DEFQRMAHON CEfiRACTERESTICS 0F SAND SUBEECTEE‘: TQ ARESGTRGPEC CYSLEC, BYfiAMEC LOASENG THESES FQR THE DEGREE G? Ph. D. MEWS-AN STATE. UNEVERSRW DAWB HARGLD TEMMERMAN 1 9 6 9 {”2315 LIBRARY 1 Michigan State University This is to certify that the thesis entitled DEFORMATION CHARACTERISTICS OF SAND SUBJECTED TO ANISOTROPIC CYCLIC, DYNAMIC LOADING presented by David H. Timmerman has been accepted towards fulfillment of the requirements for Ph.D. degree inCivil Engineering , . /"’"” " ‘_ :3 . Major professor J; Irv’bk CU Date [IE/KG" "101 I (I éi 0-169 ABSTRACT DEFORMATION CHARACTERISTICS OF SAND SUBJECTED TO ANISOTROPIC CYCLIC, DYNAMIC LOADING BY David H. Timmerman The magnitude of irrecoverable foundation settle- ment that can be expected to occur progressively with each cycle of a vibratory loading cannot be adequately determined by existing theories in most cases. This study was undertaken, therefore, in an attempt to develop a better qualitative understanding of dynamic soil deformation characteristics and to determine basic relationships which could be used to express the soil deformation in a quantitative manner. In order to obtain stress-deformation relation- ships in the laboratory which could be used for settle- ment computations, it was necessary to reproduce the in situ stress conditions in the experiments. This was accomplished with a modified triaxial cell and a specially designed loading mechanism to apply cyclic loadings to the specimens. With this apparatus, specimens of air-dried Ottawa sand were first loaded statically to a specified anisotropic stress state to represent the David H. Timmerman in situ static stress state existing under a foundation. A sinusoidal loading was then superimposed upon the existing static stress state in one of two ways. Either a cyclic axial stress or a cyclic confining pressure was applied. The frequency of the cyclic loadings varied from 0.1 to 25.0 cps. The applied loadings and the resulting axial and radial deformations were measured electronically. The variables studied during the experimental investigation were the static principal stress ratio, RS, i.e., the ratio of axial to radial stress, the maximum stress ratio, Rm, to which the specimen was subjected during the cyclic loading, the frequency of the cyclic loading, f, and the initial void ratio of the sand, e0. The axial strain resulting from 10,000 cycles of loading was considered as the dynamic strain under the given stress condition. For a sand with a given initial void ratio, e this dynamic strain was correlated with O, a stress factor F=Rm3 RDR, where RD=(Rm-RS)/Rm. This dynamic strain was not unique for a given stress state, however. It was found to have two distinct possible magnitudes; a higher strain level and a lower strain level. The frequency of having a higher strain level increased as the void ratio, e0, and the stress factor, F, increased. I[[[ls [[I[{£[( Ill IIIII‘ (Ill . l David H. Timmerman Two significant stress levels, Fc’ and Ff, were noted. The dynamic strains for loadings where F was smaller than FC, were small. When F was greater than Fe, the resulting dynamic strains were proportionately much larger. Ff denotes the stress level at which the sand specimen would fail under the dynamic loading. For loadings with F less than Ff, the resulting dynamic strains were of limited, well-defined magnitude. For loadings with F greater than Ff, the dynamic strains were much larger and of undefined magnitude. Ff was found to be a function of the magnitude of the cyclic stress as indicated by RD. The dynamic strains were found to be independent of the frequency of the load applications for frequencies between 2.5 and 25 cps; however, the strains were larger. for slow repetitive loadings at 0.1 cps. The volumetric strains were measured during the experimental investigation and were found to be a func- tion of Em and RS. For tests with RS below a certain value, RV, the soil densified during the cyclic loading with the amount of densification increasing as Rm in- creased. For tests with R8 greater than RV, the soil expanded during the cyclic loading. The expansion increased with increasing Rm. DEFORMATION CHARACTERISTICS OF SAND SUBJECTED TO ANISOTROPIC CYCLIC, DYNAMIC LOADING BY AA David H? Timmerman A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil Engineering 1969 “I .[rllff‘gi llIIlllllll |lllltl.lllifll ‘\ O-‘ ‘n ' t . .V.\ ‘\ \X\ If, ACKNOWLEDGEMENTS The author is grateful to his advisor, Dr. T.H. wu, for the time, support, encouragement, and suggestions which he provided; to the Civil Engineering Department of Ohio State University for providing laboratory space and equipment necessary for the conduct of the investi- gation; to the men of the Civil Engineering machine shop for their invaluable assistance in designing and fabricating much of the equipment for the project; and to Michigan State University for the NDEA Fellowship which enabled the author to continue his education. Special thanks go to his wife, Fay, for her en- couragement and assistance in preparing this manuscript. ii TABLE OF CONTENTS CHAPTER I. INTRODUCTION . . . . . . . . . . . General Stress—Deformation Considerations. Stress State Beneath Loaded Footing. . . Review of Literature. . . . . . . . II. EXPERIMENTAL INVESTIGATION . . . . . . Equipment and Instrumentation. . . . . 8011 O O O O O O O O O O O O 0 Testing Procedures . . . . . . . . Test series Test series Test series Test series Test series Test series "lilt-‘f-ZIUOUEHII> III. RESULTS OF EXPERIMENTAL INVESTIGATION. . . Axial Strain. . . . . . . . . . . Typical results Dynamic-vs-static axial strain Axial strain-vs-stress state Axial strain—vs-void ratio Stress path effects Strain Level Probabilities Frequency effects Failure Conditions iii IO lO 17 19 28 28 CHAPTER Volumetric Strain . . . . . . . . Contribution of Shearing and Volumetric Deformations to Axial Strain. . . . Nonsymmetrical Cyclic Stress Effects. . Static Prestress Effects. . . . . . Cumulative Axial Strain . . . . . . Particle Size and Gradation Effects . . IV. SUMMARY . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . iv PAGE 90 Table I. II. III. IV. VI. VII. VIII. Summary of data Summary of data Summary of data Summary of data axial pulse . Summary of data Summary of data medium sand . Summary of data Summary of data LIST OF TABLES for tests with loose sand . for tests with medium sand. for tests with dense sand . from tests with a non-symmetrical for prestressed medium sand . from cumulative loading tests on for fine Ottawa sand. . . . for well-graded brown sand. Page 101 102 103 10% 104 105 106 106 Figure H \OCDVOV‘l-F'UUN F4 +4 F4 n: +4 C) 13. 1h. 15. 16. 17. 18. LIST OF FIGURES Stress state beneath loaded footing. . . . . . . . . Diagram of modified triaxial cell. . . . . . . . . . Diagram of pulsating pressure cell . . . . . . . . . Diagram of radial strain measuring device. . . . . . Grain size distribution curves . . . . . . . . . . . Loading sequence . . . . . . . . . . . . . . . . . . Stress-strain curve for loading sequence . . . . . . Mohr's circles for cyclic axial loading. . . . . . . Mohr's circles for cyclic cell pressure. . . . . . . Stress-strain curve for prestress loading sequence . Stress-strain curve for cumulative loading sequence. Typical curves of axial strain-vs-number of stress eyeleSo O O O O O O O O O O O O O O O O O O O O O Idealized axial strain-vs-number of stress cycles. . Typical curves of axial strain squared-vs-number of stress cycles. . . . . . . . . . . . . . . . . . . Static-vs-dynamic stress-strain data for loose sand. Static-vs-dynamic stress-strain data for medium sand Static-vs-dynamic stress-strain data for dense sand. Dynamic axial strain-vs-stress state . . . . . . . . vi Page . 