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Stress path dependency of dilatancy and stress-strain response of sand Sasitharan, Sabanayagam 1989

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STRESS PATH D E P E N D E N C Y OF DILATANCY A N D STRESS-STRAIN RESPONSE OF SAND B y S A B A N A Y A G A M S A S I T H A R A N B. Sc. (Eng) Hon. , University of Peradeniya Sri Lanka, 1985. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R O F A P P L I E D S C I E N C E in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1989 © S A B A N A Y A G A M S A S I T H A R A N , 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of British Columbia Vancouver, Canada Date A ^ U P . f c a a . >ssq DE-6 (2/88) A B S T R A C T The drained loading behaviour of water pluviated Erksak sand is investigated in the triaxial apparatus by varying consolidation history, stress path and loading direction (compression or extension). It is shown that, under identical minor effective principal stress, anisotropically consolidated sand has a higher tangent modulus than the isotrop-ically consolidated sand in the initial stages of the shearing phase. This difference in the tangent modulus reduces as the sand approaches failure. The modified hyperbolic model, in which the increment in the deviator stress after consolidation is considered as the stress variable, is shown to represent satisfactorily the stress-strain response of anisotropically consolidated sand. The small strain response of anisotropically consolidated sand also shows a hyperbolic variation which is different from the large strain one. The elastic tangent modulus, at a given stress state, of water pluviated isotropically consolidated sand is not unique. It varies with stress path and direction of loading. Thus, the incremental elastic modeling based on hyperbola under conventional stress paths is shown not applicable for other stress paths and loading direction. The failure strength of sand is uniquely related to maximum rate of dilatancy dev/dea regardless of the relative density, minor effective principal effective stress at failure and stress path for both compression and extension loading. The failure strength depends only on the normal stresses at failure and relative density and is not affected by consolidation history or stress path. The water pluviated sand yields a higher failure strength under compression loading than under extension loading. n Table of Contents A B S T R A C T ii List of Tables vi . List of Figures vii List of Symbols xi Acknowledgement xiii 1 I N T R O D U C T I O N 1 2 R E V I E W O F P R E V I O U S I N V E S T I G A T I O N S 3 2.1 General Aspects of Drained Behaviour of Sand 3 2.2 Stress-Strain Response 5 2.3 Failure Strength 12 2.4 Rate of Dilatancy 14 3 E X P E R I M E N T A T I O N 17 3.1 Testing Apparatus 17 3.1.1 A x i a l Loading 17 3.1.2 Lateral Loading 19 3.1.3 Stress Path Control 19 3.1.4 The Triaxial Cel l 23 3.1.5 Standard Reference and Data Acquisition 23 i i i 3.2 Material Tested 24 3.3 Testing Procedure 26 3.3.1 Sample Preparation 26 3.3.2 Sample Set Up 27 3.3.3 Repeatability of Test Results 28 3.4 Testing Program 30 3.4.1 Effect of Consolidation History 30 3.4.2 Effect of Stress Path 30 3.5 Membrane penetration 32 4 T E S T RESULTS 36 4.1 Introduction 36 4.2 Stress Strain Behaviour 37 4.2.1 Behaviour Under Conventional Triatrial Compression 37 4.2.2 Effect of Stress Paths 37 4.2.2.1 Compression Stress Paths 41 4.2.2.2 Extension Stress Paths 44 4.2.2.3 Comparison of Behaviour Under Compression and Exten-sion Stress Paths 46 4.2.3 Incremental Elastic Representations 46 4.2.3.1 Effect of Consolidation History 46 4.2.3.2 Effect of Stress Path 65 4.2.3.3 Addit ional Remarks 75 4.3 Shear Resistance 77 4.3.1 Effect of Consolidation History 77 4.3.2 Effect of Stress Path 77 iv 4.3.2.1 Rate of Dilatancy at Failure 77 4.3.2.2 Angle of Dilation 81 4.3.2.3 Nature of Stress Path Dependency of Peak Friction Angle 84 4.3.3 Prediction of Peak Friction Angle 87 5 S U M M A R Y A N D C O N C L U S I O N S 92 Bibliography 94 Appendices 99 A Tangent Modulus of Anisotropically Consolidated Sand 99 B Stress Strain Response Under Various Stress Paths 102 C Stress Strain Response Under Conventional Triaxial 105 v List of Tables 3.1 Details of Stress Paths Tested 33 4.1 Hyperbolic parameters obtained from triaxial compression data base . . . 50 4.2 Hyperbolic parameters for compression and extension loading 75 vi List of Figures 2.1 Typical Stress-Strain Behaviour of Loose and Dense Sand At Low Confin-ing Pressure in Conventional Triaxial Compression Test 4 2.2 Typical Stress-Strain Behaviour of Sand A t High Confining Pressure in Conventional Triaxial Compression Test 6 2.3 Hyperbolic Stress-Strain Response in Triaxial Compression 8 2.4 Different Stress Paths and Common Point of Intersections 11 3.1 Schematic of The Stress Path testing System 18 3.2 Schematic of The Stress Paths Tested 20 3.3 Gra in Size Distribution of Erksak Sand 25 3.4 Repeatability of Test Results 29 3.5 Stress Paths Investigated 31 3.6 Membrane Penetration Per Unit Surface Area with Changing Effective Confining Pressure 35 4.1 Stress Strain Response in Conventional Triaxial Compression at Low Con-fining Stress 38 4.2 Stress Strain Response in Conventional Triaxial Compression at High Con-fining Pressure 39 4.3 Strains During Hydrostatic Consolidation 40 4.4 Effect of Loading Direction on Stress-Strain Response 42 vi i 4.5 Stress-Strain Response of Isotropically Consolidated Sand Under Various Compression Stress Paths. (Dr=26% and <73C=250 kPa) 43 4.6 Stress-Strain Response of Isotropically Consolidated Sand Under Various Extension Stress Paths.(Dr=26% and cr^=250 kPa) 45 4.7 Stress-Strain Response Under Different Consolidation History.(Dr=26%) 47 4.8 Stress-Strain Response Under Different Consolidation History.(Dr=56%) 48 4.9 Stress-Strain Response Under Different Consolidation History.(Dr=70%) 49 4.10 Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=26%) 51 4.11 Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=56%) 52 4.12 Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=70%) 53 4.13 Variation of Ei/pa with <?'&./Pa in Conventional Triaxial Compression. . . 54 4.14 Variation of KE with Relative Density. 55 4.15 Comparison of Measured and Predicted Stress-Strain Response 57 4.16 Transformed Hyperbolic stress-strain Behaviour of KD Consolidated Sand. 58 4.17 Transformed Hyperbolic stress-strain Behaviour of KQ Consolidated Erk-sak Sand in Small Strain Range 59 4.18 Variation of Et as a Function of <r& for Ka and Isotropically Consolidated Sand. (Dr=56% and <4=100 kPa.) 61 4.19 Variation of Et as a Function of crd for KQ and Isotropically Consolidated Sand. (Dr=56% and cr^=250 kPa.) 62 4.20 Variation of Et as a Function of aj for KQ and Isotropically Consolidated Sand. (Dr=26%) 63 vm 4.21 Variation of Et as a Function of crj for K0 and Isotropically Consolidated Sand. (Dr=70%) 64 4.22 Deviator Stress A x i a l Strain Response Under Various Compression Stress Paths 66 4.23 Variation of (r-2jt)/(r-l) with t o n - 1 ( - l / r ) 67 4.24 Comparison of Modul i Evaluated At Fixed Stress State (o-'d,<r3) Following Different Stress Paths in Triaxial Compression. (Dr=26%) 69 4.25 Comparison of Modul i Evaluated At Fixed Stress State (cr'd,<T3) Following Different Stress Paths in Triaxial Compression 70 4.26 Comparison of Modul i Evaluated At Fixed Stress State (crd,cr3) Following Extension Stress Paths. (KE, n and </> Obtained from o~'3 constant Triaxial Compression Tests) 72 4.27 stress-strain Response of Extension Stress Path at Different Constant <T3. (Dr=56%) 73 4.28 Variation of Ei/pa with o~'3/pa for Constant o~'3 Triaxial Extension Tests. . 74 4.29 Comparison of Modul i Evaluated A t Fixed Stress State (<r'd,<r'3) Following Extension Stress Paths. (KE, n and <f> Obtained from constant <r3 Triaxial Extension Tests) 76 4.30 Variation of Peak Friction Angle W i t h Max imum rate of Dilatancy. . . . 79 4.31 Friction Angle Mobilized in The Ring Shear Apparatus 80 4.32 Variation of Peak Friction Angle W i t h Max imum Dilation Angle 83 4.33 Variation of Peak Friction Angle W i t h o~'3f in Triaxial Compression. . . . 85 4.34 Variation of Peak Friction Angle W i t h o~'mf in Triaxial Compression. . . . 86 4.35 Variation of Peak Friction Angle W i t h <T'3J in Triaxial Extension 88 4.36 Variation of Peak Friction Angle W i t h (r'mi in Triaxial Extension 89 4.37 Comparison of Peak Friction Angle in Triaxial Compression and Extension . 90 ix B . l Stress Strain Response From Test Performed Along Various Stress Paths. (Dr=56%) 103 B. 2 Stress Strain Response From Test Performed Along Various Stress Paths. (Dr=70%) 104 C. l Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=26%) 106 C.2 Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=56%) 107 C.3 Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=70%) 108 x List of Symbols b = Intermediate principal stress parameter (<T'2 — <rz)l{°~\ D r = Relative density. D n = Initial relative density under 20 k P a confining stress. 8 = Increment. E = Youngs modulus. E i = Initial Youngs modulus. E t - Tangent modulus. ei Major principal strain. 3^ = Minor principal strain. e a - Axia l strain. e r = Radial strain. = Volumetric strain. e = Direction of stress increment. K c Consolidation stress ratio a'vJ<j'hc. K E Modulus number. K 0 Inverse stress ratio of zero lateral deformation. - Poisson ratio. n = Modulus exponent. P a Atmospheric pressure. R - Stress ratio {o~[/o~3). X I r = Incremental stress ratio. <Tj = Major effective principal stress. a'2 = Intermediate effective principal stress. erg = Minor effective principal stress. a'v = Vertical effective stress. (Tjj = Horizontal effective stress. crj = Deviator stress ((X^-cr'^. cr'm = Effective mean normal stress (a[+2o~3)/2. (j> = Friction angle. (j)cy = Constant volume friction angle. (f>'p = Peak friction angle. •0 = Angle of dilation. xn Acknowledgement In presenting this thesis, the author expresses his gratitude to his supervisor, Dr. Y . P. Vaid , for his continuous encouragement and guidance throughout this research. The author also wishes to thank Dr. P. M . Byrne for reviewing the manuscript and mak-ing valuable suggestions. The helpful discussions and constructive criticism from my colleagues Wije , Norman, Ralph and Lee is deeply appreciated. The help of the C i v i l Engineering department work shop with the development and construction of test equipment is gratefully acknowledged. Financial support provided by the University of British Columbia is acknowledged with deep appreciation. x i n Chapter 1 INTRODUCTION In deformation analysis of geotechnical problems, sand is often modelled as an incremen-tally elastic material. The stress dependent elastic constants are normally determined from the results of conventional triaxial compression tests. In these tests, the sand is initially consolidated hydrostatically and sheared under constant confining stress. The test data from a series of such tests at different constant confining stresses is used in an analytic expression to compute elastic moduli. Elastic moduli are expressed as state variables - the state of the material constituting the current values of deviator stress, o~d= (^i"""^) a n d confining pressure, <r'3 . In most geotechnical problems loading, in general, involves simultaneous changes in <rd and 1T3. In addition, field loading does not always correspond to the major principal stress <r[ coincident with the vertical deposition direction of the sand, as is the case in triaxial compression test. Field stress paths may involve loading with o~\ direction at arbitrary angle ' a ' to the vertical direction. Furthermore, init ial stress conditions in many field deposits prior to loading are anisotropic, generally K0. Considering that the deformation response of soils is dependent on both consolidation history and stress path, data based on the results of conventional triaxial test, which represents isotropic consolidation and loading with cr'x vertical under constant <r'% may not be able to represent sand behaviour under other consolidation histories and general stress paths. The failure strength, characterized by the peak friction angle <f>', is also commonly 1 Chapter 1. INTRODUCTION 2 determined using the conventional triaxial tests. The constant volume friction angle (j)cv that is mobilized at large strain is considered constant for a given sand (Rowe, 1962 ; Negussey et al. , 1988). The excess resistance ((f)p-(j>cv) above this lower bound is a con-sequence of dilation rate at failure. Therefore, unique relationships have been suggested between 4>'p-(f>cv (or <f>p) and rate of dilation at failure. It is not known if these relation-ships apply during shear under stress paths with varying o~'z as well as stress paths with o Jl oriented in directions other than the vertical. This thesis