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Stress path effect on the static behavior of Monterey no. 0/30 sand

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Title:
Stress path effect on the static behavior of Monterey no. 0/30 sand
Creator:
Goldstein, Barry
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English
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217 leaves : illustrations ; 28 cm

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Subjects / Keywords:
Sand -- Testing ( lcsh )
Soil mechanics ( lcsh )
Sand -- Testing ( fast )
Soil mechanics ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 195-198).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Civil Engineering.
Statement of Responsibility:
by Barry R. Goldstein.

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Source Institution:
|University of Colorado Denver
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Auraria Library
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Resource Identifier:
19782988 ( OCLC )
ocm19782988
Classification:
LD1190.E53 1988m .G64 ( lcc )

Full Text
STRESS PATH EFFECT ON THE STATIC BEHAVIOR
OF MONTEREY NO. 0/30 SAND
by
Barry R. Goldstein
B.S., Colorado State University, 1983
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Civil Engineering
1988


This thesis for the Master of Science degree by
Barry R. Goldstein
has been approved for the
Department of
Civil Engineering
by
Z-
Date


Goldstein, Barry R. (M.S., Civil Engineering)
Stress Path Effect on the Static Behavior of Monterey No. 0/30 Sand.
Thesis directed by Professor Nien-Yin Chang
Two of the most important characteristics of soil behavior
are stress-path dependency and principal stress rotation effect.
Actual field conditions often result in soil elements following
different stress paths. When a structure is subjected to axial and
lateral loading, the resulting transfer of load will cause the soil
elements to undergo different stress paths. The theory of isotropic
linear elasticity is sometimes applied in the analysis of soil
mechanics problems. It is well known, however, that soils do not
behave as an isotropic linear elastic material. Thus, the influence
of different stress paths on the behavior of soil should be of
importance.
The rotation of principal stress directions can occur under
in-situ conditions, such as during slope movements, excavations and
cyclic loading. Soil behavior is significantly affected by the
rotation of the principal stress direction. Behavior which is
affected includes the maximum deviatoric stress, internal angle of
friction, pore pressure generation-volume change characteristics and
plastic deformation.
In this study, Monterey No. 0/30 sand was used and nineteen
static triaxial tests on seven different stress paths were con-
ducted. The test results were used to analyze the behavior of
Monterey No. 0/30 sand on different stress paths in the Rendulic
plane. Also, using the results from the isotropic compression test,
conventional triaxial compression test, and the reduced triaxial


iv
compression test,
were calculated,
behavior of soils
the soil parameter for Lade's elasto-plastic model
These parameters can be Used to predict the
under other stress paths on the Rendulic plane can
be simulated.


ACKNOWLEDGEMENTS
This study was conducted in the Department of Civil En-
gineering at the University of Colorado at Denver. The author
wishes to express his sincerest appreciation to his advisor,
Professor Nien-Yin Chang, for his guidance and support.
Gratitude is also extended to Dr. Tzong-H. Wu for his
insightful discussions. The author is grateful to his fellow
graduate students, Mr. Hsing-Cheng Liu, Dr. Jing Wen Chen, Mr. Z. X.
You and Ms. Elaheh Kheirkhahi for their help during this study.
I would also like to thank Mr. Joseph Cesare for his support
over the last several years. Finally, I would like to express my
gratitude to Carol McAmis for her patience and assistance in the
typing and Ms. Pauline LeBlanc for finalizing this thesis. Finally,
I would like to thank my parents for all their continued support and
encouragment.


LIST OF TABLES
Table
11.1 Stress Paths on the Rendulic Plane
(after Das, 1983)........................................16
11.2 The Parameters M and N in Equation 11.34
(after Naylor, 1978).....................................42
III.3 Parameters for Lade's Model..............................63
IV. 1 Proposed Testing Program.................................65
IV. 2 Stress Paths Conducted for Study.........................66
IV. 3 Physical Properties for Monterey No. 0/30 Sand...........69
V.l Sample b-Parameters Prior to Testing.....................88
VII. 1 Initial Conditions of Triaxial Samples..................100
VII.2 Elastic Moduli of Monterey No. 0/30 Sand
from Hydrostatis Compression Tests Dr = 43.2) 103
VII.3 Initial Young's Moduli determine from CTC Tests .... 113
VII.4 Initial Young's Moduli determined from TC Tests .... 121
VII.5 Initial Young's Moduli determined from RTC Tests. . . 130
VII.6 Initial Young's Moduli determined from RTE.C Tests. . 138
VII.7 Initial Young's Moduli determined from TC Tests .... 146
VII.8 Initial Young's Moduli determined from CTE Tests. . . 154
VII.9 Initial Young's Moduli determined from TC Tests .... 163
VII.10 Initial Young's Moduli determined from TC Tests. . 164
VII.11 Volume Change Characteristics Drained Triaxial Tests 166


vii
Table
VIII.1 Parameters for Lade's Model Monterey No. 0/30
Sand CTC Stres Path.....................................185
VIII.2 Parameters for Lade's Model Monterey No. 0/30
Sand RTE Stress Path....................................189


LIST OF FIGURES
Figure
1.1 Orientation of Stress Directions at Failure.............3
11.1 Cambridge Stress Field (Roscoe, et al., 1958)
Illustrating Principal Stresses.........................7
11.2 Mohr Stress Circle Defining Stress Parameters
t' and s' (after Atkinson and Bransby, 1978)............9
II.3 Stress Paths for (a) Drained and (b) Undrained
Loading Tests (after Atkinson and Bransby, 1978) . .12
11.4 Stress Field for Axial Symmetry (after Henkel,
1960).....................................................13
11.5 Rendulic Diagram (after Das, 1983)....................... 15
11.6 Results of Drained Triaxial Tests on (a) Dense
Sand and (b) Loose Sand (after Bishop and
Henkel, 1962).............................................18
11.7 Results from Undrained Triaxial Tests and (a)
Medium Dense and (b) Loose Sand (after Bishop
and Henkel, 1962).........................................19
11.8 Deviatoric Stress-Strain Curve for Triaxial
Test Showing Alternate Shear Moduli (after
Naylor, 1978)..............................'.............30
11.9 Flow Rule in Three-Component Space (after
Naylor, 1978).............................................37
11.10 Yield Surface in Principal Stress Space
(after Naylor, 1978)......................................38
11.11 Yield Surfaces; 1 = Mohr-Coulomb,
2 = Extended VonMises, 3 = Compromise Cone,
4 = Axial Extension Cone, and
5 = Drucker-Prager (after Naylor, 1978).................40
11.12 General Static Failure Criteria (after
Horita, 1983)
45


IX
Figure
11.13 General Static Failure Criteria (after
Horita, 1983)............................................46
11.14 (a) Classical Failure Criteria in Principal
Stress Space and (b) Cross-Sections of
Mohr-Coulomb Failure Criterion Shown for
Three Different Friction Angles (after
Bishop, 1966).............................................47
111.1 Strain Components in Lade's Model (after
Lade, 1977)...............................................53
111.2 Ultimate Strength and Yield Surface in
Lade's Model (after Lade, 1977)...........................55
111.3 Yield Surface and Plastic Potential Surface
in Triaxial Plane of Lade's Model.........................59
IV.1 Grain Size Distribution of Monterey No.0/30
Sand (after Muzzey, 1976).................................18
IV.2 Material Testing System Equipment Used in
Study.....................................................72
V.l Plot of Internal Pressure Against Volumetric
Polyethylene Tube Expansion...............................90
VII.1 Isotropic Consolidatiron Triaxial Test Results .... 102
VII.2 CTC Stress Paths in p':q' Stress Space .................. 106
VII.3 Mohr Stress Circles at Failure for CTC
Triaxial Tests ......................................... 107
VII.4 CTC Triaxial Test Data for 30 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea ............................... 108
VII.5 CTC Triaxial Tests Data for 60 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea ............................... 109
VII.6 CTC Triaxial Tests Data for 90 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea ............................... 110
VII.7 Stress-Strain Curves for CTC Triaxial Tests
Conducted at Different Effective Confining
Stress Levels
111


X
Figure
VII.8 TC Stress Paths in p':q' Stress Space...................114
VII.9 Mohr Stress Circules at Failure for TC
Triaxial Tests ....................................... 116
VII.10 TC Triaxial Test Data for 30 psi Effective
Confining Pressure (a) q Against Ea and
(b) Volume Change Against Ea...........................117
VII.11 TC Triaxial Test Data for 60 psi Effective
Confining Pressure (a) q Against Ea and
(b) Volume Change Against Ea...........................118
VII.12 TC Triaxial Test Data for 90 psi Effective
Confining Pressure (a) q Against Ea and
(b) Volume Change Against Ea...........................119
VII.13 Stress-Strain Curves for TC Triaxial Tests
Conducted at Different Effective
Confining Stress Levels................................120
VII.14 RTC Stress Paths in p':q' Stress Space ................ 123
VII.15 Mohr Stress Circles at Failure for RTC
Trixial Tests..........................................124
VII.16 RTC Triaxial Test Data for 30 psi Effective
Confining Pressure (a) q Against Ea and
(b) Volume Change Against Ea..................... 126
VII.17 RTC Triaxial Test Data for 60 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea ....................... 127
VII.18 RTC Triaxial Test Data for 90 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea...............................128
VII.19 Stress-Strain Curves for RTC Triaxial Tests
Conducted at Different Effective Stress
Levels.................................................129
VII.20 RTE Stress Paths in p':q' Stress Space ................ 132
VII.21 Mohr Stress Circles at Failure for RTE
Triaxial Tests ....................................... 133
VII. 22 RTE Triaxial Test Data for 30 psi Effective
Confining Pressure (a) q Against Ea and
(b) Volume Change Against Ea ....................
134


XI
Figure
VII.23 RTE Triaxial Test Data for 60 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea..............................135
VII.24 RTE Triaxial Test Data for 90 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea ............................. 136
VII.25 Stress-Strain Curves for RTE Triaxial Tests
Conducted at Different Effeective Confining
Stress Levels..................................................137
VII.26 TE Stress Paths in p':q' Stress Space..................140
VII.27 Mohr Stress Circles at Failure for TE
Triaxial Tests ....................................... 141
VII.28 TE Triaxial Test Data for 30 psi Effective
Confining Pressure (a) q Against Ea and
(b) Volume Change Against Ea...........................142
VII.29 TE Triaxial Test Data for 60 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Sgainst Ea ............................. 143
VII.30 TE Triaxial Test Data for 90 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea...............................144
VII.31 Stress-Strain Curves for TE Triaxial Tests
Conducted at Different Effective Confining
Stress Levels..................................................145
VII.32 CTE Stress Paths in p':q' Stress Space ................ 148
VII.33 Mohr Stress Circles at Failure for CTE
Triaxial Tests ....................................... 149
VII.34 CTE Triaxial Test Data for 30 psi Effective
Confining Pressure (a) q Against Ea and
(b) Volume Change Against Ea...........................150
VII.35 CTE Triaxial Test Data for 60 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea ............................. 151
VII.36 CTE Triaxial Test Data for 90 psi Effective
Confining Pressure (a) q Against Ea and (b)
Volume Change Against Ea..........,...................152


