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UBC Theses and Dissertations

Behaviour of sand under general stress paths in the hollow cylinder torsional device Sayao, Alberto S.F.J. 1989

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BEHAVIOUR OF SAND UNDER GENERAL STRESS PATHS IN THE HOLLOW CYLINDER TORSIONAL DEVICE by ALBERTO S.F.J. SAYAO B.Sc, Catholic University of Rio de Janeiro, 1976 M.Sc, Catholic University of Rio de Janeiro, 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1989 ©ALBERTO S.F.J. SAYAO, 1989 In p r e s e n t i n g th is thesis in partial f u l f i lmen t of t h e r e q u i r e m e n t s f o r an advanced d e g r e e at the Univers i ty o f Brit ish C o l u m b i a , I agree that t h e Library shall make it f reely available f o r re fe rence and s tudy. I fu r ther agree that pe rmiss ion f o r ex tens ive c o p y i n g o f th is thesis f o r scholar ly pu rposes may b e g r a n t e d by t h e head o f m y d e p a r t m e n t o r by his o r her representat ives. It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on o f this thesis f o r f inancial gain shall n o t b e a l l o w e d w i t h o u t m y w r i t t e n pe rm iss ion . D e p a r t m e n t o f C i v i l Engineering The Univers i ty o f Brit ish C o l u m b i a Vancouver , Canada Da te 05 Mav. 1989  DE-6 (2/88) ABSTRACT A f u n d a m e n t a l i n v e s t i g a t i o n o f t h e s t r e s s - s t r a i n b e h a v i o u r o f i n h e r e n t l y a n i s o t r o p i c sands i s p r e s e n t e d . The s t u d y was c a r r i e d out i n a n e w l y d e v e l o p e d h o l l o w c y l i n d e r t o r s i o n a l a p p a r a t u s (HCT). The HCT i s t h e o n l y a p p a r a t u s t h a t p e r m i t s a s o i l specimen t o be s u b j e c t e d t o m u l t i a x i a l s t r e s s s t a t e s w i t h c o n t r o l l e d v a r i a t i o n s i n f o u r s t r e s s p a r a m e t e r s : t h e m agnitudes o f t h e t h r e e p r i n c i p a l s t r e s s e s ( o a , o2 and o 3 ) and t h e d i r e c t i o n o f one o f t h e s e s t r e s s e s . The e x p e r i m e n t a l program was aimed a t a s s e s s i n g t h e i s o l a t e d e f f e c t s o f c o n t i n u o u s c h a n g e s i n o n l y one d e r i v e d s t r e s s p a r a m e t e r , o^ (mean e f f e c t i v e s t r e s s ) , R ( p r i n c i p a l s t r e s s r a t i o ) , b ( i n t e r m e d i a t e s t r e s s p a r a m e t e r ) , o r a ( d i r e c t i o n o f o x r e l a t i v e t o t h e v e r t i c a l ) on d e f o r m a t i o n r e s p o n s e . M i n i m i z a t i o n o f s t r e s s and s t r a i n n o n u n i f o r m i t i e s , i n h e r e n l y p r e s e n t i n HCT s p e c i m e n s , was a c h i e v e d t h r o u g h c a r e f u l s e l e c t i o n o f specimen geometry and r e g i o n s o f t h e s t r e s s space t o be i n v e s t i g a t e d . A new a p p r o a c h t o d e f i n e t h e s e n o n u n i f o r m i t i e s i s p r o p o s e d i n terms o f t h e d i s t r i b u t i o n o f s t r e s s r a t i o R a c r o s s t h e specimen w a l l . A l l t e s t s were c a r r i e d o u t on s a t u r a t e d specimens o f p l u v i a t e d sands u n d e r f u l l y d r a i n e d c o n d i t i o n s . I n h e r e n t a n i s o t r o p i c b e h a v i o u r o f sand i s c l e a r l y i l l u s t r a t e d by t h e d e f o r m a t i o n r e s p o n s e t h a t i s s t r o n g l y dependent on t h e l o a d i n g d i r e c t i o n a. C o n t r a c t i v e volume changes and s h e a r d e f o r m a t i o n s a r e shown t o i n c r e a s e w i t h i n c r e a s e i n a. When compared t o t h e p r i n c i p a l s t r e s s d i r e c t i o n s , p r i n c i p a l s t r a i n i n c r e m e n t d i r e c t i o n s a l w a y s d e v i a t e towards t h e h o r i z o n t a l , w i t h t h e e x c e p t i o n o f l o a d i n g a t a=0 ( v e r t i c a l compression). Effects of induced strain anisotropy were observed to be negligible at R-levels less than 2.0. Continuous principal stress rotation at constant principal stress magnitudes induces progressive accumulation of both volumetric contractions and shear distortions. Additional cycles of rotation in the same direction result in progressively smaller incremental strains, implying hardening effects of previous rotations. Deformations are shown to increase significantly with decrease in relative density and increase in R and o'. m The e f f e c t of continuous variations in b, under constant R, o' and ' m a, has been evaluated for the f i r s t time. Deformations increase progressively as b increases from 0 to 1, regardless of the direction a. During proportional loading i n general stress space, which corresponds to changes i n o 1 alone, the s t r a i n directions remain r a m constant, and hence the nature of inherent anisotropy i s essentially preserved at R-levels up to about 2.0 and at relative densities greater than about 30%. - i i i -TABLE OF CONTENTS Page Abstract i i L i s t of Figures i x L i s t of Tables x i v L i s t of Symbols xv Acknowledgements . x v i 1. INTRODUCTION 1 2. REVIEW OF STRESS PATH TESTS AND BEHAVIOUR OF SANDS 6 2.1 LABORATORY SHEAR TESTING DEVICES 7 2.1.1 Devices with Fixed P r i n c i p a l Stress Directions.. 7 2.1.2 Devices Allowing P r i n c i p a l Stress Rotation 12 2.2 SAND BEHAVIOUR UNDER GENERAL STRESS PATHS 16 2.2.1 Anisotropy 16 2.2.1.1 Inherent Anisotropy 17 2.2.1.2 Induced Anisotropy 26 2.2.2 Sand Response Under Continuous Rotation of Pr i n c i p a l Stresses 28 2.2.3 Effect of Intermediate P r i n c i p a l Stress 3A 2.3 SUMMARY Al 3. STRESS NONUNIFORMITIES IN HCT SPECIMENS 43 3.1 INTRODUCTION A3 3.2 STRESSES AND STRAINS IN HCT SPECIMENS AA 3.3 STRESS NONUNIFORMITY IN THE WALL A7 - i v -TABLE OF CONTENTS (Continued) Page 3.A NONUNIFORMITY COEFFICIENTS FOR GENERAL STRESS STATES 51 3.5 INFLUENCE OF PRINCIPAL STRESS RATIO 51 3.6 INFLUENCE OF MEAN EFFECTIVE STRESS 55 3.7 INFLUENCE OF SPECIMEN GEOMETRY 57 3.7.1 Specimen Thickness 57 3.7.2 Specimen Diameter 59 3.7.3 Specimen Height 62 3.7.A Recommended Geometry 62 A. APPARATUS AND EXPERIMENTATION 66 A.l THE UBC-HCT TESTING DEVICE 66 A.1.1 General Description 66 A.1.2 Loading and Measuring Systems 67 A.2 DATA ACQUISITION 80 A.3 SPECIMEN PREPARATION 81 A.3.1 Reconstitution of Sand Specimens 81 A.3.2 Preliminary Preparation Steps 82 A.3.3 Specimen Preparation Steps 83 A.3.A Test Preparation Steps 87 A.A PERFORMANCE AND CONTROL 89 A.A.I Reproduction of Conventional Test Results 89 A.A.2 Stress Path Control 91 A.A.3 Repeatability of Results 93 - v -TABLE OF CONTENTS (Continued) Page 4.5 EXPERIMENTAL PROGRAM 93 4.5.1 Materials Tested 93 4.5.2 Testing Program 98 5. RESULTS AND DISCUSSION 102 5.1 INTRODUCTION 102 5.2 INITIAL ANISOTROPY 102 5.2.1 Hydrostatic Loading 102 5.2.2 Shear Loading 103 Stress-Strain Behaviour 103 Direction of Strain Increment 110 Strain Paths 112 5.2.3 Conclusions 114 5.3 CONTINUOUS PRINCIPAL STRESS ROTATION 114 5.3.1 Rotation Tests on Loose Sand 117 Stress-Strain Behaviour 117 Direction of Strain Increment 119 5.3.2 Effect of Relative Density 121 Stress-Strain Behaviour 121 Direction of Strain Increment 121 5.3.3 Effect of Mean Effective Stress 124 Stress-Strain Behaviour 124 Direction of Increment 124 - v i -TABLE OF CONTENTS ( C o n t i n u e d ) Page 5.3.A E f f e c t o f P r i n c i p a l S t r e s s R a t i o 127 S t r e s s - S t r a i n B e h a v i o u r 127 D i r e c t i o n o f S t r a i n Increment 127 5.3.5 E f f e c t o f I n t e r m e d i a t e S t r e s s Parameter 130 S t r e s s - S t r a i n B e h a v i o u r 130 D i r e c t i o n o f S t r a i n Increment 130 5.3.6 R o t a t i o n T e s t s o f E r k s a k Sand 133 E f f e c t o f R e l a t i v e D e n s i t y 133 E f f e c t o f P r i n c i p a l S t r e s s R a t i o 135 5.3.7 C o n c l u s i o n s 138 5.A INTERMEDIATE PRINCIPAL STRESS PARAMETER 1A0 5.A.1 Shear T e s t s a t V a r i o u s b 1A1 S t r e s s - S t r a i n B e h a v i o u r 1A1 D i r e c t i o n o f S t r a i n Increment 1A1 S t r a i n P a t h s 1AA 5.3.2 C o n t i n u o u s V a r i a t i o n s i n b 146 S t r e s s - S t r a i n B e h a v i o u r 146 D i r e c t i o n o f S t r a i n Increment 146 S t r a i n P a t h s 148 5.4.3 C o n c l u s i o n s 152 5.5 PROPORTIONAL LOADING IN GENERAL STRESS SPACE 153 5.5.1 I n c r e a s e i n o 1 a t v a r i o u s a 153 m S t r e s s - S t r a i n B e h a v i o u r 153 - v i i -TABLE OF CONTENTS (Continued) Page Direction of Strain Increment .... 155 Strain Paths 158 5.5.2 Behaviour of Other I n i t i a l Conditions 158 Effect of Relative Density 160 Effect of Intermediate Principal Stress 164 Effect of Principal Stress Ratio 167 5.5.3 Conclusions 170 5.6 ADDITIONAL INVESTIGATIONS ON STRESS-STRAIN BEHAVIOUR .. 170 5.6.1 Effect of Previous Stress History 171 5.6.2 Effect of Rotation Cycles on Subsequent Shearing 176 5.6.3 Conclusions 178 6. SUMMARY AND CONCLUSIONS 180 REFERENCES 184 APPENDIX 195 - v i i i -LIST OF FIGURES Page Figure 2.1 Schematic Representation of Laboratory Test Methods ... 9 2.2 Evidence of Strength Anisotropy from Various Devices .. 19 2.3 Evidence of Strength Anisotropy from HCT Device 22 2.4 Inherent Anisotropy of Pluviated Sands in HCT Device .. 23 2.5 Induced Anisotropy on I n i t i a l l y Isotropic Sand 27 2.6 Volumetric Strains Due to Principal Stress Rotation ... 30 2.7 Undrained State Boundary Surface of Sand (after Symes et a l . , 1984) 32 2.8 Influence of Stress Parameter b on Stress-Strain Behaviour in True Triaxial Tests (after Kjellman, 1936) 36 2.9 Reported Variations of <f>' with b in Sands 37 2.10 Influence of b on Strength and Strains at Failure (after Green, 1971) 38 2.11 Influence of b on Mobilized Strength and Strains at Yielding Condition (after Haruyama, 1981) 40 3.1 Load and Stress Conditions in HCT Specimens 45 3.2 Elastic Stresses Across the Wall of HCT Specimens 49 3.3 Stress-Strain Behaviour of Medium-Loose Sand in Triaxial Compression 50 3.4 . Nonuniformity Coefficient B3 at R = 3 52 3.5 Nonuniformity Coefficient at R = 3 53 3.6 Nonuniformity Coefficient 8^ at R = 2 54 3.7 Effect of R-Level on Nonuniformity Coefficients 56 3.8 Effect of Wall Thickness on Nonuniformity Coefficients 58 - ix -LIST OF FIGURES (Continued) Page 3.9 Effect of Inner Radius on Nonuniformity Coefficients .. 60 3.10 Sensitivity of R to Changes in Differential Confining Pressure as a Function of Inner Radius and 61 3.11 Specimen Geometry of Reported Hollow Cylinder Devices.. 64 4.1 The UBC-Hollow Cylinder Torsional Apparatus 68 4.2 Torque Loading System 72 4.3 Polished End Platen with Radial Ribs 74 4.4 Fluid Pressure and Volume Change Measuring Systems .... 76 4.5 Rotational Displacement Measuring System 78 4.6 Specimen Preparation by Water Pluviation 84 4.7 Levelling the Specimen's Upper Surface 86 4.8 Details of the UBC-HCT Device 88 4.9 Comparison of Results from HCT and Triaxial Devices ... 90 4.10 Experimental Control in Principal Stress Rotation Tests 92 4.11 Repeatability of HCT Test Results 94 4.12 Grain Size Distribution: Ottawa and Erksak Sands 95 4.13 Typical Examples of General Stress Path Tests 99 5.1 Strain Response Under Hydrostatic Loading 104 5.2 Strain Response Under Shear Loading 106 5.3 Anisotropy in Strain Response at Two R Levels 107 5.4 Anisotropy in Strain Response of Dense Sand (after Miura, 1985) 109 5.5 Directions of Principal Strain Increments in Directional Shear I l l 5.6 Strain Paths in Directional Shear 113 - x -LIST OF FIGURES (Continued) Page 5.7 Principal Stress Rotation Paths at Constant o m and b .. 116 5.8 Strain Development Due to Cyclic Principal Stress Rotation 118 5.9 Strain Increment Directions During Cyclic Principal Stress Rotation 120 5.10 Effect of Relative Density on Strain Development Due to Principal Stress Rotation 122 5.11 Effect of Relative Density on Strain Increment Directions During Principal Stress Rotation 123 5.12 Effect of Effective Confining Stress on Strain Development Due to Principal Stress Rotation 125 5.13 Effect of Effective Confining Stress on Strain Increment Directions During Principal Stress Rotation.. 126 5.14 Effect of Effective Stress Ratio on Strain Development 128 Due to Principal Stress Rotation 5.15 Effect of Effective Stress Ratio on Strain Increment Directions During Principal Stress Rotation 129 5.16 Effect of Intermediate Stress Parameter on Strain Development Due to Principal Stress Rotation 131 5.17 Effect of Intermediate Stress Parameter on Strain Increment Directions During Principal Stress Rotation.. 132 5.18 Strain Development Due to Principal Stress Rotation on Erksak Sand: Effect of D r 134 5.19 Strain Increment Directions During Principal Stress Rotation on Erksak Sand: Effect of D f 136 5.20 Strain Development Due to Principal Stress Rotation on Erksak Sand: Effect of Stress Ratio 137 5.21 Strain Increment Directions During Principal Stress Rotation on Erksak Sand: Effect of Stress Ratio 139 5.22 Effect of b on Strain Response Under Shear Loading .... 142 5.23 Effect of b on Strain Increment Directions During Shear Loading 143 - x i -LIST OF FIGURES (Continued) Page 5.24 Effect of b on Strain Paths During Shear Loading 145 5.25 Strain Development During Cyclic b-Tests 147 5.26 Principal Stresses and Strains in b-Tests on Medium-Loose Sand 149 5.27 Direction of Intermediate Strain Increment During b-Tests 150 5.28 Strain Paths During b-Tests 151 5.29 Stress Components During Proportional Loading in General Stress Space 154 5.30 Effect of a on Strain Development Due to Proportional Loading 156 5.31 Strain Increment Directions During Proportional Loading 157 5.32 Effect of a on Strain Paths During Proportional Loading 159 5.33 Effect of D r on Strain Development Due to Proportional Loading 161 5.34 Effect of D r on Strain Paths During Proportional Loading 162 5.35 Effect of b on Strain Development Due to Proportional Loading 165 5.36 Effect of b on Strain Paths During Proportional Loading 166 5.37 Effect of R on Strain Development Due to Proportional Loading 168 5.38 Effect of R on Strain Paths During Proportional Loading 169 5.39 Effect of Stress Path History on Strain Response Under Shear Loading 173 5.40 Effect of Stress Path History on Strain Response Under Principal Stress Rotation 175 - x i i -LIST OF FIGURES (Continued) Page 5.41 Effect of Previous Rotation Cycles on Strain Response Under Shear Loading 177 - x i i i -LIST OF TABLES Page Tables 2.1 Summary of Laboratory Stress Path Test Methods 8 3.1 Stress Path Devices Using Hollow Cylinder Specimens ... 63 4.1 Index Properties and Grain Characteristics of Ottawa and Erksak Sands 96 4.2 HCT Tests - Summary of Main Testing Program 100 - xiv -LIST OF SYMBOLS normal stress shear stress major effective principal stress intermediate effective principal stress minor effective principal stress 0 J / O 3 = principal effective stress ratio (oJ+Oj+op/S = mean effective normal stress (a'2-o'3) / (a[-a'3) = intermediate principal stress parameter direction of o1 relative to the deposition direction direction of major principal stress increment (do:) normal strain shear strain major principal strain intermediate principal strain minor principal strain e1+e2+e3 = volumetric strain ej-e 3 = maximum shear strain direction of major principal strain increment (dea) torque about the vertical axis vertical force internal and external confining pressures internal and external specimen radii specimen height friction angle nonuniformity coefficient for an individual stress component (see expression 3.8). nonuniformity coefficient in terms of stress ratio R (see expresion 3.9). relative density - xv -ACKNOWLEDGEMENTS The constant interest and guidance of my supervisor, Professor Y.P. Vaid, throughout this research are greatly responsible for the merits of this thesis. His professional attitude and example w i l l surely have a great influence on my geotechnical l i f e . I would like to thank the members of my Examining Committee, Professors P.M. Byrne, R.G. Campanella, Y.P. Vaid, D.L. Anderson, S. Calisal and P.V. Lade for their c r i t i c a l comments and suggestions. Thanks are also due to Professors W.D.L. Finn and V.K. Garga, who also kindly reviewed the text. Appreciation must be extended to my colleagues Dawit, Mustapha, Wije, Ralph, Carlos, Upul, Francisco, and John for help-ful and clarifying discussions. Mr. Fred Zurkirchen manufactured the testing equipment with Swiss precision and also gave valuable suggestions during the experimental program. The a b i l i t y and speed of Mrs. Kelly Lamb in typing the original manuscripts and the a r t i s t i c s k i l l s of Mrs. Monica Sayao in preparing several figures are sincerely acknowledged. Financial support for my doctoral program was granted by the National Research Council of Brazil (C.N.Pq.). Funding during the final months of the research was made available by the Department of C i v i l Engineering at U.B.C. Incentive from my colleagues at the Catholic University of Rio de Janeiro (PUC-RJ) i s also appreciated. The presence and support of my family and, very particularly, the continuous encouragement and dedication of my wife Monica throughout the various, stages of study, research and writing, made this thesis viable. A l l my efforts serve only to acknowledge my debt to them. - xv i -1 CHAPTER 1 INTRODUCTION Most sedimentary soils are inherently anisotropic. Their response to loading w i l l therefore depend on the directions of principal stresses in relation to the deposition direction. In nearly a l l geotechnical problems, principal stress directions gradually rotate during loading. As a consequence, deformations occur due not only to the change in the magnitudes of the principal stresses, but also to change in their direc-tions. For a given magnitude of principal stress increments, deforma-tions due to the accompanying change in directions depend on the inclinations of principal stresses to the material axes prior to the stress changes. Both the changes in magnitude and directions of principal stresses are embodied in the term "stress path". It i s widely recognized that s o i l behaviour i s stress path depend-ent. Yet, in most modelling of geotechnical materials, this dependence is ignored. This probably stems from the limited capabilities of conventional testing devices. Experimental characterization of inherent anisotropy and the effect of principal stress rotation in soils requires the a b i l i t y to control both the directions and the magnitudes of principal stresses on a test specimen. This i s indeed a formidable task to achieve in any testing device. Experimental research aimed at identifying inherent anisotropy and i t s effect on stress path dependent behaviour of sands has been in progress for almost two decades. This led to recent development of more versatile testing devices, like the Directional Shear Cell (DSC) and the Hollow Cylinder Torsional (HCT) apparatus (Arthur et a l . , 1977; Hight et 2 a l . , 1983). The DSC in i t s present version can be used for stress path testing only at low confining stresses and under plane strain conditions. The HCT device is much more versatile, as i t i s the only testing device capable of imposing generalized stress paths on s o i l specimens. In the HCT device, adequate control of four stress parameters can be achieved: mean normal stress, shear stress (or stress ratio), intermediate princi-pal stress magnitude and rotation of principal stress directions (in one plane). This allows separation of the effects of each stress parameter on s o i l behaviour. Such a separation of effects i s of utmost importance in a l l fundamental studies of stress-strain behaviour of inherently anisotropic materials. Current understanding of the stress-strain behaviour of sand under generalized stress conditions i s very limited. Previous research has focussed mainly on shear strength. In contrast, working stress levels i n sands are, in most f i e l d situations, well below failure. Very l i t t l e i s known about the isolated or combined effects of intermediate principal stress magnitude and major principal stress direction on strain response at low shear stress levels. Moreover, experimental data on principal stress rotation effects, obtained in devices other than the HCT, is influenced by unwanted variations in other stress parameters. Even in the HCT device, where the effects of principal stress rotations may be isolated, excessive stress (and hence strain) non-uniformities across the specimen wall may occur. Insufficient attention as to the selection of a suitable specimen geometry, together with a not so rational understanding of nonuniformity, contribute to this problem. As a consequence, serious questions about the valid i t y of HCT data have been raised in some cases. 3 The primary objective of this thesis is to investigate general stress path dependent behaviour of sand. This necessitated f i r s t of a l l the development of a hollow cylinder torsional apparatus. Minimization of the stress nonuniformities mentioned above, under generalized stress paths, was an essential requirement in the selection of specimen geometry together with the region of stress space to be explored. Theoretical considerations of stress nonuniformity, previously suggested by Hight et a l . (1983), have been c r i t i c a l l y reviewed. An important improvement to Hight et al.'s c r i t e r i a for defining and quantifying nonuniformity in HCT specimens is proposed. A systematic testing program was devised aimed at investigating the fundamental behaviour of sand under general stress paths. In particular, the effects of independent changes in only one stress parameter, while others are kept constant, are examined in detail. A l l tests have been carried out on reconstituted specimens of saturated sand under drained conditions. The main objectives of the experimental program were to examine: (1) the inherently anisotropic strain response of the sand; (2) the isolated effects of both monotonic and cyclic rotations of principal stresses, at various stress levels; (3) the isolated effects of the relative magnitude of the intermediate principal stress at various inclinations of principal stress directions; and (4) the isolated effects of increasing mean effective stress under constant principal stress ratios and directions. Sand behaviour is examined predominantly at i n i t i a l medium-loose relative density of about 34% and at low shear stress levels (principal stress ratio R £ 2.0, where R = a[/a'3). The effects of density and shear stress states on behaviour are also addressed. Stress-strain behaviour is 4 examined and discussed separately in terms of volumetric and shear strain responses. Whenever appropriate, the response is also examined in terms of strain paths and directional changes of the major principal strain increment. These provide indication as to possible changes in inherent anisotropy on loading. In Chapter 2 , a brief review of the currently available stress path testing devices is presented, emphasizing their capabilities and limitations. This is intended to put the HCT apparatus i n a proper perspective in i t s potential as a general stress path testing device. Important findings and conclusions, from previous investigations related to the objectives of this thesis, are then c r i t i c a l l y reviewed. In Chapter 3, definitions of average stress and strain components in HCT specimens are presented, together with the assumptions involved. A c r i t i c a l assessment of stress distribution across the specimen's wall is also presented. A new approach is suggested for delimitation of stress space regions that can be explored with acceptable levels of stress nonuniformity within the HCT specimen. In Chapter 4, the Hollow Cylinder Torsional apparatus developed at UBC is described in detail, with emphasis on the techniques adopted for minimization of experimental errors. The specimen preparation method and experimental program are also described. In Chapter 5, the results of general stress path tests are presented and discussed. The effects of anisotropy, continuous principal stress rotation, intermediate principal stress magnitude and mean effective stress are isolated and presented in a systematic manner. Observations on the effect of previous stress history on sand behaviour during subsequent loading are also presented. 5 In Chapter 6, the relevant conclusions presented in previous chapters are summarized. Emphasis is given to the experimental findings and their possible implications in geotechnical engineering. 