σ

−= + and

the shear strain 1 3γ ε ε= − for different angles of δ from plane strain compression tests of

Toyoura; the shear strains are local values averaged for a 1 cm-wide band including a shear bands (Fig. 1.3) (Tatsuoka et al. 1991).

3) The value of 1 3

1 3

arcsinmob

σ σφ

σ σ −

= + at each point in the sand bed was then obtained by

41

substituting the measured local shear strain into the relationship between the mobilised angle of friction and local shear strain presented in Fig. 4.21.

Fig. 4.22a shows the distribution of local shear strain at the moment of the peak footing load in the second test (e= 0.722) shown in Fig. 4.16. Fig. 4.22b shows the corresponding

distribution of the mobilised angle of friction 1 3

1 3

arcsinmob

σ σφ

σ σ −

= + estimated by the method

described above. It can be seen that the failure of the ground is not simultaneous at all in the sense that at any moment of loading, the peak local strength is never mobilised simultaneously

a)

b)

c) Figure 4.22. a) Shear strain field at the peak footing load state; b) the corresponding field of mobilised angle of friction (in degree); and c) stress state below footing at the peak footing load (see Fig. b) (B0= 10 cm; e= 0.66; Tatsuoka et al. 1991).

Pre-peak stress state

Post-peak stress state

Near-peak stress state

42

Figure 4.23. Comparison between the stress characteristics solutions assuming the perfect plasticity and the measured bearing capacity (Tatsuoka et al. 1991).

along the potential failure planes. At the moment of peak footing load, the following trends of be-haviour could be seen from Fig. 4.22b (see also Fig. 4.22c): 1) Some part adjacent to the

footing edges is already in the post-peak regime.

2) The peak and near peak stress states are attained in only a limited zone below the footing.

3) A small zone immediately below the footing base and the large remaining part outside the footing width are in the pre-peak regime.

The significantly progressive nature of ground failure described above was further confirmed by calculating the footing load by the stress characteristics method based on the anisotropic peak strength which is a function of the values of δ and 3σ at each point in the sand bed while using the friction angles at the footing base measured at the moment of peak footing load (Fig. 4.23). In this figure, the relationships between the measured value of Nγ and the void ratio for both a rough footing and a smooth footing with a lubricated footing base are shown. These measured bearing capacity values are compared with the respective theoretical values obtained as above for the rough and smooth footings. It may be seen that even when the strength anisotropy is considered, the use of peak strength along the potential failure planes results into a significant over-estimation of the measured bearing capacity.

Large- scale 1g model tests: The above-mentioned fact was further confirmed by performing similar model tests but in a scale that was larger by a factor of five than the above-mentioned series of model tests (Morimoto 1990; Tatsuoka et al. 1991; Siddiquee et al. 1999, 2001). Figs. 4.24, 4.25, 4.26 and 4.27 show the test configurations of the large model tests. All the test conditions (i.e., the shape of the sand bed, the relative size of the sand bed to the footing size, the plane strain conditions with the lubrication of side wall, the model sand, the air-pluviation method to prepare the sand bed, the use of eleven two-component load cells at the central third of footing, the displacement-control loading, the observation of the deformation of the latex rubber membrane used for the lubrication to obtain the deformation of sand bed and so on, except for the size of model) were made the same with those for the small-scale model tests described above.

Fig. 4.28 shows the results from two representative 1g tests using a rough footing with a physical width 0B equal to 50 cm (n.b., the results from a centrifuge test shown in this figure are discussed later in this paper). Fig. 4.29 shows a picture taken when the footing load became nearly the peak in one of these two tests. Fig. 4.30 shows a typical shear strain field that was constructed based on such a picture. It may be seen that the strain field is extremely non-

43

Lubrication of the side wallusing a thin latex membrane smeared with silicone grease

Pluviation of air-driedToyoura sandThrough air

Figure 4.25. Preparation of a lubrication layer on the side walls and preparation of sand bed by pluviating air-dried sand particle through air from a slit of a moving hopper (Morimoto 1990; Tatsuoka et al. 1991).

uniform, and the degree of non-uniformity is much larger than the one that was observed in the corresponding small-scale model test (Fig. 4.18). This was due to the fact that the ratio of the thickness of shear band, which was essentially independent of the scale of model test, to the footing size was smaller by a factor of five compared with that in the small scale model tests. This means that the degree of progressive failure (i.e., the degree of non-simultaneous mobilisation of local peak strength) becomes larger with the increase in the ratio of footing size to particle size; i.e., the bearing capacity factor Nγ becomes lower with the increase in the

footing width 0B when the footing is placed on the same type of sand. Such a decrease in the

Nγ values with the increase in the 0B value has been called the scale effect.

Rough footing (0.5 m wide & 2 m long)

Three local load cells on each 1/3 of footing

Eleven local load cells on central 1/3 of footing

Large pit (2m wide, 7 m long and 4 m deep

Figure 4.24. Large-scale model tests; a) footing (the footing base shown in this picture); and b) general view (a loading reaction frame set above the footing) (Morimoto 1990; Tatsuoka et al. 1991).

44

Fig. 4.31 shows the picture taken nearly at the end of loading, at a footing settlement ratio of about 16 %. It may be seen that a wedge has developed below the footing, but the full potential failure planes have not developed at all by this loading stage. Small- scale centrifuge model tests: It is shown below that the scale effect on the 0/N S B∼ relationship as well as the values of Nγ and “ 0/S B at the peak footing load” consists of the following two components (Tatsuoka et al. 1991; Siddiquee et al. 1999): 1) Pressure-level effect, which is due purely to the effect of pressure changes on the stress-

strain behaviour of sand. This effect can be purely observed in centrifuge tests changing the pressure level (or vertical acceleration level) keeping the same size of footing and using the same sand type.

2) Pparticle size effect, which is due purely to the effect of the ratio of sand particle size relative to footing size. This effect can be observed by comparing the bearing capacity characteristics between different tests using the same sand for the same equivalent footing width oB B n= ⋅ , but for different physical footing widths oB (n is the acceleration level; n= 1.0 means the gravitational acceleration). This is indeed a comparison of bearing capacity characteristics behaviour between the 1g tests and the centrifuge tests shown in Fig. 4.28 and Fig. 4.32. That is, Figs. 4.28 and 4.32, respectively, compare the 0/N S B∼ relationships from a set of corresponding 1g and centrifuge tests for the same (or very similar) equivalent footing width oB B n= ⋅ , but for different physical footing widths oB ; Fig. 4.28 is for the case of 50B cm= , while Fig. 4.32 is for the case of 21 23B cm= ∼ . It may be seen from these figures that, as the 0/S B becomes larger, the prototype behaviour (i.e., the 1g model tests in this case) is less satisfactorily simulated by a centrifuge test using the same sand as the prototype but the behaviour is more affected by the particle size effect: i.e., (1) the initial load-settlement curve at small footing settlements in the centrifuge test is

Fig. 4.26 Setting of footing on the surface of prepared sand bed (Morimoto 1990; Tatsuoka et al. 1991).

Figure 4.27. Side view of the model seen through the transparent side wall and the lubrication layer (Morimoto 1990; Tatsuoka et al. 1991).

45

similar to the corresponding prototype behaviour;

(2) but the peak footing load and the settlement at the peak footing load in the centrifuge test are noticeably larger than those of the prototype footing (i.e., the 1 g test in this case)..

With respect to the pressure effect, when the pressure level (i.e., the acceleration level) is changed in centrifuge tests using the same size of footing and the same type of sand, we would not observe any change in the 0/N S B∼ relationship as well as the values of Nγ and “ 0/S B at the peak footing load” if the sand has the following properties:

a) the relationship among the stress ratio 1 3/σ σ , the shear strain γ and the volumetric strain volε , together with the angle of internal friction φ , during the shearing process of the sand used in the model tests is independent of pressure level; and

Figure 4.28. Relationships between normalised footing load and normalised footing pressure from two 1g tests with 0 50B cm= and the corresponding centrifuge test

(Morimoto 1990; Tatsuoka et al. 1991).

Figure 4.29. Sideview of the model around the peak footing load state (see Fig.4.28). (Tatsuoka et al. 1991).

Figure 4.30. Shear strain field around the near peak footing load state (grid spacing= 1 cm; see Fig. 4.28) (Morimoto 1990; Tatsuoka et al. 1991).

46

Figure 4.31. Side view of the model around at the end of loading (see Fig. 4.28) (Morimoto 1990; Tatsuoka et al. 1991).

b) the relationship between the logarithm of the mean pressure

'p and volε during the compression process of the sand used in the model tests is independent of pressure level.

If the actual properties of sand were as above and when the behaviour of a prototype footing before the effects of shear banding become significant is to be evaluated, small scale 1g model tests using the prototype are more than sufficient to predict the prototype behaviour. However, the actual sand does not have such properties as described above, and therefore, as the pressure level increases, the initial slope of 0/N S B∼ relation decreases, “ 0/S B at the peak footing load” increases and the value of Nγ decreases.

