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Dynamic properties of sands under cyclic torsional shear Uthayakumar, Muthukumarasamy 1992

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D Y N A M I C P R O P E R T I E S O F S A N D S U N D E R C Y C L I C T O R S I O N A L S H E A R by M U T H U K U M A R A S A M Y U T H A Y A K U M A R B.Sc .Eng. , University of Peradeniya, Sri Lanka. 1985 M . E n g . , Nagoya University, Japan. 1990 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R S O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F CIVIL E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A September 10, 1992 © M U T H U K U M A R A S A M Y U T H A Y A K U M A R , 1992 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 ? - ' U / L grv/é»/aj(££K?/njç The University of British Columbia Vancouver, Canada Date •CS" C-.P T i^=> , l<?92. . DE-6 (2/88) Abstract Dynamic properties of soils have to be well understood in order to assure stability and acceptable performance of soil structures under seismic and wave loadings. It has been found the two important dynamic properties - shear modulus and damping factor are complex functions of many variables. In order to study the influence of various factors on shear modulus and damping factor, drained cyclic torsion shear tests were carried out in the hollow cylinder torsion device using medium Ottawa A S T M C-109 sand. Effects of shear strain amplitude, stress history, effective mean normal stress {(r'm = l/3(a-'1Jra'2 + (r'3)), principal effective stress ratio (R = 01 /03) , intermediate principal stress parameter (b = (<r2 —03 ) / (0 i — 0-3)), void ratio, number of cycles of loading are some of the factors studied in this thesis. During the application of cyclic shear stress (r'm ,R and b were kept constant at pre-selected values for each test. This technique allows to study the effect cr'm, R and b independently. For example, the effect of R on dynamic properties can be isolated by a series of tests on specimens that have identical a'm and b but different levels of R and all parameters <r'm, R and b are held constant during cyclic shear application. It is shown that shear modulus increases with number of cycles of a constant amplitude cyclic shear stress when the induced shear strain is higher than a certain threshold value. The damping, however, decreases with number of cycles even at strain amplitudes less than this threshold value. There is also a threshold value of shear strain below which zero volumetric strain occurs due to cyclic shear loading, and hence no pore pressure would develop if cyclic loading was undrained. Effects of stage ii testing and small strain history on dynamic properties is shown to be insignificant. W i t h decrease of void ratio, shear modulus increases and damping factor decreases. It is shown that for a given b, the void ratio factor F(e) = (2.17 — e ) 2 / ( l + e), collapses the modulus degradation curves obtained at different void ratios in to a single curve. For a given init ia l stress state and shear strain amplitude, shear modulus obtained at different R levels do not show any significant difference when R < 3. Damping factors, however, seems to be unaffected by the change in R at all R levels. When R < 3, shear moduli in triaxial extension condition (b = 1) are found to be less than those in triaxial compression condition (b = 0) and damping factors for b = 1 are higher than those for b = 0. Both triaxial compression and extension state of loadings yielded same values shear modulus and damping factors at large amplitude of shear strain at R = 3. Test results indicate that when b < 1, the dynamic properties are independent of intermediate principal stress. Effects of stress history due to decrease in R from 3 to 2, is significant only in the small strain range, and as the strain level increase, the effects of stress history diminishes. Table of Contents Abstract ii Acknowledgements xi 1 Introduction 1 2 Literature review 6 2.1 Introduction 6 2.2 Shear modulus 11 2.2.1 Effect of shear strain amplitude 11 2.2.2 Effect of strain history 11 2.2.3 Effect of void ratio 15 2.2.4 Effect of confining stress or mean normal stress 16 2.2.5 Effect of stress ratio 18 2.2.6 Effect of intermediate principal stress 21 2.3 Damping factor 22 2.4 Needs for research 24 3 Hollow cylinder torsional device 25 3.1 General description 25 3.2 Definition of stresses and strains in hollow cylindrical specimens . . . 28 3.2.1 Average stresses 31 3.2.2 Average strains 31 3.3 Measurement of strains 32 iv 3.4 Measurement of surface tractions 33 3.5 Stress path control and data acquisition system 33 3.6 Stress path control 35 4 Testing procedure 37 4.1 Specimen preparation 37 4.1.1 Reconstitution of sand specimen 37 4.1.2 Preliminary preparation steps 38 4.1.3 Specimen preparation steps 38 4.1.4 Test preparation steps 40 4.2 Material tested 42 4.3 Experimental program 44 5 Results and Discussion 47 5.1 Introduction 47 5.1.1 Effect of number of cycles 48 5.1.2 Stage versus no stage testing 58 5.1.3 Effect of void ratio 60 5.1.4 Effect of effective mean normal stress 67 5.1.5 Effect of effective stress ratio 73 5.1.6 Effect of intermediate principal stress 78 5.1.7 Effect of fluctuations in R and b 84 5.1.8 Effect of stress history 87 6 Conclusions 94 Bibliography 96 v List of Tables Stress states of soil specimens before the application of cyclic shear stresses VI List of Figures 2.1 Definition of shear modulus and damping 7 2.2 (a) Stresses acting on an element in a hollow cylindrical specimen dur-ing cyclic shear loading and (b) Mohr's circle corresponding to that stress state 9 2.3 Variation of (a) effective stress ratio R and (b) intermediate principal stress parameter b with shear strain amplitude during cyclic shear loading 10 2.4 Variation of shear modulus with shear strain (after Guzman et al 1989) 12 2.5 Variation of shear modulus with effective mean normal stress (after Iwasaki et al 1978) 17 2.6 Variation of m with shear strain (after Iwasaki et al 1978) 19 2.7 Variation of damping with shear strain (after Tatsuoka et al 1978) . . 23 3.1 Schematic diagram of the H C T device (after Sayao 1989) 26 3.2 Details of the H C T device (after Sayao 1989) 27 3.3 Polished end platen with radial ribs (after Sayao 1989) 29 3.4 Surface tractions and stress state in an element in the hollow cylindrical specimen (after Wijewickreme 1990) 30 3.5 Schematic diagram of the data acquisition system (after Wijewickreme 1990) 34 4.1 Specimen preparation by water pluviation (after Wijewickreme 1989) 39 4.2 Levelling the specimen's upper surface (after Sayao 1989) 41 vii 4.3 Particle size distribution of medium Ottawa and(after Wijewikreme 1990) 43 5.1 Typical shear stress - shear strain response for loose sand 49 5.2 Variation of (a) shear modulus (b) shear strain with number of cycles for loose sand 50 5.3 Variation of damping with number of cycles for loose sand 52 5.4 Variation of cumulative volumetric strain with number of cycles N for loose sand 54 5.5 Variation of volumetric strain with shear strain amplitude for loose sand 55 5.6 Pore water pressure build up after ten loading cycles for Monterey No.O sand (after Dobry et al 1982) 56 5.7 Variation of (a) shear modulus (b) damping with shear strain ampli -tude for loose sand 57 5.8 Variation of (a) shear modulus (b) damping with shear strain ampli -tude for loose sand-effect of stage testing 59 5.9 Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b=0 61 5.10 Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b=0.5 62 5.11 Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b = l 63 5.12 Variation of normalized shear modulus (G/F(e ) ) with shear strain . . 64 5.13 Variation of damping with shear strain - effect of void ratio on damping for b=0, 0.5 65 5.14 Variation of damping with shear strain - effect of void ratio on damping f o r b = l 66 vi i i 5.15 Variation of shear modulus with shear strain - effect of a'm on shear modulus for b=0 68 5.16 Variation of shear modulus with shear strain - effect of <r'm on shear modulus for b=0.5 69 5.17 Variation of shear modulus with shear strain - effect of <7'm on shear modulus for b = l 70 5.18 Variation of shear modulus with effective mean normal stress 71 5.19 Variation of m (exponent of (T'm) with shear strain 72 5.20 Variation of damping with shear strain - effect of a'm on damping for b=0 74 5.21 Variation of damping with shear strain - effect of on damping for b=0.5 75 5.22 Variation of damping with shear strain - effect of <r'm on damping for b = l 76 5.23 Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping factor for 0^=100 k P a and b=0 . 77 5.24 Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping factor for 0^=200 k P a 79 5.25 Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping for 0^=400 k P a 80 5.26 Variation of (a) shear modulus (b) damping with shear strain - effect of intermediate principal stress on shear modulus and damping for < = 1 0 0 k P a , R=2 , and I> r=60% ; 81 5.27 Variation of (a) shear modulus (b) damping with shear strain - effect of intermediate principal stress on shear modulus and damping for 0^=300 k P a , R=2 , and Dr=40% 83 ix 5.28 Variation of (a) shear modulus (b) damping with shear strain - effect of intermediate principal stress on shear modulus and damping for (7^=100 k P a , R=3 , and D r = 4 0 % 85 5.29 Variation of (a) shear modulus (b) damping with shear strain - effect of fluctuations in R and b for 0-^=100 k P a , R=3 , and Z> r=40% . . . . 86 5.30 Variation of (a) shear modulus (b) damping with shear strain - effect of fluctuations in R and b for 0-^=400 k P a , R=2 , and D P = 4 0 % . . . . 88 5.31 Stress path diagram showing the init ia l stress states used for tests (a), (b), (c), (d), (e) and (f) to study the effects of stress history 90 5.32 Variation of (a) shear modulus (b) damping with shear strain - effect of stress history at «7^=400 k P a 91 5.33 Variation of (a) shear modulus (b) damping with shear strain - effect of stress history at a^=200 k P a 92 x A ckno wledgement s I am deeply indebted to Dr . Y . P . V a i d for his support, encouragement and patience throughout the study. His advice are guidance are greatly appreciated. I would like to thank Dr . P .M.Byrne for reading this thesis and his valuable advice. Support from Dr . Wijewikreme and visiting scholars Dr . Loo and Dr. Uchida in this research study is thankfully acknowledged. I would like to thank M r . A r t Brookes and M r . John Wong in the civil engineering workshop, for their help. Discussions with Srithar has helped me very much. I take this opportunity to thank him. Help from Joyis, Ra ju , Ralph and Huaren is gratefully acknowledged. Final ly I wish to express my thanks for the financial support provided in the form of research assistantship by the department of C i v i l Engineering, University of Br i t ish Columbia. XI Chapter 1 Introduction In recent years much progress has been made i n the development of analytical methods to evaluate the response of soil structures due to earthquake and wave load-ings. The accuracy of the predicted response by these methods depends very much on the material properties used in the analysis. Therefore it is essential to estimate these properties with confidence. Shear modulus and damping are identified as the most important properties in dynamic analysis. A number of test devices have been used by various researchers to estimate shear modulus and damping factors i n the laboratory. Among them are, 1. Resonant column (e.g. Hardin and Drnevich 1972 a,b) 2. Shear wave velocity measurement (e.g. Roesler 1979, Lee and Byrne 1990) 3. Simple shear (e.g. Silver and Seed 1971, Sugimoto et al 1974) 4. Hollow cylinder torsion shear (e.g. Tatsuoka et al 1978, Iwasaki et al 1979). The first two devices are generally suited for measuring low