BEHAE’EOR MD ANALYSIS OF A MODEL SOIL - ICE BARREER Thesis for the Degree of Ph. ‘3. MECHIGAN STATE UNIVERSITY DAVID L. WARDER 1,959 {Mtetz This is to certify that the thesis entitled "Behavior and Analysis of a Model Soil-Ice Barrier" presented by David L. Warder has been accepted towards fulfillment of the requirements for Ph D degree in Qiyfl Engineering @\%,MQr\A/QATQ\ Major professor Date Akfiffii S \W‘ (“0‘ 0-169 ABSTRACT BEHAVIOR AND ANALYSIS OF A MODEL SOIL-ICE BARRIER By David L. Warder The strength and impermeability of frozen soil ren- der it useful as a temporary barrier in certain types of engineering construction. In this study, the use of ar- tificially frozen soil as a barrier around the periphery of a model shaft was investigated with regard to the stress-deformation characteristics of the soil-ice cylin— der. Two analyses are proposed and are compared with results of an experimental program. The first analysis involves the extension of a creep equation proposed by AlNouri (1969). After making certain simplifying assumptions, this constitutive rela- tionship was solved together with the equation of equilibrium, the compatibility equation, and the strain- displacement definitions. This analysis gives expressions for stresses, strain rates, and displacement rates which are applicable to the steady state portion of the creep curve. The second approach is based on the use of time dependent strength parameters, concepts from the David L. Warder Mohr-Coulomb failure theory, and the conditions for plas- tic equilibrium in the frozen soil. The experimental program consisted of loading hollow, cylindrical, frozen soil models by an outside pressure in a special test cell. Radial deformation at the inner surface was measured at various time intervals after application of the load. The axial force on the cylinder was also measured using a load cell mounted in the base of the test cell. The low temperatures neces- sary were maintained by submerging the entire apparatus in a cooled ethylene glycol-water solution. Tests were conducted on a sand-ice material and a frozen Ontonagon clay. Measured displacement rates at the inner surface were compared with those predicted by the analysis. Additional parameters were introduced in order to ob- tain agreement. A comparison of the measured and calculated values of axial force provided reasonably good agreement in most instances. The behavior of the two soil types studied was found to be vastly different. The sand-ice material showed very little deformation during primary creep and rapidly approached a well defined steady deformation state. The axial force in the sample was noted to in- crease at a decreasing rate with time. This reflects the stiff nature of the material and its delay in reaching a final stress state. By contrast, the frozen David L. Warder clay exhibited relatively large deformations during primary creep and a greater delay before approaching a more poorly defined steady state creep. Due to this greater ability to flow and a tendency to reach a final stress state quickly, the measured total axial force was found to reach a constant value almost immediately after loading. BEHAVIOR AND ANALYSIS OF A MODEL SOIL-ICE BARRIER \. By David L} Warder 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 ACKNOWLEDGMENTS The writer wishes to express his sincere appre- ciation to Dr. 0. B. Andersland, Professor of Civil Engineering, under whose direction this research was per- formed, for his patience, guidance, and encouragement throughout the preparation of this thesis. Thanks are al- so due the other members of the writer's guidance committee: Dr. C. E. Cutts, Chairman and Professor of Civil Engineering, Dr. W. A. Bradley, Professor of Applied Mechanics, and Dr. J. w. Trow, Professor of Geology. The writer owes a debt of gratitude to Dr. R. R. Goughnour, Associate Professor of Civil Engineering, for his many worthwhile suggestions, and to Mr. R. w. Laza for his assistance in the laboratory. To Mr. Leo Szafranski and Mr. Don Childs of the Division of Engineering Research Machine Shop, special thanks for their cooperation in the fabrication of experimental equipment. In addition to those acknowledged above, the writer wishes to thank the National Science Foundation and the Division of Engineering Research at Michigan State Univer— sity for the financial assistance which made this research possible. 11 ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES NOTATIO CHAPTER I. II. III. IV. V. TABLE OF CONTENTS NS 0 O O O O O O O O 0 INTRODUCTION . . LITERATURE REVIEW . . . . . . . . . . . . 2 1 Field Practice . . . . . 2.2 Mechanical Properties of Frozen Soil 2.3 Existing Methods of Analysis . . . . 2 A Previous Experimental Work . . . . . SOILS STUDIED AND SAMPLE PREPARATION . . . . 3.1 Soils Studied . . . . . . . . . . . . 3.2 Sample Preparation . . . . . . . . . 3.2.1 Equipment . . 3.2.2 Preparation of Cohesive Samples . . . . . . . . 3.2.3 Preparation of Cohesionless Samples . . . . . . . . . TESTING EQUIPMENT AND PROCEDURES . . . . . . A.1 Equipment . . . . . . . . . . . . . . A. 1. 1. Test Cell . . . . . . . . . A. 1. 2 Deformation Measuring Apparatus . . . . . . . . . A. 1. 3 Pressure System . . . . . . A.1. A Cooling System . . . . A.2 Procedures . . . . . . . . . . . A.2.1 Cohesive Soil . . . . A.A. 2 Cohesionless Soil . . . . . THEORETICAL CONSIDERATIONS . . . . . . . 5.1 Formulation of the Problem . . . . Analysis Based on a Creep Equation 5.2 111 Page ii vii ix 11 18 21 23 23 2A 27 28 31 31 31 3A 35 37 38 39 A2 A2 A6 CHAPTER Page 5.3 Analysis Based on Time Dependent Strength Parameters . . . . . . . . . .' 51 VI. EXPERIMENTAL RESULTS . . . . . . . . . . . . . 57 6.1 General . . . . . . . . . . . . . 57 6. 2 Deformation Results . . . . . . . . . . 60 60201 sand-Ice c o o o o o o o o o 60 6.2.2 Frozen Clay . . . . . . 6A 6.3 Axial Force Results . . . . . . . . . . 69 VII. DISCUSSION AND INTERPRETATION OF RESULTS . . . 73 7.1 General . . . . . . . . . . . . . . . . 73 7. 2 Sand-Ice . . . . . . . . . 75 7.2.1 Comparison with Creep _ Equation Analysis . . . 75 7. 2. 2 Comparison with 0- -¢ Analysis . 89 7. 3 Frozen Clay . . . . . . . . . . . . . . 92 VIII. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH o o o o o o o o o c o o a o o o c o o 99 8.1 Summary of Conclusions . . . . . . . 99 8. 2 Suggestions for Further Research . . . 102 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 10A APPENDICES . . . . . . . . . . . . . . . . . . . . . 107 iv Table M -q 0‘ ox cm ox L» c- u) to #4 ‘1 M b «a -q «a -q A-lO A-ll A-12 LIST OF TABLES Index properties of Ontonagon clay . . . . Mineralogical prOperties of Ontonagon clay Summary of sand-ice tests . . . . . . . . Sand-ice displacement rate results . . . . Summary of frozen clay tests . . . . . . . Frozen clay displacement rate results Summary of displacement rate comparisons using Equation (5.23'), sand-ice . . . Summary of displacement rate comparisons using Equation (5.23"), sand-ice . . . Results of total axial force calculations Values of a (From AlNouri, 1969) . . . . . Data for c-¢ analysis . ... . ... . . . Summary of comparisons for c-¢ analysis TeSt data, SA.“ 0 o o o o o o o o o o o 0 Test Test Test Test Test Test Test Test Test Test Test data, data, data, data, data, data, data, data, data, data, data, SA-6 SA-7 SA-8 SA-9 SA-ll SA-12 SA-lA SA-16 SA-18 C-3 . C-A Page 25 25 61 65 66 66 78 82 87 90 91 93 108 111 113 116 119 122 125 128 130 132 13A 136 Table A-13 B—l B—2 C—l Test data, C—5 Calibration data for measuring device Calibration data for Values of 11m and 66m vi deformation load cell Page 138 1A2 1A5 1A9 Figure 2.1 5.3 6.1 6.2 6.3 6.A 6.5 6.6 6.7 7.1 7.2 7.3 LIST OF FIGURES Page Typical Creep Curve for Frozen Soil . . . . . 15 Elasto-Plasto-Viscous Model . . . . . . . . . 15 Preparation Mold and Drilling Accessories . . 26 Preparation Mold and Accessories for Varying Sample Size . . . . . . . . . . . 29 Preparation of Clay Sample . . . . . . . . . 29 Test Cell Detail . . . . . . . . . . . . . . 32 Test Cell . . . . . . . . . . . . . . . . . . 33 Deformation Measuring Device . . . . . . . . 36 Pressure System . . . . . . . . . . . . . . . 36 Clay Sample Mounted in Test Cell . . . . . . A0 Test Set-Up . . . . . . . . . . . . . . . . . A0 Sketch of Model Soil—Ice Barrier . . . . . . A3 Distribution of u and EB for a unit rate of displacement at inner surface . . . . . . 52 Mohr Circle Representation . . . . . . . . . 53 Time-Displacement Plot for Test SA-9 . . . . 62 Time—Displacement Plot for Test SA-ll . . . . 63 Time-Displacement Plot for Test C—A . . . . . 67 Time-Axial Force Plot for Test SA-ll . . . . 70 Time-Axial Force Plot for Test SA-12 . . . . 71 Time-Axial Force Plot for Test C—5 . . . . . 72 Time-Axial Force Plot for Test C-A . . . . . 72 Range of Strain Rates, Sand-Ice . . . . . . . 7A Range of Strain Rates, Frozen Clay . . . .'. 7A Graphical Comparison, Sand-Ice, Equation (5023') 0 0'0 0 o o o o o c o o c o o c o 80 vii Figure _ Page 7.A Graphical Comparison, Sand-Ice, Equation (5.23') (Modified) . . . . . . . . . . . 81 7.5 Graphical Comparison, Sand-Ice, Equation (5023") o o o o u c o o o o o o o o o c o 83 7.6 Graphical Comparison, Sand-Ice, Equation (5.23") (MOdified) c o c o o I o o o o o 85 7.7 Distribution of or and 09 According to Equations (5.22a) and (5.22b) . . . . . . 86 7.8 Sketch Showing Development of Radial Stress Distribution . . . . . . . . . . . . . . . 87 . 7.9 PlOt Of é VS a o o o o o o o o o o o o o o o 90 7.10 Distribution of a and 0 According to Equations (5.28) and 5.25') . . . . . . . 9A ,7.11 Graphical Comparison, Frozen Clay, Equation (5.23') (Modified) . . . . . . . . . . . 96 8—1 Calibration Curve for Deformation Measuring DeVice I O O O I O 0 O O O O O O O O O I O 1’43 B-2 Calibration Curve for Load Cell . . . . . . . 1A6 viii Append (13 LIST OF APPENDICES Appendix Page A Test Data . . . . . . . . . . . . . . . . . . 107 B Calibration Data . . . . . . . . . . . . . . 1A0 C Sample Calculations . . . . . . . . . . . . . 1A7 ix C. X:Y:Z NOTATIONS inner radius of hollow, circular cylinder experimental parameters (Vialov, 1965b) outer radius of hollow, circular cylinder experimental parameters (Andersland and Akili, 1967) time dependent cohesion experimental parameters (AlNouri, 1969) total axial force height of cylinder constants of integration experimental parameters (Goughnour and Andersland, 1968) tan (u50+9) radial pressure on cylinder :(\/_3/2) N-m/2 XN-m/2 =('\f3—/2) N-Ym/2 =orthogona1 coordinates in the radial, circum- ferential and axial directions time intensity of tangential stresses (Vialov, 1965b) displacement in radial direction time derivative of u orthogonal rectangular coordinates E Y Yre’Yez’er, ny’sz’sz or,oe’oz x’ y 01 “3 QIS parameter which varies derivatoric stress component parameter which varies hydrostatic stress component ' c cos ¢ thickness of cylinder, b-a direct strain time derivative of direct strain direct strains on the r, e, and 2 planes direct strains on the x and y planes plastic direct strain time derivative of plastic direct strain effective strain rate shear strain shear strain on the plane of the first sub- script in the direction of the second subscript intensity of shearing deformation (Vialov, 1965b) time dependent friction angle direct stress direct stresses on the r, 6, and 2 planes direct stresses on the x and y planes major principal stress minor principal stress mean or hydrostatic stress effective stress xi T = shear stress Tre’Tez’Tzr’ = shear stress on the plane of the.first sub- script in the direction of the second subscript Txy’Tyz’sz xii CHAPTER I INTRODUCTION Frozen soil, because of its strength and imper- meability, may be used to advantage in certain types of engineering construction. High strength of soil-ice barriers can eliminate the necessity for temporary sup- ports; and its impervious nature prevents ground water from entering the construction area.‘ Thus, the method of artificially freezing soil to form a temporary barrier is gaining increasing acceptance in construction situations where more conventional methods are not practical. This technique was first employed in the late 1800's in Europe as an aid to shaft sinking by the mining industry. A method developed by F.H. Poetsch (Sanger, 1968) was among the first used and with minor modifica- tions is still the most widely used today. This method involves sinking pipes, closed at the bottom, around the periphery of the shaft to be sunk. Smaller diameter pipes, open at the bottom, are then placed inside them. A coolant from a refrigeration plant is then pumped down the inner pipes and up through the annular region between the pipes. As the coolant circulates, it extracts heat from the ground and a frozen cylinder of soil is formed around each freeze pipe. As the size of each frozen soil cylinder increases, the desired soil-ice barrier is com- pleted. Although this method has been used successfully in many cases, it has been used sparingly and only in situa- tions where unusual circumstances have caused other methods to fail or to appear impractical. The major disadvantages are the excessive cost of installing freeze pipes and op- erating refrigeration equipment and the time required to freeze the soil. Freezing time has been measured in weeks or even months before excavation of the shaft could even begin. Due to the large expense involved, it appears im— portant that there be a more reliable means of determining the minimum dimensions of the soil-ice barrier that has the necessary strength and deformation characteristics. At present, only empirical and rule of thumb design pro- cedures exist. In the past, frozen soil barriers have been designed using the false assumption that frozen soil is an elastic material or by applying a very large factor of safety. It is the purpose of this study to contribute knowledge toward the understanding of the behavior of a soil-ice cylinder subjected to an outside pressure. The primary reason for the difficulty in describing the behavior of a soil-ice barrier is the complex nature of the frozen soil itself. Its stress-strain properties depend upon temperature, soil structure, water and ice con- tent and are also time dependent. Frozen soil has been described as an elasto-plasto-viscous material. Models containing these various elements have been proposed to describe its stress-deformation behavior. However, none have had wide spread applicability and, thus, this behav- ior is still not well understood. Any method of designing a soil-ice barrier must con- sider the limiting conditions of strength and deformation. If the strength of the barrier is exceeded, ground water would be permitted to flow into the excavated region and the safety of workmen would be endangered. Less obvious, but also detrimental, are the consequences of excessive deformation. This could cause large deflections and even failure in the freeze pipes, as well as difficulties in the installation of permanent supports. Due to the stress relaxation and creep properties of frozen soil, each of these limiting conditions must be carefully considered. In this study, certain assumptions and simplifica- tions are made in order to solve the problem. A hollow, thick-walled cylinder of frozen soil is considered under conditions of plane strain. The loading is assumed to be a radial, uniform, compressive pressure applied to the outside surface of the cylinder. The validity of such a representation is dependent upon construction procedures. In some cases, shafts have been sunk to great depths before permanent supports have been installed. In such cases, the plane strain and uniform pressure assumptions probably closely approximate the actual conditions in the shaft at points 3 to A diameters or greater from its ends (Vialov, 1965b). In other excavations, where permanent supports have been installed as the digging proceeds, the end effects cannot be neglected and the plane strain assump- tion may not be realistic. A further simplification is that no attempt is made to describe the "primary creep" portion of the deformation. It is reasoned that this deformation occurs during excavation and