It is also possible to observe the particle size effect by comparing the bearing capacity characteristics between different tests either in 1g or in centrifuge tests using the same footing size on different model sand beds made of different sands having different particle sizes but having the same global pre-peak stress-strain properties and peak strength and also the same post-peak stress-strain properties within the shear band (so different stress ratio-shear deformation relationships in the post-peak regime of shear band). This latter type of comparison is practically very difficult. Instead, some approximated method was attempted as shown later in this paper (Tatsuoka et al. 1997b).

Fig. 4.33 summarises the Nγ values from the physical model tests, which are plotted versus the equivalent footing size, defined as nB B n= ⋅ , and the corresponding results from a series of FEM analysis (explained below). It may be seen that the Nγ values from the 1g model tests exhibits a large scale effect, which consists of the pressure level effect and the particle size effect.

4.3 FEM simulation Realistic results can be obtained by FEM analysis only when taking into account properly the actual complicated deformation and strength characteristics of

Figure. 4.32. Comparison of bearing capacity between 1g and centrifuge tests for the same sand Toyoura sand and a very similar equivalent footing width 0B B n= ⋅ of 21 – 23 cm (Morimoto 1990; Tatsuoka et al. 1991).

47

geomaterial (i.e., sand in this case) as described above. The constitutive modelling of

48

Toyoura sand that is described in Tatsuoka et al. (1993) is used in the FEM simulation described below. In this modelling, the observed non-linear stress-strain behaviour together with the observed dilatancy characteristics are formulated, while the flow characteristics are not associated with the peak frictional angle. For the FEM analysis of the model bearing capacity tests described above (Tatsuoka et al. 1991: Siddiquee et al. 1999), the simulation of strain localisation into a shear band(s) is the most difficult part. The following assumptions were adopted (Siddiquee et al. 1995b): 1) each FEM element has a specific peak

strength that is independent of boundary conditions and stress and strain histories;

2) in each FEM element, when the peak strength is reached, strain localisation starts taking place into a single shear band having a thickness that is specific to a given type of sand (i.e. Toyoura sand in the present case), while the relationship between the stress ratio and the shear deformation is specific to a given type of sand; and

3) the average stress-average strain relationship in each FEM element is the same as that of a plane strain specimen having the same size as the FEM element.

1 10 100 1000

0

100

200

300

400

500

Particle size effect

Scale effect

Pressure level effect

1g test results Centrifuge test results

(B0=3 cm)

Void ratio, e=0.66N

γ=(2q/γB)max

1g simulation Centrifuge simulation

B=n.B0 (cm) in Log

Bea

ring

capa

city

fact

or, N

γ

Scale effects+= pressure level effects* + particle size effects

+ by 1g testsfor different footing sizesand the same sand type

*by centrifuge tests underdifferent n valuesfor the same footing sizeand the same sand type

Figure 4.33. Comparison of the Nγ values from the model tests as a function of B (Siddiquee et al.

1999): nB B n= ⋅ , and the corresponding relation from FEM analysis (Tatsuoka et al. 1991).

Figure 4.34. Initial and deformed FEM meshes for 0 50B cm= in 1g (Siddiquee et al.

1999).

49

In this way, for a given type of sand, the post-peak stress-strain relationship of each FEM element becomes dependent on the mesh size, while the result of FEM analysis can become mesh size-independent. In the same way, the post-peak stress-strain relation in each FEM element becomes dependent on the particle size, and therefore, the effects of particle size can be simulated by the FEM analysis. A more detailed description of the FEM analysis is given in Siddiquee et al. (1995b; 1999, 2001), Kotake et al. (1999) and Peng et al. (2000).

Fig. 4.34 shows the mesh used in the FEM analyses, and Fig. 4.35 compares the results from the following FEM analyses using the mesh shown in Fig, 4.34 with the corresponding experimental results for

0B = 50 cm in 1 g: a) the value of φ from the plane strain compression test at 90oδ = performed at pressure

levels lower than the critical value was used (Fig. 4.10): the φ value is essential constant with the respect to the changes in the pressure at this low pressure level and the φ value at this low pressure level is the highest value for a given mass of sand.

b) For the above value of φ , the pressure-dependency of φ , shown in Fig. 4.10, was considered.

c) In addition to the above, the factor of inherent anisotropy of the deformation and strength characteristics of the test sand, as shown in Fig. 4.9, was considered.

d) In addition to the above, the post-peak strain softening was considered in such a way that the post-peak stress-strain relationship in each element did not depend on the mesh size, while it was the same with the relationship between the average stress and the average strain for the whole of a specimen (20 cm-high, 16 cm-long and 8 cm-wide in the present case); and:

e) In addition to the case c), the strain localisation into shear bands having a specific thickness and a specific relationship between the stress level and the shear deformation was considered.

It may be seen Fig. 4.35 that the peak footing pressure by analysis a) is consistent with the classical solution obtained for the same strength characteristics. It may be seen however that the peak footing load is attained at a very large (so unrealistic) footing settlement in this FEM analysis. This is because a very large settlement of footing is necessary to mobilise the peak strength fully along the potential failure planes when using the realistic pre-failure deformation characteristics of sand. It may also be seen that only the solution of analysis e) is realistic. The Nγ values from these analyses are plotted and compared with the experimental results in Fig.

4.13. These Nγ values by the solution of analysis e) are plotted against the equivalent footing

width 0B B n= ⋅ in Fig. 4.33. It may be seen from these figures that only analysis e) simulates very well the observed scale effect, pressure-level effect and particle size effect.

Figure 4.35. Comparison among FEM simulations based on different assumptions of sand stress-strain properties and experimental results ( 0 50B cm= in 1g) (Tatsuoka et al. 1991).

50

Figure 4.36. Relationships between the shear stress level nR (E. 4.1) and the post-peak shear deformation of a shear band for three types of granular materials having different particle sizes (see Fig. 4.2) (Tatsuoka et al. 1997c; Siddiquee et al. 1999).

Small-scale 1g model tests using different types of sands having different particle diameters: As the last series of physical model test, 1g and centrifuge tests were performed using a coarser sand (Silver Leighton Buzzard sand) and a small-diameter gravel (Hime gravel) (Tatsuoka et al. 1997c). Fig. 4.36 shows the relationships between the shear stress level nR and the post-peak relationships between the shear stress level and the shear deformation of shear band of the three granular materials (Toyoura sand, SLB sand and Hime gravel), obtained by Yoshida and Tatsuoka (1997).

Fig. 4.37 summarises the relationships between the experimentally obtained Nγ value and the corresponding respective value of

( 90 )oPSC atφ δ = for the three types of granular materials. A set of Nγ values for each type of granular material was obtained from 1g tests using different footing sizes (see Fig. 4.13 for Toyoura sand). When comparing these experimental relationships with the relationships between the Nγ and φ from the classical bearing capacity theories shown in Fig. 4.12, it can be seen that as the particle size increases (i.e., as the shear deformation increment that is necessary for the stress state to change from the peak to residual states increases), the Nγ value increases, approaching the relationships by the classi-cal bearing capacity theo-ries. Fig. 4.38 summa-rised the relationships between the following two quantities for the three types of granular materi-als:

Figure 4.37. Comparison of the relationships between the Nγ and φ

from the representative classical bearing capacity theories with the measured relationships between “ Nγ from 1g tests using different

footing sizes” and ( 90 )oPSC atφ δ = for three types of granular

materials (Tatsuoka et al. 1997c).

51

1) the ratio of the respective experimentally obtained Nγ value to the

corresponding ( ) theoryNγ value obtained by substituting the corresponding value of

( 90 )oPSC atφ δ = into the isotropic perfectly plastic solution by Meyerhof (1951); and

2) the logarithm of the ratio of the mean diameter

50D to the physical

footing width 0B . The experimental data

plotted in Fig. 4.38 were obtained from all the 1g and centrifuge tests reported in this paper. The variation in the value of /( )theoryN Nγ γ

for the same 50 0/D B value seen in the data is due basically to the pressure level effect. Such a general trend as that the

/( )theoryN Nγ γ value increases with the increase in the 50 0/D B value can be clearly seen. It may

also be noted that the values of /( )theoryN Nγ γ of the data from the 1g tests on Hime gravel noticeably exceed the unity. This fact is apparently not consistent with the results of Toyoura sand; i.e., with Toyoura sand, this value should be always smaller than the unity, as the factors of strength anisotropy and the progressive failure explained in the above should control the ratio

/( )theoryN Nγ γ . This result of Hime gravel can be explained by the fact that in these model tests, the potential shear band would have been very thick, compared with the footing width. Therefore, a failure mechanism consisting of distinct shear bands (i.e., an active wedge and so on) did not develop below the footing. In such a case, the analysis assuming that the material is continuous is not relevant even when strain localisation with shear banding is taken into account.

The effects of strain localisation could be taken into account in the limit equilibrium analysis although it reflects only partially the actual strain localisation phenomenon. Koseki et al. (1997) and Tatsuoka et al. (1998) showed such a method in modifying the dynamic earth pressure theory by Mononobe and Okabe, while Leshchinsky (2001) in the stability analysis of reinforced soil structures.