Xll
Figure
VII.37 Stress-Strain Curves for CTE Triaxial Tests
Conducted at Different Effective Confining
Stress Levels...................................................153
VII.38 Stress Paths for 30 psi Effective Confining
Pressure................................................157
VII.39 Stress Paths for 60 psi Effective Confining
Pressure................................................158
VII.40 Stress Paths for 90 psi Effective Confining
Pressure............................................... 159
VII. 41 Failure Envelope Defined in p':q' Space...............160
VIII. 1 Plot of Elastic Modulus Against Confining
Pressure to CTC Stress Path.............................170
VIII.2 Plot of Volumetric Strain Against Effective
Confining Pressure From an Isotropic
Consolidation Triaxial Test.............................172
VIII.3 Plot of Plastic Collapse Work, Wc, and fc
Parameter...............................................174
VIII.4 Determination of the Parameters and m
for the CTC Stress Path.................................175
VIII. 5 Plot of tj2 Aagainst fp of Various a'3 Levels
for CTC Stress Path.....................................178
VIII. 6 Plot of Intercept Against a3 for CTC Stress
Path....................................................179
VIII.7 Plot of Plastic Work Against fp for Various
cr3 Levels for CTC Stress Path..........................182
VIII. 8 Plot of q against a3 for determining a and
/3 parameters for the CTC Stress Path...................183
VIII.9 Plot of Wppeak Against a3 for Determining
1 and p Parameters for the CTC Stress Path..............184


TABLE OF CONTENTS
ABSTRACT.........................................................iii
ACKNOWLEDGEMENTS...................................................v
LIST OF TABLES............................................... .vi
LIST OF FIGURES.................................................viii
CHAPTER
I. INTRODUCTION ............................................. 1
1.1. Purpose............................................1
1.2. Scope..............................................4
II. BACKGROUND AND LITERATURE REVIEW ......................... 6
II.1. Stress paths on the Rendulic plane ............... 6
11.2 Static Behavior of Granular Materials.............14
11.3 Factors Affecting Triaxial Testing................20
11.3. a Introduction.............................20
11.3. b Inherent Soil Properties.................21
11.3. c Initial Void Ratio.......................23
11.3. d Confining Pressure.......................24
11.3. e Loading Conditions.......................25
II. 3. f Rate of Loading..........................25
11.4 External Effects on the Triaxial Test.............25
II.4.a Introduction
25


xiv
CHAPTER
II. 4.b Rubber Membrane...........................26
II. 4. c Piston Friction..........................27
II. 5 Stress-Strain Laws for Soil.......................27
11.5. a Introduction.............................27
II. 5.b Linear Elastic Laws......................28
II. 5. c Variable Elastic Laws....................31
11.5. C.1 Hyperbolic Model................31
11.5. C.2 Differential Models.............31
11.5. d Elastic-Plastic Laws.....................33
11.5. d.l Yield Surface...................34
11.5. d.2 Strain Hardening................35
II. 5 d. 3 Flow Rule....................35
11.5. d.4 Specific Forms of Yield
Surface.........................36
II. 6 Failure Criteria..................................43
11.6. a Classical Failure Criteria................44
11.6. b Three-dimensional Failure Criteria . .48
III. LADE'S ELASTO-PLASTIC MODEL..............................50
III. l Introduction....................................50
111.2 Strain Components...............................51
III. 2. a Elastic Strains........................52
111.2. b Plastic Collapse Strains................54
111.2. C Plastic Expansive Strain................57
111.3 Summary of Stress-Strain Parameters.............62


XV
CHAPTER
IV. MATERIALS AND TESTING APPARATUS............................64
IV. 1 Test Program......................................64
IV.2 Monterey No. 0/30 Sand..............................64
IV. 3 Test Equipment....................................67
IV.3.a Triaxial Test...............................67
IV. 3.b MTS Loading Machine.......................71
IV.4.c Data Acquisition Systems....................77
V. SAMPLE PREPARATION ....................................78
V. l Relative Density Control..........................78
V.2 Sample Preparation................................79
V.3 Flusing...........................................84
V.4 Connecting Pore Pressure Transducer...............85
V.5 Saturation........................................86
V.6 Determining B-Parameter...........................86
V. 7 Consolidation.....................................87
VI. TEST PROCEDURES............................................92
VI. 1 MTS Operation.....................................92
VI.2 Connecting Loading Ram to Top Cap...................93
VI.3 Conducting Tests with a Constant Radial Stress .94
VI.4 Conducting Tests with Changing Radial Stress . .95
VI. 5 Conducting High Pressure Tests....................98
VII. RESULTS AND DISCUSSION OF LABORATORY RESULTS ............ 100
VII. 1 Introduction...................................100
VII.2 Hydrostatic Compression (HC) Test................100
VII.3 Conventional Triaxial Compression
(CTC) Tests .
104


xvi
CHAPTER
VII.4 Triaxial Compression (TC) Tests ............. 112
VII.5 Reduced Triaxial Compression (RTC) Tests. . 122
VII. 6 Reduced Triaxial Extension (RTE) Tests...........125
VII.7 Triaxial Extension (TE) Tests .............. 131
VII.8 Conventional Triaxial Extension (CTE) Tests . 147
VII. 9 Summary..........................155
VIII. STRESS PATH EFFECT ON LADE'S PARAMETERS....................168
VIII. 1 Introduction ...................... 168
VIII. 2 Soil Parameter Calibration ............... 168
VIII.2.a Calibration of the CTC stress
path parameters......................169
VIII.2.b Calibration of the RTE Stress
Stress Path Parameters ............. 181
IX. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
FOR FURTHER STUDY.........................................190
IX. 1 Summary........................................190
IX. 2 Conclusions......................................193
IX.3 Recommendation for Further Study ............... 194
BIBLIOGRAPHY.....................................................195
APPENDIX.........................................................199
A. TEST DATA
200


CHAPTER I
INTRODUCTION
I.1 Purpose
Soils, in a geotechnical sense, can be regarded as engineer-
ing materials. Their physical characteristics can be determined by
experiment, and the application of methods of analysis enables these
properties to be used to predict their likely behavior under defined
working conditions. Unlike other engineering materials such as
metals and concrete, over which control can be maintained during
manufacture, soils are naturally occurring materials which most
often must be used in their natural condition. Even when some type
of processing is possible, soil can be modified only to a limited
extent and by relatively simple procedures.
The main problems associated with soil mechanics are
generally divided into two catagories: deformation and stability.
The former are considered to be governed by stress-strain and
strain-time relationships and the latter by the shear-strength
properties. The extent and accuracy to which an analysis to
determine stability and/or deformation can be achieved is often
limited by the practical problem of sampling and testing to deter-
mine inherent characteristics and that of understanding soil
behavior. In general, much emphasis has been placed on refinements


2
in sampling techniques and testing procedures. Even with these
refinements, accurate solutions can be obtained only if the soil
strata are for practical purposes, homogeneous and continuous in a
spacial aspect. To obtain more certain engineering characteristics
based on quantitative measurements of relevant soil properties to
predict and/or determine the subsequent performance of the actual
structure, further understanding of basic soil behavior under in-
situ conditions is necessary.
Two of the most important characteristics of soil behavior
are stress-path dependency and principal stress rotation effect.
Actual in-situ conditions often subject soil elements to different
stress paths. When a structure is subjected to axial and lateral
loading, the resulting transfer of load will cause the soil elements
to undergo different stress paths. The theory of isotropic linear
elasticity is sometimes applied in the analysis of soil mechanics
problems. It is well known, however, that soils do not behave as an
isotropic linear elastic material. Thus, the influence of different
stress paths on the behavior of soil should be of importance.
The rotation of principal stress directions can occur under
in-situ conditions, such as during slope movements, excavations and
cyclic loading (see Figure 1.1). Soil behavior is significantly
affected by the rotation of the principal stress direction.
Behavior which is affected includes the maximum deviatoric stress,
internal angle of friction, pore pressure generation volume change
characteristics and plastic deformation.


Fig. 1.1 Orientation of Stress Directions at Failure


4
In summary, soil behavior is stress-path dependent and is
significantly affected by the rotation of the principal stress
direction.
The purpose of this study is four-fold.
1. To determine the stress-strain relationships and volume
change characteristics of Monterey No. 0/30 sand on different stress
paths and at varying effective stress levels on the Rendulic plane.
2. To determine the influence of stress paths on the
characteristics of soil behvior and soil parameters.
3. To develop the parameters necessary to apply Lade's
elasto-plastic model. To perform triaxial tests on different stress
paths such that these tests can be used as actual comparison with
the predicted behavior generated by Lade's constitutive model.
4. To perform triaxial tests so that a comparison of the
behavior of a solid cylindrical sample tested in a standard triaxial
cell and that of a hollow cylindrical sample tested in a large
diameter triaxial cell, developed by the University of Colorado at
Denver, can be made.
I.2 Scone
The scope of this thesis is to evaluate the stress-strain
characteristics of Monterey No. 0/30 sand on varying stress paths on
the Rendulic plane. Triaxial tests on the following stress paths
were conducted:
CTC Conventional Triaxial Compression
RTC Reduced Triaxial Compression
TC Triaxial Compression


5
CTE Conventional Triaxial Extension
RTE Reduced Triaxial Extension
TE Triaxial Extension
IC Isotropic Compression
Testing was conducted at three different effective stress
levels; 30 psi, 60 psi and 90 psi. Sample density used in the
testing program was the same as that used in the testing program
with the hollow cylinder triaxial cell conducted at the University
of Colorado at Denver. This would allow for more complete com-
parison of the soil behavior under two triaxial systems.
Upon completion of the laboratory testing, stress-strain,
volume change-strain and p:q plots were developed for each stress
path. Using the HC, CTC and RTE triaxial test data, the soil
parameters for Lade's elasto-plastic constitutive model were
evaluated.
Results from this study will be used to further understand
soil behavior on the Rendulic plane, and soil behavior as it relates
to stress path dependency. Finally, this study will also assist in
the continued development of the University of Colorado at Denver's
hollow cylinder triaxial cell.