6 CHAPTER 2 REVIEW OF STRESS PATH TESTS AND BEHAVIOUR OF SANDS I n c o n v e n t i o n a l a p p r o a c h , sand r e s p o n s e i s c o n s i d e r e d t o depend o n l y i i o n t h e l e v e l s o f s h e a r s t r e s s ( o r s t r e s s r a t i o R = al/a3) and c o n f i n i n g I I I p r e s s u r e ( o r mean e f f e c t i v e s t r e s s o^ = (o1+o3+o3)/3). S p e c i f i c a l l y , t h e e f f e c t s o f i n t e r m e d i a t e p r i n c i p a l s t r e s s and o f p r i n c i p a l s t r e s s d i r e c t i o n s a r e d i s r e g a r d e d . T h i s a p p r o a c h o r i g i n a t e d n o t from a l a c k o f r e c o g n i t i o n o f t h e a n i s o t r o p i c n a t u r e o f sand d e p o s i t s and o f t h e p o s s i b l e s i g n i f i c a n c e o f o 3 ( u s u a l l y s t u d i e d by means o f t h e n o r m a l i z e d p a r a m e t e r b = ( o 3 - o 3 ) / ( O j - o 3 ) ) on sand b e h a v i o u r , b u t stemmed from t h e l i m i t a t i o n s o f commonly u s e d l a b o r a t o r y t e s t i n g d e v i c e s ( e . g . , t h e c o n v e n t i o n a l t r i a x i a l ) t o s i m u l a t e more r e a l i s t i c l o a d i n g c o n d i t i o n s . The h o l l o w c y l i n d e r t o r s i o n a l (HCT) d e v i c e i s t h e o n l y t y p e o f equipment t h a t e n a b l e s t h e c o n t r o l o f s i m p l e as w e l l as g e n e r a l i z e d s t r e s s p a t h s . S p e c i f i c a l l y , i t has t h e u n i q u e c a p a b i l i t y o f a l l o w i n g i n d e p e n d e n t c o n t r o l o f t h e magnitudes o f t h e t h r e e p r i n c i p a l s t r e s s e s t o g e t h e r w i t h t h e d i r e c t i o n o f o 1 ( u s u a l l y i n d i c a t e d by t h e r o t a t i o n a n g l e a, w h i c h o a makes w i t h t h e v e r t i c a l d e p o s i t i o n d i r e c t i o n ) . The HCT d e v i c e i s t h u s s u i t a b l e f o r a s y s t e m a t i c i n v e s t i g a t i o n o f t h e i s o l a t e d e f f e c t s o f e a c h s t r e s s p a r a m e t e r (R, a ' , b a n d a) on t h e d e f o r m a t i o n m r e s p o n s e o f san d s . A b r i e f o v e r v i e w o f t h e v a r i o u s l a b o r a t o r y s h e a r t e s t i n g d e v i c e s i s f i r s t p r e s e n t e d . T h i s i s i n t e n d e d t o h i g h l i g h t t h e d i f f e r e n t r e g i o n s o f s t r e s s s p a c e t h a t can be i n v e s t i g a t e d w i t h e a c h d e v i c e . I n a d d i t i o n , t h e n o n u n i f o r m i t i e s o f s t r e s s and s t r a i n , w h i c h a r e i n h e r e n t l y a s s o c i a t e d 7 with a l l testing devices, are pointed out. Advances in fundamental experimental s o i l mechanics research have been achieved only by accepting and minimizing these nonuniformities. The review clearly brings out the ve r s a t i l i t y of the HCT apparatus over other testing devices in conducting fundamental research on s o i l behaviour under generalized stress conditions. In the second part of this chapter, previous investigations into the drained anisotropic behaviour of sands are c r i t i c a l l y reviewed. Findings related to the nature of inherent and induced anisotropy are examined. The effects of principal stress rotation a, as well as of the intermedi-ate stress parameter b, on sand response are also summarized. Areas of research which have been the subject of controversial conclusions or not yet addressed are pointed out. 2.1 LABORATORY SHEAR TESTING DEVICES The various laboratory shear testing devices can be broadly cl a s s i -fied into two groups, based on their a b i l i t y to impose continuous rota-tions of the principal stress directions (Table 2.1). 2.1.1 Devices with Fixed Principal Stress Directions In the devices included in this group, only normal stresses can be applied to the specimen boundaries. As a consequence, the principal stresses have their directions fixed and only 90° jump rotations can be imposed. The most common example i s the standard t r i a x i a l test on cylindrical specimens (Bishop and Henkel, 1962), where only two stress components (Oj and o 3) can be controlled (Fig. 2.1a). Table 2.1. Summary of Laboratory Stress Path Test Methods Test Method Principal Stress Control Imposed Conditions Main Characteristics Selected References Fixed Principal Stress Directions Standard Triaxial °i a2=o3 & a=0(compression)or o2=o1 & a=90°(extension) most used in practice; axisymmetric stress paths; °r =°9 a s s u r a e < ^ Bishop & Henkel (1962) Bishop & Wesley (1975) Fixed Principal Stress Directions Plane Strain Triaxial °i ° 3 e2=0 a=0 (compression) or a=90°(extension) simulates many f i e l d condi-tions; f r i c t i o n effects at rigid boundaries Cornforth (196A) Campanella & Vaid (1973) Fixed Principal Stress Directions True Triaxial °i ° 2 ° 3 a=0 or 90° stress or strain controlled cubical specimens with flexible and/or r i g i d boundaries Kjellman (1936) Ko & Scott (1967) Pearce (1971) Fixed Principal Stress Directions Hollow Cylinder Triaxial ° 1 ° 2 ° 3 a=0 or 90° V p i ' Th=° n i A stress controlled assumption for o r and Og distributions; large non-uniformities near failure Kirkpatrick (1957) Broms & Ratman (1963) Rotating Principal Stress Directions Torsional Triaxial ° 1 °2 ° 3 sin2a=b cylind. specimens: non-uniform; x zg distr. and or=Og assumed hollow cylinders: P e =P^ Habib (1953) Saada & Baah (1967) Rotating Principal Stress Directions Simple Shear K 0-consolidation e2=ey=0 ; ex=0 cylind. or cubical specimen; simulates f i e l d ; no control on a; large stress non-uniformities Kjellman (1951) Roscoe (1953) Budhu (1984) Rotating Principal Stress Directions Directional Shear Cell ° 1 ° 3 a e,=0 . stress controlled flexible boundaries in the plane of strain; low stress levels; complex techniques for strain measurement Arthur et a l . (1977) Sture et a l . (1987) Rotating Principal Stress Directions Hollow Cylinder Torsional a i ° 3 a V p i ; V° usually stress controlled assumptions for o r, Og and x zg distribut.; large non-uniformities near failure; most general stress path device Broms & Casbarian (1965) Hight et a l . (1983) Sayao & Vaid (1988) 07 07 ever 07 = cr e a) Triaxial Gi < J > h I I cr, b) Plane st ra in ^^0T = CTe s in 2 oc = b e) Torsional tr iaxial o > o > c r x c) True t r i ax ia l 07 > (T2 > <T3 d) Hollow cylinder t r iax ia l 6X = fly =0 o;,cr2 , c r 3 , c < = ( ? ) f ) Simple shear g) Direct ional shear cell h) Hollow cylinder torsional Figure 2.1 Schematic Representation of Laboratory Test Methods. 10 Interpretation of t r i a x i a l test data is normally based on the assumption that = (and = e^). However, experimental observa-tions (Casbarian and Jamal, 1963; Frydman et a l . , 1971) have indicated that this assumption may be inaccurate at large strain levels due to end restraint effects. This aspect has been discussed i n detail by Saada and Townsend (1981). Recognition of the fact that many geotechnical problems can be better approximated by a plane strain condition has led to the develop-ment of plane strain testing techniques (Cornforth, 1964; Campanella and Vaid, 1973). Rectangular prismatic specimens are normally used and a condition of zero longitudinal strain i s imposed by a pair of fixed r i g i d plates (Fig. 2.1b). Consequently, no control on the magnitude of o a can be exercised. Minimization of fr i c t i o n a l forces at the r i g i d boundary plates must be ensured, in order to keep o 2 in the longitudinal direction. Improvement on the stress path testing capabilities can be further achieved with true t r i a x i a l devices, where the three principal stress magnitudes can be independently controlled (Fig. 2.1c). In particular, the influence of the parameter b on the stress-strain characteristics of soils can be properly assessed. Several true t r i a x i a l devices have been described in the literature. Depending on the boundary conditions imposed on the specimen, three basic types can be pointed out: (1) flexible boundaries, where ideally uniform principal stresses are applied to a l l 6 specimen faces through pressur-ized bags or membranes (Ko and Scott, 1967; Arthur and Menzies, 1968); (2) r i g i d boundaries, where uniform strains are imposed by r i g i d plates (Pearce, 1971; Hosseini and Cousens, 1988); (3) mixed boundaries, where 11 r i g i d plates and flexible membranes are both used to apply boundary strains or stresses (Green, 1971; Lade, 1978). The distinct advantages and limitations of each type have been indi-cated by Sture and Desai (1979). Interference between adjacent plates or membranes at the specimen's edges seems to be a common d i f f i c u l t y with most devices that gives rise to stress nonuniformities. A simpler alternative for imposing multiaxial stress paths could be achieved with hollow cylinder specimens, by independently controlling external and the internal confining pressures (P and P^, respectively) together with axial load (Fig. 2.Id). When no torque is applied, o , o and OQ can be considered as principal stresses. This configuration has been frequently u t i l i z e d in testing either sands (Kirkpatrick, 1957; Wu et a l . , 1963; Broms and Jamal, 1965; Arnold and Mitchell, 1973) or clays (Wu et a l . , 1963; Suklje and Drnovsek, 1965; Anderson et a l . , 1988). Although attractive because of i t s simplicity, criticism against the use of hollow cylinder t r i a x i a l devices has often been raised. This i s due to the inherently nonuniform distribution of stresses across the specimen's wall when the condition P g^ P^ i s imposed. In this case, average values of assumed o and an d i s t r i b u t i o n s across the wall are 6 r 0 normally used in prescribing the state of stress. More recently, i t has been shown that these stress nonuniformities can be greatly minimized by a suitable choice of specimen dimensions and avoiding certain stress paths (Hight et a l . , 1983; Sayao and Vaid, 1988b). This subject i s analyzed in detail in the next chapter. In a l l the devices mentioned so far, the principal stresses are fixed in the vertical and horizontal directions. Thus, stress paths with direction angle a different than 0 or 90° can only be achieved on 12 " t i l t e d " specimens. This has been frequently reported in investigations on inherent anisotropy (Arthur and Menzies, 1972; Oda et a l . , 1978), as reviewed i n Section 2.2.1. The results obtained from tests on t i l t e d anisotropic specimens have been the subject of serious criticism (Saada and Townsend, 1981; Saada, 1988). When the specimen's axis of symmetry does not coincide with the loading axis, highly non-uniform distortions may result due to end restraint. Preference should thus be given to test methods where the principal stress o x, rather than the specimen axis, i s inclined. 2.1.2 Devices Allowing Principal Stress Rotation Three-dimensional stress states can also be achieved by applying a t o r s i o n a l load T^ to the end platens of standard t r i a x i a l specimens. Independent control of axial load, c e l l pressure and torque about the vertical axis (Fig. 2.1e) allows loading with different magnitudes of the 3 principal stresses (Habib, 1953; Ishihara and L i , 1972). With this configuration, o x and o 3 simultaneously rotate i n the vertical plane normal to the radial direction. The rotation angle a is a direct function of o x, o 3 and o 3 and thus can not be independently controlled. Based on the usual assumption of 0^=0^, and from Mohr ci r c l e construc-tion, i t can be shown that si n 2 a = b. The main criticism against the use of torsional t r i a x i a l devices l i e s on the inherently non-uniform distribution of torsional shear strains in the radial direction (zero at the center and maximum at the external surface of the cylindrical specimen). As a consequence, tor-sional shear stresses are also radially non-uniform, the only exception being for r i g i d plastic materials. Average values of shear stress and 13 shear strain need then to be defined in order to consider the specimen as a single s o i l element. Stress non-uniformities are greatly reduced when thin hollow cylindrical specimens are used (Saada and Townsend, 1981). In addition, the assumption of uniform = Og is better approximated but not entirely guaranteed by maintaining = P g (Hight et a l . , 1985). However, the stress condition sin 2o = b s t i l l remains unchanged. Consequently, independent assessment of the effects of four non-zero stress parameters (o 1, R, b, a) can not be achieved, m Another shear device that imposes continuous rotation of principal stresses under plane strain conditions is the simple shear apparatus (Fig. 2.If). Depending on the specimen's cross-section, two basic configurations of simple shear devices have been used: (1) short cylindrical specimens (Kjellman, 1951; Bjerrum and Landva, 1966); and (2) rectangular prismatic specimens (Roscoe, 1953). The limitations of simple shear devices have been the subject of extensive discussions (Hvorslev and Kaufman, 1952; Duncan and Dunlop, 1969; Saada and Townsend, 1981; Lacasse and Vucetic, 1981; Budhu, 1984) . Complementary shear stresses x are inherently absent on the t i l t i n g boundaries of the simple shear specimen. Hence, considerations of equilibrium and boundary conditions require the distribution of both shear and normal stresses to be necessarily non-uniform on the specimen's surfaces. In addition, principal stress directions and magnitudes are either unknown or uncontrolled. The device is therefore not suitable for fundamental investigations of the effects of principal stress rotations on s o i l behaviour. However, i t s close modelling of some practical f i e l d 14 situations has made the simple shear device attractive for obtaining design strength parameters. In an attempt to overcome the limitations of simple shear devices, Arthur et a l . (1977) developed the directional shear c e l l (DSC). In this device, normal and shear stresses can be independently controlled on four faces of a cubical specimen, as illustrated in Fig. 2.1g. These stresses are applied through flexible membranes (pressure bags and shear sheets), while nominal plane strain conditions are maintained by having the speci-men constrained between smooth r i g i d end platens on the other two faces. Conceptually, the operation of the DSC is very attractive, since i t resembles an infinitesimal element under plane strain conditions. In practice, however, simultaneous application of uniform normal and shear stresses to different faces of a test specimen is not a simple task. The distribution of stresses can be expected to be reasonably uniform, although some details associated with the shear loading system are s t i l l i n . need of further improvements (Arthur et a l . , 1981a; Arthur, 1988). The distribution of strains i n the central plane has been suggested to be reasonably uniform i n dense specimens (Arthur et a l . , 1981b; Wong and Arthur, 1986). However, strain differences of about 10 to 15% of the average value on the central plane have been observed along the a3 direction, probably due to end restraint at the r i g i d boundaries (Wong and Arthur, 1986). Moreover, DSC specimens of loose sand have been reported to exhibit significant nonuniformity of strains (Arthur et a l . , 1981a). The DSC, i n i t s present stage of development, s t i l l presents several practical limitations, the most important being the restriction to very low s t r e s s l e v e l s ( u s u a l l y o 3 £ 14 kPa) . At these low confining 15 stresses, relatively large dilative volumetric strains and high f r i c t i o n angles can be expected. Accuracy i n strain measurements is also d i f f i c u l t to achieve in the DSC, because of the specimen's flexible boundaries. Radiographic and photographic techniques have been used (Arthur et a l . , 1977, 1981a) with obvious disadvantages in terms of cost and i n not providing instantaneous information for the needed corrections of specimen geometry. Alternatively, a rather complex system of 20 displacement transducers has been described (Sture, 1986; Sture et a l . , 1987) to obtain average values of boundary strains corrected for the movements of the loading frame. Since the DSC i s a plane strain device, no control of aa (or b) can be achieved. The device has been mainly used in the investigations of inherent and induced anisotropy. Improvements in the device have been recently suggested and can be expected over the near future. Wong and Arthur (1986) have reported attempts to measure o2 by replacing one of the r i g i d platens by a flexible bag f i l l e d with de-aired water. However, these attempts were only partly successful since values of e 2 up to 0.7% were observed, causing departure from the plane strain condition. Undrained tests have also been tried (Sture et a l . , 1985) but apparently without provision for pore pressure measurements. The only device suitable for general stress path tests in which 4 stress components (e.g., olt o 2, o 3 and a) can be independently controlled i s the hollow cylinder torsional (HCT) device. As illustrated i n Fig. 2.1h, HCT s o i l specimens are subjected to different internal and external chamber pressures (P. ^ P ), axial force (F ) and torque (T, ) i e z n about the vertical axis. 16 The HCT device may be seen as a combination of the hollow cylinder t r i a x i a l and the torsional t r i a x i a l devices previously discussed. As a re s u l t , nonuniform d i s t r i b u t i o n s of o , o. and t q are also present r o zo across the wall i n the HCT specimen. As suggested by Hight et a l . (1983) and Sayao and Vaid (1988b), these non-uniformities can however be greatly minimized by suitable selection of specimen's dimensions and by avoiding c e r t a i n regions of the stress space (o^, R, b, a). In particular, the HCT i s most suitable for investigating stress-strain behaviour of soils subjected to stress states substantially below failure. This topic i s discussed in detail in the next chapter. 2.2 SAND BEHAVIOUR UNDER GENERAL STRESS PATHS 2.2.1 Anisotropy It has long been recognized that the mechanical behaviour of granular materials i s influenced by their anisotropic properties. One of the earliest experimental observations of anisotropic strain response of sand was reported by Kjellman (1936). He noted significant differences in the three principal strains during hydrostatic compression of a cubical specimen. A convenient distinction between two types of anisotropy was f i r s t suggested by Casagrande and Carrillo (1944). Inherent anisotropy was considered as a physical characteristic inherently present in the material before the straining process i s initiated. Induced anisotropy, on the other hand, was defined as due exclusively to the strain associated with the applied stresses. These definitions were later extended to include past strain events as a part of inherent (or i n i t i a l ) anisotropy (Arthur and Menzies, 1972; 17 Wong and Arthur, 1985). As stated by Saada (1981), "a stress induced anisotropy is an inherent anisotropy for the next state of stress". 2.2.1.1 Inherent Anisotropy In nearly a l l natural sand deposits, the mode of deposition and the shape of individual grains introduce some form of inherently anisotropic fabric to the s o i l . Yet, in practice, soils are frequently modelled as isotropic in the interest of simplicity. Alternatively, some models consider the s o i l as possessing an inherent cross-anisotropy. This is characterized by a vertical axis of symmetry and, consequently, a hori-zontal plane of isotropy. Cross-anisotropic fabric could result natur-a l l y from vertical sedimentation in approximately horizontal layers. Experimental evidence in support of inherent anisotropy of undisturbed sand samples, obtained by a freezing technique, has been presented by Ladd et a l . (1977). Conventional hydrostatic compression in the t r i a x i a l c e l l indicated the radial strains in loose Niigata sand to be about 2.A times larger than the vertical ones. Similar conclusions have been reported by Miura and Toki (198A) from drained compression and extension t r i a x i a l tests on undisturbed sand specimens trimmed from blocks. Block samples were obtained by freezing at two sites of relatively homogeneous sand deposits. Recent research on anisotropic characteristics of sands has, how-ever, concentrated on reconstituted pluviated specimens (Arthur et a l . , 1981a; Symes, 1983; Negussey, 198A; Miura, 1985). Pluviation closely * simulates the sedimentation process occurring in many deposits in-situ. *Details on the water pluviation technique are presented in Chapter A (Item A.3). 18 Reconstitution of homogeneous specimens of controlled density by pluvia-tion thus provides a convenient manner of studying properties of sand in situ. Anisotropy in Shear Strength Experimental observations of the inherent strength anisotropy of pluviated sands in different devices have been reported by many researchers (Fig. 2.2). In each test, the rotation angle a was kept constant. In devices where the principal stress o x is fixed in direc-tion, loading at inclination a was obtained by forming the specimen in a t i l t i n g mould. The disadvantages of this technique have been already pointed out in Section 2.1.1. It can be noted, from the results presented in Fig. 2.2, that the f r i c t i o n angle cf>' varies with the direction of loading. The highest value of <j>' corresponds to a = 0 (Oj coincident with the deposition direction). This is consistent with the lower deformability observed in the vertical deposition direction in hydrostatic compression. Some controversy arises, however, in relation to the value of a at which minimum strength was observed. Some investigators have indicated that the lowest <J>' occurs when o x is perpendicular to the deposition direction (a = 90°). This was attributed to the preferred horizontal orientation of sand particles in pluviated specimens (Oda, 1972; Arthur and Menzies, 1972). Oda's investigations on sand fabric w i l l be reviewed in more detail later. Plane strain results reported by Oda et a l . (1978) indicate, how-ever, minimum <J>' values at a between 65° and 75°. Observation of failure 19 55 0 30 60 CT, _ d i r e c t i o n , oc ( d e g ) Symbol Device Sand D r(%) CTC' (kPa) R e f e r e n c e T 1 Standard Triaxial To yo Lira 8 6 <72'= 0-3=50 Oda et a l , 1978 T 2 8 3 cr2'=03=100 Oda , 1976 T 3 Plane Strain 8 9 cr3* = 50 Oda et a l , 1978 T 4 cr3'=4oo Oda et a l , 1978 LB 1 Leighton Buzzard 9 0 Arthur $ Assadi, 1977 LB2 True Triax. C2'=fJ3 = 55 Arthur 4 Menzies, 1972 A D.S.Cell 0-3'= 14 Wong 4 Arthur, 1985 Notes : ( I ) all tests dra ined; (2) DSC tests non - t i l ted specimen Figure 2.2 Evidence of Strength Anisotropy from Various Devices. 20 planes nearly parallel to the bedding plane was directly associated to * the indicated direction of lowest shear strength. Figure 2.2 suggests that, anisotropic characteristics seem to be more accentuated under plane strain conditions, than under t r i a x i a l stress conditions. This can be partly attributed to the differences in the intermediate stress parameter b in the two types of tests. Also presented in Fig. 2.2 are results consisting of three data points reported by Wong and Arthur (1985) from tests in the directional shear c e l l . In this case, the specimen's axis of symmetry is vertical and the inclination a i s controlled through the boundary shear and normal loads. Despite these differences in testing techniques and the low value of confining pressure (o 3 = 14 kPa), DSC results seem to agree with the plane strain data previously reported by Arthur and Assadi (1977). Unfortunately, the value of o 3 used in these plane strain tests was not reported. Thus, the significantly lower strength observed from true t r i a x i a l tests on the same material (Arthur and Menzies, 1972) can not be f u l l y explained. As already discussed in Section 2.1, torsional t r i a x i a l tests on solid cylinders and simple shear tests suffer from significant stress non-uniformities. Therefore, torsional tests on hollow cylinders are, together with DSC tests, the other options available for studies of anisotropy of sands. *From simple considerations of Rankine's active stress conditions, f a i l -ure planes should be inclined at i = (45° - (f>'/2) to the major principal stress direction. For dense Toyoura sand under plane strain conditions (<f>' = 46°), inclination i = 23°. Lowest fri c t i o n a l strength could be anticipated when failure and bedding planes become nearly parallel. This condition i s thus indicated by loading direction a = 90° - i = 67°, which agrees with the experimental observations. 21 The available evidence of strength anisotropy of sands from HCT tests i s summarized in Fig. 2.3. The general trends in c/>' indicated by Symes et a l . (1982, 1983, 1984 and 1988) and Miura (1985) agree with the results previously shown in Fig. 2.2. However, the surprisingly high value of <$>' = 65° at a = 45° (in contrast with cj>' = 46° at a = 0), reported by Tong (1975), clearly suggests problems with his equipment. Also, extremely high f r i c t i o n angles are indicated by Miura (1985), especially at low values of inclination a. In this case, however, unfavourable combination of stress conditions and specimen dimensions was suggested to have seriously affected the results (Shibuya and Hight, 1986). Minimum values of <J>' at a = 65 to 70° can also be noted in Fig. 2.3 for both Toyoura and Ham river sands in dense states. This observation i s i n agreement with Oda et a l . (1978) plane strain test results previously discussed. Further investigations of strength anisotropy using the DSC device have been presented by Arthur et a l . (1981a). Despite marked inherent anisotropy in stress-strain behaviour, these researchers have suggested l i t t l e influence of principal stress direction on shear strength. This i s in contradiction to the results shown in Figs. 2.2 and 2.3, and in particular to other plane strain results using t i l t e d specimens. Anisotropy in Stress-Strain Behaviour Studies on the drained anisotropic stress-strain behaviour of pluviated sands, carried out in the HCT device are reproduced in Fig. 2.4 (Miura, 1985; Symes et a l . , 1988). A strong decrease i n distortional stiffness with increase in a may be noted for both dense and medium dense 22 70 3 0 I— 1 1 1 1 1 1 L_ 0 30 60 9 0 0", — d i r e c t i o n , o c ( d e g ) Symbol Sand D r(%) CTm(kPa) Condition Reference A Ottawa 84 =^50 drained Tong, 1975 T Toyoura 82 98 Miura, 1985 H 1 Ham r iver 88 2 0 0 Symes et al.1982 H 2 Symes, 1983 H 3 44 Symes et al,l988 H 4 200(init.) undrained Symes et al,l984 Note •- all tests with b= 0.50 Figure 2.3 Evidence of Strength Anisotropy from HCT Device. b) Ham r i ve r s a n d , water pluviation ( a f t e r Symes et a l , 1988) Figure 2.4 Inherent Anisotropy of Pluviated Sands i n HCT Device. 