PROPERTIES OF SANDS UNDER CYCLIC

Transcription

1 DYNAMIC PROPERTIES OF SANDS UNDER CYCLIC TORSIONAL SHEAR by MUTHUKUMARASAMY UTHAYAKUMAR B.Sc.Eng., University of Peradeniya, Sri Lanka M.Eng., Nagoya University, Japan A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 10, 1992 MUTHUKUMARASAMY UTHAYAKUMAR, 1992

2 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 permission. be allowed without my written 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)

3 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 ASTM C-109 sand. Effects of shear strain amplitude, stress history, effective mean normal stress {(r'm = l/3(a-'1 J ra'2 + (r'3)), principal effective stress ratio (R = 01/03), intermediate principal stress parameter (b = (<r2 03)/(0i 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

4 testing and small strain history on dynamic properties is shown to be insignificant. With 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 initial 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.

5 Table of Contents Abstract ii Acknowledgements xi 1 Introduction 1 2 Literature review Introduction Shear modulus Effect of shear strain amplitude Effect of strain history Effect of void ratio Effect of confining stress or mean normal stress Effect of stress ratio Effect of intermediate principal stress Damping factor Needs for research 24 3 Hollow cylinder torsional device General description Definition of stresses and strains in hollow cylindrical specimens Average stresses Average strains Measurement of strains 32 iv

6 3.4 Measurement of surface tractions Stress path control and data acquisition system Stress path control 35 4 Testing procedure Specimen preparation Reconstitution of sand specimen Preliminary preparation steps Specimen preparation steps Test preparation steps Material tested Experimental program 44 5 Results and Discussion Introduction Effect of number of cycles Stage versus no stage testing Effect of void ratio Effect of effective mean normal stress Effect of effective stress ratio Effect of intermediate principal stress Effect of fluctuations in R and b Effect of stress history 87 6 Conclusions 94 Bibliography 96 v

7 List of Tables Stress states of soil specimens before the application of cyclic shear stresses VI

8 List of Figures 2.1 Definition of shear modulus and damping (a) Stresses acting on an element in a hollow cylindrical specimen during cyclic shear loading and (b) Mohr's circle corresponding to that stress state Variation of (a) effective stress ratio R and (b) intermediate principal stress parameter b with shear strain amplitude during cyclic shear loading Variation of shear modulus with shear strain (after Guzman et al 1989) Variation of shear modulus with effective mean normal stress (after Iwasaki et al 1978) Variation of m with shear strain (after Iwasaki et al 1978) Variation of damping with shear strain (after Tatsuoka et al 1978) Schematic diagram of the HCT device (after Sayao 1989) Details of the HCT device (after Sayao 1989) Polished end platen with radial ribs (after Sayao 1989) Surface tractions and stress state in an element in the hollow cylindrical specimen (after Wijewickreme 1990) Schematic diagram of the data acquisition system (after Wijewickreme 1990) Specimen preparation by water pluviation (after Wijewickreme 1989) Levelling the specimen's upper surface (after Sayao 1989) 41 vii

9 4.3 Particle size distribution of medium Ottawa and(after Wijewikreme 1990) Typical shear stress - shear strain response for loose sand Variation of (a) shear modulus (b) shear strain with number of cycles for loose sand Variation of damping with number of cycles for loose sand Variation of cumulative volumetric strain with number of cycles N for loose sand Variation of volumetric strain with shear strain amplitude for loose sand Pore water pressure build up after ten loading cycles for Monterey No.O sand (after Dobry et al 1982) Variation of (a) shear modulus (b) damping with shear strain amplitude for loose sand Variation of (a) shear modulus (b) damping with shear strain amplitude for loose sand-effect of stage testing Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b= Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b= Variation of shear modulus with shear strain - effect of void ratio on shear modulus for b=l Variation of normalized shear modulus (G/F(e)) with shear strain Variation of damping with shear strain - effect of void ratio on damping for b=0, Variation of damping with shear strain - effect of void ratio on damping forb=l 66 viii