CHAPTER II
BACKGROUND AND LITERATURE REVIEW
II.1 Stress Paths on the Rendulic Plane
In a general cubical element of material, there are six
independent stresses; three shear stresses arid three normal stres-
ses. If the element is rotated such that the faces become principal
planes, the shear stresses of the face become zero and the normal
stresses become principal stresses. For a soil, the state of stress
is completely defined by three principal total stresses (and their
directions) and the pore water pressure. The three principal
effective stresses may be calculated based on Terzaghi's effective
stress equation.
To define an effective (or total) stress space, a three-
dimensional plot with a1 Figure II.1. The instantaneous state of effective stress in an
element may be plotted as a point in the effective stress space and
the line joining all points of instantaneous states of stress is
defined as the effective stress path. The points represent only the
magnitude of the principal stresses. The stress path does not
indicate direction nor does it indicate any rotation of the prin-
cipal planes.


7
Fig. II.1 Cambridge Stress Field (Roscoe, et al, 1958)
Illustrating Principal Stresses


8
It may be at times, convenient to plot effective stress
paths in the two-dimensional effective stress plane o'1:o3 ignoring
the intermediate principal stress, az .
The instantaneous two-dimensional state of stress may be
represented by a Mohr's circle of stress, as shown in Figure II.2.
From the geometry of the Mohr circle, and noting that rxz = tzx;
t' = 1/2[(ctx-ctz)2+4txz]1/2
s' = 1/2(ctx-ctz)
or, in terms of principal effective stresses;
t' = 1/2 (ct-l -a3 )
s = 1/2 (o^+CTg)
II.1
II.2
II. 3
II.4
The parameters t' and s', their magnitudes for a given state
of stress are independent of the choice of reference axes, and are
known as stress invariants.
Octahedral normal stress, uQct> an<3 t^ie octahedral shear
stress, roct-, are invariants and defined as;
^oct 1/3 (x y *" CTz)
II.5


T
Fig. II.2 Mohr Stress Circle Defining Stress
Parameters t' and s(after Atkinson
and Bransby, 1978)


10
('oct)2 = V9 [ (o^-ap2 + (ffy-a^)2 +
(&2 ax)2 + 6(r2xy + r2yz + r2xy)] II.6
or in terms of principal stress:
CToct = 1/1(CTi + ^2 + 117
rOCt = 1/1 [ (CT1 "a2 )2 + (*2"a3 )2 + (^-Oi)2] 11 8
For the case where ag = cr^ and where a = 0 (a = angle of
rotation);
a^t = l/3(a{+2^) II. 9
r^ct = II.10
To avoid the recurring 1/372 term, the following in-
variants are defined, where a'z = o'z\
For a general three-dimensional state of stress, q* and p*
become;
p' = l/3(a[+2a^) = cr£ct II. 11
q' = (ol+2ai) = 3 1/72 r'ct 11.12
For a general three-dimensional state of stress, q' and p'
become:


11
p' = 1/3(a{+a^+a^) 11.13
q' = 1/J2[ (cr^ -CTg )2 + (cr^ -erg )2 + (erg -cr^ )2 ]1 /2 11.14
and the third invariant a, will be non-zero.
Roscoe, Schofield and Wroth (1958) at the University of
Cambridge, England, developed the use of the mean of the three prin-
cipal effective stresses (cr^ a^, 03) instead of the mean major and
minor principal stresses. This method of plotting, which will be
used throughout this text, is known as the Cambridge stress path
plot.
To distinguish between triaxial compression and extension,
since in both cases q is positive (by definition , the
parameters of stress (and strain) can be redefined as;
q' = 11.15
p' = l/3+2a;) 11.16
where cr' is the effective axial stress and oL is for effective
a r
radial stress. The stress paths shown in Figure II.3 are plotted
using p,p':q,q' space.
A method of representing the stress path for triaxial tests
is the plot suggested by Rendulic (1937) and later by Henkel (1960).
The Rendulic plot is a diagram of the results of triaxial tests for
the condition of axial symmetry, as shown in Figure II.4. The equal


h'b
12
P,P'
P (b)
Fig. II.3 Stress Paths for (a) Drained and
(b) Undrained Loading Tests
(after Atkinson and Bransby, 1978)


13
Fig. II.4 Stress Field for Axial Symmetry
(after Henkel, 1960)


14
stresses for a2 and ct3 on the horizontal axes are denoted by or
(radial stress), and the vertical (axial) stress by aa. The
combined representation of az and ct3 in the stress plane is equal to
72 against ax (horizontal axis).
On the Rendulic diagram (plane), as shown in Figure II.5,
the line Od represents the isotropic stress line. The direction
cosines of this line are 1/73, 1/73 and 1/73. Line Od will have a
slope of 1 vertical to Jl horizontal. The trace of the octahedral
plane (cx1 + az + a3 = constant) occurs at right angles to line Od.
In a triaxial test, if a soil sample is hydrostatically
consolidated (aa = ct-j-) it may be represented by the point 1 on line
Od. If the sample is then subjected to additional stresses, the
resulting stress paths can be traced, see Table II.1.
II.2 Static Behavior of Granular Materials
Ideally, the triaxial test should permit independent control
of the three principal stresses so that generalized states of stress
can be examined. The relatively high compressibility of the soil
skeleton and the magnitude of the shear strains required to cause
failure, lead to mechanical difficulties which make independent
control very complicated and such tests difficult to perform.
Therefore, the most common type of triaxial test conducted, is on a
cylindrical sample in which only two of the three principals stress
are independent.
Generally, the application of the all-around (hydrostatic)
pressure and of the deviator stress, form two separate stages of the


15
c
Fig. II.5 Rendulic Diagram
(after Das, 1983)


16
Table II. 1 Stress Paths on the Rendulic Plane (after Das, 1983)
Coordinates (Refer to Figure II.5) Represents
1-2 Drained axial compression test where a'a is increased and a£ is kept constant
1-3 Drained axial compression test where a^ is kept constant and Oy- is reduced
1-4 Drained axial compression test where the mean principal stress (J = Oy+o^+o?,) is kept constant
1-5 Drained axial extension test where o is kept constant and a'z is reduced
1-6 Drained axial extension test where o'a is kept constant and is increased
1-7 Drained axial extension test with J = a[+a2+cf^ constant (J = a'a+2a^ constant)
f1 1 00 Undrained compression test
1-9
Undrained extension test


17
triaxial test. These tests are therefore classified according to
the conditions of drainage allowed during each stage.
Drainage allowed during
application of stress:
Test Tvoe Hvdrostatic Deviatoric
undrained no no
consolidated-undrained yes no
drained yes yes
Typical triaxial data obtained from standard drained and
undrained compression tests on loose and dense sand samples are
shown in Figures II.6 and II.7.
Differences in behavior between the four samples are as
follows. The drained test on the loose sample (Figure II.6 (b))
results in a q':Ea curve which reaches a constant maximum after
large axial strains occur, while the sample compresses (Ev),
substantially as axial strain increases. Toward the completion of
the test, the sample appears to reach an ultimate state, resulting
in negligible changes in stresses or in volume for continuing shear
distortion (Atkinson and Bransby, 1978).
The dense sample (Figure II.6), in contrast, exhibits a
defined peak on the q':plot and q' decreases, from thereafter.
Initially, the sample contracts slightly, but then dilates waxing
until the end of the test. No ultimate level is reached. The
sample continues to dilating as q' is decreasing.


(per cent)
18

0
ea (per cent)
(a)
ea (per cent)
(b)
Dense sand
Loose sand
Fig. II.6 Results of Drained Triaxial Tests on
(a) Dense Sand and (b) Loose Sand
(after Bishop and Henkel, 1962)


19
(a) Medium dense sand
3
<
(b) Loose sand
Fig. II.7
Results from Undrained Triaxial Tests on
(a) Medium Dense and (b) Loose Sand
(after Bishop and Henkel, 1962)


20
The shapes of the q':Ea curves for the undrained tests
(Figure II.7) are similar in trend, though the values of q' at
failure are very different. The pore pressure changes at failure in
the two tests are quite different. The loose sample exhibits a
positive pore pressure at failure, while the pore pressure is
largeand negative for the medium dense sample. The major cause of
the large difference in the observed shear strength of the two
samples is due to the difference in pore water pressure at failure.
The effective radial stress at failure in the dense sample is
substantially larger than that for the loose sample (Atkinson and
Bransby, 1978).
The similarity between the E^'.Ea curves from the drained
tests and the Au:Ea curves from the undrained tests, for both dense
and loose samples can be noted. The loose sample contracts in the
drained test and generates positive pore pressure in the undrained
test, while the dense sample dilates and generates negative pore
pressure.
II.3 Factors Affecting Triaxial Testing
II.3.a Introduction
Factors which can influence the strength of cohesionless
soils include those that affect the shear strength of a given soil,
and those that cause the strength to differ (even under identical
confining pressure and void ratio). The most important factors
affecting shear strength are void ratio and confining pressure.
Factors which may cause variation in shear strength include particu-
late size, shape and texture, gradation and mineral composition.


21
Additional factors which may influence the strength and behavior of
sands tested in the laboratory under triaxial conditions are a
function of sample preparation and testing appartus. These factors
include piston and end plate friction, use of filter paper, rubber
membrane and end cap effects.
II.3.b Inherent Soil Properties
For a granular soil, in a given state of compaction and
particle orientation, the friction angle should theoretically be a
constant. It may be affected to a varying extent by: the magnitude
of the intermediate principal stress; particulate shape especially
if they depart significantly from bulky grains; previous stress
history; and by the rate of strain. Variation of the internal
friction angle (') is most affected by the state of compaction,
coarseness of grains, particle shape and roughness of grain sur-
faces, and gradation.
The shear strength of a cohesionless soil is a combination
of strength due to sliding friction, plus dilatancy effects, and
crushing and rearranging effects. The dilatancy effect may be
positive or negative depending on whether the volume increases or
decreases during shear (Lee and Seed, 1967).
Mineral composition affects the friction angle of a granular
soil in two ways. First, it affects the void ratio that is obtained
with a given compactive effort; and second, it affects the friction
angle that is achieved for that void ratio, as a function of sliding
friction.