can be compen- sated for at that time. It is the "secondary" or "steady state" creep occurring after excavation which, if not properly accounted for, will present difficulties in lining the shaft. Two separate analyses are studied in order to des- cribe stress-deformation behavior. First, the boundary value problem is formulated using the equations of equi- librium and compatibility along with a constitutive equation similar to one predicted by the rate process theory. In the second analysis, strength parameters c and ¢, corresponding to a given creep rate and tempera- ture, are used to relate the dimensions of the cylinder and the pressure to the creep rate. Both methods use ex- perimental results given by AlNouri (1969). Finally, the results of an experimental program used to check the analyses as well as to gain additional insight are reported. Frozen clay and sand-ice cylinders were prepared and loaded in a testing cell which was designed and fabricated specifically for this study. The radial displacement at the inner surface and the axial force in the sample were measured. CHAPTER II LITERATURE REVIEW 2.1 Field Practice The method of artificially freezing soil to form a barrier around a shaft has been used in mining for more than a century (Brace, 190A). Not until recently has it been widely used by the construction industry. The usual procedure is to place water tight freezing tubes, approx— imately A to 10 inches in diameter, around the periphery of the proposed shaft. These tubes, closed at the bottom, are usually spaced at 3 to A foot intervals. Smaller cir- culating pipes, open at the bottom, are then placed inside the freezing tubes. The circulating pipes are joined to- gether by a circulating ring. Similarly, the freezing. tubes are joined by a collector ring. A circuit is com- pleted as the circulating and collector rings are connected to a refrigeration plant. The coolant is pumped down through the circulating pipes, up through the freezing tubes to the collector ring, and back to the refrigeration plant. While a brine solution is most commonly used as the coolant, liquid nitrogen at a much lower temperature has been used in at least one case (Cross, 196A). A frozen soil cylinder is thus formed around each freezing tube. In most cases, these frozen soil cylinders have been allowed to expand until the entire region of the proposed shaft is frozen, forming one large solid cylinder of frozen soil. The frozen soil is then excavated out of the center of the cylinder. Brace (190A) reports that this material is about as difficult to excavate as soft lime- stone. The frozen soil is usually loosened by jack hammers or blasting and then removed by clam shells. In some cases, the excavation continues to the entire depth of the shaft, with the hollow frozen soil cylinder providing temporary support. The permanent lining is then begun at the bottom and continued upward. In other cases, the excavation is carried out in 20 or 30 foot sections. The permanent lining is then completed for each section be- fore proceding to the next. According to Brace (190A), the first example of artificial soil freezing as an aid to excavation was in England in 1852, where brine was circulated through freeze pipes to stabilize a bed of quicksand in sinking a well. F.H. Poetsch introduced the method described above in 1883 at Saxony when quicksand was encountered at a depth of 111.5 feet during excavation of the Archibald Shaft. Eighteen feet of quicksand was frozen and the excavation continued. There were several other examples of this method being used in Germany, Prussia, Sweden, and Belgium in the 1880's. The first application in the United States appears to have been at a shaft at the Chapin Mine, Iron Mountain, Michigan in 1887. Freezing pipes were placed around the periphery of a 15% by 16% foot rectangular shaft in order to freeze an unstable sand. Tsytovich and Khakimov (1961) report the use of an artificial soil—ice barrier in the excavation of an under- ground railway station in Moscow in 19A9. The excavation took place adjacent to the construction of a high frame building. The authors claim that the absence of temporary supports permitted mechanization of the construction and thus speeded construction time. Artificial freezing was used in a shaft for a salt mine near Windsor, Ontario in 195A. The 16 foot diameter shaft was sunk to a depth of 1100 feet. Only the first 720 feet, however, required freezing. Six inch freezing tubes with two inch circulating pipes were placed on a circle of 32 foot diameter. A brine cooled to -120 F. in a 200 ton refrigeration plant was pumped through the pipes. Approximately three months freezing time was required. Af- ter each 28 feet of excavation, reinforcing steel and forms for concrete were placed and concrete poured for the lining before beginning excavation of the next section. Approx- imately 15A hours and AA construction workers were required for each 28 foot section. Latz (1952) reports the sinking of a 15 foot diameter, 1000 foot deep shaft to a potash bed in Carlsbad, New Mex- ico. When exploratory borings revealed the presence of several horizons of quicksand in the first 350 feet, it was decided to freeze the soil to that depth. Twenty eight freeze pipes were placed on a diameter of 31 feet and chilled brine was circulated. Drilling of freeze holes be- gan July 15, 1950, the refrigeration plant began operating on November 19, and sinking of the shaft began January 19, 1951. Concrete lining was placed in 25 feet sections as excavation proceeded. The shaft lining was completed on September 19, 1951. When the City of New York had to sink a 318 foot shaft as part of a system carrying sewage under the East River, the most obvious method of keeping the excavation dry was to lower the ground water table by pumping. This, however, had to be ruled out since it would have resulted in excessive ground water depletion and possible differen- tial settlement and damage to nearby buildings which were as close as 56 feet from the shaft. Therefore, freeze pipes were placed around the outside of the 1A.5 foot diameter shaft to a depth of 123 feet and the material was frozen before excavation began. Silinsh (1960) reports that the entire cost of the shaft was approximately $6.5 million. Low (1960) gives an account of artificial soil freezing used in Montreal in 1960 in a railway tunnel when track rearrangement dictated that a single concrete arch replace the existing double arch system. The tunnel was located directly beneath a busy street and between two 10 buildings. Thus, an open cut was impossible since it would have cut through too many services and required heavy bracing to support the soil pressure due to the buildings. Vertical freeze pipes were placed along the entire length of the tunnel and a cooled methanol solution was circulated through them. After the material was frozen, it was exca- vated by drilling and blasting. While steel liner plates and ribs were used as temporary supports in the tunnel, Low concludes that they were not necessary and could be omitted from future jobs. There was an upward heave of the street of about 5 inches causing cracking and it had to be re- placed. In 1962, the City of New York was faced with the problem of sinking shafts at either end of a 25,000 foot water supply tunnel under the New York harbor from Brooklyn to Staten Island. Stewart gt al.(1963) reports that on the Brooklyn side, lowering of the water table could have caused movement to nearby factories which contained pre— cision equipment. Thus, the first 118 feet to bedrock of the 965 foot deep shaft was frozen using 28 freeze pipes on a 30 foot diameter. Freezing was completed A2 days after the freezing plant was put into operation. The 20 foot di— ameter shaft was sunk to its entire depth before concrete lining was begun at the bottom and proceeded upward. Cross (196A) describes a slightly different method of using artificially frozen soil in the construction of a four mile sewer tunnel in Milwaukee in 196A. While digging 11 an 18 foot diameter, 90 foot deep shaft to begin the tunnel, soft clay and running silt were encountered on one side. Six freeze pipes were placed horizontallly in the wall of the shaft and liquid nitrogen was circulated through them. Although liquid nitrogen is relatively expensive, its low temperature (--3200 F.) permitted a section of the wall to be frozen in a day and a half; whereas, a brine solution would probably have taken several weeks. Cross warns that extreme caution must be used when using nitrogen since it is heavier than air and could, thus, endanger the safety of workmen in the bottom of the shaft if leakage were to occur. 2.2 Mechanical Properties of Frozen Soil Mechanical properties of unfrozen soil depend pri- marily on internal friction and on cohesion due to internal interparticle bonds. When soil is frozen, other factors and parameters must be considered. These include tempera- ture, ice content, and characteristics of ice crystals, all of which affect ice cementation of particles (Yong, 1963). Of the factors governing the mechanical properties of frozen soil, ice cementation bonds are probably the most important (Tsytovich, 1963). These bonds appear to be the strongest as well as the most sensitive to external tem- perature change. According to Tsytovich, "The mechanical properties of frozen soil depend mainly on the number and properties of these bonds." In all frozen soil, a portion of the pore water remains in a liquid state. Tsytovich 12 theorizes that frozen soils are characterized by a dynamic equilibrium between the frozen and unfrozen water. An ex- ternal load causes weakening due to melting and slippage. Simultaneously,strengthening occurs due to denser packing of mineral particles and refreezing of water. Damped or un- damped creep results depending on whether strengthening or weakening predominates. Vialov (1965a) describes frozen soil as an elasto- plasto—viscous material. The presence of the viscous property is exemplified by the strong time dependence of its stress-deformation characteristics. Experience has shown that frozen soil exhibits both creep, or growth of deformation with time under a constant stress, and a re- duction in strength with time. Deformations in frozen soil can be divided into those that are recoverable and those that are irrecoverable (Vialov, 1965a). Recoverable deformations include elastic and structurally reversible deformations. Elastic defor- mation is associated with elastic changes in the crystal lattice of the ice and mineral particles and elastic com- pression of air and unfrozen water. This disappears immediately when the load is removed and is usually small enough to ignore in most considerations. Structurally re- versible deformation arises from the change in thickness of water films between particles. This may be considered a visco-elastic response since it grows with time and grad- ually diminishes when the load is removed. 13 Part of the irrecoverable deformation consists of structurally irreversible deformations associated with con— solidation. These include squeezing out air, regrouping of particles, and breaking up of bonds. They increase with time and are completely irrecoverable. Plastic deforma- tion is the other part of the irrecoverable portion. This represents the irreversible displacement of solid particles and the flow of ice. A typical creep curve for frozen soil under constant stress is given in Figure 2.1. While this representation is usually associated with uniaxial compression, creep curves for other stress states are similar (Sanger, 1968). Section 0A represents the instantaneous response which may be entirely elastic or elasto-plastic (Vialov, 1965b). Section AB corresponds to the first stage of creep where the deformation grows at a decreasing rate. Deformation during this stage is only partially recovered when the load is removed. The first stage continues until the slope reaches some minimum value at which time the process enters the second, or steady flow stage. Sanger (1968) states that most deformation in practice occurs during the second stage of creep. As the deformation continues, the third or progressive flow stage (CD) is reached. During this stage, deformation continues at an increasing rate. Point C is often considered as corresponding to failure. There is considerable disagreement regarding the re- lationship between stress and strain in frozen soil. While 1A it is generally agreed that such a relationship for a par- ticular frozen soil at a constant temperature is time dependent, it is not well understood how stress, strain, and time are related. Various mechanical models such as the one shown in Figure 2.2 proposed by Vialov (1965a), can qualitatively describe the various components of the deformation pro- cess. Springs represent elastic qualities, dashpots viscous qualities, and the friction element represents plastic qualities. However, due to the many peculiarities involved in the actual deformation process, all such . models have failed to quantitatively describe the process over a wide range of stresses. Vialov (1965a) reports that stress-strain curves can be expressed by the power law a = A(t)em1 (2.1) where (313 stress, a is strain, t is time, A(t) is the modulus of total deformation, and ml is known as the strengthening factor. A(t) can be determined using the Botzmann-Volterra theory of hereditary creep and is both temperature and time dependent. The strengthening factor is a positive number equal to or less than one and depends on neither time nor temperature. Vialov suggests deter- mining this relationship fcr a simplified stress state such as uniaxial compression. It can then be extended to complex stress conditions by using the intensity of tan- gential stresses, 15 Strain Time Figure 2.1. Typical Creep Curve for Frozen Soil Figure 2.2. Elasto-Plasto-Viscous Model 16 T a\fi/6 [(cx-oy)2-(ay-oz)2-(cz-ox)2I+Txy2+1y22+12x2 (2.2) and the intensity of shearing deformation, r a ’2/31(6x-ey)2-(ey-ez)2-(ez—ex)2 +~yxy2+yy22+yzx2. (2.3) Subscripted 0's and 8'8 represent direct stresses and strains, respectively and 1's and 7's are shear stresses and strains. Thus, Equation (2.1) can be written T = A(t)Fm1. (2.4) Goughnour and Andersland (1968) found that the follow- ing equation fit the behavior of polycrystalline ice: ep = Klexp(-nlep) + K2exp(n2!bdep). (2.5). After modifying this equation by the use of stress factors, it was shown to fit experimental data for a sand-ice mater- and é ial. e are the plastic portion of the strain and D p strain rate, respectively. K1, K2, n1, and n2 are experi- mentally determined parameters that depend on stress and temperature. The term lodep represents the strain energy absorbed during plastic deformation. Equation (2.5) was shown to describe the entire creep curve for the materials and stresses used. Andersland and Akili (1967) propose that creep in frozen soil is a thermally activated process and suggest an equation of the form 1? é = C1 exp (B0) (2'6) to describe it. This is based on the results of a series of uniaxial compression tests on a frozen clay at different temperatures. B and C1 are determined experimentally and are temperature dependent. Equation (2.6) was found to describe the steady state portion of the creep curve. AlNouri (1969) extended this approach by performing a series of differential creep tests on two frozen soils under triaxial stress conditions. He was thus able to in- clude the effect of the hydrostatic part of the stress as well as the derivatoric. AlNouri's equation for a frozen soil at a constant temperature takes the form a = C exp[N(al-03)]-exp(—mom). (2.7) C, N, and m are experimentally determined parameters, a is m the mean stress, and 01 and 03 are the major and minor principal stresses, respectively. Equation (2.7) pre- dicted the axial strain rate during steady state creep for a frozen clay and a sand-ice material under a constant axial load with varying confining pressures. AlNouri (1969) further showed how Equation (2.7) can be used to give the cohesion (c) and angle of internal friction (d) for a given strain rate and temperature. He found that different combinations of stresses produced the same axial strain rate. After sketching the Mohr circle corresponding to each set of stresses, he showed that a 18 straight line could be drawn tangent to each of the cir- cles. Defining the line by the intercept, c, and the slope, ¢. a relationship can be established between these par- ameters and the principal stresses for a particular strain rate. It was found that the angle of internal friction appeared to remain constant for the Ottawa sand and was in- dependent of temperature but that the cohesion varied with strain rate and temperature. 