Summary: It has been shown how progressive the failure of ground could be in the sense that the local peak strength is not mobilised simultaneously along the full potential failure planes. It was also shown that the scale effect observed in the bearing capacity of a footing on an unbound granular material is due not only to the effect of confining pressure on the peak friction angle φ and deformation characteristics (i.e., the pressure level effect), but also the effect of the particle size relative to the footing size (i.e., the particle size effect). It was also shown that the failure of a mass of dense granular material can be numerically simulated reasonably by the FEM (and

Figure 4.38. Relationship between /( )theoryN Nγ γ and log. of

50 0/D B value for three types of granular materials having different

particle sizes; these ( )theoryNγ values were obtained by substituting

the respective corresponding value of ( 90 )oPSC atφ δ = into the isotropic perfectly plastic solution by Meyerhof (1951) (Tatsuoka et al. 1997c).

52

others) only when the pressure-level dependency of φ and the deformation characteristics, the inherent anisotropy in the strength and deformation characteristics, the deformation characteristics of shear band (as a function of particle size), among other parameters, are taken into account. All the results of the physical model tests and the FEM analysis of the experimental data indicate that geotechnical engineers should be very careful when using classical bearing capacity theories in engineering practice.

In ordinary engineering practice, a low value of φ , such 30 - 35 degrees, is used, which is rather equivalent to the residual angle of friction. When the design Nγ is obtained by substituting this value into the classical bearing capacity theories, a very conservative result would be obtained, as seen from Fig. 4.13. The use of such a design value of Nγ as above would be too conservative (i.e. not economical), in particular when a foundation structure is constructed on a large-particle granular material and/or on a granular material that has a peak strength that is much large than the residual strength (such as very dense well-graded gravels). That is, the design could not be consistent among different types of granular materials that are compacted to different relative densities. 5 TIME-DEPENDENT DEFORMATION PROPERTIES OF GEOMATERIALS

5.1 Introduction The last topic of this paper is also one of the oldest and most classical topics of geotechnical engineering, but it seems that this topic is still a new and challenging topic one. I discussed on this issue in my three recent keynote lectures in Hamburg in 1997 (Tatsuoka et al. 1999a), Napoli in 1998 (Tatsuoka et al. 2000) and Torino in 1999 (Tatsuoka et al. 2001a). It was shown in these three lecture notes that this issue is still full of many topics that are important in both geotechnical engineering practice and research but only poorly understood.

0 250 500 750 1000 1250 1500 1750 2000-70

-60

-50

-40

-30

-20

-10

0

10

3P

The 1995 Hyogo-ken Nambu earthquake

Settl

emen

t, S

(mm

)

Elasped time (days)

0

2

4

6

8

10

b)

End of tower construction

26th Jan. 1990

App

lied

pres

sure

,

(p) av

e (kg

f/cm

2 )

Figure 5.1. Time histories of the applied average footing pressure and the settlement of Pier 3 of Akashi Kaikyo Oh-hashi bridge (Tatsuoka et al. 2001a).

52

5.2 Engineering needs It is often required to predict the long-term residual deformation of ground and displacements of a completed structure subjected to sustained static loads and dynamic load, such as traffic load. Fig. 5.1 shows the time history of the settlement of Pier 3 of the Akashi Kaikyo Oh-hashi Bridge, which was open to service in 1998 (n.b., similar time histories of Pier 2 are reported in Tatsuoka et al. (2001a). Fig. 2.8 shows the relationship between the footing settlement and the footing average pressure constructed using the data presented in Fig. 5.1. The prediction of the settlement during construction and the post-construction residual settlement of this and other footings was of the important geotechnical engineering issues with this bridge. It may be seen from Fig. 5.1 that the construction speed was not constant. Perhaps for this reason, the tangent modulus of the relationship between the footing pressure and the settlement shown in Fig. 2.8 is not smooth. For example, there were three relatively long periods where the footing load was kept nearly constant (i.e., the creep stages) at an intermediate construction stage where the footing pressure was about a half of the final value and before and after the construction of the tower. During these periods, the footing settlement increased despite a nearly constant footing pressure. When the construction was restarted at a normal construction speed following the respective creep stage, the tangent modulus of the footing pressure and settlement relationship was relatively high. And generally, the tangent modulus was larger when the construction speed was larger. This behaviour was, at least partly, due to the viscous deformation property of the ground. It may also be seen from Fig. 5.1 that the time period before the opening of the bridge to service is quite long. In contrast, the time period used for the loading stage until reaching the creep loading stage in laboratory creep tests that are performed to predict such a residual settlement of foundation as described above is substantially shorter. Although this difference should be properly considered when predicting the field behaviour based on results from such laboratory creep tests (Tatsuoka et al. 2001a), it is usually not the case. Summarising the above, it is often required to predict: 1) load-deformation behaviour at different rates of construction or loading; 2) creep deformation and stress relaxation during a period following continuous construction or

loading at different rates (as the case described above); 3) load-deformation behaviour after construction or loading is restarted at a certain rate

following a long period of intermission; and 4) creep (residual) deformation and stress relaxation at unloaded conditions (e.g. Uchimura et al.

1996). For such a prediction as above, the characterisation of the time-dependent (viscous) stress-

strain behaviour of geomaterial is essential, in particular the following aspects: a) effects of constant strain rate on the stress-strain behaviour, including those on the peak

strength; b) changes in the stress-strain behaviour when the strain rate is suddenly or gradually increased

or decreased from a certain value to another, c) creep deformation and stress relaxation; d) stress-strain behaviour when loading is resumed at a constant strain rate after a stage of

creep or strain relaxation; and e) time-dependent stress-strain behaviour in the course of unloading and reloading,

or more generally the stress-strain-time behaviour for a arbitrarily general stress history, including cyclic loading. A number of different constitutive models have been proposed to simulate the behaviour described above.

Described below is one type of constitutive modelling, which has been developed recently by the author and his colleagues. This modelling procedure is still on a long way towards the final goal: i.e.. the development of a three-dimensional model which can simulate and predict the time-dependent behaviour (ageing effects and loading rate effects) of geomaterials in general subjected to arbitrary loading histories.

5.3 Experimental issue

53

It has been considered that effects of bedding errors at the top and bottom ends of specimen be negligible in triaxial creep tests, considering that they could be significant only when the effective axial stress increases (Tatsuoka et al. 1999b, 2000; Hayano et al. 2001). This is not the case at least with sedimentary soft rock, however, as shown in Fig. 5.2. In this figure, significant effects of bedding error can be noted as differences in the axial strain between; a) the external measurement from the axial displacement of the specimen cap detected by

means of a proximity transducer (or gap sensor), denoted as AGS (cap), and b) the local measurement by means of a pair of LDTs, denoted as CLDT, and two pairs of

proximity transducers, denoted as BGS (local). It may be noted that the effects of bedding errors increase not only during monotonic loading stages but also during creep stages. This behaviour could be attributed to extra time-dependent deformations of a thin disturbed zone that should have been formed at the specimen ends during specimen preparation. Large differences in the axial strain between the external gauge (denoted as @EXT) and the proximity transducer (denoted as AGS (cap)) is due to the deformation of the triaxial apparatus (i.e., the system compliance). This result indicates that the use of local axial strain gauge is imperative in such triaxial creep tests.

5.4 Constitutive modelling-1 (Isotach type modelling) One of the relevant frameworks for constitutive modelling for the present purpose is the general three component model (Di Benedetto et al. 2001a). A simplified version that is used herein is shown in Fig. 5.3. The heart of the model is as follows: 1) Strains are first defined in terms of increments, and each strain rate ε& is decomposed into

elastic and inelastic (or irreversible) components, eε& and irε& . The irreversible strain increment irε& cannot be decomposed linearly into plastic and viscous components (Tatsuoka et al. 2000).

2) Stress σ is decomposed into the inviscid and viscous components, fσ and vσ .

ＬＤＴ

External dial gauge

Load cell

Cap

Gap sensors

TargetPedestal

Specimen

Loading piston

High-pressure cell

Gap sensor(proximeter)

hi nge

0.0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6

‡C ‡B

‡A

‡@

(a)

Creep

Creep

‡CLDT‡BGS(local) ‡AGS(cap) ‡@Ext.

Dev

iato

r stre

ss, q

(MPa

)

Axial strain, εv (%)

Figure 5.2. Drained triaxial creep test on sedimentary soft mudstone; each creep period is three days (Tatsuoka et al. 1999b,2000; Hayano et al. 2001); the details of the testing method is described in Hayano et al. (1977).

54

According to the conventional isotach model, the stress σ is a unique function of instantaneous total strain ε and its rate

tεε ∂=

∂& . Tatsuoka et al. (1999d, 2000)

showed the limitations of this type of model: for example, this model cannot simulate the stress relaxation process. This is because when the condition of stress relaxation, 0ε =& , is given to a geomaterial element, the stress σ becomes suddenly and discontinuously the value corresponding to 0ε =& , which is not realistic.