1h Figure Page 19. Frequency of high strain level-vs-F . . . . . . . . . .A5 20. Frequency of high strain level-vs-F'. . . . . . . . . .h7 21. Time ofcmcurrence of strain path transitions. . . . . .48 22. Number of strain path transitions-vs-number of cycles .99 23. Dynamic strain rate as function of frequency. . . . . .53 2%. Rm-vs-RD at failure . . . . . . . . . . . . . . . . . .56 25. Stress-strain curves from static triaxial tests . . . .58 26. Minimum RD for a given F for which failure does not occur 0 O O O O O O O O O O O O O O O O O O O O O O .60 27. Static-vs-dynamic volume strain for loose sand. . . . .61 28. Static-vs-dynamic volume strain for medium sand . . . .62 29. Static-vs-dynamic volume strain for dense sand. . . . .63 30. Dynamic volume strains for loose sand . . . . . . . . .66 31. Dynamic volume strains for medium sand. . . . . . . . .67 32. Dynamic volume strains for dense sand . . . . . . . . .68 33. Axial strain: shear-vs-volume deformation. . . . . . .71 3%. Axial strain: shear-vs-volume deformations . . . . . .72 35. Axial strain: shear-vs-volume deformation. . . . . . .73 36. Axial-vs-volume strain. . . . . . . . . . . . . . . . .7h 37.. Axial strain for symmetrical-vs-nonsymmetrical cyclic loadings. O O O O O O O O O I O O O O O O .. O O O I 077 38. Influence of ACE) on dynamic strain. . . . . . . . . . .78 39. Dynamic strain-vs—stress state for prestressed sand . .80 #0. Dynamic strain-vs-stress state for cumulative loading teStSo O O O O O O O O O O O O O O 0 O O O O O O O .83 vii Figure Page 41. Apparent volume change-vs-temperature change . . .97 A2. Calculated-vs-measured volume changes. . . . . . .99 viii LIST OF APPENDICES APPENDIX PAGE A. Volumetric Strain Measurements . . . . . . . 95 B. Summary of Test Data . . . . . . . . . . 100 ix CHAPTER I INTRODUCTION In recent years, considerable attention has been given to the field of soil dynamics and, in particular, to the problems relating to the soil-foundation response character- istics for various types of dynamic loadings. These dynamic loadings may be impulse loadings such as those caused by explosions, or vibratory loadings such as those produced by earthquakes, reciprocating engines, etc. In particular, it is often important to determine the amount of irrecoverable settlement that can be expected to occur progressively with each cycle of vibratory loading. Since the magnitude of such settlements resulting from vibratory loading cannot be adequately determined by existing theories in most cases, this study was undertaken in an attempt to develop a better qualitative understanding of dynamic soil deformation characteristics and to determine basic relationships which can be used to express the soil deformation behavior in a quantitative manner. With such information, it should be possible to analyze the general problem of foundation settlements under vibratory loads. 2 1.1 General Stress-Deformation Considerations In order to determine the amount of settlement that would occur under a foundation subjected to a vibratory loading, it is first necessary to determine the static state of stress existing under the foundation plus the superimposed dynamic stress state due to the dynamic loading. The static stress state can be estimated from existing stress distribution theories. Obtaining the dynamic stress state is another problem. However, as a result of research in this area, a reasonable estimate of the dynamic stress state can be made. Assuming that the complete stress state existing under the foundation can be determined to a satisfactory degree of accuracy, knowledge of the dynamic stress- deformation relationships of the soil is required to estimate the settlement resulting from the dynamic loads. Settlement under cyclic loading may be due to a combination of two factors. First, foundation settle- ment can result from densification of the soil in a zone under the foundation. Second, settlement can occur due to shearing deformations within the supporting soil mass. In addition to general shear deformations, shear surfaces could be developed or sections of the soil mass could be set in motion along developed shear surfaces. 3 Shear zones or zones of weakness could also be developed as a result of the shear and compression waves being trans- mitted through the supporting soil media from the vibrating foundation. Since these two types of waves to not propagate at the same velocity through the soil, there would be loca- tions (at periodic distances from the foundation) where the two waves would be out of phase, i.e., the shearing stress (from the shear wave) would be at a maximum when the com- pressive stress (from the compression wave) would be at a minimum. If one or more of these locations falls within the zone of significant pressure under the foundation, the prin- cipal stress ratio at that location could lead to a local shear failure. As a simplification, the foundation settlement can also be considered as the deformation of a block of soil that supports the foundation. (See Figure 1.) This deformation would include both volumetric strains (densification) and shear strains within the block. This latter approach to the analysis of foundation settlement can be more easily compared with laboratory stress- deformation data, and is therefore taken in this study. 1.2 Stress State Beneath Loaded Footing In order to obtain stress-deformation relationships in the laboratory which can be used for settlement computations, it is necessary to reproduce the in situ stress conditions in the experiments. p //////////A | l l supporting I I soil I 1 block I I l | l I | L_ _____ __l Fig. 1. "Stress state beneath loaded footing 5 The general stress state beneath a footing subjected to a vertical static and cyclic dynamic loading is shown in Figure l. The vertical static load exerted by the footing is shown as P, and the cyclic load pulsating at a frequency, f, is shown as izfiP. At a distance, 2, below the footing, an element of soil is subjected to a static stress plus a superimposed dynamic stress. The static stress state con- sists of a vertical stress,CR’, a lateral stress,C§‘, and a shear stress,’T . These stresses are a combination of the at rest stress state of the soil plus the stress state from the static load on the foundation. The dynamic stress state consists of the cyclic vertical stress,:l:Ao;’, the cyclic lateral stress,;tAO'H , and the cyclic shear stress,:tA’r . These stresses are the ones induced by the foundation under the action of the cyclic load, :leP . By restricting the investigation to the stress state existing under the center line of a vertically loaded cir- cular footing, the stresses on the soil element shown in Figure 1 reduce to a radially symmetric state of stress with.’r andlflfl'being zero. Since this stress condition can be reproduced in a triaxial cell, the desired streSs-deforma- tion characteristics can be determined by this means in the laboratory. 