2L3 Existing Methods of Analysis Until recently, methods used in designing soil-ice retaining structures have failed to account for the time- dependent behavior of frozen soil. For example, the formula V (2.8) has been used to determine the wall thickness, 6, of a frozen soil cylinder of inner radius a, loaded by an outside pressure, p. ad is the maximum permissive compressive stress of the frozen soil. This formula is based on Lamé's solution for an elastic material. Since frozen soil does not behave elastically, Equation (2.8) could not be expected to give results which are consistent with the actual behavior. Further, this formula only considers the _ strength of the frozen soil. It does not provide for the determination of deformations which often control the design of such a structure. Other empirical formulas have also 19 been proposed but none have proved to have widespread applicability. The only attempt to incorporate the creep and strength reduction properties of frozen soil in the anal- ysis of soil-ice cylinders was made by Vialov (1965b). The results of his work are briefly outlined in the following paragraphs. Vialov points out that any analysis of a frozen soil barrier must take into account two limiting con- ditions——strength and deformation. Stress must not be allowed to exceed the shear strength of the soil. Neither can excessive deformations be permitted. Experience has shown that intolerable deformations can occur at stresses well below those necessary to produce failure. Vialov considers a thick-walled frozen soil cylinder of inner radius a and outer radius b loaded by a uniform outside pressure, p. He assumes that a plane strain con— dition exists in the cylinder. For the strength limiting condition, Vialov simultaneously solves the equation of equilibrium and a yield criterion for the portion of the cylinder where the yield strength has been exceeded. This produces expressions for stresses in the "plastic zone". He then uses the Lame’ solution for the zone in which the stresses are less than the yield strength. Since the radial stress, Or’ must be the same on either side of the "elastic-plastic boundary", the two expressions for Or are 20 equated. The limiting condition is when the stresses in the entire section have exceeded the yield strength. Vialov solves the problem for several different yield criteria. In each case, he arrives at an expression which relates the strength parameters and the geometry of the cylinder to the outside pressure. Thus, for a given soil and outside pressure, the required wall thickness can be calculated for fixed inner diameter. In considering deformation, Vialov uses one stress- strain relationship for the instantaneous case and a second relationship for deformation during creep. In both cases, he uses the equation of equilibrium, the incompress- ibility condition, the strain-displacement definitions, Hencky's equations, and a constitutive equation to arrive at a result. For the instantaneous case, he uses the con- stitutive equation given in Equation (2.1), using the value of A at t - O. The result is u ..(mlpfi. , a _1_ a A 1, 2[1_(%)2m]Jm1 (2.9) This equation gives the instantaneous deformation at the inner surface, ua, in terms of the pressure, the dimensions of the cylinder, and the experimental parameters A and m1. For the condition during creep, Equation (2.1) is again used, this time using the law of hereditary creep. The resulting equation is 21 1 mip 2ml (2.10) a B 7 1-A t 2ua(t)]nu_ I a 2.A Previous Experimental Work Due to the specialized equipment and techniques re- quired in a study of this type, experimental work in this area has been very limited. A testing program conducted in the USSR (Vialov, 1965b) is apparently the only one reported in the literature where cylindrical frozen soil models have been used. Information is also lacking from actual con- struction projects. Contracting companies using this method "have much proprietary information which cannot be 'published" (Sanger, 1968). Soil properties and other valu- able data are often omitted from reports. In order to determine the relationship between par- ameters in the prototype and a model, Vialov (1965b) uses the criteria of similitude. Based on this, he finds that the following relationships must hold: b b 8i am h h _l . .2 (2.12) 81 am “1 nm a . a (2.13) 22 a and b are as defined above, h is the height of the cylinder, and u is the radial displacement. The subscripts i and m indicate "in situ" and model, respectively. He ' further points out that since the properties of the material must be preserved, any parameters which describe material behavior must be the same in the prototype and model. That is, the same material should be used. In addition, pressure, temperature, and time must be the same in each to insure similarity. Vialov indicates that if the criteria of similitude outlined above is satisfied, the Reynolds' criterion is automatically satisfied. This makes it impossible to sat- isfy the Froude criterion. Vialov points out that this causes no significant error since it only indicates that the soil weight has been neglected. Vialov used a testing cell similar to the one used in the current study to test frozen soil cylinders. The model was placed inside a rubber sheath and loaded by an outside pressure. Deformation was measured using a lever device and dial gages. Tests were performed on a sandy loam and a clay ma- terial at various temperatures. Various dimensions were used and pressures varied from 20 to 80 kilograms per square centimeter. The results are given only in terms of total displacement at a fixed time after loading. Thus, strain rates are not available. CHAPTER III SOILS STUDIED AND SAMPLE PREPARATION 3.1 Soils Studied In an attempt to arrive at results which have applicability for all soils, two soil types were used in this study: one cohesive and one cohesionless. The co— hesive soil was an Ontonagon clay which occurs naturally in Northern Michigan. This soil was taken from a roadside site midway between Rudyard and Kinross, Michigan at a depth of approximately 2A inches. The soil was air dried and then ground to a fine powder. The index properties of this material are given in Table 3.1. The clay content of the Ontonagon soil was about 70%. The clay fraction was found to contain the following clay minerals in the approximate amounts indicated: Illite A5% Vermiculite 20% Kaolinite 15% Chlorite 10% The remaining 10% is made up of montmorillonite, quartz, feldspar, and amorphous material. The illite, vermiculite, 23 2A and chlorite appeared to be randomly interstratified within the soil. This data was derived from x-ray diffraction, differential thermal analysis, infrared absorption, and var- ious other tests performed on the clay fraction. Further data on the clay fraction are given in Table 3.2. The cohesionless soil studied was a standard Ottawa sand purchased from Soiltest, Inc. (CN-501 Density Sand). Only that portion which passed a No. 20 sieve and was re- tained on a No. 30 sieve was used. 3.2 Sample Preparation 3.2.1 Eguipment Preparation techniques differed for the two soil types studied and will be discussed separately. The basic appar- atus, common to both, consisted of a split cylindrical mold used to form the outside of the cylindrical soil samples (See Figure 3.1). It was cut from a steel pipe (5 3/A" outer diameter, 5" inner diameter) which was split length- wise and then machined to preserve a cylindrical shape. Two punched steel flanges were welded to each half so that they could be joined using l/A" bolts. The split portion of the mold was 12 inches long. Unsplit extensions, 3 inches long, which fit into grooves in the split part were provided for the top and bottom of the mold. In order to study samples of different diameters, liners of varying sizes were placed inside the mold. These were cut from steel pipe of approé priate sizes, split, and machined in the same manner as the outside of the mold (See Figure 3.2). 25 Table 3.1 Index properties of Ontonagon clay W I r _n-a Plastic Limit 23;6% Liquid Limit 60.5% Plasticity Index 36.9% Table 3.2 Mineralogical properties of Ontonagon clay Surface Area 215m2/g Cation Exchange Capacity Ca/Mg A8.5 meq/lOOg K/NHu 17.7 meq/lOOg Potassium Content 3.7% 26 Disassembled ) a ( Assembled (b) Preparation Mold and Drilling Accessories Figure 3.1. 27 3.2.2 Preparation of Cohesive Samples A weighed amount of air dried clay was first placed in a metal pan. Enough distilled water was then added to bring the water content to 27%. The water was added slowly and carefully mixed by hand. When it appeared that the water had been uniformly distributed throughout the clay, the mixture was placed in an airtight container until used. The inside of the split mold was lubricated with a thin coat of oil and covered with a sheet of polyethylene in order to reduce friction. The mold was assembled by joining the flanges with the l/A" bolts. The three inch extensions were placed at the top and bottom. A solid cylindrical metal plug, 3 inches long and the same diameter as the mold was placed inside the bottom extension. The clay was then placed in the mold. This was done by placing small amounts at a time and then compacting by hand to guard against void spaces. The amount of soil added was that necessary to give a density of 100 pounds per cubic foot for the prescribed volume of the sample. Both density and water content were chosen to agree with AlNouri (1969). The clay was then statically compacted to the desired density using a Tinius Olsen testing machine. A three inch solid cylindrical plug was attached to the driving head of the machine to compact the sample from the top. By suspen- ding the bottom of the mold, compaction from the bottom plug was also achieved, thus assuring a more uniform den- sity. 28 After compaction, the sample was in the form of a solid cylinder. A hole was then drilled in the center using a 1 1/2" auger. This was done by first placing a one inch thick circular plate on tOp of the mold. It fit into a groove and was secured by tie rods from studs in the plate to the flanges of the mold. A six inch long pipe with a 1 1/2" inside diameter was threaded vertically into the center of the plate. This served as a guide for the auger (See Figure 3.1). The auger was placed in the guide and rotated to make the center hole in the sample. It was necessary to withdraw the auger frequently to clean the loose soil from it. The mold was then stripped by removing the bolts. Figure 3.3 shows a clay sample which had been prepared in this manner. 3.2.3 Preparation of Cohesionless Samples A thin coat of lubricating oil was applied to the in- side of the mold and covered with a polyethylene sheet. In the bottom of the mold was placed a 13/16 inch thick, cir- cular, stainless steel plate with a smoothly machined 1 1/2 inch hole in the center. A smoothly machined 1 1/2 inch stainless steel round bar, 18 inches long, was placed into the hole in the plate. The bar was tapered one ten thousandth of an inch over its length in order to permit its removal after the sample had frozen. A thin coat of oil and polyethylene sheets were also applied to the plate and bar. As the mold was assembled, vacuum grease was applied to all joints to seal them against leakage. 29 r u I :- I . ’ a o v -‘-.v.1li Figure 3.2. Preparation Mold and Accessories for Varying Sample Size Figure 3.3. Preparation of Clay Sample 30 The dry sand was placed in the mold in four layers, each layer being tamped 25 times. The amount of sand used was that necessary to give 6A% sand by volume, using 2.65 as its specific gravity. This is in agreement with AlNouri (1969). Distilled water was then added slowly from the top until the sample was saturated. It was then placed in a freezer at -18° C. for A8 hours. After freezing, the center rOd was removed by placing the mold in the Tinius Olsen testing machine and extruding it with a hydraulic jack. The mold was then returned to the freezer and stripped from the sample by removing the bolts. Since an irregular cap of ice usually formed at the top of the sample, it was necessary to smooth the top using coarse sand paper until the end was square. 'CHAPTER IV TESTING EQUIPMENT AND PROCEDURES A.l Equipment A.1.1 Test Cell A cross sectional sketch of the test cell is shown in Figure A.l and photographs of it are given in Figure A.2. The cell consists of a hollow cylinder which, at its ends, fits into grooves cut into square plates. Rubber 0- rings are placed in the grooves in order to seal the cell. The plates are held by tie rods at their corners which are tightened using nuts at each end. A pedestal, upon which the soil sample rests, is built into the bottom plate. The base of the pedestal is connected to a flat load cell which measures the axial force in the sample. Calibration data for the load cell are given in Appendix B. At the top of the sample is a piston which fits through the top plate. The purpose of the piston in this 'study was to enforce a plane strain condition by permitting no axial movement of the sample. With minor modification it could be used to transmit an axial load to the sample. A third square plate at the top is fixed by 8 nuts to 31 32 \g 2% ”7.; Plate 9" x 1" ~ SW2 x 0'-9' /////// l1" i /\ = 1% x 2'—o 3/u" -= (1@ ea. corner) :- = = = I: = = =- = = = = A 1/2" “$5. Sign — a ’ h R . R <§§f Rubber 0- Bin; 3 - \ \ \ g s S\ \\ s o S 13A"J_11/2 i137." § J H \l N ‘. 5 ~ \\ § §; . A. :r \ > A .—4 % Cylinder 7 1/A" J § 1D X 0 ID X .4 § 1 -1 1/2 ”’5" §i _13/.” T 1"T1/2 1%" §i A! \ A ‘ \ //// §f P1ate09"x l” gi— 7? V // / 74/,"1U X ' a g" §§§i§ /S\\I\\§§§§ l|'1-/! Load Cell E- \s \=’ s t.) g Figure A.1. Test Cell Detail 33 C ¢ \ o I . Jo a o It a 9st! I ll‘.’ 1 ‘ 0 ~ a I ... an! my“... ..I .0. ..a. 1 v‘nIOAI . 9...... I s on c . r a“! a' r! n.- c v m . I ' p A . Ago—- 3 .. .L»- _- K '1 '7." v d I ... Disassembled (a) Assembled (b) Test Cell Figure A.2. 3A rigidly hold the piston. A collar containing a rubber 0- ring is fastened to the top plate at the point where the piston fits through the plate to seal it at that point. The cell is made entirely of stainless steel and was fabricated in the Division of Engineering Research Machine Shop at Michigan State University. Dimensions are as given in Fig— ure A.l. A.l.2 Deformation Measuring Apparatus A sketch of the device used to measure the deformation of the inner surface of the frozen soil cylinders is shown in Figure A.3. It was made of thin steel strips pinned to- gether to form an unstable, trapezoidally shaped structure. At the bottom of the trapezoid, the steel strips are ex- tended and equipped with rounded brass pieces which are pinned at their ends. The brass pieces rest against the inner surface of the sample. At the top of the trapezoid is pinned a round bar which was placed parallel to the axis of the cylinder. A steel pipe encases the round bar and has a collar at its top which is fastened to the top of the piston by A screws. The collar is equipped with a set screw so that the position of the steel pipe can be adjusted. Steel discs which act as guides for the round bar are located in- side the pipe at the top and bottom. This device was placed through the center hole of the piston and sample and then secured. As the inner surface of the sample moved radially inward, the rounded brass pieces were displaced, thus causing an upward movement of the round 35 bar. The top of the round bar was connected to a linear differential transformer which measured its vertical move- ment. The linear differential transformer and the load cell in the base of the testing cell were electronically con- nected to a two channel recorder which recorded the displacement of the bar and the force on the load cell. Due to the nonlinearity of the relationship between the inward movement of the brass pieces and the vertical movement of the round bar, it was necessary to calibrate the device over the range of its use. This was done by mounting the device in a vertical position and placing a micrometer, secured in a vise, over the brass pieces. Then by adjusting the micrometer and noting the recorder reading, the rela- tionship was obtained. Calibration data are given in Appendix B. A.l.3 Pressure System Figure A.A shows a sketch of the pressure system. The source of the pressure was a cylinder of compressed nitro- gen. The pressure in the cylinder was reduced to the desired pressure by the use of a high pressure regulator placed at the cylinder outlet. This pressure was applied to the top of a liquid in a high pressure cell which served as a reservoir for the fluid in the test cell. The high pressure cell consisted of two flat steel plates at either end of a hollow steel cylinder. The plates were held tightly by tie rods at the four corners. A valve was loca- ted near the inlet to the high pressure cell in order to T. 41% -\. 