10 5.15 Variation of shear modulus with shear strain - effect of a'm on shear modulus for b= Variation of shear modulus with shear strain - effect of <r'm on shear modulus for b= Variation of shear modulus with shear strain - effect of <7'm on shear modulus for b=l Variation of shear modulus with effective mean normal stress Variation of m (exponent of (T'm) with shear strain Variation of damping with shear strain - effect of a'm on damping for b= Variation of damping with shear strain - effect of on damping for b= Variation of damping with shear strain - effect of <r'm on damping for b=l 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= Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping factor for 0^=200 kpa Variation of (a) shear modulus (b) damping with shear strain - effect of R on shear modulus and damping for 0^=400 kpa Variation of (a) shear modulus (b) damping with shear strain - effect of intermediate principal stress on shear modulus and damping for <=100 kpa, R=2, and I>r=60% ; 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 Dr=40% 83 ix

11 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 kpa, R=3, and Dr=40% Variation of (a) shear modulus (b) damping with shear strain - effect of fluctuations in R and b for 0-^=100 kpa, R=3, and Z>r=40% Variation of (a) shear modulus (b) damping with shear strain - effect of fluctuations in R and b for 0-^=400 kpa, R=2, and DP=40% 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 Variation of (a) shear modulus (b) damping with shear strain - effect of stress history at «7^=400 kpa Variation of (a) shear modulus (b) damping with shear strain - effect of stress history at a^=200 kpa 92 x

12 A ckno wledgement s I am deeply indebted to Dr. Y.P.Vaid 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 Mr. Art Brookes and Mr. 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, Raju, Ralph and Huaren is gratefully acknowledged. Finally I wish to express my thanks for the financial support provided in the form of research assistantship by the department of Civil Engineering, University of British Columbia. XI

13 Chapter 1 Introduction In recent years much progress has been made in the development of analytical methods to evaluate the response of soil structures due to earthquake and wave loadings. 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 in 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

14 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 + <t3)/3, as the sole representative of the initial stress state. Later researchers have identified that the initial 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, Yu 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 initial 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 initial stress state was axisymmetric {(T2 = <r'3 or a^) and cyclic shear stress was applied holding boundary stresses constant. This imposes cyclic fluctuations in R and b about their initial 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 initial stress

15 Chapter J. Introduction 3 state and dynamic stresses to which in-situ soil elements are subjected due to earthquake 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 initial stress state in this test is also axisymmetric and shear deformation occurs under plane strain conditions. It is therefore not possible in this test to simulate a generalized initial stress state nor does it enable isolation of the effects of initial 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 in 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 in 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 Hz, depending on the type of material and initial 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

16 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 stressstrain 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, initial 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 UBC hollow cylinder torsion device were carried out on medium Ottawa ASTM C-109 sand in both loose and medium dense states. Soil specimen were consolidated under the specified initial 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 initial 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 initial 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 UBC hollow cylinder torsion

17 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.

18 Chapter 2 Literature review 2.1 Introduction This chapter reviews the findings of previous researchers on the dynamic properties (shear modulus and damping) of sands. Since soils have non linear stress-strain relationship as shown in Fig 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 OA of shear loading and the loop is defined by ABCDA as shown in Fig.2.1. By 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, Wd = 2irXkX 2 (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.= kx 2 (2.2) 2 From Eqs.2.1 and 2.2, A can be expressed as A = Wd (2.3) 6

19 Figure 2.1: Definition of shear modulus and damping

20 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

21 Chapter 2. Literature review 9 (a) Figure 2.2: (a) Stresses acting on an element in a hollow cylindrical specimen during cyclic shear loading and (b) Mohr's circle corresponding to that stress state.