A great deal of experimental results indicate that, with geomaterials, the stress σ is a unique function of instantaneous

irreversible strain irε and its rate irε& (Tatsuoka et al. 2000). Tracking this line and following the framework of the three-component model described in Fig. 5.3, Tatsuoka et al. (1999d, 2000, 2001b) proposed the so-called New Isotach model (Fig. 5.4). According to the New Isotach model, we obtain different stress-strain relationships for monotonic loading at different constant strain rates. The creep deformation is a process where irε& continuously decreases towards zero at a constant σ . Similarly, the stress relaxation is the process where irε& continuously decreases towards zero at a constant e irε ε ε= + with negative edε and positive irdε . The elastic strain increment is obtained by a relevant hypo-elastic model, which is described in Chapter 2, for which the tangent stiffness is a function of instantaneous stress. Fig. 5.5 is a typical example of this strain decomposition for a sedimentary soft rock (Tatsuoka et al. 2000; Hayano et al. 20001). Note that the irreversible strain can be obtained only after such strain decomposition.

The other important features of this model, as summarised in Fig. 5.4 are: 1) fσ is a function of the instantaneous value of irε ;

Non-linear inviscid component; 　　　　　　　　　　　　　　　　 Hypo-elastic component: σ

&ε 　　　　　　　　　　　 　 Non-linear viscous component; 　　　　　　　　　　　　

&εe

&εvp

fσ

vσHypo-elastic model

irε&Figure 5.3. Framework used in the development of models described in this paper (Di Benedetto et al. 2001a & b; Tatsuoka et al. 2001b).

σ

010ε ε= ⋅& &

0ε ε=& &

0 /10ε ε=& &

Lower bound at 0i rε =& :

( )f irσ ε

Creep 0

ε

　New Isotach Model（the simplest three-component model）

( ) ( ),f ir v ir irσ σ ε σ ε ε= + &

( ) ( ), ( )v ir ir f ir irvgσ ε ε σ ε ε= ⋅& &

The stress is always a uniquefunction of instantaneous valuesof and .

Different stress-strain relationsdevelop by loading at differentstrain rate; and corresponding tothe above, creep and stressrelaxation take place.

( ) [1 exp{1 ( 1) }] ( 0)ir

ir mv ir

r

gε

ε αε

= ⋅ − − + ≥&& &

fσ

vσ

Non-linear inviscid component; 　　　　　　　　　　　　　　　　 Hypo-elastic component: σ

&ε

　　　　　　　　　　　

　 Non-linear viscous component; 　　　　　　　　　　　　

&εe &εvp

irε irε&

Figure 5.4. New Isotach model (Tatsuoka et al. 1999d, 2000, 2001b).

55

2) vσ is always proportional to fσ , which is one of the specific features of this model; and 3) ( )ir

vg ε& is the viscous function, which is a highly non-linear function of instantaneous value of irε& (this function is explained in Fig. 5.13). Note also that the elastic strain, which is obtained by integrating elastic strain increments for a closed loop of stress path, could be irreversible (Puzrin and Tatsuoka 1998). On the other hand, the relationship between σ and irε , which is obtained by integrating

irε& , may form a closed loop. In this sense, reversibility and irrversibility of strain is

0.00 0.05 0.10 0.15 0.20 0.250

20

40

60(a)

C: Drained creep for one day

C

C

C

C

É0

É0

É0

É0

É0

É 0

É0

É0/100É0/100

É0/100Irreversible

Elastic

Total

Axial strain, εv (%)

Dev

iato

r stre

ss, q

(kgf

/cm

2 )

Sedimentary soft rock (mudstone; Kazusa group)

Figure 5.5. Separation of total axial strains into elastic and irreversible parts for the stress-strain relation from a drained TC test on sedimentary soft rock ( 0ε& = 0.01 %/min) (Tatsuoka et al. 2000;

Hayano et al. 2001).

0.00 0.05 0.10 0.15 0.20 0.250

10

20

30

40

50

.

.

.

.

.

.

.

.

.

(ε)0/100

(ε)0/100

(ε)0/100

(ε)0

(ε)0

(ε)0

(ε)0

(ε)0

(ε)0

(e) C

Measured

Calculated

C: Drained creep for one day

C

CC

C

(ε)0

Total axial strain, εv (%)

Dev

iato

r st

ress

, q (

kgf/c

m2 )

0.00 0.02 0.04 0.06 0.08 0.100

10

20

30

40

50(f)

n=1

Calculated q-(εir)

static

Calculated q-εi r

C: Drained creep for one day

C

C

C

C

C

É0

É0

É0

É0

É0/100

É0/100

Calculated irreversible axial strain, εvir (%)

Dev

iato

r stre

ss, q

(kgf

/cm

2 )

Simulation by the New Isotach model

Viscous effects

Figure 5.6. a) Simulation of the test result presented in Fig. 5.5 by the New Isotach model; and b) decomposition of the stress component in the simulation (Tatsuoka et al. 2000; Hayano et al. 2001).

a)

b)

56

defined only with respect to strain increments. Fig. 5.6 shows the simulation of the test result presented in Fig. 5.5 by the New Isotach model (Tatsuoka et al. 2000; Hayano et al. 2001). According to this model, the relationship between

fσ and irε is obtained by monotonic loading at an infinitively slow strain rate, which will therefore be called the reference curve. In this simulation, the reference curve was obtained so that the entire test result can be simulated by using the same viscous function

( )irvg ε& . It may be

seen that this model can simulate well the entire stress-strain behaviour, including the creep and unloading behaviour. In particular, the decrease in the axial strain with time at the creep stage in the course of global unloading (i.e., the creep recovery phenomenon) is well simulated. The model is also capable to simulate the effects of step increase and decrease in the constant strain rate as well as the behaviour during and immediately two creep stages (Fig. 5.7). Fig. 5.8 shows the simulation by the New Isotach Model of the relationship between the average footing pressure and the footing settlement of Pier 3P, Akashi Kaikyo Bridge, shown in Fig. 2.8 (also refer to Fig. 5.1). In this considerably simplified simulation, the supporting ground was treated as one soil element, and the parameters of the model were determined to obtain the best fit. Despite the above, it is seen that the overall effects of loading rate are well captured: i.e., the fluctuation of the relation is not due to simple measurement errors, but it is

0.0 0.3 0.6 0.9 1.2 1.50

1

2

3

4

5

.

.

.. .

.

..

..

..

....

ε0/100

ε0/100

ε0/100

ε0/100

ε0/10

ε0/10

ε0/10

ε0/10

ε0

ε0ε0

ε0ε0Measured(a)

Silt-sandstone σ'c=1.29MPa ε0=0.01%/min

C: Drained creep

Simulated

ε0

C

C

Total axial strain, εvt (%)

Dev

iato

r stre

ss, q

=σ' v-σ

' h (MPa

)

Sedimentary soft rock (mudstone; Kazusa group)

Figure 5.7. Simulation by the New Isotach model of the stress-strain behaviour in CD triaxial compression test on sedimentary soft rock (Tatsuoka et al. 2000; Hayano et al. 2001).

0 10 20 30 40 50

0

2

4

6

8

10

b-1)

Load control

3P

Field data Simulation

Ave

rage

con

tact

pre

ssur

e, (p

) ave (

kgf/c

m2 )

Irreversible settlement, Sir (mm)

Simulation of the foundation behaviour by the New Isotach model:(the ground is treated as a single element)

Figure 5.8. Approximated simulation by the New Isotach Model of the relationship between the average footing pressure and the footing settlement of Pier 3P, Akashi Kaikyo Bridge; the ground is treated as one element (Tatsuoka et al. 2001a).

57

due to changes of loading rate. More discussion on this simulation is given in Tatsuoka et al. (2001a). Of course, the final goal of this study is to simulate the observed behaviour by relevant FEM analysis.

Results from PSC tests - 1

Very small differenceamong the behaviourat constant strainrates differing by afactor up to 500.

The stress value changeswhen the strain rate isstepwise changed by afactor of 100, and thestress change decays withstrain.

Apparentcontradiction！

Test name e0.05 dεv / dt (%/min) HOS01 0.6146 variable H302C 0.6153 10ε0

H303C 0.6162 ε0/10 H304C 0.6149 ε0/10 H305C 0.6160 10ε0

H306C 0.6155 ε0/10 H307C 0.6164 ε0/50

0 1 2 3 4 5 6 7 8 9

3.0

3.5

4.0

4.5

5.0

5.5

6.0

.

.

Fig. 2(graph 11)

.

..

..

.

.

.

.

.

.Saturated Hostun sand (Batch A)

ε0= 0.0125 %/min

Test HOS01c1-e1 ε0/10 e1-f1 10ε0 f1-h1 ε0/10h1-i1 10ε0i1-k1 ε0/10k1-l1 10ε0

d1, g1, j1 5 times small cyclic loading

H307C ε0/50

H304C ε0/10

H305C 10ε0

H306C ε0/10

H303C ε0/10

H302C 10ε0

l1

k1j1

i1

g1

h1

f1

e1

d1c1

Shear strain, γ = εv - εh (%)

Stre

ss ra

tio, R

= σ

' v/σ' h

Figure 5.10a. Plane strain compression tests on Hostun sand by monotonic loading at different constant strain rates and loading with step changes in the constant strain rates (Matsushita et al. 1999; Tatsuoka et al. 1999b; Di Benedetto et al. 2001a).

Plane Strain Compression Tests on Sand

'vσ

PSC

' ' 3.0v hR σ σ= =

(not to scale) Initial state 0 'hσ (kPa) 29 392

20cm

8cm 16cm

Figure 5.9. Plane strain compression test conditions to study the viscous properties of sands (Di Benedetto et al. 2001b and Tatsuoka et al. 2001b); the details of the plane strain compression testing method are described in Yasin et al. (1999a & b) and Yasin and Tatsuoka (2000).