3/8" ¢ Bar 1" ¢ Pipe Brass Guides Figure A.3. Deformation Measuring Device linder 0) $4 :3 1.) X -r-1 2 rd . :0 cur—I (5 $4 Q) . p.12: (64-) 3b.] Figure A.A. Pressure System 37 bleed the pressure in the system. A high pressure hose transmitted the pressure from the pressure cell to the test cell. The pressure entered the test cell through a hole in the bottom plate. A gage placed near the inlet to the testing cell was used to determine the pressure in the cell. Pressures ranging from 100 to 900 pounds per square inch were used in this study. The liquid used in the system was a mixture of 50% water and 50% ethylene glycol. A.l.A Cooling System In order to maintain the frozen soil samples at the low temperatures required, the entire test cell was sub- merged in a coolant maintained at the desired temperature. The testing cell was placed in a galvanized steel tank (1A" x 1A" x 1'-9 1/2") through which the coolant was circu- lated. The tank was open at the top and covered with an insulating material (styrofoam). The coolant used was a solution of 50% water and 50% ethylene glycol. The fluid was cooled in a low temperature bath equipped with a thermoregulator. By setting the thermoregu- 1ator at the desired temperature, the low temperature bath alternately heated and cooled the fluid in order to main- tain it at that temperature. The fluid was circulated from the low temperature bath into the bottom of the galvanized steel tank and then back into the bath through the top of the tank. Temperature control of 0.10 C. was easily attainable using this apparatus. 38 A.2 Procedures A.2.1 Cohesive Soil In order to reduce friction at the ends of the sample, a thin coat of oil was applied to the top of the pedestal and the bottom of the piston. A sheet of polyethylene, cut to the required annular shape, was then placed over the ped- estal and piston. Due to the stiffness of the clay used, it was not nec- essary to freeze the sample prior to mounting it in the testing cell. Thus, the hollow cylindrical sample, prepared as described in Chapter III, was placed on the pedestal and 3the piston placed on top of the sample. Two rubber mem- branes were then placed over the sample and stretched over the pedestal and piston. Circular clamps were tightened, around the pedestal and piston to seal the sample. Strips of heavy rubber were placed between the clamps and the outer membrane in order to protect the membrane. A photograph taken at this stage of preparation is shown in Figure A.5. Since the weight of the fullyassembled testing cell made handling difficult, it was placed in the galvanized steel tank at this point. The stainless steel cylinder was then positioned, followed by the top plate. After the top plate was secured by tightening the nuts, the collar on the outside of the piston was placed and the plug in the top plate tightened. The remainder of the cell was then assem- bled and the temperature of the fluid was brought to -18°C. .....w. n. .. ..Wll 131.1 39 This temperature was maintained for approximately A8 hours. Twenty four hours before loading the sample, the temperature was raised to —12°C. This temperature history was used in order to agree with AlNouri (1969). Thirty minutes before beginning the test, the deforma- tion measuring device, which had been precooled to eliminate the possibility of thawing where it contacted the sample, was positioned and secured. The linear differential trans- former was then assembled and the recorder balanced. The pressure was applied to the sample by opening the nitrogen cylinder valve and adjusting the regulator to the desired pressure. A pressure of 100 pounds per square inch was ini- tially applied in most tests. The pressure was increased in increments of 100 p.s.i. Each increment was applied until the steady state portion of the creep curve was determined. For each increment, the deformation and the force on the load cell were recorded on the two channel recorder. At the end of the test, the pressure was released and any recovery of deformation noted. The cell was then dis— assembled and the sample examined. Figure A.6 shows a photograph of the test set-up. A.2.2 Cohesionless Soil Since the cohesionless samples were frozen before being placed in the testing cell, a slightly different pro- cedure was used. The pedestal and piston were kept in the freezer at -l8°C. for several hours before mounting to guard against the sample thawing at points of contact. Friction A0 Figure A.6. Test Set—Up Al reducers, as described for the clay samples, were placed on the pedestal and piston. The sample was mounted and mem- branes placed while the apparatus remained in the freezer. In early tests, it was found that there was a greater occurrence of membrane failures in the sand-ice samples. Therefore, between four and six membranes were used on each sample. After the clamps were tightened at the ends, the cell was transferred to the cooling tank which had been cooled to a temperature of -18°C. The pump was temporarily turned off to lower the level of the liquid in order to allow placement of the testing cell. The remainder of the assem- bly and testing procedure are as described above for the cohesive soil. CHAPTER V THEORETICAL CONSIDERATIONS 5.1 Formulation of the Problem Vialov (1965b) indicates that the limiting strength condition for soil-ice barriers can be adequately handled by solving the equation of equilibrium together with the Mohr- Coulomb failure criterion and the appropriate boundary conditions. The strength parameters can be determined from a series of triaxial tests on undisturbed soil samples fro- zen to the desired temperature. It is the objective of this study to gain additional insight into the deformation characteristics of frozen soil barriers, thus contributing knowledge toward the understanding of their limiting defor- mation condition. By way of approximating the actual conditions in a soil-ice barrier surrounding a circular shaft, consider a thick-walled cylinder of inner diameter 2a and outer diam- eter 2b, loaded at the outside surface by a uniform compressive pressure, p (See Figure 5.1). The inner surface is unloaded. The following assumptions and simplifications are made: A2 AA (1) A plane strain condition exists in the section being considered. Thus, the strain inthe vertical direc- tion must be everywhere zero; and all other strains and (stresses must be independent of the vertical coordinate. This approximation is reasonable at sections distant from the ends of the shaft. (2) There is no volume change in the frozen soil dur- ing deformation. Thus, 6 + e = 0, (5.1) where er and 80 represent the direct strains in the radial and tangential directions, respectively. It follows from the incompressibility condition that Poisson's ratio is equal to 0.5. Experiments on frozen soils have shown that this is valid "for sufficiently developed creep deforma- tions" (Vialov, 1965b). Therefore, it is necessary that = 1 oz 2 (or + 06). (5.2) where the subscripted o's denote direct stresses in the in— dicated orthogonal directions.) Taking compressive stresses as positive, a is the major principal stress, or the 6 minor, and oz the intermediate. (3) Displacements are small enough that the initial dimensions and coordinates can be used throughout the defor- mation process. (A) Only the steady state portion of the creep curve need be considered. Adjustments for deformation occurring A5 during primary creep can be made during excavation. Stresses necessary to produce progressive flow during the construction period may be ruled out by the limiting strength condition. However, the results of this study are not sufficient to insure this. Therefore, care should be taken in this regard. Due to assumption (1) and the radial symmetry in- volved, the equilibrium equations reduce to the following single equation for this case. (101, (Jr-0'6 (5.3) flee Ee-Er (5J4) 'The strain —displacements relationships are u (5.5) 86 — '1: 23nd _d_u (5.6) 8r 3 dr ' ivhere u is the radial displacement and is considered pos- lrtive in the inward direction. The boundary conditions on Stresses are A6 0 at r (5-7) 0 II ll 9) and c = p at r = b. (5:8) The solution involves using the conditions given above along with a constitutive relationship for the mater- ial. This is done in the following paragraphs for frozen soil using two relationships given by AlNouri-(1969). 5.2 Analysis Based on a Creep Equation While it has often been assumed that the mean normal stress has no effect on creep in frozen soil, AlNouri (1969) has shown that the following equation appears to describe the creep behavior for two frozen soil types: 61 = C° exp [N(al-c3)].exp(-mam). (2.7) The mean normal stress, am, is equal to one—third the sum of the direct stresses, while C, N, and m are parameters which must be determined experimentally. N and m were found to vary only with soil structure, while C depends on both temperature and soil structure for the range of stresses and temperatures studied by AlNouri. Equation (2.7) is based on the results of differential creep tests, during which the hydrostatic pressure was varied in increments while the applied axial stress remained constant. Tests were conducted in a standard triaxial testing cell. A7 Since Equation (2.7) was derived for a uniform stress condition in which two of the principal stresses were always equal, it is necessary to modify it in order to apply it to a more general stress situation. It is proposed that Equa- tion (2.7) be extended as follows: ml- = C: exp(N3) .exp(-mom) (5.9) where o is the effective stress given by Q I ={l/2[(o me)2 + (ore-oz)2 + (oz-cr)2] + 3(Tr22 + r 2 + Ire?) 1/2. (5.10) 26 This expression seems a reasonable one to use for the deriv- -0.) 1 3 for the stress condition used in deriving Equation (2.7) atoric portion of the stress since it reduces to (c (i.e. dz = 01, 0r = 09 = 09, all T's = 0). Similarly, : is the effective strain rate given by 1' = 2/9[(e°:r-ée)2 + (ea-632)? + (éz-érfl] ° 2 ' 2 ' 2 1 2 ' + 3(er + Yze + Yr6 ) / ‘ (5.11) For the case of ér = e E + e + é = 0, E = El, and all shear-strains equal to zero, 2 becomes a}, thus making it a A8 reasonable choice for the strain rate element. The mean normal stress is o = 1 (o + o + o ) . ' ‘ m 3 r 9 z . . (5012) Using Equations (5.2), (5.3), and (5.10) to calculate 8 for the thick-walled cylinder, it becomes .5 3 3/2)r§f£. ' (5.13) dr Similarly, based on the plane strain assumption (sz 8 0) and the incompressibility condition (5.1), the effective strain rate is e '(2‘5’73‘9‘ (5.114) The calculation of am yields do 2: r. (5.15) Substituting the strain-displacement relationships (5.5) and (5.6) into Equation (5.A) and taking the time derivative, the equation g; +% a 0 (5.16) r is obtained. Solving Equation (5.16), k1 (5.17) A9 where k1 is a constant for the steady state portion of the creep curve. Thus, from Equation (5.5), . k E -: 1 9 r7 (5.18) and substituting this for £9 into Equation (5.1A), .3 g .2551. ' (5.19) 3 r2 The governing differential equation is formed by sub- stituting expressions (5.13), (5.15), and (5.19) into Equation (5.9). After doing this, taking the natural logarithm of both sides, and rearranging terms, the dif- ferential equation becomes 932.-(_m)o =.__1_ 1,, .2431 (5.20) dr Qr I‘ Qr 301-2 , where Q = £31" - m. Solving Equation (5.20) for or’ 2 1 35-h) 2 ' '3 (5.21) a —— 1n'———§- +._Q Q or m (3Cr m2 + k2(r) Using boundary conditions (5.7) and (5.8) to solve for the constants of integration, k1 and k2, they are found to be a k =fli§ex9§+pm+21nkl l 2 m m -m (be- (mi-1 and 50 p + 31 111(3) k2 = a in. Q Q (b) -(a) The expressions for stresses, radial displacement.rate, and strain rates are then ”Q Q Q L(b) (a) ‘1; p 1 + 21 (a, m n - p b m 19. T<;__‘,___/ \—-—\r-—’ P Q Q (b) (a) -1 L. .- exp gfi'+ m T? a ’ pm + ZlnL-J Q Q ’ L(b) (a) -1, Q Q L“) (a) -1 13 2 j? * _1 a 2 p + a ln‘s or = a 1n ; + (a -1 E _E' Q Q (b) (a) -1 _ m " i - 2 a 2 Q p + - ln(-) .9 . - .3, ing.) . [(g) (1 . 3+1] E m Q Q Eb) (a) -1 m _ T ' 2 a 2 D +- 1n - Oz 7 EE'ln (%’] + [(E) 1- 2%)-1] 1!: ~mlb' (5- (5. (5. 22a) 22b) 22c) .23) .2Aa) .2Ab) 51 The variation of u and as across the section is shown in Figure 5.2 for a unit rate of displacement at'r - a. Equation (5.23) can be used to predict the deformation of a cylindrical soil-ice barrier which satisfies the pre— ceding assumptions at any time after the initiation of secondary creep. Therefore, the dimensions of the shaft could be determined to fit the size of the permanent sup— ports at the time of their placement. The parameters C, N, and m could be determined by conducting differential creep tests on undisturbed samples frozen to several different temperatures. Based on this, the temperature and the size of the soil-ice barrier could be chosen so that displace- ments would remain within tolerable limits. Stresses calculated according to Equations (5.22) can be used to check the strength of the barrier. Stress com- binations must be compared with the failure criterion (e.g. Mohr-Coulomb) to insure that conditions necessary to pro- duce failure exist nowhere in the barrier. 5.3 Analysis Based on Time Dependent 7— Strength Parameters AlNouri (1969) reports that time dependent strength parameters cohesion, c, and angle of internal friction, e, can be used to describe creep behavior in cylindrical sand- ice samples. Using the results of differential creep tests, AlNouri plotted Mohr circles for various stress conditions, each of which produced the same strain rate (See Figure 5.3). He showed that for a sand-ice material one straight 52 1.0-- H I s ...-q t m 0.5 1 0 w 0 i i i A 1.00 1.50 2.00 2.50 Radius-in 1.0 - .fi 8 0.5 u \. c .H l o: 0 t i : i 1.00 1.50 A 2.00 2.50 Radius-in _ b=2. 5" K Figure 5.2. Distribution of u and £9 for a Unit Rate of Displacement at Inner Surface 53 coapmpcomomaom oHoaHu ago: .m.m oasmflm o .mmoppm pooafim 66 @«b Gab emb one 6.6 O 6, AU \\ \\ \V\\ .\ S u m (\\\\\\1 a ewx\\ % \\ a 8 S S l pampmcoo u manpmaoQEoe .opma Gammon meow map oosooaa zoom "opoz 6 w .6 .6 moaonfio ago: an. ooudomoanou momwonpm 5A line could be drawn tangent to each of the circles. Thus, this straight line, defined by its intercept d, and slope angle ¢. is characteristic of a given strain rate. The use of such a geometrical representation immedi- ately suggests an analogy to the Mohr-Coulomb failure criterion. It should be noted here, however, that the straight line described above is not related to failure in the conventional sense. Rather, it defines stress combina- tions necessary to produce a particular strain rate. Failure need not be approached anywhere in the soil-ice mass. 3 By considering the geometry of the Mohr plot, the principal stresses can be related to the time dependent strength parameters as follows: a -N20 = 2cN. (5.25) l 3 01 and 03 are the major and minor principal stresses, re- spectively, and N = tan (A50 + ¢/2). As noted above, 06 corresponds to 01 and or to 03 in the problem under consid- eration. Thus, after rearranging terms, Equation (5.25) can be rewritten 00 = Nzor + 2cN. (5-25') Substituting this expression for a into the equation 8 of equilibrium (5.3), the resulting differential equation is 55 333 °r(1'-'N2) , 20N. 03-26) dr + r r' Solving Equation (5.26) for c , r N2-1 7 (5.27) N -1 where k is a constant of integration. The value of k can 3 . 