24 sands. As already pointed out, excessive stress nonuniformity renders results of Miura (1985) questionable. It may be noted that although the test performed with o x in the vertical direction was terminated at R = 20 (mobilized <j>' = 65°), indication of failure had not yet been observed. At this stress condition, nonuniformities within the hollow cylinder specimen would be unacceptably high (see Chapter 3). Although to a much smaller scale, similar criticism may be raised against Symes et a l . (1988) HCT tests on medium dense sand. When the test with a = 0 condition was terminated, mobilized f r i c t i o n angle was already close to 47° and high levels of stress nonuniformities could be expected. In addition, a l l tests reported by Symes and his co-workers are not on virgin pluviated specimens, as noted by Sayao and Vaid (1989a), since a drained " t r i a x i a l " compression cycle was routinely applied to every HCT specimen. Dense specimens were vertic a l l y preloaded to a principal stress ratio R = 2.6, and loose to R = 2.0 (Symes, 1983). These R levels were shown to correspond to more than 50% of the maximum shear strength. This load-unload cycle may have significant hardening effects on the strain response upon subsequent reloading stages. It could also cause alteration in the inherent anisotropic fabric of the sand. Fabric Anisotropy Also in agreement with the experimental results presented in Figures 2.2 to 2.4 are the concepts of fabric anisotropy introduced by Oda (1972 and 1976). The anisotropic response of homogeneous sand deposits is considered to be dictated by two factors: (1) preferred alignment of non-spherical particles; and (2) preferred orientation of contact normals. 25 During deposition under the action of gravity, sand grains tend to have their long axis aligned in a nearly horizontal direction. Examina-tion of this preferred alignment has been made possible by preparing vertical and horizontal thin sections of both natural and reconstituted sand samples. This was done after i n f i l t r a t i o n of a resin binder into the s o i l voids (Oda, 1972). Experimental evidence indicating that anisotropy in shear strength i s significantly affected by the geometric shape of sand particles has been presented by Oda (1976). Anisotropy was in this case defined by the ratio between shear strengths of specimens loaded at a = 90 and 0°. The lower values of strength ratio obtained with sands containing elongated grains was attributed to a pronounced horizontal orientation of particles. It i s interesting to point out that sand deposits consisting of perfect spheres would show, by definition, no preferred orientation of grains. However, anisotropic response of specimens made of pluviated spherical grains has been demonstrated by several investigators (Oda, 1981; Haruyama, 1981; Shibuya and Hight, 1987). This has been attributed to the second factor, already mentioned, affecting anisotropy of sands: preferred orientation of the normals to the planes of contact between any two particles. Oda (1976) has indicated a marked preferred orientation of these contact normals in a direction parallel to deposition, for specimens composed of either elongated or spherical grains. The most important conclusion from Oda's work seems to be that any granular s o i l , when subjected to vertical gravitational deposition, tends to develop a significantly anisotropic fabric. As a result, inherently anisotropic 26 stress-strain-strength chara'cteristics are to be expected, with s t i f f e s t response occurring for loading with o x in the deposition direction. This has been already indicated in Fig. 2.4. It i s important to note, however, that other methods of specimen preparation may result in less marked degrees of inherent anisotropy as a consequence of a more random orientation of contact normals. An example of this has been recently described by Chen et a l . (1988). Hollow cylindrical specimens of glass spheres, prepared by compaction in five layers, were suggested to be nearly isotropic. 2.2.1.2 Induced Anisotropy The mechanism of fabric changes associated with shearing of a granular mass i s significantly dependent on the i n i t i a l (inherent) geometric fabric (Oda, 1976). Induced anisotropy is more conveniently studied from an i n i t i a l l y isotropic fabric. When subjected to shearing stresses, the spatial arrangement of solid particles and the associated voids of a granular mass progressively change (Oda et a l . , 1985). As a result, new geometric fabric gradually evolves and the sand becomes increasingly anisotropic. Elongated grains tend to become aligned in a d i r e c t i o n p e r p e n d i c u l a r to the major p r i n c i p a l s t r e s s o x. Simultaneously, increasing concentration of contact normals along the direction of o x is produced (Oda et a l . , 1985). Experimental observations of induced anisotropy have also been carried out in different shear devices. Fig. 2.5 reproduces the results reported by Arthur et a l . (1981a) and Mould et a l . (1982) respectively from the DSC and the cubical t r i a x i a l devices. In both cases sand specimens were i n i t i a l l y preloaded by increasing o x in the original plane DRY LEI6HT0N BUZZARD SAND ( D r = 9 5 % ) ,0 RELOADING o * O 4 / / / f j £ 14 KPa P - AVERAGE PRE-LOADING CURVE ( o c = 0) J I I L_ A) PLANE STRAIN LOADING IN THE DIRECTIONAL SHEAR CELL ( a f t e r Arthur et al , 1981a ) DRY LEIGHTON BUZZARD SAND ( D r = 95 % ) " m a x = ° i - <-3 B) AXI -SYMMETRICAL LOADING IN THE CUBICAL TRIAXIAL APPARATUS ( af ter Mould et al , 1982) Figure 2.5 Induced Anisotropy on I n i t i a l l y I sotropic Sand. 28 of isotropy (normal to the deposition direction) and then unloaded to a hydrostatic stress condition. Pronounced decrease in stiffness with increasing inclinations a of o x direction can be noted for subsequent reloading. Mould et a l . (1982) have also carried out an additional test in which the specimen was again unloaded and then reloaded in the original preloading direction. The observed stress-strain behaviour in this second reloading was nearly similar to the f i r s t reloading, although the loading direction was changed by 90°. This was simply suggested by Mould et al to be "caused by isotropic processes taking place as the s o i l fabric dilates", but no further discussion was presented. It i s clear.however that, after two successive loadings in different directions, producing relatively high shear s t r a i n l e v e l s (Y = 3% i n each loading), a rather complex 'max b r geometric fabric should result. This induced fabric i s probably very much dependent on the previous al loading directions and strain levels and thus no simple explanation of the observed behaviour can be offered. 2.2.2 Sand Response Under Continuous Rotation of Principal Stresses As already discussed in the previous section, sand deposits are known to be inherently anisotropic. Yet, they are normally treated as isotropic i n most deformation or st a b i l i t y analyses. Consequently, the results of such analyses are independent of the directions of the principal stresses. As correctly pointed out by Arthur et a l . (1980), principal stress rotations are a major feature of nearly a l l f i e l d stress paths and yet l i t t l e i s known about the effects of these rotations. This was a major incentive to the development of the directional shear c e l l (Arthur et a l . , 1977) and of recent versions of the HCT apparatus (Hight et a l . , 1983). 29 Drained plane strain DSC tests on dense sand, with continuous controlled rotations a, were reported by Arthur et a l . (1979 and 1980). A steady accumulation of the major principal strain (e x) was noted, even after 50 cycles of o x rotation, between a = -35° and +35°, at high values of R (> 4.0). Unfortunately, variations in other strain components were not reported. Thus, a complete picture of the strain response of the dense sand cannot be obtained, although weakening due to dilation was e x p l i c i t l y suggested. Since o^, R and b could not be kept constant during rotations, effects of rotation alone on deformations cannot be isolated. Tests where the magnitudes of the principal stresses are kept constant and only a is continuously varied can only be performed in the HCT apparatus. Very l i t t l e information exists on this topic. Symes et a l . (1982) have reported such a test, but at a relatively high stress ratio (R = 3.5). Drained principal stress rotation on dense sand starting from a = 0 was shown to i n i t i a l l y induce moderate volumetric contraction. After a = 35°, the specimen started to undergo strong d i l a -tion and eventually failed at a = 65°. The above result i s reproduced in Fig. 2.6, together with entirely contrasting results of similar HCT tests reported by Miura (1985). In Miura's tests, a steady moderate dilation of dense sand was observed up to a = 120°, regardless of the i n i t i a l direction of al prior to rotation (a. .^. , = 0 or 15°). Further increase i n rotation angle a was i n i t i a l 6 accompanied by a steady volumetric contraction. No failure resulted from continuous rotation of a, even though the mobilized stress ratio was even higher (R = 4.0) than in the test reported by Symes et a l . (1982). 0.2 HCT t e s t s , b = 0.50 J I L 0 30 60 90 C K ( d e g ) 120 150 180 Curve Sand D r (%) (TrnUPa) R ^ i n i t i a l R e f e r e n c e H Ham river 88 2 0 0 3.5 0 Symes et a l , 1982 T l Toyoura 8 2 9 8 4.0 0 Miura , 1985 T2 15° Figure 2.6 Volumetric Strains Due to P r i n c i p a l Stress Rotation. u> o 31 Differences in and R during continuous rotation a of o x can only p a r t i a l l y explain the contrasting HCT results shown in Fig. 2.6. Different levels of stress non-uniformities across the specimen's wall were probably also important for the different strain responses. As discussed in Chapter 3, non-uniformity effects are li k e l y to be more severe in Miura's HCT results, because of smaller specimen dimensions and higher values of stress ratio R. It should be pointed out that very l i t t l e i s known about the effects of principal stress rotation on sand response at moderate levels of stress ratio. Accordingly, this constitutes one of the main objectives of the experimental investigation of this thesis (section 5.3). Under undrained conditions, a progressive increase in porewater pressure of a medium loose sand was shown to occur as a result of continuous rotation a of a1 (Symes et a l . , 1984). In these HCT tests, the magnitudes of the three total principal stresses were kept constant. A simple framework for a qualitative understanding of the effects of both principal stress rotation and i n i t i a l anisotropy has been proposed by Symes et a l . (1984). The concept of a bounding surface (BS), origin-a l l y introduced by Roscoe et a l . (1958), was extended to include a as an additional parameter. The BS could be visualized in a 3-D space, by keeping b constant in undrained HCT tests. The contractant region of the BS obtained for medium loose Ham river sand i s reproduced in Fig. 2.7a. If the sand specimen i s subjected to a stress state corresponding to the post-peak region of the BS, unstable response under stress controlled conditions would occur. This i s normally followed by a tendency to dilate, with corresponding reduction of pore pressures. The complete °L°L(kPa> p k 2 P 0 S , ' p e i ! l|Pre-pe ak a) c o n t r a c t a n t reg ion °L°L(kPa> 2 AO 100 Ham r iver sand D r = 4 4 % b = 0 .50 90 0 b) complete s u r f a c e Figure 2.7 Undrained State Boundary Surface of Sand (after Symes et a l . , 1984). bounding surface, composed of both the contractant and the dilatant regions, i s reproduced in Fig. 2.7b. Qualitative predictions of the pore pressure response during general stress path tests can be inferred from the position of the observed effective stress path (ESP) relative to the bounding surface. Only small pore pressures are expected to develop when the ESP lie s beneath the BS On the other hand, large pore pressures would develop when the ESP travels on the BS It should be mentioned that the existence of the bounding surface has been suggested only i f the effects of induced anisotropy can be neglected, (i.e., the induced shear strains are small). Experimental evidence in support of the BS has been also indicated for a different granular material (pluviated glass spheres) in similar undrained HCT tests (Shibuya and Hight, 1987). More recently, the concept of bounding surface has been further expanded to include variations in void ratio (e) under drained loading conditions (Symes et a l . , 1988). HCT test results were reported to substantiate the existence of a four-dimensional BS, defined in terms of q = (o 1-o 3)/2, o^, a and e. This BS was suggested to qualitatively explain the behaviour of a cross-anisotropic medium loose sand undergoing drained principal stress rotations. A l l experiments were however limited to constant values of o^ and b (200 kPa and 0.5, respectively). Although very attractive because of i t s simplicity, the concept of bounding surface should be applied with caution. As indicated by Sayao and Vaid (1989a) , not only induced anisotropy, but also previous stress history effects are i n t r i n s i c a l l y neglected when the BS is considered unique in i t s position and/or shape. Apparent contradiction between 34 conclusions regarding the effect of prior stress path on the subsequent deformation response from two similar HCT investigations by Hight et a l . (1983) and Symes et a l . (1988) seems to confirm that the concept of a bounding surface should not be widely applied to a l l combinations of and R (Sayao and Vaid, 1989a). 2.2.3 Effect of Intermediate Principal Stress The effect or a3 on the stress-strain-strength characteristics of soils i s frequently disregarded in geotechnical engineering. This fact can be explained by two traditionally strong reasons. F i r s t , the Mohr-Coulorab failure criterion, usually incorporated in analysis of limit equilibrium, i s formulated i n terms of o1 and o 3 only. Second, the standard laboratory and in-situ testing methods, commonly employed in the determination of design parameters, do not permit o 2 to be independently monitored or controlled. As suggested by Bishop (1966), the influence of o a on s o i l response can be more readily appreciated i n terms of parameter b = (o 2-o 3)/ (o 1-o 3). Being non-dimensional, b directly represents -the relative magnitude of o 2 in relation to the. major and minor principal stresses. Furthermore, b has a fixed range of variation (between 0 and 1) and is independent of the drainage conditions imposed. Although being sometimes the subject of discussions (Saada and Bianchini, 1978; Saada, 1981; Saada and Puccini, 1985), the use of b, rather than o 2, as a controlling stress parameter in multiaxial testing of so i l s seems to be well accepted. It was introduced by Habib (1953) , who was investigating the shear strength characteristics of clays and sands i n torsional t r i a x i a l tests. Habib's conclusions however do not 35 reflect the influence of b alone, because the rotation angle o also varied (sin 2a = b) in his testing program. The true t r i a x i a l device has been frequently used in investigations of the effect of b or a2 on the shear response of reconstituted sand. It is interesting to note that, as early as 1936, Kjellman was already reporting results emphasizing the importance of taking the intermediate principal stress into consideration. Re-interpretation of his results in terms of more usual stress and strain parameters is presented in Fig. 2.8. The significant differences that can be noted in both volumetric and shear strain responses in the two tests can be mostly attributed to the differences in b values. More recently, several detailed investigations on the variation of <j>' with b have been reported. These are schematically shown in Fig. 2.9. Most studies seem to indicate that c/>' increases from axisymmetric (b = 0) to plain strain conditions (b = 0.25 to 0.50, at failure). This is also suggested by Kjellman's results in Fig. 2.8. However, for larger values of b, conflicting trends of cf>' = f(b) have been reported. Several factors can explain these conflicting observations: (1) differences in anisotropic fabric; (2) loading directions (a = 0 or 90°); (3) variable °m* ^ testing techniques (loading or unloading shear); (5) experimental errors (e.g., interference or restraint at the loading boundaries; stress or strain nonuniformities). In cases where dispersion of results is accentuated, some degree of personal judgement may be required before fi n a l conclusions are drawn. This is illustrated in Fig. 2.10, where the individual test results and the average trend suggested by Green (1971) are reproduced. Figure 2.8 Influence of Stress Parameter b on Stress-Strain Behaviour in True T r i a x i a l Tests (after Kjellman, 1936). 37 Curve R e f e r e n c e Relat. Density ( a ) Bishop ( 1966) Loose l b ) ' Lade 4- Duncan ( 1973) Reades 4 Green (1976) Loose Loose to Dense ( c ) Green (1971) Lade 4 Duncan (1973) Loose to Dense Dense ( d ) Ergun (1981) Haruyama (1981) Dense Loose ( e ) Sutherland 4 Mesdary (1969) Ramamurthy 4- Rowot (1973) Loose to Dense Dense ( f ) Shankariah 4 Ramamurthy(1980) Medium to Dense Figure 2.9 Reported Variations of <J>' with b i n Sands. 38 Ham river sand, D r = 9 0 % Figure 2.10 Influence of b on Strength and Strains at Failure (after Green, 1971). 39 It should be pointed out that the investigations described above, including those in Fig. 2.9, have a l l been limited to stress paths corresponding to increasing shear stresses with a = 0 or 90°. For the particular case of a = 45°, HCT tests reported by Symes et a l . (1982) also indicate a higher value of <f>* at b = 0.5 than at b = 0. Standard t r i a x i a l and plane strain tests on " t i l t e d " specimens, at a = 30° and 60°, are also i n agreement with the conclusion that cj>' is the lowest at b = 0 stress condition (Lam and Tatsuoka, 1988). In contrast with the relatively larger number of studies on shear strength, the effect of b on s o i l response at pre-failure stress conditions has received l i t t l e attention. The results of Haruyama (1981) on glass spheres indicate, however, that similar variations of mobilized f r i c t i o n angle with b could be expected at both yielding and failure conditions (Fig. 2.11). Yielding was, in this case, defined as corres-ponding to the point of maximum curvature in the octahedral stress-strain curve (T ./a' . vs r . ). As this definition can lead to an imprecise oct oct oct r or subjective determination of yield conditions, Haruyama's conclusions should.also be accepted with some caution. Earlier i n this chapter, attention was drawn to the limitations against investigations of failure characteristics using either the HCT apparatus or t i l t e d specimens in t r i a x i a l or plane strain devices. In both cases, significant errors in results may have been introduced by non-uniformities in stresses or strains within the test specimens. The HCT device i s most suitable for studies of the stress-strain behaviour of soils with different values of b and a, at low levels of the mobilized shear strength. In particular, the effect of b alone could be directly assessed from HCT stress path tests in which a l l other stress parameters 40 0> o - E 5* 36 32 28 24 20 0.2 GLASS SPHERES , LOOSE f a i l u re -y ie ld ing _ » 0.1 a> o o -0.1 -0.2 o» 0.4 0 s " — g » 0.3 E >» o 0.2 0 0.2 0.4 0.6 0.8 1.0 b * ( c r ; - c r ; > / < < r ; - c r ; ) Figure 2.11 Influence of b on Mobilized Strength and Strains at Yielding Condition (after Haruyama, 1981). 41 (° m» R and a) are held constant and only b i s allowed to vary between 0 and 1 under drained conditions. Such tests have, not yet been reported in the literature. 2.3 SUMMARY It i s apparent from the review presented in this chapter that a l l laboratory stress path devices f a i l to meet the fundamental and ideal requirement of uniform and well defined distributions of stresses and strains within the test specimen. Furthermore each device i s ideally suitable for investigations of only specific regions of the general stress space. From a l l the devices currently available, the HCT is the only one that offers the possibility of independent control of 4 stress components (R, o^, b and a). Consequently, studies on fundamental beha-viour of i n i t i a l l y anisotropic materials and, in particular, isolation of the effects of continuous rotation a or intermediate principal stress parameter b can only be performed with a HCT apparatus. Although major advances have recently been made in understanding the effects of b and a on cross-anisotropic s o i l s , most studies have been performed either on t i l t e d specimens or concentrated on behaviour at high shear stress levels. Thus, some of the conclusions from these studies are question-able due to relatively large levels of stress nonuniformity. Further-more, many investigations have been restricted to dense sands. Also, contradictory effects of principal stress rotation on s o i l behaviour have been reported. The effect of b on deformations has not been directly investigated. Only comparisons of tests with fixed b and increasing R values have so far been reported with several contradictory trends. 42 Therefore, a more comprehensive and systematic fundamental investi-gation on the effects of b and a, at lower R levels, on anisotropic sands is necessary. In order to accomplish this a^  new HCT apparatus was developed at the University of B.C. (see chapter 4). Due to the inherent nonuniformities of stresses associated with HCT specimens, a c r i t i c a l assessment of the available definitions of nonuniformity has been made. Accordingly, the experimental program was devised with proper considera-tion as to the stress regions where nonuniformity levels are minimized. Nonuniformity aspects are detailed in the next chapter. The testing program was designed to investigate the deformation response of sand under general stress paths. Particular emphasis was placed in delineat-ing how each stress parameter R, a, and b influences response when others are held constant. 43 CHAPTER 3 STRESS NONUNIFORMITIES IN HCT SPECIMENS 3.1 INTRODUCTION Apart from those due to end restraint, stress nonuniformities arise i n the wall of a hollow cylinder specimen as a result of curvature of the wall. These nonuniformities develop when a torque or a difference in external and internal pressures are applied. Traditionally, the effects of end restraint have been minimized by techniques that reduce radial shear at the ends together with the use of sufficiently long specimens. This approach of minimizing effects of end restraint was also used in the design of the U.B.C. HCT apparatus (see Chapter 4). No systematic analysis of stress nonuniformities due to curvature of the wall in HCT specimens was made prior to the work of Hight et a l . (1983). The analysis enabled them to select suitable specimen dimensions that were considered to minimize the effect of stress nonuniformities in the wall. 'No go' regions of the stress space (R, b, a) were delineated that would result in unacceptable levels of nonuniformities. In this chapter, Hight et al.'s c r i t e r i a of stress nonuniformity are c r i t i c a l l y examined. It is pointed out that their approach of acceptable level of nonuniformity in individual stress components i s not satisfac-tory. It can lead to serious and unacceptable nonuniformities in the distribution of stress ratio R across the specimen wall. Since R i s generally recognized as the most important stress variable controlling response of fr i c t i o n a l granular materials, concepts of acceptable nonuni-formity are advanced in terms of R. Based on these concepts, new 'no go' 44 regions of the stress space are identified together with their inter-relationship to specimen geometry. 3.2 STRESSES AND STRAINS IN HCT SPECIMENS The four surface tractions - v e r t i c a l force F , torque T^, and external and i n t e r n a l pressures P g and P^ respectively, acting on a hollow cylindrical specimen are illustrated in Fig. 3.1(a). These induce stresses a , a , a. and T' i n an element in the wall of the specimen z' r ' 0 z0 r (Fig. 3.1(b)). Similarly the four non-zero strain components are e , e , Z IT e0 a n d rzQ-Interpretation of results from HCT test is made by considering the entire specimen as a single element deforming as a right circular cylinder. Since the stresses vary across the wall of the cylinder for a variety of loading conditions, i t becomes necessary to work in terms of average stresses and strains. The following expressions are used for calculating average stresses: F + TT(P R 3 - P.R?) z e e i i , 0 , v Z TT(R - R.) e l °r (P R2 - P.RJ + 2(P - P.)RJR2£n(R /R.) = ^ e £ i _ e _ i e _ j L _ ( 3 > 2 ) °0 ( Re " V Figure 3.1 Load and Stress Conditions i n HCT Specimens. A6 i n which Rg and are respectively the external and internal specimen r a d i i . Only o z i s not dependent on the material constituive law and is obtained by equilibrium considerations only. The remaining stress components correspond to the assumption of a linear elastic isotropic material. The expressions for o , o Q and x q used above are obtained by r u zo averaging over the volume of the specimen. Hight et a l . (1983) and others use slightly different expressions for these stresses. The differences arise partly on account of averaging across the wall instead of the volume of the specimen as well as assuming plastic constitutive law for evaluating x Q . These differences, however, are minor and Zo usually do not exceed 2% among different expressions proposed. For the sake of consistency, however, a l l stress components should be computed by assuming a single constitutive law and not by a hybrid combination of elastic for some and plastic for other components, as done by Hight et a l . The r a d i a l stress o i n HCT tests i s usually the intermediate r • J principal stress a2. Application of torque therefore causes rotation (a) of stresses a1 and o 3 in the vertical plane perpendicular to the radial direction (Fig. 3.1(c)). Mohr's ci r c l e can be used to compute o x, o 3 and a from the known average stress components o , o Q and x Q. Z U Zu The various average strain components are calculated using the following expressions: -AH (3.A) e H z -(AR - AR.) e l (3.5) e R - R. e I r where AG is the angular displacement at the base. These definitions stem from considerations of compatibility of displacements together with the assumption of a linear variation of displacement across the specimen wall. Expressions (3.4) to (3.7) are i d e n t i c a l to those used by Hight et a l . (1983). AR^ is obtained from measured volume change of the inner chamber and ARg can be computed from the measured values of AR., e and e ,. Likewise for stresses, l z vol ' principal strains e x and e 3 can be computed from the known average strain components e , e Q and y Q using Mohr's circ l e construction. The radial strain corresponds to e 2. 