58

5.4 Constitutive modelling-2 (TESRA model)

TESRA stands for “temporary effects of strain rate and strain acceleration”. The reason why this mew model was to be developed is explained below. Fig. 5.9 shows the PSC (plane strain compression test) procedure by which the viscous properties of sand were investigated (Matsushita et al. 1999; Tatsuoka et al. 2001a; Di Benedetto et al. 2001a). Fig. 5.10a shows

results from a series of PSC tests on six saturated specimens of air-pluviated Hostun sand (from France), in which monotonic loading at different constant strain rates were applied after anisotropic consolidation. In the other test, the axial strain was changed stepwise several times during otherwise monotonic loading at a constant strain rate. The following trends of behaviour can be seen: 1) It may be seen from Fig. 5.10a that the overall stress-strain curves from the monotonic

loading tests at different constant strain rates are nearly the same (n.b., rigorously, the initial part is different for different strain rates).

2) Despite the above, the stress-strain curve from the other test exhibits very stiff behaviour immediately after the strain rate increases stepwise. Then, after having exhibited clear yielding, the stress-strain curve tends to rejoin the stress-strain curve that would have been obtained if the loading had continued at the strain rate before a step change.

3) The behaviour that is opposite to the above takes place after the strain rate decreases stepwise. These two behaviours apparently contract each other.

Note that such noticeable viscous effects were observed also in similar PSC tests on air-dried specimens and in triaxial compression tests on saturated and air-dried sand (Matsushita et al. 1999; Di Benedetto et al. 2001a). Fig. 5.10b shows results from another PSC test on Hostun sand, in which creep and relaxation stages are included during otherwise monotonic loading, and the test result is compared with those from the PSC tests performed at different constant strain rates, presented in Fig. 5.10a (Matsushita et al. 1999; Tatsuoka et al. 20001a; Di Benedetto et al. 2001a). It may be seen Fig. 5.10b that the specimen exhibits noticeable creep deformation and stress relaxation,

Noticeable creepdeformation and stressrelaxation take place.

0 1 2 3 4 5 6 7 8 9

3.0

3.5

4.0

4.5

5.0

5.5

6.0

.ε0= 0.0125 %/min.

..

..

..

.

.

.

.

.

.

Fig. 3 (graph 8)

Test HOSB1c2-d2 10ε0 j2 -k2 creepd2-e2 creep k2-m2 10ε

0

e2-g2 10ε0 m2-n2 relaxationg2-h2 creep n2-o2 10ε0

h2-i2 accidental pressure drop, o2-p2 creep followed by relaxation stage p2-q2 ε0/10i2 -j2 ε

0/10

H307C ε0/50

q2

p2

H304C ε0/10

H305C 10ε0

H306C ε0/10

H303C ε0/10H302C 10ε0

o2

n2

m2

l2k2j2

i2

g2

h2

f2

e2d2

c2

Shear strain, γ = εv - εh (%)

Stre

ss ra

tio, R

= σ

' v/σ' h

Very small differenceamong the behaviourat constant strainrates differing by afactor up to 500.

Apparentcontradiction！

Results from PSC tests - 2

Figure 5.10b. Plane strain compression tests on Hostun sand by monotonic loading at different constant strain rates and loading with creep and relaxation stages (Matsushita et al. 1999; Tatsuoka et al. 1999b; Di Benedetto et al. 2001a).

59

which is again apparently inconsistent with the behaviour during loading at different constant strain rates.

To simulate the above peculiar behaviour, a new model was developed as follows. Fig. 5.11 shows the procedure at the preparation stage for the above. That is, the viscous stress vσ of the

TESRA (temporary effect ofstrain rate and acceleration) model　１

( )f ir vσ σ ε σ= +

1 1

1

( , ) ( )

( )

( )

{ ( )}

( )( )

ir ir

irir i r

ir

ir

v f irv

v f irv

irf irir f v

vir ir ir

g

d d g

gg d

ε ε

τ ε ττ ε τ ε

ε

τ ε τ

σ σ ε

σ σ ε

εσ εε σ τ

ε ε ε

= =

=

= ⋅

= = ⋅

∂∂= ⋅ + ⋅ ⋅ ⋅ ∂ ∂

∫ ∫

∫

&

&

& &&& & &

( 0)vσ ≥

fσ

vσ

Non-linear inviscid component; 　　　　　　　　　　　　　　　　 Hypo-elastic component: σ

&ε 　　　　　　　　　　　 　 Non-linear viscous component; 　　　　　　　　　　　　

&εe

&εvp

the New Isotach Model:

Figure 5.11. Reforming of the New Isotach model into an integral form as a preparation for the development of the TESRA model; τ is the value of irε at a certain moment before the current state (Di Benedetto et al. 2001a).

1

1

( )

( )

( )

{ ( )}

( )

ir

ir

ir

ir

fir

ir irf v

i r

v f irv

vir ir

g d

g

d

gd

ε

ττ ε

ε

τ ε τ

ε εσ

ε

σ σ ε τ

σε τ

εε

=

=

∂⋅ ⋅

= ⋅ ⋅

= + ⋅

∂⋅ ∂

∂

∫

∫&& &

& &

&

& ( 0)vσ ≥

σ b

010irε ε= ⋅& & 0i r

vi r

ird

dε

σ εε =

∂ ⋅ ∂ &&

vdσ

0irε ε=& &

0ir

vir

ird

dε

σε

ε =

∂⋅ ∂ &

Parallel a vσ

{ ( ) ( )}v f ir irvd d gσ σ ε ε= ⋅ &

fσ

f fdσ σ+

0 ε

1irε&

iraε&

e ird d dε ε ε= +

Effect of irreversible strain rate

Effect of irreversible strain acceleration

The New Isotach Model:

Figure 5.12. Illustration of the effects of irreversible strain rate and acceleration with the New Isotach Model (Di Benedetto et al. 2001a).

60

New Isotach Model is expressed in the integral form. That is, the increment of vσ consists of “its derivative with respect to irε ” times “irreversible strain increment irdε ”, showing the effect of irreversible strain rate, and “its derivative with respect to irε& ” times “irreversible strain rate increment irdε& ”, showing the effect of irreversible strain acceleration (see Fig. 5.12). Note that the viscous function ( )ir

vg ε& should be defined so that its derivative with respect to irε& is always smooth (Fig. 5.13).

In the TESRA modelling, the effects of irreversible strain rate and acceleration that take place at a certain loading stage where irε τ= are considered to be temporary and decay with the increase in the strain difference “ irε τ− ” until the current state where ir irε ε= (Fig. 5.14). This property is expressed by the decay function ( )ir

decayg ε τ− , which is explained in Fig. 5.15. The introduction of the decay function is the heart of the TESRA model. The decay function represents such a property of sand as that the sand tends to gradually forget what happens in the past due to gradual changes in the structure (i.e., rearrangement of relative locations of particles) with the increase in the irreversible strain (not with time). Fig. 5.16 illustrates how the model behaves.

Fig. 5.17 shows a typical simulation by the TESRA model of the PSC test result of Hostun sand in which the strain rate was changed stepwise several times. Note that the specimen was air-dried, so the observed viscous effect was not due to the partial and delayed drainage of pore water. It may be seen that the model simulates very well the test result. Note that the details of the behaviour after the strain rate is changed stepwise are simulated surprisingly well.

Fig. 5.18 shows the simulation of the result of another PSC test, in which the strain rate was increased and decreased at a constant rate respectively for some strain range and this sequence was repeated two times. One creep stage was included between the above mentioned two

( ) [1 exp{1 ( 1) }] ( 0)ir

ir mv ir

r

gε

ε αε

= ⋅ − − + ≥&& &Viscosity function:

This function can be determined experimentally　！

1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01

1

1.0

.

.

1.000.99

1.10

Fig. 15b

b

Hostun sand

Toyoura sand

Irreversible strain rate, εir (%/sec)

1+ g

v(εir )

0.000 0.002 0.004 0.006 0.008 0.0100.00

0.02

0.04

0.06

0.08

0.10

Fig. 15a.

. Hostun sand

Toyoura sand

Irreversible strain rate, εir(%/sec)

g v(εir )

Theconditionsto besatisfied:

1) 0.0 ( )irvg ε α≤ ≤& for any value of

irε& between −∞ and ∞ ;

2) ( 0) 0irvg ε = =& ; and

( )( 0)

irirv

ir

g εε

ε ∂

= ∂

& && = irr

mαε⋅& (a finite positive value).

Figure 5.13. Structure of the viscous function (Di Benedetto et al. 2001a).

61

sequences. It may be seen that the model is able to simulate very well the whole stress-strain behaviour, including the behaviour during the strain rate was changed gradually, the creep behaviour and the post-creep behaviour during loading at a constant strain rate. It is one of the characte ristic features of the TESRA model that the stress-strain state can be located below the reference curve when the negative effects of negative strain acceleration on the vσ value in the recent strain history become dominant.