1.3 Review of Literature There has been considerable research performed in the area of soil dynamics. Much of this research has been con- cerned with general soil-foundation response characteristics 6 such as the work done by Alpan (l)? Barkan (2), Lysmer (25) (26), Richart (30), Richart and Whitman (32), and Sung (39), and the stress distribution in the soil mass under dynamic loading such as the works of Bernhard (5) (7) (8) and Richart et.a1 (31). Also having received considerable attention are the effects of dynamic loadings on soil properties. This line of research includes the investigations of dynamic shear strength and failure conditions by Johnson and Yoder (21), Linger (23), Mogami (27), Seed et.al (33) (35) (36) (37) (38), and Whitman and Healy (M6), and the investigations of damp- ing properties or energy absorption of the soil subjected to low amplitude vibrations by Hall and Richart (17), Hardin (18), Hardin and Black (19), and Weissmann and Hart (45). Some information relating to stress-deformation characteristics has come from vibratory compaction studies. Early investigations along this line, such as those done by Bernhard (H) and Converse (11), were concerned primarily with determining what vibratory loading conditions at the soil surface would give maximum soil compaction. Although these works have shown the general effect of such variables as the magnitude of the applied loads, the area being loaded, *Numbers in parentheses refer to the reference numbers of specific works listed in the bibliography. 7 and the cyclic frequency of the load repetitions on soil densification or deformation, they have not defined clearly the stress-deformation relationships of the soil. Later vibratory compaction research by Gomes and Graves (16) indicated that the maximum compaction or den- sification occured at a given distance below the vibrating surface load with the relative density decreasing upward and downward from this location. In comparing this relative density vs. depth with the theoretical shear stress vs. depth, there appears to be a possible correlation between the magnitude of the shear stresses induced in the soil mass and the resulting relative density or deformation. Since foundation settlement is related to changes in void ratio or porosity of the supporting soil, studies con- cerning the effects of dynamic loading upon the soil's void ratio are useful for studying foundation settlements. Barkan (2), Watanabe (4%), and others have attempted to relate the acceleration of the imposed vibratory loading to the result— ing void ratio attained by the soil. Each of these investi- gations has obtained a correlation between imposed accelera- tion and resulting void ratio. However, the various correla- tions are not in good agreement. Bazant and Dvorak (3) suggested that besides accelera- tion, the frequency of the cyclic loading and the height and volume of the vibrating soil mass should also be considered as important factors in compaction. 8 As a result of the apparent discrepancies in the results of this type of research, the question was raised and invest- igated by Luscher (24) as to whether acceleration or some other variable such as stress should be considered the primary controlling variable. From this investigation (currently in progress) it appears that stress is probably the more impor- tant factor of the two, at least for accelerations less than 1 g. In addition to laboratory research concerning soil be- havior under cyclic loadings, there have been field and model investigations which measured the actual settlement of found- ations subjected to dynamic loadings. Murphy (28) and Okamoto (29) measured the settlements of small, statically- loaded plates resting on sand when the entire system (soil and loaded plate) was subjected to a horizontal vibratory motion on a shaking table. Both investigators correlated the plate settlement with the acceleration of the shaking table. Settlements have also been recorded for larger founda- tions. The Corps of Engineers (12) measured the settlements of 5 to 10 ft. diameter bases subjected to vertical static and cyclic loads. All of the previously cited research contributes general information to the foundation settlement problem. However, the only studies concerning the irrecoverable de- formation that can be expected to occur progressively with each cycle of vibratory loading are those by Luscher et.a1. 9 (2H), and Seed et.al. (36) (37) (38). Luscher's studies have not yet been completed and Seed's studies are limited to the condition of undrained loading of saturated sands. CHAPTER II EXPERIMENTAL INVESTIGATION 2.1 Equipment and Instrumentation The most suitable type of test in general which allows the stresses to be controlled to the desired degree is the triaxial test. Therefore, it was decided to modify a standard triaxial cell so that the dynamic loading could also be applied. In order to have enough flexibility to study various aspects of the stress- deformation problem, a hydraulic loading system was devised so that the axial load and the cell pressure on the soil sample could be pulsed independently. The phase angle between the axial load and cell pressure could also be varied. This loading system is shown 9 schematically in Figures 2 and 3. A Geonor triaxial cell for 1.h” diameter samples was used with additional open- ings tapped into the bottom to allow for a connecting line from the pulsating pressure cell, for electric outlets from the force transducer, LVDT, and thermo- couple, and for the insertion of a pressure transducer. 10 To constant is pressure cell \ 11 To pulsating / pressure cell Oi "-Fluid L’Steel ball l l; a To recorder e LVDT (Fluid) l C Rubber Radial membm strain~ measuring device LVDT Thermocouple Soil fi J1 j '**t__ cel Triaxial _Force transducer _ b-‘ I I l W To constant To volume pressure C9“ change measuring device \ \To p —-=> A —-D a “if ulsating pressure cell To recorders Fig.2.-- Diagram of modified triaxial cell 12 fAdjustable pivot point 1M— [ ‘JJ-I fl 0 U Cam 0 o ...-. :- o o 1: 1:1 o u... _ O 0 ~: :3" o o 2; y: (pew-Compression spring 0 fEL—Z: ° ° 5 luid)? 0 Ct .’ -- ‘L- Q ’ \ \ To triaxial cell / Fig. 3. -—Diagram of pulsating pressure cell 13 The applied cell pressure was measured by means of a Dynisco Pressure Transducer (Model No. TCAPT25-lC) inserted into the base of the triaxial cell. A Dynisco Force Transducer (Model No. FT2-2C) was connected to the loading head (Figure 2) to measure the axial load on the sample. The force transducer was put inside the triaxial cell to measure the axial load at the top of the sample so that any friction between the piston and the bushing would not affect the load readings. The lateral deformation of the sample was measured by means of the device shown in Figure A. A Sanborn Linearsyn Differential Transformer (Model No. 595DT-100) was used. The axial deformation was measured with a Daytronic LVDT (Model No. 1030-200) mounted on the top of the cell as shown in Figure 2. The Electronic com- ponents and leads inside the triaxial cell were coated with silicon rubber for waterproofing. The transducers that measure the axial load and the cell pressure were connected to a Tektronix Dual- Beam Oscilloscope (Type RM-565) equipped with two Tektnx- nix Carrier Amplifiers (Type 3066). This allowed the magnitude and frequency of the loads, and the phase angle between the two loads to be monitored continually. Any variations in the input loading could be noted and the necessary adjustments of the loading system could be made. 11+ Compression "ZZZ-.3 Tension spring Pivot Fig.4.