is an attempt to study sand behaviour as a function of consolidation history and stress path and compare it with predictions based on results of conventional triaxial tests. Both stress-strain behaviour and failure strength are considered. The triaxial test is used for this study. As such, <r\ during shear loading is constrained to either vertical (triaxial compression a = 0) or horizontal (triaxial extension a = 90°) directions only. For each of these a[ directions, the sand behaviour is studied following a series of linear stress paths (constant as well as changing o~'3) starting from a given end of consolidation state. Comparative behaviour in a given mode (compression or extension) under a given stress path is also studied for isotropic and anisotropic consolidation histories. The response of sand is studied over a range of relative densities and confining pres-sures up to 2500 kPa. These high confining pressures are relevant to problems of high earth dams. In order to study both prepeak and postpeak behaviour of sand under stress paths other than conventional triaxial (cell pressure constant), a special loading system was designed that enabled strain controlled testing under any stress path. The system essen-tially involved a stepper motor set regulator coupled to a computer based data acquisition system. A n y stress path in the triaxial stress plane could be followed under stress or strain controlled loading including switch, if necessary, from one to the other mode part way into the test. Chapter 2 REVIEW OF PREVIOUS INVESTIGATIONS 2.1 General Aspects of Drained Behaviour of Sand The drained behavior of a sand depends on confining pressure and relative density. Fig-ure (2.1) shows idealized stress-strain behaviour during a conventional triaxial (constant o~'h) compression test for loose and dense states at identical low <r3 (< about 400 - 500 kPa). For dense sand, the deviator stress (CTJ) peaks and then decreases to an essentially constant value at large strains. The volume of sand initially decreases slightly and then increases to a stage beyond which it remains constant. For loose sand, the deviator stress increases steadily until it reaches an essentially constant value. The volume of sand grad-ually decreases all the way until it reaches a constant value. The ultimate strength and void ratio reached at any density: loose or dense are essentially identical. This state is called critical state. The characteristic stress-strain behaviour in Fig . (2.1) is observed at low confining pressure (generally < 400 to 500 kPa) when grain slippage is the dominating mechanism of deformation. However, at high confining pressure, the grain crushing becomes the dominant mechanism in addition to grain slipping. A t such pressures all relative densities show similar pattern of deviator stress-strain and volume change behaviour (Fig. 2.2). 3 Figure 2.1: Typical Stress-Strain Behaviour of Loose and Dense Sand At Low Confining Pressure in Conventional Triaxial Compression Test. Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 5 The deviator stress and volume change response is similar to that observed for the loose sand at low a'3. The above observation are confined to behaviour in the conventional triaxial com-pression test. The stress strain response along the other stress paths where o~'3 changes during shearing and under the extension mode of loading has not been investigated as extensively as that under conventional triaxial compression. 2.2 Stress-Strain Response Unt i l recent past the behaviour of sand was predominantly evaluated in terms of failure using limit equihbrium analysis. In most cases the deformation associated with the failure got little attention. However, the introduction of new high technology computing method made numerical solution possible for complex geotechnical problems. For a realistic numerical solution of deformation problems of sands, it is important that the stress-strain characteristic be represented in a reasonable manner. The characterization of stress-strain response (modeling) of sand is complex due to its nonlinear inelastic and stress dependent behaviour. Often the stress-strain behaviour is modeled by curve fitting methods such as non linear elastic as well as plastic or visco elastic theories. Incremental elastic hyperbolic model developed by Duncan and his co workers (1970) has been widely used as a representation of nonlinearity of sand. This is based on Kondner's (1963) finding that the triaxial compression stress-strain behaviour of sand that is isotropically consolidated can be approximated by a hyperbola. Kondner proposed the following hyperbolic expression for the stress strain behaviour. (To = a + bea (2.1) Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 6 Figure 2.2: Typical Stress-Strain Behaviour of Sand At High Confining Pressure in Con-ventional Triaxial Compression Test. Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 7 in which <J[, <r'3 are major and minor effective principal stresses, 'a' and 'b' are constants that can be determined experimentally by plotting test data in transformed axis (Fig. 2.3). It can be shown that the reciprocal of 'a' is the init ial tangent modulus Ei and the reciprocal of 'b' is the asymptotic value of <T[-CT'3. Thus, a = 1/Ei ; b = l/(<rj - cr^U In most cases, asymptotic (a[ — cr3)uu exceeds the measured peak (a[ — a'3)f (failure) strength. In order to achieve the best fit, a third hyperbolic parameter Rf is introduced to replace the theoretical asymptotic strength in terms of actual soil strength. Defining Rf — (°"i ~ ° 3 ) / ~ °3)uft> the hyperbolic stress-strain relation can now be written in the following form by substituting &=1/Ei, b=Rf/(cr[ — cr 3) uj t and crd = {<T[-(T3) in Eqn. (2.1): - T T f e (2-2) Duncan et al . , (1980) found that in the transformed plot (Fig. 2.3) the stress-strain data diverges from the straight line in the small and large strain range. They proposed, by examining a variety of soils, the best fit straight line passes through 70% and 95% stress levels. Departure of data from the straight line implies that Eqn. (2.2) cannot define the stress-strain behaviour satisfactorily in the region of small strains. Negussey (1984) proposed dual hyperbolic representation of the stress-strain behaviour as an improvement in incremental elastic modeling of sand over the entire strain range. Duncan et al . , (1980) took into account the stress dependency of the stress-strain behaviour by considering dependence of Ei on cr'3 in the form which was suggested by Janbu (1963). Chapter 2. R E V I E W OF PREVIOUS INVESTIGATIONS 8 Figure 2.3: Hyperbolic Stress-Strain Response in Triaxial Compression. Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 9 Ei = KEpa .Pa (2.3) in which pa = atmospheric pressure expressed in the same units as a'3. KE and n are constants which can be determined by plotting Ei/pa verses o~'3/pa from a series of constant a'3 tests on a log-log scale. On this plot KE equals the intercept and n equals the slope of the straight line. The failure strength (a[ — o~'3)f of sand can be expressed by 2(T3Sin<f> (2.4) (1 — Sincj)) The friction angle (f> generally varies with confining pressure. Duncan et al. , (1980) suggested that this variation may be represented by an expression of the form (j) = 4>0 — A ^ > l o g (£T 3 / p a ) . Here </>„ is the value of cf> when <r3 equal to pa and A<p is the reduction in 4> for a 10 fold increase in cr3. Most quartzitic sands, however, do not show a discernible dependence of <j> on cr'3 for cr'3 < 400 to 500 k P a (Hettler et al . , 1984; Bishop et al., 1953). The instantaneous slope of the stress-strain curve, the tangent modulus Et, can now be derived by differentiating Eqn. (2.2) and substituting Eqns. (2.3) and (2.4) yielding : Et = 1 Rf(l — sin<p)(<r[ — cr'3) KEP<, (2.5) 2(r'3sin(/> E q n 2.5 expresses tangent modulus Et as a state variable dependent only on current levels of (T3 and <T\-<J'3. Implicitly no dependence of Et on either consolidation history or stress path is implied. The conventional <r'3 constant triaxial compression data base used in formulating Eqn. (2.5) is thus used to compute Et under stress paths in which (i) <T3 continuously increases or decreases (ii) direction of o~\ not coincident with deposition direction and (iii) init ial consolidation history different from isotropic. Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 10 Little research has been carried out that is aimed at investigating the effect of factors outlined above on the value of Et. Most natural sand deposit exist in a state of anisotropic consolidation (e.g. state F in F ig . 2.4). In the conventional hyperbolic representation, the stress-strain response of sand from the init ial anisotropic consolidation state would be treated as if it had reached this state by following stress path B F . The tangent modulus Et at state F would be determined by simply using ( o ^ t r ^ ) in Eqn. (2.5). Vaid (1985) has demonstrated large differences between the stress-strain response predicted from the isotropically consoli-dated conventional triaxial test data base and that measured from tests on anisotropically consolidated Haney clay. He proposed a modified hyperbolic relation (Eqn. 2.6), which uses the increment in the deviator stress after consolidation, rather than the deviator stress in Eqn. (2.5) for expressing stress-strain behaviour under any value of anisotropic consolidation stress ratio, <r'le/ak'3c. o-d-o-do = x  €° e a ~ (2.6) in which a^o deviator stress at the end of consolidation. He emphasized the need to duplicate anisotropic in situ consolidation stress history in order to obtain representative model parameters Ei, Rj and (<TJ — <Td0)f. Stress increments due to field loading comprise paths that represent both changes in o~[ and er3 as well as their direction. By use of the conventional hyperbolic Eqn. (2.5), tangent modulus at stress point G for example would be computed by simply using <r[G and OT'ZG relevant to state G . This process implies that the stress state G was reached by following the conventional stress path O D G and stress increment direction along the same conventional triaxial stress path G M . Considering that the sand deformation is stress path dependent (e.g. Lade et al. , 1976), it is unlikely that behaviour along stress Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS Figure 2.4: Different Stress Paths and Common Point of Intersections. Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 12 path such as G J at stress state G could be treated as i f the sand had followed stress path D G all along. Negussey (1984) investigated such stress path dependency of the tangent modulus Et for a selected number of stress paths in compression. He showed that the modulus evaluated from the conventional triaxial test results could not predict the moduli measured in other stress paths at a common stress states (c^o-g). His investigation was confined to compression loading mode only. The validity of the predicted stress-strain response using Eqn. (2.5) under a variety of stress paths that include arbitrary stress increment direction 6 (Fig. 2.4) as well as involves both compression and extension modes of loading is of paramount practical interest. 