amplitude shear strain modulus. The last two, however, have been used to measure shear modulus over larger shear strain amplitudes typically experienced during strong motion shaking due to earthquakes. These laboratory studies have revealed that the important factors influencing dynamic properties of sand are 1 Chapter 1. Introduction 2 1. Shear strain amplitude 2. Void ratio and 3. Initial state of stress. Earlier studies mostly on resonant column tests, considered effective mean normal stress a'm = {<r[ + <r'2 + < t 3 ) / 3 , as the sole representative of the init ial stress state. Later researchers have identified that the in i t ia l stress ratio also influences the dynamic properties in addition to tr'm (e.g. Tatsuoka et al 1979). Others have attempted to relate shear modulus not to <T'm but to individual stress components (e.g. Roesler 1979, Y u and Richart 1984). These studies, however have been confined to assessment of low strain modulus only, and their validity in the region of larger strain amplitudes is not known. The init ia l state of stress on a soil element can be prescribed by the principal stresses cr[,(T2 and a'3 or the derived stress parameters (r'm, R = c[/cr3 and b = [cr'2 — v'aïli?! ~~ a'i)i which have been recognized to affect sand behavior. In most resonant column and torsional shear studies of dynamic properties, the init ia l stress state was axisymmetric {(T2 = <r'3 or a^) and cyclic shear stress was applied holding boundary stresses constant. This imposes cyclic fluctuations i n R and b about their init ial prescribed values during cyclic shear stress applications. Thus the measured dynamic properties could have been influenced by the nature of these R and b fluctuations. The effect of R on dynamic properties, for example, can be isolated only by a series of tests on specimens that have identical a'm and b but different levels of R and all parameters <r'm, R and b are held constant during cyclic shear stress application. It is possible to make such studies in stress path controlled hollow cylinder torsional shear tests. Simple shear test has been considered to simulate most closely the init ial stress Chapter J . Introduction 3 state and dynamic stresses to which in-situ soil elements are subjected due to earth-quake shaking in horizontally layered sites. It is however, difficult to assess small strain modulus and damping in this test because of low measurement resolution. The possible init ial stress state in this test is also axisymmetric and shear deformation oc-curs under plane strain conditions. It is therefore not possible in this test to simulate a generalized init ia l stress state nor does it enable isolation of the effects of init ial stress parameters <r'm and b on dynamic properties, because both a'm and b fluctuate during the application of cyclic shear stress. In resonant column tests dynamic properties are expressed in terms of average shear strains across the cross section of the specimen. In a solid cylindrical specimen shear strain at the center is zero while it is maximum at the periphery. Use of average shear strain i n that case, especially at larger strains over which shear modulus suffers significant degradation, may not be appropriate for it 's relationship to the average measured modulus. To minimize this problem hollow cylindrical specimens are preferable. To bring the soil specimen to resonance i n a resonant column test, frequency of excitation is increased from a low value, with a specified amplitude of shear stress. Then from the resonant frequency and the dimensions of the specimen, dynamic properties of the soil are evaluated. Resonant frequencies are usually in the range of 20 - 200 Hz , depending on the type of material and init ia l stress range of interest. These frequencies cause the specimens to be subjected to a few thousands of loading cycles before readings are taken to evaluate the dynamic properties. As the shear strain amplitude increases, the effect of the number of loading cycles on the dynamic properties of soils may increase. In certain dynamic problems, interest may lie in specification of modulus and damping over a small (10 to 20) number of cycles. In such events, properties assessed from resonant column tests which apply to large Chapter 1. Introduction 4 number of cycles may not be appropriate. For the sake of time economy, stage testing method is commonly used in the evaluation of dynamic properties. In this technique, first the specimen is subjected to the lowest possible value of cyclic shear stress for a specified number of cycles. The test is then repeated with unchanged normal stresses and a slightly higher value of shear stress amplitude. This procedure is repeated five or six times at successively higher stress amplitudes and dynamic properties are evaluated from the shear stress-strain curve for each amplitude. It is not very clear from the current literature if the previous small strain history affects the dynamic properties of the soil during subsequent cycles. This research addresses the effects of previous strain history, number of cycles of loading, init ial levels of effective mean normal stress, principal effective stress ratio and intermediate principal stress parameter on shear modulus and damping in sands. To perform this study drained tests using the U B C hollow cylinder torsion device were carried out on medium Ottawa A S T M C-109 sand in both loose and medium dense states. Soil specimen were consolidated under the specified init ia l stress state <r'm, R and b, followed by the application of cyclic shear stress, during which a'm, R and b were held constant. Low frequency of loading (about 0.05 Hz) was used to ensure fully drained condition during cyclic shear of saturated specimens. Several tests with two of the three init ial stress parameters identical, enabled assessment of the effect of the third parameter on the dynamic properties of soils. Studies of this type which allow isolation of the effects of init ia l stress parameters <r'm, R and b on dynamic properties of sand, have so far not been carried out. In chapter two, literature review emphasizing the effects of various factors on shear modulus and damping of sand is given. Description of the U B C hollow cylinder torsion Chapter 1. Introduction device is given in chapter three, followed by the description of testing procedures chapter four. In chapter five test results and discussion are presented. Finally, chapter six important conclusion are documented. Chapter 2 Literature review 2.1 Introduction This chapter reviews the findings of previous researchers on the dynamic properties relationship as shown i n F i g 2.1, shear modulus is expressed as the secant modulus determined by the extreme points on the hysteresis loop. Hysteresis loop is assumed to begin after the specimen has been subjected to a quarter cycle O A of shear loading and the loop is defined by A B C D A as shown in Fig.2.1. B y relating work done during a cycle (hysteresis loop area) Wd, to the stored elastic energy (area of triangle) W,, a simple expression for damping can be developed. For a single degree of freedom system it can be shown that, (shear modulus and damping) of sands. Since soils have non linear stress-strain Wd = 2irXkX2 (2.1) where A = damping factor, k = spring constant of the system and X = the peak displacement. Also it can be shown that the stored energy W.= kX2 (2.2) 2 From Eqs.2.1 and 2.2, A can be expressed as A = Wd (2.3) 6 Figure 2.1: Definition of shear modulus and damping Chapter 2. Literature review 8 Equation 2.3 enables the calculation of damping factor from the measured areas Wi and W, of the shear stress-strain loops. From a number of studies on both cohesive and non cohesive soils, it has been concluded that the following parameters have very important effect on shear modulus and damping of soils (e.g. Seed and Idriss 1970, Hardin and Drnevich 1972a, Park and Silver 1975). 1. Shear strain amplitude 2. Shear strain history 3. Void ratio and 4. Effective mean normal stress In all resonant column and torsional shear studies reported in the literature, cyclic shear stresses were applied to the soil specimen by holding axial and radial stresses constant. In this configuration, although o~'m remains constant, R and b undergo changes with respect to their initial preselected values. In Fig.2.2a, average elastic stresses in the specimen for which axial and radial stresses are kept constant during cyclic shear are shown. Mohr's circle corresponding to this initial stress state (o-J = o-'z,o~'3 = <t't = cr'e) is shown as the solid circle in Fig.2.2b. The stress state corresponds to R ^ 1 and b = 0. When cyclic shear stress Tcy is applied, c[ increases and a'3 decreases (c^ remains constant) as shown by the dashed circle in the same figure. Principal stress ratio R and b will therefore increase with shear stress. These variation of R and b with cyclic shear stress level for initial stress states a'm = 100 kPa and R = 1, 2 and 3 are shown in Figs.2.3a, b. (The assumed modulus reduction curve with shear strain is shown as in inset). It can be observed in Fig.2.3a that R undergoes substantial increases in the region of larger strain amplitudes. The magnitude of increase relative to its selected value Chapter 2. Literature review 9 (a) Figure 2.2: (a) Stresses acting on an element i n a hollow cylindrical specimen during cyclic shear loading and (b) Mohr 's circle corresponding to that stress state. Chapter 2. Literature review Figure 2.3: Variation of Chapter 2. Literature review 11 increases with decrease in init ial R level. Very small change in R occurs for shear strains less than 10~ 2 %, the domain generally investigated for low strain modulus. The increase in b also is most dramatic for R = 1 that is spread over the entire range of shear strain. For this hydrostatic stress state, b simply jumps from an init ial undefined value to 0.5 as soon as is applied. A t higher levels of R (2, 3), increase in b is small and is confined to shear strain levels in excess of about 10~ 2 %. The smallest increase is associated with the largest init ia l R level for a given shear strain amplitude. The typical fluctuations i n R and b illustrated in Figs.2.3a and 2.3b for a specific a'm and Dr wi l l also apply at other levels of (T'm and Dr, i f identical modulus degradation curves are assumed. 2.2 Shear modulus 2.2.1 Effect of shear strain amplitude Fig . 2.4, shows typical variation of shear modulus with shear strain from Guzman et al (1989). They obtained these curves from resonant column and torsional shear tests using dry Ottawa sand under isotropic stress condition. Hollow cylindrical specimens were used for all tests. It can be observed that for strains greater than about 1 0 _ 2 % , modulus decreases rapidly, and becomes as low as 20% of the original value at 0.1% shear strain at all levels of in i t ia l confining stress. 2.2.2 Effect of strain history When evaluating dynamic properties for a given loading cycle, cyclic strains applied previously, are termed as strain history in this thesis. Strain history can be of two types: Chapter 2. Literature review 12 180 1 0 " « 1 0 _ a 10 " a 10.