22 Chapter 2. Literature review Figure 2.3: Variation of

23 Chapter 2. Literature review 11 increases with decrease in initial R level. Very small change in R occurs for shear strains less than 10~ %, the domain generally investigated for low strain modulus. 2 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 initial undefined value to 0.5 as soon as is applied. At 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 initial R level for a given shear strain amplitude. The typical fluctuations in R and b illustrated in Figs.2.3a and 2.3b for a specific a'm and Dr will also apply at other levels of (T'm and Dr, if identical modulus degradation curves are assumed. 2.2 Shear modulus 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 10 _ 2 %, modulus decreases rapidly, and becomes as low as 20% of the original value at 0.1% shear strain at all levels of initial confining stress 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:

24 Chapter 2. Literature review "«10 _ a 10 " a 10.-» 1 10 SHEAR STRAIN, Figure 2.4: Variation of shear modulus with shear strain (after Guzman et al 1989)

25 Chapter 2. Literature review 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 8xl0 % to 0.35%. From the axial stress-strain - 3 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~ % to 0.5%. From these studies they found 2 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.

26 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 will 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 initial values on measured shear modulus and damping cannot be assessed. To evaluate the effects of cyclic shear strain history on dynamic properties of Toyoura 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 5xl0 % to 0.3%. In group one, stage _ 3 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 amplitude cyclic shear stress were applied, followed by decreasing stress amplitudes in subsequent stages. By 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~ % results in an increase in shear modulus of about 40% over that obtained from 2 a virgin sand specimen. They obtained their results from resonant column tests on

27 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 if dry sand test specimens are used, as has been the case in most investigations 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 Hi chart (1963). Hardin and Ri 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 10 %), shear modulus was evaluated. When metric system units are used, Hardin recommended A = 700 and void _4 ratio factor,

28 Chapter 2. Literature review 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 (10 % to 1%), drained tests on saturated Toyoura sand were performed by Iwasaki _ 4 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 10 % to 10~ %. Then using hollow cj'linder torsion device, specimens _ 4 2 with dimensions identical to that used in the resonant column device, dynamic properties were evaluated for shear strains from 10" % to 1%. They confirmed the validity 2 of the void ratio function F(e) given by equation 2.7, for a large range of strains from 10 % to 1%. However, it is not known whether the normalizing void ratio function _ 4 F(e) also applies under initially nonhydrostatic stress state Effect of confining stress or mean normal stress As can be seen from Fig.2.4, increase in effective mean normal stress results in 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.

29 Chapter 2. Literature review Mean Principal Stress. Figure 2.5: Variation of shear modulus with effective mean normal stress Iwasaki et al 1978) (after

30 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~ % or less to 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 10 % to 1.0 at shear strain 6xl0 % 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, if 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 initial 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 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

31 Chapter 2. Literature review

32 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 initial 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. Yu 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. Yu and Richart used resonant column tests on solid cylindrical specimens. During the shearing 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, Yu 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 = AF(e)(P B) w (^p ) M (l - 0.3iC 5 ) (2-10) where o~'a and o~'p are the normal effective stresses in the directions of wave propagation and particle motion respectively. This equation implies that a maximum of 30% reduction in shear modulus will occur when the effective stress ratio is at the maximum value.

33 Chapter 2. Literature review Effect of intermediate principal stress It was reported that the effect of intermediate principal stress is relatively insignificant by Yu and Richart (1984). However, they did find difference in shear modulus between triaxial compression and extension cases. It was suggested that the difference in 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 Yu 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 in 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 Ni (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 him to simulate true triaxial stress states on the soil specimen. For the shear strain range from 10-4 % to 0.1% Ni 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.

34 Chapter 2. Literature review 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 initial shear stress level particularly at the lower strain amplitudes. However, Tatsuoka et al (1979) found that the effects of initial 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.