2.3 Failure Strength The failure strength can be represented by a single parameter - the angle of internal friction. There are two important values for the drained friction angle: the peak friction angle <f>' and the constant volume friction angle (j)cv. The constant volume friction angle depends only on the mineral comprising the sand grains and is constant for a given sand (Rowe, 1962 ; Negussey et al., 1987). O n the other hand, the peak friction angle <j>'p can be influenced by the relative density Dr , normal stress a', stress path and physical characteristic of the sand. Confining stress a'3, \/2{a[-\-a3) and mean normal stress <r'm = l/3(cr( + a'2 + o~'3) have been used has alternative measures for expressing normal stress dependence of <f>'p. In most cases these normal stresses are referred to the failure condition. In the range of normal stress of common engineering interest (cr3 = 30 to 400 kPa) rounded and subround quartz sands do not seems to show significant normal stress dependency of <f>' at any relative density (Hettler et al. , 1984 ; Bishop et al. , 1953). In contrast, other quartz sand do show significant reduction in <j>'p over the same pressure range (<r3 = 30 to Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 13 400 kPa) . This reduction can amount up to 3° for loose and 5° for dense states. Variation in mineral composition together with difference in particle size, angularity and gradation of the sand accounts for the observed difference in behaviour. For a given sand, the increase in <f>' between Dr = 0 and 100 % can range between 12° to 15° at low o-'m. This increase becomes smaller as <rm increases (De Bear, 1965). The change can depend on average particle size, gradation and angularity. At very large stresses grain crushing becomes the dominant mechanism of deformation. The strength envelope at these high stresses then becomes a straight line and passes through the origin of the Mohr diagram (Bishop, 1966). Many researchers have found that the friction angle in triaxial compression loading to be 3° - 4° different from that in triaxial extension loading. This difference in friction angle can be due to the difference in intermediate stress parameter 'b' =(<7'2"' c r3)/( ai" < T3) a n c ^ ' a ' values i n these two shearing modes (compression b=0,a=0 ; extension b = l , a = 90°) . Friction angles measured in triaxial extension tests seems to be subjected to considerable variation in contrast to those measured in compression tests. Both prismatic samples having square cross section and cylindrical samples have been used for the measure-ment of friction angle (Reades and Green, 1976; Bishop et al. , 1953; Haythornthwaite, 1960). Variability in extension test results has been attributed to difference in several test conditions, such as sample slenderness, boundary condition (flexible or rigid), sample size and sample shape. L a m and Tatsuoka (1988) conducted a series of drained triaxial compression and extension tests on both prismatic and cylindrical samples. The sam-ple slenderness, boundary condition (flexible or rigid), sample size, sample shape were carefully controlled. They found that in triaxial extension test the failure strength was a function of the failure mode. The failure mode can be one of the following (1) a shear band develops between two flexible boundaries (2) a shear band intersects one rigid load-ing platens (top or bottom) (3) a shear band intersects both top and bottom rigid loading Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 14 platens. These modes of failures were controlled by the sample slenderness and boundary conditions (flexible or rigid). They concluded that the large scatter in triaxial extension friction angle which is often reported in literature is likely due to the differences in the mode of failure in various test set ups. Bishop et al. , (1953) and Haythornthwaite (1960) performed a limited number of compression loading and unloading tests together with extension loading and unloading tests. Their results show that the effect of stress path dependency on failure condition was very small, and was generally overshadowed by the scatter in the test results. Most of these tests were performed under stress controlled loading. In such loading, the post peak behaviour cannot be measured. Thus, confident measurement of peak friction angle becomes rather difficult. If the loading was strain controlled the post peak behaviour could be measured and the peak friction angle wi l l also be confidently defined. This may lead to reduced scatter in results of friction angle reported by the previous researchers. 2.4 Rate of Dilatancy The variation of (j>'p with failure o~'3 and relative density D r can be explained by the associated variation in rates of dilatancy. The maximum dilatancy rate (dev/dea)max is normally associated with the instant at which peak friction angle is mobilized in the conventional triaxial test. Both D r and <r'3 together affects (dev/dea)max. For a given sand in conventional triaxial compression tests, <f)'p appears to be linearly related to {dev/dea)max regardless of D r and a'3 (Bishop, 1972). This relation implies that difference in Dr and cr'3 can be explained solely in terms of their effect on dilatancy rates. The existence of a unique relation between 4>'p and (dev/dea)ma3. for stress paths other than conventional triaxial compression and in particular triaxial extension paths has not been investigated. Zero intersection of the linear relationship (<f>' - (dev/dea)max) with <f>'p axis Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 15 should yield 4>cv of the sand (Bishop, 1972). Experimental <ficv values compared with such a zero intersection have been generally determined by dry heap method (Cornforth, 1973). The ring shear apparatus is believed to be a more confident method to obtain <£TO of a sand (Wijewickreme, 1985) and thus a proper comparison of extrapolated ^ „ should be made with determined from such a test. Bolton (1986) has proposed useful empirical relation for estimating peak friction angle of a sand at any Dr and oJm level. The predictions are made from the empirical relations which were derived from a critical study of drained behaviour of 17 sands. The empirical relations based on results of triaxial compression test are: ft-**, = ZIR (2.7) ~ ^ = 0.31* (2.8) u J max where IR = Dr(Q - Ina'm) - 1 and Q is a constant. These relationships correlate drained extra friction angle in excess of <f>cv and dilatancy rate to IR a relative density index (0< IR < 4). These relationship were shown valid over o~m range between 100 - 1000 kPa . IR is a function of both D r and o~m, and constant Q depends on the mineralogy and on the units for a'm. For many quartzite and felspathic sand, when o~'m is in kPa , Q was found to be equal to 10. This equation takes account of the fact that zero dilatancy is achieved at a critical o~m for a given Dr . <f>a, for this expression were determined by dry heap method, a method that cannot be considered very desirable. The experimental data used to derive empirical relations (Eqns. 2.7 and 2.8) were obtained from triaxial compression tests. This test condition represents only one of the many field loading conditions. It would be of considerable practical importance to check the applicability of the above relations for other stress paths under both compression and Chapter 2. REVIEW OF PREVIOUS INVESTIGATIONS 16 extension loading modes. The literature review presented herein suggests that no comprehensive studies have been performed to address the general field loading condition which involves simultaneous changes in o~d and <r'3. Most previous investigations have been confined to conventional compression tests where <x'3 is held constant during loading. Due to this, the stress strain response along stress paths where o~'3 changes during shearing or under an extension mode of loading, has not been extensively investigated. Furthermore, the validity of the conventional hyperbolic equation (2.5) for the prediction of stress strain response under a variety of stress paths has not been investigated, although this equation (2.5) is used regardless of field loading conditions. The limited number of tests which have been performed under changing o~<i and <r'3 have been conducted under stress controlled loading conditions. In such loading, the post peak behaviour cannot be measured. Thus, peak friction angle cannot be measured with confidence. The maximum rate of dilatancy which occurs at peak friction angle is difficult to measure. Hence, the variation of </>' and (dev/dea)max with stress paths other than conventional stress path has not been investigated in detail. Therefore, comprehensive investigation of the effect of stress path on stress strain response and failure strength is necessary. In order to accomplish this, a special loading system was designed such that any arbitrary stress path may be simulated under strain controlled conditions. The details of this system are described in the next chapter. B y the use of this system many difficulties experienced by previous researchers have been over come. Thus, the stress stain response and failure strength under a variety of stress path is investigated in detail. Chapter 3 EXPERIMENTATION In this chapter, the details of the experimental aspects are discussed. The testing ap-paratus is described first, followed by the description of the material tested and sample preparation techniques. The details of the testing program is then outlined and finally the membrane penetration correction to the measured volume change are discussed. 3.1 Testing Apparatus A l l tests were conducted using the triaxial apparatus. A schematic layout of the testing apparatus is shown in Figure (3.1). Important features of the loading system are : 3.1.1 Axial Loading The axial loading system is capable of applying compression or extensional loads to the triaxial specimen under stress or strain controlled conditions. Coupling of a double acting air piston in series with a constant speed drive SDi allows for change from stress con-trolled to strain controlled loading and vice versa part way into the test. In addition to stress controlled shear loading, the air piston serves to consolidate specimen anisotrop-ically as well as provides compensation for vertical uplift on the loading ram during hydrostatic consolidation. The stress controlled shear loading does not allow measure-ment of post peak behavior. One the other hand, the strain controlled feature permits 17 /////////////// Legend: CPT DPT PPT R SMR Cell Pressure Transducer Differential Pressure Transducer Pore Pressure Transducer Regulater Stepper Motor Set Regulater O "a n ••1 Co ft X ft § ft s H >—i O Figure 3.1: Schematic of The Stress Path testing System. oo Chapter 3. EXPERIMENTATION 19 study of both pre peak and post peak behavior. 3.1.2 Lateral Loading The lateral principal stress cr'h can also be applied either under stress controlled or strain controlled conditions. Stress control is exercised by controlling air pressure. Strain control is achieved by a constant speed drive SD2 driven piston in a saturated water cylinder connected to the triaxial cell. Such strain controlled loading enables determination of post peak behaviour for extension loading (o~'h increasing) and compression unloading (cr'h decreasing) stress paths. 3.1.3 Stress Path Control Such a control is required for all shear tests in which cr'h varies. It is also needed if consolidation prior to shear loading is anisotropic. The stress path control is achieved by a stepper motor set regulator. The stepper motor responds to commands in the from of discrete pulses from a personal computer ( P C ) . Each pulse turns the stepper motor and hence the regulator through a certain angle. Thus controlled changes in pressure can be applied by a string of pulses to either chambers of the vertical loading piston or the cell pressure. For the P.C and the stepper motor set regulator chosen, each pulse represents a change in pressure of 0.2 kPa . The period of the square pulse can be varied as necessary so that loads are applied slowly in order to ensure fully drained conditions as required in drained tests. A sand specimen after set up in the triaxial tests is under a small hydrostatic state of stress (Point 0 in Fig . 3.2). Conventional triaxial tests which represents hydrostatic consolidation (path O A ) and shear loading with <r'h constant (compression path A B and extension path A H ) do not require any stress path control. For these tests, cell pressure Chapter 3. EXPERIMENTATION 20 Figure 3.2: Schematic of The Stress Paths Tested. Chapter 3. EXPERIMENTATION 21 control is provided by regulator R3 (see Fig . 