-» 1 10 S H E A R S T R A I N , Figure 2.4: Variation of shear modulus with shear strain (after G u z m a n et a l 1989) Chapter 2. Literature review 13 1. Loading the soil specimen to a higher shear strain amplitude first and then subjecting it to smaller shear strain amplitudes or, 2. Stage testing. This is a method commonly used to obtain dynamic properties of soils over a wide range of shear strains. In this method first a specimen is subjected to the lowest possible value of cyclic shear stress for a specified number of cycles. Then a slightly higher shear stress amplitude is applied for the same number of cycles. This procedure is repeated several times at successively higher strain amplitudes. To study the dynamic properties of dry Silica sand, Park and Silver (1975), Silver and Park (1975) performed strain controlled cyclic triaxial tests. Stage testing using 300 cycles for each strain amplitude, was used. The range of axial strain ea (single amplitude) investigated was from 8 x l 0 - 3 % to 0.35%. From the axial stress-strain response, dynamic Young's moduli E were evaluated. Shear modulus G and shear strain 7 were then evaluated using the theory of elasticity, ie; G = WT7) . . (2-4) 7 = ( l + ^ ) e - (2-5) where fi is the Poisson's ratio and was assumed to be 0.4. The range of shear strain investigated therefore was from 10~ 2 % to 0.5%. From these studies they found that for upto about 25 cycles of loading, modulus and damping at a given shear strain amplitude for virgin and stage tested dry sand specimens are approximately the same. For a given amplitude of shear stress, only a slight increase in shear modulus with the number of loading cycles was found, while damping was found to be independent of the number of loading cycles. Chapter 2. Literature review 14 The assumption that Poisson's ratio is constant in the large strain domain which is required to derive G from E is questionable. Application of cyclic stresses in the triaxial test causes variations of both a'm and R. As the shear strain amplitude increases, fluctuations of <r'm and R wi l l also increase. Moreover b fluctuates between 0 and 1 for compression and extension modes of cyclic loading. The effect of these changes in <T'm, R and b over their init ia l values on measured shear modulus and damping cannot be assessed. To evaluate the effects of cyclic shear strain history on dynamic properties of Toy-oura sand, Tatsuoka et al (1979) performed a series of drained cyclic torsional shear tests on hollow cylindrical specimens. Since equal external and internal pressures were used, only axisymmetric stress states could be simulated. Two groups of tests were performed for shear strains ranging from 5 x l 0 _ 3 % to 0.3%. In group one, stage testing with increasing stress amplitudes in successive stages was used. Ten cycles of loading were applied in each stage. For group two, first ten cycles of a high am-plitude cyclic shear stress were applied, followed by decreasing stress amplitudes in subsequent stages. B y comparing results from these two groups of tests Tatsuoka et al found that the effect of strain history on dynamic properties is not significant. It should be noted that both group of tests were affected by previous strain history ie; group one by small strain history and group two by large strain history. It must be emphasized that in order to study the effects of strain history as opposed to no strain history it is necessary to compare the two group of test results with those on fresh specimen tested at each stress amplitude. In contrast to the findings of Silver and Park (1975) and Tatsuoka et al (1979), Guzman et al (1989) found that stage testing involving cyclic shear strain equal to 10~ 2 % results in an increase in shear modulus of about 40% over that obtained from a virgin sand specimen. They obtained their results from resonant column tests on Chapter 2. Literature review 15 hollow cylindrical specimens under drained conditions. Experiments were performed under both isotropic and anisotropic stress conditions. In a discussion to Guzman et al's finding Tatsuoka et al (1991) presented test results on Toyoura sand that showed no effects of stage testing and thus questioned the results of Guzman et al. Studies by Hardin and Drnevich (1972a) Tatsuoka et al (1979) and Guzman et al (1989) have shown that dynamic properties of sand depend on the number of loading cycles applied to the specimen in the range of larger strain. There is a need to establish a threshold shear strain level above which significant change in the dynamic properties may occur. It is also desirable to examine the volumetric strain accumulation due to cyclic loading, since development of volumetric strain is an index of stiffening of soil. Volumetric strains cannot be recorded in tests i f dry sand test specimens are used, as has been the case in most investigations. 2.2.3 Effect of void ratio In the state of the art report on stress-strain behavior of soils, Hardin (1978) proposed an empirical equation for low shear strain modulus Gmax, in the form, Gmax = AF(e)(Paf-m(cr'mr (2.6) in which A is a dimensionless constant, <r'm effective mean normal stress and Pa is the atmospheric pressure. He developed this equation from test results of Hardin and H i chart (1963). Hardin and R i chart used resonant column device to measure shear wave velocity of dry and saturated clean sands under drained conditions. Solid cylindrical specimens were tested under isotropic stress condition. From the resonant frequency at very low shear strain levels (less than 1 0 _ 4 % ) , shear modulus was eval-uated. When metric system units are used, Hardin recommended A = 700 and void ratio factor, Chapter 2. Literature review 16 1 + e for round grained sand. For angular grained sand A = 326 and the void ratio factor, 1 T 6 The exponent m = 0.5 for both cases. To study the effect of void ratio on shear modulus for a wide range of strain ( 1 0 _ 4 % to 1%), drained tests on saturated Toyoura sand were performed by Iwasaki et al (1978). They used hollow cylindrical specimens under isotropic stress conditions. First resonant column tests were performed to evaluate dynamic properties for shear strains from 1 0 _ 4 % to 10~ 2 %. Then using hollow cj'linder torsion device, specimens with dimensions identical to that used in the resonant column device, dynamic prop-erties were evaluated for shear strains from 10" 2 % to 1%. They confirmed the validity of the void ratio function F(e) given by equation 2.7, for a large range of strains from 1 0 _ 4 % to 1%. However, it is not known whether the normalizing void ratio function F(e) also applies under initially nonhydrostatic stress state. 2.2.4 Effect of confining stress or mean normal stress As can be seen from Fig.2.4, increase in effective mean normal stress results i n increase of shear modulus at a given shear strain amplitude (e.g. equation 2.6 for Gmax). In Fig.2.5, variation of shear modulus with effective mean normal stress for Toyoura sand from Iwasaki et al (1978) is shown. The method of testing and range of shear strain investigated is given in the previous section. These results have been obtained from a number of tests with different effective mean normal stress under isotropic stress states. Chapter 2. Literature review Mean Principal Stress. Figure 2.5: Variation of shear modulus with effective mean normal stress (after Iwasaki et al 1978) Chapter 2. Literature review 18 It can be seen from Fig.2.5 that for a wide range of shear strain there is a linear relationship between shear modulus and effective mean normal stress on a log-log scale. It can be shown that the exponent m in equation 2.6 is the slope of the lines shown in Fig.2.5. Iwasaki et al found that with increasing shear strain amplitude the exponent m increases from about 0.4 at shear strain about 10~ 4 % or less to 0.9 at shear strain of about 0.5% (see Fig.2.6). Similar tests by Drnevich and Richart (1970) using resonant column technique on dry Ottawa sand also indicate that the exponent m increases from 0.5 at shear strain 1 0 - 3 % to 1.0 at shear strain 6 x l 0 - 2 % . It should be noted that in tests carried out by both Iwasaki et al and Drnevich and Richart, effective stress ratio R was not constant during cyclic shear. As explained in section 2.1, i f R is not held constant, it's variation during each cycle may influence dynamic properties. It is necessary to isolate the effect of R from other parameters, which can only be achieved by keeping R constant at the in i t ia l value during cyclic shear. From a study of existing experimental data, Seed and Idriss (1970) recommend 0.5 for the exponent m regardless of shear strain amplitude. This may be due to the fact that most of the data analyzed by Seed and Idriss was obtained using resonant column tests, and hence corresponded to the small strain range. For the small strain range the use of a single value 0.5 for the exponent m may be reasonable. However for large strain range this may not be appropriate. More experimental data is necessary to evaluate the exponent m at large strain range. In particular, its dependence on strain level for initially nonhydrostatic stress state is not known. 2.2.5 Effect of stress ratio To study the effects of stress ratio on dynamic properties of soils Tatsuoka et al (1979) performed a series of drained cyclic torsional shear tests on hollow cylindrical Chapter 2. Literature review Chapter 2. Literature review 20 specimens. Saturated Toyoura sand was used for these tests. During the application of cyclic shear, effective mean normal stress was held constant, but R and b were not constant. Only two dimensional axisymmetric init ial stress states were investigated. From this study Tatsuoka et al (1979) reported that the effects of stress ratio on shear modulus are notable but only in the triaxial extension case. For the triaxial compression case the effect is negligible for stress ratios less than 4. Y u and Richart (1984), however, suggest that the effect of stress ratio on shear modulus is significant if the stress ratio is higher than about 2.5 regardless of the loading mode. Y u and Richart used resonant column tests on solid cylindrical specimens. During the shear-ing phase, mean normal stress was held constant but R and b would not be constant. As discussed in the previous sections, allowing R and b to fluctuate during each cycle, may affect measured dynamic properties. To take into account the effect of stress ratio, Y u and Richart (1984) defined a parameter Kn as, in which (a[/o-'3)max is the maximum effective stress ratio possible, or the failure criterion of sand. The empirical equation developed for G m a x using the parameter Kn is, Gmax = A F ( e ) ( P B ) w ( ^ p £ ) M ( l - 0 . 