22
The value of the internal friction angle varies relatively

little, even with differing mineral compositions. These apparent
differences, for a given initial void ratio, result primarily from
different degrees of interlocking (Lambe and Whitman, 1969). This
interlocking and resulting friction angle, is a function of the
materials initial void ratio.
Unless a sand contains mica, it makes little difference
whether the sand is composed primarily of quartz or one of the
feldspars. A micaceous sand will often have a large void ratio,
hence little interlocking, and a low friction angle (Lambe and
Whitman, 1969).
Hennes (1952) concluded from the results of direct shear
tests that, for uniformly graded aggregates, the effect of particle
shape is only of moderate importance; but the particle size has a
strong influence on frictional resistance, especially up to a grain
diameter of about 1/4 inch.
For comparable compactive efforts, a well graded sand has
both a smaller initial void ratio and a larger friction angle. A
wider distribution of particle sizes produces better interlocking
and consequently a higher friction angle. A well graded soil
experiences less breakdown than a uniform soil of the same particle
size, as in a well graded soil, there are many interparticle
contacts and the load per contact is less than in the uniform soil.
Therefore, a decrease in void ratio causes an increase in the angle
of internal friction of a granular material and, at equal void


23
ratios, well graded aggregates have higher strength characteristics
than a uniform material.
It would be expected that angular particles would interlock
more thoroughly than rounded particles, and hence, sands composed of
angular particles would have the larger friction angle. In gravels,
the effect of angularity is less pronounced because of particle
crushing.
For a given value of void ratio, the angle of internal
friction of a granular material appears to increase considerable if
the angularity of the particles is increased by crushing, although
part of this increase may be due to an increase in surface roughness
(Vallerga et al., 1957).
The effect of a varying coefficient of uniformity and its
effect on soil strength were investigated by Koener (1970). Test
results indicated that a change in the coefficient of uniformity had
a negligible effect on the friction angle in quartz predominated
soils, but on other soils in which feldspar and calcite were the
main constituents, an increase in the coefficient of uniformity
resulted in higher friction angles.
II.3.c Initial Void Ratio
The shear resistance of sands is made up of two components;
one whose magnitude is controlled by the undrained internal angle of
friction, and a second component whose magnitude is related to
particulate interlocking. The greater the degree of interlocking,
the greater the overall shear resistance and therefore, the greater
the friction angle.


24
For most of the typical ranges of void ratio three com-
ponents control the strength of a granular material: (1) strength
mobilized by frictional resistance; (2) 'strength developed by energy
required to rearrange and reorient soil particles; and (3) strength
developed by energy required to cause expansion or dilation of the
material (Lee and Seed, 1967).
A decrease in void ratio causes an increase in the angle of
internal friction of a granular material and at equal void ratios,
well graded soils have higher strength than a uniform material.
II.3.d Confining Pressure
The effect of confining pressure on the friction angle of a
soil can be neglected within the pressure ranges of practical
interest (Taylor, 1948). With the development of higher earth dams,
however, the need arose for information on the strength of cohesion-
less soils at very high confining pressures. Triaxial tests
performed on sands and sandstone using confining pressures up to
1,000 psi obtaining data indicating a significant curvature and
progressive flattening of the failure envelope (Bishop, 1966).
Axial and volumetric strains at failure increased as the
confining pressure increased, and the angle of internal friction
decreased as the confining pressure increased (Marachi, 1972).
Samples tested at very low confining pressures illustrate
that the more dilatant deformational behavior during shear at
extremely low pressures produces an increase in the principal stress
ratio of failure as a result of the energy required for expansion
(Ponce and Bell, 1971). Consequently, for loose sands, shear can be


25
associated with dilative volume changes if the normal pressures are
low enough. Thus, the behavior of loose sands at extremely low
pressures is similar to that of dense sands at moderate pressures.
11.3. e Loading Conditions
The effect of differing loading conditions on the friction
angle of a soil using compression and extension tests has been
conducted by Roscoe, et. al. (1962). Results indicate that the
friction angle is the same for both cases. It has been also noted,
however, the friction angle was greater, by several degrees, if
a2 = 1969).
11.3. f Rate of Loading
The friction angle of sand, as measured in triaxial compres-
sion, is essentially unaffected regardless of the loading rate. The
increase in tan resulting from a slower to faster loading rate
change, is at most 10%, and probably is only 1 to 2% (Whitman and
Healy, 1963). It is possible that the effect might be somewhat
greater if the sand is sheared in plane strain or if the confining
pressure is in excess of 100 psi (Lambe and Whitman, 1969) .
II.4 External Effects on the Triaxial Test
11.4. a Introduction
When considered individually, corrections to triaxial test
results for the loads carried by filter paper drains, rubber
membranes, and chamber piston friction, may have negligible values.
Since their effects are cumulative, however, their influences may


26
become quite appreciable, especially when the consolidation pres-
sures and strengths are low (Duncan and Seed, 1967). Failure to
correct for the load-carrying capability of the filter paper drains,
rubber membranes, and piston friction results in an overestimation
of the axial stress in the test specimens at all stages of testing
and influences the values of the consolidation pressure, strength,
strength parameters, pore-pressure parameter A, and effective
principal stress ratio (Duncan and Seed, 1967).
II.4.b Rubber Membrane
The rubber membrane that encloses the specimen in the
triaxial cell contributes a small amount of strength that becomes
significant when testing specimens at very low confining pressures.
Corrections for the added strength depend on two physical charac-
teristics of the membrane; the thickness and the elastic modulus
(Ponce and Bell, 1971).
A minimum end membrane thickness is required for a friction-
less test on sand to prevent effective bearing of particles through
the end membranes and into the cap and base. It has been recognized
that beyond some optimal thickness, a Poisson's effect dominates
causing an artificial reduction in sand strength. Therefore, sand
strength decreases gradually as end membrane thickness increases,
first as end restraint is reduced and then as the Poisson's effect
takes over (Norris, 1981).


27
11.4. c Piston Friction
Appreciable friction can arise as a result of lateral forces
applied to the loading ram against guide bearings. External lateral
forces can be limited by the proper alignment of the triaxial
assembly. Chamber piston friction may be measured using a load cell
to measure the force required to push the piston through the
surrounding guide and seal. .
II.5 Stress-Strain Laws for Soil
11.5. a Introduction
The solution of any load-deformation boundary value problem
requires a known relation between stress and strain of the form:
Act = DAe 11.17
or
Ae = D'1Act 11.18
where Act, Ae, are vectors of the components of stress and strain
increment respectively, and D is a modulus matrix. The increments
may be large (as in linear analysis) or almost infinitely small (as
in incremental plasticity). The finite element formulation for
displacement usually requires the law in the form of equation 11.17.
For certain non-linear applications however, such as the analysis of
elasto-plastic materials using the visco-plastic method, the inverse
form of equation 11.18 is required (Naylor, 1978).
Most of the stress-strain laws described in this chapter may
be defined in terms of either total or effective stress. Effective
stress is usually more appropriate, and in the case of the critical
state model it is essential.


28
II.5.b Linear Elastic Laws
Elastic isotropic stress-strain laws are fully defined by
two independent parameters. These are conventionally taken to be
Young's modulus, E, and Poisson's ratio, fj.. In soils, however,
there are advantages in using the bulk modulus, K, and the shear
modulus, G, as the behavior of soil in the separate modes of volume
change and shear are reasonably well understood (Naylor, 1978) The
moduli K and G may be expressed in terms of E and /z as follows:
K------------- 11.19
3(1 2/i)
2(1 + /0
11.20
Given the effective stress moduli, E' an /z' or K' and G',
the total stress equivalents for the undrained analysis of a
saturated soil can be obtained. The pore water is almost incompres-
sible compared with the soil skeleton (except in a few, very dense,
well-graded soils) so that /zu tends to 0.5 and Ku is nearly infinite
(Naylor, 1978). Also, since changes in mean stress cause no
distortion in elastic isotropic materials, the shear moduli Gu and
G' are equal. Then
q = Eu_____ = p i = _____E' ...
u 2(1 + /zu) 2(1 + /z')
E
u
1.5
1 + M'
E'
and


29
The parameters needed to define stress-strain laws are most
commonly based on the results of laboratory tests. Initial moduli
(i.e. moduli at very small strain) can, however, be obtained from
measurements of the speed of propagation of shock waves through the
soil. Laboratory measurements of stiffness very often under-
estimate actual in-situ values. This is particularly true of
measurements in unconfined undrained triaxial shear tests (Naylor,
1978).
The models described here will require knowledge of the
strength parameters c, or c', ' Also, a shear stress-strain
curve (refer to Figure II.8) will be needed to obtain the shear
stiffness. A mean stress-volume change relation will be required to
determine the bulk stiffness in terms of effective stress. For clay
soils the latter will be measured by Cc or Cs, the slopes of the
virgin consolidation and swelling curves respectively on a plot of
void ratio against log10a' (either from odometer or triaxial
consolidation tests). A further requirement for clays is a
knowledge of the preconsolidation pressure, OpC. It is usually
assumed that Cc and Cs are constants, so that the bulk modulus
increases linearly with stress:
Kt =
P(1 + e)
0.434C
11.22
where KT is the tangential bulk modulus at a mean effective stress,
p, and void ratio, e, and C stands for Cc or Cs as appropriate.
The undrained Young's modulus of granular soils is even
more variable than that of clays. It depends principally on the


30
Gt= Tangential shear modulus
G = Average
Gq = Initial *
Fig. II.8 Deviatoric Stress-Strain Curve for Triaxial
Test Showing Alternate Shear Moduli
(after Naylor, 1978)


31
relative density, the gradation, and the nature of the particles
(Naylor, 1978).
The effective stress Poisson's ratio (/*') varies from near
zero under initial loading to 0.5 as failure is approached. Under
repeated loading of granular soils it will typically lie in the
range 0.3 to 0.4 (Naylor, 1978).
11.5. c Variable Elastic Laws
11.5. C.1 Hyperbolic Model
A model for undrained saturated clays need only involve one
variable modulus since Poisson's ratio is 0.5. The hyperbolic model
originally attributed to Kondner (1963) defines the stress-strain
relation in the form
- ff3> =r^b77 n-23
where a and b are constants when the equation is applied to conven-
tional (constant ct3) triaxial tests. 1/a is the initial tangential
Young's modulus and 1/b would be the failure value (ax a3)f; were
it not for a refinement (Naylor, 1978). This is to allow (ox a3)
to become asymptotic to a stress in excess of (a1 or3)fj the curve
being cut off at this value.
The model is not restricted to the undrained analysis of
saturated soils. For general applications, a and b vary with a3 <
11.5. C.2 Differential Models
A differential model defines a tangential modulus. It is
suited to incremental methods of analysis in which the steps are


32
sufficiently small that the increment approximates a differential
(Naylor, 1978).
In general, the bulk modulus of soil increases with confin-
ing stress, and the shear modulus reduces with shear stress becoming
zero at failure (Naylor, 1978). It is therefore logical to define
the tangential values of K and G separately. 'K-G' models can then
be defined in which K and G are assumed to vary linearly with stress
invariants. According to Naylor, (1978):
K = Kx + akp' 11.24
0 = 0]^+ QGp' + /3Gq 11.25
in which
P' = 1/3 ( and
q2 = (a1 a2) + a2(az a3) + a3 (ct3 a1)
q is related to the second deviatoric invariant, J2(=i/3q2), by
q = /(3J2) and to the octahedral shear stress, roct, by
q = (3//2)7-oct. The expression for q is the same whether the stress
is effective or total. An alternative is