3.1) and axial load applied by regulator Ri or i2 2 (stress control) or strain drive SDi (strain control). Compensation for the uplift on the ram is provided by pressure using regulator i 2 x . A n i s o t r o p i c C o n s o l i d a t i o n This is achieved by following the paths O J B (Fig. 3.2). Stress changes represented by O J are applied by the vertical piston using the regulator Ri. The stepper motor set regulator R4 is now set at the cell pressure as delivered by R3. The cell pressure control is now transferred to regulator R4 by isolating R3. Along anisotropic consolidation path J B , changes in <r'v and a'h are related by in which S = consolidation stress ratio Kc=cr'vclo''hc. Consolidation along this path is initiated by monotonically increasing the pressure in the upper chamber of the loading piston. This is conveniently achieved by coupling regulator R\ directly to a motor. The speed of the motor is chosen such that fully drained conditions are ensured. As the motorized regulator is turned on, experimental data is continuously scanned by the data acquisition system (Fluke) which is interfaced with the P .C . Following each scan the change in Aa'v is computed. This determines the required adjustments Ao~'h = Ao~'v/S to the cell pressure in order to follow the prescribed consolidation path. The P.C then sends out the necessary number of pulses to the stepper motor set regulator causing the change A<r'h. Another data scan is now taken and the process is repeated until the target value of o~'ve is reached. Vertical loading rates are so selected that the necessary adjustments in <r'h during the control operation do not lag behind by more than about SAa'h 0.2 kPa . Chapter 3. EXPERIMENTATION 22 Shear Loading Shear loading along any linear stress path can be carried out under either stress or strain controlled loading. Strain controlled loading in particular is a unique capabil-ity of the loading system under changing a'h. During shear loading along an arbitrary stress path, the stresses a'v and <r'h change continuously in a manner similar to that for anisotropic consolidation, (i.e) Aa'v = S Aa'h The Constant 'S' is characteristic of the desired stress path. For example, this con-stant 'S' is equals -2 for the constant mean normal stress path A F (Fig. 3.2). A < = -2 ACT; Prescribed stress paths are followed more closely if the lesser of the two stress change is controlled by the stepper motor set regulator R4. Therefore, stress paths for which I S I > 1 and thus Ao~'h < Acr'v, are followed by increasing or decreasing cr'v using the strain drive SD\ and changing cr'h by the stepper motor set regulator R$. For stress paths for which | S | < 1, Acr'v < Ao~'h, these stress path tests are therefore carried out by increasing or decreasing a'h using the strain drive SD2 and controlling cr'v by the stepper motor set regulator # 4 . It may be pointed out that the loading need not be restricted to linear stress paths. A n y nonlinear stress path can also be applied by the system by prescribing the loading function Ao~'h = f(Acr^) specific to the desired path. Although the above discussion is confined to strain controlled loading, same testing arrangements can also be used to perform stress controlled tests. During stress controlled loading, the strain drives SDi, SD2 are replaced by motor coupled regulators R\ or R2 and R3 respectively. Chapter 3. EXPERIMENTATION 23 3.1.4 The Triaxial Cell Triaxial test specimens were 63 mm diameter x 130 m m long. Smooth anodized alu-minium end platens with centrally located 20 mm diameter porous discs were used in order to minimize end restraint. Lubricated end platens were not used because of the associated large bedding errors (Sarsby et al., 1980). This is crucial for confident mea-surements of axial deformations particularly in the region of small strain. The selected sample geometry and steps taken to minimize apparatus compressibility, bedding errors and improved membrane penetration corrections (Vaid and Negussey, 1984) together with the use of high resolution data system enabled confident and consistent measurements of both axial and volumetric deformation increments in the order of 1 0 - 4 . 3.1.5 Standard Reference and Data Acquisition A l l test variables were measured using electronic transducers. A l l transducers were ex-cited by a common power supply that was set to 5 volts. The transducers were selected with careful consideration as to their stability and sensitivity. High resolution data sys-tem was used which eliminated need to amplify signal before measurements by the A / D converter. Two L V D T ' s one with sensitivity 10 times that of the other were used in parallel. This was necessary to monitor small deformations to a high degree of resolution as well as large deformations. The standard references of cell and pore pressure transducers were taken correspond-ing to the water level at sample mid height. The load cell output when suspended from the air piston rod was taken as zero reference load. During both anisotropic consolidation and shearing phases, test data was scanned continuously at 0.4s interval. A total of eight channels were monitored for each scan. The first channel read the time and the subsequent six channels read the transducer data. Chapter 3. EXPERIMENTATION 24 The eighth channel monitored the excitation voltage. The collected data were reduced and simultaneously displayed on the video monitor of the computer. This facilitated the comparison between the stress path desired and that which was followed. The exper-imental data acquired by the Fluke was stored in a floppy diskette at prescribed axial strain increments. This axial strain increment was adjusted manually during shearing process depending on the strain interval at which data was desired. 3.2 Material Tested Erksak sand, obtained from the near surface of the molikpaq core in Beaufort sea, was used in this study. The sand was obtained through the courtesy of Gul f Canada Resource L t d , Calgary. A bulk clean sand sample was prepared by first washing it to remove the clay and silt particles. The sand was air dried and the fraction passing #10 sieve and retained on #100 sieve was used for testing. The bulk sample was mixed thoroughly to prevent particle segregation. The sample prepared by the above procedure ensured sample homogeneity among all tests. Erksak sand is uniform brown, medium sand with subangular particles. The mineral composition is primarily quartz ( 80% quartz, 10% feldspar, 10% other) . The grain size distribution of the sand is shown in Figure (3.3). The median grain size D50 is 0.34 m m and specific gravity = 2.66. The maximum and minimum void ratio in accordance with standard test method A S T M D2049, are 0.775 and 0.525 respectively. < Ld (J or LU Q_ 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 G R A V E L S A N D S ILT COARSE MEDIUM FINE COARSE MEDIUM FINE COARSE \ \ \ \ \ i irksak ir IC \ \ \ \ > 9176 5 4 3 2 1 0 0 98 7 6 5 4 3 2 1 0 98 7 6 5 4 3 2 9 8 7 6 5 4 3 2 1 0.1 0 . 0 1 8 •8 to 8 g o 55 GRAIN SIZE, m m Figure 3.3: Grain Size Distribution of Erksak Sand. t o Chapter 3. EXPERIMENTATION 26 3.3 Testing Procedure 3.3.1 Sample Preparation The basic method of sample preparation by water pluviation adopted in this study has been long used at U B C . Further refinements were introduced by Negussey (1984) to enhance replication of structure and density. Samples were formed with extreme care to ensure repeatability and consistency of test results. A fixed mass (approximately 612 gms) of oven dried sand was used to reconstitute each specimen. The sand was mixed with water in a flask and was boiled for about 30 minutes. After cooling it to room temperature, the boiled sand was kept under vacuum unti l sample formation. A l l the drainage lines on the plexiglass reservoir attached to the triaxial cell base were saturated with de-aired water. The porous stones were boiled in water and cooled to the room temperature before the sample was about to be prepared. A height reference was taken by placing an aluminum dummy sample of known height between the bottom pedestal and the top cap. The diameter of the split mold, thickness of the membrane and the dry sand weight were constant. Therefore, the desired relative density could be obtained by controlling the specimen height. The required sample height to obtain the desired density was then calculated and the dial target reading required for the sample height was calculated. The 63 m m diameter sample was formed in a rubber membrane lined split mold which was filled with de-aired water. The membrane was sealed at the base pedestal and stretched to the mold wall by applying a small vacuum suction. A rubber stopper with a glass nozzle was fitted to the flask containing the boiled sand and de-aired water was added until full to the nozzle tip. The flask was then inverted and was supported on a Chapter 3. EXPERIMENTATION 27 stand with a clamp. The deposition of the sand was started by penetrating the nozzle tip in the water surface. The sand was deposited by continuously moving the nozzle in a circular motion while maintaining the top of the nozzle submerged below the water surface. When all of the sand was deposited, the flask was removed from the stand and the top of sand surface was leveled. The top cap with the attached loading ram was then placed on top of the deposited sand. The level of the top cap was checked by using a level tube in two perpendicular directions. After placing the dial gauge on the top of the loading ram the sand specimen was densified by vibrating along the cell base with a soft hammer. During densification, the top and bottom drainage lines were kept open. Densification was stopped at the predetermined dial gauge reading for the desired density of the sand. The membrane was then pulled over the cap. After sealing the top cap by an 'o' ring, the top drainage line was plugged. A vacuum of approximately 20 k P a was applied to the sample through the bottom drainage line. The split mold was now dismantled leaving the sample with confining pressure of about 20 kPa . The base drainage was now closed to maintain confinement and the vacuum line was dismantled. After filling the cell with water, the final dial gauge reading was taken to determine the sample height. This completed the sample preparation phase. 3.3.2 Sample Set Up The triaxial cell was placed within the loading frame. The init ial readings of the pressure transducers were taken corresponding to water level at the mid height of the sample. The cell pressure line was then connected to the cell pressure system and a pressure of about 20 k P a was applied to release the vacuum confinement. After connecting the sample drainage line to the pipettes, the reference scans of the other transducers were Chapter 3. EXPERIMENTATION 28 taken. The ' B ' values were measured by incrementing the cell pressure and observing the corresponding undrained increase in pore pressure. The sample was then brought to an effective confining pressure of 50 k P a with a back pressure of 100 kPa . Isotropic consolidation to the desired o~'h was carried out by incrementing the cell pressure by 50 kPa. Uplift on the loading ram during this increment phase was compensated by loading with the air piston. In the case of anisotropic consolidation under a given Ke, the sample which was consolidated hydrostatically at 50 k P a was first loaded drained following conventional triaxial path to the Kc state with a'k= 50 kPa . Thereafter the sample was Kc consolidated continuously up to the desired stress level. After reaching the final consolidation stress level, the sample was allowed to sit for about 15 min at this stress level in order to allow most of the secondary consolidation (creep) to be completed (Mejia et al. , 1988). A t this point a second reference scan of al l transducers was taken and the shearing process was initiated. 3.3.3 Repeatability of Test Results Confidence level in experimental observation and results is enhanced by the ability to repeat tests. Repeatability depends on consistent reproduction of relative density, soil structure, measurement accuracy and exact duplication of the test routine and the stress path followed during shearing. The test procedure described previously ensured that this was achieved to the highest degree. Typical results of repeated tests on identical samples (Dr=26%) hydrostatically consolidated to 250 k P a and sheared drained in extension are shown in Fig. (3.4). Excellent repeatability in results showing stress strain response may be noted. Chapter 3. EXPERIMENTATION 29 Figure 3.4: Repeatability of Test Results. Chapter 3. EXPERIMENTATION 30 3.4 Testing Program The effect of consolidation history and stress path on stress strain and the failure strength were studied by performing drained triaxial tests. The effect of consolidation history was studied by consolidating samples of a given relative density under anisotropic and hydrostatic conditions and shearing them in compression at constant confining pressure. The effect of stress path was studied by consolidating samples hydrostatically and then shearing under a variety of stress paths, both in compression and extension. 3.4.1 Effect of Consolidation History Triaxial compression tests with constant confining pressure were performed on loose, medium dense and dense specimens (Dr = 26%,56% and 70%). Specimens of a given D r were first anisotropically and isotropically consolidated to identical confining pressure prior to shear. Test were carried out at <r'3 = 100,250 and 400 k P a on loose, 100 and 250 k P a on medium dense and 250 k P a on dense specimens. 3.4.2 Effect of Stress Path Specimens were sheared using several stress paths. These are illustrated in Fig . (3.5). A series of specimens at a given D r was first isotropically consolidated (point A in Fig . 3.5). Shearing was carried out under both compression and extension modes. This was intended to evaluate the effect of inherent anisotropy in pluviated sand on deformation and shear strength behaviour. Compression loading corresponds to o~[ during shear co-inciding with the deposition direction (a = 0) and in extension loading at 90° to the deposition direction (a = 90°) . Chapter 3. EXPERIMENTATION 31 Figure 3.5: Stress Paths Investigated. Chapter 3. EXPERIMENTATION 32 For each loading modes (extension and compression) shearing paths followed, encom-passed both increasing and decreasing mean normal stress. In terms of stress components o~'v, <r'h, the stress path represented (i) either o~'v or a'h constant (ii) <r'v, <r'h simultaneously increasing or decreasing or one increasing and the other decreasing. A t a given relative density, tests were performed over a range of hydrostatic confining pressures. Not all stress paths were imposed at each level of consolidation stress. Con-solidation stresses ranging from 100-2400 k P a were used. Higher stress levels represent conditions under high earth dams. The behaviour of Erksak sand was investigated over a range of relative densities; loose, medium dense and dense. Table (3.1) outlines the details of all stress path tests carried out. A additional compression stress path, in which 5cr'v and 6o~'h decreases as 6<r'v/6<r'h=0.5 was performed at Dr=26% and (T 3 c =250 kPa . This stress path is referred as (5-6). 