3 i C 5 ) (2-10) where o~'a and o~'p are the normal effective stresses in the directions of wave prop-agation and particle motion respectively. This equation implies that a maximum of 30% reduction in shear modulus wi l l occur when the effective stress ratio is at the maximum value. Chapter 2. Literature review 21 2.2.6 Effect of intermediate principal stress It was reported that the effect of intermediate principal stress is relatively insignif-icant by Y u and Richart (1984). However, they did find difference in shear modulus between triaxial compression and extension cases. It was suggested that the difference i n the results is due to the intermediate principal stress which equals major principal stress in the extension case and minor principal stress in the compression case. It should be noted that the stress states simulated in Y u and Richart's testing program is also two dimensional axisymmetric. The effect of intermediate principal stress was not studied for cases other than triaxial compression and extension loadings. Wave propagation studies on dry sands under triaxial stress states, carried out by Stokoe et al (1985) have shown that the shear wave velocity depends equally on the principal stresses in the directions of propagation and particle motion. The third principal stress was found to have no effect on the shear wave velocity. As noted i n the previous chapter, wave propagation techniques give shear modulus values at extremely small strains. The influence of the third principal stress on shear modulus at large strain may or may not be insignificant. Using a combination of resonant column and torsional shear device N i (1987) carried out an extensive study on dynamic properties of dry a sand. Hollow cylindrical specimen with different internal and external pressures were used in his experiments. Independent control of these two pressures together with axial stress allowed h im to simulate true triaxial stress states on the soil specimen. For the shear strain range from 1 0 - 4 % to 0.1% N i found that shear modulus is dependent only on normal stresses in the wave propagation and particle motion directions and the effect of third normal stress is not significant. Chapter 2. Literature review 22 2.3 Damping factor Unlike shear modulus, not much data on damping have been published. Damping of soils increase with strain amplitude as shown in Fig.2.7 and at large strains damping appears to take on values well over 25%. These results were obtained by Tatsuoka et al (1978) for saturated Toyoura sand using torsional shear tests under isotropic stress conditions. The range of damping values from Seed and Idriss (1970) is also shown in this figure by dashed lines. Seed and Idriss obtained their results from resonant column, triaxial and simple shear tests using a number of different sands at different confining stress levels. As Hardin and Drnevich (1972a) and several other researchers found, damping depends on the confining stress level. Lumping results from different soils at different stress levels is bound to yield a wide envelope such as that proposed by Seed and Idriss. Hardin and Drnevich (1972a) reported that damping decreases with the number of cycles of loading for a given strain amplitude. However Park and Silver (1975) and Silver and Park (1975) found that damping values for sands are relatively insensitive to the number of cycles of shear strain. Hardin and Drnevich (1972a) reported that damping factor increases with the init ia l shear stress level particularly at the lower strain amplitudes. However, Tat-suoka et al (1979) found that the effects of init ial shear stress or stress ratio are not significant. The effect of void ratio on damping is not clear from the available literature and in all cases, it was concluded that damping is insensitive to void ratio (Hardin and Drnevich 1972a, Park and Silver 1975, Shérif et al 1977, Tatsuoka et al 1978, 1979, Saxena and Reddy 1989). However, all researchers found that damping factor increases with effective mean normal stresses. Figure 2.7: Variation of damping with shear strain (after Tatsuoka et al 1978) Single Amplitude Shear Strain Chapter 2. Literature review Chapter 2. Literature review 24 2.4 Needs for research Despite the large number of investigations in the literature, the effects of each parameters on shear modulus and damping factors does not appear to have been isolated. Also, conflicting results have been reported by various researchers regarding the effects of factors such as strain history, effective stress ratio and stage testing on dynamic properties of sands. Results from wave propagation studies at extremely small strain range can not be generalized for a wide range of strain. There is a need to establish more thoroughly the manner in which shear modulus and damping factors vary with factors such as effective stress ratio and intermediate principal stress. In resonant column tests several thousand of constant amplitude shear stress cycles are applied to the soil specimen before it reaches resonance. In addition stage testing technique has been commonly used by most researchers. There is little data to suggest that the dynamic properties at such large number of cycles apply to much smaller number of cycles, typical in earthquake shaking. In this research it is intended to address some of the above mentioned problems in a systematic manner. Chapter 3 Hollow cylinder torsional device The U B C hollow cylinder torsional device used in this research was fabricated in the civil engineering workshop in 1986 and a detailed description of the device is given by Sayao (1989) and Vaid et al (1990). In order to carry out complex stress path tests, a fully automated test control and data acquision system was added later (Wijewickreme 1990). 3.1 General description A schematic diagram of the U B C hollow Cylinder torsional device is shown in Fig.3.1 and 3.2. This device is capable of applying axial load, torque about the vertical axis and independent internal and external pressures for a hollow cylindrical soil specimen. Independent control of these four tractions enables the specimen to be loaded along a prescribed stress path in the four dimensional stress space - R , a'm, b, aa. etc is the direction of major principal stress with respect to the vertical deposition direction and the other parameters were denned in the previous chapter. The specimen is approximately 30 cm high and internal and external diameters are 10.2 cm and 15.2 cm respectively. Sayao (1989) describes the selection of these d i -mensions in detail from the considerations of minimizing stress nonuniformities across the specimen wall . The specimen is fixed at the top and laterally confined by internal and external water pressures acting on flexible 0.3 m m thick rubber membranes. A double-acting air piston mounted at the bottom of the supporting table is used 25 Chapter 3. Hollow cylinder torsional device 26 P O S m O N I N G BOLT LVDT ( A H ) EXTERNAL PRESSURE PORE PRESSURE SUPPORTING TABLE " UNEAR-ROTARY BEARING TORQUE CABLES r.rr THRUST BEARING AXIAL LOAD CELL c ii • • -r—*• • TOP CROSS BEAM - CHAMBER TOP .TOP CAP • TOP PLATEN • PLEXIGLASS CELL • LOADING FRAME •SOIL SPECIMEN • RIGID ROD BASE PLATEN BASE PEDESTAL PRESSURE TRANSDUCER •INTERNAL PRESSURE LVDT ( 8 ) LOADING SHAFT TORQUE CELL CENTRAL PULLEY TORQUE PULLEY - TORQUE PISTONS (TWO PAIRS) AXIAL LOAD PISTON 0 20 SCALE (cm) Figure 3.1: Schematic diagram of the H C T device (after Sayao 1989) Chapter 3. Hollow cylinder torsional device 27 TOP CROSS BEAM EXTERNAL | PRESSURE Il CELL BASE J PORE 1 PRESSURE LOADING SHAFT INTERNAL PRESSURE I I I—I I I 0 5 SCALE (cm)' Figure 3.2: Details of the H C T device (after Sayao 1989) Chapter 3. Hollow cylinder torsional device 28 to apply vertical normal stress to the sample in either compression or extension. The load is transmitted through a 25 m m diameter polished stainless steel ram that has its vertical alignment guaranteed by two frictionless Thompson combination bush bearings. Torsional shear stresses are applied by two pairs of identical single-acting air pistons and a system of cables and pulleys. Diametrically opposite pistons are inter-connected to a common regulated pressure supply. This configuration is necessary to apply torque i n either direction and to eliminate horizontal side forces on the loading ram. Transfer of both torsional shear and vertical normal stresses from the loading ram to the soil specimen requires prevention of slip between the soil specimen and the end platens in the tangential direction and minimal frictional restraint in the radial direction. This is achieved by using polished, anodized aluminum platens having 12 thin radial ribs (1 mm thick and 2.3 m m deep), as shown in Fig.3.3. Drainage from the specimen is achieved by six 12.8 m m diameter porous discs set 60° apart, flush with each platen surface. 3.2 Definition of stresses and strains in hollow cylindrical specimens In order to achieve a specified stress state in a hollow cylindrical specimen, four surface tractions have to be varied in a prescribed manner. They are: vertical load {Fz), torque (Th), external and internal pressures ( P e , P , ) . F i g 3.4 shows these trac-tions together with the stress state in an element in the wall. The four stress compo-nents a2,ar,(Tg and rz8 induce the four strain components ez,eT,ee, and ezg in the soil element. Chapter 3. Hollow cylinder torsional device Figure 3.3: Polished end platen with radial ribs (after Sayao 1989) Chapter 3. Hollow cylinder torsional device 30 Figure 3.4: Surface tractions and stress state in an element in the hollow cylindrical specimen (after Wijewickreme 1990) Chapter 3. Hollow cylinder torsional device 31 3.2.1 Average stresses Interpretation of results from hollow cylinder torsion tests is made by considering the entire specimen as a single element, deforming as a right circular cylinder. Stresses crT,(Te and rzg are obtained by averaging stresses over the volume of the specimen and assuming soil to be linear elastic (Sayao 1989, Wijewickreme 1990). Vertical stress crz is assumed to be distributed uniformly across the cross section, and thus obtained using equilibrium considerations only. The resulting expressions are, 3.2.2 Average strains Chapter 3. Hollow cyhnder torsional device 32 - ARj £r = 5 —^ (3.6) ARe + ARj 66 = —rTTrT ( 3 J ) 2A9{Rl - j g ) 7 2 0 " ~ZH{Rj:rW) ( 8 ) where H , A H are the height and height change and A# angular displacement of the specimen. The change in the inner radius ARi is obtained from the measured values of A H and volume change of the inner chamber. The external radius change ARe can then be computed from the measured values of A H , volume change of the sample and ARi. 3.3 Measurement of strains Four measurements are needed for computing the four components of strain in the hollow cyhnder specimens. Two displacement transducers (linear variable differential transformers L V D T ) are used to monitor vertical and angular displacements of the specimen's base pedestal. From these, average axial and shear strains (e2 and jzg) can be obtained. Bo th L V D T ' s can detect movements in the order of 1 0 - 3 m m . This results in a resolution of about 5 x l 0 _ 4 % in both ez and jzg. Two differential pressure transducers ( D P T ) are used to register volume changes of the saturated soil specimen and of the inner pressure chamber. These are required for the evaluation of radial and tangential normal strains. Measured volume changes are corrected to account for membrane penetration, i f any, as proposed by Vaid and Negussey (1984). The D P T used for measuring sample volume change can detect volume changes in the order of 3 m m 3 , resulting in a resolution of about 5x 1 0 - 4 % in Chapter 3. Hollow cylinder torsional device 33 volumetric strain e„ after correcting for membrane penetration effects. The D P T for the inner chamber can detect volume changes in the order of 10 m m 3 . 3.4 Measurement of surface tractions Four surface tractions - vertical load, axial torque, internal and external chamber pressures are monitored during tests. Pore water pressure and chamber pressures are measured using sensitive pressure transducers having a resolution in the order of 0.25 k P a . Vertical load is measured using a load cell placed outside the cell chamber. Ver-tical stress as low as 0.2 k P a can be accurately measured using this load cell. Torque measurements are made with a torque transducer placed just above the central pulley (see Fig.3.1). As low as 0.05 N m of torque can be measured using this transducer. This corresponds to a resolution of shear stress rzg in the order of 0.1 k P a . The torque cell has a negligible amount of cross talk between vertical load and torque. 