Note that for the simulation shown above, the model parameters were determined so that the test results were best fit. However, the same parameters are used for all the tests performed under otherwise the same test conditions (i.e., the PSC tests referred in Figs. 5.17 and 5.18). So, it can be concluded that the TESRA model captures the basic characteristic feature of the viscous property of sand.

Fig. 5.19a shows the results from two special PSC tests on air-pluviated Toyoura sand (Tatsuoka et al. 2001b). The two specimens were loaded first at constant axial strain rates that

TESRA (temporary effect ofstrain rate and acceleration) model 2　

( )f ir vσ σ ε σ= +

vσ =1 1

( , ) ( ){ }( ) ( )

ir ir

ir

ir ir

f i irdeca

rv y

v gd gdε ε

τ ε ττ ε τ ε

σ εσ ε τ= =

= −⋅ ⋅∫ ∫ &

( 0; 0)v orσ ≥ ≤

Decay function: ( )

1( )irir

decayg r ε τε τ −− =

fσ

vσ

Non-linear inviscid component; 　　　　　　　　　　　　　　　　 Hypo-elastic component: σ

&ε 　　　　　　　　　　　 　 Non-linear viscous component; 　　　　　　　　　　　　

&εe

&εvp

the same as the New Isotach model

Figure 5.14. Introduction of the decay function for the TESRA model; r1 is a constant lower than unity (Di Benedetto et al. 2001a).

Sand gradually forgetsthe viscous effectsthat took place in the pastat τ with subsequentirreversible straining.

vdσ σ

Current state Event of ( )

vdτ

σ

vσ

Strain difference

i rε τ− fσ

σ

1irε

i rε τ=

irε

0 ( )

irirdecayg rε τε τ −− =

irε

1.0

r1

( )irdecayg ε τ−

for

( )( )vd τσ

i rε τ− i rε τ− 1.0 0 ( )i r

decayg ε τ−

( )1( )

irir

decayg r ε τε τ −− =Decay function:

Fig. 5.15 Explanation of the decay function (Di Benedetto et al. 2001a).

62

were different by a factor of 100 as vε& = 0.25 and 0.0025 %/min. The stress-strain relationships of the two tests gradually collapse into a single relationship, as simulated by the TESRA model. In both the tests, creep tests were then performed at two stages, each stage lasting for 24 hours. The following important trends of behaviour may be seen from Fig. 5.19a: 1) Despite nearly the same stress and strain states at the start of creep in the two tests, the time

history of creep strain is significantly different between the two tests. This fact should be explained by the fact that despite the same stress and nearly the same strain at the start of each creep stage in the two tests, the initial creep axial strain rate at the start of creep is very different (by a factor of about 100).

2) Immediately after loading is restarted at constant but different vε& values following each creep stage, the difference in the stress-strain relations between the two tests becomes large again, which is to a larger extent than it is immediately after the start of loading. As loading continues at a constant vε& , after having exhibited clear yielding, the difference becomes gradually smaller again. These facts above indicate that, even when limiting to the monotonic loading case, the stress-

strain state is not a unique function of instantaneous irreversible strain rate, but it is also signif icantly affected by the recent strain history. The new isotach model is, and perhaps most of the existing elasto-viscoplastic models are, not able to properly simulate these behaviours. The TESRA model is able to simulate this behaviour, as shown above and also below.

Fig. 5.19b compares the measured and simulated time histories of vε and vR ε∼ relationships. It may be seen that the TESRA model simulates the measured behaviour well, in particular the following aspects: 1) the creep strain is larger in the test in which vε& at the start of creep is larger, despite that the

stress-strain state at the start of each creep stage is nearly the same in the two tests; and 2) the stress range in which the stiffness is very high that appears immediately after loading was

restarted following each creep stage is larger when loading is restarted at a higher vε& , while

Stress is a specific function of instantaneous and andstrain history.

The stress value could be the same for the different instantaneous and , while creep deformation and stress relaxation can take place.

The viscous stress could be either positive, zero or negative depending on the strain history.

σ

010irε ε= ⋅& & Reference relation:

( )f irσ ε

0irε ε=& &

0 /10irε ε=& & 　

Creep 0

ε

vσ

Non-linear inviscid component; 　　　　　　　　　　　　　　　　 Hypo-elastic component: σ

&ε

　　　　　　　　　　　

　 Non-linear viscous component; 　　　　　　　　　　　　

&εe

&εvp

TESRA (temporary effect ofstrain rate and acceleration) model　3

irε&irε

irε irε&

Figure 5.16. Behaviour of TESRA model (Di Benedetto et al. 2001a).

63

the stress-strain curves in the two tests become gradually similar as loading continue at a constant vε& after having exhibited clear yielding.

It is likely that only the TESRA model, and other models having the similar basic structure including the viscous evanescent model (Di Benedetto et al. 2001b), can simulate these aspects.

In engineering practice, three or five-component models have often been used to predict the residual settlement of a footing. A large amount of laboratory creep tests were performed to determine the model parameters. However, this method has the following serious drawbacks: 1) The model is not adequate because of the linear property of the three components. 2) In the analysis, it is often assumed that the creep starts after a sudden instantaneous loading.

Furthermore, it is often assumed that all the ground deformation before the start of creep phase or the instant settlement of footing is elastic. Corresponding to the above, specified creep load is applied to the specimen in a rather sudden manner in usual laboratory creep tests. It is readily seen that the residual settlement of a footing should be predicted by a model that

can take into account the effects of recent loading history, such as the New Isotach and TESRA models.

Another important issue is the simulation of the time-dependent behaviour during unloading process. An example of a test result and its simulation dealing with this issue is shown in Fig. 5.20. An air-dried specimen of Toyoura sand was used in this PSC test for the easiness of testing lasting for such a long period, while based on the fact that air-dried and saturated specimens of Toyoura sand exhibited essentially the same stress-strain behaviour in drained TC tests (Tatsuoka et al., 1986b). The specimen was isotropically consolidated to ' 'v hσ σ= = 392 kPa prior to the start of drained PSC loading. The primary PSC loading was made at an axial strain rate of 0.125 %/min. Two full unload/reload cycles were applied between R= 1.0 and 4.0 or 5.0. The shear stress was decreased at a constant stress rate, q& = - 98 kPa/hour (i.e., R& = - 0.25/hour= - 0.0042/min). Reloading was made by strain control as the primary loading. By such stress unloading, it becomes possible to closely observe the development of positive irreversible axial strain (and shear strain) for some shear stress range immediately after the start

PSC test on sand 3

0 1 2 3 4 5 6 7

3.0

3.5

4.0

4.5

5.0

5.5

6.0

.

Fig. 12b(grpah3)

Test Hsd03

Experiment Simulation Reference curve

(in terms of total strain)

α= 0.25; m=0.04;

εrir=10-6 (%/sec); and

r1= 0.1 (for strain difference in %)

Stre

ss ra

tio, R

=σ v'/

σh'

Shear strain, γ (%)

3.5 4.0 4.5 5.0 5.54.6

4.7

4.8

4.9

5.0

5.1

5.2

5.3

5.4

5.5

5.6

Fig. 12d(graph14)

Test Hsd03

Experiment Simulation Reference curve

(in terms of total strain)

Stre

ss ra

tio, R

=σv'/σ

h'

Shear strain, γ (%)

Figure 5.17. Simulation of the behaviour of air-dried Hostun sand in plane strain compression test by the TESRA model (Di Benedetto et al. 2001a).

64

of decreasing the shear stress. In Fig. 5.20, this behaviour can be seen during the stress unloading process starting from point U. The neutral condition is then reached, where ir

vε&

becomes zero and a switching from positive to negative irvε& values takes place. Such behaviour

as described above is often observed in load (or pressure)-controlled plate loading and pressuremeter tests. It is usually very difficult to evaluate the elastic property by analysing load (pressure)-displacement curves observed immediately after the start of decreasing load or pressure (Tatsuoka et al. 2001a). Creep tests, each lasting for four hours, were performed at every increment of q= 196 or 392 kPa (i.e., every increment of R of 0.5 or 1.0) in the course of primary loading, unloading and reloading.

The following trends of behaviour can be seen from Fig. 5.20: 1) For some stress range immediately below point U, where the shear stress was decreasing, the

total axial and shear strains (and the irreversible axial and shear strains) are still increasing. 2) As the shear stress further decreases, the neutral state, where ir

vε& = 0, is reached; then,

unloading with negative irvε& values starts. However, the exact location of the neutral state is

not obvious. 3) At the creep stages where the shear stress is lower than that at the neutral state, the sign of

creep axial strain (and shear strain) is negative, while the absolute value of negative creep strains becomes larger at a lower shear stress (i.e., the phenomenon called “creep recovery”). Tatsuoka et al. (2001a) discussed on this phenomenon based on data from a number of field loading tests and laboratory stress-strain tests on soft clay, sand and gravel and sedimentary soft rock, while showing the importance of this behaviour in many geotechnical engineering issues.

4) The sign of axial strain increments (and shear strain increments) at the creep stages during reloading becomes positive again as it is during primary loading, and the amount of creep strain increases as the shear stress increases.

0.0 0.5 1.0 1.5 2.0 2.5 3.03.0

3.5

4.0

4.5

5.0

5.5

6.0

.