3.3 STRESS NONUNIFORMITY IN THE WALL The degree of stress nonuniformity in the wall of the HCT specimen depends on the stress state, specimen dimensions and constitutive law of the s o i l . o i s usually assumed uniform. The remaining stress compon-ents o , o Q, T Q w i l l a l l vary across the wall for generalized loading r u z b conditions, with torque T^ f 0 and confining pressures P^ P •. Hight et a l . (1983) defined the nonuniformity coefficient B3 for each individual stress component as follows: R P a = 1 1 (3.8) R - R. S e l 48 i n which S = stress component in question (o , o Q or x Q) , S = average r o zo av value of the stress component and S^ = stress level, taken as ( T Z Q ) a v f° r T, n and 4[(o ) + (o-J ] for o and an. The level of stress nonuni-z6 2 r av 0 av r 9 formity was considered acceptable i f B3 < 0.11. In their analysis, non-uniformity c o e f f i c i e n t s were considered for o r and only, since the assumption of uniform izQ i m p l i e d B 3 = 0 f o r t h i s shear stress component. Figure 3.2 shows elastic distributions of o , o. and x n across the & r' 0 z9 wall of a HCT specimen for two ar b i t r a r i l y selected stress states: (a) = 300 kPa, R = 3, b = 0, a = 45°; and (b) o^ = 300 kPa, R = 3, b = 0.5, a = 0. The r a d i i R g and R^  are considered to be 7.6 and 5.1 cm respec-tively (the geometry adopted for the UBC-HCT apparatus). Both stress states r e s u l t i n B, (o or o Q) = 0.07, implying an acceptable level of J r o nonuniformity by the c r i t e r i a of Hight et a l . (1983). Figure 3.2 also shows the computed distribution of stress ratio R across the specimen wall. For the stress state (a), R ranges between 2.64 and 3.61 and for (b) between 2.54 and 3.92. Such nonuniformities in R are clearly unacceptable for a fr i c t i o n a l material whose deformation response i s intimately linked to the level of R. Figure 3.3 shows the results of a conventional t r i a x i a l test on medium loose sand, carried out in the HCT device. It may be noted that the highest level of stress r a t i o (R r a a x) i n Figure 3.2 could possibly represent a state of failure, whereas the lowest (R . ) could correspond to a state substantially below mm r J failure. Large differences between maximum and minimum R values would thus imply very serious strain gradients across the specimen wall. A more logical index of stress nonuniformity for a fr i c t i o n a l material would be expressed i n terms of R, i n preference to B3 of Hight et a l . (1983). This i s proposed as: Figure 3.2 E l a s t i c Stresses Across the Wall of HCT Specimens. R 1.0 I . . . + , , , , , 1 - 4 - 2 0 2 4 6 STRAIN ( % ) Figure 3.3 Stress-Strain Behaviour of Medium-Loose Sand i n T r i a x i a l Compression. 51 R - R max mm (3.9) R av It i s suggested herein that stress nonuniformity could be considered acceptable i f £ 0.20. This normally corresponds to a maximum of 10% variation in R across the wall from the average value R f l V» Based on this criterion, the levels of stress nonuniformity for the stress conditions in Figure 3.2 (B^ = 0.32 and 0.A6 for cases (a) and (b)) would be clearly unacceptable. 3.A NONUNIFORMITY COEFFICIENTS FOR GENERAL STRESS STATES Contours of B coefficients at mobilized R=3 in b,a plane are shown in Figures 3.A and 3.5. The specific stress states considered earlier in Figure 3.2 correspond to points A and B in these figures. Figure 3.A shows contours of B3 together with the B3 surface and Figure 3.5 i l l u s t r a t e s s i m i l a r information for the proposed coefficient BD. For a given stress state (R, b, a), B contours do not depend on the level of o^. It may be noted i n Figure 3.A that, but for near the corners (b=l, <x=0) and (b=0, a=90°), B3 values are within the acceptable limits of B3 < 0.11; This implies that v i r t u a l l y entire (R = 3, b, a) space could be open for investigation of sand response without introducing serious stress nonuniformities. B D contours, on the other hand (Figure 3.5) show that only lim i t e d regions of stress space could be explored i f B D is to be limited to 0.20. 3.5 INFLUENCE OF PRINCIPAL STRESS RATIO The level of nonuniformity decreases with decreasing R value. This may be noted in Figure 3.6 where BR contours and surface similar to those ° (a) UBC-HCT : / ? R con tours ; R = 3 (b) UBC-HCT : (3R su r face ; R = 3 Figure 3.5 Nonuniformity Coefficient 8 at R = 3. R Figure 3.6 Nonuniformity Coefficient B at R = 2 55 in Figure 3.5 are presented at mobilized R = 2. The region in b,a space where B £ 0.20 i s now much more extensive than in Figure 3.5. Thus a larger region of stress space could be explored at lower R levels with acceptable levels of stress nonuniformity For any given R level, largest nonuniformities are confined in the v i c i n i t y of corners (b=l, a=0) and (b=0, a=90°) and, unlike for B3, also along a ridge near a = 45° where large torque i s required. These rigdes appear because of elastic assumption for T Q , as opposed to plastic used by Hight et a l . (1983). Z o As already pointed out, i t seems inconsistent to assume elastic constitu-tive law for some stress components and plastic for others. It would be in order to mention that for elastic T n , HCT tests with P = P.(in which z0 e l b = s i n 2 a ) , can also be subject of unacceptable nonuniformities in terms of B^ i n the neighbourhood of a=45°. For t h i s a range, the usual consideration of P g = P^ as being superior for i t s minimization of stress nonuniformities i s thus a misconception. The manner in which the nonuniformity coefficients vary with R i s shown in Figure 3.7 for two stress states (b=0.5, a=0) and (b=0, a=45°). Both B 3 and (^increase with R. If stress nonuniformity was assessed in terms of B3, f u l l range of R levels (in excess of 4) could be explored. In terms of B^ however, unacceptable nonuniformities occur for R in excess of about 2.0 to 2.2 for the specimen geometry under consideration (Rg = 7.6 cm, R^ = 5.1 cm). Most HCT devices may thus be not suitable for generalized stress path testing beyond R = 2.0 to 2.5 i f stress nonuni-formities are to be kept within acceptable levels. 3.6 INFLUENCE OF MEAN EFFECTIVE STRESS It can be shown that for a given stress state (R, b and a), B D does K not depend on the l e v e l of a' . This is a consequence of assuming o1 = 56 R = o"! / cr3. Figure 3.7 E f f e c t of R-Level on Nonuniformity C o e f f i c i e n t s . 57 constant across the wall together with elastic isotropic constitutive law. Thus, the l e v e l of o^ does not impose any new limitations on the exploration of stress space than already dictated by (R, b, a) states. 3.7 INFLUENCE OF SPECIMEN GEOMETRY The discussion of stress nonuniformity so far has been limited to a fixed specimen geometry. As already pointed out, nonuniformity for a given stress state (R, a, b) depends on the specimen dimensions. Wall thickness, diameter and height constitute the three geometrical components of hollow cylinders. 3.7.1 Specimen Thickness Stress nonuniformity increases with wall thickness for a given average specimen radius (R. + R )/2. This is illustrated in Figure 3.8 l e for the stress state (b=0, a=45°) at R=2 and 3. It may be noted that, for a given R level, acceptable nonuniformities in terms of B3 become clearly unacceptable using $ u criterion, when larger wall thicknesses are considered. However, a minimum wall thickness must be provided, even though B D could be reduced further using thinner specimens. This minimum thickness i s dictated by essentially two considerations (Hight et a l . 1983): (i) Wall thickness sufficiently large in relation to the specimen's maximum grain size so as to ensure uniform sand density across the wall; ( i i ) Wall thickness sufficiently large so as to minimize relative significance of potential volume change corrections due to membrane penetration. 58 10 2 0 30 Thickness , R e - R, ( mm ) Figure 3.8 E f f e c t of Wall Thickness on Nonuniformity C o e f f i c i e n t s . 59 A wall thickness of 20 to 26 mm i s considered to meet the above c r i t e r i a for tests on sands. 3.7.2 Specimen Diameter Figure 3.9 shows the nonuniformity coefficients B3 and as func-tions of the inner radius of the specimen, for a fixed wall thickness of 25.A mm. The results are shown for the stress state (b=0, ot=45°) at R=2 and 3. At both R l e v e l s , B^ may be seen to be much more sensitive to changes i n inner radius than B3. Inner r a d i i smaller than even 20 mm would be satisfactory i f B3 i s considered as a measure of nonuniformity, but would c l e a r l y be unacceptable i f B^ is considered. Figure 3.9 also indicates that not much gain i n reducing B^ i s accomplished for inner r a d i i in excess of about 40 to 50 mm. Although B D decreases at both levels of stress ratio with increasing R^ , a very large inner radius may not be a practical solution in terms of stress path control. This i s shown in Fig. 3.10(a). For loading with a given increase in R, while o^, b, and a are held constant, the magnitude of d i f f e r e n t i a l pressure P -P. decreases as the inner radius increases. r e l For l a r g e R^ (of about 120 mm), P e~P^ equals a mere 20 kPa for an increase i n stress ratio from 1.5 to 3.0. This extreme sensitivity of R to small changes i n Pe~P^ would be d i f f i c u l t to control experimentally. Such a control w i l l become even more d i f f i c u l t i f needed to be accomplished at lower values of o' (Fig. 3.10b). m Based on the above considerations and considering a wall thickness within the range of 20 to 26 mm, adequate geometry of HCT specimens (in terms of both nonuniformity and experimental control criteria) may be obtained i f R./R is within 0.65 and 0.82. I e 60 Figure 3.9 Effect of Inner Radius on Nonuniformity Coefficients. 61 Figure 3.10 S e n s i t i v i t y of R to Changes i n D i f f e r e n t i a l Confining Pressure as a Function of Inner Radius and a'. m 62 3.7.3 Specimen Height Radial f r i c t i o n a l restraint at the specimen boundaries causes stress nonuniformities in addition to those due to specimen curvature. These have been considered to be minimal i f length (H) to diameter (2R ) ratio e of the specimen is within the range 1.8 to 2.2. This conclusion is supported by theoretical (Saada and Townsend, 1981; Hight et a l . , 1983) as well as experimental studies (Lade, 1981; Fukushima and Tatsuoka, 1982). Techniques that aim to reduce radial f r i c t i o n at the end boundaries w i l l further minimize stress nonuniformities. Such a technique was adopted in the design of UBC-HCT device (see Chapter 4). 3.7.4 Recommended Geometry Geometrical characteristics of hollow cylinder shear devices used by various researchers are lis t e d in Table 3.1. It may be noted that many of these devices were used only for simulating multiaxial stress states, with no rotation of principal stresses. It is clear that the hollow cylinder device i s not superior to a true t r i a x i a l device for multiaxial stress paths with a=0 due to stress nonuniformities (see Figures 3.5 and 3.6 for values corresponding to a=0). Consequently, the use of a HCT apparatus should be primarily to simulate principal stress rotation effects or to assess directional properties of inherently anisotropic s o i l s . The three geometrical characteristics of hollow cylinders in Table 3.1 are illustrated in Figure 3.11 in the form of specimen thickness (Rg-R^) versus R^/Rgas well as H/2Rg. Desirable geometry is suggested by the two boxed areas in the figure. The suggestions are based on both practical and theoretical considerations of limiting stress nonuniformi-Table 3.1 - Stress Path Devices Using Hollov Cylinder Specimens Ho. Reference Institution Speciaen Dimensions (ma) Q.J 1 Control Restrictions Applications H R. R i SOU Type 1 Cooling & Saith 1936 Building Research Station (ESC) 19-38 50.8 41.3 Clay w pi-° Undrained shear strength 2 Norton 1938 M.I.T. (USA) 50.8 11.1 7.9 Clay Torsional deforraability ceraaic clays 3 Geuze & Kie 1953 S.H. Laboratory, Delft (HOD 80.0 19.0 13.0 Clay Undrained creep 4 Kirkpatrick 1957 university of Glasgov (SCOT) 152.4 50.8 31.8 Sand vv° o, effect on f a i l u r e condition 5 Haythornthvaite 1960 Brom University (USA) I t t S i l t o, effect on f a i l u r e condition 6 Whitman & Luicher 1962 H.I.T. (USA) 76 or 127 25 or 19 12.7 Sand V 0 : V ° Soil-structure interaction at f a i l u r e 7 Wu et «1. 1963 Michigan State University (USA) 127.0 50.8 38.1 CSS v° o, effect on f a i l u r e condition 8 Brorai t Ratntn 1963 Cornell University (USA) 114.3 76.2 38.1 Clay v° 3-D consol. effects on strength 9 Broms & Casbarian 1965 Cornell University (USA) 254.0 63.5 38.1 Clay o a and a effects on strength 10 Brora & Jaaal 1965 Cornell University (USA) 304.8 76.2 38.1 Sand v° V a l i d i t y of °r'°g assumption 11 Esrlg i Beaben 1965 Cornell University (USA) 203.2 50.8 38.1 Sand v° o, and t , effects on strength 12 Suklje & Drnovsek 1965 University of Ljubljana (TOG) 80.0 32.0 20.0 Clay V 0 : V ° Deformability under plane stress 13 Jamal 1966 Cornell University (USA) 203.2 50.8 13-38 Sand V 0 : P . - p i Wall thickness effect on strength 1* Saada h Baah 1967 Case W.R. University . (USA) 151.1 35.1 25.4 Clay p . - p i Influence of anisotropy IS Proctor 1967 University of Manchester (EKG) 152.4 50.8 19.1 Sand v° Drained shear strength 16 lomize et a l . 1969 C i v i l Engineering Institute (USSR) 180.0 155.0 125.0 Clay Drained creep under 3-D stress state 17 Frydman et a l . 1971 Israel Institute of Tech. (ISRL) 203.2 50.8 25.4 Sand v° End r e s t r a i n t ; membrane penetration 18 Drnevich 1972 University of Kentucky (USA) 100.0 25.0 20.0 Sand p . - p i Torsional resonant column tests 19 Arnold & Mitchell 1973 University of Adelaide (AUS) 142.0 76.0 51.0 Sand v° 3-D stress effect on strength 20 Ishibaahi 4 Sherif 1974 University of Washington (USA) 13 to 25 50.8 25.4 Sand Liquefaction characteristics 21 Tong 1975 University of Waterloo (CAN) 203.2 50.8 31.8 Sand Yiel d and f a i l u r e c r i t e r i a 22 Lade 1975 U.C.L.A. (USA) 50.0 110.0 90.0 Sand p . - p i a effect on stress-strain 23 Ivaiaki et a l . 1978 Inst. Ind. Science (JAP) 100.0 50.0 30.0 Sand p . - p i S tress-strain for y > 10-'X 2* Ude 1981 U.C.L.A. (USA) 400.0 110.0 90.0 Sand p . - p i Influence of specimen's height 25 Dusseault 1981 University of Alberta (CAN) 200-240 50.8 25.4 Sand v° Tune11ing and pressuremeter paths 26 Fukuehiaa & Tatsuoka 1982 Inst. Ind. Science (JAP) 200.0 50.0 30.0 Sand p . - p i Deformation k strength behaviour 27 Symes et a l . 1982 Imperial College (EHG) 254.0 127.0 101.5 Sand b and a effects on strai n response 28 Ishihara & Tovhata 1982 University of Tokyo (JAP) 104.0 50.0 30.0 Sand p . - p i a e ffects; l i q u e t , characteristics 29 Ishibashi et a l . 1985 Cornell University (USA) 142.0 35.5 25.4 Sand p . - p i Liquefaction characteristics 30 Miura et a l . 1986 Hokkaido University (JAP) 200.0 50.0 30.0 Sand -J a effect on stress-strain & strength 31 Alarcon et a l . 1986 Purdue University (USA) 203.0 35.5 19.0 Sand v p l Stress-strain for r > 10">t 32 Anderson et a l . 1988 University of Sheffield (EHG) 150.0 75.0 12.5 Clay v° Pressuremeter paths; undrained creep 33 Chen et a l . 1988 Cornell University (USA) 193.0 51.0 35.5 Sand p . - p i Dynamic shear mod. of glass spheres 34 Sayao & Vaid 1988 University of B.C. (CAN) 304.8 76.2 50.8 Sand Effects of o.R.b & o' on strain resp. is 8 = 10 I3c 40 15 16 27 19 -9 J3b = l7 m20:25 22=24 23=26=28=30 "4=21 .33 .31 7=11 = 13a 6c = 6d * • 0 •|2 29 14 6a = 6b 18' 35-f or 30 + 25 20 15 10 -5 -E E 9 or 20 22 121 (FVRj=62) J5 J6 .27 19 23 28 7 ,6c •|2 I 10=13c 9=l3b 25 17=34 24 26=30 M • '21 .33 6d^ll = l3a 29. 18' 14 _6a refer to Table 3.1 .31 ,6d 1.0. 0.8 0.6 0.4 Ri 0.2 0.2 0.6 1.0 1.4 1.8 2.2 2.6 VR e 3.0 3.4 H/2RE Figure 3.11 Specimen Geometry of Reported Hollow Cylinder Devices. 65 t i e s due to specimen curvature (B^) and end r e s t r a i n t to acceptable levels. The suggested geometry i s also consistent with the a b i l i t y to explore a reasonably l a r g e r e g i o n of (R, b, a, o^) space. The recommended dimensions for HCT specimens are: a) wall thickness: b) inner radius: c) height: R - R. = 20 to 26 mm e l 0.65 * R./R * 0.82 l e 1.8 «; H/2R <; 2.2 66 CHAPTER 4  APPARATUS AND EXPERIMENTATION 4.1 THE UBC-HCT TESTING DEVICE In this section, the hollow cylinder torsional device developed at the University of British Columbia (UBC-HCT) is described in detail. The preliminary design started early in 1984 and the f i r s t successful experi-ment was performed about two years later. The development of the UBC-HCT apparatus was directed towards achieving an optimum combination of three distinct requirements: (a) simplicity i n design and construction, (b) v e r s a t i l i t y in operation, and (c) r e l i a b i l i t y i n measurements at small strain levels. In detailing loading and measuring systems, f u l l use was made of the experience gained at UBC through the development of earlier research testing devices (Vaid, 1968; Finn et a l . , 1971; Carapanella and Vaid, 1972; Bosdet, 1980; Vaid and Negussey, 1983). 4.1.1 General Description In i t s present configuration, the UBC-HCT device i s suitable for controlled stress path testing of reconstituted sand specimens under drained or undrained, monotonic or cyclic stress conditions. Hollow cylindrical specimens can be subjected to axial load, different external and internal confining pressures and torque about the vertical axis. Any pre-failure region i n the four-dimensional stress space (o x,Oj,o 3,a) is thus open for investigations. For the chosen geometry, however, certain 67 regions of stress space cannot be investigated without significantly compromising stress uniformity, as already discussed in Chapter 3. A schematic layout of the UBC-HCT apparatus i s shown in Figure 4.1. The specimen dimensions are: height = 30.2 cm, outer diameter =15.2 cm, and wall thickness = 2.5 cm (cross-sectional area = 100 cm2). Selection of relatively large dimensions permitted improved resolution of strains together with acceptable levels of stress non-uniformities i n a l l tests reported in Chapter 5. The test specimen i s laterally confined by internal and external f l u i d pressures acting on flexible rubber membranes 0.3 mm thick. Torsional and vertical loads are transmitted through annular ribbed aluminum platens at the top and bottom boundaries. Drainage from the specimen i s provided by six interconnected small porous discs inbuilt i n the end platens. The specimen i s fixed at the top. Normal and shear loads are applied at the bottom. This configuration has two main advantages: (1) f a c i l i t a t e s test preparation procedures; and (2) avoids corrections i n o due to the weight of loading ram and load c e l l . The -steel loading z frame provides a r i g i d reaction for vertical and torque loads. A total of nine transducers is needed to measure the pore pressure and four stress and four strain components in general stress path tests with the HCT device. In most situations, interactive use of a micro-computer i s required for a proper stress path control. 4.1.2 Loading and Measuring Systems Normal and shear tractions to the HCT specimen are applied by means of controlled air pressures, using precision pneumatic regulators. 68 LVDT ( A H ) POSITIONING BOLT NOTE : SEE FIG. 4.8 FOR DETAILS EXTERNAL PRESSURE PORE PRESSURE SUPPORTING TABLE LINEAR - ROTARY BEARINGS TORQUE CABLES I f THRUST BEARING AXIAL LOAD CELL TOP CROSS BEAM CHAMBER TOP TOP CAP TOP PLATEN PLEXIGLASS CELL LOADING FRAME SOIL SPECIMEN RIGID ROD BASE PLATEN BASE PEDESTAL PRESSURE TRANSDUCER INTERNAL PRESSURE LVDT (6) i LOADING SHAFT TORQUE CELL CENTRAL PULLEY TORQUE PULLEY TORQUE PISTON AXIAL LOAD PISTON 0 20 SCALE (cm) Figure 4.1 The UBC-Hollow Cylinder Torsional Apparatus. 69 Supply pressure to the test frame was regulated to a maximum of 750 kPa, that was below the operating range in the laboratory's main line. Continuous monitoring of loads, pressures and deformations in the UBC-HCT tests is achieved by using electronic transducers placed outside the c e l l chamber. During the i n i t i a l stages in the development of the apparatus, consideration was given to the possible use of internal instrumentation, as recommended by Hight et a l . (1983). However, a c r i t i c a l evaluation of recent developments in laboratory instrumentation reveals that both external and internal approaches have advantages and limitations. There i s no consensus for or against either approaches, as exemplified by Atkinson and Evans (1985) and Jardine et al (1985). This i s probably due to the differences i n external measurement system configurations, leading to different amounts of bedding, seating and t i l t i n g errors in displacement and loads. Moreover, installation of internal transducers poses additional d i f f i c u l t i e s i n the specimen preparation procedures and i n the interpretation of test results, as discussed later in this chapter. Preliminary experimental comparisons between internal and external measurements of axial displacements in dense t r i a x i a l specimens were performed at UBC. Sensitive displacement transducers were used to monitor relative displacements between two reference footings mounted over a central gage length on the specimen's membrane. This technique is similar to that described by Daramola and Vaughan (1982) and Costa Filho (1985). The results indicated no significant differences between ver t i c a l strains derived from external or internal measurements i f appropriate care was taken to eliminate bedding, seating and/or t i l t i n g errors. 70 It was therefore decided to benefit from UBC's previous experience on improved techniques of external intrumentation (Negussey, 198A). The philosophy adopted has been to recognize potential errors associated with external measurement systems and take suitable measures to eliminate or minimize these errors. Integrated Bottom Loading System A double acting air piston i s mounted at the bottom of the support-ing table (Fig. 4.1) to apply compressive or extensional axial load to the HCT specimen. The load i s transmitted through a 25 mm diameter polished steel shaft, that has i t s vertical alignment guaranteed by two frictionless Thompson combination bushing bearings. These bearings permit linear and rotary motions of the loading shaft and are placed in series at the base of the c e l l chamber. The specimen's base platen and pedestal are mounted on the upper end of the loading shaft. A 25 mm diameter 0-ring provides seal for the cell's f l u i d pressure at the location where the shaft emerges from the c e l l chamber. Friction between the 0-ring and the shaft may be neglected, since i t was found to be less than 2 N, regardless of c e l l pressure. Vertical stress o as low z as 0.2 kPa can be accurately measured with a force transducer placed outside the c e l l chamber. It i s interesting to note that, although the internal load c e l l developed at Imperial College (El-Ruwayih, 1975; Hight, 1983) eliminates the problem of loading ram f r i c t i o n , i t s large vertical deformability introduces significant errors i n external measurements of axial displacements of dense specimens (Costa Filho and Kupper, 1983; Campos, 1984). 71 The integrated bottom loading system encompasses torque loading to the central vertical shaft underneath the supporting table, as shown in Figures 4.1 and 4.2. Torsional loads are applied by two pairs of identical single-acting air pistons and a system of cables and pulleys. Diametrically opposite pistons are interconnected to a common regulated pressure supply. This configuration i s necesary to eliminate horizontal side forces on the central shaft and also allows application of torque in either direction., A dual thrust bearing i s placed below the central torque pulley in order to prevent rotational movements of the vertical load piston rod. Placement of lateral pulleys was found necessary to avoid side load on the torque piston rods due to vertical movements of the torque pulley when the specimen deforms vertically. This integrated bottom loading system i s an improved version of the system described by Lade (1981). Torque measurements as low as 0.05 Nm can be reliably obtained by a bonded strain gage transducer placed above the central pulley. This corresponds to an average shear stress T i n the order of 0.1 kPa. Z D Cross talk between axial load and torque was made negligible by careful placement of the strain gages at 45° to the transducer's axis. This torque transducer i s in fact a double load c e l l , since i t was also gauged for measuring vertical force. This was particularly useful i n permitting comparisons with vertical force measurements obtained from the load c e l l below the torque system. No significant differences could be observed between the two measurements. Adequate transference of both torsional shear and vertical normal stresses from the loading shaft to the s o i l specimen requires prevention of relative displacement between the specimen and the end platens in the scale (cm) Figure 4.2 Torque Loading System. 