3.5 Membrane penetration The sample volume change during drained triaxial test are normally measured by the quantity of water entering or leaving the pipette attached to the saturated sample. The measured volume change due to a stress change is composed of (1) The sample volume change (2) Penetration of the membrane enclosing the specimen into peripheral voids of the sand particles due to change in cell pressure (Newland and Allely, 1956). The mea-sured volume change in the pipette must therefore be corrected for membrane penetration effects in order to compute the actual volume change of the sample. When stress paths with changing cell pressures are followed, the measured volume changes are subjected to error due to this membrane penetration. These volume changes were corrected for membrane penetration effects using the method proposed by Vaid and Chapter 3. EXPERIMENTATION Table 3.1: Details of Stress Path: s Tested. 33 Relative Density (%) Consolidation pressure < (kPa) Stress paths tested 26 100 2 (loose) 250 1 to 8 400 2 600 2 1200 2,5,7 2400 2,7 56 50 2 (medium dense) 100 2,9 150 2,9 250 rto 9 350 2,9 450 2 1200 2,7 2400 2 70 250 2 to 8 (dense) 600 2 1200 2,5,7 2000 2,7 Chapter 3. EXPERIMENTATION 34 Negussey (1984). Accordingly, for Erksak sand and the membrane used in the testing program, the membrane penetration correction curve was determined. This is shown in Fig . (3.6). 6^, the unit membrane penetration represents volume correction per unit surface area of the specimen covered by the rubber membrane. It may be noted, in Fig . (3.6) that the best fit straight line does not passes through the init ial reference hydrostatic pressure 50 kPa as observed by other researchers. Since, all confining pressures during stress path tests reported herein were greater than 100 k P a , the straight line relationship between e m and logcr3 was used for computing membrane penetration corrections due to changes in cell pressures. E o \ E o o o o E to Q ti-es "a Co tq X tq O 10 100 o-'3c(kPa) Figure 3.6: Membrane Penetration Per Unit Surface Area with Changing Effective Con-fining Pressure. CO Chapter 4 TEST RESULTS 4.1 Introduction In this chapter, the drained loading behaviour of Erksak sand is discussed under a variety of stress paths in both compression and extension modes of loading. The stress-strain response is considered first, followed by the description of the shearing resistance. The init ial state of sand tested covered a wide range of relative density (Dr=26% to 70%) and confining stress from 50 to 2400 kPa . Considerable densification was observed under large confining stress for the loosely deposited Erksak sand. Hence, very loose densities were not accessible to study the behaviour under high confining stress. Thus, relative density is not an independent parameter at high confining stress (e.g. loosest pluviated Erksak sand under cr'^ of 2400 k P a cannot possess relative density less than approximately 31% ). In the following discussion, compressive strains are assumed to be positive and the deviator stress <TJ is defined as (<r'v — a'h) instead of (o~[ — a'3). This enables proper distinction between compression and extension modes of loading. A positive (o~'v — cr'h) represents compression loading and negative (o-'v — cr'h) implies extension loading. 36 Chapter 4. TEST RESULTS 37 4.2 Stress Strain Behaviour 4.2.1 Behaviour Under Conventional Triaxial Compression Typical stress-strain response of loose (Dr=26%) and dense (Dr=70%) Erksak sand at a confining stress (<r3c) of 250 k P a is shown in Fig . (4.1). The dense sand exhibits a higher peak deviator stress than the loose sand. Furthermore, the dense sand shows more dilative volume change response than the loose sand. On the other hand, Figure (4.2) shows typical stress-strain response of initially loose (.Dr;=20%) and medium dense sand (Dr=56%) at (r3 c=2400 kPa . The loose and medium dense sands show similar pattern of deviator stress axial strain behaviour. The volumetric strain of loose sand at confining stress of 2400 k P a gradually increases with increase in axial strain unti l it reaches a constant value. The medium dense sand at confining stress 2400 kPa also shows more contractive behaviour than that of loose sand at the lower confining stress 250 kPa . These results indicate that Erksak sand has stress-strain characteristic typical of other sands and its use as a representative material in the study of sand behaviour is thus justified. It may also be noted that the stress-strain response is governed by both relative density D r and confining stress a'^. Thus, the effect of these two parameters cannot be considered in isolation i n the study of sand behaviour. 4.2.2 Effect of Stress Paths Most pluviated sands are inherently anisotropic. Figure (4.3) depicts the radial and axial strain response of Erksak sand during hydrostatic consolidation. It can be observed that the sand is more compressive in the horizontal direction than the vertical direction. This clearly shows the inherent anisotropy of the pluviated sand. Implication of this results is that in a triaxial test stress-strain response would be different in compression and Chapter 4.^ TEST RESULTS 38 1000 e. (*) Figure 4.1: Stress Strain Response in Conventional Triaxial Compression at Low Con-fining Stress. Chapter 4. TEST RESULTS 39 Chapter 4. TEST RESULTS 0.25 0.25 e„ (*) Figure 4.3: Strains During Hydrostatic Consolidation. Chapter 4. TEST RESULTS 41 extension modes of loading under identical a3. Figure (4.4) shows the stress-strain behaviour of medium dense sand (Dr=56%) loaded under compression and extension modes of loading at identical a'3 (=250 kPa) . To provide a better comparison of the stress-strain response the absolute values of deviator stress and axial strain have been used in this figure. It may be noted that the stress-strain response is much softer in the extension loading than in the compression loading. Furthermore, the extension mode of loading shows more contractive volume change than compression mode. Some difference in these behaviours can be attributed to the value of intermediate principal stress parameter b = l in extension mode as compared to the value of b=0 in compression mode. However, it has been reported by Hight et al . , (1983) that the effect of b on stress-strain response is relatively less significant. The results presented in Fig . (4.4) suggest that the stress-strain response does depend on the mode of loading i.e. compression or extension. Hence, the dependence of stress-strain response on stress path under compression and extension modes of loading wi l l be considered separately. 4.2.2.1 Compression Stress Paths Typical compression loading behaviour of loose sand (Dr=26%), isotropically consoli-dated to a confining stress of 250 k P a , for a variety of stress paths is shown in Figure (4.5). These stress paths represent constant a'v or cr'h, increasing a'v and a'h, increasing o~'v and decreasing o~'h. A clear distinction can be made between the stress-strain response under stress paths with varying cr'3 versus that of constant a'3. Volume change response during shear is a function of both the change in mean normal stress and shear stress. At low stress ratio, increase in mean normal stress and shear stress causes contractive vol-ume change response. On the other hand, decreasing mean normal stress causes dilative volume change response. In stress paths 1 to 3, the increase in mean normal stress and Chapter 4. TEST RESULTS 43 Figure 4.5: Stress-Strain Response of Isotropically Consolidated Sand Under Various Compression Stress Paths. (Dr=26% and 0-^=250 kPa) Chapter 4. T E S T RESULTS 44 shear stress cause initial volume change to be contractive. On the other hand, in paths 4 and 5 the reduction in mean normal stress at low stress ratios ( R < ~ 1.5) causes the init ial dilative volume change response. In other words, the dilative volume change com-ponent due to reduction in mean normal stress outweighs the shear induced volumetric contractive tendency at low stress ratio in stress paths 4 and 5. Similar stress-strain behaviour to that of loose sand (Dr=26 %) was observed at other relative densities (Dr=56% and 70%) also that was isotropically consolidated to the same confining stress of 250 kPa . 4.2.2.2 E x t e n s i o n Stress P a t h s Typical stress-strain response for loose sand (Dr=26%) initially isotropically consolidated to a confining stress 250 k P a for a variety of stress paths is shown in Fig . (4.6). The stress paths presented include decreasing o~'v and a'h, constant o~'h and decreasing a'v with increasing er'h. Deviator stress axial strain response is different for each of these paths because of changing <r'h. The volume change behaviour of extension stress paths 6 to 8 can also be explained by considering changes in mean normal stress and shear stress. Stress paths 6 and 7 represent changes in stresses such that mean normal stress reduces and shear stress increases during loading. The init ial volume change response of these paths are expansive due to reduction in mean normal stress at low stress ratio (R=sl.5). When the stress ratio exceeds approximately 1.5, the shear induced contractiveness over shadows the expansion due to the reduction in mean normal stress and causes contractive volume change. Hence, the volume change response changes from dilative to contractive. A t high stress ratio the volume change response changes from contractive to dilative. Similar volume change behaviour was also observed for other relative densities (Dr=56% and 70%) and at various confining stress levels. These observations are consistent with init ial negative excess pore pressure observed Figure 4.6: Stress-Strain Response of Isotropically Consolidated Sand Under Various Extension Stress Paths.(Dr=26% and 0-^=250 kPa) Chapter 4. TEST RESULTS 46 by Chern (1984) and Chung (1985) in unloading extension undrained tests (stress path 7). Similar type of response i.e. initially dilative and then contractive was also observed by Negussey (1984) for constant shear stress paths under extension loading. 4.2.2.3 Comparison of Behaviour Under Compression and Extension Stress Paths It may be noted from Figures (4.5) and (4.6) that the deviator stress axial strain response of all extension stress paths are softer than compression stress paths at any given stress state (<7d, <r3). The extension stress paths also show more contractive volume change than the compression loading paths. Sand formed by water pluviation possesses inherent anisotropy because of the major principal axis of grain orientation tends to be inclined disproportionately towards horizontal. Because of this inherent anisotropy and orienta-tion of potential slip planes (compression 45+^/2 ; extension 45-^/2 to horizontal), the extensional shearing encourages closer packing and hence more contractive (Negussey, 1984). 4.2.3 Incremental Elastic Representations 4.2.3.1 Effect of Consolidation History Figures (4.7) to (4.9) show the comparison of deviator stress vs axial strain response of KQ and isotropically consolidated sand at identical er3 for Dr=26%, 56% and 70%. It can be observed that at fixed stress level (cr^, <r3), the tangent modulus Et of K0 consolidated sand is much stiffer than that of isotropically consolidated sand. This difference becomes smaller as the failure stress is approached during shearing. However, it is not convenient to compare tangent modulus quantitatively from the stress-strain curves. Tangent mod-ulus for isotropically consolidated sand can be evaluated from hyperbolic representation Chapter 4. TEST RESULTS 300 - r -100 kPa U"T i 1 1 1 r 1 , °-° 1-0 2.0 3.0 4.0 1200-1 — Figure 4.7: Stress-Strain Response Under Different Consolidation History.(Dr=26%) Chapter 4. TEST RESULTS 48 Figure 4.8: Stress-Strain Response Under Different Consolidation History.(Dr= 56%) Chapter 4. TEST RESULTS 49 Figure 4.9: Stress-Strain Response Under Different Consolidation History.