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Analysis of cone tip resistance in sand Ahmadi, Mohammad Mehdi 2000

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A N A L Y S I S O F C O N E T I P R E S I S T A N C E I N S A N D by M O H A M M A D M E H D I A H M A D I B . S c , Sharif University of Technology, Tehran, Iran, 1978 M . S c , Sharif University of Technology, Tehran, Iran, 1988 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E S T U D I E S (Department of C i v i l Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A M a y 2000 © Mohammad Mehdi Ahmadi, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l £ v M \ v ^ e - r \ i The University of British Columbia Vancouver, Canada Date H DE-6 (2/88) A B S T R A C T The cone penetration test (CPT) has been used for decades in in-situ geotechnical engineering practice. The reliability and repeatability of the C P T measurements has increased its acceptance as a predominant tool in this field. The cone is pushed into the soil at a standard rate of 20 mm/s, and different measurements such as pore water pressure, sleeve friction, and most importantly cone tip resistance can be made. These measurements are then used to obtain information regarding stratigraphy of the site. Over the years there has been a high demand for validated correlations between cone resistance and engineering properties of soil. The correlations for sands are mostly obtained from experiments in calibration chamber tests with specified boundary conditions. The correlations for clays are mostly obtained from laboratory tests on undisturbed samples. During the course of this study, several approaches to analyze the cone penetration process were investigated; and different codes using the computer program F L A C were written. Only two of these approaches are worth mentioning in this thesis. In the first approach, the cone is placed in a predetermined location in the grid, and is given a downward vertical displacement. Analysis is carried out to seek stresses that remain constant with continued increase in displacement. In this approach, the analytical results show that with continued penetration, the soil stresses around the cone tip do not reach a constant value. This is especially true for sand; and it is unacceptable. This approach was not pursued further in this study. In another approach, the complete process of cone penetration is modeled as the cone starts to penetrate the soil from the ground surface to deeper layers. The results obtained in this approach are reasonable. Based on this approach, numerical results are compared with experimental values from calibration chamber tests on clean, non-cemented, unaged sands. The proposed model is verified by comparing the numerical values with the published experimental results obtained on Ticino sand at the E N E L -CRIS calibration chamber. The results from all four different boundary condition types (BC1 to B C 4 ) used in the experiment are compared numerically. It is shown that the second approach gives numerical values of tip resistance that are in agreement with calibration chamber test results. The agreement is, in general, in the range of ±25%. The Mohr-Coulomb elasto-plastic soil model with stress dependent parameters is used for both approaches. Several applications of the proposed simulation are then presented. The importance of horizontal effective in-situ stresses on the cone tip resistance is addressed in some length in this thesis. It is shown that with the proposed simulation, horizontal stresses play a major role in affecting the magnitude of tip resistance in sand. This is supported by measured experimental results. Numerical simulation is also carried out to investigate the calibration chamber size effect. This study is important to correlate the calibration chamber test results with iii field measurements for the sand of the same relative density and horizontal and vertical stresses. These simulations are performed for chambers of different sizes under all four different boundary conditions. The simulation is carried out for loose as well as dense sands. The numerical simulation clearly shows that for loose sand, calibration chamber size effect is not significant. For dense sand, however, the effect can be substantial. This is in agreement with the experimental observations. Analysis of cone penetration in layered soil is also addressed. In the analysis, the soil layers consist of sands with two different relative densities, i.e. loose and dense, or layers of two different soil types; i.e. sand and clay. The interface distance that the cone "senses" the approaching new layer is predicted in the numerical analysis. The predictions agree with those measured during experimental tests. Another application of the proposed model is to investigate what property of soil affects the tip resistance to a larger extent. To this end, a sensitivity analysis is carried out. It is seen that the deformational properties of soil, i.e., modulus and dilatancy properties