3.5 Stress path control and data acquisition system A schematic diagram of the stress path control and data acquisition system is shown in Fig.3.5. The system consists of four motor set (stepper motor) precision regulators that control surface tractions. Nine transducers monitor various loads, pressures and displacements through a multi-channel scanner and an A / D converter together with a personal computer. A l l transducers are exited with a stable D . C . power supply of 6 V . The L V D T ' s being high output devices do not need any signal amplification. Other transducers are of bonded strain gauge type with a full scale output of about 3 m V / V . Signal outputs of these transducers are scaled to 2 V full scale using variable gain amplifiers. Chapter 3. Hollow cylinder torsional device 34 HCT DEVICE - o n — 1 <î> PC DATA ACQUISITION SYSTEM Q STEPPER MOTORS I I TRANSDUCERS Figure 3.5: Schematic diagram of the data acquisition system (after Wijewickreme Chapter 3. Hollow cylinder torsional device 35 Scanning is triggered simultaneously on all channels at specified instances depend-ing on the stress path being followed. Signal from each transducer over a 50 mS period is integrated by an analog circuit and the average value is obtained. This average value is then digitized by the A / D converter and temporarily stored. The digital output from the A / D conversion yields a decimal number ranging from 1 to 40000 16 bits in binary) corresponding to an analog range of 0 to 4 V . This corresponds to a resolution of the transducer outputs to be in the order of 0.1 m V . A t the end of conversion, the digital voltages are retrieved by the computer in a sequential manner. 3.6 Stress path control Stress path control is achieved by controlling the four surface tractions by the four precision motor set regulators (stepper motors). In response to an input to the computer, a series of square pulses are sent to the stepper motors. The input contains information about the number of pulses and their directions for achieving a prescribed increment in traction. Approximately 12 to 13 pulses are needed to obtain a pressure change of 1 k P a , thus enabling pressure changes of as low as 0.1 k P a . A computer program which allows the user to apply a generalized stress path loading to the soil specimen has been written (Wijewickreme 1990). Details about the stress path loading and the number of steps needed to apply stresses i n incremental form from one stress state to another are specified in a data file. The program allows incremental tractions to be applied simultaneously to all channels in each step. After the application of tractions, sufficient time is allowed for deformation equilibrium before channels are scanned. Stresses and strains are calculated using the scanned data and the applied stresses are compared with the target stresses for each step. Each surface traction is ad-justed until the applied stresses are within a specified tolerance from the target value. Chapter 3. Hollow cyHnder torsional device 36 Stresses, strains and deformations are then stored as acquired data and the next load increment is applied. This type of feed-back system allows the user to follow any specified stress path accurately. Chapter 4 Testing procedure 4.1 Specimen preparation 4.1.1 Reconstitution of sand specimen Since soil properties are significantly dependent on the method of specimen prepa-ration, special attention should be paid in the selection of specimen preparation tech-niques. A laboratory specimen is assumed to represent an element in the soil mass. Therefore it is essential to prepare the specimen as homogeneous as possible. Water pluviation followed by vibration was selected as the most suitable technique for re-constituting soil specimens. This technique enables preparation of homogeneous sat-urated specimens of uniform sands with controlled density (Vaid and Negussey 1988). Pluviat ion is considered to duplicate the sedimentation process and, hence, the fabric of many natural or artificial sand deposits (fluvial and hydraulic fills etc.). Laboratory studies on pluviated sands should therefore give a close indication of the behavior of these deposits (Oda et al 1978, M i u r a and Told 1984). The main limitation of water pluviation is the segregation of particles during sedimentation of well-graded and silty sands. For these materials, an alternative slurry deposition technique has been developed by Kuerbis and Vaid (1988) for preparing homogeneous saturated specimens. In this research a uniform sand was used as the test material and specimen reconstitution was done by water pluviation. 37 Chapter 4. Testing procedure 38 4.1.2 Preliminary preparation steps A known dry weight of sand (about 5 kg) is boiled for about 10 minutes in several flasks and left overnight under vacuum. A l l porous stones are boiled and allowed to cool in water at room temperature. A l l drainage Unes are flushed through by de-aired water and saturated Prior to sample preparation, a dial gauge mounted on a removable stand, is used to obtain a reference reading on the top platen. This is used later for determining the height of the sand specimen after deposition. First the inner rubber membrane is positioned and sealed to the inner surface of the base platen. De-aired water is flushed through the base drainage line, and saturated porous stones are placed in position. The four piece inner split mold is then assembled and the inner membrane is stretched around it . The four pieces of the inner mold are held together by two internal metallic discs, the annular base platen and one 0 ring at the top. The outer membrane is then positioned and sealed to the outer surface of the base platen, and outer mold is assembled subsequently. The outer mold has its inner surface lined with porous plastic, through which vacuum is applied for stretching the outer membrane. 4.1.3 Specimen preparation steps The flasks with boiled sand are filled with de-aired water upto the top of the tubes which stick out of the stoppers. The annular cavity formed by the two molds is then filled with de-aired water. Once the sand flask is inverted and has its tube tip submerged in the specimen water cavity, sedimentation of the sand proceeds under gravitational influence and mutual displacement with water as shown in F ig 4.1 During the pluviation process, flasks are slowly traversed over the annular area to deposit sand with an approximately level surface at all times. Water pluviation results in a loose density regardless of the height of particle drop (Vaid and Negussey 1988). Chapter 4. Testing procedure 39 Figure 4.1: Specimen preparation by water pluviat ion (after Wijewickreme 1989) Chapter 4. Testing procedure 40 Sedimentation is continued unti l an excess of sand over that required for the final grade has been deposited. The upper surface is then carefully levelled by siphoning off excess sand using a suction of about 2 k P a as shown in Fig.4.2. This causes minimal disturbance of sand grains below the surface. The excess sand is oven-dried to allow determination of the dry weight of sand used in the specimen. The top platen containing saturated porous stones is then carefully seated on the levelled sand surface. Since the pluviated specimen is loose, no further loosening of the top layers wi l l occur because of penetration of the platen's ribs into sand (Vaid et al 1990). De-aired water is then percolated upwards through the specimen under a very small gradient. This is done to remove entrapped air bubbles, i f any, between the rubber membranes and the vertical face of the top platen. After sealing both mem-branes to the top platen with 0 rings, the top drainage is closed and a vacuum of about 20 k P a is applied to the bottom drainage line. This provides an effective confinement to the specimen prior to dismantling forming molds. A t this stage, the dimensions of the specimen are recorded. Corrections to the specimen dimensions because of thickness of membranes are taken into consideration. The top loading cap is now installed, thus completing the specimen preparation process. Relative density of the specimen as deposited has been found to be about 35%. 4.1.4 Test preparation steps The cell chamber is placed in position, and de-aired water is allowed to fill the inner and outer chambers slowly. Care is taken to ensure full saturation of the inner chamber. The top cross beam is then swivelled in position and firmly bolted to the reaction frame. The central rod used for monitoring vertical displacement is then installed thus sealing the inner chamber. The specimen is then moved upwards by er 4. Testing procedure LEVELLING BAR •+— L EXTENSION CONTAINER OUTER FORMER SAND SPECIMEN INNER FORMER 3 EXCESS SAND Figure 4.2: Levelling the specimen's upper surface (after Sayao 1989) Chapter 4. Testing procedure 42 pressurizing the bottom chamber of the vertical loading piston unti l the top cap contacts the cross beam of the reaction frame. The top cap is secured against the cross beam by a bolt. This set up ensures coincidence of the specimen's vertical axis and the frame's center axis. A locating p in , protruding from the cross beam, is then inserted into the loading cap at 30 m m from the specimen axis as shown in Fig.3.2. This pin serves to arrest any rotational movement of the loading cap in tests requiring application of torque. Measurement of Skempton's B value for checking specimen's saturation then pro-ceeds in several increments of confining pressure under undrained conditions. Ful l saturation of the soil specimen is ensured by insisting on a B value greater than 0.98. The specimen is then isotropically consolidated to a reference effective stress <Tm = 50 k P a against a back pressure of 150 k P a . In all tests reported in this study, samples were isotropically consolidated from the reference effective stress o~'m = 50 k P a to the desired effective mean normal stress level, followed by simultaneous increments of effective stress ratio and intermediate principal stress parameter to the desired values. This state of stress prior to the application of cyclic shear stress, about the vertical axis is referred to as the init ial state of stress. 4.2 Material tested Medium Ottawa sand A S T M - C - 1 0 9 was used as the test material. Ottawa sand is a rounded quartz sand with a specific gravity of 2.67. The sand is uniformly graded ( C u = 1.9) and has average particle size D 5 0 of 0.4 mm. The particle size distribution is given in Fig.4.3. Research at U B C has shown that for Ottawa sand the amount of particle crushing under moderate stresses is negligible and thus the sand could be recycled conveniently Chapter 4. Testing procedure 43 Figure 4.3: Particle size distribution of medium Ottawa and(after Wijewikreme 1990) Chapter 4. Testing procedure 44 for testing purposes (Chern 1985). A S T M maximum and minimum void ratios for the sand are 0.82 and 0.50 respectively. 