(Dr=70%) Chapters TEST RESULTS 50 (Eqn. 2.5.) with conventional triaxial data base. The conventional triaxial data are plot-ted in Figs. (4.10) - (4.12) together with transformed plots. For the purpose of clarity only the data corresponding to a limited number of a'3 levels are shown. The straight line variation in the transformed plot in (Figs.4.10 to 4.12) show excellent conformity to the hyperbolic representation. Figure (4.13) shows the variations of Ei/pa wi th o-'zjpa for loose , medium and dense sand. In this plot E{ was obtained from the intercept of the best fit straight line in Figs. (4.10) to (4.12). For a range of confining stress from 50 to 2500 k P a excellent linear plots are observed in Fig . (4.13). Hyperbolic parameters Kg and n can be obtained from the intercept and slope of the straight line in F ig . (4.13). Table (4.1) provides the values of KE and n thus obtained for each relative density. Table 4.1: Hyperbolic parameters obtained from triaxial compression data base KE n Rf Loose (Dr=26%) 1169 0.45 0.94 Medium dense (Dr=56%) 2200 0.46 0.89 Dense (Dr=70%) 3156 0.41 0.89 It is observed from the above table that the values of n closely match the commonly used value of 0.5. Further, the KE values obtained are in agreement with the values computed by Duncan et al., (1980) for Monterey No.O sand, which is also composed of subangular to subrounded grains. The KE values also show approximately a linear variation with relative density (Fig.4.14). Chapter 4. TEST RESULTS 2000 51 1600- <r'3o-600 kPa Figure 4.10: Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=26%) Chapter 4. TEST RESULTS 52 1600 Figure 4.11: Stress-Strain Response in Triaxial Compression at Different Confining Stress. (Dr=56%) Chapter 4. TEST RESULTS 53 Chapter 4. TEST RESULTS 54 Figure 4.13: Variation of Ei/pa with a'^/pa in Conventional Triaxial Compression. Chapter 4. TEST RESULTS Figure 4.14: Variation of KE with Relative Density. Chapter 4. TEST RESULTS 56 The predicted behaviour in conventional triaxial compression using the value of KE and n obtained from table 4.1 is compared with the measured stress-strain response in Fig . (4.15) for medium dense sand at <r3c=250 kPa . The predicted stress-strain response compares closely with the measured except for the small strain range where the large strain hyperbolic representation normally cannot represent the stress-strain response (Negussey, 1984). Vaid (1985) has shown that KD consolidated stress-strain response is also hyperbolic if the increment in the deviator stress over the end of consolidation value (Tdo is considered. i.e a + bea Erksak sand conforms to such hyperbolic representation only at larger strains (Fig. 4.16). However, the expanded transformed plot (Fig.4.17) shows that the small strain region can be modelled by two additional hyperbolic representations. Hence, the entire range of strain can be represented by three separate hyperbolic curves so as to follow the measured data satisfactorily. However, each of these hyperbolic representation is only valid over a finite range of strain. For example, the analytical expression to compute the tangent modulus at any stress point for all the three regions for Dr=56% at 0-^=100 k P a can be written as Et Et Et 146.0 103.2 74.7 1 _ ~ Vdo) 91.7 1 - {o~d - <Tdo) 153.1 179.4 when 0 < eQ < 0.063 when 0.063 < ea < 0.3 when 0.3 < ea (4.1) (4.2) (4.3) in which crd and Et are in k P a and M P a respectively. Similar analytical expressions Chapter 4. TEST RESULTS 57 1000 Figure 4.15: Comparison of Measured and Predicted Stress-Strain Response Chapter 4. TEST R E S U L T S 58 Figure 4.16: Transformed Hyperbolic stress-strain Behaviour of KQ Consolidated Sand. Chapter 4. TEST RESULTS 59 Figure 4.17: Transformed Hyperbolic stress-strain Behaviour oi KD Consolidated Erksak Sand in Small Strain Range. Chapter 4. TEST RESULTS 60 for oJ3c=25Q k P a at Dr=56% and for other relative densities are given in Appendix A . Comparison of tangent moduli Tangent moduli that were calculated as a function of crj from Eqn.2.5 for isotropic and Eqns. (4.1) to (4.3) for anisotropic consolidated samples at Dr=56% and 0-^=100 k P a are shown in F i g . (4.18). It may be seen that in i t ia l tangent modulus of K0 is about 2 to 3 times that predicted by the hyperbolic representation based on the results of conventional tr iaxia l test. In the region of strain where deformation response is of interest to the analyst, K0 tangent modulus is greater than that of isotropic. Thus, the use of isotropic hyperbolic representation w i l l overpredict deformation of K0 consolidated sand. Figure (4.19) shows similar comparison at (7^=250 k P a . Similar differences in Et are also observed between isotropic and K0 consolidated states. For this medium dense sand, both confining stresses show approximately equal amount of difference in tangent moduli between KQ and isotropic consolidations. Comparison of isotropic and K0 tangent moduli at Dr=26% (Loose) for two value of cr'3c and at Dr=70% (dense) for one value of <r'3c are shown in Figs. (4.20) and (4.21). In all cases the tangent modulus for K0 consolidation is grater than that for isotropic con-solidation. The difference in tangent modulus of K0 and isotropic consolidation appears to reduce wi th increasing relative density. However, for the loose sand (Dr=26%), the difference in tangent modulus between K0 and isotropic states increases wi th increasing confining stress. Chapter 4. TEST RESULTS 61 250 200 H 400 or'v-(r'h (kPa) Figure 4.18: Variation of Et as a Function of crd for K0 and Isotropically Consolidated Sand. (Dr=56% and 0-^=100 kPa.) Figure 4.19: Variation of Et as a Function of <rd for KQ and Isotropically Consolidated Sand. (Dr=56% and ^ = 2 5 0 kPa.) Figure 4.20: Variation of Et as a Function of ad for K0 and Isotropically Consolidated Sand. (Dr=26%) Chapter 4. TEST RESULTS 64 500 «r'v-tr'h (kPa) Figure 4.21: Variation of Et as a Function of <rd for K0 and Isotropically Consolidated Sand. (Dr=70%) Chapter 4. TEST RESULTS 65 4.2.3.2 Effect of Stress Path Compression Paths Deviator stress axial strain response under a variety of stress paths following hydro-static consolidation upto cr3 c=250 kPa of loose sand (Dr=26%) are shown in Fig . (4.22). Stress paths presented include changing o~'h with u'v increasing, constant or decreasing. Elastic tangent modulus Et at any stress point P (Fig. 4.22) along a given changing a'3 stress path can be shown to be related to the slope of the stress-strain curve E't (see Fig . • 4.22) by Negussey (1984) E, = ( 4 . 4 ) in which the stress path direction r = 6a'v/6o~'h, and fi = Poissons ratio. Poisson's ratio /x can be computed from the slope of the volumetric axial strain curves of conventional triaxial tests as follows: H = -der dea J _ dea dev — dea dea de„ (4.5) Poisson ratio fi was found to range from 0.2 to 0.3 for Dr=26%. For the range of r in stress path tests, the value of the factor (r — 2fi)/(r — 1) in Eqn. (4.4) as a function of tan~ l{ — 1/r) for the extreme limits of yi values (0.2 and 0.3) is shown in Fig . (4.23). In this figure tan' 1 ( — 1/r) = 0 represents r= oo (conventional triaxial compression) and tan~ 1( — l/r)= 90° represents r=0 (active compression). It can be observed that the Chapter 4. TEST RESULTS 66 BOO «. (*) Figure 4.22: Paths Deviator Stress Axia l Strain Response Under Various Compression Stress Chapter 4. TEST RESULTS 67 1.2 0.0 H 1 1 1 1 1 1 —20 20 60 100 tan~ 1(-l/r), degrees Figure 4.23: Variation of (r-2^)/(r-l) with ten_1(-l/r). Ciiapter 4. TEST RESULTS 68 selection of an average value u=0.25 would not change the computed value of elastic modulus Et by more than 10%. The measured values of Et along several compression stress paths in Fig . (4.22) are plotted as a function of <r'3 in Fig . (4.24). The value of ad along the stress path is related to cr'3 by " * = W - O ( r - l ) (4.6) Also plotted in this Figure (4.24) are the Et values predicted at identical stress points {<7d,o~3) based on hyperbolic representation (Eqn. 2.5) that utilizes constant a'3 data base. Figure (4.24) clearly shows that the conventional hyperbolic approximation overestimates elastic modulus Et for decreasing cr'3 stress paths (paths 3,4,5,5-6) and underestimates Et for increasing o~3 stress path (path 1). The magnitude of the difference between measured and predicted Et values tends to increase with decreasing r or when the stress path changes direction progressively from the right of conventional path towards the left. However, for a given stress path, the difference in Et decreases as the stress path progresses towards failure, as indicted by converging measured and predicted Et curves. Similar comparison of elastic moduli along several stress paths for medium dense (Dr=56%) and dense (Dr=70%) sand are shown in Fig. (4.25). Stress strain data for these tests are given in Appendix B. The Poisson ratio ' /J ' was found to he between 0.25 and 0.5 for medium dense sand and between 0.3 and 0.5 for dense sand. Average values 0.33 and 0.40 were used to calculate Et values for medium dense and dense sands respectively. The nature of underprediction or overprediction of elastic moduli for medium dense and dense sand may be seen similar to that observed for loose sand. The results presented above clearly show that deformation would be overpredicted by conventional hyperbolic representation when o~3 increases during a stress increment. On Figure 4.24: Comparison of Modul i Evaluated A t Fixed Stress State (<r'D,(T3) Following Different Stress Paths i n Triaxial Compression. (Dr=26%) Chapter 4. TEST RESULTS 70 500 Figure 4.25: Comparison of Modul i Evaluated A t Fixed Stress State (<r'd,<r3) Following Different Stress Paths in Triaxial Compression. Chapter 4. TEST RESULTS 71 the other hand, for <r'3 decreasing stress increments conventional hyperbolic representation would underpredict the deformation. E x t e n s i o n P a t h s Elastic moduli were predicted for extension paths in an identical manner to that for compression paths for relative density= 56%. The KE and n values for these predic-tions corresponded to compression test data base as well as done in routine deformation analysis. These predicted elastic modulus are compared with experimental values in Fig . (4.26). It can be clearly observed that the predicted Et is much larger than that mea-sured for all extension stress paths for any given stress state (<TJ, <T3). Thus, the use of compression data base to predict deformation under extension loading would lead to gross under prediction of deformations. Since sand is anisotropic in mechanical behaviour (Fig.4.4) an improvement in the predicted and observed values of elastic moduli along extension stress path would likely result if data base from constant <r'3 extension test was used for prediction. The existence of hyperbolic representation of sand behaviour under constant <r3 paths in extension load-ing has not been explored in the literature. Figure (4.27) shows results of a series of such tests on Erksak sand at Dr=56% together with the transformed hyperbolic plots. The straight line transformed relationships emerging in Figure (4.27) indicates that deviator stress axial strain variation in constant a'3 extension paths is also hyperbolic. Figure (4.28) shows a plot of Ei/pa with o~'3/pa in order to evaluate KE and n values under extension loading. In Figure (4.28) results of compression loading constant cr'3 tests are also superimposed for the purpose of comparison. In table (4.2) values of constants KE and n under compression and extension loading at Dr=56% are shown. The table clearly illustrates the wide difference between extension and compression loading parameter, and justifies the separate treatment of these two modes of loading in the prediction of deformation behaviour. Chapter 4. TEST RESULTS 72 350 50 100 150 200 250 t^(kPa) Figure 4.26: Comparison of Modul i Evaluated A t Fixed Stress State f » i ) Following Extension Stress Paths. (KE, n and <f> Obtained from <r> constant Triaxial Compression Chapter 4. TEST RESULTS -1000 Figure 4.27: stress-strain Response of Extension Stress Path at Different Constant (Dr=56%) Chapter 4. TEST RESULTS 74 Dr=56* 100-0.1 T-TT 1 T — r ' ' i | 10 -i—i—i i i Figure 4.28: Variation of Ei/pa with <r'3/pa for Constant <r'3 Triaxial Extension Tests. Chapter 4. TEST RESULTS 75 Table 4.2: Hyperbolic parameters for compression and extension loading. Dr=56% KE n Rf compression extension 2200 1196 0.46 0.37 0.89 0.88 Using the hyperbolic data base from constant <T'3 extension loading (Table 4.2), Et for Dr=56% can be written as: E t = [l °- 8 9 ( 1 ~ S i n ^ < - ga) 2cr35int/> In Fig . (4.29) a comparison between the measured elastic modulus in extension stress paths 6,7 and 8 is made with predicted values based on Eqn. (4.7) utilizing the data from constant <T3 extension tests. It may be noted that the difference between the measured and predicted Et for any stress path is greatly reduced compared to that of Fig . (4.26). Nevertheless, even using the extension data base, loading along all the extension paths overestimate Et. It may also be noted that all these extension stress paths represent cr3 decreasing stress increments. The overestimation of elastic moduli is therefore similar to that observed under compression loading, (i.e. in cr3 decreasing stress increments the conventional hyperbolic representation over estimates the elastic modulus.) 