are the most influential in affecting the cone tip resistance values. Finally, a preliminary analysis of pore pressure response during cone penetration in sand is carried out to investigate whether the proposed model can predict, in a reasonable way, the generation of pore pressure around the cone. The preliminary analysis gives results that seem to be promising. However, more work is needed to fully clarify all the intricacies of the pore pressure analysis during cone penetration in sand. iv T A B L E O F C O N T E N T S A B S T R A C T ii T A B L E O F C O N T E N T S V L I S T O F S Y M B O L S A N D A B B R E V I A T I O N S viii L I S T O F T A B L E S X L I S T O F F I G U R E S xi A C K N O W L E D G E M E N T S xiii C H A P T E R 1. L I T E R A T U R E S U R V E Y 1 1.1. Introduction 1 1.2. Literature survey 3 1.2.1. Bearing capacity theory 3 1.2.2. Cavity expansion theory 6 1.2.3. Strain path method 9 1.2.4. Finite element method 12 1.3. Scope of the present work 15 C H A P T E R 2. C A L I B R A T I O N C H A M B E R T E S T I N G 19 2.1. Introduction 19 2.2. Calibration chambers 20 2.3. Limitations of calibration chamber tests 22 2.3.1. Sample age and cementation 22 2.3.2. Types of sand tested 23 2.3.3. Chamber size and boundary effect 24 2.4. Selected database 26 C H A P T E R 3. A N I N T R O D U C T I O N T O F L A C 28 3.1. Introduction 28 3.2. Finite difference method 28 3.3. Equations of motion 29 3.4. Explicit, time-marching scheme 29 3.5. Lagrangian analysis 30 3.6. Notes on grid elements 31 3.7. Disadvantages of F L A C 32 3.8. Why F L A C ? 33 v C H A P T E R 4. S I M U L A T I O N O F C P T T I P R E S I S T A N C E I N S A N D 3 5 4.1. Introduction 35 4.2. Constitutive law for sand 36 4.3. Size of the numerical grid 41 4.4. First approach in cone penetration 41 4.5. Second approach in cone penetration 47 4.6. Variation of horizontal stress ahead and behind the tip 56 4.7. Comparison of numerical results with experimental values 59 4.7.1. Comparison for BC1 type boundary condition 60 4.7.2. Comparison for B C 3 type boundary condition 63 4.7.3. Comparison for B C 4 type boundary condition 64 4.7.4. Comparison for B C 2 type boundary condition 66 4.8. A review of the experimental results 67 4.9. Summary of all data 70 C H A P T E R 5. M O D E L A P P L I C A T I O N S 73 5.1. Introduction 73 5.2. Which stress affects the tip resistance: horizontal or vertical? 74 5.3. Effects of boundaries 77 5.4. Effects of vertical (bottom) boundary condition 81 5.5. Analysis for layered soil 84 5.5.1. Loose sand over dense sand 86 5.5.2. Dense sand over loose sand 89 5.5.3. Sand on clay 92 5.6. Sensitivity analysis 94 5.6.1. Effect of soil friction angle on tip resistance 95 5.6.2. Effect of soil dilation angle on tip resistance 97 5.6.3. Effect of soil modulus on tip resistance 98 5.7. Pore pressure analysis 100 C H A P T E R 6. S U M M A R Y , D I S C U S S I O N , A N D F U T U R E R E S E A R C H 110 6.1. Summary of the present work 110 6.2. Discussion 113 6.2.1. Model used 113 6.2.2. Model parameters for over-consolidated sand 114 6.2.3. Limitations of the large strain analysis 115 6.2.4. A note on imposed displacement boundary 115 6.2.5. Sand type: field versus chamber 116 6.3. Future studies 117 6.3.1. Friction resistance along sleeve 117 6.3.2. Other sand types 118 6.3.3. Pore pressure analysis 119 vi B I B L I O G R A P H Y 120 A P P E N D I X A 125 A P P E N D I X B 130 vii LIST OF SYMBOLS AND ABBREVIATIONS B Bulk modulus B C 1 Boundary condition type 1 in calibration chamber, ah = Constant, o v = Constant B C 2 Boundary condition type 2 in calibration chamber, Sh = 0, s v = 0 B C 3 Boundary condition type 3 in calibration chamber, Sh = 0, a v = Constant B C 4 Boundary condition type 4 in calibration chamber, s v = 0, Oh = Constant C u Coefficient of uniformity C P T Acronym for "Cone Penetration Test" C P T U Piezocone sounding with pore pressure measurement D R Relative density of the sand in the chamber before penetration, and after consolidation Dio Diameter for 10% finer by weight D 5 0 Diameter for 50% finer by weight e m a x Maximum void ratio e m in Minimum void ratio F L A C A computer program, acronym for "Fast Lagrangian Analysis of Continua" G Shear modulus of soil G s Specific gravity of soil I r Rigidity index Irr Reduced rigidity index K B Bulk stiffness number. K G Shear stiffness number K M Constrained modulus number Ko Coefficient of earth pressure at rest, or coefficient of lateral stress before penetration M Constrained modulus m Exponent N c Cone factor in clay N q Cone factor in sand n Exponent O C R Over-consolidation ratio P A Atmospheric pressure p ' 0 Mean effective stress q c Cone tip resistance q C i C C Cone tip resistance measured in calibration chamber qc,fieid Corrected cone tip resistance expected to obtain in the field su Undrained shear strength U l Pore pressure on the mid cone face U 2 Pore pressure behind the tip a Angle which describes the curvature of the failure envelope s v Volumetric strain viii Drained friction angle of soil Constant volume friction angle <t>f Friction angle at failure Vo Secant angle of friction at a'ff = 2.72 P A Yd Dry density after consolidation Yd.max Maximum dry density Yd,min Minimum dry density <P Dilation angle a'ff Effective normal stress on the failure surface at failure a ' h Effective horizontal stress 0 " ' m Mean effective stress a v Total vertical stress a ' v Effective vertical stress iff Shear stress on the failure surface at failure ix L I S T O F T A B L E S Table 2.1. Boundary conditions available in Italian calibration chambers. 