3 be determined by using the boundary condition that or = 0 at r = a. N2-l 2c§ k3 N -1 (a) Thus, [ N2-1 ] ( 8 2cN g -l 5.2 ) 9' a N2-1 (a) Since the pressure, p, is applied at r = b, the following relationship results: 2 N -1 p - asks '4]- (5.... Equation (5.29) relates the outside pressure and the geome- try of the soil-ice barrier to the time dependent strength parameters. 56 The use of Equation (5.29) is complicated by a diffi- culty in obtaining reliable values for c and ¢~ Little is known regarding the nature of these time dependent parame- ters. By using a trial and error method, it would be necessary to conduct a large number of creep tests in order to determine them for a range of strain rates. However, AlNouri's (1969) work indicates that only a few tests are needed to evaluate the constants C, N, and m. Then Equa- tion (2.7) can be used for the determination of C and e. For a particular strain rate, various values of 01 could be specified and the resulting values for 03 calculated according to Equation (2.7). Thus, a series of Mohr cir- cles could be drawn and the strength parameters would then be determined by drawing the straight line tangent to them. 3 A further limitation is that it is not known whether this approach is applicable to'a wide range of soil types. Available data (AlNouri, 1969) are only for a sand-ice system. Assuming that satisfactory values of c and d could be obtained, Equation (5.29) would be a useful tool for de- signing frozen soil barriers. The radius of the shaft, a, would be fixed in most cases and the pressure could be es- timated based on earth pressure theory. The size of the barrier necessary to limit the strain rate to an acceptable value at a given temperature could then be determined by calculating the outer radius, b, from Equation (5.29). CHAPTER VI EXPERIMENTAL RESULTS 6.1 General The experimental program consisted of 18 tests on the sand-ice material and 6 on frozen clay. In each test (with two exceptions), a lateral load of 100 pounds per square inch was initially applied to the model. This produced little or no deformation in the sand-ice samples but served to account for any seating difficulty caused by shifting of the sample. After a brief period of time, the load was in— creased to 200 p.s.i. and then in 100 p.s.i. increments for the remainder of the test. Each load increment was allowed to remain until the steady state portion of the creep curve was established. The only exception to this was at low stress levels in the sand-ice samples where such stresses produced insignificant deformation. The rate of deformation for each load increment was then calculated by using the calibration information for the deformation measuring de— vice (See Appendix B). Due to the new design of the testing equipment and procedures, a number of difficulties were encountered in the 57 58 experimental work. The results of early tests are probably questionable since preparation techniques and testing pro- cedures were being formulated, and it was uncertain exactly what methods would provide the desired effect. These tests served to define the capabilities and limitations of the equipment as well as to establish procedures. Modifications in both the equipment and techniques were made throughout the entire testing program in order to produce more reliable results. In addition, mechanical problems eliminated results from several tests and rendered results invalid on others. Five tests were aborted due to failure of the membranes en- casing the samples. This allowed leakage of the ethylene glycol into the sample, causing an immediate loss of pres— sure in the system and melting of the sample. Membrane failure was believed to be related to difficulty in obtain- ing flat, square ends, particularly in the sand-ice samples. This caused small gaps between the sample and the piston, thus allowing the membranes to be punctured as they were stretched into these irregularities by higher pressures. Greater care was exercised during sample preparation and more membranes were used on each sample to correct this. However, the problem was not completely solved. The deformation measuring device was the source of another difficulty. The brass pieces had a tendency to ro- tate approximately 900 as the device was installed. This caused the wrong side of the brass pieces to be in contact ..sl 59 with the sample and, thus produced erroneous readings on several tests. When this problem was recognized, re- straints which limited the range of rotation of the brass pieces were placed on them. Lateral shifting of the sample both during the prepa- ration and during testing created difficulties in aligning the center hole of the model with the hole in the piston. To rectify this problem during preparation, a precooled steel rod, 2A inches long and 1 7/16 inches in diameter, was placed through the center hole of the model and into the hole in the pedestal. The piston was then guided into place by sliding it over the steel rod. The rod remained in place until the cell was assembled and placed in the cooling tank. To eliminate shifting of the sample during testing, a styro— foam plug was placed in the hole in the pedestal and allowed to extend 1/2 inch into the center hole of the sample. This stabilized the sample but did not interfere with its defor- mation. The two piece.piston was initially fabricated by "force fitting" the bottom, flat cylindrical portion onto the shaft sportion. It was soon discovered that the downward force due to the pressure on the flat portion was causing it to slip slightly on the shaft. A single 1/8 inch pin was then placed through the two pieces and it was later replaced by three, l/A inch pins. This slipping induced an axial load in the sample in excess of that necessary to enforce the plane strain condition. It also appeared that a small 1' 60 amount.of ethylene glycol leaked into the sample through the area of contact between the two pieces. This caused weaken- ing of the sample and excessive deformation in several tests. A silicon rubber sealer was placed at the interface to eliminate leakage. Both the slippage and leakage ap- peared to occur in the samples with a 5 inch outer diameter and at high pressure. This condition created the greatest downward force on the piston. Thus, while a total of 2A tests were conducted, only a somewhat smaller number can be considered valid. All re— sults are reported in the following sections. Comments are made regarding the relative validity of them. 6.2 Deformation Results 6.2.1 Sand-Ice A summary of the tests conducted on sand-ice samples is given in Table 6.1. An attempt was made to determine displacement rates for various pressures and wall thick— nesses. Using an inner diameter of 1 l/2 inches, samples with outer diameters of 3 1/2, A, A 1/2, and 5 inches were - tested. The length of nearly all samples was approximately 9 inches. The only exception was Test SA-lO where damage near the end of the sample caused its length to be reduced to 7 1/2 inches. Typical displacement results are given for two tests in Figures 6.1 and 6.2 in the form of time vs. radial dis— placement at the inner surface plots. These results show 61 Table 6.1 Summary of sand—ice tests Test Outer 'Height Water .Designation Diameter (inches) Content (inches) (g) Remarkg SA-l 5 9 21.0 Membrane failed SA-2 5 9 18.5 Membrane failed. Reused sand changed proper- ties. SA-3 5 9 21.6 Low pressures used produced insignif- icant deformation. SA-A A 9 5/16 20.6 ——— SA-5 3 1/2 9 20.2 Poor alignment of sample and piston SA—6 3 1/2 9 20.2 Contaminated sand used SA—7 3 1/2 9 1/8 21.0 -—— SA-8 u 9 5/16 21.6 Cooled to —25.A°c prior to test SA—9 A 9 3/16 21.2 -—— SA-10 3 1/2 7 1/2 20.7 Brass piece ro- tated SA-ll 3 1/2 9 1/8 21.A -—— SA-l2 A 1/2 9 3/16 21.0 -—— SA-13 3 1/2 9 1/8 20.8 Sample shifted SA-lA 5 9 l/A 21.6 Piston slipped. Ethylene glycol may have reached sample. SA-15 3 1/2 9 -—- Membrane failed SA-l6 5 9 l/A 21.3 Small amount of ethylene glycol may have reached sample. SA-l7 3 1/2 9 21.8 Membrane failed SA-18 5 9 21.A Small amount of ethylene glycol may have reached sample. Note: Data on all samples Inner diameter 1 1/2 inches Test temperature -12.OOC Sand density 6A% by volume mlm III III com om.HH om\MH om.HH om.m mm.0H m>.m m=.m om.m III III com om.m om.m mb.s mo.m oo.m o=.w m>.w mm.m mm.m oo.m cos oo.m oa.w mm.w mm.a m:.= om.m m=.a ou.m mm.m om.m com 05.: om.m mm.m m=.m mm.m mH.m om.m mw.m ~>.H mm.H com III III III mm.m wN.H wo.m NH.N mm.m =N.H om.H oo: =muao =muao emuao =N\H :uoo =N\H muao eanOIkhmao :N\H muno #M\H muao tango mHI¢m mHIf'the stress (3) is separated from the hydrostatic portion (<3m). The relative effect of these portions on deformation 21:3 not well understood. In fact, it has often been assumed that creep is completely independent of the hydrostatic Stress. It is, therefore, proposed that an adjustment be made in the relative contributions of the two parts of the E3tzress. This adjustment can be made in two ways. First, the hyclrostatic component can be allowed to remain as in 77 Equation (5.9) while the deviatoric portion is adjusted. After doing this, Equation (5.23) becomes - q 2 ' pm + 2 1n (3) 1'1 =1 30a 2Q + __ b 2r ex? m m L (b) (a) -1J (5.23') 46?"?! where Q' = XN-m/2. The value of x, which was1/372 in the original formulation, can then be chosen to fit the experi- mental data. I In order to aid in the selection of an X value, a program was written for the CDC 3600 computer. This allowed a large number of possibilities to be tried with a minimum computation time. The results of these computations for various values of X are given in Table 7.1. Also listed in Table 7.1 are experimental values for displacement rates from correspond- ing test results. The experimental results given are those from the tests which appear to be most reliable. Since Equation (5.23') yields a straight line when the natural logarithm of the displacement rate is plotted against the applied pressure, such a plot is useful for com- parison purposes. This is shown in Figure 7.3 for three sizes of cylinders at varying pressures. The straight lines represent the relationship of Equation (5.23') for the in— dicated values of X . Experimental values are plotted for several different pressures from each of three tests. 78 m\Enzx n .0 "opoz ea.oa ma.m om.c mm.s. ec.c ao.c ee.m me.oa oom cc.c mm.» sc.c no.0 mm.m mc.m ac.: cm.c ccc a=.m mm.m mz.m mm.: cc.: :m.z am.m mc.m cos ca.m =>.: a=.= ca.: mm.m cm.m mm.m mm.: ccc cc.= am.m mm.m nm.m ua.m mm.m aw.m m:.m ccm mm.m cc.m am.m an.m :c.m am.m mm.m mm.m cc: mau¢m =m\a sumo m>.ma m~.:a sc.ma sm.aa mm.ca ca.m cc.w mc.ma cm.za com am.ma cm.aa mc.ca mm.w sc.m mm.» mm.c cc.m m~.m ccw mm.m ca.c cs.s mc.c mm.c c>.m sm.m cm.c c:.c cos mm.c m=.c am.m ::.m cc.m ac.z mm.: uc.m cm.m ccc mm.m sm.: mm.: mm.: :m.m mc.m m:.m m>.m ma.m ccm am.m cc.m cz.m mm.m ca.m mm.m ss.m cc.m sc.m co: m a m|< mum mm.am ~:.>m a~.mm ma.cm m~.>a =m.ma mn.am m:.ca ccc Na.mm mm.am um.ma mm.ca mm.:a mc.ma m:.aa mm.ca mm.ca ccm cc.~a wa.ma am.ma :c.ma m~.ca wm.m mm.m c~.c cc.m con mm.aa m:.ca mz.m mm.m nu.» :c.~ mm.c no.3 m:.= com um.» mm.~ mc.c aa.c mc.m sa.m 55.: mm.m mm.m ccm am.m cc.= cc.: mm.= no.3 cm.m cm.m cc.m mm.a cc: ac.a mc.a cc.a mm.a cm.a am.a mm.a n a aa|< aau.=H mm.ma 0m.=H 000 mm.» 0H.m 00.0 mm.m H:.0H Hm.HH 00.0 m>.m 000 :0.0 03.0 00.0 0:. m0.0 00.0 0m.0 0:.0 00» mm.: 0H.m ma.m zm.m mm.0 m0.0 Nm.m 0m.m 000 mm.m 0a.: mm.= 0m.: Hm.= 00.m mw.m mH.m 00m ma.m mm.m 0:.m mm.m m~.m mm.m 00.m ~0.m 00: m a m|< mu.ma 05.0 00.0. 00> m=.m Hm.m mm.> mm.~ 03.0 HH.0 50.: m=.= 000 hm.a mfi.m 0:.m m>.m =H.m mm.0 mm.m mm.m 00m >0.m mm.m no.3 mm.= 03.: 00.: 00.m mm.a 00: m=.0 32.0 ma.0 m:.0 H=.0 03.0 n a HHn< Hau ] + [(a) (1 + 2Qv)"] m by (5-226 ) m —m 6' ‘Q’v _b a l_ (where Q' = XN-m/2) over the cross section of the cylinder. (See Appendix C for details of the calculations). Results of these calculations are given in Table 7.3. Comparing these values with those shown in Figure 6.5, it can be seen that they fall within or near the range of the axial forces measured. In most cases, the measured values appear to be asymptotically approaching the calculated ones. Thus, the assumption that 0 = %(or + 09) appears to be reasonable 2 for the advanced stages of creep. 89 7.2.2 Comparison With c—¢ Analysis Equation (5.29) was used to compare the analysis based on time dependent strength parameters with eXperimental results. For a given measured strain rate, values of c and 0 were determined based on data reported by AlNouri (1969). The pressure was then calculated according to Equation (5.29) and compared with the actual pressure that produced the strain rate. Values of a for 5 different axial strain rates meas- ured by AlNouri are given in Table 7.4. The corresponding value of 0 as determined from the series of differential creep tests is 35.2°. A series of constant strain rate tests (é = 3 x 10'3 min‘l) produced a friction angle of 25°. Since the strain rate varies across the transverse section of the cylinder, it was necessary to determine what strain rate would give the prOper value of a before carrying out the calculations. It was assumed that the average value of E would approximate AlNouri's axial strain rate. (See 6 Appendix C for details of calculations). Values of a were selected from the E vs a plot shown in Figure 7.9. Values of c were calculated according to the formula 0 = a/cos ¢ (Lambe and Whitman, 1969) for the two values of ¢. Results of the calculations of c from measured displacement rates are summarized in Table 7.5. A comparison of pressures calculated according to Equation (5.29) from measured strain rates and the actual pressures that produced these strain rates is given in Table 1‘ 90 Table 7.h. Values of a (From AlNouri, 1969) é a (min‘lx 10-5) (psi) u 79 6 96 8 112 10 125 20 155 ¢ = 35.2° (Creep test) 1 ¢ = 25° (Constant strain rate test, 8 = 3 X 10-3 min- 3 c a uu3 psi) . 160 -- 120 4» ‘5'. cm 80 +- l s 40 .. 0 t # i t % 0 5 10 15 20 25 E - min x 10'5 Figure 7.9. Plot of E vs a 91 TABLE 7.5 Data for c-¢ analysis Outside Pressure 500 600 700 800 900 (psi) . u (in/minax 10‘5) 2.98 A.A5 8.00 10.55 16.u5 fr: 69a t—i A(min'} x 10'5) 3.98 5.9M 10.67 lu.08 20.9 33 6010 :;(min'1 x 10-5) 1.70 2.5a u.57 6.03 9.u0 \ a H (psi) 57 65 8A 97 122 m "C, ¢=35.2° Q (psi) 69.8 79.6 102.7 118.7 1h9.3 C90, ¢=25O (psi) 63.0 71.7 92.7 107.0 l3u.8 Ufa (in/min x 10'5) 2.15 3.50 6.u0 9.75 1h.30 P. 88a ‘T(mi“-} x 10‘5) 2.87 u.67 8.5a 13.00 19.07 < e ;"3(min-19’£ 10-5) 1.08 1.75 3.20 £4.87 7.15 a- a u (psi) 50 58 71 87 106 39. <1>=35.2O 61.2 71.0 87.0 106.14 129.8 (psi) 9. ¢=25° (psi) 55.1 69.0 78.u 96.0 117.0 0 23(1n/m4nax 10-5) 3.u5 A.35 6.05 8.20 10.75 H 80 $(nnn-4 7 10‘5) u.60 5.80 8.07 10.92 lu.33 a; :0m _5 ' ' ::(min“ x 10 ) 1.53 1.93 2.69 3.6a u.77 a '7 (psi) 55 59 67 76 87 "C, ¢=35.20 0 (psi) 67.u 72.2 82.0 93.0 106.u Oc, ¢=25° (psi) 60.7 65.1 7u.0 83.9 96.0 91 = rate of radial displacement at r = a E20a E0m circumferential strain rate at r = a mean value of circumferential strain rate 92 7.6. It can be seen that the predicted pressures are from 1 1/2 to nearly 3 times the actual pressures when the value 35.20 is used for 0. For the constant strain rate friction angle of 25°, the agreement is considerably better. The distribution of or and 09 as calculated by Equa- tions (5.28) and (5.25'), respectively, is shown in Figure 110 for a cylinder with an outside diameter of N 1/2 inches and an inside diameter of 1 1/2 inches. The value of c used was 7U.0 p.s.i. This value was calculated using a dis— placement rate at the inner surface of 6.05 x 10"5 in/min, the measured value for an outside pressure of 700 p.s.i. iNote the close agreement between this distribution and that xoredicted by Equations (5.22a) and (5.22b) (See Figure 7.7). 