33
K = Kx + aka,?
11.26
G = Gi + QG a's + /fc CTd
11.27
in which
a' = 1/2 (a[ + a3')
CTd = [i - az]
Either alternative requires values to be assigned to five
parameters. The ctK and aGare positive and /3G is negative. A
suitable choice of Gx, aG and /UG will make G zero when the stresses
satisfy a failure criterion (Naylor, 1978). Equations 11.24 and
11.25 can satisfy a yield criterion of the conical type, whereas
equations 11.26 and 11.27 can satisfy a Mohr-Coulomb criterion
(Bishop, 1966) equations. Soils, in general, adhere more closely to
a Mohr-Coulomb criteria, hence equation 11.26 and 11.27 are
preferred.
A different form of K-G model has been proposed by Nelson
and Baron (1971). Their specification of G is essentially the same
as equation 11.25, but they assume K to be a quadratic function of
the volumetric strain (Naylor, 1978).
II.5.d Elastic-Plastic Laws
Elastic-plastic stress-strain laws relate small increments
or rates, of stress and strain. In soils applications it is usual


34
to set aside (or ignore) true time effects so that the rate of
loading has no effect. Thus, d( )/dt can be interpreted as d( )
when it appears to the same power on either side of the equation
(Naylor, 1978).
II.5.d.l Yield Surface
The yield surface is a function of stress which when
evaluated for the stress components of a typical point must have a
value less than or equal to zero.
F(ct) <0 11.28
It can be viewed as a surface in a stress space having as
axes either the components of stress or some functions of the
components. The requirement of equation II.28 in geometric terms is
that the point representing the state of stress must lie on or
within the yield surface.
If the point lies within the yield surface the stress-strain
law is assumed to be elastic usually linear and isotropic. If,
during loading, the point reaches the yield surface and tries to
cross it, it is constrained on the surface and plastic strains
become superimposed on the elastic (Naylor, 1978). These are
governed by a flow rule and a strain hardening law.
Yield surfaces must be convex (although they can have local
flat areas) and must contain the stress origin (Prager, 1959).


35
11.5. d.2 Strain Hardening
The yield surface may change in size when plastic yielding
occurs. If it gets larger strain hardening occurs, if smaller,
strain softening. It is usually assumed that the shape of the yield
surface remains the same as it hardens or softens, and that it
expands or shrinks about the origin. This type of hardening is
called isotropic. An alternative is kinematic hardening (Prevost
and Hoeg, 1975).
Strain hardening is usually assumed to be controlled by a
single parameter, h, although there is no theoretical objection to
there being more than one (Naylor, 1978). Equation 11.28 may be
generalized to include h,
F(a,h) < 0 11.29
Sometimes the yield function will be expressed in the form
of equation 11.28 even when there is hardening. This will be done
when variation of F with a is considered so that h can be treated as
a constant (Naylor, 1978).
11.5. d.3 Flow Rule
The flow rule fixes the proportions of the components of the
plastic strain rates. To do this the assumption is made that the
directions of principal plastic strain rate coincide with the
principal stress directions. This is called the assumption of
coaxiality of the stress and plastic strain rate tensors, (Note:


36
controversy exists about the extent to which it is valid in soils.),
according to Naylor, (1978).
The flow rule is expressed as
deP = dAaq 11.30
in which deP is the vector of plastic strain rate components, dA is
a parameter related to a if there is strain hardening (or soften-
ing) otherwise it is determined by the boundary conditions, and aq
is the gradient vector to a scalar function of stress, Q(cr) known
as the plastic potential.
The assumption of coaxility allows the flow rule to be
interpreted geometrically (Naylor, 1978). A constant Q defines a
surface in stress space which passes through the point representing
the current state of stress. If the components of plastic strain
rate are also assigned to the stress component axes, an outward
pointing normal to the surface originating from the current stress
point has the plastic strain rates as its components, (Figure II. 9).
Normality is said to apply Q = F the resulting flow rule is
then associative. Otherwise the flow rule is non-associative.
II.5.d.4 Specific Forms of Yield Surface
Yield surfaces relevant to soil mechanics may be divided
into those which are open ended cones in principal stress space, and
those which are capped cones (refer to Figure II.10, Naylor,


Fig. II.9 Flow Rule in Three-Component Space
(after Naylor, 1978)


38
Fig. II.10 Yield Surface in Principal Stress Space
(after Naylor, 1978)


39
1978). In both classes the cones may or may not be right circular.
Those which are right circular are fully defined by the stress
invariants p and q. Of various possible non-circular cones only
those representing the Mohr-Coulomb criterion will be examined. It
is a hexangular pyramid, and is fully defined by the stress in-
variants cts and crd .
Figure II.11 illustrates the Mohr-Coulomb yield surface and
four of the right circular family of conical yield surfaces. They
are shown as intersections of the three-dimensional surface with the
pi plane, the plane in principal stress space at right angles to the
line a1 = a2 = ct3. There is a three-fold symmetry in the pi plane.
Consequently the surfaces are fully defined in the 120 segment
shown. Figures II.11(b) and II.11(c) show alternative plots for the
surfaces in terms of the relevant pairs of invariants (Naylor,
1978) .
The open yield surfaces are generally assumed not to strain
harden (that is, the cone angle does not increase) although they may
strain soften. Consequently yielding implies local failure. The
yield surface becomes a failure criterion (Naylor, 1978).
According to Naylor, 1978 the equations defining the yield
surface may be expressed in different ways. For Mohr-Coulomb, it


(b) Conical surfaces
flow rule
(c) Mohr Coulomb
Fig. II.11 Yield Surfaces; 1 = Mohr-Coulomb, 2 = Extended VonMises,
3 = Compromise Cone, 4 = Axial Extension Cone, and
5 = Drucker-Prager (after Naylor, 1978)
o


41
may be written in terms of the shear and normal stress (fp.Op) on
the plane of failure as
F(*p,crp) = 7p crp tan c < 0 11.32
or in terms of as and as
FC^S'^d) = ^d Scrs T < 0 11.33
in which S = 2 sin and T = 2 c cos , or yet again as
F(p,q,0) =q Mp N < 0
11.34
in which M and N are functions of c, $, and a third stress in-
variant, 6. 6 is an angle in the pi plane (Figure II.11) whose
tangent is 1/73 times Lode's parameter (Hill, 1950),
tan
a\ ~ g3
73(a1 ct3)
11.35
9 measures the relative value of the intermediate principal stress.
The conical yield surfaces (2,3,4,5, in Figure II.11) may
also be represented by equation 11.34 but with M and N containing c
and only. The expressions based on Naylor for M and N are given
in Table II.2 for the five cases.
Associative flow rules will cause excessive dilatancy with
yield surfaces of the type considered here (Naylor, 1978). This is


42
TABLE II.2
THE PARAMETERS M AND N IN EQUATION 11.34
(after Naylor, 1978)
Criterion M N
1. Mohr-Coulomb 3 sin ^ 3c cos
3cos0-sin0sin$ 3cos0-sintf;
2. Extended von Mises 6 sin 6c cos
3 sin 3 sin 0
3. Compromise Cone 2 sin 2c cos
4. Axial Extension Cone 6 sin 6c cos
3 + sin 3 + sin
5. Drucker-Prager 3 sin ^ 3c cos
3 + sin 3 + sin


43
illustrated in Figure II.11(c). The solid arrow represents an
associative flow rule. Its direction is given by
ded/de| = ?S = -2 sin 11.36
in which de§ = l/2(de5-de§) and de| = (de^-deg). This means that
the plastic deviator strain rate de§ is associated with very
significant negative plastic volumetric strain rate (-def).
"...This causes more dilatancy than actually occurs. An
alternative way is to introduce a non-associative flow rule in
which S in equation 11.36 is replaced by S = 2 sinV>. rp, the
dilatancy angle, will be less than 0, typically in the range 0-
20. . according to Naylor, 1978.
II.6 Failure Criteria
Failure of a material most often involves three-dimensional
stress conditions within the material. Knowledge of the stress-
strain behavior and the states of stress which constitute failure
are necessary to analyze such failures. To determine the stress
conditions which govern fracture of intact materials, a general
three-dimensional failure criterion is required. Most of the
failure criteria proposed for three-dimensional stress states
involve relatively complex expressions for which more than three
material parameters are required. These criteria have been
developed to mimic the experimentally determined shape of the
failure surfaces as observed in the principal stress space.


44
II.6.a Classical Failure Criteria
Classical failure criteria for static shearing include Tresca's,
von Mises' and Mohr-Coulomb's. Drucker and Prager (1952) extend the
Tresca's and the von Mises' failure criteria to include the hydros-
tatic pressure effect. Generalized static failure crit-eria are
summarized in Table II.3 and are shown in Figures 11.12 and 11.13.
The Extended Tresca criterion and the Extended von Mises
(Drucker-Prager) criterion are employed in several existing con-
stitutive models to describe the failure condition of frictional
materials (Horita, 1983).
In the principal stress space, the shapes of corresponding
failure surfaces are conical and they have cross-sections of a
regular hexagon (Tresca) or a circle (von Mises) with thin center
lines coinciding with the hydrostatic axes. Both criteria yield
that the material has the same strength in compression and extension
for a given magnitude of mean and normal stress.
The Extended Tresca criterion and the Extended von Mises
criterion are in principal unable to represent the behavior of
cohesionless material (Bishop, 1966). The failure surfaces may
extend outside part of the principal stress space where all stresses
are compressive and positive, as shown in Figure 11.14. For higher
friction angles the states of stress near triaxial extension are
located in parts of the stress space where one of the principal
stresses is negative. Without the effect of cohesion, this is not
reasonable (Lade, 1972).