24 Table 2.2. Properties of Ticino sand. 27 Table 4.1. Parameters used for deformation and shear strength of Ticino sand. 40 Table A l Data base for calibration chamber tests on Ticino sand at E N E L 127 CRIS, B C 1 type boundary Table A 2 Data base for calibration chamber tests on Ticino sand at E N E L 128 CRIS, B C 3 type boundary Table A3 Data base for calibration chamber tests on Ticino sand at E N E L 129 CRIS, B C 4 type boundary Table A 4 Data base for calibration chamber tests on Ticino sand at E N E L 129 CRIS, B C 2 type boundary Table B l Comparison of F L A C solution with van den Berg solution 134 X L I S T O F F I G U R E S Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 2.1 Fig. 3.1 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig. 4.13 Fig. 4.14 Fig. 4.15 Fig. 4.16 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Assumed failure mechanisms for deep penetration 5 Steady state deformation around "simple cone model" 11 Schematic view of Eulerian approach of cone penetration problem 14 General arrangement for E N E L - C R I S calibration chamber 21 Overlayed quadrilateral elements used in F L A C 31 Variation of constrained modulus number with sand relative density 38 The shape of the grid in the vicinity of the cone tip (first approach) 43 A reasonable variation of cone factor with penetration of tip in clay 44 soil (first approach). Unacceptable variation of tip resistance with penetration of tip in 46 sandy soil (first approach) The deformation pattern around the cone tip obtained in analysis 48 The deformation pattern around the cone tip obtained in experiments 50 The location of cone tip after a penetration of about 0.35 m 52 The location of cone tip at mid grid depth, after a penetration of 0.75 53 m Contours of vertical stresses around the cone tip 54 Contours of horizontal stresses around the cone tip 55 Variation of horizontal stresses in the soil elements ahead and behind 57 the cone tip. A typical profile of penetration resistance versus depth. 61 Agreement of predicted and measured tip resistance for B C 1 type 62 boundary condition (second approach) Comparison between predicted and measured tip resistance for B C 3 65 type boundary condition (second approach) Comparison between predicted and measured tip resistance for B C 3 71 type boundary condition after removing the suspicious measurements (second approach). Comparison of measured and predicted values of tip resistance for all 72 types of boundary conditions. Effect of effective horizontal stress on tip resistance 75 Effect of effective vertical stress on tip resistance 76 Effect of chamber size and boundary condition on tip resistance 78 (dense sand). Effect of chamber size and boundary condition on tip resistance 80 (loose sand). Profile of tip resistance with penetration depth for B C 1 and B C 4 82 boundary condition (loose sand). xi Fig. 5.6 Profile of tip resistance with penetration depth for BC1 and B C 4 85 boundary condition (dense sand). Fig. 5.7 Result of drained cone penetration analysis for loose sand overlying 87 dense sand Fig. 5.8 Experimental measurement of penetration resistance in a layered soil. 88 Fig. 5.9 Result of drained cone penetration analysis for dense sand overlying 90 loose sand Fig. 5.10 Result of drained cone penetration analysis in sand into undrained 93 clay Fig. 5.11 Effect of variation of friction angle on tip resistance 96 Fig. 5.12 Effect of variations in dilation angle on tip resistance 97 Fig. 5.13 Effect of variation of soil modulus on tip resistance 99 Fig. 5.14 Numerical prediction of variation of excess pore pressure during 101 penetration on the cone face ( U l ) and behind the tip (U2) with soil permeability. Fig. 5.15 C P T U sounding from Alex Fraser Bridge research site. 104 Fig. 5.16 Contours of excess pore pressure around the cone tip, 107 permeability=5e-7 m/s. Fig. 5.17 Contours of excess pore pressure around the cone tip, 108 permeability=5e-5 m/s. Fig. B . 1 F L A C zone geometry for bearing capacity of a strip footing. 131 Fig. B.2 Model boundary conditions used in F L A C . 132 Fig. B.3 Comparison of theoretical solution of Prandtl with F L A C response. 