35 Chapter 2. Literature review Single Amplitude Shear Strain Figure 2.7: Variation of damping with shear strain (after Tatsuoka et al 1978)

36 Chapter 2. Literature review 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.

37 Chapter 3 Hollow cylinder torsional device The UBC 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 UBC 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 dimensions 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 mm thick rubber membranes. A double-acting air piston mounted at the bottom of the supporting table is used 25

38 Chapter 3. Hollow cylinder torsional device 26 LVDT ( AH) POSmONING BOLT TOP CROSS BEAM r.rr c ii -r * - CHAMBER TOP.TOP CAP TOP PLATEN PLEXIGLASS CELL LOADING FRAME SOIL SPECIMEN RIGID ROD BASE PLATEN BASE PEDESTAL EXTERNAL PRESSURE PORE PRESSURE SUPPORTING TABLE " UNEAR-ROTARY BEARING TORQUE CABLES PRESSURE TRANSDUCER INTERNAL PRESSURE LVDT ( 8 ) LOADING SHAFT TORQUE CELL CENTRAL PULLEY TORQUE PULLEY - TORQUE PISTONS (TWO PAIRS) THRUST BEARING AXIAL LOAD PISTON AXIAL LOAD CELL 0 20 SCALE (cm) Figure 3.1: Schematic diagram of the HCT device (after Sayao 1989)

39 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 HCT device (after Sayao 1989)

40 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 mm 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 interconnected to a common regulated pressure supply. This configuration is necessary to apply torque in 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 mm deep), as shown in Fig.3.3. Drainage from the specimen is achieved by six 12.8 mm 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 (Pe,P,). Fig 3.4 shows these tractions together with the stress state in an element in the wall. The four stress components a2,ar,(tg and rz8 induce the four strain components ez,et,ee, and ezg in the soil element.

41 Chapter 3. Hollow cylinder torsional device Figure 3.3: Polished end platen with radial ribs (after Sayao 1989)

42 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)

43 Chapter 3. Hollow cylinder torsional device 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, Average strains

44 Chapter 3. Hollow cyhnder torsional device 32 - ARj r = 5 ^ (3.6) ARe + ARj 66 = rttrt ( 3 J ) 2A9{Rl - jg) ~ZH{Rj W) " :r ( 8 ) where H, AH 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 AH and volume change of the inner chamber. The external radius change ARe can then be computed from the measured values of AH, 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 LVDT) 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. Both LVDT's can detect movements in the order of 10 mm. This - 3 results in a resolution of about 5xl0 % in both _ 4 ez and jzg. Two differential pressure transducers (DPT) 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, if any, as proposed by Vaid and Negussey (1984). The DPT used for measuring sample volume change can detect volume changes in the order of 3 mm 3, resulting in a resolution of about 5x 10-4 % in

45 Chapter 3. Hollow cylinder torsional device 33 volumetric strain e after correcting for membrane penetration effects. The DPT for the inner chamber can detect volume changes in the order of 10 mm 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 kpa. Vertical load is measured using a load cell placed outside the cell chamber. Vertical stress as low as 0.2 kpa 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 Nm of torque can be measured using this transducer. This corresponds to a resolution of shear stress rzg in the order of 0.1 kpa. 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. All transducers are exited with a stable D.C. power supply of 6 V. The LVDT'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 mv/v. Signal outputs of these transducers are scaled to 2 V full scale using variable gain amplifiers.

46 Chapter 3. Hollow cylinder torsional device 34 PC HCT DEVICE <î> DATA ACQUISITION SYSTEM -on 1 Q STEPPER MOTORS I I TRANSDUCERS Figure 3.5: Schematic diagram of the data acquisition system (after Wijewickreme

47 Chapter 3. Hollow cylinder torsional device 35 Scanning is triggered simultaneously on all channels at specified instances depending 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. value is then digitized by the A/D converter and temporarily stored. This average The digital output from the A/D conversion yields a decimal number ranging from 1 to 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 mv. At 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 kpa, thus enabling pressure changes of as low as 0.1 kpa. 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 in 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 adjusted until the applied stresses are within a specified tolerance from the target value.

48 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.