7.3 Frozen Clay Although the amount of experimental data is less for ‘the frozen clay, a comparison of the analytical and experi- Inental results similar to the one given for the sand-ice rmaterial was attempted. The following values for experimen- tal parameters were taken from the results of AlNouri (1969): c = 2.77 x 10"“ min-1 N = 2.68 x 10"3 in2/1b m = 1.0N9 x 10‘2 in2/1b 1\s shown in Figure 7.2, the strain rates measured in this Estudy are at the lower end of the range used by AlNouri. TPhis is due to the relatively large deformation which 93 TABLE 7.6 Summary of comparisons for c-¢ analysis ...—...-.- Actual fi gem Predicted Pressure a Pressure (psi) (psi) (in/min x 10‘5) (min-l x 10-5) 0 = 35.2O 0 = 250 500 2.98 1.70 801 331 S 600 14.45 2.514 9113 376 «fi? 700 8.00 A.57 1179 _ u87 0:9 800 10.55 6.03 1363 - 562 C) . 900 16.u5 9.u0 171a 708 500 2.15 1.08 1035 379 fi 600 3.50 1.75 1201 the 7:5; 700 6.u0 3.20 1u71 539 <1: 833 800 9.75 n.87 1799 660 900 lu.30 7.15 2195 8AA 500 3-“5 1-53 1575 519 E 600 14.35 1.93 1687 557 HA asifl 700 6.05 2.69 1916 633 I £53, 800 8.20 3.61: 2173 717 Q 900 10.75 u.77 2u87 821 ca = rate of radial displacement at r = a E = mean value of circumferential strain rate em 94 16004 Eq (5.25') 2 a. 800- I 74.0 psi Dd) X 1.0-5 O 7 11, % 1.00 1.50 2.00 Radius-in 800-- / /' E (5 22afy”/ H Q- . ‘/ w / Q ./ I 400-- ‘/’// Eq.(5.28) o“ ’,/” //’ 2’ 0 : 7 i 1.00 1.50 2.00 Radius-in b = 2.25" \\ I : a 3 .75" Sand-Ice Cylinder p = 700psi Figure 7.10. Distribution of 0 r and 06 According to Equations (5.28) and (5.25') 95 occurred during primary creep. This made it impossible to obtain larger steady state displacement rates before the to- tal deformation became excessive. Using the parameters given above, direct calculations according to Equation (5.23) produce displacement rates con- siderably larger than those measured. It was, therefore, raecessary to modify Equation (5.23) by adjusting the rela- 'tive contributions of the deviatoric and hydrostatic parts c>f the stress as in Section 7.2.1 for the sand-ice. Using [aquation (5.23') with an X value of 6.40, and dividing the (coefficient to the exponential by 100, the comparison illus- txrated in Figure 7.11 was obtained. It was not possible to fit the experimental data by using Equation (5.23") which ‘vardes the hydrostatic portion. Figures 6.6 and 6.7 show that the total axial force in tale frozen clay samples remained nearly constant for each libad increment. Thus, it is probable that, unlike the sand- 113e»samples, the stresses are constant throughout the derformation process. This tends to indicate that the devel- OFunent of the stress distribution illustrated in Figure 7.8 fxxr the sand-ice material occurs much more rapidly in the Case of frozen clay. The difference in axial force development between the thud materials may be explained by considering the difference 1ri the deformation characteristics of them. The small pri- "Ntby creep deformation observed in the case of the sand—ice -5 a - in/min x 10 u Rate of Radial Displacement at Inner Surface, 3.0 2.0 .15 Figure 7.11. 96 By Analysis (X = 6.40) (D C - 4 B C - 3 AL 47-- 1 f ) .. a 21n — - _GCa )20' p’“ (b) “ ua ‘ 200 'exP ‘5‘ + m. _p_ > I V \ bQ a Q -l J .. ' = _ B Q XN 2 -- c = 2.77 x 10’“ min"1 N = 2.68 x 10"3 in2/lb m = 1.0u9 x 10"2 in2/lb a = 0.75 in b = 2.50 in I i i I 0 100 200 300 u00 Outside Pressure, p - psi Graphical Comparison, Frozen Clay, Equation (5.23') (Modified) 97 indicates that this material is quite stiff and, thus, is slow in reacting to a change in external pressure. On the other hand, the frozen clay exhibited relatively large def— ormation during primary creep. This tends to indicate that the material flows more readily, reacts more quickly to an external change in pressure and, thus, reaches a final state of stress more rapidly. A related result was reported by Goughnour (1967). He found that, in constant strain rate tests, the peakstress was reached at approximately 2.5% strain for sand—ice sam- ples; whereas, in the case of frozen clay the peak stress ‘was not approached until a strain of 10% or greater had been realized. This tends to substantiate the argument given above regarding the relative stiffness and flow characteris— tics of the two materials. This may be explained by considering the development of the frictiOnal component of shearing resistance. Indications are that in the case of the sand-ice material, the sand particles are initially either in contact with one another or very nearly so. This allows the frictional component to be mobilized almost imme— diately. However, for clay, where the particles are widely dispersed, the frictional component cannot be fully developed until the occurrence of relatively large strains. This sug- gests the use of an analysis where 0 is taken as zero and a strain rate dependent cohesion used. There are, however, in- sufficient data available to verify this. 98 In order to check the values of total axial force meas- ured, Equation (5.22c') was integrated across the section. Using the experimental parameters given above for the frozen clay and X = 6.40, the values for total axial force were found to be 1958, 3925, and 5888 pounds for outside pres- sures of 100, 200, and 300 p.s.i., respectively. Comparing these with the measured values illustrated in Figures 6.6 and 6.7, it can be seen that the calculated ones are higher in each case. One possible explanation for this discrepancy, as well as for the inconsistency in displacement results, is that the experimental parameters may be strain rate depen- dent. Therefore, since it was impossible to achieve the range of strain rates for which the parameters were estab- lished (See Figure 7.2), the use of these parameters may not be Justified. A second explanation concerns the fact that the constitutive equation used was established for steady state creep. Since this stress distribution appears to have developed during primary creep, it may be improper to use the constitutive relationship for this case. CHAPTER VIII CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 8.1 Summary of Conclusions Based on the experimental results for the two frozen soils used and their comparison with analytical calculations, the following conclusions can be drawn with regard to hollow frozen soil cylinders, loaded by an outside pressure: 1. Displacement rates calculated according to Equa- tion (5.23), which is an extension of a creep equation suggested by AlNouri (1969), do not agree with actual dis- placement rates when experimental parameters reported by AlNouri are used. After modifying this approach by the in- troduction of additional parameters, the analytically predicted displacement rates compare favorably with those experimentally measured. Therefore, it can be concluded that, while the form of Equations (5.22), (5.23), and (5.24) appears to be valid, more experimental data are re- quired to elucidate the nature and the use of the experimental parameters. 2. The relationship presented in Section 5.3 based on time dependent strength parameters (AlNouri, 1969) appears to be valid for the sand-ice material, provided the 100 correct value of friction angle is used. The friction angle which provided the best check with experimental data in this study was that determined from constant strain rate tests, rather than the value from creep tests. Since this result is unexpected and difficult to explain, no generalization re- garding it can be made without further experimental work. 3. The deformation characteristics of sand-ice cylin— ders under constant radial load differ considerably from those of frozen clay, although both are time dependent. In the case of the sand-ice material, the magnitude of the def- ormation during primary creep is small, the length of time necessary to reach secondary creep is small, and the steady state portion of the creep curve is well defined. For frozen clay, both the length of time and the amount of deformation are much greater for primary creep and'the steady state part of the creep curve is less well defined. 4. The behavior of the total axial force in hollow frozen soil cylinders under conditions of plane strain also differs for the two materials used in this study. The axial force in sand—ice cylinders is time dependent. It increases during the early stages of creep and then approaches a con- stant value. This increase in axial force is probably due to a continuing redistribution of stresses during the time imme- -diately following loading. As creep continues, the stress diStribution stabilizes, producing a constant axial force. 101 By contrast, the total axial force in the frozen clay cylinders reaches a constant value shortly after loading. Thus, any redistribution of stresses occurs rapidly. 5. The phenomena discussed in Conclusions 4 and 5 appear to be compatible. They can be explained by consider- ing the relative stiffness and flow characteristics of the two materials. The stiffer sand—ice exhibits little defor— mation during primary creep, reacts slowly to an external load, and, therefore, experiences considerable delay in reaching its final state of stress. The frozen clay, how- ever, shows a greater ability to flow by its large primary creep deformation and is able to reach its final stress state more quickly. 6. The total axial force, as calculated by integrating Equation (5.22c') across the section, was in reasonable agreement with the actual axial force for most cases in- volving the sand-ice material. This supports the approximation that the axial stress is equal to one half the sum of the major and minor principal stresses. The above relationship is based on the argument that frozen soil is in- compressible during the deformation process, at least for the advanced stages of creep. These results are in agreement with Vialov (1965b). Discrepancies occurred regarding the total axial force in the frozen clay samples. Calculated values (Equation 5.220') were 50 to 100% higher than those measured. This may Iv 102 be due to a difference between the strain rates used in this study and the ones from which AlNouri (1969) determined the experimental parameters. 8.2 Suggestions For Further Research This study represents a small contribution toward the understanding of the stress-deformation characteristics of soil-ice barriers used in shaft sinking. Many areas require further study before the complete problem can be solved. Several of these areas are listed in the following para- graphs: 1. Further investigation is required into the nature of the experimental parameters used in the analyses. Several of these were well established by AlNouri (1969) for a cer- tain range of strain rates and for the uniform stress condition existing in a triaxial test. It is yet unclear how they should be extended for use in situations where stresses are non uniform. More information is also neces- sary regarding the parameters X and Y introduced in this study. 2. Studies similar to this one should be carried out on different soil types at varying temperatures to determine the general application of the conclusions drawn above. 3. The temperature was uniform throughout the frozen soil models tested in this study. Under actual field con- ditions where freeze pipes are used, such a condition would 103 be impossible to achieve. Therefore, it would be desirable to conduct a model study in which a more realistic approx- imation of temperature distribution is used. BIBLIOGRAPHY BIBLIOGRAPHY AlNouri, I. "Time Dependent Strength Behavior of Two Soil Types at Lowered Temperatures", Ph.D. Thesis, Michigan State University, East Lansing, Michigan, 1969. Andersland, O. B., and W. Akili, "Stress Effect on Creep Rates of a Frozen Clay Soil", Geotechnique, Vol. XVII, No. 1, March 1967, pp 27-39. Brace, J. H., "Freezing as an Aid to Excavation in Unstable Material", Transactions, ASCE, Vol. 52, pp 365-436, 1904. Cross, B., "Liquid Gas Freezes Bad Soil", Construction Methods and Equipment, July, 1964. Goughnour, R. R. "The Soil-Ice System and the Shear Strength of Frozen Soils", Ph.D. Thesis, Michigan State Univer- sity, East Lansing, Michigan, 1967. Goughnour, R. R. and O. B. Andersland. "Mechanical Proper- ties of a Sand-Ice System", Journal of the Soil Mechanics angFoundations Division, ASCE, Vol. 94, No. 8M4, July, 1968. "Freezing Makes Shaft Sinking Easier", Construction Methods and Equipment, Oct., 1964. Lambe, T. W. and R. V. Whitman. Soil Mechanics. John Wiley &Sons, Inc., N. Y., 1969. Latz, J. E. "Freezing Method Solves Problem in Carlsbad N. M. Shaft", Mining Engineering, Oct., 1952. Low, G. J. "Soil Freezing to Reconstruct a Railway Tunnel", Journal of the Construction Division, ASCE, Vol. 86, No. CO3, Nov., 1960. Sanger, F. J. "Ground Freezing in Construction", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. 3M1, Jan., 1968} Silinsh, J. "Freezing Keeps Shaft Dry and Holds Dirt in Plzce", Construction Methods and Equipment, Jan., 19 0. Stewart, G. C., W. K. Gildersleeve, S. Janpole, and J. E. Connolly. "Freezing Aids Shaft Sinking", Civil En- gineering, ASCE, April, 1963. 105 106 Tsytovich, N. A. "Instability of Mechanical Properties of Frozen and Thawing Soils", Proceedings of the Permae frost Internationaliponference, National Academy of Sciences——Nationa1 Research Council Publication No. 1287. pp 325-331. 1963. Tsytovich, N. A., and K. R. Khakimov. "Ground Freezing Applied to Mining and Construction", Proceedings of the Fifth International Conference on§gil Mechgnics and Foundation Engineering, Vol. 2, pp 737-741, 1961. Vialov, S. S. "Plasticity and Creep of a Cohesive Medium", Proceedings of the Sixth International Conference on Soil Mechanics and Foundation Engineering, Vol.1, University of Toronto Press, 1965. Vialov, S. S. Rheological Properties and Bearing Capacity of Frozen Soils, Cold Regions Research and Engineering Lagoratory Translation 74, Hanover, New Hampshire, 19 5a. Vialov, S. S. "Rheology of Frozen Soils", Proceedings of the Permafrost International Conference, National Academy of Sciences-—Nationa1 Research Council Publi- cation No. 1287, pp 332- -337, 1963. Vialov, S. S. (Ed). The Strength and Creep,of Frozen Soils and Calculations for Iceeggil Retaining Structures, Cold Regions Research and Engineering Laboratory Translation 76, Hanover, New Hampshire, 1965b. Yong, R. N. "Soil Freezing Considerations in Frozen Soil Strength", Proceedings of the Permafrost International Conference, National Academy of Sciences-—Nationa1 Re— search Council Publication No. 1287, pp 315-319, 1963. APPENDIX A TEST DATA. 1* 108 TABLE A-l. Test data, SA-4 Materia1-Ottawa Sand Initial Sample Dimensions Sand Density—~64% by Volume Outside Diam. 4" Water Content——20.6% Inside Diam. 1 1/2" Test Temperature-—-12.0°C. Height 9 5/16" Date of Test-—7 Jan 69 Time Outside Pressure Change in Total Axial min) (psi) Diameter (in) Force (lb) 0 O 0 '0 5 100 0.00038 -132‘ 10 100 0.00038 -122* 12 200 0.00038 ~75“ 30 200 0.00072 ~64* 33 303 0.00072 408 35 303 0.00081 408 38 304 0.00128 418 45 305 0.00140 472 55 307 0.00174 483 65 308 0.00202 525 75 309 0.00212 558 90 310 0.00246 53} 93 398 0.00302 12 96 398 0.00319 1294 110 400 0.00387 1434 125 402 0.00451 1563 135 404 0.00494 1638 145 405 0.00523 1713 155 406 0.00562 1788 165 407 0.00587 1863 175 408 0.00625 1906 185 409 0.00647 1981 196 409 0.00685 2045 206 410 0.00723 2099 216 411 0.00748 2153 226 411 0.00766 2206 236 412 0.00792 2260 246 413 0.00817 2303 256 405 0.00851 2367 266 405 0.00864 2410 276 405 0.00885 2464 286 405 0.00898 2496 300 405 0.00936 2539 303 503, 0.00978 2825 305 503 0.00987 2911 315 504 0.01047 3125 325 505 0.01098 3276 *Apparent negative force probably due to shifting of the sample. 