133 xn A C K N O W L E D G E M E N T I would like to thank my supervisors, Dr. P. M . Byrne and Dr. R. G. Campanella, for their assistance during the length of this thesis. I appreciate valuable discussions I had with Dr. Byrne, who pushed this study to new boundaries. I am also grateful to Dr. Campanella who defined an interesting area for research, and who gave me the opportunity to do this work. Their advice and suggestions during the preparation of this thesis are very much appreciated. Special thanks and sincere gratitude are due to Dr. D . L . Anderson whose comments and suggestions have also been very helpful in shaping this research program. During my years of studies at U B C , I was fortunate to have also stimulating discussions with several other distinguished professors such as Dr. Vaziri , Dr. Vaid, Dr. Foschi, and Dr. Gadala. I would like to thank the committee members Dr. Howie and Dr. Atukorala who, along with Dr. Byrne and Dr. Campanella reviewed this thesis. This research is funded by the University of British Columbia through U G F awards and by the Natural Science and Engineering Research Council of Canada ( N S E R C ) . This support is gratefully acknowledged. Finally, this thesis could not have been finished without the extensive support of my mother, Sedigheh Tavakolian. M y mother patiently tolerated all the problems of my difficult life and time at U B C . I was, am, and shall be indebted to her for what she has done for me. The love of my intelligent daughter, Noora, has always been a xiii strong motivation and encouragement for whatever I have accomplished throughout these years. This thesis is dedicated to these loving individuals as well as to two people who have left this life: my beloved departed wife, Maryam Sheikh Bagher Mohajer, whose good memories are with me constantly; and my father, Modjtaba Ahmadi, who always encouraged me to pursue my education and who made considerable sacrifices to that end. xiv In memory of my beloved wife, Mary am, and of my dear father, Modjtaba, with thanks to my devoted mother, Sedigheh, and my loving daughter, Noora. XV C H A P T E R 1: L I T E R A T U R E S U R V E Y 1.1. Introduction Due to the difficulties in retrieving undisturbed samples, in-situ testing techniques are widely used to characterize the engineering properties of granular materials. The repeatability and reliability o f the cone penetration test (CPT) has increased its acceptance as a predominant choice in in-situ testing techniques. The cone penetrometer is cylindrical in shape having a cone in front with a base area of 10 cm 2 and 60 degree tip apex angle. The friction sleeve, located behind the conical tip, has a standard area of 150 cm 2 . The addition of pore pressure measurements during C P T has enhanced the interpretation of geotechnical parameters particularly in loose or soft, saturated deltaic deposits. The continuous measurement of pore pressures along with tip resistance and friction along the sleeve has given a new status to C P T as a premier tool for stratification logging of soil deposits. The excess pore pressure measured during penetration is a useful indication of the soil type and provides an excellent means for detecting details in stratigraphy. In addition, when the steady penetration is stopped, the excess pore pressure decay with time can be used as an indicator of the coefficient of consolidation. Also, the 1 equilibrium pore pressure value after complete dissipation is reached, provides important data on the existence of any gradients causing ground water movements. The most important limitation of the C P T for site characterization and in-situ property determination is that no samples are obtained for positive soil identification. Although soil classification charts have been developed in an attempt to overcome this limitation, care must be taken in their use, and soil samples should always be taken to validate the chart interpretations, unless prior experience is available for the site under investigation. The recently developed vision cone penetrometer (Raschke and Hryciw, 1997) may help significantly in overcoming the lack of sample retrieval in CPT. Other sensors can also be incorporated into the cone penetrometer system to measure seismic velocity, electrical resistivity, pH, temperature, specific ion concentration, fluorescence, gamma radiation, and oxygen reduction potential, to name a few. These are useful for detection of pollutants, their concentrations, and their distributions (Mitchell et al. 1998). Improvements in equipment and the variety of tests performed by C P T , however, are not always complemented by a theoretical understanding of interpretation techniques. Over the years, there has been a great demand for validated correlations between cone resistance and engineering properties of soil. The correlations are mostly obtained from experiments in calibration chamber tests with specified boundary conditions. 2 These correlations still rely to a great extent on empiricism. Thus, it is not surprising that the main application of the C P T in site investigation is for soil profiling. The large strain and material non-linearity associated with the cone penetration problem together with the complicated boundary conditions make the analysis an extremely difficult task. However, despite inevitable simplifications and idealizations introduced, theoretical approaches to this problem can provide a better understanding of C P T results. A proper formulation o f the theories can shed light on the significance of various parameters and helps the selection of the appropriate form of correlation equations. 