4.3 Experimental program The experimental program was designed to investigate the influence of the follow-ing on shear modulus and damping, 1. Number of cycles of loading - N 2. Stage testing - ie; strain history as a result of previous loading cycles under smaller amplitudes of strain 3. Initial state of stress, as prescribed by c^o^ and cr'z or the equivalent stress parameters a'm, R and b 4. Stress history. Investigations as to the influence of init ia l stress parameters were carried out which ensured that all init ial stress parameters remained constant during the application of cyclic shear stress. Several specimens at a given relative density and init ia l state <rm and b but different R , for instance, enabled isolation of the effect of R alone on dynamic properties. The effect of other in i t ia l stress parameters a'm and b were investigated in a similar manner. These effects of each init ia l stress parameters on dynamic properties were assessed on both loose and dense states. Also a number of tests were carried out in the conventional manner (ie, by holding boundary stresses constant during cyclic shear). The influence of fluctuations in R and b can be studied by comparing these test results with tests for which init ia l stress parameters a'm, R and b were held constant during cyclic shear. Chapter 4. Testing procedure 45 The foregoing objectives were accomplished by carrying out the tests summarized in Table 4.1. The table documents the init ial stress parameters {cr'm, R and b) and void ratio of each specimen, the minimum cyclic shear stress amplitude of loading and the number of cycles under each amplitude. Each specimen was stage tested under successively increasing amplitudes of shear stresses, unti l a maximum shear strain amplitude approximately 0.2% to 0.5% was obtained. Thus, shear modulus and damping were assessed over a shear strain range of about 5 x l 0 _ 3 % to 0.3%, the smaller value being the lowest that can be confidently assessed with the measurement resolution of the hollow cyhnder torsion device used. Chapter 4. Testing procedure 46 Table 4.1: Stress states of soil specimens before the application of cyclic shear stresses Test Number R b DT T(°)cy N (kPa) (kPa) held constant 1 100 2 0.0 40 2.5 50 Yes 2 100 2 0.0 40 2.5 3 j) 3 100 2 0.0 40 7.5 3 ,, 4 100 2 0.0 40 12.5 3 ,, 5 100 2 0.0 40 20.0 3 „ 6 100 2 0.0 40 30.0 3 7 100 2 0.5 40 5.5 3 „ 8 100 2 1.0 40 5.5 3 „ 9 100 3 0.0 40 5.5 3 „ 10 100 3 0.5 40 5.5 3 11 100 3 1.0 40 5.5 3 12 300 2 0.0 40 5.5 3 „ 13 300 2 0.5 40 5.5 3 „ 14 300 2 1.0 40 5.5 3 „ 15 100 2 0.0 60 5.5 3 16 100 2 0.5 60 5.5 3 17 100 2 1.0 60 5.5 3 18 200 2 0.0 40 3.0 3 „ 19 400 2 0.0 40 5.5 3 „ 20 400 1 0.0 40 5.5 3 No 21 400 2 0.0 40 5.5 3 „ 22 400 1 0.0 40 3.0 3 „ 23 200 2 0.0 40 3.0 3 24 200 3 0.0 40 3.0 3 „ 25 200 2 0.0 40 3.0 3 „ 26 200 1 0.0 40 3.0 3 „ 27 200 2 0.0 40 3.0 3 28 100 3 0.0 40 2.5 3 where, r ( o ) c y and N denotes the minimum cyclic shear stress and number of loading cycles respectively. Chapter 5 Results and Discussion 5.1 Introduction In this chapter, dynamic properties of Ottawa sand A S T M C-109 obtained under a variety of init ial stress states are discussed. A l l tests were carried out drained on saturated specimens. The loading frequency used was 1/20 Hz which ensured fully drained condition under cyclic shear loading. Initial stress states prior to the application of cyclic shear stress are given in table 4.1. During the application of cyclic shear stresses, effective mean normal stress a'm, effective stress ratio R and intermediate principal stress parameter b were kept con-stant. This eliminated possible influence on modulus and damping due to variations in any of these stress parameters during cyclic shear loading. In the following sections, detailed examination of the effects of 1. Number of cyclic shear stress 2. Cyclic strain history 3. Void ratio 4. Effective mean normal stress 5. Effective stress ratio and 6. Intermediate principal stress 47 Chapter 5. Results and Discussion 48 on shear modulus and damping in Ottawa sand are presented. Direct comparisons are also made with properties determined using conventional technique in which principal effective stress ratio R and intermediate principal stress parameter b are allowed to fluctuate with respect to the ambient values. In addition, the effect of stress history on dynamic properties is also considered, though on a limited scale. Typical shear stress-strain response curve from a cyclic torsion test for loading cycles 3, 10 and 48 are shown in Fig.5.1. The example illustrated corresponds to large shear strain amplitude of the order of 0.15%. As discussed in chapter two, shear modulus was determined as the secant slope between the compression and extension peaks within the same cycle and hysteretic damping was measured from the area of the loop. It can be seen from Fig.5.1, that for a given shear stress amplitude, as the number of loading cycles increases, shear strain amplitude decreases, shear modulus increases and damping decreases. 5.1.1 Effect of number of cycles In Fig.5.2a, variation of shear modulus with number of cycles for a typical stage test is shown. For this test dm, R and b were kept constant at 100 k P a , 2, 0 respectively, while shear stress was cycled approximately 50 times in each stage starting from an amplitude of 2.5 k P a . Initial relative density of the specimen was 40%. Independence of shear modulus with the number of cycles at lower amplitudes of cyclic shear stress (from 2.5 k P a to 25 k P a for this case) may be noted. However for higher amplitudes of shear stress, modulus values initially increase with the number of cycles but tend to level off after a certain number of cycles. Both the final value and the number of cycles to reach it , depend on the shear stress amplitude. Chapter 5. Results and Discussion Figure 5.1: Typical shear stress - shear strain response for loose sand Shear strain amplitude Chapter 5. Results and Discussion No. of cycles N Figure 5.2: Variation of (a) shear modulus (b) shear strain with number of cycles for loose sand Chapter 5. Results and Discussion 51 For shear stress amplitudes of 30 and 37.5 k P a , the increase in modulus be-tween first and fiftieth cycle are 19% and 59% respectively. Corresponding amplitude of shear strain for stress amplitude 30 k P a decreased from 0.103% (first cycle) to 8 . 8 x l 0 ~ 2 % (fiftieth cycle), and for shear stress amplitude 37.5 k P a from 0.182% to 0.118%. In F ig 5.2b, variation of shear strain amplitude with number of cycles in the same test is shown. It can be observed that if the amplitude of shear strain is less than 6 x l 0 _ 2 % it remains constant irrespective of the number of cycles of loading. On the other hand, if the shear strain is greater than 6 x l 0 _ 2 % during the first cycle, it attenuates with the number of cycles and like modulus values, levels off after a certain number of cycles. It is clear from Figs.5.2a and 5.2b that as long as the shear strain amplitude is less than about 6 x l 0 _ 2 % , shear modulus is independent of the number of cycles of loading. Large strain modulus values however, obtained from devices such as triaxial , simple shear and torsion shear are likely to be significantly affected by the number of cycles of loading. Variation of damping factor with number of cycles for shear stress amplitudes 12.5, 20, 30 and 37.5 k P a is shown i n Fig.5.3. Unlike shear modulus, damping is much more affected by the number of cycles at al l shear stress levels. For example, for shear stress amplitude 12.5 k P a corresponding to a strain amplitude of 6 x l 0 _ 2 % , modulus remains unchanged with the number of cycles. Damping however, drops to low values, about 50% between first and fiftieth cycles. Use of damping values obtained after a few thousand of cycles of loading in resonant column tests, in the dynamic analysis, may therefore give results which are extremely conservative. It is therefore recommended to use devices such as hollow cylinder torsion with high resolution measurements, and apply the anticipated number of cycles during a strong motion earthquake, to the undisturbed soil sample. Chapter 5. Resulis and Discussion 30 Figure 5.3: Variation of damping with number of cycles for loose sand Chapter 5. Results and Discussion 53 The occurrence of volumetric strains during the application of cyclic shear stress is a reflection of inelastic material behavior. Fig.5.4 shows cumulative volumetric strain development with number of cycles. No measurable strains develop for shear stress amplitudes less than 7.5 k P a , corresponding to a T/a'm = 0.075. At higher stress amplitudes, volumetric strains accumulate with number of cycles, and their magnitude increase with the amplitude of shear stress. Fig.5.5 shows the relationship of volumetric strain to shear strain, for 1st, 3 r d and 48 t h cycles for each stress amplitude stage. It may be noted that volumetric strain is zero until shear strain exceeds about 10~ 2 %. This threshold value of shear strain below which volumetric strain is essentially zero was observed for all tests carried out in this study. Dobry et al (1982) carried out a number cyclic triaxial tests using Monterey No. 0 sand under undrained conditions. They found the existence of a threshold shear strain of l . l x l 0 - 2 % , below which no measurable residual pore water pressure developed (see Fig.5.6). This observation was made for test conditions with relative density varying from 45 - 80% and effective confining pressure from 533 - 4000 psf (25.5 -192 kPa) . Development of volumetric strain in a drained test is equivalent to pore pressure development in an undrained test. Therefore the threshold value of shear strain obtained from this study is in agreement with the findings of Dobry et al. Fig.5.7a shows the variation of shear modulus with shear strain obtained from cycles 1, 3, 10 and 50. It should be remembered that fifty cycles of constant amplitude cyclic shear stresses were applied in each stage for this test. A unique relationship between modulus and strain, that is independent of the number of cycles, may be noted. Eventhough at higher stress levels, the modulus increased with number of cycles, the corresponding shear strain decreased so as to result in a unique modulus degradation relationship. Fig.5.7b shows the variation of damping with shear strain for 1st, 3rd, 10th and 50th Chapter 5. Results and Discussion 54 1.50 No. of cycles N Figure 5.4: Variation of cumulative volumetric strain with number of cycles N for loose sand Chapter 5. Results and Discussion Figure 5.5: Variation of volumetric strain with shear strain amplitude for loose sand Shear strain Chapter 5. Results and Discussion 56 Figure 5.6: Pore water pressure build up after ten loading cycles for Monterey No.O sand (after Dobry et al 1982) Chapter 5. Results and Discussion 57 Figure 5.7: Variation of (a) shear modulus (b) damping with shear strain amplitude for loose sand Chapter 5. Results and Discussion 58 loading cycles. It can be seen that unlike shear modulus, damping decreases with the number of loading cycles N , for a given shear strain amplitude. Therefore N is a very important parameter for damping than for shear modulus. For the sake of time economy, tests were carried out with three loading cycles in each stage. Further discussion in this thesis are based on dynamic properties evaluated for the third cycles in each stage. As noted in the previous paragraph, this would not influence modulus degradation relationships. Damping factors would however be specific to cycle three only. 5.1.2 Stage versus no stage testing Five tests with identical init ial stress state (cr^ = 100 k P a , R = 2, b = 0 and Dr — 40%) were conducted. In the first test, three cycles starting with r(0) = 2.5 k P a amplitude shear stress were applied, followed by 5.5, 7.5, 12.5, 20.0, 30.0 and 37.5 k P a amplitude shear stress, each with three cycles. In the second test, 3 cycles with r(0) = 7.5 k P a shear stress amplitude only were applied. Similar procedure was followed for tests 3, 4 and 5 with shear stress amplitudes r(0) = 12.5, 20, 30 k P a respectively. The relationship between shear modulus and shear strain from these tests is shown in Fig.5.8a. It should be noted that stage testing technique was used for test 1, while fresh specimens were used for tests 2 to 5 with increasing shear stress amplitude. Existence of a unique modulus versus shear strain curve implies that stage testing does not have any significant effect on shear modulus for the range of shear strain in Fig.5.8a. Fig.5.8b. shows the variation of damping with shear strain. Similar to the unique modulus degradation relationship, a unique damping - shear strain relationship emerges, suggesting that dynamic properties of soils are insensitive to stage testing, provided that the strain amplitude in the previous stage is less than the current one. Chapter 5. Results and Discussion Figure 5.8: Variation of (a) shear modulus (b) damping with shear strain amplitude for loose sand-effect of stage testing Chapter 5. Results and Discussion 60 For all tests discussed in the following sections, stage testing was adopted. 