-- Diagram of radial strain measuring device 15 The electronic devices used to measure the sample deformations were connected to a Brush, H-channel, 6K0 Carrier Amplifier (Model 13-5493-00) with the output signals being recorded on a Brush Light Beam Oscillograph (Model 16-2308-00). The temperature inside the soil sample was measured with an iron-constantan thermocouple and recorded on a Varian Graphic Recorder (Model G-IH). The pulsating pressure for the triaxial cell pressure and axial load was supplied by two pulsating pressure cells as shown schematically in Figure 3. Each of these consisted of a piston and cylinder which were linked mechanically with a rotating cam. The pivot point of the mechanical linkage could be shifted to provide different pulse magnitudes. The cams for both cells were connected to a common drive axle by means of electric, magnetic clutches. This enabled the relative phase of the two pulses to be varied. Either one could be dis- engaged if only one of the loads were to be pulsed. The common drive axle was driven by a % H.P., D.C. motor with a variable frequency control. The loading system described produced a sinusoidal cyclic stress with a usable cyclic frequency range from approximately 2 to 30 cps; however, due to the dynamics of the system, the best results could be obtained at approximately 10 cps. The theoretical natural frequency of the soil sample and its loading system was estimated and found to be much 16 higher than the frequencies to be used. This was also confirmed from the force-displacement-frequency relation- ships observed during the testing program. For a given soil sample and a given pulsating axial force amplitude, the resulting axial strain amplitude of the sample was independent of the pulse frequency for the frequency range of 2-30 cps used in these tests. This is what would be expected if the imposed forcing function has a frequency much lower than the natural frequency of the system. Since air-dry sand was used in the tests, the sample volume change was determined from the amount of air enter- ing or leaving the sample. This was accomplished by means of a device described by Bishop and Henkel (10) with which the air pressure is maintained at atmospheric so that the compressibility of the air need not be con- sidered. This device did not permit measuring the volume change occuring during a given cycle of loading, but it did give the average volume change of the sample over a given period of time. The sample volume changes could also be calculated from the measured lateral and axial sample deformations. This not only provided a check on the measured volume change, but provided a means to deter- mine the volume change occuring during a given cycle if this were desired. 17 2.2 S911 Air-dry Ottawa sand was used in the majority of the tests because of its widespread availability. It had a gradation as shown by curve A in Figure 5. Three different initial relative densities were used in the testing pro- gram which are called loose, medium, and dense sand in subsequent discussions. The loose sand was at the maximum void ratio which could be consistantly obtained, and the dense sand was formed by compacting the sand with a rod and tapping the sample mold. This resulted in the minimum void ratio attainable for the material. The specimens considered as loose had a range of void ratios from 0.599 to 0.620 with the average being 0.607, those of medium relative density had a void ratio range of 0.560 to 0.575 with the average being 0.567, and those considered as dense had a void ratio range of 0.521 to 0.540 with the average being 0.533. The maximum and minimum void ratios which could be obtained by placing this material in a rigid metal container were 0.700 and 0.520. Using these values for emax and emin’ and defining relative density as 100(emax-e)/(emax-emin), the relative densities for the average void ratios given above become 52%, 7%%, and 93% for the loose, medium, and dense specimens respectively. l8 moiso coSJQTSmE mum 58010. .911 AEEV wN_w 27310 8 NO 8 to QC Nd OJ QN QM Qv O m HBNld 1N3333d O In 00.. 19 Later in the testing sequence, two other sands were used for various reasons. One of these was a fine Ottawa sand with a gradation shown by curve B, and the other was a well graded, clean, brown sand as shown by curve C in Figure 5. Only cohesionless soils were used in this study since this type of soil is more sensitive to vibratory loading than cohesive soils. 2.3 Testing Procedures If the results of the experimental investigation are to be applied to a dynamically loaded foundation, it is necessary to have the soil specimens subjected to a stress history similar to that in the field. To meet this requirement, the following loading sequences were used in the testing program. All tests were performed on 1.H" diameter by approximately 2.8" high cylindrical specimens. Test series A In the primary phase of the testing program (test series A), specimens of air-dried Ottawa sand (gradation A) were first loaded hydrostatically in the triaxial cell to the desired confining pressure,C§:. Throughout the test- ing program a confining pressure of 20 psi was used. This is shown as stage I of the loading in Figure 6. After application of the hydrostatic pressure, the axial stress on the specimen was increased until the desired static principal stress ratio, Rs=(O1-/OE3)5= (cg/0;), 20 H. D s I... ii i (I) (II) (III) (III) Fig. 6.—-— Loading sequence iiiivq .q OquCq T'r AXIAL STRESS 0? PERCENT AXIAL STRAIN Fig. 7.-- Stress-strain curve for loading squence 21 was attained as shown in Figure 6 as stage II. The speci- men was allowed to remain in this condition until there was no appreciable increase of deformation with time. In the third stage, a cyclic stress was superimposed upon the existing static stress state. For the first phase of the experimental investigation, a cyclic axial stress, :tAO1‘, was superimposed On the existing static stress, 0; . This loading sequence, or stress history, resulted in the combined static and dynamic stress state shown in Figure 6 as stage IV. Figure 7 shows the axial stress-strain curve for this type of loading condition. During the first stage of loading, the application of the confining pressure, CE , caused an axial strain, €91 . The application of the de- viator stress, giving a total axial stress,, was of primary concern in this investigation. A third representation of the imposed stress con- ditions is shown by the Mohr's circles in Figure 8. Dur- ing the first stage of loading stress path 0A was followed 22 ,r \ l A l I (7 ° <7. 0". 0|. Tu AC1; AUfJ Fig.8.--Mohr's circles for cyclic axial loading 0 ' OE: - s U. IAUEIAOEI lMé AT Fig. 9.——Mohr’s circles for cyclic cell pressure 23 until the hydrostatic pressure,cfi , was reached. During the second stage of static loading, stress path AB was followed until the desired static principal stress ratio, Rs=C§/@Z} was attained. The state of stress on the specimen at the end of this stage is shown by circle 1. During the cyclic loading, the state of stress in the specimen cycled between circles 2 and 3 as indicated by the cyclic stress path CD. This gave a maximum principal stress ratio of Rm: Ol'J/O;=(O; +qu)/O; and a minimum principal stress ratio of Rn: O;/O;=(O;—A0;)/Oé. Test series B A variation in the stress history was introduced to examine the effects of stress path on the dynamic strain response. In this series of tests, (seriesB), stages I and II remained the same as in series A, but stage III was changed. This is indicated by the Mohr's circles in Figure 9. The application of the hydrostatic stress (stage I) is shown by CA and the stress path for stage II is given by AB as before. The cyclic stress was then superimposed by pulsing the cell pressure by an amountztafig . This is indicated by the cyclic stress path CD in Figure 9. The stress state in the sample at the end of the static load- ing is given by circle 1. Circles 2 and 3 indicate the application of the cyclic stress. This gave a maximum CT-13CT principal stress ratio, m=E§E:§E§-and a minimum principal C C stress ratio, Rn=(O;+A0; )/(O;+AO;). 