1.2. Literature survey Different approaches have been used in the analysis o f cone penetration. These approaches include: (i) bearing capacity theory, (ii) cavity expansion theory, (iii) steady state approach, and (iv) finite element method. This section deals with the basic ideas behind these approaches and their capabilities as well as limitations. 1.2.1. Bearing capacity theory The bearing capacity theory in foundation design is based on the plasticity approach developed by Prandtl. The success of this theory in predicting the bearing capacity o f 3 shallow foundations is widely acknowledged. A number of attempts have been made to extend this approach to deep foundation and cone penetration problems. In the bearing capacity theory, the cone tip resistance is assumed to be equal to the collapse load of a deep circular foundation in soil. The extension of the bearing capacity approach to penetrometer analysis required the assumption of a failure mechanism. Some o f the mechanisms which have been proposed are shown in Fig . 1.1, and are described as follows [Durgunoglu and Mitchell (1975)]. (a) : The slip surface ends at the cone base level. The effect of embedment depth is replaced by a surcharge pressure acting at the level of the cone base, and the shear strength of the overburden is neglected [Terzaghi, (1943)]. (b) : The slip surface curves back to the shaft of the penetrometer [Meyerhof (1951)]. (c) : The radial slip surface ends prior to reaching the base level o f the penetrometer [Berezantzev et al. (1961), Vesic (1963)]. (d) : The radial slip surface reaches a vertical tangency [Hu (1965)]. The analysis involved for the above failure mechanisms is based on a plane strain model, and the failure mechanisms cannot be extended directly to an axially symmetric problem. The use of shape factors to convert the plane strain solution to the axisymmetric solution is questionable. Also, a major difficulty with this approach is that the stress-strain behavior of the soil is not embodied in this type of analysis. The soil is assumed to be a rigid plastic 4 ^ ^ ^ ^ (a). Terzaghi (1943) (b). Meyerhof (1951) Fig. 1.1: Assumed failure mechanisms for deep penetration. 5 material, and the compressibility of the soil is not accounted for. To address this limitation, other solutions have been investigated. 1.2.2. Cavity expansion theory The analogy between cavity expansion and cone penetration was introduced by Bishop et al. (1945). They suggested that the pressure required to produce a deep hole in an elastic-plastic medium is proportional to the pressure required to expand a cavity of the same volume under the same conditions. In order to use the cavity expansion approach to predict cone resistance the following steps need to be followed [Yu and Mitchell (1998)]: (1) : Develop theoretical (analytical or numerical) limit pressure solutions for cavity expansion in soils. (2) : Relate cavity expansion limit pressures to cone resistance. Vesic (1972) developed solutions in a Mohr-Coulomb material model for both cohesive and frictional soils. He also suggested a procedure to account for the volume change in the plastic region. The cone tip resistance can then be estimated by determining the limit pressure during the expansion of a spherical cavity. The cone factor, defined as N c = (qc-CTv)/su for a clay material, was found to be: N c = 3 . 9 0 + ^ ln( —) (1.1) where N c is the cone factor in clay, q c is cone tip resistance, a v is total vertical in-situ stress, G is the shear modulus of clay, and su is its undrained shear strength. Vesic (1972) has also shown that the following cone factor can be used for sand. N „ = L . ° )exp[( x/2-f )tanf ]tan2(45 + ^ / 2 ) ( I J n (1.2) 3 - sine? In the above relation, N q is the cone factor in sand defined as qja'v. The rigidity index I r is defined as: I r = G / p'o tan and the reduced rigidity index is defined as = I r / (1+Ir Sv), where s v is the average volumetric strain in the plastically deformed region. Here also <]>' is the effective friction angle of sand, p'o is the mean normal effective stress, a ' v is the effective in-situ vertical stress, Ko is the coefficient of earth pressure at rest, and n = 4 sin cj)7[3(l+sin (j)')]. The cone penetration process has also been modeled using cylindrical cavity expansion theory. Carter et al. (1986) presented analytical solutions for cavity expansion in a non-associated Mohr-Coulomb material. Y u and Houlsby (1991) extended the method by incorporating large strains in the plastic region, and Salgado et al. (1997b) presented solutions based on variable friction and dilation angles around the cavity and the dependence of shear modulus on pressure and void ratio. The