109 TABLE A-l Continued Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in), Force (lb) 335 505 0.01145 3383 345 506 0.01187 3469 355 506 0.01230 3544 367 508 0.01280 3619 375 508 0.001310 3673 385 509 0.01349 3726 395 509 0.01434 3812 405 510 0.01434 3812 415 510 0.01455 3866 425 511 0.01502 3909 435 511 0.01528 3962 ,445 511 0.01566 4005 455 512 0.01604 4048 465 512 0.01638 4091 467 601 0.01668 4301 475 603 0.01796 4623 485 605 0.01868 4805 495 605 0.01932 4945 505 605 0.02000 5052 515 606 0.02051 5149 525 606 0.02124 5234 535 606 0.02182 5294 545 606 0.02240 5374 555 606 0.02298 5385 564 606 0.02351 5428 574 605 0.02404 5470 584 604 0.02485 5613 594 603 0.02520 5567 604 602 0.02573 5599 718 547 0.03023 5478 720 604 0.03036 5428 729 606 0.03114 5642 739 605 0-03173 5749 749 604 0.03236 5803 759 603 0.03295 5846 762 709 0.03350 5810 766 711 0.03445 6057 769 711 0.03491 6175 779 711 0.03623 6422 789 711 0.03750 6572 799 711 0.03860 6636 809 710 0.03964 6711 819 710 0.04068 6776 829 ~710 0.04164 6829 839 709 0.04268 6872 TABLE A-l Continued 110 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 849 709 0.04372 6915 859 709 0.04468 6947 869 709 0.04564 6980 879 709 0.04660 7001 889 709 0.04760 7023 899 709 0.04869 7044 909 708 0.04968 7055 919 707 0.05073 7055 929 707 0.05164 7065 111 TABLE A-2 Test data, SA—6 Material——Ottawa Sand Initial Sample Dimensions Sand Density-—64Z by Volume Outside Diam. 3 1/2" Water Content—~20.2% Inside Diam. 1 1/2" Test Temperature-—-12.0°C. Height 9" Date of Test-—30 Jan 69 Time Outside Pressure Change in (min) (psi) Diameter (in) 0 0 0 5 101 0.00004 10 101 0.00004 15 200 0.00013 20 200 0.00030 30 200 0.00043 40 200 0.00055 50 200 0.00072 60 200 0.00081 65 300 0.00132 70 300 0.00153 80 300 0.00183 90 300 . 0.00217 100 300 0.00243 110 300 0.00264 120 300 0.00285 130 299 0.00302 140 299 0.00323 145 401 0.00383 150 402 0.00409 160 401 0.00460 170 401 0.00502 180 400 0.00528 190 400 0.00562 200 400 0.00587 210 400 0.00621 220 400 0.00643 230 399 0.00672 240 399 0.00689 250 399 0.00715 260 399 0.00736 270 399 0.00757 280 399 0.00779 290 399 0.00800 295 500 0.00872 300 500 0.00906 310 500 0.00953 320 500 0.01000 112 TABLE A-2 Continued Time Outside Pressure Change in (min) (psi) Diameter (in) 330 500 0.01038 340 500 0.01081 350 500 0.01119 360 500 0.01157 370 500 0.01191 380 500 0.01223 390 500 0.01260 400 500 0.01289 410 500 0.01323 420 500 0.01357 430 500 0.01387 440 500 0.01417 445 600 0.01519 450 600 0.01557 460 600 0.01617 470 600 0.01677 480 599 0.01740 490 599 0.01796 500 599 0.01855 510 599 0.01906 520 599 0.01953 530 599 0.02004 550 599 0.02098 560 599 0.02142 570 599 0.02191 580 598 0.02244 590 598 0.02289 595 703 0.02356 600 703 0.02404 610 703 0.02502 620 703 0.02591 630 703 0.02676 640 703 0.02760 650 703 0.02844 660 703 0.02916 670 703 0.02982 680 703 0.03050 690 702 0.03118 700 702 0.03186 710 702 0.03250 720 702 0.03314 730 702 0.03373 740 702 0.03432 Note: Sand was reused for this test. Contam- inated sand may have influenced results. 113 TABLE A-3 Test data, SA-7 ....- -1... Matcria1-Ottawa Sand Initial Sample Dimensions: Sand Density—~64z by Volume Outside Diam. 3 1/2" Water Content-—2l.0% Inside Diam. 1 1/2" Test Temperature--12.0°C. Height 9 1/8" Date of Test-—13 Feb 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (lb) 0 0 0 0 3 100 0 -122* 10 100 O -122* 13 205 0 -72* 20 205 0 -61* 25 301 0.00013 167 30 302 0.00013 173 40 302 0.00042 200 50 302 0.00081 232 60 302 0.00123 253 70 302 0.00157 280 80 302 0.00196 275 90 302 0.00221 301 100 302 0.00251 328 110 302 0.00277 355 113 400 0.00315 744‘ 120 400 0.00379 760 130 400 0.00451 819 140 400 0.00511 878 150 399 0.00574 937 160 399 0.00634 996 170 399 0.00719 1066 180 399 0.00766 1136 190 399 0.00830 1211 200 399 0.00868 1286 215 399 0.00953 1393 230 399 0.01034 1495 240 399 0.01064 1559 250 399 0.01123 1618 260 399 0.01157 1683 265 500 0.01255 1964 270 500 0.01311 2103 280 500 0.01362 2296 290 500 0.01434 2436 300 500 0.01502 2554 310 499 0.01570 2661 g*Apparent negative axial force probably due to shift- ing of the sample. 1 1.1L _- 1__‘ a. TABLE A-3 Continued 114 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 320 499 0.01613 2758 330 499 0.01677 2849 340 499 0.01740 2929 350 499 0.01813 3004 360 499 0.01847 3063‘ 370 499 0.01911 3133 380 499 0.01974 3184 390 499 0.02004 3230 400 499 0.02076 3273 410 499 0.02138 3315 413 602 0.02191 3543 420 600 0.02271 3843 430 599 0.02400 3993 440 602 0.02507 4025 450 601 0.02618 4127 460 600 0.02702 4229 470 600 0.02782 4320 480 600 0.02844 4379 490 600 0.02924 4444 500 600 0.03005 4497 510 600 0.03073 4551 520 600 0.03155 4583 530 600 0.03218 4626 540 600 0.03295 4648 550 _ 600 0.03368 4685 560 600 0.03441 4712 563 700 0.03486 4929 570 700 0.03600 5198 580 599 0.03736 5359 590 698 0.03836 5477 600 703 0.03955 5477 610 703 0.04077 5584 620 703 0.04200 ———— 630 702 0.04291 -——— 640 702 0.04386 ———— 650 701 0.04468 -——— 660 700 0.04573 ———— 670 700 0.04682 -——- 680 700 0.04800 -——- 690 699 0.04932 -——— 700 699 0.05047 -——- 710 ' 699 0.05163 -——— 713 804 0.05284 6648 720 804 0.05377 6304 730 802 0.05581 —-—— TABLE A-3 Continued 115 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 740 801 0.05772 —-—— 750 800 0.05972 -——— 760 800 0.06186 ——9— 770 800 0.06372 -——- 773 905 0.06465 6747 780 905 0.06772 7026 790 905 0.07177 -——— 800 905 0.08095 —— 810 905 0.08667 -- 820 905 0.09156 -——— 830 905 0.09698 __1. 840 905 0.10215 -- 116 TABLE A-4 Test data, SA-8 Material-—Ottawa Sand Sand Density-—64X by Volume Initial Sample Dimensions: Outside Diam. 4" Water Content——21.6% Inside Diam. 1 1/2" Test Temperature-—-12.0°C. Height 9 5/16" Date of Test——20 Feb 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) ForCe (1b) 0 0 0 0 3 100 0.00013 14 10 100 0.00021 14 13 200 0.00055 97 20 200 0.00077 140 30 200 0.00094 172 33 301 0.00145 1062 40 301 0.00200 1062 50 299 0.00247 1148 60 299 0.00289 1201 70 299 0.00336 1244 80 299 0.00366 1287 90 299 0.00396 1330 100 299 0.00430 1373 110 299 0.00464 1405 120 299 0.00498 1448 130 299 0.00511 1470 133 400 0.00579 2153 140 399 0.00634 2217 150 399 0.00689 2324 160 399 0.00753 2410 170 399 0.00804 2475 180 399 0.00855 2539 190 399 0.00902 2604 200 399 0.00949 2657 210 399 0.00996 2711 220 399 0.01038 2754 230 399 0.01080 2807 240 399 0.01123 2861 250 399 0.01170 2904 260 399 0.01204 2947 270 399 0.01243 2990 280 399 0.01281 3022 283 500 0.01374 3319 290 501 0.01443 3517 300 501 0.01532 3716 310 500 0.01613 3844 320 500 0.01681 3962 TABLE A-4 Continued 117 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) ‘Force (lb) 330 500 0.01753 4059 340 500 0.01826 4145 350 500 0.01889 4220 360 500 0.01962 4295 370 500 0.02031 4395 380 500 0.02084 4413 390 500 0.02138 4477 400 500 0.02213 4509 410 500 0.02276 4564 420 500 0.02347 4606 430 500 0.02404 4649 433 604 0.02444 4880 440 603 0.02533 5116 450 603 0.02658 5331 460 603 0.02769 5492 470 603 0.02876 5610 480 603 0.02978 5707 490 603 0.03077 5792 500 604 0.03168 5857 510 604 0.03259 5932 520 604 0.03350 5895 530 604 0.03436 6039 540 604 0.03523 6082 550 604 0.03614 6146 560 604 0.03695 6168 570 604 0.03759 6189 580 604 0.03850 6232 583 701 0.03900 6465 590 704 0.04018 6701 600 703 0.04182 6894 610 704 0.04318 7023 620 704 0.04455 7119 630 705 0.04600 7216 640 705 0.04727 7280 650 705 0.04863 7345 660 705 0.04995 7409 680 705 0.05270 7495 690 705 0.05395 7538 700 705 0.05530 7580 710 705 0.05665 7602 720 705 0.05781 7634 730 705 0.05916 7656 735 800 0.06060 8017 740 800 0.06172 8156 750 800 0.06377 8339 760 800 0.06586 8457 TABLE A-4 Continued ...-.... 118 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 770 800 0.06786 8553 780 800 0.06977 8608 790 800 0.07163 8682 800 800 0.07349 ~——- 810 800 0.07535 -- 820 800 0.07725 —-—— 825 903 0.07949 9084 830 903 0.08133 9289 840 903 0.08471 9503 850 903 0.08810 9642 860 903 0.09146 9738 870 902 0.09488 ———— 880 902 0.09829 ———— 890 902 0.10165 -- 900 902 0.10510 -——— 910 902 0.10865 -—- 119 TABLE A-5 Test data, SA-9 I. Material—~Ottawa Sand Initial Sample Dimensions Sand Density-—64% by Volume Outside Diam. 4" Water Content——2l.2% Inside Diam. 1 1/2" Test Temperature—--l20°C. Height 9 3/16" Date of Test—-6 Mar 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 0 0 0 0 3 100 0.00004 -——— 10 100 0.00009 -—— 13 201 0.00009 -—- 20 200 0.00042 -——- 30 200 0.00064 -—— 33 302 0.00077 697 40 301 0.00132 688 50 301 0.00200 735 60 301 0.00246 794 70 300 0.00294 842 80 300 0.00332 890 90 299 0.00370 933 100 299 0.00396 976 110 298 0.00425 1019 120 297 0.00451 1051 123 402 0.00472 1563 130 401 0.00545 1659 140 401 0.00621 1740 150 401 0.00694 1809 160 401 0.00758 1901 170 401 0.00817 1981 180 400 0.00873 2056 190 400 0.00916 2121 200 400 0.00962 2174 210 400 0.01000 2217 220 400 0.01042 2271 230 400 0.01081 2324 240 400 0.01119 2367 250 399 0.01153 2416 260 399 0.01187 2459 270 399 0.01213 2496 273 501 0.01268 2782 280 501 0.01327 3034 290 501 0.01417 3222 300 501 0.01495 3361 310 500 0.01561 3469 320 500 0.01621 3571 330 500 0.01676 3662 120 TABLE A-5 Continued Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (lb) 340 500 0.01741 3742 350 500 0.01783 3842 360 500 0.01825 3876 370 500 0.01872 3941 380 499 0.01911 4000 390 499 0.01953 4059 400 499 0.01996 4113 410 499 0.02040 4166 420 499 0.02085 4214 423 601 0.02209 4462 430 601 0.02267 4719 440 601 0.02355 4934. 450 601 0.02413 5084 460 601 0.02485 5208 470 601 0.02556 5310 480 600 0.02618 5295 490 600 0.02689 5481 500 600 0.02752 5551 510 600 0.02809 5615 520 600 0.02894 5674 530 600 0.02956 5723 540 600 0.03073 5776 550 600 0.03109 5825 560 600 0.03168 5867 570 600 0.03225 5900 573 702 0.03304 6100 580 701 0.03577 6406 590 701 0.03727 6615 600 701 0.03864 6802 610 701 0.03978 6878 620 701 0.04082 6969 630 701 0.04227 7006 640 703 0.04359 7087 650 703 0.04486 7151 660 703 0.04636 7200 670 703 0.04736 7253 680 703 0.04863 7302 690 703 0.04986 7334 700 703 0.05116 7366 710 703 0.05251 7403 720 703 0.05363 7430 723 802 0.05740 1609 730 802 0.05916 7909 740 802 0.06098 8124 750 802 0.06279 8253 760 802 0.06502 8339 770 802 0.06684 8414 I— 121 TABLE A—5 Continued Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) ‘Force (lb) 780 802 0.06842 8467 790 802 0.07070 8510 793 904 0.07391 8753 800 900 0.07581 9118 810 900 0.07893 9311 820 900 0.08176 9435 830 900 0.08500 9521 840 900 0.08728 9590 850 900 0.08990 9633 860 900 0.09302 9676. 122 TABLE A-6 Test data, SA-ll .- Materia1-—Ottawa SAnd Initial Sample Dimensions Sand Density——64% by Volume Outside Diam. 3 1/2" Water Content-21.4% Inside Diam. 1 1/2" Test Temperature-—-l2.0°C Height 9 1/8" Date of Test——20 Mar 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 0 0 0 0 3 100 0 -255* 10 102 0.00009 -234* 13 200 0.00009 -71* 20 200 0.00013 —26* 23 303 0.00196 125 30 300 0.00226 243 40 300 0.00243 334 50 300 0.00247 423 60 300 0.00251 452 70 300 0.00251 500 80 300 0.00251 554 83 401 0.00353 696 90 401 0.00374 899 100 401 0.00374 1097 110 401 0.00383 1264 120 401 0.00404 1457 130 401 0.00421 1634 140 401 0.00460 1779 150 401 0.00498 1908 160 401 0.00519 2015 170 401 0.00549 2117 180 401 0.00574 2198 190 401 0.00600 2283 193 500 0.00706 2428 200 500 0.00762 2758 213 500 0.00860 3026 220 500 0.00902 3144 230 500 0.00970 3267 240 500 0.01081 3375 250 500 0.01102 3466 260 500 0.01153 3541 270 500 0.01208 3621 280 500 0.01230 3680 290 500 0.01315 3734 300 500 0.01391 3788 310 500 0.01472 3830 320 500 0.01485 3873 330 500 0.01532 3906 *Apparent negative force probably due to shifting of sample. 123 TABLE A-6 Continued ‘l Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) FOrce (1b) 340 500 0.01549 3943 343 601 0.01664 4053 350 601 0.01751 4348 360 601 0.01864 4552 370 601 0.01983 4690 380 601 0.02062 4788 390 601 0.02178 4798 400 601 0.02253 4873 410 600 0.02316 4938 420 600 0.02396 4938 430 600 0.02506 4959 440 600 0.02573 5002 450 600 0.02622 5034 460 600 0.02711 5088 470 600 0.02791 5120 480 600 0.02889 5120 490 600 0.02973 5142 493 702 0.03068 5262 500 702 0.03195 5519 510 702 0.03332 5713 520 702 0.03477 5782 530 702 0.03618 5906 540 702 0.03741 5970 550 702 0.03886 6045 560 702 0.04032 6077 570 702 0.04155 6099 580 702 0.04255 6110 590 702 0.04382 6110 600 702 0.04518 6120 610 702 0.04627 6142 620 702 0.04759 6142 623 803 0.04895 6283 630 803 0.05079 6541 640 803 0.05307 6712 650 803 0.05502 6820 660' 803 0.05726 6906 670 803 0.05916 6916 680 803 0.06116 6927 690 803 0.06335 6948 700 803 0.06530 6959 703 904 0.06712 7133 710 904 0 06977 7444 720 904 0.07372 7573 730 903 0.07702 7680 740 903 0.08010 7712 750 903 0.08295 -—- 124 TABLE A-6 Continued Time Outside Pressure Change in. Total Axial (min) (psi) Diameter (in) Force (1b) 760 903 0.08629 -——- 770 . 903 0.08905 -——- 780 902 0.09205 TI— 125 TABLE A-7 Test data, SA-12 —" Material-—Ottawa Sand Initial Sample Dimensions Sand Density-—64% by Volume Outside Diam. 4 1/2" Water Content-—21.0% Inside Diam. 1 1/2" Test Temperature--1£2.0°C. Height 9 3/16" Date of Test——27 Mar 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 0 0 0 0 3 100 0.00026 -45* 10 99 0.00051 -40. 13 200 0.00191 204 20 200 0.00247 247 23 300 0.00591 1151 30 300 0.00715 1087 40 300 0.00779 1087 50 299 0.00842 1114 60 299 0.00894 1140 70 299 0.00983 1173 80 298 0.01013 1199 90 298 0.01060 1226 100 298 0.01076 1258 103 400 0.01319 2157 110 400 0.01396 2146 120 400 0.01502 2187 130 400 0.01583 2243 140 400 0.01660 2297 150 400 0.01715 2356 160 400 0.01770 2409 170 399 0.01847 2463 180 399 0.01940 2511 190 399 0.02004 2565 200 399 0.02080 2602 210 399 0.02089 2645 220 399 0.02138 2688 223 501 0.02302 3372 230 501 0.02342 3496 240 501 0.02413 3614 250 501 0.02493 3710 260 501 0.02618 3796 270 501 0.02707 3887 280 501 0.02755 3957 290 501 0.02849 4032 300 500 0.02933 4102 310 500 0.03005 4172 *Apparent negative force probably due to shifting of sample. 126 TABLE A-7 Continued v Time Outside Pressure Change in .Total Axial (min) (psi) Diameter (in) Force (1b) 320 500 0.03018 4241 330 500 0.03100 4311 340 500 0.03214 4376 350 500 0.03286 4440 360 500 0.03341 4494 363 601 0.03486 4931 370 601 0.03568 5199 380 601 0.03695 5414 390 601 0.03791 5564 400 600 0.03914 5693 410 600 0.03977 5800 420 600 0.04068 5907 430 600 0.04191 5993 440 600 0.04300 6079 450 600 0.04414 6159 460 600 0.04486 6218 470 600 0.04595 6282 480 600 0.04668 6342 490 600 0.04741 6401 500 600 0.04813 6454 510 600 0.04936 6503 513 701 0.05140 6908 520 701 0.05163 7203 '530 701 0.05330 7417 540 701 0.05442 7573 550 701 0.05605 7697 560 700 0.05726 7804 570 700 0.05860 7922 580 700 0.05991 7986 590 700 0.06135 8061 600 700 0.06228 8136 610 700 0.06372 8201 620 700 0.06465 8244 630 700 0.06605 8297 640 700 0.06721 8351 643 800 0.06707 8650 650 801 0.07028 9067 660 800 0.07237 9293 670 800 0.07433 9459 680 800 0.07619 9588 690 800 0.07805 9690 700 800 0.07953 9777 710 800 0.08138 9861 720 800 0.8276 9936 730 800 0.08438 10001 127 TABLE A-7 Continued 1 . Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) F0rce'(1b) 733 900 0.08624 10438 740 901 0.08767 10728 750 901 0.09093 10975 760 901 0.09341 11146 770 901 0.09571 11275 780 901 0.09790 11382 790 901 0.10000 11468 800 901 0.10210 11532 810 901 0.10380 11586 820 901 0.10625 11640 128 TABLE A-8 Test data, SA-14 . . ..