Test Combi1 (Toyoura sand)α=0.25; m=0.05;

εrir= 10-6 (%/sec); and

r1= 0.1 (for strain difference in %)

Fig. 16 b-1(graph7)

Experiment

Reference curve(in terms of total strain)

Simulation

Vertical (axial) strain, εv (%)

Stre

ss ra

tio, R

=σv'/σ

h'

PSC test on sand 4

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.04.7

4.8

4.9

5.0

5.1

5.2

5.3

Fig. 16 b-3graph14

ExperimentReference curve

Simulation

Vertical (axial) strain, εv (%)

Stre

ss ra

tio, R

=σv'/

σh'

0 10000 20000 30000 40000 50000 600001.25

1.30

1.35

1.40

1.45

1.50

1.55

1.60

1.65

Test Combi1

Start of creep stage

Fig. 16e(graph 11)

Simulation

Experiment

Elapsed time (sec)

Ver

tical

stra

in, ε

v (%)

Figure 5.18. Simulation of the behaviour of Hostun sand in plane strain compression test by the TESRA model (Di Benedetto et al. 2001a).

65

It can be seen from Fig. 5.20 that the TESRA model simulates very well all the details of the stress-strain-time behaviour from loading phase towards unloading phase. It is particularly important that the following aspects are simulated very well: 1) the increase in the positive total and irreversible axial strains continues for some stress range

immediately after the start of decreasing the shear stress at a constant negative rate;

PSC test on sand 5

0.0 0.5 1.0 1.5 2.0 2.53.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Creep

②,③

Low strain rate（ei=0.740）

High strain rate (higher by a factor of 100)（ei=0.742）

Axial strain, εv (％)

Stre

ss r

atio

, R=

σv/

σh

Different creep strain rates; despite having started from nearly the same stress and strain state !

0.0 0.5 1.0 1.5 2.0 2.5 3.03.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Experiment (thin curve)

Low strain rate

High strain rate

Axialstrain, εv (%)

Str

ess

ratio

, R=

σ v/σh

Figure 5.19. Simulation of the behaviour of Hostun sand in plane strain compression test by the TESRA model (Tatsuoka et al. 2001b).

a)

b)

66

2) creep recovery takes place at the creep stages in the course of unloading below the neutral state, and the amount of creep recovery increases as the shear stress at the creep stage decreases.

Summary: It has been demonstrated that the issue of time-dependent stress-strain behaviour

of geomaterials (ageing effects and loading rate effects) is one of the fresh topics in geotechnical engineering, requiring developments of new constitutive models.

Careful and systematic laboratory tests are required to understand the time-dependent deformation properties of geomaterial, and a comprehensive series of experimental study on various types of geomaterials are necessary to understand the general framework of this issue. It was shown above that erratic results could be obtained from conventional creep tests on stiff geomaterials measuring axial strains externally and local axial strain measurement is imperative for such tests. It was also shown that one of the relevant testing methods to validate a constitutive model developed to simulate the time-dependent deformation properties of geomaterial and to obtain the parameters of a model include: 1) stepwise or gradual changes in the strain rate; and 2) switching between strain- and stress- control tests; between constant rate loading and creep and stress relaxation tests. Constitutive models are required to simulate this behaviour for various stress histories, including unloading. It was shown that three-component models are relevant for this purpose. It was also shown that the viscous stress-strain behaviour of some types of geomaterials (such as sedimentary softrocks) can be simulated by the New Isotach model (without a decay in the viscous stress), which is an extension of the conventional three-component rheology framework. One new model (the TESRA model), with a decay in the viscous stress, was introduced herein to simulate the peculiar viscous behaviour of sand.

PSC test on sand 6

Increase in the strain after the stress has started decreasing !

1.0 1.2 1.4

1

2

3

4

Fig. 18b

UTest Ulcrp3

Experiment Simulation

Stre

ss ra

tio, R

=σ'

v/σ' h

Vertical (axial) strain, εv (%)

8.5 9.0 9.5 10.0 10.5 11.01.30

1.35

1.40

1.45

Start of creep stage

Ver

tical

(ax

ial)

stra

in, ε

v (%

)

Fig. 19b

U

Test Ulcrp3

Experiment Simulation

by the TESRA model

Elapsed time (hour)

35 36 37 38 39 40 411.00

1.05

1.10

1.15

1.20

Start of creep stage

Ver

tical

(ax

ial)

stra

in, ε

v (%

)

Fig. 19c

Test Ulcrp3

Experiment Simulation by the TESRA model

Elapsed time (hour)

Figure 5.20. Simulation of the behaviour during stress-controlled unloading and the creep in the course of unloading (Tatsuoka et al. 2001b).

67

More study is necessary to generalise these models to apply to more general stress paths with and without cyclic loading. In particular, more experimental study is necessary on viscous effects on the yield locus and flow rule, as discussed by Tatsuoka and Ishihara (1973, 1974a & b) for sand, and the hardening function, as discussed by Tatsuoka et al. (2001c) for sand.

6 CONCLUDING REMARKS Impacts on the theories and practice of geotechnical engineering of several findings obtained from recent advanced laboratory stress-strain tests on geomaterials have been demonstrated. The topics discussed in this lecture note are only some of similarly important issues in geotechnical engineering. The main objective of this lecture is to demonstrate that geotechnical engineering is still very young.

It seems that laboratory stress-strain tests have recently become less popular. It is perhaps because: 1) it is very difficult to retrieve high-quality undisturbed samples in many occasions; 2) laboratory stress-strain tests are often considered to be less direct (so less useful) than field

loading tests for design purposes, while laboratory stress-strain tests are just painstaking and time-consuming. In many cases, however, the following is also true:

1) More proper characterisation of the stress-strain property of geomaterials becomes possible with a help of relevant laboratory stress-strain tests, which could result into more rational (so safer and more cost-effective) design.

2) Proper understanding of the stress-strain-time behaviour of geomaterials, which is also essential for rational (safer and more cost-effective) design, is not possible only by field loading tests and back-analysis of full-scale behaviour. Relevant laboratory stress-strain tests can play an essential role for this purpose.

Acknowledgements: The materials that were referred to in this lecture note were obtained by many previous and present colleagues of the author at the Institute of Industrial Science, University of Tokyo, where the author spent nearly twenty years, and the Department of Civil Engineering, University of Tokyo, where the author is presently working. The co-operation and help from these colleagues are deeply acknowledged.

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STP No.977, 7-67. Tatsuoka,F., Pradhan,Tej B.S. and Horii,N. (1988), “Discussion on the paper by Jewell and Wroth”,

Géotechnique, 38-1, 148-153. Tatsuoka,F., Tani,K., Okahara,M., Morimoto,T., Tatsuta,M., Takagi,S. and Mori,H. (1989a), “Discussion

on the Paper by Hettler and Gudehus,” Soils and Foundations, 29-4, 146-154. Tatsuoka,F., Huang,C-C., Morimoto,T. and Okahara,M. (1989b), “Discussion on the Paper by Graham et

al.”, Canadian Geotechnical Journal, 26-4, 748-755. Tatsuoka,F., Nakamura,S., Huang,C-C., and Tani,K. (1990a), “Strength anisotropy and shear band

direction in plane strain tests on sand”, Soils and Foundations, 30-1, 35-54. Tatsuoka,F., Shibuya,S., Goto,S., Sato,T. and Kong,X.J. (1990b), “Discussion on the Paper by Clayton et

al.”, Geotechnical Testing Journal, 13-1, March, 63-67. Tatsuoka,F., Shibuya,S., Teachavorasinskun,S. and Park,C-S. (1990c), “Discussion on the Paper by

Bolton and Wilson”, Géotechnique, 40-4, 659-663. Tatsuoka,F., Okahara,M., Tanaka,T., Tani,K., Morimoto,T. and Siddiquee,M.S.A. (1991), “Progressive

failure and particle size effect in bearing capacity of a footing on sand”, Proc. ASCE Geotech. Engineering Congress, 1991, Boulder, ASCE GSP, 27, 788-802.

Tatsuoka,F. and Huang,C.-C. (1991), Discussion of “Bearing capacity of foundations in slopes” by Shields et al., Journal of Geotechnical Engineering, ASCE, 117-12, 1970-1975.

Tatsuoka,F. and Shibuya,S. (1991), “Deformation characteristics of soils and rocks from field and laboratory tests”, Keynote Lecture for Session No.1, Proc. of the 9th Asian Regional Conf. on SMFE, Bangkok, 1, 101-170.

Tatsuoka,F., Siddiquee,M.S.A., Tanaka,T. and Okahara,M. (1992), “A new aspect of a very old issue: Bearing capacity of footing on sand”, Panel Dis cussion, Proc. of the 9th Asian Regional Conf. on SMFE, 2, 358-359.

Tatsuoka,F., Siddiquee,M.S.A., Park,C.-S., Sakamoto,M. and Abe,F.(1993), “Modeling stress-strain relations of sand”, Soils and Foundations, 33-2, 60-81.

Tatsuoka,F., Sato,T., Park,C.-S., Kim,Y.-S., Mukabi,J.N. and Kohata,Y. (1994a), “Measurements of elastic properties of geomaterials in laboratory compression tests”, Geotechnical Testing Journal，ASTM, 17-1, 80-94.