73 tangential direction only. A good approximation of these ideal boundary conditions i s achieved by using polished anodized aluminum platens having twelve thin radial ribs (1 mm thick and 2.3 mm deep), as illustrated i n Figure A.3. Six 12.8 mm diameter porous discs are set 60° apart, flush with each platen surface, for drainage i n and out of the specimen. Ribbed boundaries have been considered necessary to ef f i c i e n t l y transmit shear stresses to sand specimens i n simple shear (Finn et a l . , 1971), i n ring shear (Hvorslev, 1939; Wijewickreme, 1986) and in hollow cylinder torsional devices (Miura et a l . , 1986; Tatsuoka et a l . , 1986). Many HCT devices use porous discs over the f u l l platen surface and thus impose undesirable f r i c t i o n a l restraint in the radial direction. This may introduce significant stress and strain non-uniformities close to the specimen ends, with considerable effects on short specimens (Lade, 1981; Fukushima and Tatsuoka, 1982). Some researchers have used annular end platens with a layer of glued sand grains for transmitting tangential shear stresses. While Lade (1981) has reported success with their use, Tatsuoka et a l . (1986) have noted significant slippage between the specimen and the end -platens. Broms and Casbarian (1965) , and Hight et a l . (1983) used enlarged outer diameter end platens. This was intended to provide an increased f r i c t i o n a l contact area with f u l l annular porous stones. The same a r t i f i c e had also been reported by Habib (1953) in torsion tests on f u l l cylinders. The resulting irregular specimen shape, however, could only be acceptable i f a l l stresses and strains were referred to a central portion of the test specimen. The need for localized internal instru-mentation then becomes mandatory. 74 A) Perspective I •—' n r-i I - I C) Side view ; 1 1 1 u J I — I — I 0 5 s c a l e ( c m ) Figure 4.3 Polished End Platen with Radial Ribs. 75 Fluid Pressures Backpressure and internal and external confining pressures are applied through air-water interfaces, as illustrated i n Figure 4.4. Air diffusion into the specimen or into the cell's de-aired water i s preven-ted by using 1.5 m long, 4 mm i.d. saran tubings as air intercepting diffusion loops. Control of the air pressure is obtained by precision regulators. Pressure monitoring i s done on the water side by pressure transducers placed close to the cell ' s base plate. This minimizes compliance effects. Resolution of a l l pressure measurements was in the order of 0.24 kPa. The air-water interface cylinder of the external pressure system was made large enough to accommodate expected variations i n the confining chamber volume. Air-water interfaces of both internal pressure and pore pressure systems are located i n graduated pipettes of suitable sizes. These pipettes are required for volume monitoring systems, as described later. Strain Measurements Four transducers are required for measurements of the four non-zero strain components in general stress path tests. Two displacement transducers (LVDT) are used for monitoring vertical and angular displacements of the specimen's base pedestal. From these, average axial and shear s t r a i n s (e and y .) can be computed. In addition, two dif f e r e n t i a l pressure transducers are used to register volume changes of the s o i l specimen and of the inner pressure chamber. These are required 76 DPT WS SOIL SPECIMEN t LEGEND PT •« gage pressure transducer DPT * di f ferent ial pressure t ransducer WS B d e - a i r e d water supply B non - displacement valve ^fc1 - d i f f us ion spiral ( Z | . 5 m ) ( b ) BACKPRESSURE Figure 4.4 Fluid Pressure and Volume Change Measuring Systems. 77 for d e f i n i t i o n of average r a d i a l and tangential normal strains (e r and e Q ) , as explained in Chapter 3. The vertical LVDT is mounted on the loading frame's top cross beam, as shown in Fig. 4.1. The transducer's core reacts against the top of a thin vertical steel rod (10 mm o.d.), r i g i d l y attached to the centre of the specimen's pedestal. In this manner, vertical movements of the specimen base only are transmitted to the transducer's core. An 0-ring placed in the top cap seals the inner chamber pressure around the vertical rod. With this configuration, compliance errors frequently asociated with external axial displacement measurements (Daramola and Vaughan, 1982; Costa Filho, 1985) are greatly minimized. Displacements are measured at the "unloaded" side, where there i s no influence of compliance of the loading components. In addition, the top cross beam constitutes a fixed reference unaffected by variations i n chamber pressure. Further improvement in the accuracy of axial displacement measure-ments i s obtained by minimization of bedding and seating errors (Jardine et a l . , 1984; Atkinson and Evans, 1985). This has been achieved by adopting several improvements in the testing technique: (a)~use of small diameter porous stones flush with the platens surface; (b) careful levelling of the specimen's top surface prior to placement of the top platen; (c) densification of the specimen after placement of the top platen; and (d) application of 50 kPa i n i t i a l hydrostatic stress state that eliminates subsequent bedding movements. Items (b) to (d) are described i n detail i n section 4.3. Rotation of the specimen's base platen is converted to linear tan-gential displacement by using the system schematically shown in Fig. 4.5. 78 counter weight Figure 4.5 Rotational Displacement Measuring System. 79 The LVDT i s placed at a distance large enough to avoid need for corrections i n the measured tangential displacement due to vertical movements of the rotation arm when the specimen deforms verti c a l l y . Both displacement transducers were calibrated against a precision micrometer. They can reliably detect displacements of the order of 1 um. This gives a resolution i n both vertical and shear strain measurements of about 5x10"*%. Volume changes of the inner chamber and s o i l specimen are obtained by monitoring the height of water column in graduated pipettes. Membrane penetration corrections were made to the measured volume changes when stress paths involving variations in Pe and/or Pi were applied. Such corrections are detailed i n the Appendix. Illustrations of both systems are presented i n Figure 4.4. Sensitive differential pressure transducers (DPT) are used to electronically register the water level i n the pipettes, as described by Campanella and Vaid (1972) and Tatsuoka (1981). The specimen's drainage system i s provided with two 5.4 mm bore pipettes placed i n parall e l . In tests where small volume changes are anticipated, one of the pipettes i s isolated from the drainage line. A backpressure of 200 kPa was normally used to ensure f u l l saturation of the s o i l specimen. With this system, variations of 3 mm3 i n the specimen's volume can be reliably detected. This corresponds to a resolution i n volumetric strain measurements better than 5x10"*%, after correcting for membrane penetration effects (see Appendix). In monitoring changes in the inner chamber's volume, only one pipette (7.9 mm bore) i s needed. Variations of about 10 mm3, equivalent to 2x10"*% of the inner chamber volume, can be reliably detected. Several calibrations of both DPT's under backpressures i n the range 0-500 kPa were performed, with no significant differences in calibration factors being detected with change in backpressure. 80 It i s interesting to note that the use of internal instrumentation does not necessarily imply a better resolution i n strain measurements. Hight et a l . (1983) have reported resolution values of 2 p and 6x10"3%, re s p e c t i v e l y for l o c a l i z e d a x i a l displacement and shear s t r a i n measurements i n HCT tests, using the electrolytic level transducers described by Symes and Burland (1984). Another aspect to be considered i s that, in order to obtain representative average values, internal measurements must be made at several locations. For example, with the HCT device described by Hight et a l . (1983), axial and shear strains are recorded i n , respectively, two and six different locations on the hollow specimen's external wall and six internal proximity transducers are used for obtaining average radial displacements near the centre of the specimen's internal and external walls. The use of a large number of internal displacement transducers introduce severe practical d i f f i c u l t i e s i n terras of preparation time, physical installation space, and data acquisition/processing requirements. 4.2 DATA ACQUISITION A micro-computer based data acquisition system was used in this research. The system has two main components: a HP-3497A data-logger consisting of multi-channel scanner and analog/digital converter, and an IBM-PC micro-computer controlled by a BASIC program. A l l transducers were excited by a common power supply which was set at 6.00 volts. Control of the laboratory temperature to 20±1°C made thermal effects insignificant. Data scanning i s initiated by a single instruction on the computer's keyboard. Output voltages from 10 channels (excitation voltage and 9 81 transducer signals) are monitored in each scan. After transformation from analog to di g i t a l form, the readings are transmitted to the computer for processing into stress and strain units. Voltages, stresses and strains are then lis t e d on a video display monitor, and the operator i s given three options: storing the data on magnetic floppy discs, monitoring one channel's output voltage, or triggering another scan. For each selected scan, output voltages and processed test data are stored in separate disc f i l e s for further analysis upon test completion. Test data i s also simultaneously output to a line printer for ready comparison with previous scans. Printer output serve as a duplicate test record in the unlikely event of system failure. Pressure gages, graduated pipettes and dial gages are also used as backup measuring devices, i n addition to the electronic transducers. 4.3 SPECIMEN PREPARATION 4.3.1 Reconstitution of Sand Specimens Standard undisturbed sampling techniques affect significantly the mechanical properties of sands (Seed et a l . , 1982). Therefore, funda-mental property characterization of sands i s frequently carried out using reconstituted specimens. Several techniques of reconstituting sand specimens i n the laboratory have been developed, the most common being moist tamping (Lambe, 1951; Ladd, 1978) and pluviation (Kolbuszewski, 1948; Chaney and Muli l i s , 1978; Miura and Toki, 1982). Comparative reviews of the methods of sand specimen preparation have been presented by Mulilis et a l . (1977), Mortensen (1982) and Vaid and Negussey (1988). 82 Water pluviation followed by vibration was selected as the most suitable technique for reconstituting HCT specimens. This technique enables preparation of homogeneous saturated specimens of uniform sands with controlled density (Vaid and Negussey, 1984), and has been frequently used at UBC over more than twenty years (e.g., Vaid, 1968; Finn et a l . , 1971; Negussey, 1984). Pluviation i s considered to duplicate the sedimentation process, and hence the fabric, of many natural or a r t i f i c i a l sand deposits (fluvial and lacustrine sediments, hydraulic f i l l s , etc.). Laboratory studies on pluviated sands should therefore give a close indication of the behaviour of these deposits (Oda et a l . 1978; Miura and Toki, 1984). The main limitation of water pluviation i s the segregation of particles during sedimentation of well-graded and s i l t y sands. For these materials, an alternative technique has been developed recently by Kuerbis and Vaid (1988) for preparing homogeneous saturated t r i a x i a l specimens. Simple modifications i n procedures would be necessary for preparation of HCT specimens of well-graded sands. 4.3.2 Preliminary Preparation Steps The i n i t i a l step for specimen preparation i s to establish a reference height. This is done by placing an aluminum dummy sample of known height between the bottom and top platens. A dial gage mounted on a movable stand i s then used to obtain a reference reading on the top platen. This reading i s later used for determining the height of the sand specimen after deposition and densification. The two rubber membranes are positioned and sealed to the inner and outer surfaces of the Base platen. De-aired water i s flushed through the 83 base drainage line and saturated porous discs are placed i n position. The inner and outer s p l i t molds are then assembled. The outer mold has i t s inner surface lined with porous plastic, through which vacuum i s applied for stretching the outer membrane. The inner membrane is stretched v e r t i c a l l y around the four-piece inner s p l i t mold. These four pieces are held together by two internal metallic discs, the annular base platen and one 0-ring at the top. A known amount (about 5 kg) of sand i s placed i n several flasks containing de-aired waterj boiled for 10 minutes and l e f t to cool overnight under 70 kPa vacuum. A.3.3 Specimen Preparation Steps A special tapered rubber stopper with a glass tube nozzle is f i t t e d to the sand flasks. De-aired water i s added to f i l l the flasks up to the top of the tubes. The annular cavity formed by the molds i s then f i l l e d with de-aired water. Once the sand flask i s inverted and has i t s tube tip submerged in the specimen's water cavity, sedimentation of the sand proceeds under gravitational influence and mutual displacement with water (Fig. 4.6). During the pluviation process, the "flasks are slowly traversed over the annular area in order to deposit sand with an approximately level surface at a l l times. Control of the height of sand drop i s not necessary, because sand grains reach a constant terminal velocity almost instantly during pluvia-tion i n water (Vaid and Negussey, 1984). No vibrations are imparted to the specimen u n t i l after placement of the top platen. Exception i s made only to dense specimens, which are gently vibrated during pluviation. 84 Figure 4.6 Specimen Preparation by Water Pluviation. 85 t This is done to reduce an excessive descent of the top platen during f i n a l densification. Sedimentation i s continued un t i l an excess of sand over that required for the fi n a l grade has been deposited. The upper surface is then carefully levelled by siphoning off excess sand using a suction of about -2 kPa (Fig. 4.7). This causes minimal disturbance of sand grains below the surface. The excess sand is oven-dried to allow determination of the dry weight of sand used in the specimen. The top platen containing saturated porous discs is then carefully seated on the levelled sand surface. Disturbance of the top layers due to penetration of the platen's ribs has insignificant effects because of the specimen's large dimensions (Tatsuoka et a l . , 1983) and also because the specimen i s s t i l l i n a loose state (Vaid, 1983). The desired density i s then achieved by applying high frequency low amplitude vibrations to the specimen's pedestal and by monitoring the changes in specimen height with the reference dial gage. During densification, the specimen i s maintained f u l l y drained by keeping top and bottom drainage lines open to the same height of water column (corresponding to the" top platen elevation). Densification under small vertical confinement provided by the top platen has been found effective in obtaining a uniform void ratio d i s t r i -bution throughout the specimen (Vaid and Negussey, 1984). The top platen follows the settlement of the top surface and any tendency for t i l t i n g i s promptly corrected. This technique proved to be very effective in practically eliminating subsequent bedding errors which plague external deformation measurements. 86 I level l ing bar ::r::''.\". .•••v.i;\-v extension container outer former sand specimen inner former p.- •; ••»; v. •** E u O CM tl excess water excess sand Figure 4.7 Levelling the Specimen's Upper Surface. 87 De-aired water i s then percolated upwards through the specimen under a very small gradient (about 5 to 10 cm of water). This is done to remove entrapped a i r bubbles between the rubber membranes and the top platen. After sealing both membranes to the top platen with 0-rings, the top drainage is closed and a vacuum of about 20 kPa is applied to the bottom drainage line. This provides an effective confinement to the specimen prior to dismantling the s p l i t molds. At this stage, the specimen dimensions are recorded. Height and outer circumference are determined respectively with the reference dial gage and a measuring tape. Inner diameter i s obtained indirectly from the volume needed to raise the water level by 50 mm in the inner chamber. Corrections to the specimen dimensions due to thickness of membranes are taken into consideration. The top loading cap is now installed, thus completing the specimen preparation. 4.3.4 Test Preparation Steps The c e l l chamber i s placed i n position and de-aired water i s slowly introduced into the inner and outer confining chambers. Care i s exercised to ensure f u l l saturation of the inner chamber. The top cross beam i s then swivelled to i t s position and firmly bolted to the reaction frame. The central rod used for monitoring vertical displacements i s then installed, thus sealing the inner chamber (Figure 4.8). The specimen i s then moved upwards by pressurizing the vertical loading piston u n t i l the top cap contacts the top cross beam of the reaction frame. The top cap i s secured against the cross beam by a bolt. This set-up ensures coincidence between the specimen's vertical axis and the frame's centre axis. A locating pin, protruding from the cross bar, 88 TOP CROSS BEAM ( LOADING FRAME) BOLT CELL TOP T - : t PEXI GLASS CELL il LOCATING PIN ir* ii u OUTER LVEMBRANE II il '.•.v\'-.*.; TOP DRAINAGE INNER MEMBRANE -o-CENTRAL ROD fit TOP PLATEN SOIL SPECIMEN TOP LOADING CAP POROUS DISC | DRAINAGE GROOVE BASE PLATEN •.V . V . V -.J';' i'.;-i.Cf>A' I I I i i i IT 11 f PEDESTAL CELL BASE EXTERNAL PRESSURE PORE PRESSURE LOADING SHAFT INTERNAL PRESSURE I I I I I I 0 5 SCALE (cm) Figure 4.8 Det a i l s of the UBC-HCT Device. 89 is then inserted into the loading cap at 30 mm from the specimen axis. The pin serves to arrest any rotational movement of the loading cap in tests requiring application of torque. A hydrostatic confining pressure of 30 kPa i s then applied to the specimen under undrained conditions. This i s sufficient to raise the porewater pressure to a small positive value. Measurement of Skempton's B-value for checking specimen's saturation then proceeds i n several increments of confining pressure. Full saturation of the test specimen i s ensured by insisting on a B-value of at least 0.98. The specimen i s allowed to consolidate over-night under an e f f e c t i v e hydrostatic stress o' = 50 kPa, with a back-m pressure of 200 kPa. The hydrostatic stress conditions (o' = 50 kPa) m together with the updated geometry constitute the i n i t i a l state of every test specimen. 4.4 PERFORMANCE AND CONTROL 4.4.1 Reproduction of Conventional Test Results A basic requirement of the newly developed HCT apparatus was to prove i t s a b i l i t y to duplicate results previously obtained with other testing devices. A series of preliminary experiments was then designed to reproduce conventional t r i a x i a l and hydrostatic compression tests on Ottawa sand reported by Negussey (1984). Medium dense specimens of water pluviated sand were used. Compara-tive results using the two test devices are presented i n Figure 4.9. Very l i t t l e difference in the stress-strain behaviour obtained with the two devices may be noted for similar i n i t i a l consolidation and density states. 90 Figure 4.9 Comparison of Results from HCT and T r i a x i a l Devices. 91 4.A.2 Stress Path Control Simultaneous and independent control of four stress parameters i s required for general stress path tests with the HCT device. In order to follow the prescribed stress path precisely, smooth changes in the control pressures must be made, either continuously or i n small increments. Moreover, for the study of drained behaviour, the rate of loading should be slow enough so as to allow for f u l l pore pressure dissipation together with measurement of time independent strain response. A l l HCT specimens were loaded in gradual, slow increments. Creep deformations of sand were noted to become increasingly significant for higher levels of stress ratio R. This i s i n agreement with experimental observations i n the t r i a x i a l device (Mejia and Vaid, 1988). Accordingly, the time corresponding to each load increment increased with R. A l l strain measurements reported i n this study refer to equilibrium conditions, in which creep effects can be neglected. In low permeability s o i l s , this would require an automatic servo-system for controlling long duration tests. However, in sands, reliable results were obtained by manual operation of the precision pressure regulators and interactive use of a computer based data acquisition system. A typical example of stress path control i s shown i n Figure 4.10. In this test, a l l four surface tractions had to be smoothly varied in order to impose a continuous rotation a of o x direction (Figure 4.10a). During rotation, since the magnitudes of the three principal stresses were held constant (Figure 4.10(b)), derived stress parameters o^, R and b were also constant. The prescribed variations i n tractions ( f u l l lines) may be seen to be closely followed by the actual data points. The remarkable stress path control capability of the UBC-HCT device is C T = 300 kPa ; : R * 2.0 ; b - 0.5 2000T — — ' Figure 4.10 Experimental Control in Principal Stress Rotation Tests. vo 93 illustrated i n Figure 4.10(c). The maximum excursion in any of the stress parameters a^, R and b, from the prescribed constant values, may be noted to be less than 1%. 4.4.3 Repeatability of Test Results Repeatability of test results i s an important requirement for the consistency of conclusions to be derived from UBC-HCT test results. St r i c t adherence to identical specimen preparation technique and test control routines i s central to achieving repeatable test results. Good repeatability can be noted i n the results of hydrostatic com-pression of three identical specimens (Figure 4.9(a)). Repeatable results on identical specimens may also be noted for tests with principal stress rotations (Figure 4.11). Two specimens at D r = 25% were f i r s t s e quentially brought to a stress state of R = 2, b = 0.5 and = 300 kPa, before being subjected to a continuous principal stress rotation under constant values of ax, a2 and o s. Considering the small magnitude of induced strains, excellent repeatability may be noted in both volumetric and shear strain response. 4.5 EXPERIMENTAL PROGRAM 4.5.1 Materials Tested HCT tests were conducted on two granular materials: Ottawa sand (ASTM C-109) and Erksak sand. Both are uniform, medium, predominantly quartz sands, with similar grain size distribution curves (Figure 4.12). Table 4.1 summarizes their index properties. Erksak sand was made available through the courtesy of Gulf Canada Resources Ltd. This sand has been used as a hydraulic f i l l material in the Molikpaq caisson island in the Canadian Arctic. As received, the 0'm= 300 kPa ; R = 2.0 ; b = 0.5 Figure 4.11 Repeatability of HCT Test Results. 9 5 SIEVE SIZE (meshes/inch) 4 10 20 40 60 100 200 GRAIN SIZE (mm) COARSE | MEDIUM | FINE G R A V E L SAND S I L T Figure 4.12 Grain Size Distribution: Ottawa and Erksak Sands. Table 4.1 Index Properties and Grain Characteristics of Ottawa and Erksak Sands PROPERTY OTTAWA SAND ERKSAK SAND Median Size D s o (mm) 0.39 0.34 Uniformity Coeff. C u 1.9 1.8 Specific Gravity G g 2.67 2.66 Limiting Void Ratios e max 0.82 0.82 e . min 0.50 0.51 Roundness 0.60 0.35 Sphericity 0.85 0.83 Mineral Content .,, , 100% quartz . ,t80% quartz io% feldspar (plagioclase) 10% others 97 sand contained about 1% material passing #200 sieve. In order to prevent segregation of fines during water pluviation, a l l HCT tests on Erksak sand were performed after removing the fines by washing with a large quantity of water. As indicated in the next item, most of the tests i n this study were carried out on Ottawa sand and only a limited number on Erksak sand. Both sands were recycled among the various HCT tests. This was done on the presumption that the predominantly quartz grains would not suffer degradation under the moderate stress levels used in the testing program. It can be noted from Table 4.1 that the grain shape of both sands has been characterized by two geometrically distinct parameters: (1) Roundness, defined as the ratio between the curvatures of the corners (and edges) and the average curvature of the grain (Wadell, 1935); and (2) Sphericity, defined as the cube root of the ratio between the volume of the grain to the volume of the smallest circumscribing sphere (Krurabein, 1941).* Based on the above definitions, sand grains can be characterized by values between zero and unity for both roundness and sphericity. Systematic determination of both parameters for several gradations of Ottawa and other sands is described by Edil et a l . (1975). Apart from the minor distinctions in mineralogy, the main difference between the two sands used is the greater angularity (lower values of *This definition of sphericity i s practically identical to the one earlier suggested by Wadell (1935). A simpler definition, which also yields essentially the same results, was proposed by Rilley (1941) as the square root of the ratio of the inscribed to the circumscribed c i r c l e diameter. 98 roundness) of Erksak sand (Table 4.1). The implications of this difference on the mechanical properties are addressed in the next chapter. 