4.2.3.3 Additional Remarks Incremental elastic representation is conveniently used in majority of geotechnical defor-mation problems. This is a consequence of not only of its convenience and simplicity but 1196pa (4.7) Chapter 4. TEST RESULTS 200 a3(kPa) Figure 4.29: Comparison of Modul i Evaluated A t Fixed Stress State (a'a') Following Extension Stress Paths. (KE, n and <f> Obtained from constant <r3 Triaxial Extension Tests). Chapter 4. TEST RESULTS 77 also because of its performance which is often comparable to other constitutive models for sand. Even with further development of more reliable comprehensive models, site conditions in many cases are not well defined to justify added complexity in the models. Thus further improvements in terms of consolidation history, stress path effects and also effect of loading direction (compression or extension) would make incremental elastic representation more effective and enhance its current wide usage. 4.3 Shear Resistance 4.3.1 Effect of Consolidation History Results of compression tests on isotropically and anisotropically consolidated Erksak sand at constant values of confining stress cr3 c are shown in Figs. (4.7) to (4.9). It may be noted that peak deviator stress ad or the peak friction angle <f>' for given value of relative density Dr and confining stress a'^ is not dependent on consolidation history. This conclusion applies to a given relative density at each confining stress and for a given confining stress to any relative density. 4.3.2 Effect of Stress Path 4.3.2.1 Rate of Dilatancy at Failure It may be noted in Figs. (4.5) and (4.6) that the maximum rate of dilatancy (dev/dea)max occurs at the instant of peak in the stress-strain curve (i.e. when <j>' is mobilized) regard-less of the stress path or the mode of loading (compression or extension). For a given value of Dr and o~'3 at failure, maximum rate of dilatancy (dev/dea)max may be noted to be smaller in extension than that in compression (Fig.4.4). This is apparently a consequence of the inherent anisotropy in pluviated sand. Chapter 4. TEST RESULTS 78 Figure (4.30) shows peak friction angle <f>' plotted against maximum rate of dilatancy (devIdea)max for all stress path tests on Erksak sand. The data consists of compression and extension tests at loose, medium dense and dense relative densities over a range of cr3 values at failure. It can be observed that a unique relationship exists between peak friction angle <j>'p and maximum rate of dilatancy (devjdea)max which is independent of stress path, a'3 at failure and relative density. The data from extension tests on loose sand deviate somewhat from the straight line relationship. As already pointed out in chapter (2), extension test results usually suffer from larger scatter than compression tests (Lam and Tatsuoka, 1988). Relationship similar to that in Fig . (4.30) was also found by Bishop (1972). However his results were confined to conventional compression tests only. Results presented in Fig . (4.30) indicate that the uniqueness of peak friction angle <j>'p with maximum rate of dilatancy ( d e v / i e „ ) m M applies regardless of stress path as well as extension or compression loading. Triaxial test enables stress paths in which <r[ direction is either vertical (compression, a=0) or horizontal (extension, a = 9 0 ° ) . Whether the unique <j>'p - (dev/dea)max relation-ship of Fig. (4.30) applies to stress paths where cr[ is arbitrarily inclined at a to vertical can only be investigated using devices permitting principal stress rotation such as the hollow cylinder torsional apparatus. Bishop (1972) also suggested that the extrapolation of cf)'p - (dev/dea)max relationship to zero rate of maximum dilatancy yields constant volume friction angle <f>cv of the sand. For Erksak sand (bcv=Z2° can be read from Fig. (4.30) as the intercept value. A n independent measurement of (f)cv of Erksak sand was carried out using the U B C ring shear apparatus (Negussey et al. , 1988) and the variation of <j> with horizontal displacement is shown in Fig. (4.31). A cf>cv value almost identical to that obtained by extrapolation in Fig. (4.30) may be noted. T r 0.2 0.4 ( d C v / d C 0 ) m o x (58) Figure 4.30: Variation of Peak Friction Angle W i t h M a x i m u m rate of Dilatancy. 40 E R K S A K SAND 35. 30 25 cn 0) D_ 20 15 10 ERKSAK SAND, D r = 2 1 * . a=1200 kPa 0 — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — r 0 5 10 15 20 HORIZONTAL DISPLACEMENT (mm) i—r 25 Figure 4.31: Friction Angle Mobilized in The Ring Shear Apparatus. Chapter 4. TEST RESULTS 81 4.3.2.2 Angle of Dilation ftp ' 4>cv is the friction angle in excess of (pcv that is mobilized under constant volume conditions and attributable to dilation. For plane strain condition (2-D strain), angle of dilation -0 is defined as V> = Sin - l dei + de3 dti — de3 (4.8) Introducing dev = dex + de3, ip = Sin  1 dev (4.9) dev — 1dt\ The above equation can now be used for both triaxial (3-D strain) and plane strain (2-D strain) conditions. In triaxial compression, de\ dea (4.10) Substituting (4.10) in (4.9) yj = Sin  1 = Sin ~ 1 de„ dev — 2dea 1 i — 2 1 ( d e „ / d * a j (4.11) In triaxial compression, dea is positive and when the sand is dilating at failure, dev is negative. Therefore, (dev/dea) is negative and Eqn. (4.11) can be written as Chapter 4. TEST RESULTS 82 tp = Sin  1 Similarly in triaxial extension, \(de, /dea + 1 (4.12) dei = deT dev = dea + 2der Substituting (4.13) and (4.14) in (4.9) (4.13) (4.14) ip = Sin  1 = 5 m - 1 = Sin' 1 de„ de„ 2der de„ dtv - 2(dev - dea) 1 - 1 (4.15) In triaxial extension, dea is negative and when the sand is dilating at failure, dev is negative. Therefore, (dev/dea) is positive and Eqn. (4.15) can be written as ip = Sin  1 1 Hdc./de - 1 (4.16) Both equations (4.12) and (4.16) show that ipmax occurs when (dev/dea) is maximum and since (dev/dea) is maximum at failure, xpmax coincides with failure conditions, i.e. at peak friction angle, the dilation angle is maximum. Figure (4.32) shows peak friction angle plotted against maximum dilation angle com-puted using Eqns. (4.12) and (4.16). It may be noted that a unique linear relationship emerges between peak friction angle and maximum dilation angle. It is important to Chapter 4. TEST RESULTS 83 Figure 4.32: Variation of Peak Friction Angle W i t h M a x i m u m Dilat ion Angle. Chapter 4. TEST RESULTS 84 note that this relationship is independent of stress path, relative density, <r3 at failure and mode of loading (compression or extension). Data from extension tests on loose sand suffers a little larger scatter for reasons explained earlier. Furthermore, extrapolation of this relation to V>m a x=0 yield cf>cv « 31.5° as expected form such a relationship. The linear relation in Fig . (4.32) can be written as <f>'p = tn + O.Mtp max (4.17) Similar to (Eqn.4.17), Bolton (1986) has also suggested a linear relation based on the data from plane strain tests and is given by: max (4.18) Cornforth (1973) observed that excess friction angle (</>' - 4>cv) in triaxial condition is approximately two third of that in plane strain condition for Brasted river sand. Ex-amination of Eqns. (4.17) and (4.18) for Erksak sand not only corroborates Cornforth's observation but generalizes its validity to any stress paths or mode of loading. 4.3.2.3 Nature of Stress Path Dependency of Peak Friction Angle Peak friction angle as a function of cr3 at failure is shown in Fig . (4.33) and as a function of o~'m at failure in Fig.(4.34) for all compression stress paths at three relative densities. Degradation of peak friction angle with <r'3 and cr'm at failure may be noted at each relative density, with larger degradation associated with increasing density. Decrease in peak friction angle per ten fold increase in <r3 equals approximately 1.4°, 2.8° and 3.2° for relative densities 26%, 56% and 70% respectively. These values are in agreement with those reported by Duncan et al., (1980) for Monterey No.O sand. Chapter 4. TEST RESULTS 85 Figure 4.33: Variation of Peak Friction Angle With <r3/ in Triaxial Compression. Chapter 4. TEST RESULTS Figure 4.34: Variation of Peak Friction Angle W i t h in Triaxial Compression. Chapter 4. TEST RESULTS 87 Plots similar to Figs. (4.33) and (4.34) for extension loading failure are shown in Figs. (4.35) and (4.36). Similar variation of peak friction angle with cr3 or cr'm at failure can be observed as for compression tests. For a given value of relative density and <r'3 or o~'m at failure the peak friction angle is larger in compression than in extension (Fig. 4.37). This difference is approximately 3 — 4° over the range of relative densities and o~'z investigated. This is likely due to the inherent anisotropy of pluviated sand as described in section (4.2). It may also be noted from Fig. (4.32) to (4.36) that the peak friction angle for a given relative density depends only on the stresses at failure and is found to be independent of the stress path under a give mode of loading. 4.3.3 Prediction of Peak Friction Angle The peak friction angle <b'p for Erksak sand can be related to relative density and cr'm at failure using data shown in Figs. (4.30) and (4.34). These relations are: = 2.467* (Fig. 4.34) (4.19) ^ = 0.1 (Fig. 4.30) ° J max = 0.2467* (4.20) where IR = />(12.2 - Ina'm) - 1 These equations were proposed by Bolton (1986). The equations predict the peak friction angle reasonably well at any relative densities and a'm at failure for all compression paths for Erksak sand. It is to be noted that coefficients in these equations are slightly different from those suggested by Bolton (1986) (see Eqns. (2.8) and (2.9)). This implies that Bolton's equations (2.8) and (2.9) should be recalibrated based on the availability of Chapter 4. TEST RESULTS -40-A (deg) 35 -\ * D (Dn=70«) -» • • '— M (Dr=56») • ; — -30-• " L (Dr=26*) 25 - 1 T 1—1 1 1 1 1 1 1 1 1—1 M i l l 1 1 1— a 1 100 1000 ff'j, (kPa) Figure 4.35: Variation of Peak Friction Angle W i t h tr 3 / i n Triaxial Extension. Chapter 4. TEST RESULTS 89 Figure 4.36: Variation of Peak Friction Angle W i t h crmf in Triaxial Extension. Chapter 4. TEST RESULTS 90 -e- 35 T 1 1 I I I I | <r'3f (kPa) 45 40 " M (Dr=56») ^ " " ^ - - ^ D (Dr=70x) " D (Dr=7Q*) L (Dr=26*) M (Dr=56«) L (Dr=26*) . , Com i " i — i — Ext i i i i i i i 1—i—i—i i i • • — . . v * 35 H 30' 25' 10 100 1000 Figure 4.37: Comparison of Peak Friction Angle in Triaxial Compression and Extension Chapter 4. TEST RESULTS 91 experimental data (Bolton, 1986). Furthermore, due to inherent anisotropic behaviour of sand Eqns. (4.19) and (4.20) for compression loading may not be applicable for extension loading. Equations similar to compression loading for Erksak sand can be derived for extension loading paths using data shown in Figs. (4.30) and (4.36). These relations are: <t>'p - (j)cv = 1.97* (Fig. 4.36) (4.21) ^ = 0.1 (Fig. 4.30) °J max = 0.197* (4.22) where IR = Dr(10.44 - ln<r^) - 1 Equations (4.19) - (4.22) can be used to predict peak friction angle under any stress path for a given value of relative density and o~'m at failure depending on the direction of loading; compression or extension. However, a reasonable estimation of constant volume friction angle 4>cv would be necessary to predict the peak friction angle <f>'. It was shown in section (4.3.2.1) that constant volume friction angle can be obtained either from ring shear test or extrapolated value of <j>' - (dev/dea)max relationship at zero rate of dilatancy. Alternatively cj)cv can be estimated from an undrained triaxial compression test as it equals angle of phase transformation <f>pr (Negussey et al. , 1988). Chapter 5 SUMMARY AND CONCLUSIONS The objective of the research was to investigate the drained loading behaviour of Erksak sand in the triaxial apparatus. The drained loading behaviour was investigated by varying the consolidation history, stress path and loading direction (compression or extension). The init ial state of the sand tested encompassed loose to dense states and confining pressures from 50 to 2400 kPa . Based on the results of this investigation, the following conclusions can be drawn: The hyperbolic representation was found satisfactory to model the stress strain be-haviour of isotropically consolidated sand for confining stress from 50 to 2400 k P a under conventional triaxial test. However, for a increment of stress, the tangent modulus eval-uated from isotropic consolidation data base was found to underestimate the tangent modulus of anisotropically consolidated sand, thus