45
Fig. 11.12 General Static Failure Criteria
(after Horita, 1983)


46
Fig. 11.13 General Static Failure Criteria
(after Horita, 1983)


(a)
(b)
Fig. 11.14 (a) Classical Failure Criteria in Principal Stress Space and
(b) Cross Sections of Mohr-Coulomb Failure Criterion Shown
for Three Different Friction Angles (after Bishop, 1966)


48
The Mohr-Coulomb failure surface is also conical in shape,
and its cross-section is an irregular hexagon as shown in
Figure 11.14. For materials without effective cohesion all prin-
cipal stresses remain positive, even for very high friction angles.
The shape of the cross-section of the Mohr-Coulomb failure surface
resembles a regular hexagon for very small friction angles and it
approaches an equilateral triangle for friction angles approaching
90. Thus, this failure surface exhibits some of the characteris-
tics necessary for correct modelling of failure of frictional
materials (Lade, 1972).
The intermediate principal stress does not appear in the Mohr-
Coulomb failure criterion. The failure surfaces are pointed in
octahedral planes and their traces in planes containing the hydros-
tatic axis are straight lines. The intermediate principal stress,
however, has been experimentally shown to influence the strength of
frictional materials (Horita, 1983). Also, experimental results
indicate that the failure surfaces are curved in planes containing
the hydrostatic axis (Lade, 1972). Consequently, the Mohr-Coulomb
criterion does not model these significant aspects of failure in
frictional materials.
II.6.b Three-dimensional Failure Criteria
Lade (1977), presented a three-dimensional failure criterion for
frictional materials without effective cohesion for soils with
curved failure envelopes. The criterion is expressed in terms of
the first and third stress invariants of the stress tensor as
follows:


49
(I?/I3 27) (Ii/pa)m = n
11.37
where
II <7} + 11.38
I3 1 2 3
I3 (ax CTy CTz) + (Txy Tyz Tzx ryx Tzy Txz)
11.39
and pa is the atmospheric pressure
The value of I31/I3 is 27 at the hydrostatic axis where
ai = ct2 =ct3 parameters ri1 and M in equation 11.38 can be
determined by plotting (I3/I3-27) versus (P^I^ at failure in a
log-log plot. The intercept of this line with (P^^) = 1 is the
value of ri1 and M is the slope of the line.
In the principal stress space the failure surface defined by
equation 11.38 is shaped as an asymmetric bullet with the pointed
apex at the origin of the stress axes (Lade, 1977). The failure
surface is concave towards the hydrostatic axis. Lade's Model is
discussed in detail in Chapter III.


CHAPTER III
LADE'S ELASTO-PLASTIC MODEL
III.1 Introduction
An elasto-plastic constitutive model for sands was developed
by Lade and Duncan (1975), based on the results of laboratory
cubical triaxial tests. The theory is applicable to general three-
dimensional stress conditions, but the parameters involved can be
derived entirely from the results of a series of conventional
triaxial compression tests. This stress-strain theory is capable of
modelling several aspects of the behavioral characteristics of
cohesionless soils, including the effects of the intermediate
principal stress, shear dilantancy effects, and the stress-path
dependency effects (Lade, 1972),.
Involved in the original theory (1972) are some simplifying
assumptions. Experimental test results indicate that failure
envelopes for sand are most often curved in the Mohr diagram. They
are, however, assumed to be straight in the model. Consequently,
the model which assumes a straight line failure envelope can only be
applied to cases in which a limited range of confining pressure is
of interest, hence the curvature of the failure envelope can be
neglected. In addition, the model is limited to predicting elastic
strain under a proportional loading condition in which the stress
path moves along a yield surface, but experimental results indicate


51
that plastic strain are also generated (Lade, 1977). Because the
work hardening law of the model is a monotonic increasing function,
the model can not simulate the softening behavior of sands. Because
of the limitations of the original theory, Lade (1977) incorporated
additional aspects of the real behavior of cohesionless soils in his
modified theory. All aspects of the previous theory were retained,
and the previous theory became merely a special case (straight-line
failure envelopes) which is contained within the structure of the
new theory (Lade, 1977). Modifications to the original theory
include a curved failure surface and a spherical yield surface
centered at the origin of the principal stress space was capped on
the conical yield surface.
The model has subsequently been modified and expanded for
use with materials other than cohesionless soils. Lade (1978)
modified his model to simulate the behavior of normally consolidated
clays. The model was then expanded to include the effect of
cohesive properties and tensile strength, to predict the behavior of
concrete and rock by Lade 1981, and Kim and Lade, 1984. Lade and
Boonyachut (1982) modified the model to simulate the behavior of
sand in triaxial tests during large stress reversals and changes in
stress involving unloading and reloading. The following portion of
this chapter contains a cursory summary of Lade's model.
III. 2 Strain Components
To model the stress-strain behavior of soils by an elasto-
plastic theory, the total strain increment (de-jj), is divided into


52
an elastic component (dee£j), a plastic collapse component (dec^j),
and a plastic expansive component (dePjj), such that
de-jj = dee£j + deCij + deP^j III.l
The elastic, plastic collapse and plastic expansive
components of strain in a drained compression test are shown in
Figure III.l.
III.2.a Elastic Strains
Using Hooke's Law, the elastic strain increments, which are
recoverable upon unloading, are calculated using the unloading-
reloading modulus of elasticity (identical to the one used with the
hyperbolic relation), as
Eur = Kur Pa (CT3/Pa)n III. 2
where the modulus number, Kur, and the exponent, n, are material
parameters determined from convential triaxial compression tests
performed with varying confining pressures. The atmospheric
pressure, pa, is expressed in the same units as Eur and ct3 The
value of Poisson's ratio, y, is assumed to be constant for the
elastic parts of unloading and reloading. The elastic strain
increments can be calculated from the generalized Hooke's Law as:


Principal'Stress Difference, c|-oj
Fig. III.l Strain Components in Lades Model
(after Lade, 1977)


54
dex 1/Eur ^-/Eur -1/Eur 0 0 0 ' dcrx
d£y -1/Eur 1/Eur -1/Eur 0 0 0 day
de z -1/Eur -1/Eur 1/Eur 0 0 0 do z
d7Xy 0 0 0 Eur/2( 1+/00 0 drxy
d7yz 0 0 0 0 Eur/2(1+aO 0 dryz
d7xz . 0 0 0 0 Eur/2(1+/0 . drxz .
III.3
III.2.b Plastic Collapse Strains
Part of the strains occurring during isotropic compression
are irrecoverable, that is, they are plastic in nature. The plastic
collapse strain increments are calculated from a plastic stress-
strain theory in which an associated flow rule is applicable. The
theory includes a cap-typed yield surface with the center at the
origin of the principal space is shown in Figure III.2 (a) and III.2
(b), (Lade, 1977). The equation for the yield surface is written in
terms of the first and the second stress invariants, 1^ and I2 as
follows:
fc = Sc = I? + 2I2 HI.4
where fc is an yield function, and gc is a plastic potential
function.
Ii x + ay + z
I2 ^xy*7yx+ryz*rzy+rzx*rxz'(<7x*<7y+cry*£Tz+CTz*crx)


Fig. III.2 Ultimate Strength and Yield Surfaces in Lade's Model (after Lade, 1977)
Ui
Ln


56
Yielding according to the yield function (Equation III.4) does not
result in failure. As the value of fc exceeds its current maximum
value, work hardening occurs and collapse strains generated (Lade,
1977).
A work hardening relationship can be defined experimentally
as follows:
for cohesionless soil
Wc=cpa.(fc/pa2)P III. 5
for cohesive soil
Wc/Pa=c+p,V(fc/Pa2)
III.6
in which c and P are material parameters and Wc is the collapse
work. The increment of Wc is expressed as:
dwc=aij deijC
III.7
The associated flow rule is expressed as:
de j[j dAc* (3gC/92CT^j ) dAc (dfc/d(jj_j )
III.8
By substituting Equation III.8 into Equation III.7, the dAc can be
solved and expressed as follows:
dWr
dWr
dAc ~
<7ij (dfc/da^) 2f
III.9
ij'


57
where dWc can be obtained by differentiating the Equations III.5
(and III.6) and III.7 as:
for cohesionless soil
dWc = c*P*pa*((pa/fc)1'P).d(fc/pa2)
III.10
for cohesive soil
dWc = (p/2).pa.((fc/pa2)-lj).d(fc/pa2) III.11
By substituting Equation III.9 into Equation III. 8, the
plastic collapse strain increments can be calculated as follows:
d6xC ^x
d£yC ay
de2c _ dWc # z
^7xyC fc fc- CM
^7yzC 2ryz
. ^7xzC . . 2rxz
III.2.C Plastic Expansive Strain
The plastic expansive strain increments are calculated from
a plastic stress-strain theory in which the non-associated flow rule
is employed (Lade, 1977). The failure surface is a symmetric bullet
with the pointed apex at the origin of the stress space as shown in
Figure III. 2. The yield surface is assumed to have the same shape
as the ultimate strength surface. The yield surface is expressed in
terms of the first and third stress invariants, 1^ and I3 as
follows:
fp = (I31/I3-27).(I1/Pa)m
III.13


58
where I3 -
CTx*ffy,crz + rxy*ryz*rzx+ryx*^zy*rxz
" (CTxryz*rzy+CTy*rxz*rzx+<7z*7'xy*ryx) m 14
fp = rh. ultimate strength surface
where m and r^1 are material parameters. It should be noted that the
ratio of I3/I3 provides the smoothly rounded edges of yield and
failure surfaces (Figure 111.2(c)) and yield exponent, m, accounts
for the curvature of the failure surface. The above give Lade's
constutitive model results which reasonably conform to experimental
evidence.
The plastic potential function, gp, was formulated through
observation of the directions of the plastic expansive strain
increments on a specified yield surface. The plastic potential
function was expressed as:
gp=I3i (27-i72 (Pa/Ii )m) I3 111 15
where rfz is constant at given values of a3 and fp
V2 = S.fp+R.yCog/pa)+t III.16
in which S, R, and t are material parameters. The comparison
between the yield and plastic potential surface is illustrated in
the triaxial plane as shown in Figure III.3.


59
Fig. III.3 Yield Surface and Plastic Potential Surface
in Triaxial Plane of Lade's Model


60
The relationship between fp and Wp can be expressed by an
exponential function as follows:
fp = ae('b*WP).(Wp/pa)1/q q>0 III.17
where
a Ji*((e pa)/(Wp peak))Vq III.18
b = 1/(qWp peak) III.19
^p peak = P*Pa*(CT3/Pa)^ III.20
q = a + ^*(o3/pa) III.21
p,Z,a and @ are material parameters.
The non-associated flow rule gives the relation between
stress and strain increments by the following expression
dejj P=dAp (ao-p/atTij) dAp>0 III. 22
The increment of plastic work is as
dWp=(7ij .deijp III. 23


61
By substituting Equation III.23 into Equation III.22, the dAp can be
solved as
d^p dWp/(aij (3gp/3cJij ))
III.24
or
dAp = dWp/(3*gp+mr72 ((Pa/Ii )m) *I3 )
III.25
where dWp is the increment of expansive plastic work and can be
derived from Equation III.17 and expressed as follows:
dWp
dfp
fp~
1
(1/(q wP))-b
III.26
Substituting Equation III. 26 into Equation III.25 gives dAp as
follows:
(dfp/fp) 1/(1/(q*Wp)-b)
dAp = --------------------------- III. 27
3*gp+m*j72*(pa/I1)m*I3
By substituting Equation III.27 into Equation III.25 and by
differentiating gp with crjj the plastic expansive stress-strain
relationship can be expressed as follows:
dexP
dfyP
dezp
d7xyP
d7yzP
. d7xzP .
((dfp)/fp) l/((l/q.Wp)-b)
3gp+M *?2-(pa/I1)m-I3


62
ay'az Tyz
azax Tzx2
x* rxy2
-[27+r,2 (Pa/Ii )m] " (t7z*Txy-rxz*ryz) (CTx*Tyz'Txy,Tzx)
-(<7y TZK-TyZ tyX)
[ (31? )+(I3/I1) *in*?2 (Pa/Ii )m]
1
1
1
0
0
0
III.28
III.3 Summary of Stress-Strain Parameters
The total strain increments are the summation of elastic
strain increments (Equation III.3), plastic collapse strain
increments (Equation III.12) and plastic expansive strain increment
(Equation III.28). To obtain the total strain increments, a total
of fourteen soil parameters are required. Table III.l lists these
parameters. The procedures used in obtaining the soil parameters
are presented in Chapter VIII.