4.5.2 Testing Program The main objective of this thesis i s to investigate the behaviour of sands subjected to general stress paths. In particular, the anisotropic stress-strain characteristics and the effects of continuous and independent changes in a and b are intended to be studied. Accordingly, the experimental program was composed of three main groups of tests. • R-tests: principal stress ratio R increased from 1.0 to R , with max constant values of o', b and a. The value of R is m max dictated by considerations of stress non-uniformity within the specimen (see Chapter 3). • a-tests: rotation angle a cycled between 0 and a m a x (usually 60°), with constant values of o', R and b. m* • b-tests: intermediate principal stress parameter b cycled between 0 and 1.0, with constant values of o^, R and a. The i n i t i a l stress state of a l l specimens was hydrostatic (.a^ = 50 kPa, see section 4.3.4). This condition, corresponding to R = 1.0, has no defined values of b and a. Starting with this stress state, each test i s then performed in various sequential phases, i n which only one stress v a r i a b l e (o^, R, b or a) is allowed to vary. Fig. 4.13 illustrates the stress paths followed in typical tests. The detailed manner in which the loading phases for each test were applied is presented in Table 4.2. CTm= 300 kPa N max A 50 B 3 0 0 or ( k P a ) o a) Po- test AB =Cf'm— phase BC = R - phase b) 25 a b - test AB = R - phase BC = or - phase CD =G'm- phase DED = b - phase c) OC - test AB = R - phase BC = b - phase CD =(J'm- phase DED -<X- phase 0"m (kPa) Figure 4.13 Typi c a l Examples of General Stress Path Tests. Table 4.2 HCT Tests - Summary of Main Testing Program R-Tests o^-phase (Consolidation) R-phase (shear) b a(deg) R - o|/o) Rl 0.0 45 R2 0.3 g R3 o' - 50 to ID 0 R'1-0 to R ^ < C/l RA 300 kPa 15 < < R5 34% 0.5 20 P © R6 (R - 1.0) 30 R7 45 R8 55 to R9 0.8 . 45 b-Tests D r R-phase o-phase o^-phase b-phases bl b2 o Bl R-1.0 to 1.3 a-0 to 25" — TAWA SA B2 34% — o m - 50 to b-0^1-0 b«0 to 0.5 TAWA SA B3 R=1.0 to 2.0 o=0 to 25* 300 kPa (-o B4 a-0 to 45° a-Tests D r R-phase b-phase o^-phase a-phases al a2 al R-1.0 to 1.3 b-0.0 to 0.5 a-O-oO'-O* a-0-90o-0* a2 — a-O»60*-0° OKVS a3 34% R-1.0 to 2.0 b«0.0 to 0.3 o' « m a-O-eO'-O" OKVS OKVS a4 50 to a«0-.60*-0° o5 R-1.0 to 3.0 300 kPa a=0-556-0° a-0-55"-0c OTT a6 60% b-0.0 to 0.5 a-0—45' a.7 42% R-1.0 to 2.0 a-0-60°-0° a-0-45* a8 20% a=o—eo^-o 0 a l l 50% a=0->-45* al2 R=1.0 to 2.0 um AND al3 25% b-0.0 to 0.5 50 to a-0-60o-0 a-0—45* u> al4 300 kPa a-0-45* IT. tui et al5 50% R-1.0 to 3.0 a-0—45° U al6 (*) 35% R-1.0 to 1.8 b-0.0 to 0.5 a-0 to 35° o'-50 to 300 in kPa a-35-45-35' }8 cycles R-1.8-2.B-1.8 (*) Note: Test al6 - after R-phase, simultaneous variations occurred in more than one stress parameter 101 Additional tests were performed in order to elucidate specific aspects related to the investigation program. Examples of these aspects are: (i) comparison of axial compression tests performed on identical specimens i n the HCT and standard t r i a x i a l devices; ( i i ) effect of stress history (from HCT tests with different sequences of loading phases); ( i i i ) direction of cyclic a rotation (positive or negative); (iv) strain response with further rotation cycles (after 2 cycles); (v) rotation of principal stresses at increasing R-levels. A l l loading phases were performed under f u l l y drained, stress controlled conditions, with a back pressure of 200 kPa. Loading steps within each phase were made small (e.g., changes in a or R were con-trolled to about 1° or 0.05, respectively). Only time independent readings were associated with corresponding loading steps. 102 CHAPTER 5  RESULTS AND DISCUSSION 5.1 INTRODUCTION The deformation characteristics of sands that reflect an inherently cross-anisotropic fabric w i l l be examined i n this chapter. The discussions are based on HCT test results in the general stress space a^t R, b and a. A systematic assessment of the isolated influence of each stress variable i s presented for medium-loose Ottawa sand. The effects of other variables, like relative density, stress path history and sand type are also investigated, with attention focussed mainly on principal stress rotation. The behaviour of sand under general stress path tests i s presented in terms of both volumetric and maximum shear strain responses. Varia-tions of the observed direction of major principal strain increment (a^ £), i n relation to the direction of applied principal stress increment (a^) and the d i r e c t i o n of p r i n c i p a l stress (a) , are presented and discussed i n a systematic manner. Strain paths i n principal strain space ( e l f e3, e 3) are also examined. These provide a significant insight into any changes i n inherent anisotropy on loading. 5.2 INITIAL ANISOTROPY 5.2.1 Hydrostatic Loading The degree of inherent or i n i t i a l anisotropy i s reflected by the relative values of volumetric and axial strains under hydrostatic loading. I d e n t i c a l principal strains e1=e2=e3 (or e ,= 3 e x) would be 103 observed i f sand was isotropic. Hydrostatic compression tests were performed on medium loose (D r = 34%) Ottawa sand, s t a r t i n g from an i n i t i a l hydrostatic stress state of = 50 kPa and proceeding up to o^ = 300 = kPa. Typical results are shown in Fig. 5.1. A linear strain path e v Q ^ = 4.5 el may be noted. This implies higher deformability i n the horizontal than i n the v e r t i c a l deposition direction (i.e. e = e. = r r 8 1.75 e ), which i s typical of cross-anisotropic granular materials (Oda, z 1972; Arthur & Menzies, 1972). Due to the linearity of the strain path, i t may be pointed out that the degree of i n i t i a l anisotropy i s not altered by hydrostatic compression at moderate stress levels (o^ < 300 kPa). The behaviour in Fig. 5.1 is consistent with the observations in the t r i a x i a l device reported by Negussey (1984) on the same material. 5.2.2 Shear Loading  Stress-Strain Behaviour Detailed observations of the directional dependence of stress-strain behaviour can be made by shearing the s o i l with different directions a of O j i n relation to the deposition direction. A series of such direc-tional shear tests was performed on medium-loose Ottawa sand. The i n i t i a l condition of a l l specimens was o' = 300 kPa, R = 1.0 and D = r ra ' r 36%. During shear, the values of o 1 and b were held constant at 300 kPa m and 0.50 respectively, while the stress ratio R was increased i n incre-ments of about 0.05 under drained conditions. As indicated i n Table 4.2 (tests R3 to R8), the range of a selected for this investigation was 0 to * 55°. Only p o s i t i v e values of a are considered, since symmetry of the In t h i s t h e s i s , p o s i t i v e values of a correspond to an anti-clockwise application of torque Tjj at the base of the HCT specimen. 104 Figure 5.1 Strain Response Under Hydrostatic Loading. 105 mechanical properties about the vertical axis of deposition can be assumed valid for cross-anisotropic sand (Symes et a l . , 1985; Miura et a l . , 1986). At R-levels of about 2.5 or greater, equilibrium strains were not recorded for every load increment, in order to cut down the test duration. At these higher R levels, both volumetric and shear creep deformations became significant and each loading increment would typic-a l l y take more than 20 rain for strain equilibrium to be reached. Never-theless, time-independent equilibrium data was obtained under sufficient R values. Figure 5.2 presents the stress-strain behaviour observed in these directional shear tests. A very profound effect of i n i t i a l cross- aniso-tropic fabric of medium-loose sand is apparent i n both volumetric and shear strain responses. The s t i f f e s t response can be noted to correspond to shear loading in the vertical deposition direction (i.e., a=0). This i s i n agreement with the observations from hydrostatic loading (section 5.2.1). Increasingly softer strain response is observed with increasing values of a. In Figure 5.3, the strains developed at R = 2 and R = 3 are compared for loading along different a directions. Both stress states correspond to contractive volumetric strain conditions. Stress ratios in excess of 3 (and, i n particular, failure stress conditions) are not considered because of serious stress non-uniformities within the specimen (see Chapter 3). Two aspects may be pointed out from the data i n Figures 5.2 and 5.3. F i r s t , the effect of anisotropy on shear and volumetric strains becomes more accentuated with increasing levels of stress ratio. Second, the direction a corresponding to the largest shear deformability seems to increase with R. At R = 2, the specimen sheared at a = 45° showed the 106 Dr = 36 % : <Tm = 300 kPa : b = 0.5 (a) Figure 5.2 Strain Response Under Shear Loading. 108 softest shear strain response. However, with increasing R levels, the largest shear strains were observed in specimen R8 (a = 55°). It may be pointed out that similar behaviour has been observed by Miura (1985) in HCT tests on dense pluviated sand (Fig. 5.4). In this figure, the strains are normalized by a coefficient K, which i s a func-tion of R only. Miura explained his observations by invoking Matsuoka's concept of mobilized planes (Matsuoka, 1974). These planes are defined as those corresponding to the maximum ratio between shear and normal *m stresses and are thus inclined at an angle i = 45° - • j - to the direction R-l of o, . i s the mobilized f r i c t i o n angle (sin 4' = zr-r). Due to the 1 m ° Tra R+l cross-anisotropic fabric of pluviated sands, maximum deformability i s expected to be observed when the mobilized and the bedding planes are coincident. In the early stages of the shearing process, i = 45° and thus maximum deformability occurs at a = 45°. With increasing R-levels, the angle i progressively decreases and consequently the inclination a, corresponding to the softest shear strain response, increases. Accord-ingly, shearing at a = 60 to 70° results in the lowest values of shear stiffness. Other investigations on inherent anisotropy with the HCT device (Symes et a l . , 1982) and the DSC (Arthur et a l . , 1981a) have focussed mainly on strength characteristics. They do not permit any reliable conclusion to be drawn as to deformations at low stress levels. The observed rate of volumetric contraction (Figure 5.2b) agrees well with the results of HCT tests reported by Symes et a l . (1988). For the medium-loose sand, the rate of contraction i s seen to depend not only on R-level but also on the direction a. At any R, progressively larger contractions may be noted for shearing at higher values of a. If undrained conditions prevailed, shearing would be accompanied by large 109 Figure 5.4 Anisotropy i n Str a i n Response of Dense Sand (after Miura, 1985). 110 reductions i n effective stresses. Thus, susceptibility to liquefaction or contractive deformations would increase with shearing at increasing a. Undrained HCT results presented by Symes et a l . (1984, 1985) give support to such a conclusion. Direction of Strain Increment Another characteristic feature of inherently anisotropic materials when subjected to shear loading at constant direction a, i s the non-coincidence of the directions of the major principal strain increment (a^) and major principal stress (a). The experimental data presented in Figure 5.5 illustrates this aspect. Behaviour of pluviated Ottawa sand i s such that a^ £ > a in a l l HCT tests reported herein. This implies that the major principal strain increment (de x) always tends to deviate towards the horizontal direction of the bedding plane, the only excep-tions being for directional shearing at a = 0 or 90°. In these cases principal stress axes coincide with axes of cross anisotropy and thus directions of principal stresses and strain increments coincide. It may be of interest to note that this important characteristic behaviour of cross-anisotropic materials has been the subject of contra-dictory observations. Arthur et a l . (1981a) reported no differences between a and i n DSC te s t s , despite marked differences in stress-strain behaviour at different a-values. Miura et a l . (1986) presented results in agreement with those i n Fig. 5.5 only for a < 45°. For shear tests with a > 45°, they observed an opposite trend, with < a even at small shear stress levels. This i s in contradiction with their observa-tions of softer shear strain response at these larger inclinations of a[, already discussed. On the other hand, Symes et a l . (1982, 1988) have I l l 90 80 70 V) QJ if 60 H 8 50 -40 -S 30-20 -10 -1.0 Figure 5.5 Directions of Principal Strain Increments in Directional Shear. 112 reported HCT results on dense and medium-loose pluviated sand, with conclusions very much similar to those suggested by Figure 5.5. It is interesting to note that, in most cases, the magnitude of differences between a^ g and a tends to decrease with increasing R levels. This may be explained partly by a continuous alteration of the i n i t i a l anisotropic characteristics of the sand on account of induced shear strains. Strain Paths The strain paths observed i n the directional shear tests of medium-loose pluviated sand are presented i n Fig. 5.6a. It i s interesting to note that regardless of a direction, i n i t i a l l y linear strain paths (gj vs e s) were followed i n the small strain (or stress ratio) region. This can be better appreciated in Fig. 5.6b, where the strain paths up to R = 1.8 are replotted for 3 selected tests with an enlarged scale. The i n i t i a l slope of the strain path increases somewhat with a. Similar i n i t i a l l y linear strain paths were observed i n the other two principal strain spaces ( e l t e a) and (e a, e 3 ) . Figure 5.6(c) shows the paths in the (e a, e,) space up to R=1.8 for the same 3 tests i n Fig. 5.6(b). The observed lin e a r i t y i n strain paths indicates that, i n this small R region, the degree of i n i t i a l anisotropy i s preserved, regardless of the loading (o x) direction. Triaxial compression (a=0, b=0) data presented by Rowe (1971) and Negussey (1984) on sand also shows i n i t i a l l y linear strain paths regardless of o' level. m At values of R higher than 1.8 to 2.0, strain paths become progress-ively nonlinear, suggesting a gradual evolution of the i n i t i a l anisotropy due to induced strains. As shown in Fig. 5.6a, at any given stress ratio R, the minimum strain increment ratio {del/de3) i s associated with the ver t i c a l shear loading (o=0). Figure 5.6 Strain Paths in Directional Shear. 114 Further experimental observations regarding the i n i t i a l linearity of the strain paths up to R = 1.80, at different values of b 0.5), are presented later (section 5.4). 5.2.3 Conclusions Based on the experimental evidence described in this section, i t may be concluded that the stress-strain behaviour of medium-loose sand deposits i s highly dependent on the loading direction. Even materials with nearly rounded grains, such as Ottawa sand, reflect a marked inher-ent cross-anisotropy. The s t i f f e s t response occurs in the vertical direction of deposition. The degree of inherent anisotropy is preserved during increases i n mean effective stress (a'). The effects of induced • m anisotropy become significant only for R-levels in excess of about 1.8 to 2.0. One important practical implication of these observations i s that deformations predicted based on conventional testing procedures (triaxial tests on "vertical" specimens) would be on the unconservative side. 5.3 CONTINUOUS PRINCIPAL STRESS ROTATION The experimental findings presented i n the previous section indicate a marked inherent anisotropy of pluviated medium loose sand. As a consequence of this anisotropy, significant deformations may be expected when the sand i s subjected to a rotation of the principal stress directions. Several principal stress rotation tests on Ottawa sand were carried out under drained conditions (Sayao and Vaid, 1989b). Throughout each test, the principal stresses alf a2 and o 3 are kept constant i n magni-tude, while the rotation angle a is varied i n a continuous manner. 115 These tests included both monotonic and cyclic rotations, over a range of relative densities and diverse i n i t i a l stress conditions. A series of rotation tests on a different sand (Erksak sand) was also performed to explore rotation effects as a function of sand type. In a l l tests, the stress state prior to principal stress rotation was achieved by sequential application of individual changes in each stress parameter (R, b and o'), with o=0 (see Table A.2). m The characteristic features of principal stress rotation paths may be better visualized in the modified stress space (Ishihara and Towhata, 1982; Miura et a l . , 1986) illustrated in Fig. 5.7(a). The corresponding path i n (R, a) space i s shown in Fig. 5.7(b). Point A corresponds to the i n i t i a l hydrostatic stress conditions (o' = 50 kPa). From A to B, the m desired changes i n stress state (R, b, a^, a=0) prior to rotation are applied to the specimen. From B to C, principal stress o x i s rotated from 0 to 60° i n a continuous manner, under condition of constant R, b and o'. m It i s noted that such rotation tests are represented by a circular path in Fig. 5.7(a). Three important geometric features can be seen in this figure. F i r s t , the radius AS (denominated as stress vector) numer-i c a l l y equals s i n e of the m o b i l i z e d f r i c t i o n angle (sin 4>mo^ = (R-1)/(R+1)). Second, the stress vector at any given stress condition is inclined at 2a to the vertical axis. The third feature i s that the -> s t r e s s increment vector (ST), i n c l i n e d at 2a^ o to the v e r t i c a l , i s tangent to the circular path. This i s because the stress ratio R i s held constant during r o t a t i o n . It also implies that = a + 45° during forward r o t a t i o n (increasing a). S i m i l a r l y , a ^ o = a - 45° when a is decreased towards the vertical direction (backward rotation). 116 R - I R +1 ( a ) M o d i f i e d s t r e s s s p a c e (b) ( R , o c ) s p a c e Figure 5.7 Principal Stress Rotation Paths at Constant a' and b. m 117 5.3.1 Rotation Tests on Loose Sand  Stress-Strain Behaviour Typical s t r a i n response of a loose specimen (Dr = 20%) to cyclic changes in a is shown in Fig. 5.8. During rotation the specimen was under a constant stress state of o' = 300 kPa, b = 0.5 and R = 2. Principal stress rotation may be noted to induce contractive volume changes regardless of whether a i s increased or decreased on either side of the v e r t i c a l d i r e c t i o n , e , tends to be more s i g n i f i c a n t for vol 6 increasing phases of a than for the decreasing phases, with the largest contraction being associated with the f i r s t time increasing rotation phase. Contractive strains become progressively smaller with further rotation phases, regardless of direction. This behaviour i s in accordance with the interpretations suggested by Symes et a l . (1988). After the f i r s t forward rotation phase, the stress path moves beneath the bounding surface. As a consequence, irrecoverable strains become much less dominant, giving rise to smaller contractions with further rotations. This cumulative contraction would clearly imply progressive pore pressure build-up under cyclic undrained conditions. Like volumetric contractions, maximum shear strains accumulate pro-gressively under cyclic changes i n a. The strains increase, however, only on increasing rotation. Decrease in a towards the vertical direction tends to result in some recovery in the magnitude of shear strains. Residual shear strains nevertheless increase with each cycle (a=0 -» ±60 -* 0) of principal stress rotations, though at a decreasing rate per cycle i n the same direction. 118 Figure 5.8 Strain Development Due to C y c l i c P r i n c i p a l Stress Rotation. 119 Direction of Strain Increment Figure 5.9 shows the observed directions of stress and strain incre-ments ( a ( j a and a ^ g t respectively) during three rotation cycles on loose Ottawa sand. It may be noted that the requirement of = a ± 45° during forward or backward rotation was followed very closely. Forward (loading) rotation i n the f i r s t cycle i s seen to result in progressively larger deviations between directions of strain increments and stress increments. This i s probably associated with the progressive accumula-tion of predominantly irrecoverable strains as forward rotation proceeds, as already indicated i n Fig. 5.8. On the other hand, decreasing rotation from 60° to 0° (i.e., "unloading") results i n closer to <*ja« This may be considered asso-ciated with a predominance of recoverable strains. These observations agree with the concept of a drained bounding surface (BS), suggested by Symes et a l . (1988). Similar trend i n the r e l a t i v e values of and o^ o may also be noted i n Fig. 5.9 for the second and third rotation cycles. Increasing rotation (a from 0 to ±60°) seems to always result i n large deviations between and regardless of prior rotations on the opposite side. Decreasing a towards zero from i t s maximum amplitude on either side tends to make closer to o^ o instead of a. It appears that alternating positive and negative rotation cycles significantly reduces the hardening effects usually associated with one-way rotation cycles (Sayao and Vaid, 1989a). The results in Fig. 5.9 suggest that in this case the position and/or shape of the BS should have been greatly affected by previous principal stress rotation cycles. This would be an indication that the i n i t i a l anisotropic fabric of the sand i s permanently changed due to 120 Figure 5.9 Strain Increment Directions During Cyclic Principal Stress Rotation. 121 induced strains during cyclic rotation. However, this i s not in agree-ment with the indications of Symes et a l . (1988) who maintain that the BS remains symmetric about the a=0 plane, after the sand has been subjected to principal stress rotation on either side of the ver t i c a l . 5.3.2 Effect of Relative Density  Stress-Strain Behaviour A l l specimens for this test series were f i r s t brought to an identi-ca l i n i t i a l stress state (o^ = 300 kPa, b = 0.5 and R = 2) prior to i n i -tiating principal stress rotation. The results of f i r s t cycle rotation are illustrated i n Fig. 5.10. Both e , and Y induced due to rotation vol max decrease progressively with increase in relative density. This would be expected because the degree of inherent anisotropy in pluviated sands, which i s primarily responsible for principal stress rotation effects, decreases as the relative density increases (Negussey and Vaid, 1986), Only sand with high relative density (60%) responds with small dilation at i n i t i a l stages of rotation. In a l l other cases, principal stress rotations induce progressive volume contractions, regardless of the relative density and the direction of rotation. Similarly, induced shear strains decrease with increasing relative density under s i m i l a r changes i n a. Except for the dense sand (Dr = 60%) , Tmax continues increasing somewhat even after a starts decreasing from i t s maximum amplitude. Recovery in fmax on decreasing a also varies with relative density. Direction of Strain Increment The above observations of smaller volume changes and irrecoverable shear strains i n dense sand under principal stress rotation are reflected 122 Figure 5.10 Ef f e c t of Relative Density on S t r a i n Development Due to P r i n c i p a l Stress Rotation. 123 Figure 5.11 E f f e c t of Relative Density on S t r a i n Increment Directions During P r i n c i p a l Stress Rotation. i n smaller deviations between a, and a, than for the loose sand (Fig. do de 5.11). The results i n this figure show directions of stress and strain increments for the f i r s t rotation cycles on specimens at D r = 20 and 60°. In particular, during backward rotation (from 60 to 0°), the results for the dense sand suggest a predominance of recoverable strains (i.e., = "do 5' 5.3.3 Effect of Mean Effective Stress  Stress-Strain Behaviour Figure 5.12 shows the effects of forward principal stress rotation on two medium loose (D r = 34%) specimens that have identical R=2 and b=0, but d i f f e r e n t a' of 50 and 300 kPa. These tests are l a b e l l e d as B4 (a-phase) and cx2 i n Table 4.2. Test results are shown for forward rota-tion (a=0 to 45°). Much large volumetric and shear strains are induced in sand at higher confining stress for a given rotation angle. The nature of the differences are qualitatively similar to those between loose and dense specimens at identical o^, illustrated in Fig. 5.10. The similar effects of decreasing relative density at constant confining stress, and increasing confining pressure at constant relative density, are a well recognized characteristic of granular materials. Direction of Strain Increment The strain increment directions are shown in Fig. 5.13. For rota-tion angles smaller than about 35°, deviation of and may be noted to be larger at the lower confining stress. At larger rotation angles, the data i n Fig. 5.13 suggests that the directions of the strain incre-ments become insensitive to the level of o'. 125 Figure 5.12 E f f e c t of E f f e c t i v e Confining Stress on Strain Development Due to P r i n c i p a l Stress Rotation. 126 Ottawa sand 1 I 1 1 1 1 f 0 20 40 60 Rotation Angle oc ° Figure 5.13 E f f e c t of E f f e c t i v e Confining Stress on Strain Increment Directions During P r i n c i p a l Stress Rotation. 