Tatsuoka,F., Teachavorasinskun,S., Dong,J., Kohata,Y. and Sato,T. (1994b), “Importance of measuring local strains in cyclic triaxial tests on granular materials”, Proc. of ASTM Symposium Dynamic Geotechnical TestingⅡ , ASTM, STP 1213, 288-302.

Tatsuoka,F., Siddiquee,M.S.A. and Tanaka,T. (1994c), “Link among design, model tests, theories and sand properties in bearing capacity of footing on sand”, Panel Discussion, Proc. of the 13th Int. Conf. on S.M.F.E., New Delhi, 5, 87-88.

Tatsuoka,F. (1994), “Measurement of static deformation moduli in dynamic tests”, Panel Discussion on Deformation of soils and displacements of structures, Panel Discussion, Proc. of the 10th European Conf. on S.M.F.E., Florence, 4, 1219-1226.

Tatsuoka,F and Kim,Y.-S. (1995): Deformation of shear zone in sedimentary soft rock observed in triaxial compression, Localisation and Bifurcation Theory for Soils and Rocks (Chambon et al., eds.), Balkema, 181-187.

Tatsuoka,F. and Kohata,Y. (1995), “Stiffness of hard soils and soft rocks in engineering applications”, Keynote Lecture, Proc. of Int. Symposium Pre-Failure Deformation of Geomaterials (Shibuya et al., eds.), Balkema, 2, 947-1063.

Tatsuoka,F., Lo Presti,D.C.F. and Kohata,Y. (1995a), “Deformation characteristics of soils and soft rocks under monotonic and cyclic loads and their relationships”, SOA Report, Proc. of the Third Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St Louis (Prakash eds.), 2, 851-879.

Tatsuoka,F., Kohata,Y., Ochi,K. and Tsubouchi,T. (1995b), “Stiffness of soft rocks in Tokyo metropolitan area - from laboratory tests to full-scale behaviour”, Keynote Lecture, Proc. Int. Workshop on Rock Foundation of Large-Scale Structures, Tokyo, Balkema, 3-17.

Tatsuoka,F., Kohata,Y., Tsubouchi,T., Murata,K., Ochi,K. and Wang,L. (1995c), “Sample disturbance in rotary core tube sampling of softrock”, Conf. on Advances in Site Investigation Practice, Institution of Civil Engineers, London, 281-292.

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Tatsuoka,F., Kohata,Y., Tsubouchi,T. and Ochi,K. (1995d), “Stiffness of sedimentary soft rocks evaluated by triaxial compression tests”, Proc. of the 8th Int. Congress on Rock Mechanics, Tokyo, 3, 1201-1204.

Tatsuoka,F., Ochi,K., Tsubouchi,T., Kohata,Y. and Wang,L. (1997a): Sagamihara experimental underground excavations in sedimentary softrock Geotech. Eng., Proc. ICE, 125, Oct., 206-223.

Tatsuoka,F., Uchida,K., Imai,K., Ouchi.T. and Kohata,Y. (1997b), “Properties of cement-treated soils in Trans-Tokyo Bay Highway project”, Ground Improvement, Thomas Telford, 1-1, 37-58.

Tatsuoka,F., Goto,S., Tanaka,T., Tani,K., and Kimura,Y. (1997c), “Particle size effects on particle size effects on bearing capacity of footing on granular material”, Proc. Int. Conf. on Deformation and Progressive Failure in Geomechanics, IS Nagoya ’97 (Asaoka, Adachi and Oka eds.), Pergamon Press, 133-138.

Tatsuoka,F., Koseki,J., Tateyama,M., Munaf,Y. and Horii,N. (1998), “Seismic Stability Against High Seismic Loads of Geosynthetic-Reinforced Soil Retaining Structures”, Keynote Lecture, Proc. 6th Int. Conf. on Geosynthetics, Atlanta, 1, 103-142.

Tatsuoka,F., Jardine,R.J., Lo Presti,D., Di Benedetto,H. and Kodaka,T. (1999a), “Characterising the Pre-Failure Deformation Properties of Geomaterials”, Theme Lecture for the Plenary Session No.1, Proc. of XIV IC on SMFE, Hamburg, September 1997, 4, 2129-2164.

Tatsuoka,F., Modoni,G., Jiang,G.L., Anh Dan,L.Q., Flora,A., Matsushita,M., and Koseki,J. (1999b): Stress-Strain Behaviour at Small Strains of Unbound Granular Materials and its Laboratory Tests, Keynote Lecture, Proc. of Workshop on Modelling and Advanced testing for Unbound Granular Materials, January 21 and 22, 1999, Lisboa (Correia eds.), Balkema, 17-61.

Tatsuoka,F., Correia,A.G., Ishihara,M. and Uchimura,T. (1999c): Non-linear Resilient Behaviour of Unbound Granular Materials Predicted by the Cross-Anisotropic Hypo-Quasi-Elasticity Model, Proc. of Workshop on Modelling and Advanced testing for Unbound Granular Materials, January 21 and 22, 1999, Lisboa (Correia eds.), Balkema, 197-204.

Tatsuoka,F., Santucci de Magistris,F. and Momoya,M. and Maruyama,N. (1999d): Isotach behaviour of geomaterials and its modelling, Proc. Second Int. Conf. on Pre-Failure Deformation Characteristics of Geomaterials, IS Torino ’99 (Jamiolkowski et al., eds.), Balkema, 1, 491-499.

Tatsuoka,F., Santucci de Magistris,F., Hayano,K., Momoya,Y. and Koseki,J. (2000): “Some new aspects of time effects on the stress-strain behaviour of stiff geomaterials”, Keynote Lecture, The Geotechnics of Hard Soils – Soft Rocks, Proc. of Second Int. Conf. on Hard Soils and Soft Rocks, Napoli, 1998 (Evamgelista and Picarelli eds.), Balkema, 2, 1285-1371.

Tatsuoka,F., Uchimura,T., Hayano,K., Di Benedetto,H., Koseki,J. and Siddiquee,M.S.A. (2001a); Time-dependent deformation characteristics of stiff geomaterials in engineering practice, the Theme Lecture, Proc. of the Second International Conference on Pre-failure Deformation Characteristics of Geomaterials, Torino, 1999, Balkema (Jamiolkowski et al., eds.), 2 (to appear).

Tatsuoka,F., Ishihara,M., Di Benedetto,H. and Kuwano,R., (2001b): Time-dependent deformation characteristics of geomaterials and their simulation, Soils and Foundations (submitted).

Tatsuoka,F., Masuda,T. and Siddiquee,M.S.A. (2001c): Modelling the stress-strain behaviour of sand in cyclic plane strain loading, Soils and Foundations (submitted).

Teachavorasinskun,S., Shibuya,S., Tatsuoka,F., Kato,H. and Horii,N. (1991a), “Stiffness and damping of sands in torsion shear”, Proc. Second Int. Conf. on Recent Advances in Geotech. Earthquake Engnrg. and Soil Dynamics, March, St. Louis, I, 103-110.

Teachavorasinskun,S., Shibuya,S. and Tatsuoka,F. (1991b), “Stiffness of sands in monotonic and cyclic torsional simple shear”, Proc. ASCE Geotechnical Engineering Congress, Boulder, Geotechnical Special Publication, Vol.27, 863-878.

Uchimura,T., Tatsuoka,F., Sato,T., Tateyama,M. and Tamura,Y. (1996), “Performance of preloaded and prestressed geosynthetic-reinforced soil”, Proc. Int. Symposium Earth Reinforcement, Fukuoka (Ochiai et al., eds.), Balkema, 1, 537-542.

Woods,R.D. (1991), “Field and laboratory determination of soil properties at low and high strains, Proc. Second. Int. Conf. On Recent Advances in Geotechnical Earthquake Engineering and Soil Dymanics, St. Lous, MO, 2, 1727-1741.

Yasin,S.J.M., Umetsu,K., Tatsuoka,F., Arthur,J.R.F. and Dunstan,T. (1999a), “Plane strain strength and deformation of sands affected by batch variations in two different types of apparatus”, Geotechnical Testing Journal, ASTM, 22-1, 80-1000.

Yasin,S.J.M. and Tatsuoka,F. (1999b): Stress history-dependency of sand deformation in plane strain, Proc. Second Int. Conf. on Pre-Failure Deformation Characteristics of Geomaterials, IS Torino ’99 (Jamiolkowski et al., eds.), Balkema, 1, 703-711.

Yasin,S.J.M. and Tatsuoka,F. (2000): Stress history-dependent deformation characteristics of dense sand in plane strain; Soils and Foundations, 40-2, 77-98.

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Yoshida,T., Tatsuoka,F., Siddiquee,M.S.A. and Kamegai,Y. (1995): Shear banding in sands observed in plane strain compression, Localisation and Bifurcation Theory for Soils and Rocks (Chambon et al., eds.), Balkema, 165-179.

Yoshida,T. and Tatsuoka,F. (1997): Deformation property of shear band in sand subjected to plane strain compression and its relation to particle characteristics, Proc. 14th ICSMFE, Hamburg, 1, 237-240.