5.1.3 Effect of void ratio Fig.5.9 presents the variation of shear modulus with shear strains at two void ratios. The ini t ia l stress state corresponded to a'm = 100 k P a , R = 2 and b = 0. Similar data at b = 0.5 and 1.0 but with identical cr'm and R are presented in Figs.5.10 and 5.11. It can be observed that as the void ratio decreases, the modulus increases at all b levels. The increase in modulus for a given decrease in void ratio does not seem to depend on the level of b. As the specimen becomes denser, resistance to shear deformation increases, resulting in higher modulus values. Effect of intermediate principal stress on shear modulus wil l be addressed in a later section. In Fig.5.12, shear modulus, normalized by F(e) = (2.17 — e ) 2 / ( l -f e) is shown against shear strain for b = 0, 0.5 and 1. As wi l l be discussed later, shear modulus for b = 1 is less than that for b = 0 and 0.5. For b = 0 and 0.5, normalizing by F(e) tends to eliminate the effects of void ratio for a given value of R and b. This confirms the findings of Hardin (1978), that for round grained sand the void ratio factor F(e) is given by the function (2.17 — e ) 2 / ( l + e). It may be noted that Hardin's finding was limited to the case of b = 0 only. In Fig.5.13, variation of damping values with shear strain for b = 0 and 0.5 are shown. Similar data at b =1 are shown in Fig.5.14. For comparison purpose damping factors for saturated Toyoura sand from Tatsuoka et al (1979) are also shown in Fig.5.13. Initial stress state for their experiment was <r'm — 100 k P a and R = 2 with void ratio 0.636. Damping was evaluated from the tenth shear stress strain loop for each shear strain amplitude and thus could be somewhat lower than computed for N = 3 for Ottawa sand. Comparable values of damping factors for Ottawa and Toyoura sand may be seen in Fig.5.13. However in contrast to the finding of Tatsuoka et Chapter 5. Results and Discussion 61 Figure 5.9: Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b=0 Chapter 5. Results and Discussion 62 Figure 5.10: Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b=0.5 Chapter 5. Results and Discussion Figure 5.11: Variation of shear modulus with shear strain - effect of void ratio shear modulus for b=l Chapter 5. Results and Discussion Figure 5.12: Variation of normalized shear modulus (G/F(e)) with shear strain Shear strain Chapter 5. Results and Discussion Figure 5.13: Variation of damping with shear strain - effect of void ratio on damping for b=0, 0.5 Shear strain Chapter 5. Results and Discussion 66 Figure 5.14: Variation of damping with shear strain for b=l - effect of void ratio on damping Chapter 5. Results and Discussion 67 al - that damping is essentially independent of void ratio - Figs.5.13 and 5.14 show a definite decrease in damping with decrease in void ratio at all b levels. W i t h decrease of void ratio, more inter-grain contacts occur, resulting in stiffer material which dissipate lesser energy during the application of cyclic shear stress and gives lower damping values. 5.1.4 Effect of effective mean normal stress Fig.5.15 shows the variation of shear modulus with shear strain, for effective mean normal stress a'm = 100, 200, 300 and 400 k P a at R = 2 and b = 0. Small differences in void ratios among individual tests were accounted for by normalizing modulus by the void ratio function F(e). Similar data for b = 0.5 and 1 at <j'm = 100 and 300 k P a at R = 2 are shown in Figs.5.16 and 5.17. Increase in shear modulus with effective mean normal stress can be observed from these three figures at all b levels. Shear modulus at several fixed amplitudes of shear strain for b = 0 is shown plotted against a'm in Fig.5.18. The slope of the Une for a given shear strain amplitude in this log-log plot equals the exponent m of the effective mean normal stress i n equation 2.6. The exponent m increases with the amplitude of shear strain. In Fig.5.19, variation of exponent m with shear strain is shown for al l b levels. The solid line in this figure is obtained from Iwasaki et al (1978). It should be noted that Iwasaki et al performed their tests under the init ial stress state of R = 1 and b = 0 as opposed to R = 2 and three different levels of b (0, 0.5 and 1) in this study. Increase of exponent m with shear strain for al l levels of b is evident from this figure with a trend similar to that noted by Iwasaki et al. Therefore a single value of m(=0.5) as suggested by Seed and Idriss (1970) and Shewbridge and Sousa (1991) would clearly under-estimate shear modulus in the large strain range. Variation of damping with shear strain are shown in Figs. 5.20 to 5.22 for b = 0, Chapter 5. Results and Discussion 68 Figure 5.15: Variation of shear modulus with shear strain - effect of <rm on shear modulus for b=0 Chapter 5. Results and Discussion 69 Figure 5.16: Variation of shear modulus with shear strain - effect of a'm on shear modulus for b=0.5 Chapter 5. Results and Discussion 70 Figure 5.17: Variation of shear modulus with shear strain - effect of <r'm on shear modulus for b=l Chapter 5. Results and Discussion Effective mean normal stress Figure 5.18: Variation of shear modulus with effective mean normal stress Chapter 5. Results and Discussion Figure 5.19: Variation of m (exponent of a'm) with shear strain Chapter 5. Results and Discussion 73 0.5 and 1 respectively. Decrease in damping factors with increase in a'm may be noted, and damping factors for b = 1 are much higher than those corresponding to b = 0 and 0.5 at a given strain amplitude. Decrease in void ratio and increase i n a ' m have the same effect on damping, ie; a decrease in damping. A large scatter in the reported damping values in the literature is clearly a consequence of lumping data from many soils with no regard to the effects of u'ml b and void ratio as revealed in this study. 5.1.5 Effect of effective stress ratio To study the effect of effective stress ratio R on shear modulus, tests at R = 2 and 3 were carried out at 40% relative density. For both tests an effective mean normal stress of 100 k P a was applied. Similar to previous tests, during the application of cyclic shear loading a ' m , R and b were held constant. In Fig.5.23a, variation of shear modulus with shear strain is shown for the case b = 0. It may be noted that shear modulus for a given shear strain amplitude, at R = 3 is less than that corresponding to R = 2. The difference between the modulus values for R = 2 and 3 seems to decrease as the shear strain amplitude increases. Variation of damping with shear strain from the same tests is shown in Fig.5.23b. It may be noted that unlike shear modulus, damping is not sensitive to the change i n effective stress ratio. Shear modulus values corresponding to R = 3 are less than those of R = 2, implying that the soil specimen become somewhat weaker as the effective stress ratio increases to 3. However damping seems to be unaffected by this change in the stress ratio. Fig.5.24a shows the variation of shear modulus with shear strain at init ia l R = 2 and 3 at a ' m = 200 k P a under triaxial compression condition (b = 0). Initial relative density for these tests were 40%. These results were obtained from tests performed by holding boundary stresses a z , <jT and ere constant during the application of cyclic Chapter 5. Results and Discussion 74 Figure 5.20: Variation of damping with shear strain - effect of o~'m on damping for b=0 Chapter 5. Results and Discussion 75 Figure 5,21: Variation of damping with shear strain - effect of a'm on damping for b=0.5 Chapter 5. Results and Discussion 76 Figure 5.22: Variation of damping with shear strain - effect of a'm on damping for b=l Chapter 5. Results and Discussion Figure 5.23: Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping factor for 0-^=100 kPa and b=0 Shear strain Chapter 5. Results and Discussion 78 shear stresses (conventional test). It should be noted that R and b are allowed to fluctuate in this test. Variation of damping with shear strain is shown in Fig.5.24b. Results from tests conducted in similar manner at o~'m = 400 k P a and init ial stress ratio 1 and 2 are shown in Figs.5.25a and 5.25b. It may be noted from Fig.5.24a, for <r'm = 200 k P a shear modulus at R = 3 is less than that at R = 2. However, damping does not appear to be influenced by R. This is in agreement with the test results obtained by holding <r'm, R and b constant during the application of cyclic shear stress. It may be observed from Figs 5.25a and 5.25b, that for R = 1 and 2, both shear modulus and damping factors do not depend on the R level. From these results it can be concluded that shear modulus does not depend on the R level i f R is less than three and damping does not depend on R at any level. Tatsuoka et al (1979) reported that for Toyoura sand, shear modulus is independent of R level for R less than 4. They also reported that damping is insensitive to the change in stress ratio. 5.1.6 Effect of intermediate principal stress To study the effect of intermediate principal stress on shear modulus, tests were carried out at intermediate principal stress parameters b = 0, 0.5 and 1. It should be noted that b = 0 and 1 correspond to triaxial compression and extension conditions respectively. Similar to other tests, o~'m, R and b were held constant during the application of cyclic shear stress. In Fig.5.26a variation of shear modulus with shear strain for different b values at a'm — 100 k P a , R = 2 and Dr = 60% is shown. First of al l it can be observed that there is a marked difference between shear modulus values for 6 < 0 and b = 1. However there seem to be no difference in shear modulus for b = 0 and 0.5. Modulus values for b = 1 are less than those for b = 0 and 0.5. Chapter 5. Results and Discussion Figure 5.24: Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping factor for 0-^=200 kPa Shear strain I Chapter 5. Results and Discussion Figure 5.25: Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping for <r'm=400 kPa Shear strain Chapter 5. Results and Discussion Figure 5.26: Variation of (a) shear modulus (b) damping with shear strain - effect of intermediate principal stress on shear modulus and damping for 0-^=100 kPa, R=2, and I>r=60% Shear strain Chapter 5. Results and Discussion 82 Variation of damping factors with shear strain amplitude is shown in Fig.5.26b. Similar to modulus values, damping values for b = 0 and b = 0.5 are not different for a given strain amplitude. Damping factors for b = 1 are much higher than that for b = 0 and 0.5. For b = 0 and 0.5, the major principal stress <r\ acts in the vertical direction while intermediate and minor principal stresses cr'2 and <j'z act i n the two horizontal directions. For b = 1, on the other hand, cr[ and a'2 which are equal act in the horizontal directions while a'3 act in the vertical direction. For b < 1, <T[ rotates about the vertical axis, while for, b = 1, it rotates about the horizontal axis during cyclic shearing. For a vertically deposited material due to cross-anisotropic effects, resistance to deformation due to cyclic rotation of principal stress about the horizontal axis is less than that about the vertical axis (Sayao 1989, Wijewickreme 1990). This is the primary reason for a smaller shear modulus and higher damping for b = 1 than that for b < 1. This also implies that the intermediate principal stress does not have any influence on the dynamic properties of sand except when ini t ia l stress state corresponds to triaxial extension (when a[ and cr2 which are equal act in the horizontal direction and b = 1) as observed from experimental results. In Fig.5.27a and 5.27b, variation of modulus and damping with shear strain at b = 0, 0.5 and 1 with a higher a-'m = 300 k P a and R = 2 at a looser init ial relative density (40%) are shown. These figures too show that modulus values for a given value of shear strain corresponding to b = 1 are much less than those corresponding to b < 1. Therefore it can be concluded that dynamic properties of soils are independent of b except at b = 1, for which the modulus values are lesser than and damping values are higher than those for other b values. Shear modulus for loose sand at Dr = 