127 5.3.4 Effect of Principal Stress Ratio Stress-Strain Behaviour The three medium loose (D r = 34%) specimens i n this test series were subjected to identical o' = 300 kPa and b = 0.5, but different values of m R = 1.3, 2.0 or 3.0 prior to imposing principal stress rotations. The results of f i r s t principal stress rotation cycle, illustrated i n Fig. 5.14 show that, for a given a, both e , and r induced increase with ' 6 ' vol 'max the level of R. Even at low R = 1.3, significant strains are induced as a consequence of rotation. At high R = 3, principal stress rotation leads to substantial contractions. These findings are i n good agreement with recent investigations reported by Symes et a l . (1988). Decrease i n a, following i t s peak value in the forward direction, i s associated with some recovery i n Y similar to that observed in pre-J 'max vious test series. The magnitude of recovery as a percentage of forward developed strains, however, decreases substantially with increase i n R. Direction of Strain Increment The observed directions of the strain increments are shown in Fig. 5.15. The results indicate that the position of relative to a or is highly dependent on the level of stress ratio. At high stress ratio (R=3) , larger deviations occur between a^ £ and , which i s consistent with larger irrecoverable strains (Fig. 5.14). Unloading (i.e., decreas-ing a) , on the other hand, results in closer to a^0t particularly at low level of R = 1.3, which now suggests a predominance of recoverable strains. Two factors could contribute to the large difference between beha-viour under principal stress rotations at low and high stress ratios. 128 Figure 5.14 E f f e c t of E f f e c t i v e Stress Ratio on S t r a i n Development Due to P r i n c i p a l Stress Rotation. 129 Figure 5.15 E f f e c t of E f f e c t i v e Stress Ratio on Strain Increment Directions During P r i n c i p a l Stress Rotation. 130 F i r s t , significant changes i n the inherent anisotropy of the sand are l i k e l y to result from increasing R to a level in excess of 2.0 prior to rotation. This has been already suggested in section 5.2.2. Second, principal stress rotation from a=0 to 55°, at stress conditions b = 0.5 and R = 3, is l i k e l y to be significantly affected by stress nonuniformi-ties (see Chapter 3). Values of B D in excess of 0.20 are expected during most of the rotation. It i s , however, very d i f f i c u l t to assess the extent of the effects of these high stress nonuniformities on the results shown i n Figures 5.14 and 5.15. 5.3.5 Effect of Intermediate Stress Parameter Stress-Strain Behaviour In t h i s test s e r i e s , three medium loose specimens (Dr = 34%) were subjected to o' = 300 kPa, R = 2.0, a = 0 and different values of b = J m ' ' 0.0, 0.3 and 0.5, prior to imposing principal stress rotations. Volumetric and maximum shear strains induced during the f i r s t rotation cycle are presented i n Fig. 5.16. It may be noted that the value of b does not appear to influence the strain response of sand to principal stress rotations i n a major way. Direction of Strain Increment Figure 5.17 shows the directions of strain increments. It i s noted that the r e l a t i v e p o s i t i o n of a ^ , during either forward or backward rotation, seems to be independent of the b-value. This may have contributed to the relative insensitivity of 'b' level on principal stress rotation effects. 131 Figure 5.16 E f f e c t of Intermediate Stress Parameter on Strain Development Due to P r i n c i p a l Stress Rotation.. 132 Figure 5.17 E f f e c t of Intermediate Stress Parameter on Strain Increment Directions During P r i n c i p a l Stress Rotation. 133 The conclusions drawn from Figures 5.16 and 5.17 have important implications, since the range of b selected represents the majority of practical loading situations ranging from axisymmetric to plain strain. Previous experimental investigations aimed at isolating the effects of principal stress rotation (drained or undrained) have concentrated on a single arbitrary value of b = 0.50 (Symes et a l . , 1982,1984,1985,1988; Miura et a l . , 1986). As suggested by the experimental results presented in this section, the conclusions from these previous investigations may thus be extended to other intermediate stress conditions with b in the range 0.0 to 0.5. 5.3.6 Rotation Tests on Erksak Sand Like the tests on Ottawa sand, a l l specimens of Erksak sand were sequentially brought to the desired i n i t i a l state (stress ratio R = 2.0 or 3.0, mean effective stress o' = 300 kPa and intermediate stress para-m meter b = 0.50) prior to in i t i a t i n g principal stress rotations. Specimens were tested at two relative densities (D r = 25% and 50%). Effect of Relative Density The volumetric and shear strain responses of both loose and medium dense sands are compared in Fig. 5.18. The f i r s t principal stress rota-tion cycle may be seen to induce significant contraction and shear defor-mations at both densities, although of smaller magnitudes for the denser state. The results of Erksak sand are very similar to those observed on Ottawa sand (Fig. 5.10), despite some differences in mineralogy and angularity of the grains (see Table 4.1). 134 C T M = 300 kPq ; R = 2.0 ; b = 0.5 0.20 i Rotation Angle OC ° Figure 5.18 Str a i n Development Due to P r i n c i p a l Stress Rotation on Erksak Sand: E f f e c t of D . 135 After the f i r s t rotation cycle, the specimens were subjected to a second forward rotation phase (a=0 to 45°). Significant hardening effects may be seen to result from previous rotation cycle i n the same direction. The magnitude of contraction i s drastically reduced i n the loose, whereas a small dilation may be noted in the medium-dense specimen. These experimental observations are also in accordance with the concept of bounding surface (Symes et a l . , 1988) previously discussed. The observed strain increment directions are shown in Fig. 5.19. Only small differences may be noted between the two densities, tends to be closer to a^ o for the denser than the looser state, during forward ro t a t i o n . The small deviation between a, and a, at low values of a de do (corresponding to small strain magnitudes) was also observed in rotation tests on Ottawa sand. During backward rotation, a closer coincidence between the directions of strain and stress increments may be associated with the predominantly recoverable strains i n both medium-dense and loose sand (Fig. 5.18). Effect of Principal Stress Ratio It i s interesting to note that the marked effects of stress ratio on the volumetric strain response of Ottawa sand during principal stress rotation (Fig. 5.14) are not observed with Erksak sand (Fig. 5.20). The different range of relative densities of the two sands can be a major factor contributing to the differences in behaviour. Figure 5.20a shows that contractive volumetric strains differ only slightly for rotation tests at R=2 and R=3. Small dilation i s observed at the i n i t i a l stages of both forward and backward rotations on the sand subjected to high 136 Figure 5.19 Strain Increment Directions During P r i n c i p a l Stress Rotation on Erksak Sand: Ef f e c t of D . 137 0 20 40 Rotation Angle OC ° Figure 5.20 Strain Development Due to P r i n c i p a l Stress Rotation on Erksak S and: E f f e c t of Stress Ratio. 138 stress ratio. Nevertheless, shear strains induced by principal stress rotation increase significantly with R-level (Fig. 5.20(b)). Also only small differences in directions may be noted in Figure 5.21 for rotation tests on medium-dense Erksak sand under different levels of stress ratio. 5.3.7 Conclusions Continuous principal stress rotation tests on two sands (Ottawa and Erksak sands) show no major differences in strain response, despite differences i n grain angularity and, to a smaller extent, in mineralogy (see Table 4.1). Progressive accumulation of both volumetric contraction and shear d i s t o r t i o n was observed. At constant stress state (o^, R, b), these deformations increase with decrease in relative density. Similarly,for a given r e l a t i v e density, deformations increase with the level of o" , and 6 J m * with the level of R. The level of principal stress parameter b between 0.0 and 0.5 seems to have no influence on the strain response of medium-loose sand during principal stress rotations. Both volumetric contractions and shear strains were found to be more significant i n the f i r s t rotation cycle. Progressively smaller cumula-tive straining was observed in subsequent cycles. The direction of the major principal strain increment deviated more from the direction of the major stress increment during the f i r s t forward rotation on either side of the v e r t i c a l . This was associated with strains that were predomi-nantly irrecoverable. During backward rotation and with subsequent 139 Figure 5.21 Strain Increment Directions During P r i n c i p a l Stress Rotation on Erlcsak. Sand; E f f e c t of Stress Ratio. 140 cycling, the directions of the strain and stress increments tend to coincide (a^ e = ), indicating predominance of recoverable strains. Under drained loading conditions, this implied hardening effects of the previous rotation cycles. This type of behaviour, under undrained conditions, would lead to progressive softening due to accumulation of positive excess pore pressure. For certain states of stress and relative density, this could lead to liquefaction (contractive deformation). 5.4 INTERMEDIATE PRINCIPAL STRESS PARAMETER The influence of the intermediate principal stress parameter b = (Oj-OjJ/tOj-Oj) on the strain response of sands is assessed in two basic manners. The f i r s t one i s through a series of tests i n which the sand i s sheared at several constant values of b. Such experiments have been carried out in true t r i a x i a l devices under a = 0 condition, with contradictory conclusions as to the effect on peak strength characteris-t i c s (see Fig. 2.9). The results of a series of drained shear tests with a = 45° i n the HCT apparatus are discussed i n detail i n this section. In the second and more direct manner, the effect of the intermediate stress parameter on strain response can be evaluated in tests in which b i s the only stress parameter that i s varied. Virtually no information i s so far available on the behaviour of sands under such continuous varia-tions i n b at constant values of R, and a. Symes et a l . (1985) report a test of this type in which focus was on pore pressure response under undrained conditions only. In the second part of this section, the effect of drained cyclic variations in b at several inclinations a is examined in detail. 141 5.4.1 Shear Tests at Various b Stress-Strain Behaviour Four tests on medium-loose sand were carried out, each with a different b-value. A l l specimens were f i r s t brought to a hydrostatic stress state o' = 300 kPa. The stress ratio R was then increased with m constant values of b (0.0, 0.3, 0.5 and 0.8), a = 45° and o' = 300 kPa. ' ' ' m The shear and volumetric strain responses are shown in Figures 5.22(a) and (b). It may be noted that the deformation response of sand due to increasing shearing stresses i s significantly dependent on the value of b. Stiffest response i s obtained when b = 0.3. This b value is usually considered to be within the range associated with plane strain conditions. Except at small levels of shear strain (Y < 0.3%), softest shear max response and largest volumetric contractions may be noted to correspond to b = 0. At larger strains (r > 1.5%), the mobilized shear strength max (R) tends to become nearly independent of b, for b values in excess of 0.3, but considerably lower for b = 0. This behaviour i s consistent with previous investigations at a = 0 (section 2.2.3), although at these high levels of stress and strain the degree of nonuniformity in HCT tests may be significant (see Chapter 3). The development of shear strains (Fig. 5.22(a)) i s in agreement with drained HCT results reported by Symes et a l . (1982) for dense sand at a = 45°. However, they report the s t i f f e s t volumetric response corresponding to b = 0, as opposed to b = 0.3 to 0.5 in Fig. 5.22(b). Direction of Strain Increment Figure 5.23 shows the directions of the principal strain increments (a, ) during these d i r e c t i o n a l shear tests with a = 45°. Deviations 143 55 - r 53 H 51 H 49 Co •8 4 7 H 45 43 1.0 Ottawa sand CTm = 300 kPa D r = 34 % v/ — ^ V v . b = 0 b = 0.3 -x- -x- b = 0.5 b = 0.8 X..A a = 45 1.4 — I — 2.2 1.8 R - C T ; / a3 2.6 —r— 3.0 3.4 Figure 5.23 E f f e c t of b on St r a i n Increment Directions During Shear Loading. 144 between a^ e and a may be noted to be similar, regardless of the b-value. The magnitude of the deviation tends to reduce on shearing. This may be attributed to a progressive alteration of the i n i t i a l anisotropic fabric of the sand due to induced shear strains. The inherent (or i n i t i a l ) cross-anisotropy of the material tested (see section 5.2) i s again indi -cated by the deviation of the strain increments towards the horizontal direction ( a ^ > a, in Fig. 5.23). Strain Paths The strain paths observed i n shear tests with different b-values are presented in Fig. 5.24. It is interesting to note that the strain paths are i n i t i a l l y linear in the small stress ratio region (R £ 1.8), regard-less of b-value (Figures 5.24(b) and (c)). This i n i t i a l linearity in the strain paths is an indication that changes i n the i n i t i a l anisotropy of the sand, due to induced strains, are only significant when R exceeds 1.8. Experimental evidence in support of this conclusion at several i n c l i n a t i o n s a has been shown in Fig. 5.6. Similarly, the level of (within the range 50 to 500 kPa) has been reported not to affect the i n i t i a l linearity of strain paths i n t r i a x i a l shear tests (Negussey, 1984). Figure 5.24c also indicates that shearing at b = 0.3 corresponds closely to the condition of plane strain, since very small strains e3 develop i n the sand specimen throughout the test. This has significance in that the s t i f f est response was shown to result when b = 0.3 (Fig. 5.22). Thus, unwanted conservatism may be introduced i f s o i l parameters obtained from conventional t r i a x i a l tests (b = 0) are used in analysis of plane strain problems. 146 5.4.2 Continuous Variations in b A s e r i e s of b-tests on medium-loose Ottawa sand (D r = 34%) was carried out under drained conditions. Throughout each test, the stress parameters R, a and were kept constant, while b was cycled between 0 and 1 i n a continuous manner. As indicated in Table 4.2 (tests B2 to B4) the values of a selected were 0, 25° and 45°. Stress-Strain Behaviour Figure 5.25 presents the volumetric and shear strain responses of the sand due to one cycle variation in b. Although small i n magnitude, both e , and Y increase progressively as b increases. Differences in vol 'max J inclination a do not seem to be effective in producing significant differences i n strain response. Volumetric contractions may be noted to become significant only when b exceeds about 0.40, regardless of the direction o. It may be noted i n Fig. 5.25 that about 50% of the maximum shear strains induced during b-loading (b =0 to 1) i s recovered on b-unloading (b = 1 to 0). Volumetric contractions, on the other hand, continue occurring, although at a reduced rate, when b i s decreased back to zero. The rates of volumetric contraction and shear distortion during b-loading have been observed to be lower for denser sand (Sayao and Vaid, 1988a). As a consequence, cyclic variations in b may be expected to have more significant effects on looser sand. Direction of Strain Increment Examination of the principal strain increment directions could perhaps permit further insight into the nature of strains induced by a 147 148 continuous variation i n b. D i f f i c u l t y arises, however, due to some inherent features of stress and strain paths in b-tests. Figure 5.26 shows magnitudes of stresses and strains observed in test B3 (a = 25°), plotted as functions of b. It may be noted that the i n i t i a l stress ( = O j ) i s the only stress component that increases with b, from a'2 = a'3 (at b = 0) to oj = a[ (at b = 1). As a consequence, the major principal stress increment occurs i n the radial direction (&a[ = Aop , while Aoa and Ao 3 are confined i n the vertical circumferential plane (specimen wall). Accordingly, the major principal strain (e t) and the major prin-cipal strain increment (Aex) are fixed i n the radial direction, while e2 and e 3 are confined in the plane of the specimen wall (Fig. 5.26b). The directions of the intermediate strain increments for a = 0, 25 and 45° are shown in Fig. 5.27. It is interesting to note that, except when a = 0 (axes of loading ( O j ) and anisotropy are coincident), e2 and e s do rotate in the plane of the wall, even though a l l three principal stresses are fixed in direction. This i s probably related to the i n i t i a l a n i s o t r o p y of the sand (see section 5.2), since continuously increases (i.e., e 3 rotates towards the horizontal direction) during b-loading. Strain Paths Figure 5.28 presents the strain paths observed when b was increased from 0 to 1 under various constant a directions. Significant nonlinear-i t y i n the strain paths may be noted for the tests with a > 0. This is particularly accentuated i n the strain space (e2, e 3 ) , which corresponds to the vertical circumferential plane (Fig. 5.28b). The rotation of principal strains in this plane, associated with the i n i t i a l l y cross-149 Figure 5.26 P r i n c i p a l Stresses and Strains i n b-Tests on Medium-Loose Sand. 150 Figure 5.2 7 D i r e c t i o n of Intermediate Strain Increment During b-Tests. 151 0.4 0.3 H 0.2 -i 0.1 -A 0.0 0.0 -0.02 H -0.04 H -0.06 -4 -0.08 D r = 34 % ; CTm = 300 kPa ; R = 2 • a = 0 0 a = 25 a = 45 I 1 1 1 — -0.26 -0.22 -0.18 Ottawa sand i i i i 1 1 1 — -0.14 -0.10 -0.06 -0.02 £ o (%) Figure 5.28 Strain Paths During b-Tests. 152 anisotropic fabric of the sand, contributes to the observed nonlinearity of the strain paths. Strain path nonlinearity i s much less evident in the specimen tested under a = 0 condition. In this case, principal strains are a l l fixed in direction due to the coincidence of principal axes of loading and aniso-tropy, as already pointed out in the previous section. The slight nonlinearity observed i n the f i n a l stages of this test (Fig. 5.28), however, corresponds to a stress region where b i s greater than 0.5. In this region ( R = 2 , a = 0 , b > 0.5), greater stress and strain nonuni-formities within the specimen wall (see chapter 3) may have contributed to this observed nonlinearity in the strain paths. 5.A.3 Conclusions Significant differences in strain response were observed in shear t e s t s at constant values of b, o' and a. The experimental r e s u l t s m suggest that plane strain problems (b = 0.3) may have their deformations largely overpredicted, when data from conventional t r i a x i a l tests (b = 0) i s used. Regardless of b-value, however, the i n i t i a l cross-anisotropy of the sand i s maintained for loading below R = 1.8 to 2.0. At higher levels of stress ratio, effect of strain-induced anisotropy becomes progressively more evident. Continuous increase i n b-value under constant R, a and o' induces m progressive accumulation of volumetric contraction and shear strains i n the medium-loose sand. These strains are, however, relatively small in the range b = 0.0 to 0.4, which corresponds to most loading conditions i n geotechnical practice. 153 5.5 PROPORTIONAL LOADING Along a proportional loading path in general stress space, the mean e f f e c t i v e stress o' varies under constant values of R, b and a. This m corresponds to loading paths in which proportional increments of stress components are imposed to the s o i l . Typical examples of proportional loading at R = 2 are illustrated i n Fig. 5.29. It may be noted that a l l three principal stress ratios (a[/a2, o\/a\ and R = a[/a3) stay constant, together with the direction a. This implies that the overall obliquity of the applied stress vector remains fixed. Study of proportional loading paths i n general stress space i s of fundamental importance for anisotropic f r i c t i o n a l materials, whose strain response i s governed mainly by the values of R and a. Previous investi-gations on proportional loading of sands have, however, been confined to conventional t r i a x i a l stress conditions (e.g., El-Sohby, 1969; Negussey, 1984). A systematic study of proportional loading paths in general stress space ( 0 £ a £ 90°; 0 £ b £ 1) i s presented herein. The strain response of medium-loose Ottawa sand i s examined f i r s t under selected inclinations a. Observations are based on stress conditions correspond-ing to b = 0 and R = 2, i n order to minimize effects of non-uniformity within the HCT specimen. The study i s then extended to other i n i t i a l density and stress conditions. 5.5.1 Increase in o' at various a m • Stress-Strain Behaviour I n i t i a l conditions for this test series were D r = 34%, R = 2, b = 0 and o' =50 kPa. Three different directions a were selected: a = 0, 25 m ' and 45°. During each test, was monotonically increased from 50 to 300 kPa, under constant R, a and b. 154 500 —i , C T ' ( W o ) Figure 5.29 Stress Components During Proportional Loading i n General Stress Space. 155 Figure 5.30 shows the stress-strain behaviour of pluviated Ottawa sand under proportional loading. For a given increment i n o^, both volumetric contractions and shear distortions may be noted to increase with the loading (o x) direction a. In particular, the response at o = 0 is much s t i f f e r than at other directions. These observations are consistent with shear and hydrostatic compression results previously reported (section 5.2), which indicated a marked i n i t i a l (or inherent) cross-anisotropy of the sand. It i s interesting to note that, at higher values of a, large shear s t r a i n s occurred under changes i n o^ only. This would imply, under undrained conditions, positive excess pore pressures much larger than A o . m Direction of Strain Increment The observed directions (a d g) of the major principal strain incre-ments are compared to corresponding constant principal stress directions a i n F i g . 5.31. Behaviour under monotonic increase in a' i s such that m a^gis nearly constant and slightly greater than a, for a-values greater (*) than zero . This may again be explained by the i n i t i a l cross-anisotropy of the sand, which causes a small deviation of e x towards the hori z o n t a l direction. Constancy of asunder constant inclination of the applied stress vector suggests that effects of strain induced anisotropy may be considered insignificant in proportional loading paths, at the i n i t i a l stress and density conditions considered in this investigation. (*)as already pointed out in earlier sections, there i s a coincidence of axes of loading and anisotropy when pluviated sands are loaded v e r t i c a l l y . Thus, a^e = a = 0 under this condition. 156 Figure 5.30 E f f e c t of a on Str a i n Development Due to Proportional Loading. 157 90 -80 -Dr = 3 4 % ; R = 2.0 ; Ottawa sand b = 0 70 -"ees ) 60 -degi 50 - o Q 75 0 £> 40 - a = 4 5 ° U J T J 30 - • ~~ B • n 20 -10 -0 -a = 2 5 ° u c I . I i . i -100 200 1 I 300 (J ' ( kPa ) Figure 5.31 Strain Increment Directions During Proportional Loading. 158 Strain Paths Any induced anisotropy i n proportional loading may be further investigated by examining the strain paths followed in principal strain spaces ( e l t e,) and (e a, e 3 ) . It is apparent, from the results shown in Fig. 5.32, that proportional loading paths result in linear strain paths, regardless of inclination a. These observations give support to the above suggestion that l i t t l e change in inherent anisotropy occurs on straining under proportional loading at R = 2 and b = 0. Similar conclusion, based on proportional loading results i n the t r i a x i a l device (o = 0), has been presented by Negussey (1984). Marked differences i n strain response as a function of a are also apparent in Fig. 5.32. Under vertical loading conditions (o = 0), a l l principal strains are contractant and are of relatively small magnitudes. Under inclined loading conditions (a = 25° and 45°), a substantial increase i n major principal strains (£j) may be noted (Fig. 5.32a). As already pointed out, this i s due to the inherently anisotropic character-i s t i c s of the sand. Associated with these larger contractant strains e x, negative (dilatant) strains e 3 are observed in the plane of the specimen wall, at a = 25 and 45°. These significant differences in principal strains e x and e s explain the occurrence of much larger shear strains (r ) at o > 0, shown in Fig. 5.30. 'max ' 6 5.5.2 Behaviour at Other I n i t i a l Conditions The results of proportional loading tests presented in Figures 5.30 to 5.32 correspond to medium-loose sand at stress conditions R = 2 and b = 0. Additional series of proportional loading tests were carried out at other selected values of D , R and b, i n an attempt to verify the 160 extension of the experimental observations pointed out in the previous section. Interpretation of results is made simpler by keeping a1 in the vertical direction in a l l remaining tests. As a consequence, the direc-tions of principal stress and strain increments were coincident with the axis of i n i t i a l anisotropy i n the pluviated sand (a = = 0). Effect of Relative Density Figure 5.33 shows the volumetric and shear strain responses of loose (D r = 20%) , medium-loose (D r = 33%) and dense (D r = 60%) Ottawa sand. Monotonic increase i n o' from 50 to 300 kPa was imposed at constant m r stress parameters R = 2, b = 0.5 and a =0. As expected, induced con-tractions and distortions are noted to be larger with decrease in relative density. The observed strain paths in principal strain space are plotted i n Fig. 5.34. It i s important to point out that behaviour in proportional loading, at the selected stress conditions described above, i s such that the major principal strain e x occurs in the radial direction, regardless of the value of D r. Accordingly, e, i s vertical and e s occurs in the circumferential direction. This i s directly related to the inherent cross-anisotropy of the sand tested. If the sand was isotropic, strain response under a = 0 condition would be necessarily such that e1 = e.