Table III.l Parameters for Lade's Model
Parameter Strain
Modulus No., Kur
Exponent, n Elastic
Poisson's Ratio, /j,
Collapse Modulus,.c Plastic
Collapse Constant, P Collapsive
Yield Constant, 17 x
Yield Exponent, m
Plastic Potential
Constant, R
Plastic Potential
Constant, S
Plastic Potential Plastic
Constant, t
Expansive
Work Hardening
Constant, a
Work Hardening
Constant, /3
Work Hardening
Constant, p
Work Hardening
Exponent, i


CHAPTER IV
MATERIALS AND TESTING APPARATUS
IV.1 Test Program
The test program was designed to determine the following:
(i) The static behavior of Monterey No. 0/30 sand on
different stress paths on the Rendulic plane.
(ii) Provide sufficient data so that the parameters
necessary to implement Lade's model can be generated.
(iii) Provide test data such that a comparison of the soil
behavior as tested in a large diameter hollow cylinder triaxial cell
developed by the University of Colorado at Denver, and a standard
solid cylindrical triaxial cell, can be made.
The program consisted of conducting nineteen static triaxial
tests on seven different stress paths and at three varying stress
levels. The test program is summarized in Tables IV.1 and IV.2
IV.2 Monterey No. 0/30 Sand
The Monterey No. 0/30 sand used in conducting all triaxial
tests was processed by Lone Star Industries of California to meet
the California Air Resources Board specification for open air sand
blasting abrasive. The use of Monterey No. 0/30 sand as a perfor-
mance specification testing standard, which replaced Monterey No. 0
sand (also produced by Lone Star Industries), is discussed in detail
by Muzzy (1983).


TABLE IV.1
Proposed Testing Program
Test (psi) Sample Relative. Density (%) Sample Diameter (in.)
Drained 30
Compression 60 39 2.0
Tests 90
Isotropic 10-100 39 2.0
Compression
Drained 30
Extension 60 39 2.0
Tests 90


66
TABLE IV.2
Stress Paths Conducted for Study
Stress Path o'a = constant a£ = constant
Conventional Triaxial Compression (CTC) No Yes
C onvent i onal Triaxial Extension (CTE) Yes No
Reduced Triaxial Compression (RTC) Yes No
Reduced Triaxial (RTE) No Yes
Triaxial Compression (TC) No No
Triaxial Extension No No
Hydrostatic Compression (HC) No No


67
Monterey No. 0/30 sand is uniformly graded and is classified
as SP in accordance with the Unified Soil Classification System
(USCS). Figure IV.1 contains a typical grain size distribution
representative of Monterey No. 0/30. Typical physical properties of
the Monterey No. 0/30 include a specific gravity of 2.65, a maximum
dry density of 105.8 pcf (pounds per cubic foot) and a minimum dry
density of 91.7 pcf. Refer to Table IV.3 for a summary of typical
Monterey No. 0/30 physical properties.
IV.3 Test Equipment
All tests for this study were conducted in a conventional
triaxial test cell using 2-inch by 4-inch solid cylindrical sample.
The loading machine used in performing the triaxial test was a
closed-loop servo valve electro-hydraulic MTS system. A separate
data acquisition system was used in addition to the MTS systems own
X-Y plotter.
IV.3.a Triaxial cell
The triaxial cells used for conducting all tests were
equipped with 2-inch bottom pedestals and 2-inch top caps. A slight
difference between cells consisted of a top cap design. One was
equipped with a top cap designed exclusively for conducting
compression tests. This cap was manufactured with a concave,
recessed assembly located on the upper surface of the cap such that
a round stainless steel loading ball seats within it. The loading
(ram) piston, the lower end of which is also concave, is
manufactured to securely contact the upper portion of the loading
I


PERCENT FINER
68
U. S. STD. SIEVES
Fig. IV.1 Grain Size Distribution of
Monterey No. 0/30 Sand
(after Muzzey, 1976)


TABLE IV.3
Physical Properties for Monterey No. 0/30 Sand
Unified Soil Classification Symbol SP
Specific Gravity Particle Size Data: 2.65
D50(mm) 0.45
cc 1 1.00
cu 2 Dry Unit Weight Data 1.60
Maximum, pcf 105.80
Minimum, pcf 91.70
1 Cc = (d30)2/(60 x d10)
2
cu = d60/d10


70
ball, and hence, transfer a uniform load through the ball and on to
the sample cap. The ball also facilitates alignment during initial
seating.
The second cell was equipped with a top cap from which a
threaded bolt (male) was attached to the upper surface. The loading
piston had within the concave end a tapped hole containing female
threads. The piston could be threaded onto and securely attached to
the top cap via the bolt. This cap permitted the application of
both extension and compression loads.
Both the top cap and bottom pedestal were designed to allow
drainage from the specimen or measurement of internal pore pressure.
The bottom pedestal connected directly through an internal channel
to the external valve. The top cap was connected to the external
valve via polyethylene tubing which attached to the base assembly
and lead to the external valve.
External valves allowed for the measurement of chamber
pressure/volume change, sample-bottom pressure/volume change and
sample-top pressure/volume change. The cells were equipped with two
outlets to measure sample-bottom pressure/volume change. This
additional connection allowed a pore pressure transducer to be
attached to a five way valve was used to monitor cell pressure, and
sample pore water pressure at both the top and bottom of the sample.
The valve, however, was designed such that only one pressure locale
at a time could be monitored.
A pressure control panel equipped with two graduated
burettes was used to apply and control cell and back pressures


71
simultaneously to the cell. An air compressor connected to the
pressure control panel furnished a maximum pressure of 175 psi
through a regulator (range 0 to 250 psi) onto a column of water
within graduated burettes. These calibrated burettes were used to
measure pore water movement during saturation and consolidation
periods (approximate accuracy range 0.1 cc).
Several tests required pressure higher than the compressor
was capable of producing. For these tests, a high pressure (2500
psi) nitrogen cylinder was plumbed into the pressure control panel.
A regulator on the nitrogen cylinder limited the pressure into the
panel regulator to a maximum of approximately 300 psi.
IV.3.b MTS Loading Machine
A series 810, Materials Test System (MTS Systems
Corporation, Minneapolis, Minnesota) shown in Figure IV.2 was used
to apply load to the sand specimens. The machine was capable of
applying both monotonic and cyclic loading. Maximum load capacity
of the system was 20,000 lbs., with maximum piston stroke being 5
inches. Hydraulic pressure was supplied by a fixed volume hydraulic
pump, with a pump capacity on the order of 3 gallons per minute.
Cyclic wave forms could be generated with a frequency range
of 1.0 x 10"-* to 990 Hz from the function generator. Available wave
forms include sine, harversine and harversquare. Wave forms could
be inverted from extension to compression. The system was also
capable of producing ramp functions including straight-line ramp,
dual slope ramp, triangle, saw-tooth, and trapezoidal with ramping
time ranging from 1.0 x 10seconds to 11.45 days.


72
Fig. IV.2 Material Testing System Equipment
Used in Study


73
Load was measured by means of a load cell located beneath
the base pedestal. Piston travel was measured by a linear variable
differential transformer (LVDT) connected directly to the piston
within the MTS housing. The MTS systems internal recording system
consisted of an X-Y plotter directly connected via coaxial cables to
both the load cell and LVDT. An additional data acquisition system
was incorporated during testing. The following contains a summary
of the main feature of the MTS system.
(1) Hydraulic pressure
Fluid pressure is furnished to the MTS system by a fixed-
volume pump. Requirements in the system necessitate that the pump
be capable of furnishing a minimum flow of 3 gallons per minute.
The hydraulic power supply may be operated locally, through use of
its own controls, or via the MTS remote control panel.
The system has two levels of operation. An output pressure
of 300 psi for the low or bypass condition, and an output pressure
of 3000 psi for the high (operational) condition. A safety pressure
control valve protects the power supply from possible excessive
pressures.
A fluid-to-water heat exchanger is used by the hydraulic
power supply to maintain the reservoir hydraulic fluid at a desired
operating temperature. The hydraulic pump is equipped with a
temperature-sensitive switch mounted on the reservoir which auto-
matically disengages the hydraulic power supply if the hydraulic
fluid temperature exceeds operational limits.


74
(2) Hydraulic Actuator
The hydraulic actuator is the force-generating and/or
positioning device in the system. Movement of the loading piston is
the direct result of the application of fluid pressure to one side
of the piston.
(3) Servovalve
The hydraulic actuator is controlled by the opening and
closing of the servovalve in response to a control signal from the
valve driver or controller. The servovalve can open in either of
two positions, thereby permitting high pressure fluid to enter into
either side of the piston. This alternating application of hydrau-
lic pressure to either side of the piston allows the application of
smooth cyclic tensile and compressive loads. When the servovalve is
opened allowing fluid flow into one end of the cylinder, the valve
on the opposite end of the cylinder is opened to provide a channel
for return flow back to the hydraulic power supply.
The rate of fluid flow through the servovalve is in direct
proportion to the magnitude of the control signal. The polarity of
the control signal determines the end of the actuator cylinder which
will receive additional fluid, thereby determining the direction of
the piston stroke.
(4) Transducers
Transducers on the MTS system sense the pressure generated
by the hydraulic actuator. As a result of the pressure, the
transducer provides an output voltage directly proportional to the
measured quantity.