predictions of tip resistance were compared with a large number of chamber tests. Typically, the measured cone resistance is predicted to within ±30%. Collins et 7 al. (1992) presented solutions for cylindrical and spherical cavity expansion theory in sand using a constitutive model based on critical state theory. Shuttle and Jefferies (1998) have used the spherical cavity expansion to predict the tip resistance, and consequently interpret the sand state. B y using the realistic soil stress-strain models significant progress has been made in developing accurate cavity expansion solutions for both clay and sand. Cavity expansion theory is expected to provide a more accurate prediction of cone resistance than can be obtained using the bearing capacity theory. This is because the influence of soil stiffness, compressibility (or dilatancy), and penetration induced horizontal stress increase can all be properly taken into account. Using a spherical cavity solution combined with critical state soil mechanics, Mayne and Chen (1994) derived expressions for penetration pore water pressures at the cone face ( U l ) and shoulder (U2) in terms of the effective stress friction angle, the rigidity index (I r), and the over consolidation ratio. Though significant progress has been made in recent years to refine the cavity expansion approach, still this approach seems to model the penetration inadequately. Cone penetration is not identical to expanding a sphere or cylinder inside a soil mass. Cylindrical cavity expansion theory appears to model the penetration reasonably well, but the movements in this model are horizontal (radial). The experimental observations show that vertical displacements of the soil particles around the tip of the cone are not negligible. 8 1.2.3. Strain path method Experimental observations o f deep penetration problems indicate that soil deformations due to the penetration of piles and penetrometers are similar even though the properties of the soils may be different [Vesic (1963)]. Ladanyi (1963) also showed that the strain field around an expanding cavity is independent of soil properties and is uniquely determined by the geometry of the problem. He suggested that i f the strain path of the soil around the cavity can be reproduced in a suitable laboratory test, the soil stresses could be determined. The above concepts led Baligh (1985) to suggest that deep steady penetration problems in soil are basically strain-controlled, and that the associated deformations are not sensitive to material properties. He also argued that, due to the kinematic constraints that exist in deep foundation problems, soil deformations can be estimated with a reasonable degree of accuracy from kinematic considerations alone. This approximate analytical procedure is known as the "strain path method". In isotropic homogenous soil, cone penetration may be treated as a steady state problem. For an observer moving with the penetrometer, the deformation pattern in the soil does not vary with time. Accordingly, by changing the reference coordinate system, the penetration process can be modeled by a steady flow o f soil past a stationary penetrometer. 9 In this approach, soil strains are estimated from an approximate velocity field. A constitutive soil model is then introduced to calculate soil stresses. Equilibrium equations are used to determine improved estimates of soil velocities; and the iteration continues until the resulting stresses satisfy equilibrium equations. Acar and Tumay (1986) presented the strain fields during penetration in clay. A further development was to couple this approach with a finite element model. In this regard, Teh and Houlsby (1991) used their strain path solutions as the initial stresses for a large strain finite element collapse load calculation. Levadoux and Baligh (1986) introduced a kinematic model, known as "simple cone model", which can be used as the initial deformation field for a strain path approach. Fig. 1.2 demonstrates the steady state deformation field around the cone. The idea behind the simple cone model originates from fluid mechanics. The velocity field of expanding a spherical cavity is identical to the velocity field o f a point source at a fixed location in a fluid at rest. The simple cone model aims to reproduce an approximate penetrometer geometry by a combination of sources and sinks in a uniform flow field. The strain path method can be considered to be superior to the cavity expansion theory because it takes into account the two dimensional nature of the penetration process. However, the strain path method, though promising in theory, has not been entirely satisfactory. The equations of equilibrium are not completely satisfied, and therefore, 10 Fig . 