24 Test series C A second variation in stress history was used to further explore the stress-deformation response. In these tests (series C), stages I and II of the loading sequence remained the same; however, during stage III, the cyclic axial stress was not symmetrical with respect to the static axial stress, 0;. That is, AOL and A00 (as shown in Figure 7) were not equal as they were in series A. These tests indicated the relative importance of AG;J and ACE) and their combined effect upon the dynamic strain, €D. Test series D In test series D, the effects of a static prestress were investigated. This involved altering Stage II of the loading sequence. After the application of the hydrostatic stress (stage I), the deviator stress equal to the total axial prestress of CE,was applied (Figure 10). After the specimen had reached equilibrium under this static stress condition, the axial stress was reduced to CE. Stage III was then initiated. A symmetrical cyclic axial stress was applied as in series A. Most of the tests in this series were performed with the prestress, CF , being equal to the P maximum stress, CK], of stage III. An additional test was performed in this series with C§,being greater than CE. The conduct of these tests deviated from that of previous tests in one additional manner. After 10,000 25 a; ...... UUI' A6" mgat. 6"! ....... EUD‘I' ‘0 O-S’_ ........ m {:1 is- 65-11. So X -) < q... PERCENT AXIAL STRAIN Fig.10.--Stress-strain curve for prestress loading sequence 01331 (0 RES AXIAL S PERCENT A XIA L STRAIN Fig.ll.-- Stress-strain curve for cumulative loading sequence 26 stress cycles, the static axial stress was again reduced to <3: , and a larger cyclic stress was applied so that the maximum axial stress,<3b, remained the same. This stress change was repeated several times. For the last loading of the sequence, the deviator stress was reduced to zero during the unloading part of the stress cycle. The dynamic strain for each loading of the sequence was considered to be the strain for that loading plus the strains from the preceding loadings. Test seriesgfl During the early part of the testing program, a series of tests was performed in which several loading conditions were applied to the same soil specimen. This was done in order to obtain as much preliminary informa- tion as possible from a single specimen so that the characteristic response trends could be determined. This series of tests used the same loading sequence as test series A. However, after the axial strain under the cyclic stress approached a stationary value (at about 6,000 cycles), the magnitude of the cyclic stress was increased and the resulting axial strain again determined. This process was continued through several stages. The axial strain for a given cyclic stress was considered to be the total axial strain accumulated through all preceding cyclic loadings plus the strain that occurred under the given cyclic stress. This is shown in Figure 11. For the stress O'U the strain is €03. 3 ’ 27 Test series F Tests were also performed on sands with gradations shown by curves B and C in Figure 5. These tests were carried out to further explore certain strain character- istics observed in test series A. The loading sequence was the same as the one used in series A. Cyclic stress frequencies In the primary phase of the testing program (test series A) and the initial exploratory phase (test series E), tests were conducted using cyclic stress frequencies ranging from 0.1 cps to 25 cps. This was to investigate possible frequency effects on the dynamic strain response. In subsequent test series, frequencies were used that were most compatible with the type of test being con- ducted. In test series B, a frequency of 6 cps was used, and 10 cps was used in test series 0, D, and F. CHAPTER III RESULTS OF EXPERIMENTAL INVESTIGATION 3.1 Axial Strain As previously indicated, the experimental investi- gation of the dynamic stress-deformation characteristics was primarily based upon test series A and B on air-dried standard Ottawa sand. In these tests, the specimens were first stressed hydrostatically to a cell pressure, CE , of 20 psi. A deviatoric stress was then applied to produce the desired static principal stress ratio, RS=CI§AOE. After the soil reached equilibrium under this static stress condition, a sinusoidal pulsating stress was applied in two ways. In series A, the cyclic axial stress was symmetrical about the static axial stress. In series B, a cyclic cell pressure was superimposed on the static cell pressure. The loading and material variables studied during this phase of the investigation were the static principal stress ratio, R3, the pulsating axial stress amplitude, iAO; , the pulsating cell pressure amplitude,i—AG; , the pulse frequency, f, and the initial void ratio, e0, of the sand. 28 29 All tests were conducted so that the axial, or major principal stress was never less than the lateral, or minor principal stress. This was due to physical restrictions of the loading mechanism. Typical results The axial deformations of the soil specimens were measured with a LVDT and recorded on an oscillograph as described previously. This provided a continuous plot of the axial deformation as a function of time or number of stress cycles which resulted from a given stress state. Curves A, B, C, D, and E, in Figure 12 show actual test data obtained in this manner. For convenience, the axial deformation is given as percent axial strain and the number of stress cycles, N, is plotted on a logarith- mic scale. These curves show only the accumulated axial strain at the end of each cycle and not the recoverable strain during each individual stress cycle. These curves indicate that, with increasing time or increasing number of cycles, the soil appeared to stabilize and the axial strain ceased to increase. For this investi- gation, the strain measured after 10,000 cycles of axial stress was considered as the dynamic strain,l€D, for the given stress state. Several tests carried to 60,000 cycles of stress showed no appreciable increase in axial strain beyond 10,000 cycles. 30 OF 865 mmobm Co Lonccjcum>i Eobm _o_xo .6 mo>Lao _oo_a>ei.m_..m_u Z . mmJU>U no mumZDZ _. Ow Ihdi 4w>m4 Z_<~:.m amImz II. 0 mmem hmMFl m Ihdd 4w>w.._ Z_m.. Z_w4 Z_>04 I 4 mmem hmwh l< NIVHIS 'lVIXV l N3383d 31 It was found that the strain for a given stress condition was not unique; furthermore, the dynamic strain occuning for a given material and stress state had two distinct possible magnitudes. This is illustrated by the idealized E-N curves in Figure 13. The two possible strain values are shown in Figure 13 as a lower strain level, €DL’ and a higher strain level, 60H. In addition to the two distinct possible dynamic strain levels, there were three possible strain—vs-time curves, or strain paths, which could be followed to reach these strain levels. As shown in Figure 13, the axial strain path could follow ABC which resulted in a lower strain level, € it could also DL’ follow ADEF which resulted in a higher strain level, 60H, or it could follow ABEF which started as a lower strain level and at some time made a transition to the higher strain level path. It was observed during the conduct of the tests that if the axial strain was following the lower strain path, and a transition to the higher level had not taken place by approximately 1,000 stress cycles, a transition would not occur at all. Rather, the strain state would remain stable at the lower level for the remainder of the test. This does not prove, however, that a transition could not occur later if the test had continued. Further study of the strain-time curves revealed that a definite distinction between the lower and higher PERCENT AXIAL STRAIN 32 1.o~~ p c 5 80H DL 0 a. + : i 1 10 102 103 104 NUMBER OF CYCLES, N Fig.13.