40%, (7^=100 k P a and R = 3, is shown in Fig.5.28a. Modulus values at a given shear strain are now not influenced by the b value. A t 40% relative density, cr'm = 100 k P a and R = 3 represents a state very close Chapter 5. Results and Discussion Figure 5.27: Variation of (a) shear modulus (b) damping with shear strain - effect of intermediate principal stress on shear modulus and damping for 0-^=300 kPa, R=2, and Z>P=40% Shear strain Chapter 5. Results and Discussion 84 to the failure state. Accordingly, its inherent cross-anisotropic character might have been obliterated. Because of that, the specimen shows no apparent difference in its behavior for rotations of a\ about vertical or horizontal axes. Fig.5.28b shows the variation of damping factor for the same tests. Unlike shear modulus, damping appears to have somewhat scatter in the large strain range. How-ever for shear strains less than 1 0 _ 1 % , there seems to be no dependence of damping on b value. Final ly it can be concluded that dynamic properties of soils are independent of intermediate principal stress <T'2. The only exception being when b = 1, and the effective stress ratio not very close to the failure state. In that event shear modulus values are less and damping factor larger than those for other values of b. 5.1.7 Effect of fluctuations in R and b As described i n chapter 2, i n the conventional cyclic torsion tests (where the boundary stresses <rz,ar, and <7e are held constant during the application of cyclic shear stress), effective stress ratio R and intermediate principal stress parameter b fluctuate during cyclic shear stress application. In this section possible effects of these fluctuations are examined. Tests were performed in the conventional manner and the results are compared with test results on identical specimens in which cr'm, R and b were held constant during the application of cyclic shear stress. Fig.5.29a shows the variation of shear modulus with shear strain from tests con-ducted by both methods described above. Initial state for these tests was D P = 40%, <r'm = 100 k P a , R = 3 and b = 0. Damping factors are shown in Fig.5.29b. As can be observed from these two figures, there seem to be little difference in dynamic properties obtained from conventional tests and tests for which <r'm, R and b were held constant during cyclic shear. Similar conclusions are apparent for init ial state Chapter 5. Results and Discussion Figure 5.28: Variation of (a) shear modulus (b) damping with shear strain - effect of intermediate principal stress on shear modulus and damping for a'm=100 kPa, R=3, and JDp=40% Shear strain Chapter 5. Results and Discussion Figure 5.29: Variation of (a) shear modulus (b) damping with shear strain - effect of fluctuations in R and b for 0^=100 kPa, R=3, and D r=40% Shear strain Chapter 5. Results and Discussion 87 Dr = 40%, (r'm = 400 k P a , R = 2 and b = 0, as illustrated in Figs.5.30a and 5.30b. A reference to Fig.2.3 shows that for R levels of 2 and 3, fluctuations in R at 1 0 _ 1 % shear strain is 0.4 and it is nearly zero at 1 0 - 2 % shear strain. The simultaneous fluctuation in b at 1 0 _ 1 % shear strain for R = 2 is only 0.08 and for R = 3 it is 0.04. At 1 0 - 2 % shear strain fluctuation in b for both R = 2 and 3 reduces to zero. Since for shear strains less than 1 0 _ 1 , fluctuations in R and b for a given init ial stress state are very small, the two methods yield essentially identical dynamic properties. From Figs.5.23 and 5.24, it may be noted that for shear strains greater than 1 0 _ 1 % , difference in shear modulus for R = 2 and 3 seems to become smaller and smaller, and it seems at large strains shear modulus tends to be independent of R level. Also it should be remembered that damping is insensitive to changes in R level (Figs.5.23b, 5.24b, and 5.25b). Also, dynamic properties have been shown to be rather insensitive to changes in b other than when 1. Therefore, it can be said that at large strain level, variation in R and b do not have much effect on the dynamic properties of sand and the test results in Figs.5.29 and 5.30 lend support to this contention. 5.1.8 Effect of stress history The term stress history herein is defined as follows, 1. First a specimen is subjected to cyclic shear loading at a given init ia l stress state o-m, R and b for the purpose of dynamic property evaluation. 2. The stress parameters o~m, R and b are now changed to new values, and dynamic properties are determined again. Since this specimen in state (2) has been subjected to stress history corresponding to state (1), its dynamic properties could be influenced on account of its excursion through stress state (1). Chapter 5. Results and Discussion Figure 5.30: Variation of (a) shear modulus (b) damping with shear strain - effect of fluctuations in R and b for 0^=400 kPa, R=2, and Dr=A0% I U I Shear strain Chapter 5. Results and Discussion 89 Six cyclic torsion tests (a, b, c, d, e and f) on a single specimen with init ial stress states as shown in Fig.5.31 were performed to study the effects of stress history. Results from these tests are compared with those tests which were subjected to no stress history. A l l tests were performed in the conventional manner, ie; by holding boundary stresses constant during cyclic shear. A fresh specimen with DT = 40% was used for cyclic torsion test (a) with init ial stress state a'm = 400 k P a , R = 1 and b = 0. Following the conclusion of test (a), a new stress state marked as (b) in Fig.5.31 was reached and cyclic torsion test repeated. Then for test (c), the stress state was brought back to cr'm — 400 k P a , R = 1 and b = 0, and cyclic stresses were applied. After finishing test (c), <r'm was reduced to 200 k P a and then R was increased to 2 at b = 0 and test (d) was performed. Test (e) was carried out after increasing R from 2 to 3 and finally test (f) was performed after reducing R back to 2 at <r'm = 200 k P a and b = 0. Fig.5.32a and 5.32b show the variation of shear modulus and damping respectively with shear strain from tests (a), (b) and (c). It should be noted that results of test (a) have no stress history effects while tests (b) and (c) have different levels of stress history. However, modulus and damping seem to be unaffected by the history of increasing R from 1 to 2 and then reducing back to 1 at a'm = 400 k P a . As explained in section 5.1.4, i f the maximum value of R during cyclic shear loading is below 3, the level of R has no influence on the dynamic properties. For tests (a) and (b), the highest value of R including fluctuations was less than about 2.5 (see Fig.2.3). Therefore increasing R from 1 to 2 and reducing it back to 1 did not make any difference in the dynamic properties. Figs.5.33a and 5.33b show the variation of shear modulus and damping respec-tively with shear strain from tests (d), (e) and (f). Shear modulus and damping obtained from test conducted on a fresh specimen at <r'm = 200 k P a , R = 2 and b = Chapter 5. Results and Discussion 90 R 200 400 <a.c) «r'm (kPa) Figure 5.31: Stress path diagram showing the initial stress states used for tests (a), (b), (c), (d), (e) and (f) to study the effects of stress history Chapter 5. Results and Discussion Figure 5.32: Variation of (a) shear modulus (b) damping with shear strain - effect stress history at 0-^=400 kPa Shear strain Chapter 5. Results and Discussion Figure 5.33: Variation of (a) shear modulus (b) damping with shear strain - effect of stress history at a'm=200 kPa Shear strain Chapter 5. Results and Discussion 93 0 at DT = 40% are also shown in these figures. Test (d) which has stress state iden-tical to the fresh specimen but has been subjected to a sequential stress history from tests (a), (b) and (c) does not appear to show any effect of stress history on dynamic properties. As explained in the previous paragraph, as long as the highest value of R applied to the specimen during cyclic shear remain below 3, dynamic properties are unaffected by the R level. For test (d), in addition to an R history, cr'm was reduced from 400 k P a to 200 k P a . The test results thus indicate that it is the current <r'm which affects the dynamic properties regardless of the past history of cr'm. Test (e) was conducted after increasing R from 2 to 3 at (r'm — 200 k P a . Similar to the test results reported in section 5.1.4, increasing R from 2 to 3 reduced the shear modulus significantly but the damping was unaffected by the change in R. After test (e), R was reduced back to 2 and test (f) was carried out. From test results in Fig.5.33a, it can be observed that the shear modulus at small strains are identical to the modulus values corresponding to R = 3. However as the shear strain amplitude increases, the modulus - strain curve bounces back to the curve with no stress history. This implies that if there is any effect of stress history due to R , it wi l l only be in the small strain range, and as the strain increases this effect wi l l diminish. Chapter 6 Conclusions In order to study the dynamic properties of Ottawa sand, drained tests were car-ried out in hollow cyhnder torsion device. Effects of shear strain amplitude, number of cycles of loading, stage testing, void ratio, effective mean normal stress dm, effective stress ratio R, intermediate principal stress and stress history on dynamic properties were studied. Intermediate principal stress parameter b = {c^ — <J-i)l{cr\ — «73) was used to study the effect of intermediate principal stress. a ' m , R and b were kept constant during the application of cyclic shear stress. B y keeping two of the above three parameters constant for a number of tests and using a different value of the third parameter for each test, effects of the third parameter on dynamic properties of soils were studied. This technique enabled isolation of the effects of each parameter on dynamic properties. It has been found that shear modulus increases with the number of cycles of a constant amplitude cyclic shear stress when the induced shear strain is higher than a certain threshold value. The damping factor, however decreases markedly with num-ber of cycles at strain amplitudes even less than this threshold value. Due to increase in shear modulus and decrease in shear strain amplitude for a given shear stress am-plitude, shear modulus - shear strain curves (modulus degradation relationships) are virtually independent of the number of loading cycles. However, damping factor for a given shear strain amplitude shows significant decrease with the number of cycles. A threshold value of shear strain of about 1 0 - 2 % has been found below which zero volumetric strain occurs due to cyclic shear loading. This would imply no pore 94 Chapter 6. Conclusions 95 pressure development under the corresponding undrained test. Effects of stage testing and small strain history on dynamic properties of soils are found to be insignificant. It has been shown that the shear modulus increases with decrease in void ratio. When the modulus values were normalized by the void ratio factor shear modulus - shear strain amplitude curves with different void ratios reduced to a single curve even for R values ^ 1. Also it has been found that damping increases with void ratio for a given shear strain amplitude. For all b levels the exponent 711(7) °f the effective mean normal stress, increases from about 0.5 to 0.9 when the shear strain amplitude increases from about 1 0 _ 3 % to 1%. For a given shear strain amplitude, shear moduli obtained at different R levels do not show significant difference when R < 3. Damping factors, however, do not depend on R at all R levels. It has been found that the level of intermediate principal stress does not have any effect on the dynamic properties except for triaxial extension case. 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