^. In addition, the stress system (R = 2, b = 0.5) imposed to the specimen also contributes to the dilatant response in the circumferential direction. This i s discussed in the next section on the effect of b. A detailed examination of Figs. 5.34(a) and (b) reveals that, at a given l e v e l of v e r t i c a l s t r a i n (e = e a ) , larger contractant radial 161 0.5 0.8 - Dr = 20% 0.6 - (t>) 33% X D 0.4 -^ E /O-^ 60% 0.2 -0.0 -C W i 100 • i 1 1 1 • 200 300 Cm ( kPa ) Figure 5.33 Effect of D r on Strain Development Due to Proportional Loading. 162 F i g u r e 5 .34 E f f e c t o f D r on S t r a i n P a t h s D u r i n g P r o p o r t i o n a l L o a d i n g . 163 s t r a i n s ( e r = e x) and slightly less dilatant circumferential strains (e^ = e 3) are associated with lower relative densities. As a result, overall deformations i n the horizontal plane are greatest for the loose (Dr = 20%) sand. This behaviour suggests that a higher degree of i n i t i a l anisotropy i s present in the loose sand. Such conclusion is consistent with the greater effect of principal stress rotation observed on loose sand (section 5.3) and with hydrostatic compression data reported by Negussey (1984). Another important aspect to be pointed out in Fig. 5.34 i s related to the linearity of the strain paths. The dense sand exhibits essen-t i a l l y linear strain paths, which are an indication that the i n i t i a l anisotropy i s maintained throughout the proportional loading. On the other hand, a small curvature may be noted in the strain paths followed by the loose (D r = 20%) sand. This suggests that the larger strains experienced by the loose sand tend to induce a small but gradual change in the i n i t i a l anisotropy. Consistent with the above observations, a very small curvature may be noted i n the strain paths corresponding to an intermediate relative density (D r = 33%), between loose and dense. But, the strain paths followed by the medium-loose sand may be considered as nearly linear, for a l l practical purposes. Small changes in inherent anisotropy under proportional loading may therefore be confined to very loose sands only. This important characteristic response of loose sand has apparently not been detected i n previous investigations on proportional loading. This i s probably due to the fact that these studies were carried out in the t r i a x i a l device, where uniform sand specimens looser than about 30% are d i f f i c u l t to prepare. Moreover, the stress system in t r i a x i a l 164 compression is limited to b = 0 conditions. The effect of b on the strain response to proportional loading i s examined in the following section. Effect of Intermediate Stress Parameter In t h i s test s e r i e s , three medium-loose specimens (D r = 33%) were subjected to R = 2.0, a = 0 and different values of b = 0.0, 0.3 and 0.5, p r i o r to increasing o' from 50 to 300 kPa. The range of b selected m represents the majority of practical loading situations ranging from axisymmetric to plane strain. Volumetric and maximum shear strains are shown in Figs. 5.35(a) and (b), respectively. It is apparent that, even though volumetric contractions are nearly similar, shear distortions are highly dependent on b-value. The observed differences i n shear strain response to proportional loading may be explained by examining the strain paths followed in principal strain space (Fig. 5.36). Under axisymmetric conditions (b = 0, a = 0), a l l three principal strains are contractant and the major strain e x occurs in the vertical direction. On the other hand, under more general stress conditions (b = 0.3 and 0.5), dilatant minor principal strains e 3 are noted. Moreover, major principal strains e1 are developed in the radial direction. This observed switch i n e1 from vertical to radial direction, when b changes from 0.0 to either 0.3 or 0.5, may be attributed to two main factors: the stress system imposed to the specimen and the inherent cross-anisotropy of the sand. As already i l l u s t r a t e d i n F i g . 5.29, i n c r e a s e i n o^ . (= a'2) i s greater when proportional loading is imposed at b = 0.5 than at b = 0. In addition, Figure 5.35 Effect of b on Strain Development Due to Proportional Loading. 166 Figure 5.36 E f f e c t of b on St r a i n Paths During Proportional Loading. 167 increase i n a'3 i s also smaller at b = 0.5. Consequently, i f the stress r a t i o i s r e l a t i v e l y low, becomes the major p r i n c i p a l s t r a i n i n inherently anisotropic sand. It may be noted in Fig. 5.36 that e t and e 3 increase i n magnitude (with opposite signs) with an increase i n b-value. This explains the differences i n shear strain response under different b-values already pointed out (Fig. 5.35). The strain paths shown in Figs. 5.36(a) and (b) are essentially linear. Thus, the i n i t i a l cross-anisotropy of the medium-loose sand may be assumed unaltered during proportional loading at the selected stress conditions, R = 2.0 and a = 0. Effect of Stress Ratio A l l proportional loading tests reported i n the previous sections correspond to only one R-level (R = 2.0). In this test series, three medium-loose sand specimens were subjected to different values of R = 1.3, 2.0 and 3.0, and to identical b = 0.5 and a = 0, prior to imposing a monotonic increase i n o' from 50 to 300 kPa. m Figure 5.37(a) shows the stress-strain behaviour of the medium-loose sand in terms of volumetric deformations. Larger contraction is induced i n sand at R = 3.0, for a given l e v e l of o^. At R i 2.0, volumetric contractions are seen to be nearly similar. In contrast, much larger differences i n shear distortion may be noted (Fig. 5.37(b)) as R increases from 1.3 to 3.0. The strain paths observed in this test series are shown in Fig. 5.38. In a l l tests, el = e r and e 3 = e^. At R = 1.3 and 2.0, principal strains induced i n the medium-loose sand follow nearly linear paths. At 168 Figure 5.37 Effect of R on Strain Development Due to Proportional Loading. 169 0.0 -0.8 -0.6 -0.4 -0.2 0.0 ( % ) Figure 5.38 Ef f e c t of R on S t r a i n Paths During Proportional Loading. 170 higher stress ratio (R = 3.0), however, principal strain paths are clearly nonlinear. This suggests that large strains may have induced gradual changes in the i n i t i a l anisotropy of the sand tested at R = 3.0. It should be pointed out, however, that significant stress nonuniformity may have also partly contributed to the differences i n strain response observed at this higher R-level, even though the degree of nonuniformity (B R) stays constant during proportional loading paths (see Chapter 3). 5.5.3 Conclusions Experimental results on proportional loading in general stress space indicate that volumetric contractions and shear distortions induced in Ottawa sand are highly dependent on the i n i t i a l stress and density condi-t i o n s . At a given o'-level, deformations are observed to increase with ra decrease in D and increase i n a, b and R. r ' Examination of principal strain paths suggest that the sand's inher-ent anisotropy is gradually affected by induced strains at low relative density (D r = 20%) or at high stress ratio (R = 3.0). Under a l l other i n i t i a l conditions investigated, strain paths were shown to be essenti-a l l y linear during proportional loading, implying preservation of the nature of inherent anisotropy. 5.6 ADDITIONAL INVESTIGATIONS ON STRESS-STRAIN BEHAVIOUR The investigations described in previous sections focussed on the strain response of sands subjected to diverse loading paths in general stress space. In each test series, a l l specimens were f i r s t brought to the desired i n i t i a l state (Dr, o^, R, b, a) in an identical manner, as a part of their preparation and set-up phase. During loading only one 171 stress parameter was varied while others were held constant. Observed differences i n strain response to identical loading paths may thus be attributed only to differences in i n i t i a l state. Stress-strain characteristics of sands are, however, known to depend not only on the i n i t i a l state, but, at a current stress state, they depend on both the previous stress history and the subsequent stress path. A preliminary assessment of the influence of these last two factors was the primary objective of two additional series of HCT tests on medium-loose Ottawa sand. In the f i r s t test series, identical specimens were subjected to a common loading paths after being brought to the same i n i t i a l state through different preparation stress paths. Differences in stress-strain behaviour during subsequent shear loading and principal stress rotation are examined independently. In the second test series, two specimens were sheared by increasing R under constant o^, b and a, aft e r identical preparation stages to a common i n i t i a l state. In one test, however, shear loading was interrup-ted at selected R-levels and a cyclic rotation of principal" stresses was applied before shearing was resumed. Effects of these cyclic rotations on strain response during subsequent shearing stages are examined. Comparisons of deformation response are made with the second test, in which no such rotations of principal stresses were imposed. 5.6.1 Effect of Previous Stress History  Shear Loading Stress conditions at the commencement of each test represent o' = 300 kPa, R = 2.0, b = 0.5 and a = 45°. This i n i t i a l stress state m 172 corresponds to point I (=1') in Fig. 5.39(a). Two medium-loose (Dr = 36%) specimens (T2 and R7) were brought to this common i n i t i a l stress state by following different stress paths (OABI and O'l', respectively). Identical shearing stress paths (IF = I'F') were then imposed, under constant o', b and o. It should be pointed out that 0 and 0' are points m representing the same state of hydrostatic stresses (R = 1.0, = 300 kPa, b and a = undefined). Shear and volumetric strain responses from both tests are compared in Fig. 5.39(b) and (c), respectively. Values of Ymax were computed with reference to specimen dimensions at i n i t i a l points 0 and 0'. That represents a hydrostatic state = 50 kPa. This is to avoid problems with possible interchange in £j and e, directions, as observed in b and o m loading (see sections 5. A and 5.5, respectively). It may be noted that strain response during i n i t i a l shearing from R = 1.0 to 2.0 is highly dependent on (b, a) state. Strains developed under axisyraraetric conditions (b = 0, a = 0, i n test T2) are much smaller than when the stress condition is more general (b = 0.5, a = 45°, i n test R7). This i s clearly related to inherent anisotropy of sand. On subsequent shearing from R = 2.0 to 2.4 along the common path (IF or I'F'), specimen T2 again shows the s t i f f e s t response. The results suggest that significant hardening may have occurred in the sand specimen subjected to the particular sequential stress path OABI, as opposed to the more direct path O'l'. This s t i f f er response shown by specimen T2 may also be due partly to a sudden change in the direction of stress increment vector in general stress space (R, o^, b, a) at point I. In contrast, the stress path imposed on the sand in test R7 i s continuous. Consequently, the stress increment vector in general stress space does not suffer a change 173 tTJ^  = 3 0 0 kPa D r = 36 % b) Shear S t r a i n Response c) Vo lumet r ic S t ra in Response Figure 5.39 E f f e c t of Stress Path History on St r a i n Response Under Shear Loading. 174 in i t s direction at point I'. Less severe effects on deformations may be associated with sudden changes in stress increment vector. This occurs in spite of the different strain states (larger distortion and smaller contraction i n specimen T2) i n i t i a l l y present at the stress state represented by the common stress points I and I 1 in Fig. 5.39. Principal Stress Rotation Two medium-loose sand specimens (Al and o4) followed different stress paths to a common stress state (o^ = 300 kPa, R = 2.0, b = 0.5, a = 0), represented by point P (Fig. 5.40 insert). From P, a continuous rotation of principal stress directions from a = 0 to 45° was imposed to both specimens, under constant R, and b conditions. The strains accumulated at stress state P i n specimen Al represent y = 0.23% and E , = 0.28%. At the same i n i t i a l state P, much larger 'max vol e shear d i s t o r t i o n and volumetric contraction (y = 0.75% and e , = max vol 0.34%) were observed i n specimen a4. This i s primarily due to the proportional loading stage (o^ = 50 to 300 kPa) imposed on specimen a4 under more severe stress conditions (b = 0.5, R = 2) than i n the case of specimen Al (R = 1, b = undefined), as suggested by the results presented in section 5.5. Volumetric and shear strains observed during principal stress rota-tion only are shown in Fig. 5.40. Volumetric contraction at any given rotation angle a i s again larger in specimen a4. In contrast, s t i f f e r shear response may be noted at i n i t i a l rotation stages (a < 20°) in specimen o4, when compared to that i n A l , but no ready explanation may be suggested for this observation. As principal stress rotation proceeds, specimen o4 shows a softer shear strain response than Al. 175 Figure 5.40 Ef f e c t of Stress Path History on Str a i n Response Under P r i n c i p a l Stress Rotation. 176 Results i n Fig. 5.40 ill u s t r a t e the effect that previous stress path history may have on subsequent strain response of sand subjected to rotation of principal stress directions. 5.6.2 Effect of Rotation Cycles on Subsequent Shearing In t h i s test s e r i e s , two medium-loose (D r = 35%) specimens were i n i t i a l l y subjected to a monotonic increase in R under constant = 300 kPa, b = 0.5 and a = 0 . At R = 1.3, shearing was interrupted i n one specimen (Tl) and two cycles of drained principal stress rotations from 0 to 90° were imposed i n a continuous manner, under constant principal stress magnitudes. Shearing was then resumed under stress conditions identical to those before rotation. Similarly, at higher selected R values (R = 2.0, 2.5 and 3.0), one drained rotation cycle (a = 0 •+ 60° -* 0) was imposed to specimen T l under constant R, and b. Figure 5.41 shows the shear and volumetric strain responses of specimen T l . For comparison, response of specimen R3, in which R was increased under no history of rotation cycles, i s also shown. Values of r and e , i n specimen T l refer to shearing phases only, a f t e r max v o l discounting the shear distortion and volumetric contraction which occurred during previous rotation cycles. Significant hardening may be noted to result from previous rotation cycles. Immediately after each cycle, a marked increase in shear stiffness i s observed i n specimen T l (Fig. 5.41(a)). Moreover, at any given R-level, the tangent shear modulus i s always greater i n specimen T l than i n R3. Similarly, after each rotation cycle, dilatant volumetric strain response i s seen in specimen T l (Fig. 5.41(b)). This i s in contrast with the contractant response immediately before each rotation 177 Figure 5.41 E f f e c t of Previous Rotation Cycles on Strain Response Under Shear Loading. 178 cycle was applied. Again, at any given R-level, volumetric response of specimen T l i s much s t i f f e r than specimen R3. 5.6.3 Conclusions Based on the experiental data presented in the two previous sections, stress path history may be considered to have a profound i n f l u -ence on the strain response of medium-loose sand during subsequent loading paths. This conclusion confirms the need for systematic prepara-tion and set up of sand specimens up to the selected i n i t i a l stress state, when laboratory investigations on fundamental stress-strain behaviour are to be carried out. Also, in practical applications of stress path testing, sand specimens should be brought to the i n i t i a l i n-situ conditions through a stress path representing as close as possible the past stress history i n the f i e l d . Significant hardening effects are noted to result from previous cycles of drained principal stress rotation on subsequent shearing. The results suggest that cyclic rotation, under drained conditions, has beneficial practical effects. It makes the medium-loose, sand behave effectively as a dense sand immediately after each rotation cycle, regardless of the R-level. It should be pointed out that i n most loading conditions i n the f i e l d , simultaneous variations i n two or more stress parameters are li k e l y to occur. In fact, situations where a l l four parameters (R, o^t b and a) vary simultaneously, over a limited loading range, may represent the majority of practical problems. Research on this broad topic i s currently under way using the HCT apparatus at UBC. Preliminary results show differences in strains 179 induced in sand specimens by simultaneous, as opposed to sequential, variations in R and a under drained conditions (Wijewickreme, 1989). 180 CHAPTER 6  SUMMARY AND CONCLUSIONS The behaviour of sands which are inherently anisotropic has been studied under general stress paths. A hollow cylinder torsional (HCT) apparatus was developed as a part of the research program. The HCT i s the only device that enables independent control of four stress para-meters: stress r a t i o R, mean normal stress a^, intermediate principal stress o 2 (or b) and direction a of the major principal stress. These constitute four parameters that influence behaviour of inherently aniso-tropic materials. Very l i t t l e amount of research has been previously done on the independent effect of principal stress rotation under multiaxial stress conditions. In devices other than the HCT, controlled principal stress rotations are either impossible or are accompanied by uncontrolled changes in other stress parameters. Where HCT devices have been used, investigations have often focussed on stress states near or at failure conditions. At such states, unacceptable levels of stress and strain nonuniformities are l i k e l y to be present across the wall of hollow cylindrical specimens. Minimization of these nonuniformities can however be achieved through careful selection of specimen geometry and avoiding certain regions of the stress space to be investigated. It is argued in this thesis that evaluation of nonuniformity levels in HCT specimens should not be done on the basis of the distribution of individual stress compon-ents. Rather, these nonuniformities should be defined in terms of stress ratio, which controls deformation response of f r i c t i o n a l materials. 181 The experimental program was designed to examine systematically the response of anisotropic sands to individual changes in each stress para-meter (R, a, b and o * ) . A l l tests were ca r r i e d out on saturated m specimens of pluviated sand under f u l l y drained conditions. Additional investigations on the effect of stress history during shear loading and principal stress rotations were also performed. Based on the experimental evidences presented in this thesis, i t i s concluded that the strain response of medium-loose sand to shear loading i s strongly dependent on the inclination a of the major principal stress relative to the deposition direction. The observations suggest that the sand tested (Ottawa sand) i s inherently cross-anisotropic, i n spite of i t s relatively rounded grains. Larger volumetric and shear deformations are shown to be associated with loading in which a > 0. This implies that deformations predicted based on conventional testing procedures (tr i a x i a l tests on "vertical" specimens, a = 0) may be on the unconserva-t i v e side. P r i n c i p a l s t r a i n increment directions (a^) always deviate towards the horizontal, when compared to principal stress directions (a = a^ o) , for a l l loading conditions other than the vertical compression. The magnitude of this deviation tends to reduce at high shearing levels, due to possible effects of induced strain anisotropy. Strain path linea r i t y i n principal strain space at R-levels lower than about 2.0 implies preservation of inherent anisotropy during shear loading. During continuous principal stress rotation at constant R, b and o^, progressive accumulation of both volumetric contractions and shear distortion occurs. These deformations increase significantly with decrease in relative density. Similarly, for a given density, larger deformations r e s u l t at higher l e v e l s of R and o'. On the other hand, 182 principal stress rotation effects are found to be relatively insensitive to differences in b-value within the range 0.0 to 0.5, which corresponds to most practical loading situations. The strain response of Erksak and Ottawa sands to principal stress rotation are found to be similar, despite some differences in mineralogy and grain angularity. During cyclic principal stress rotation, deformations are more significant i n the f i r s t rotation cycle. In addition, larger deviations between a^ e and occur in the f i r s t forward rotation cycle on either side of the ver t i c a l . These observations indicate the predominance of irrecoverable strains during principal stress rotation. The value of the intermediate stress parameter b was found to affect strain response to directional shear loading. At b = 0.3, which may be closely related to plane strain conditions, the response was much s t i f f e r than at b = 0.0. The effect of continuous cyclic variations in b at constant R, o ' and o has been evaluated for the f i r s t time. Test results m on medium-loose sand show that, regardless of loading direction a, both volumetric and shear strains increase progressively as b increases from 0 to 1. Changes in b alone w i l l thus add deformations to those already induced by changes in R, offi and a, under general loading conditions. The strains caused by change in b in the range 0.0 to 0.4 were found to be relatively small i n magnitude. Changes in o^ at several levels of constant R, b and a correspond to proportional loading paths i n general stress space. Experimental results on medium-loose Ottawa sand indicate that, at a given o 1 level, both m shear distortion and volumetric contraction increase with increase i n R, b and a. During proportional loading, the nature of inherent anisotropy was essentially preserved at a l l stress conditions investigated, the only 183 exception being at high R = 3. Similarly, induced anisotropy tend to become noticable at low relative density states. A l l findings summarized above are related to comparisons between specimens with identical stress history prior to loading along a selected stress path. The effects of different stress histories on the strain response to identical subsequent loading are shown to be very significant in both shear loading and principal stresses rotation. 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Ph.D. Thesis, in preparation, Department of C i v i l Engineering, University of British Columbia, Canada. Wong, R.K.S. and Arthur, J.R.F. (1985). "Induced and Inherent Anisotropy i n Sand", Geotechnique, Vol. 35, No. 4, pp. 471-481. Wong, R.K.S. and Arthur, J.R.F. (1986). "Sand Sheared by Stresses with Cyclic Variations i n Direction", Geotechnique, Vol. 36, No. 2, pp. 215-226. Wu, T.H., Loh, A.K. and Malvern, L.E. (1963). "Study of Failure Envelope of Soils", Journal of the Soil Mech. and Found. Div., ASCE, Vol. 89, No. SMI, pp. 145-179. 195 APPENDIX MEMBRANE PENETRATION CORRECTIONS The experimental studies described in this thesis required adequate control of boundary stresses and precise determination of boundary displacements (or strains). As indicated i n section 3.2, strain components e and e Q i n UBC-HCT tests are computed from volume changes of the internal confining chamber (AV.) and of the s o i l specimen (AV ). X s In the HCT device, both the internal and the external surfaces of the hollow cylindrical specimen are covered by flexible rubber membranes. When drained loading paths of changing c e l l pressures are followed, measured volume changes of coarse grained soils are subjected to error, caused by changes in penetration of each membrane into or out of the soil's interstices. In a l l experimental results presented i n this thesis, membrane penetration corrections were applied i n accordance with the method (*) proposed by Vaid and Negussey . It i s considered that the correct volume change of the s o i l specimen (AV ) i s equal to the total volume s change recorded (AV^,) minus the volume change caused by membrane penetration (AV ). The value of AV i s defined as: m m AV «= Ae • A + Ae • A m m. s. m s 1 1 e e where Ae amd Ae are the variat i o n s i n unit membrane penetration, m. m l e respectively on the internal and the external boundary surface areas (A s * 1 (*)Geotechnical Testing Journal, ASTM, Vol. 7, No. 2, June 1984, pp. 70-76. 196 and A ). For a given variation i n the effective confining pressure (AP! S X e (or A P ' ) , the value of Ae (or Ae ) was estimated from the linear log e m. m 1 e relationship reproduced i n Fig. A l . This relationship was obtained for Ottawa sand, with 0.3 mm thick membranes similar to the ones used in the investigations with the HCT device. Essentially identical relationship for membrane penetration corrections was obtained for Erksak sand, as described i n Golder Associate Report No. 862-2089 (Feb. 1987). S i m i l a r l y , the volume change of the i n t e r n a l chamber ( A V ^ ) i s obtained by correcting the total recorded value for the penetration of the inner membrane (AV ) . Thus AV = Ae * A , where Ae i s also m. m. m. s. m. I i i i I estimated from Fig. A . l , for a given change in AP£. 50 100 200 3 0 0 Ef fect ive conf ining pressure, kPo 5 0 0 Fig. A . l . Membrane Penetration Correction Curve (after Vaid and Negussey, 1984). 

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