--Idea|ized axial strain -vs-number of stress cycles 33 strain paths could be made by plotting the square of the axial strain-vs-number of cycles. Curves A, B, C, D, and E, in Figure 1% show the typical strain paths from actual data plotted in this manner. On this type of plot, the lower strain path plotted as a series of three (or occasionally four) straight line segments, as shown by curve A. The higher strain path, on the other hand, plotted as a curve with a continuous change in curvature as shown by curves C and D in Figure 1%. Curve B shows a test which had a transition from the lower path at the beginning to the higher path later in the test. The transition phase can easily be seen by the departure from the characteristic straight line plot of the lower strain path when the strain rate increased during the transition to the higher strain path. This strain-time characteristic made it possible to distinguish whether the dynamic axial strain from a given test was at the high or low strain level without direct comparison with other strain data. It should be emphasized that although three possible strain paths could be followed in a given test, there were only two possible values for the dynamic strain for that test. This was because both the higher strain path and the path with a transition resulted in the same axial strain. Therefore, these two strain paths may be considered to be- long in the same category; the higher strain path represents a transition to the high strain level at the beginning of the test. 31+ 0.. Moe 0288 mmobm Co LonEnclm? ooconom :6ch 638 do mo>hao _oo_a>el.:.m_u z .3366 no mum—>52 mo. 9 Ihw._ Z_w4 ZEN—hm ”57.02.. Im mewm bmmklo I._.m1. Z_<~:m mum-10.1 I< mwzumm hmwhlU ZO_._LwZmu Z_<~:w aw>>O.._ |< wwEwm kwmh I< v.0 (‘l o 8 z( NIWLS 1V|Xv 1N3383d) v.0 0.0 35 Dynamic-vs-static axial strain Figures 15, 16, and 17 show a comparison between the static and dynamic axial strains for the loose, medium, and dense sands, respectively. The solid curves in these figures show the static stress-strain relation- ships. The results of the dynamic tests are shown by the vertical lines in these figures. The mark near the middle of each line indicates the static principal stress ratio, RS, of that test. The top and bottom of each line represent the maximum and minimum principal stress ratios, Rm, and Rn, under the cyclic loading. The percent axial strain indicated by each line represents the total axial strain at the end of the test. It is the sum of the strains which occured during the application of the static deviatoric stress and the cyclic stress. The dynamic axial strain,€k), is the strain difference between the vertical line and the abscissa of a point on the static stress-strain curve at the same value of RS. Most of the data show, as would be expected, that the total strain was greater than that which would have been obtained if the principal stress ratio had been in- creased statically to Rm. There were, however, some exceptions to this. In tests with relatively low Rm values, and particularly in the denser sands, some of 36 N.m 0N ocom omoo_ cod Boo Eobmummobm o_Eoc>oim>lo:on Imrmi 2.43% ._<_X< hzwumwa mm 0.N w 0;. NA 0.0 ‘0 0... 0.m OIIVH $53813 1VdJDNléld 0.? % 0d 37 ocom E269: co.— Boo Eobmimmobm o_Eoc>olm>lo:3ml.©e.mE Z _U-m>uo:3mI§.mE Z_uc6cfi _o_xo o_Eoc>QI.mr.mE ow.z_<»m 44:2 Azmomma 3 N; 0; ad 90 v.0 . «.0 o 4 mmem hmm._.lozU I._O ammZDZ Now . Or 0240 mmzmo I O 0240 220m: I 2 024.0 M0004 I 4 1.9 Or mo_o>o .6 LonE3CIm>I mcofmcob Eon. Eobm .o LonEale.mm.m_n_ Z .0m..U>U I..O mumZDZ 0.. or m9 m SNOIl ISNVHJ. H lVd NIVUlS :IO UBBWON 50 correlation between the stress level and the strain path followed. For both the loose and medium sands, 5 out of 8 high strain levels occurred as a strain path transition, and, for the dense sand 1 out of 3 was a transition. Figure 22 shows the number of transition points that occurraiat various numbers of cycles after the initiation of cyclic loading. As can be seen, there was a slight preference for transitions to occur from about 30 to 100 cycles after the start of the cyclic loading, but the transitions occurred anywhere from approximately 10 to 1000 cycles with about equal probability. As pointed out previously for tests which followed the lower strain path, if no transitions occurred during the first 1,000 cycles, the sand appeared to remain stable at the lower strain level under continued cyclic loading. Frequency Effects A cyclic stress frequency of 10 cps was used for the majority of the tests in this series since the loading system devised for this project operated most satis- factorily at this frequency. However, as mentioned pre- viously, several researchers have attempted to relate soil deformation as a function of the acceleration. In order to investigate this further, other tests were con- ducted at 2.5, 5, and 25 cps. At all three frequencies the dynamic axial strain 51 obtained for a given stress factor, F, and a given initial void ration, e , was the same as those obtained at 10 cps. o The data plotted in Figures 15 through 18 include the data for all four frequencies. Since the characteristics shown in these figures were independent of the frequency, the various frequencies are not identified in these plots. The maximum input acceleration in a system can be expressed as A602 or A(2’n"f)2 , where A is the deforma- tion amplitude of the system, 0) is the frequency in radians/time, and f is the frequency in cycles/time. Analysis of the cyclic strain amplitudes of the soil specimens for given cyclic stress amplitudes indicated that the ratios of stress to strain were independent of the frequency through the range of 2.5 to 25 cps. Therefore, the acceleration of the soil particles for a given cyclic stress was directly proportional to the square of the frequency. It is thus obvious that since the axial strain, € , was 0 independent of the frequency, it was also independent of the imposed acceleration. Instead, the soil strain was simply a function of the stress state imposed on the soil mass for the stresses and frequencies used in this investi- gation. . This does not, however, prove that deformation is independent of acceleration for all accelerations. The maximum acceleration in these tests was less than 1/10 the acceleration of gravity, whereas in much of the previous research, the accelerations ranged from 52 approximately 0.5g to 3.0g. It should not be concluded that all deformation characteristics were independent of frequency. One characteristic which was definitely frequency dependent was the strain rate, or strain increment.per cycle of load. The higher the pulse frequency, the less was the axial strain per cyc1e. Figure 23 compares the strain rate for a sand of medium initial density tested at five different frequencies. Approximately %8% of the dynamic strain, E , was obtained during the first 10 cycles of D loading at 25 cps; at 10 cps, 67% was obtained in 10 cycles; at 2.5 cps, 75%; and at 0.1 cps, 82%. The data for 0.1 cps were obtained from two special tests in which a slow repeated deviator stress was applied manually. These tests were conducted to compare the strain under slow repeated stress with that under a dynamic stress. The axial strains during the cyclic loading for these two tests were 20% and H0% higher than those predicted from the F- 60 plot for this sand, although the tests were only continued for 200 and 500 cycles respectively. Another noticeable difference was that under the slow loading the specimens underwent very little volu- metric strain while the dynamic tests conducted at the same stress level showed significant densification. (Volumetric strains, in general, are discussed in a sub- sequent section.) Since the sand densified in the >ocozooa. .o c0325. mo 9.6.. £me o_Eoc>QI.mm.m.I.. Z .0u.._U>U I..O meZDZ .. 0 N0. 0.. r mao 0.0.0. I 36 0.9 Iol $0 9.. Ile $0 3 Ial N 0 V NIVI‘JLS OIWVNACI .:IO 1N3383d .0MJU>U 000.0. ..