75
The load-cell is a force-measuring transducer that provides
an output voltage directly proportional to the applied load.
Compressive and tensile forces are distinguished by the polarity of
the output voltage.
The linear displacement of the loading ram is measured by a
linear variable differential transformer (LVDT). The LVDT requires
a-c excitation and provides an a-c output. The amplitude of the
output varies in direct proportion to the amount of displacement of
the LVDT core.
(5) Transducer Conditioners
Transducer conditioners supply excitation voltages to their
respective transducers and control the transducer output voltages to
d-c levels suitable for use in the control portion of the system.
Output of each transducer conditioner is 10 volts, positive or
negative, when the mechanical input to the transducer equals 100% of
the selected operating range.
(6) Feedback Selector
To perform either stress or strain controlled tests, a feed-
back selector is used to select the output from a particular trans-
ducer conditioner for use in controlling the hydraulic actuator.
(7) Servo Controller
The servo controller provides an error signal proportional
to the difference between the two inputs based on a comparison of
the command and feedback inputs. The command and feedback signals
are continuously monitored and if not equivalent, the resultant
error signal opens the servovalve in a direction and by an amount


76
which causes the hydraulic actuator to correct the error. When
command and feedback signals are equal the error signal is reduced
to zero, the servovalve closes, and the system is maintained in a
state of equilibrium.
(8) Control Panel
The control panel regulates console power on/off, hydraulic
pressure on/off, and the simultaneous starting and stopping of
programmers and recorders.
An interlock circuit controls the normal/abnormal condition
sensors located throughout the system. If sensors detect an
abnormal condition, the interlock circuit opens and hydraulic
pressure is released from the system. The interlock circuit also
stops all programmers and recorders. If the interlock is opened by
an abnormal condition, it must be reset manually, either during
normal system start up or after correcting the abnormal condition.
(9) Digital Function Generator
Dynamic loadings can be created by use of the dynamic gen-
erator. Both cyclic wave forms and ramp functions can be produced.
Cyclic wave forms include sine, haversine, and haversquare. Wave
forms can, in addition, be inverted. Available system frequencies
range from 1.0 x 10"^ to 990 Hz. System ramp functions include
ramp, dual slope, triangle, saw tooth, and trapezoid. Ramp function
time periods are adjustable from 0.001 seconds to 11.45 days.
(10) Counter Panel
Electro-mechanical counters are incorporated in the counter
panel. A register indicates the number of test events completed and


77
the value is retained internally enabling the system to stop testing
after a predetermined count has been achieved.
IV.4.c Data Acquisition Systems
(1) X-Y Plotter
For recording axial load and vertical deformation, the MTS
system is equipped with an X-Y recorder. Axial load and vertical
deformation are recorded on the Y axis and X axis, respectively.
Adequate resolution can be acquired by use of range switches located
on the plotter.
In addition to the MTS systems data acquisition plotter, an
external digital data acquisition system was used. This system was
a Fluke 2240C programmable datalogger. The datalogger can monitor
axial load, axial deformation and pore pressure simultaneously at
prescribed intervals. The datalogger yields a printed copy of
analog output readings in strip chart form.
(2) Pore Pressure Measurements
To measure hydraulic pressures, both cell and pore
pressures, a Genisco Tech (IBS 103G-32) transducer was used. The
transducer was rated for 0 to 300 psiG. A Tektronix Power Supply
and Digital Multimeter (TM 506) was used to supply power to the
pressure transducer. Voltage displayed as an electronic digital
readout from the multimeter was converted to readings in psi.


CHAPTER V
SAMPLE PREPARATION
V.1 Relative Density Control
To ensure that representative and accurate results were
obtained from the triaxial tests, a specified procedure was deter-
mined and followed.
A relative density of 39 percent was chosen as the target
density for the testing program. Reasoning for this value being
chosen, was that it corresponds to the average relative density
achieved in conducting triaxial tests with the large diameter hollow
cylinder cell. This would allow for a more complete comparison of
test results from the two triaxial cells.
Relative density is expressed as:
'frnax "Yd 'Ymin
Dr ----- x ------------ x 100 V.l
'Ymax"'Ymin
where:
Dr = relative density (in percent)
7max = maximum dry density (unit weight) of the soil
7min = minimum dry density (unit weight) of the soil
7,5 = in-place dry density (unit weight) of the soil


79
To obtain a final relative density of 39 percent after
saturation and consolidation, it was determined (by trial and error
based on the following sample preparation procedures) that an
initial target relative density of 35 percent was necessary.
To determine the air dry weight of soil necessary to create
the sample, the following equation was used:
7d = WsA(Ds2)(Hs)/4 V. 2
where:
73 = unit dry weight of the sample at desired relative
density
Ws = dry weight of specimen
Ds = diameter of specimen
Hs = height of specimen
The diameter and height of the specimen are the theoretical
final measurements and were determined by calculating the mold size
and subtracting the membrane thickness. Several trial samples were
prepared initially to ensure proper density was being achieved.
V.2 Sample Preparation
The following sample preparation procedures were used for
this study:
(1) Initially, all triaxial cell parts were wiped clean,
connectors and lines purged with air to flush remaining water from
the lines.


80
(2) Two filter papers were cut to a diameter slightly less
than that of the porous stones. The filter paper was cut slightly
smaller to prevent necking of the sample at the sample-porous stone
interface. The bottom porous stone, the two filter papers, the top
porous stone and the top cap were placed on the bottom pedestal of
the triaxial cell.
(3) Using a (free standing) vernier caliper and a level,
the height of those components were measured. Three readings were
made at approximately one-third distances apart, and the average
height computed. The thickness of the rubber membrane was measured
using vernier calipers. Measurements were taken at each end and at
the middle, the average thickness was then determined. These values
were needed to determine the actual dry density of the final sample.
All measurements were to the nearest 0.001 inch.
(4) A thin film of lubricant grease (Dow Corning No. 4) was
placed around the exterior of the bottom pedestal. The rubber
membrane was stretched and placed over the pedestal with the end
forced flush against the triaxial cell base. All air was pressed
out from between the membrane and base pedestal. Elastic bands were
seated over the membrane into the groves of the pedestal. This
provided a water tight seal between the membrane and the base
pedestal of the triaxial cell. It should be noted, most often
rubber 0-rings are used in place of the elastic bands. It was
found, however, that the elastic bands did not protrude out as far
as the 0-rings, thus providing a more even fit for the surrounding,
split mold.


81
(5) One porous stone (air-dried) was placed into the
membrane and seated on top of the base pedestal. Next, a filter
paper was positioned on top of the porous stone.
(6) To form the sample, an aluminum split mold was used.
The molds dimensions were; 2.020-inches inside diameter by 5-inches
in height. To prepare the mold, a thin film of grease was placed
along the connecting edges to assist in providing an air-tight seal.
The split mold was positioned on the base of the triaxial cell, and
held together using a circular clamp.
(7) Next, the membrane was stretched over the top of the
mold, and the inner surface fitted to make it smooth. A 20-inch
mercury vacuum (approximately 9.2 psiG) was applied to the void
between the mold and membrane via two lines which lead to each half
of the mold. The membrane was vacuumed securely and evenly to the
inside of the mold providing a perfectly cylindrical mold for the
sand to be placed into.
(8) A zero raining device was set inside the mold and
filled with the proper amount of sand, as previously calculated.
Gradually and smoothly, the zero raining device was lifted upward,
allowing no space to occur between the sand in the mold and the
bottom of the device. The zero raining device in general, provided
a density less than that desired, therefore, some slight tapping of
the filled mold was necessary to cause uniform settlement and hence
the desired density was achieved.


82
(9) Using a porous stone and rotating it slightly, the
surface was leveled. A bubble level was used to adjust the upper
surface was perpendicular to the sample axis.
(10) Once level, the porous stone was removed, the filter
paper placed on the sand surface, and the porous stone replaced.
(11) Next, a thin film of grease was applied around the
exterior of the top cap, and the cap placed on top of the porous
stone. While holding the top cap in position, the membrane was
pulled from around the mold and stretched over the top cap. The
membrane was pulled upward, and squeezed to remove any entrapped
air. Elastic bands were placed over the membrane into the grooves
on the top cap. Any excess membrane was folded down, so that none
extended beyond the upper surface of the top cap. The top cap was
once again checked to ensure it was level.
(12) A vacuum was applied to the sample via the bottom back
pressure valve. A 10-inch mercury vacuum (approximately 4.6 psiG)
was applied slowly, as not to shock the sample. Once the vacuum had
been applied to the sample, the vacuum leading to the split mold was
released, and the mold removed from around the specimen.
(13) Once complete and removed from the mold, the sample
was measured. Accurate measurements of the diameter and height were
made to 0.001 inch. A thin steel circumferential type measuring
tape (pi tape) and/or vernier calipers were used to measure specimen
diameter. The diameter was measured at three locations, on the
sample. At each location, two readings, 90 degrees apart, were
taken. The following equation was used to determine the average


83
diameter of the sample as outlined by the U.S. Army Corps of
Engineers:
Dp + 2Dc + Djj
D0 = ---------------
4
V. 3
where:
D0 = average diameter of sample
Dt = measured diameter of sample at the top
Dc = measured diameter of sample at the center
= measured diameter of sample at the bottom
Measurements of the specimen height were made using a free
standing vernier caliper, with an extension arm. Three measurements
were taken at approximately 120 intervals. The three height
measurements were averaged to determine the height of the sample.
Using the initial measurement (of the two porous stones, filter
papers, top cap and bottom assembly made prior to creating the
sample), the actual sample height was calculated.
The actual sample diameter was determined by subtracting the
average membrane thickness from the average sample diameter. Using
the actual sample diameter, and the average height, the sample
volume was calculated. The unit weight of the sample was determined
by dividing the weight of Monterey No. 0/30 sand used by the
calculated specimen volume. The relative density of the sample was
then calculated using equation V.l.


84
(14) To assemble the triaxial cell, the base was wiped
clean and a thin layer of grease applied. The steel ball was placed
on the top cap, if applicable. A rubber O-ring was placed and
seated on the triaxial base. The triaxial cell was placed over the
sample and pressed flush against the base. At this time, a visual
check of the alignment between the loading ram and top cap (steel
ball or bolt) was made to assure a proper connection was attainable.
The cell collar was then attached around the exterior of the cell.
(15) To fill the cell with water, a water line was attached
to the confining pressure valve of the cell. The cell was vented
through a quick connect on the top of the cell chamber. Once the
chamber was filled with water, the valve was shut off, the water
line disconnected, and the quick connect vent unplugged. The vacuum
line leading to the back pressure was then shut off, and the line
disconnected.
(16) Next, the cell confining pressure was connected to the
pressure panel. Simultaneously, while applying a confining pressure
of 10 psi to the chamber, the 5 psi vacuum was released. This
maintained the sample under a 10 psi effective stress at all times.
V.3 Flushing
To saturate the sample, de-aired water was used. A con-
tainer located a few feet above the sample was used to induce a
hydraulic gradient through the sample. The hydraulic gradient
across the specimen during saturation was kept to a minimum, never
exceeding 5. The water line was attached to the bottom back
pressure valve, and an open line to the top. Prior to saturation,