— Materia1-—Ottawa Sand Initial Sample Dimensions Sand Density—~64% by Volume Outside Diam. 5" Water Content—-21.6% 0 Inside Diam. l l/2" Test Temperature-—-12.0 C- Height 9 1/4" Date of Test-—10 Apr 69 Time Outside Pressure Change in Total Axial (min) (psi), Diameter (in) ‘Force (lb) 0 0 0 0 3 100 0 70 10 100 0 38 13 200 0 763 20 200 0 739 23 300 0 1605 30 300 0 1573 40 300 0 ' 1573 50 300 0 1573 60 299 0 1583 70 299 0 1594 80 299 0 1594 .83 400 0 3317 90 400 0 3478 100 400 0 3488 110 401 0 3478 120 401 0.00013 3478 130 401 0.00038 3488 140 401 0.00043 3499 143 500 0.00047 5565 150 500 0.00081 5629 160 500 0.00174 5737 170 500 0.00285 5769 180 500 0.00366 5812 190 500 0.00472 5855 200 500 0.00574 5876 210 500 0.00677 5898 220 500 0.00766 5908 230 500 0.00842 5919 240 500 0.00915 5930 250 500 0.01009 5951 260 500 0.01068 5962 263 600 0.01234 8296 270 600 0.01268 8221 280 600 0.01481 8189 290 600 0.01655 8168 300 600 0.01706 8168 310 .600 0.01872 8168 320 600 0.02036 8168 . - “'h'l; 15_ i it. 129 TABLE A-8 Continued Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) ForCe (lb) 330 600 0.02253 8168 340 600 0.02320 8168 350 600 0.02542 8168 360 600 0.02569 8178 370' 600 0.02715 8200 380 600 0.02898 8210 383 700 0.03036 10148 390 700 0.03177 10126 400 700 0.03250 10094 410 700 0.03473 10083 420 700 0.03723 10083 430 700 0.03895 10083 440 700 0.04045 10083 450 700 0.04205 10094 460 700 0.04341 10105 470 700 0.04482 10115 480 700 0.04650 10126 490 700 0.04768 10137 500 700 0.04968 10158 510 700 0.05074 10169 520 700 0.05256 10180 530 700 0.05395 10201 533 801 0.05637 11901 540 801 0.05712 11891 550 801 0.05991 11880 560 801 0.06233 11880 570 801 0.06456 11880 580 801 0.06698 11891 590 801 0.06856 11901 600 801 0.07116 11912 610 801 0.07270 11923 620 801 0.07544 11944 630 801 0.07721 11955 640 801 0.07958 11966 650 801 0.08200 11987 653 901 0.08424 13742 660 902 0.08610 13753 670 902 0.08933 13753 680 901 0.09239 13764 690 901 0.09600 13764 700 901 0.09917 13774 710 901 0.10240 13785 720 901 0.10535 13796 730 901 0.10835 13806 740 901 0.11235 13817 750 900 0.11535 13828 760 900 0.11850 13849 Note: Bottom portion of shaft slipped. This may have caused excessive axial force and deformation. It also may have allowed small amount of ethylene glycol to reach sample 130 TABLE A-9 Test data, SA-16 Material—-Ottawa Sand Initial Sample Dimensions Sand Density——64% by Volume Outside Diam. 5" Water Content-2l.3% Inside Diam. 1 1/2" Test Temperature-—-12.0°C Height 9 1/4" Date of Testc-24 Apr 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (lb) 0 0 0 0 3 100 0 49 10 100 0 44 13 200 0 184 20 200 0 216 23 300 0 468 30 300 0 521 40 300 0 586 50 299 0 634 60 299 0 682 63 400 0 940 70 400 0 1043 80 400 0 1160 90 400 0 1268 100 400 0 1375 110 400 0.00034 1461 120 400 0.00081 1546 123 500 0.00136 2035 130 500 0.00221 2196 140 500 0.00331 2400 150 500 0.00460 2582 160 500 0.00574 2754 170 500 0.00698 2915 180 500 0.00791 3046 ‘ 190 500 0.00889 3226 200 500 0.01004 3371 210 500 0.01191 3516 220 500 0.01306 3623 230 500 0.01404 3730 240 500 0.01498 3838 243 599 0.01608 4520 250 600 0.01694 4777 260 600 0.01830 5035 270 600 0.01953 5271 280 600 0.02089 5399 290 600 0.02280 5550 300- 600 0.02400 5689 310 600 0.02529 5807 320 600 0.02644 5925 131 TABLE A-9 Continued Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 330 600 0.02760 6032 340 600 0.02880 6038 350 600 0.02995 6204 360 600 0.03114 6290 363 700 0.03214 6940 370 700 0.03350 7283 380 700 0.03532 7530 390 700 0.03750 7701 400 700 0.03927 7852 410 700 0.04118 7959 420 700 0.04282 8045 430 700 0.04445 8131 440 700 0.04623 8216 450 700 0.04795 8281 460 700 0.04986 8345 470 700 0.05149 8410 480 700 0.05312 8474 485 800 0.05512 9176 490 800 0.05637 9380 500 802 0.05916 9573 510 799 0.06191 9831 520 798 0.06456 10002 530 796 0.06730 10110 540 798 0.07047 10170 550 796 0.07256 10271 560 798 0.07512 10271 570 798 0.07767 10367 580 797 0.08029 10442 590 796 0.08300 10496 593 902 0.08486 11038 600 902 0.08810 11403 610 901 0.09229 11682 620 900 0.09639 11854 630 900 0.10065 11940 640 900 0.10475 12036 650 900 0.10885 12096 660 899 0.11295 12165 670 898 0.11711 12219 680 896 0.12154 12272 690 903 0.12615 12261 Note: Small amount of ethylene glycol may have reached sample. 132 TABLE A-10 Test data, SA-l8 Material-—Ottawa Sand Initial Sample Dimensions Sand Density-—64% by Volume Outside Diam. " .1 Water Content——2l.4% Inside Diam. 1 1/2" Test Temperature-—-12.0°C. Height 9" Date of Test-9 May 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (lb) 0 0 0 0 3 100 0 70 10 100 0 70 13 201 0 312 20 204 0 302 23 299 0 747 30 299 0 832 40 300 o 832 50 300 o 897 60 299 0 1004 64 399 0 -——— 70 399 0 -- 80 398 0 -——— 90 400 0 ———— 100 400 0 ~—-— 110 398 0.00047 ~——— 120 397 0.00094 -——— 123 500 0.00132 2561 130 500 0.00225 2786 140 499 0.00332 3055 150 499 0.00421 3291 160 498 0.00510 3505 170 499 0.00595 3677 180 499 0.00690 3848 190 498 0.00774 4008 200 498 0.00890 4170 210 497 0.01013 4321 220 497 0.01077 4460 223 599 0.01191 5163 230 600 0.01323 5528 240 603 0.01532 5775 250 599 0.01651 6011 260 601 0.01787 6226 270 601 0.01915 6386 280 601 0.02031 6558 290 601 0.02169 6698 300 601 0.02289 6826 310 601 0.02396 6944 320 602 0.02507 7052 133 TABLE A-lO Continued Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force'(10) 323 700 0.02654 7798 330 702 0.02840 8056 340 702 0.03036 8333~ 350 703 0.03209 8581 360 705 0.03382 8699 370 705 0.03573 8839 380 703 0.03736 9085 390 700 0.03877 9139 400 710 0.04014 9171 410 696 0.04141 9450 420 685 0.04236 9515 423 800 0.04386 10174 430 801 0.04623 10410 440 802 0.04886 10646 450 801 0.05121 10904 460 801 0.05349 11054 470 805 0.05814 11075 480 805 0.05860 11344 490 792 0.06014 11462 500 802 0.06233 11419 510 802 0.06460 11504 513 900 0.06702 12304 520 898 0.06926 12519 530 901 0.07260 12776 540 901 0.07581 12959 550 901 0.07953 13098 560 902 0.08286 13227 570 898 0.08657 13367 580 898 0.09063 13452 590 899 0.09390 13506 600 898 0.09756 13603 Note: Small amount of ethylene glycol may have reached sample. ations in pressure. Leak in cell caused fluctu- 134 TABLE A-ll Test data, C-3 Materia1-—Ontonagon Clay Initial Sample Dimensions Density——97.3 1b/ft3 Outside Diam. 5" Water Content-—26.2% Inside Diam. 1 1/2" Test Temperature-—-l2.0°C. Height 9 1/4" Date of Test——2l Nov 68 Time Outside Pressure Change in (min) (psi) Diameter (in) 0 0 O 4 300 0.1381 8 300 0.1586 16 300 0.1806 24 300 0.1924 32 300 0.2012 40 300 0.2075 48 300 0.2119 56 300 0.2159 64 300 0.2188 72 300 0.2212 80 300 0.2234 88 300 0.2253 96 300 0.2269 104 300 0.2281 112 300 0.2297 120 300 0.2309 128 300 0.2319 136 300 0.2331 144 300 0.2338 156 300 0.2353 164 300 0.2362 172 300 0.2369 180 300 0.2378 188 300 0.2384 200 300 0.2394 208 . 300 0.2400 216 300 0.2406 224 300 0.2412 232 300 0.2419 240 300 0.2428 248 300 0.2431 256 300 0.2431 264 300 0.2438 272 300 0.2444 278 300 0.2444 288 300 0.2447 296 300 0.2453 304 300 0.2459 135 TABLE A-ll Continued Time Outside Pressure Change in (min) (psi) Diameter (in) 312 300 0.2462 320 300 0.2462 328 300 0.2469 336 300 0.2472 344 300 0.2475 352 300 0.2475 360 300 0.2478 368 300 0.2484 376 300 0.2484 384 300 0.2488 392 300 0.2488 400 300 0.2491 404 300 0.2494 408 400 0.2643 412 400 0.2721 420 400 0.2829 428 400 0.2911 436. 400 0.2970 444 400 0.3022 452 400 0.3063 460 400 0.3104 468 400 0.3135 476 a 400 0.3165 484 400 0.3192 492 400 0.3215 136 TABLE A-l2 Test data, 0-4 Material—~Ontonagon Clay Initial Sample Dimensions Density-—98.2 lb/ft3 Outside Diam. 5" Water Content——25.0% Inside Diam. 1 1/2" Test Temperature——-12.0°C. Height ' 9 3/16" Date of Test—-12 May 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 0 0 0 0 4 101 0.01838 521 10 100 0.02356 682 15 100 0.02578 730 20 100 0.02769 757 30 100 0.02964 778 40 100 0.03109 795 50 100 0.03209 800 60 100 0.03286 800 70 100 0.03364 790 80 100 0.03423 800 90 100 0.03495 800 100 100 0.03523 805 110 100 0.03582 778 120 100 0.03645 800 130 100 0.03682 789 140 100 0.03718 789 150 100 0.03764 789 160 100 0.03795 789 170 100 0.03818 795 180 100 0.03850 800 190 100 0.03877 800 200 100 0.03914 795 210 100 0.03927 795 220 100 0.03959 795 230 100 0.03973 795 240 100 0.03982 789 250 100 0.04014 789 260 100 0.04023 784 270 100 0.04036 784 280 100 0.04059 784 290 100 0.04068 789 300 100 0.04077 784 304 200 0.07126 2501 310 200 0.08181 2522 315 200 0.08638 2511 320 200 0.08929 2501 330 200 0.09371 2479 340 200 0.09668 2458 137 TABLE A-l2 Cont inued I, Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 350 200 0.09883 2447 360 200 0.10050 2426 370 200 0.10185 2414 380 200 0.10290 2404 390 200 0.10385 2393 400 200 0.10480 2383 410 200 0.10560 2372 420 200 0.10640 2361 430 200 0.10695 2351 440 200 0.10765 2351 450 200 0.10800 2340 460 200 0.10860 2340 470 200 0.10890 2329 480 200 0.10925 2318 490 200 0.10950 2318 500 200 0.10980 2318 510 200 0.11030 2308 520 200 0.11065 2308 530 200 0.11080 2308 540 200 0.11105 2297 550 200 0.11125 2297 560 200 0.11175 2297 570 200 0.11205 2286 580 200 0.11225 2286 590 200 0.11245 2286 600 200 0.11265 2286 138 TABLE A-13 Test data, C-5 Materia1-—Ontonagon Clay Initial Sample Dimensions Density——98.2 lb/ft3 Outside Diam. 5" Water Content-—24.8% Inside Diam. 1 1/2" Test Temperature--12.0°C. Height 9 3/16" Date of Test——22 May 69 Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in) Force (1b) 0 O 0 0 4 200 0.06567 2190 10 200 0.08105 2275 15 200 0.08605 2254 20 200 0.08910 2254 30 200 0.09380 2233 40 200 0.09639 2222 50 200 0.09800 2211 60 200 0.09932 2195 70 200 0.10095 2179 80 200 0.10160 2168 90 200 0.10310 2168 100 200 0.10435 2157 110 200 0.10465 2157 120 200 0.10495 2136 130 200 0.10595 2125 140 200 0.10635 2093 150 200 0.10755 2082 160 200 0.10785 2082 170 200 0.10905 2072 180 200 0.10915 2061 190 200 0.10945 2050 200 200 0.10975 2050 210 200 0.10985 2050 220 200 0.11005 2050 230 200 0.11015 2039 240 200 0.11180 2029 250 200 0.11180 2029 260 200 0.11190 2029 270 200 0.11200 2039 280 200 0.11215 2039 290 200 0.11225 2039 300 200 0.11235 2039 304 300 0.13316 3901 310 300 0.14705 3998 315 300 0.15153 3998 320 300 0.15521 3998 330 300 0.15889 4008 340 300 0.16297 3976 139 TABLE A-l3 Continued Time Outside Pressure Change in Total Axial (min) (psi) Diameter (in), ‘Force (lb) 350 300 0.16546 3966 360 300 0.16584 3998 370 300 0.16622 3998 380 300 0.16762 3987 390 300 0.16789 3987 400 300 0.16940 3998 410 300 0.16973 3987 420 300 0.17081 3987 430 300 0.17097 3998 440 300 0.17130 3998 450 300 0.17151 3987 460 300 0.17184 3998 470 300 0.17331 3998 480 300 0.17319 3998 490 300 0.17351 3998 500 300 0.17373 3998 510 300 0-17395 3998 520 300 0.17497 3998 530 300 0.17503 4008 540 300 0.17503 4008 550 300 0.17513 3998 560 300 0.17535 4008 570 300 0.17557 4008 580 300 0.17562 4008 590 300 0.17573 4008 600 300 0.17703 4008 APPENDIX B CALIBRATION DATA 141 B-l CALIBRATION OF DEFORMATION MEASURING APPARATUS The deformation measuring apparatus described in Sec- tion 4.2.1 was calibrated by placing it in a vertical position and moving the brass pieces by measured increments using a micrometer. The micrometer was secured in a vise and was re—centered periodically. The movement of the brass pieces caused movement of the vertical rod which was meas- ured by a Linear Differential Transformer (Sanborn Linearsyn Differential Transformer Model No. 575 DT-500). The LDT was connected electronically to a two-channel recorder (Sanborn Recorder Model 7702B with a Sanborn Carrier Preamplifier Model 8805A). The left channel of the recorder was used with a calibration factor of 337. This provided for a stylus deflection of one division on the chart for each movement of 0.00025 inches through the LDT. Calibration data are given in Table B-1 and the calibration curve is shown in Figure B-l. 142 TABLE B-l Calibration data for deformation measuring device Hole Diameter Recorder Rdg Hole Diameter Recorder Rdg (in) (Divisions) (in) (Divisions) 1.50 0 1.24 507.5 1.49 23.5 1.23 523.5 1.48 47.0 1.22 539.5 1.47 69.5 1.21 555.5 1.46 91.5 1.20 571.0 1.45 113.5 1.19 586.0 1.44 135.0 1.18 600.0 1.43 156.5 1.17 614.0 1.42 178.0 1.16 627.5 1.41 199.0 1.15 641.0 1.40 219.5 1.14 654.0 1.39 239.5 1.13 667.0 1.38 259.5 1.12 680.0 1.37 279.0 1.11 693.0 1.36 298.0 1.10 706.0 1.35 317.0 1.09 719.0 1.34 336.0 1.08 731.5 1.33 354.5 1.07 744.0 1.32 373.0 1.06 755.5 1.31 391.0 1.05 767.0 1.30 408.5 1.04 778.0 1.29 425.5 1.03 789.0 1.28 442.5 1.02 799.5 1.27 459.0 1.01 810.0 1.26 475.5 1.00 820.0 1.25 491.5 Recorder Reading - Divisions 700 600 500 400 300 200 100 1.5 143 l U I U 1.4 1.3 1.2 Hole Diameter - Inches Figure B-l Calibration Curve for Deformation Measuring Device 144 B-2 CALIBRATION OF LOAD CELL The load cell (Strainsert Universal Flat Load Cell, 25000 lb. capacity) was calibrated by loading it in 1000 pound increments using a Tinius Olsen Testing Machine. The load cell was connected electronically to the right channel of the two channel recorder (calibration factor 123.5). Re- sults of the calibration are presented in tabular form in Table B-2 and graphically in Figure B—2. 145 TABLE B-2 Calibration data for load cell Load Recorder Reading (lb) (Divisions) 0 0 1000 94.5 2000 186.0 3000 278.0 4000 372.0 5000 464.0 6000 558.0 7000 653.0 8000 745.0 9000 840.0 10000 931.0 Recorder Reading - Divisions 146 1000 w 800 w 600 -- 400 " Calibration Factor 200 w = 10.729 lbs/div 0 % i ‘r t 7 0 2000 4000 6000 8000 10000 Figure B-2 Load — Pounds Calibration Curve for Load Cell 1w. __. APPENDIX C SAMPLE CALCULATIONS 148 C-1 CALCULATION OF um AND 66m To find the mean rate of radial displacement from the measured value at the inner surface, recall from Equation (5.23) that Thus, for a = 0.75 It) '75ua 753; b .75 dr ' b .75ua [1n r .75 [1'] I775 .75ua[1n b-ln.75] b-.75 .75ua b-. 75 [In b-ln. 75] Similarly, from Equation (5.24a), 149 b '75 = .115. l .. 1,333lfia 69 I [0 b-.75 b .75 dr Table C-l gives values of um and 56m for various values of b. TABLE 0-1 Values of am and é 9m Outside Radius, Mean Rate of Mean Rate of b (in) Radial Displacement, Tangential Strain, o g _1 um(in/min) 62m‘(min ) 1.75 0.636ua 0.571ua 2.00 0.589ua 0.500ua 2.25 0.550ua 0.444ua 2.50 0.515ua 0.40011a Ill-ll 150 C-2 CALCULATION OF 0a FROM TEST DATA To find 0a from test results, it was first necessary to calculate the displacements at the inner surface for the various times. This was done by using the calibration data given in Table B-1. The relationship between the change in hole diameter and the recorder reading was assumed to be linear between the measured values at 0.01 inch increments. SAMPLE CALCULATION: On Test SA-9, a recording reading of 73.2 would result I in a change of diameter, A, of A = 0.03 + 2175 (.01) = 0.03168 in. The radial movement at the inner surface, ua, was then found by halving this value. The value of 08 was found by plotting u against time and measuring the slope; a 151 C-3 CALCULATION OF TOTAL AXIAL FORCE FROM TEST DATA To find the total axial force, Fz’ from the load cell readings, first calculate the force on the pedestal due to the pressure in the test cell. This was done by computing the area on the bottom of the pedestal, subtracting the ex— posed area on the top for the size sample in question, and multiplying the result by the pressure. The difference be- tween this result and the force on the load cell as computed according to the calibration data in Figure B-2 represents the total axial force in the sample. SAMPLE CALCULATION: For Test SA—12, at a time of 513 minutes, the load cell reading was 278.5. The pressure was 700 p.s.i. and the outer radius was 2.25 inches. Force on Pedestal 7OO17[(2.52-l.52) - (2.52 — 2.252)] = 9896 lbs. Force on Load Cell = 278.5 d1V(lO.729le/div) = 2988 lbs. Total Axial Force = 9896-2988 6908 lbs. 152 0—4 CALCULATION OF ua, or, and 66 BY EQUATIONS The calculation of ua, Or, and 06, according to Equa- tions (5.23), (5.22a), and (5:22b), respectively was accomplished by merely substituting the appropriate values for the parameters, dimensions, and pressure directly into these expressions. Due to the long calculation time and the large number of operations, a computer program was written for the CDC 3600 computer to carry out the operations of Equation (5.23) for 8a. 153 C-5 CALCULATION OF TOTAL AXIAL FORCE BY EQUATION (5.220) The total axial force in the sample was calculated by in- tegrating Equation (5.22c) across the section. SAMPLE CALCULATION: Take: p = 700 p.s.i. a = 0.75 in b = 2.25 in m = 0.01206 N = 0.00808 0 = 9.103 x 10"6 X = 1.66 ‘Substituting these values into Equation (5.22c'), the result is 27.4 + 166.0 In r + 300.319”633 0 ll ‘11 II 217 2.25 I I [27.4 + 166.01n r + 300.3r1'633errd8 o .75 '11 II 2.25 ' 27‘I [27.4r + 166.0r ln r + 300.3r2°633]dr .75 2.25 .75 2"[13.7r2 + 83.0r21n r + 41.5r-2 + 82.7r3'533] 154 = 2n{13.7(2.252—.752) + 83.0[2.252(.811)—(.75)2(-.288)] +41.5(2.252—.752) + 82.7(14.6-.4)} 2n(1406) 8828 lbs. 155 C-6 CALCULATION OF PRESSURE BY EQUATION (5.29) For a given measured u it was necessary to determine a, the proper é value from which to choose a 0 value. This was done by calculating the value of 69m by the coefficients given in Table C-1. The value of a was then taken from Fig- ure 7.9 and the apprOpriate value of c calculated. SAMPLE CALCULATION: From Test SA-12, a = .75 in b = 2.25 in p 8 700 psi 6.05 x 10‘5in/min :0 I 66m = 6.05 x 10-5 (.444) = 3.58 x 10"5min'l = 67 (From Figure 7.9) For 0 = 25°, cos 0 = 0.906 c = 58%53 = 74.0 Substituting into Equation (5.29), 2 4.0 1. 1. 461 p ' 2. W‘% = 633 p.s.i. 78 0509 3 1293 03 H u u "u u u I I‘ll“ .|||| H I AHII I“ HI