< z., were from 25% to 180% greater than those indicated by the higher strain level curves of Figure 18. Furthermore, there was no correlation between F and 6%) for this data. It was considered, therefore, that failure had occurred in these tests and this data was considered separately and not included in Figure 18. It is important, from a practical point of view, to know whether a soil mass when subjected to a given stress state will have a limited strain as shown in Figure 18, or will have a large, finite strain of indefinite magni- tude. In this investigation, therefore, failure was defined as having occurred when the dynamic strain,€D , was greater than that indicated by the appropriate curve in Figure 18. 55 The plot of axial strain-vs-number of stress cycles for a test in which failure took place looked like the typical curves given previously, except that the dynamic strain was much larger. Analysis of the failure data indicated that a second critical stress factor, Ff, could be fairly well defined such that for values of F larger than F the correlations f, of Figure 18 were no longer applicable, i.e., failure would occur. In static triaxial tests, the primary factor controlling failure is the maximum principal stress ratio to which the soil is subjected. For the dynamic loadings, failure also depended upon the value of RD. This can be seen in Figure 24. This plot shows all the tests in which failure occurred and all the tests in the same range of stress for which failure did not occur. The stress conditions at which failure occurred appear to be about the same for the dense and medium sand and somewhat lower for the loose sand. This figure also indicates that the sand could be subjected to higher values of maximum principal stress ratio, Rm, when the value of RD was also relatively high. The stress ratio at failure could be as high as 4.35, 4.35, and 3.65 for the dense, medium, and loose sands, respectively, if the value of RD was greater than about 0.3, but the respective values of Rm at failure were 3.80, 3.80, and 3.00 for values of R below about 0.125. D 56 83:2 S 0.1-9.; Exam .9“. m mm. em. 0m. 8. mm. m... 3. 9. oo.. ._4...._...44__ ON 0240 M000... I I U 023.. .228: I o 0 mm 0240 w02m0 I 4 4 MEI—m... mmzmzn. 02 F 0.0 \\\ I \ 2 A 1. a «m 30.3 3 an A... I. I \ 4 . 04 « Ex 4. o l. a 2 us; mzmo 0..» 57 These maximum principal stress ratios for the dynamic tests may be compared with the static, stress- strain curves. (Stress-strain curves for static, stress controlled, triaxial tests are shown in Figure 25.) The values of Rm at failure in dynamic tests may be related to the stress ratios in static tests at which plastic deformations become important. For example, in dynamic tests, the maximum values of Rm were equal to 4.35, 4.35, and 3.60 for dense, medium, and loose sands (Figure 24). It can be noted that the stress ratios corresponding to a tangent modulus of 700 psi in the static tests were 4.35, 4.25, and 3.65 for the dense, medium, and loose sands, respectively. Also, the stress ratios corresponding to a modulus of 2,500 psi in the static tests were 3.80, 3.70, and 3.00 for the dense, medium, and loose sands, respectively. These values are comparable with the corresponding values of Rm equal to 3.80, 3.80, and 3.00 for dynamic tests (Figure 20, for values of RD below 0.125). If yielding of the soil in static tests is considered to be represented by some specified value of the tangent modulus, then the preceding examples would suggest that failure may occur under dynamic loading when Rm exceeds the value necessary to produce yielding. It should be remembered, however, that the value of RD has considerable influence on the dynamic failure value PRINCIPAL STRESS RATIO as 58 'o 1.5 1 .0 O 1.0 2.0 3.0 4,0 5.0 PERCENT AXIAL STRAIN Fig.257-Stress—strain curves from static triaxial tests 59 of Rm. This is further illustrated by Figure 26 which is computed from the data of Figure 24. Figure 26 shows the minimum values of RD for a given stress factor, F, for which failure does not occur. For example, when a dense sand was subjected to a combined stress factor, F, of 30.0, if the value of R D was less than 0.20, failure did occur, and if R was D greater than 0.20, failure did not occur and the resulting axial strain from the cyclic stress could be predicted from the F-vs- ED plot of Figure 18. 3.2 Vglumetric Strain Throughout the testing program,in addition,to measuring the axial strain of the cylindrical samples, the volumetric strain was also determined. This was accomplished by the two methods described in Chapter II. Further details are discussed in Appendix A. The average volumetric strains from these two methods of determination have been used in the following discussions. Figures 27, 28, and 29 show the volumetric strain data from the static and dynamic tests for the loose, medium, and dense sands, respectively. The continuous, solid line in each figure shows the percent volumetric strain-vs-principal stress ratio, C§///bg , from a static, stress-controlled test. The arrows show the results of the cyclic loading tests. The tail of each arrow gives the value of RS for that test and the 60 .3660 Ho: moon ocnzo. £6.53 co. I. .52.. 6 Lo. om E:EE..>.I.©m.m_n. 61 ocom omoo_ Lo. Sofim oE:_o> o_Eoc>o-m>I 63.8.0 I Nm .9“. ow d>u._ :0... I o 0W JM>MJ >>O.. l0 m0} "0:4”. 00 “#30 44a .02 .mn. 401 N.0 I No+ V. O + 0.0 + NIVHIS BWO'TOA 1N3083d (NOISNVdXB) (NOII OVEINOO) 62 Ucom E369: Lo. :6ch oE:_o> o_Eoc>o-m>I 6330:-.wm .9... m x w So no". I am d>u4 :91. n. w «3 no". I 00 4u>u._ 26.. I I v.oim N m 3.5m 5”: I 00 d5... 10:. I e ( m mwawm Sm: .. ow ._H._>u._ 26.. I x a 3 < mmEum 5m: I 00 ..u>u. rail 0 w 4. SEE Emu. I 0.0 .33”... 30.. .. o «o um. I. mD\I_Uuo: 0_Eoc>oum>uo_.0.mi.mm.mfi. m d V N S tolo N o o M w 305.. I0} I «.0 ..w 00 $54 30.. I o w 3 l 6 mU\I_Uuo: o.E0:>QII.Om.m_II_ “I O I (NOISNVdXE) mm m.“: 10.1 I o 00 ..u>m._ 30.. .. o 0‘ l 0 E0 0 6' + ‘vaais 3 wmoA 1N3383 66 N d 4- 0A '3 0.0 + OVHINOO) 4.0 + (NOIl 0.0 + 67 000m E2008 Lo. 0.0.30 0E:_o> 0.0.055 I..m.m_n. M. 03052. 300232 m 35.3 5”: l 000.65. 10.: I 0 m 35.5 Bu. | w .030. 303 I < 3.03 50. I 00 00>“: 10:. < mmEum Sm. I ow. din 304 E0 “I O I ( NOIS NVdXEI) o' l 0‘ + ‘ vaais BWO'IOA l Naoaad |O/\3 + O' + (NOIIOVULNOD) l“. O + 68 000m 0900 00. 0.0.5... 0E:_o> 0:00.30 I.mm .mfi. . m.) Nola V N 0 S a 020.02. 300232 m 00 ..u>m._ 10:1 I 0 BEN. ow ._u>w._ 30.. I o d E0 0 m C N l A o m ..0+W 3 S m ~.0+M 3 A m 00+ 2. ( NOIJ. oval NOD) 0.0+ 69 3.3 Contribution of Shearing and Volumetric Deformations to Axial Strain In order to gain more insight into the stress-strain phenomenon, an attempt was made to separate the axial strain into two components; one due to volumetric deforma- tion and one due to shearing deformation. The volumetric and shear strains of the soil can be written as, Evol = Gov+ 26w: AV and, E —E = 2 Y or L?’ ‘ where €vol= AV =volumetric strain 2)’ =shear strain €DV =axial strain due to AV QDY =axial strain due to 2V ELV =lateral strain due to AV EL), =lateral strain due to 2X . If it is assumed that the volumetric strains are isotropic, i.e., that Em]: €LV a then €vo. 2‘ AV 2' GOV + 2€LV=3€LV, or, 6W = €Dv = AV/B . Also, AV 2 €D+2€L , where 6L =total lateral strain. _ __1_ _ 3 _ Therefore, €Dr—€D 3(€D+2€L)— 3(60 €L) 1 and, €Dv=-3-(€D+2€L) . Using the assumption that the average lateral strain of the sand specimen, €L , was equal to % the lateral strain measured at the specimen mid-height (Appendix A), ED , €03“ , and EDV were determined from the data 70 and plotted against the number of cyclic stress repeti- tions. Typical plots are shown in Figures 33, 3h, and 35. From these figures, it can be seen that for all densities and strain levels, a high percentage of the total axial strain (75%-95%) was due to shearing deforma- tion. It can also be seen that in the tests with a transi- tion from the low strain path to the high strain path, the transition is reflected in both 602, and €Dv . Since the volumetric strain is simply 360V , the transitions would also be noticeable in a plot of V0 Figure 36 shows the total axial strain-vs-the volumetric strain for the data shown in Figures 33, 34, and 35. The strain path transitions are not reflected in this plot. Hence, the ratio of the volumetric strain to the axial strain appears to be independent of the strain path. This ratio, however, changes with the number of load cycles, with the ratio increasing as the number of cycles increases. It is also dependent on the stress conditions to which the sample was subjected, but appears to be independent of the sand density. For comparative purposes, data from static loading tests are also shown in Figure 36. As can be seen, the shape of these curves is very different from that of the dynamic tests. Under static stress, as the axial strain increased, there was a tendency of the volumetric strain 71 cochcoEU mE:_o>..m>I Lomzm "£me _o_x<.ul.mm .9... z .mme>U no mmmZDZ 0. m9 «9 or F o «.0 v.0 0.0 >ow llull. xow llll mo ow 024m 2282qu d>m._ 264 . osmuu - x 024m 2282 now .65.. 26.. I Nun"... . o O... N IVELS lNEIDHI-H 72 Or mcoSoELoEB oE3_o>|m>|Loocm "Eobm _o_x< IKVM .mi 2. mmmU>U LO mum—232 r m9. Now 0 >OW lullll XDW II III ow zIm>ILUmLm ":6me _U_X Im>I _o_xm. 10.: I mmoo. I admqun. owdfij 10.... I umoou I 0.9qu o ow ..m.>m. 10.1 I mmoo. I 0.9qu 4 ow .55.. 30.. I 220sz 3nd I o ow $3... 10.: I 2282 I 0.8"... I < ow .53: 2,0. I 2232 -95"... I x ow .65.. 10.: I mmzmoémmuu I0 v. o + (NOIlDVbllNOO) 0.0 + 75 toward expansion. On the other hand, under dynamic stress, as the axial strain increased, there was a tendency toward increased densification of the sand. From the above observations concerning Figures 33 through 36, a general statement can be made concerning the dual level strain phenomenon. Since the strain path transi- tions appear in the curves relating both axial strain ((50 , € \l, and Ecn,) and volumetric strain to the number of stress D cycles, and since the ratio of volumetric strain to axial strain remained essentially constant before, during, and after the transitions occurred, it appears that the dif- ference between the high level and low level strains was not exclusively a result of either shear deformations or volumetric deformations. Rather, the dual strain level characteristic was a strain rate phenomenon with both types of deformation contributing proportionately at both strain levels. 3.H Nonsymmetrical Cyclic Stress Effects The preceding sections concern tests in which the soil was first subjected to a static stress ratio, RS, with a cyclic stress superimposed symmetrically on the static stress. ‘In addition to these tests, several other tests were conducted in which the cyclic axial stress was not symmetrical, i.e., AOL #AO’D. From this data, the relative contribution to deformation of the cyclic stress increase, AOL, and the cyclic stress decrease, AO'D , can be observed. 76 In analyzing these results, it was first determined whether the axial strain was a high level or low level strain. (See p. 33.) This observed strain was then compared with the data in Figure 18Hfor symmetrical loading. These results are shown if Figure 37 which is a plot of the ratio of the observed strain under nonsymmetri- cal loading, €35 , to the corresponding strain under symmetrical loading, E: , versus the ratio of AOL to ACID . Although there is considerable scatter and only a limited amount of data, there is an obvious trend which indicates that for a given Ao'u , the ratio of €35 to €50 is inversely proportional to AC}, - Another way to consider the implications of this is shown in Figure 38. This figure shows two typical tests. In bothitests the samples were subjected to the same confining pressure, <3E , the same static axial stress, (3; , and both were subjected to the same cyclic stress increase, ZXCKJ ; therefore, both tests had the same stress factor, F. The dynamic strain, EC) , was greater for test 2 than for test 1, although the total cyclic stress amplitude in test 2 was less than that in test 1. It is quite probable that the parameters used in Figure 18 could be modified to take into account the effects of nonsymmetrical loading, but available data is insufficient for detailed analysis. 77 oI—HIIICFH LEVEL QC, I o-Lok cva CD I 1.3 i 4.2 I 1 / . I // I // 06 0.8 12 14 1.6 A 1/ I $4 | ’ » D O / I / | .8 / I Is! I o/ I / Eo / '7 Fig. 37.--Axial strain for symmetrical -vs—nonsymmetrical cyclic loadings 78 :6me 2.853 :0 mod .0 ooco:_EHII.®m.m_I._ 4m I F mmI Eobm o_Eoc>Q Ikmm .mE ow . 225m .<.x< hzuoau; NF 9. Q0 0.0 V0 «.0 IIIDZdzw EBDwZII o. ow .63.. so. I 8.? am mo.muEml x I. ow In..>m._ 30.. I o.muEmI o D .E ON mace W 4w>w4 10.1 I 0.9: m I G. ow d5. 30. I oeuEm I 4 077 0m 0? 81 occur in this stress range for sand without prestressing. Also, the value of FC for the prestressed sand was depen- dent upon the value of Rm whereas it was independent of Rm for sand not prestressed. For both Rm=3.0 and Rm=h.0, the value of Afijbequals approximately 8 psi for values of Fc equal to 9 and 20 respectively. For prestressed sand, therefore, there was no significant strain as long asIACRJ was less than a specified value (8 psi in this case) which was apparently independent of the maximum stress level (providing the maximum stress level was below the failure stress level). The axial strain for an additional test in which Rp was 15% larger than Rm is also shown in Figure 39. As can be seen, the strain was comparable to the tests with Rp=Rm. For the prestressed sand of medium density, a high strain level was obtained in only one test. It is there- fore apparent that prestressed sand had a much greater tendency toward following the lower strain path than sand which was not prestressed. Not only was the dynamic strain less for the pre- stressed sand, but there was a delay of 10 to 15 stress cycles before significant strains appeared. In comparison, a high percent of the dynamic strain for soils which were not prestressed had taken place by the end of 15 stress cycles. This can be seen from curve E in Figures 12 and 14. 82 This characteristic could be a significant factor for a soil mass which is subjected to only a small number of stress cycles. 3.6 Cumulative Axial Strain During the early part of the experimental testing program, a series of tests was performed in which several loading conditions were applied to the same soil specimen. (Test series E.) The dynamic strain under a given loading was considered to be the cumulative strain from that loading and previous dynamic loadings. The data from these tests are shown in Figure #0. Also shown in this figure are the F-vs-(ElD curves for the high level and low level strains as given in Figure 18. The data does not divide itself into well-defined low level and high level strains; however, the magnitudes of the strain agree quite well with the data from Figure 18. One possible reason that the data do not show the two well—defined strain levels is that the control of the stresses in these tests (which were performed early in the testing program) was not up to the standards used in the main part of the testing program. Also, it is possible that one strain level could occur under one dynamic load and the other strain level occur under a subsequent loading; thus, the cumulative strain would be a mixture. The general good agreement in strain measured in these tests and those from test series A and B indicates 83 mommy 9:002 6383an Lo. 33m mmobm Im>I c 6ch o_Eoc>Q Itov .9“. 0 W . Z_