resulting in overestimating deforma-tions. Further, the modified hyperbolic representation, in which the increment in the deviator stress is considered as a stress variable, was found to satisfactorily represent the stress-strain response of anisotropically consolidated sand but in the large axial strain range. The stress strain response of anisotropically consolidated sand in the small strain range nevertheless showed a hyperbolic variation, which was different from the large strain response. The stress strain response of Erksak sand was also found to vary with loading di-rection (compression or extension) due to the inherent anisotropic fabric resulting from 92 Chapter 5. SUMMARY AND CONCLUSIONS 93 pluvial deposition of sand. The extension loading, at a given stress state, showed softer response than the compression loading. The extension loading also showed more con-tractive volume changes. The hyperbolic model (conventional) was found to overestimate the elastic tangent modulus Et values for stress paths involving a'3 decrease. O n the other hand, Et values were underestimated by the hyperbolic model for stress path representing stress increment involving increasing a'3. The constant <r3 extension stress-strain behaviour was also found hyperbolic, though it was different from the compression behaviour. Different KE and n values were observed for extension and compression when loaded under constant a'3 conditions. The use of constant a'3 extension test data base for hyperbolic representation was found to reduce the difference in the predicted and measured elastic moduli values for various extension paths investigated. The failure strength <j>' was found uniquely related to the maximum rate of dilatancy (dev jdea)max regardless of relative density, failure a'3 and stress path for both directions of loading. Extension loading, at identical <r3y, exhibited a smaller (dev/dea)max than the compression loading. Furthermore, extrapolation of the <f>'p-(dev/d€a)max relationship to zero rate of dilatancy yielded a peak friction angle which was almost equal to the constant volume friction angle ^ determined using the ring shear apparatus. M a x i m u m dilation angle was found uniquely related to <j>' that was independent of the loading direction and stress paths. The study also showed that the failure strength $ at identical Dr and a'3f was not affected by the consolidation history or the stress path leading to failure. The failure strength was only a function of the normal stress at failure and Dr . However, it was found to change with the direction of the loading (compression or extension). A t identical failure a'3, the failure strength in compression loading was higher than that in extension loading. Bibliography [1] Arthur, J. R. F. and Menzies, B. K. (1972) , " Inherent Anisotropy in a Sand ", Geotechnique, Vol. 22, No.l, 115-128. [2] Barden, L. and Khayatt, A . J. (1966) , "Incremental Strain Rate Ratios and Strength of Sand in the Triaxial Test " , Geotechnique, Vol. 16, No. 4, 338-357. [3] Bishop, A . W. and Eldin, A . K. G . (1953) , " The Effect of Stress History on The Relation Between (p and Porosity of Sand " , Proceedings of the 3rd International Conference on Soil Mechanics and Foundation Engineering, Switzerland, Vol. 1, 100-105. [4] Bishop, A . W . (1966) " The Strength of Soil as Engineering Material " , Geotech-nique, Vol. 16, No. 4, 89-130. [5] Bishop, A . W. (1971), " Shear Strength Parameter for Undisturbed and Re-moulded Soil Specimens " , Proceedings of the Roscoe Memorial Symposium, Cam-bridge University, 3-58. [6] Bolton, M . D . (1986), " The Strength and Dilatancy of Sand " , Geotechnique, Vol. 36, No.l, 65-78. [7] Chem, J. C. (1984), " Undrained Response of Saturated Sand W i t h Emphasis on Liquefaction and Cyclic Mobil i ty " , Ph.D. Thesis, The University of British Columbia, Vancouver, Canada. 94 Bibliography 95 [8] Chung, E. K. F. (1985), " Effect of Stress Path and Prestrained History on The Undrained Monotonic and Cyclic Loading Behaviour of Saturated Sand " , M.A.Sc. Thesis, The University of British Columbia, Vancouver, Canada. [9] Cornforth, D. H . (1973), " Prediction of Drained Strength of Sand From Relative Density Measurements " , In Evaluation of Relative Density and Its role in Geotech-nical Projects Involving Cohesionless Soils. ASTM Spec. Tech Publication 523, 281-303. [10] De Beer, E . E. (1965), " Influence of The Mean Normal Stress on The Shear-ing Strength of a Sand " , Proceedings of the 6th International Conference on Soil Mechanics and Foundation Engineering, Montreal, 165-169. [11] Duncan, J. M . and Chang, C. Y . (1970), " Nonlinear Analysis of Stress and Strain in Soils " , ASCE, Journal of Soil Mechanics and Foundation Division, Vol. 96, No. SMS, 1629-1653. [12] Duncun, J. M . , Byrne, P., Wong, K. S. and Mabry, P. (1980), " Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analyses of Stress and Movements in Soil Masses " , Report No.UCB/GT/80-01, College of Engineer-ing, Office of Research Services, University of California, Berkeley. [13] Green, G . E . (1969), " Strength and Compressibility of Granular Materials Under Generalized Strain Conditions " , Ph.D Thesis, University of London. [14] Green, G . E. (1971), " Strength and Deformation of Sand Measured in an In-dependent Stress Control Cel l " , Proceedings of the Roscoe Memorial Symposium, Cambridge University, 285-323. Bibliography 96 [15] Haythornthwaite, R. M . (I960), " Mechanics of the Triaxial Test for Soils " , ASCE, Journal of Soil Mechanics and Foundation Division, Vol. 86, No. SM5, 35-62. [16] Hettler, A . and Vardoulakis, I. (1984), " Behaviour of Dry Sand Tested in a Large Triaxial Apparatus " , Geotechnique, Vol. 34, No.2, 183-198. [17] Hight, D.W. (1983), " Laboratory Investigation of Seabed Clays " Ph.D. Thesis, University of London, England. [18] Janbu, N . (1963), " Soil Compressibility as Determined by Oedometer and Tri-axial Tests " , European Conference on Soil Mechanics and Foundation Engineering, London, Vol. 1, 19-25. [19] Kondner, R. L. and Zelasko, J. S. (1963), " A Hyperbolic Stress-Strain For-mulation of Sands " , Proceedings of The Second Pan American Conference on Soil Mechanics and Foundation Engineering, Brazil, Vol. 1, 289-324-[20] Lade, P. V . and Duncan, J. M . (1976), " Stress-Path Dependent Behaviour of Cohesionless Soil " , ASCE, Journal of Geotechnical Engineering Division, Vol. 101, No. GT1, 51-68. [21] Lam, W . K. and Tatsuoka, F. (1988), " Triaxial Compressive and Extension Strength of Sand Affected by Strength Anisotropy and Sample Slenderness " , Ad-vanced Triaxial Testing of Soils and Rocks, ASTM STP 977, 655-666. [22] Lambrechts, J. R. and Leonards, G . A . (1978), " Effect of Stress History on Deformation of Sand " , ASCE, Journal of Geotechnical Engineering Division, Vol. 104, No. GT11, 1371-1387. Bibliography 97 [23] Law, K. T . (1981), " Effect of Stress Path Geometry on Soil Brittleness " , Geotech-nique, Vol. 31, No.2, 279-287. [24] Medeiros, L. V . and Eisenstein, Z. (1983), " A Deep Retaining Structure in T i l l and Sand: Part I Stress Path Effects " , Canadian Geotechnical Journal, Vol. 20, No.l, 120-130. [25] Medeiros, L. V . and Eisenstein, Z. (1983), " A Deep Retaining Structure in T i l l and Sand: Part II Performance and Analysis " , Canadian Geotechnical Journal, Vol. 20, No.l, 120-130. [26] Mejia, C .A , Viad, Y .P . and Negussey, D. (1988), " Time Dependent Be-haviour of Sand " , International Conf. on Rheology and Soil Mech., Coventry, Eng-land, Elsevier Publications. [27] Negussey, D. (1984), " A n Experimental Study of Small Strain reponse of Sand ", Ph.D. Thesis, The University of British Columbia, Vancouver, Canada. [28] Negussey, D., Wijewickreme, W. K. D. and Vaid, Y . P. (1988), " Constant Volume Friction Angle of Granular Materials " , Canadian Geotechnical Journal, Vol. 25, 50-55. [29] Newland, P. L. and Allely, B. H . (1959), " Volume Change During Undrained Triaxial Tests on Saturated Dilatant Granular Materials " , Geotechnique, Vol. 9, No.4, 174-182. [30] Oda, M . (1972), " Initial Fabrics and Their Relation to Mechanical Properties of Granular Material " , Soils and Foundations, Vol. 12, No.l, 17-36. [31] Reades, D. W. and Green, G . E. (1976), " Independent Stress Control and Triaxial Extension Tests on Sand " , Geotechnique, Vol. 26, No.4, 551-576. Bibliography 98 [32] Rowe, P. W. (1962), " The Stress-Dilatancy Relation for Static Equil ibrium of an Assembly of Particles in Contact " , Proceedings of The Royal Society, A269, 500-527. [33] Rowe, P. W . (1969), " The Relation Between The Shear Strength of Sands in Triaxial Compression, Plane Strain and Direct Shear " , Geotechnique, Vol. 19, No.l, 75-86. [34] Sarsby, R. W . , Kaltesiotis, N . and Haddad, E . H . (1980), " Bedding Error in Triaxial Test on Granular Media " , Geotechnique, Vol. 30, No.3, 302-309. [35] Tatsuoka, F. (1987), " Discussion on The Strength and Dilatancy of Sand " , Geotechnique, Vol. 37, No. 2, 219-225. [36] Vaid, Y . P. and Negussey, D. (1984), " A Crit ical Assessment of Membrane Penetration in The Triaxial Test " , ASTM, Geotechnical Testing Journal, Vol. 7, No.2, 70-76. [37] Vaid, Y . P. (1985), " Effect of Consolidation History and Stress Path on Hyper-bolic Stress-Strain Relations " , Canadian Geotechnical Journal, Vol. 22, 172-176. [38] Vaid, Y . P. (1987), " Friction Angle of Sand " , Internal Report NGI. [39] Varadarajan, A . , Sharma, K. G. , Mishra, S. S and Kuberan, R. (1983), " Effect of Stress-Path on Stress Strain Volume Change Behaviour of Jumna Sand " , Proceedings of the International Conference on Constitutive Laws for Engineering Materials, Tucson, Arizona, 289-296. [40] Wijewickreme, D., (1986), " Constant Volume Friction Angle of Granular Ma-terials " M.A.Sc Thesis, The University of British Columbia, Vancouver, Canada. Appendix A Tangent Modulus of Anisotropically Consolidated Sand The stress strain response of anisotropically consolidated sand can be represent by (Vaid, 1985) - °-do = — n r - ( A . l ) in which crdo is the deviator stress at the end of consolidation. The constants,'a' and 'b' can be determined from the intercept and the slope of the straight line. B y differentiating the above equation ( A . l ) , the tangent modulus Et, the slope of the stress-strain curve can be obtained as Et = a[l-b{<Td-<Tdo)}2 (A.2) Analytical expression of tangent modulus obtained from tests on anisotropically con-solidated (# o=0.0427) Erksak sand at Dr=26% , 56% and 70% are as follows: 6r3=100 kPa and Dr=26% Et = 105.82 53.10 99 when 0 < ea < 0.092 (A.3) Appendix A. Tangent Modulus of Anisotropically Consolidated Sand Et = 85.25 Et = 38.46 <73=400 k P a and Dr=^D7o 1 (gj ~ <r<to) 95.23 (<Td — ado) 122.18 when 0.092 < ea < 0.3 when 0.3 < ea Et = 200 1 -Et = 115.5 = 74.6 o-;=100 k P a and Dr=56% 1 -182.78 302.57 {** - <?do) 501.76 when 0 < ea < 0.146 2 when 0.146 < e a < 0.3 2 when 0.3 < e„ 1 2 Et = 146 1 Et = 103.2 Et = 74.7 o-3=400 k P a and Dr=56% 1 -(<Trf — <Tdo) 91.68 153.14 ~ <Tdo) 179.40 when 0 < ea < 0.063 when 0.062 < ea < 0.3 when 0.3 < e„ Et Et Et 241.9 164.4 149.5 1 _ (*d ~ <?do) 181.48 466 when 0 < ea < 0.058 when 0.058 < eQ < 0.3 1 -(<Td — (Tdo) 501.3 when 0.3 < ea Appendix A. Tangent Modulus of Anisotropically Consolidated Sand 101 tr3=250 k P a and Dr=70% Et = 331.1 Et = 232.6 1 - ~ ^ 255.6 1 _ ( a d ~ 698.9 1 2 when 0 < ea < 0.04 1 2 when 0.04 < ea (A.15) (A.16) Appendix B Stress Strain Response Under Various Stress Paths This appendix contains the deviator stress axial strain response of medium dense (Dr=56%) and dense (Dr=70%) sand under a variety of stress paths. Figures ( B . l ) and (B.2) show the deviator stress axial strain response of Dr=56% and 70% respectively. 102 Appendix B. Stress Strain Response Under Various Stress Paths 103 fT'Ll'l' S t r 6 S S S t r a i D R e s P ° n s e F r o m T e s t Performed Along Various Stress Paths. Appendix B. Stress Strain Response Under Various Stress Paths -2.0 -400-4 -800-1 ^70%)' S t f e S S S t i a i n R e S p ° n S e F r o m T e s t Performed Along Various Stress Paths Appendix C Stress Strain Response Under Conventional Triaxial This appendix contains the stress strain response of loose (Dr=26%) , medium dense (Dr=56%) and dense (Dr=70%) under conventional triaxial test. Figures ( C . l ) , (C.2) and (C.3) show the stress strain response of Dr=26%, 56% and 70% respectively. 105 Appendix C. Stress Strain Response Under Conventional Triaxial 106 Appendix C. Stress Strain Response Under Conventional Triaxial 107 Figure C.2: Stress-Strain Response in Triaxial Compression at Different Confining (Dr=56%) Stress. Appendix C. Stress Strain Response Under Conventional Triaxial 108 6000 «. (*) f n ^ n S f S t r e S S - S t r a i n R e s P ° n s e in Triaxial Compression at Different Confining Stress. [L)T— / 0 /oj 

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