1.2. Steady state deformation around "simple cone model", (after Levadoux and Baligh, 1986) the method is an approximate solution. In addition, the method has only been used with limited success for undrained clays. The application of the strain path method for analysis o f the cone penetration in sandy soils is not found in the literature. l l 1.2.4. Fini te element method The finite element method has also been used in the analysis o f cone penetration. Both small strain and large strain formulations have been investigated by different researchers. In the small strain analysis given by de Borst and Vermeer (1982) and Griffiths (1982), the cone is introduced into a pre-bored hole, with the surrounding soil still in its in-situ stress state. A plastic collapse calculation is performed, and the collapse load is assumed to be equal to the cone resistance. However, it can be argued that during cone penetration, high lateral stresses develop next to the shaft of the cone resulting higher cone resistance than that predicted by small strain analysis in a pre-bored hole. To consider the effects of cone penetration on the initial stress conditions, a large strain analysis is required to model the penetration of the cone by a vertical distance equal to several times its diameter. Results for large strain analysis have been presented by Budhu and W u (1992) for penetration in cohesive materials, and by Cividini and Gioda (1988) for penetration in frictional materials. Zero-thickness elements are used to model the interface behavior between the cone and the soil. In their analysis, the roughness o f the cone-soil 12 interface can be varied. Kiousis et al. (1988) have also used a large strain formulation for penetration o f a smooth cone. The Updated Lagrangian formulation is adopted for these analyses. A Lagrangian approach implies that the nodal points of the finite element mesh are coupled with the material points. Updated Lagrangian means that after each incremental calculation, the mesh is modified. To obtain the new coordinates of the nodal points, the incremental nodal displacements are added to the previous coordinates o f the nodes. The disadvantage in this formulation is that i f large incremental nodal displacements occur, the elements may become distorted or even turn inside out, causing inaccurate solutions. Using an Eulerian formulation, van den Berg (1994) presented a large strain analysis of the cone penetration in both frictional and cohesive soils. In the Eulerian formulation, the finite element mesh is fixed in space, while the material points stream through the mesh. The cone was modeled as a fixed boundary, and interface elements were introduced between the soil and the cone. For a given initial stress state, the penetration process was started by applying incremental material displacements at the lower boundary o f the mesh. A s is shown in Fig . 1.3, a material point A , originally located underneath the cone tip, moves upward and is compressed around the shaft o f the cone, arriving at B . The corresponding stress and strain field around the cone are then calculated. The calculation is stopped when a steady state is reached with respect to the stress and strain distribution in the soil. 13 Fig. 1.3: Schematic view of Eulerian approach of cone penetration problem. (after van den Berg, 1994) Though progress has been made in recent years by introducing the finite element technique to analyze the cone penetration phenomena, a common shortcoming is noticed: the results of finite element analysis have not been verified or compared by experimental values in the field or in the chamber. 14 In addition to the previous methods, some other alternative approaches have also been introduced in the literature. The moving point dislocation of Elsworth (1993) and the discrete-element method of Huang and M a (1994) are examples of novel techniques to analyze the cone penetration process. 1.3. Scope of the present work In the previous section a number of different approaches for modeling the cone penetration process were discussed. The advantages and disadvantages for each type of modeling were also presented. The cone penetration is a steady state process. The cone penetrates the soil with a constant speed o f 20 mm/s. The soil located on the cone path is pushed aside as the cone moves downward through the ground. The stresses around the tip, both horizontal and vertical, increase significantly. This is especially true for penetration in sandy material. Also, the deformations in the soil mass around the cone tip are large. In this study, two different approaches to model cone penetration are discussed. In the first approach, the cone is placed at a prescribed location inside the soil mass. Interface elements are introduced in the model to separate the cone from the surrounding soil; and a collapse load calculation is performed. Only small strain analysis can be performed. Since the numerical analysis with this approach did not 15 seem to provide an accurate prediction of tip resistance, a second approach was developed. In the second approach, however, the penetration process is simulated in a realistic manner. Soil initially located underneath the cone tip is pushed aside. In this approach, the penetration starts from the ground surface (top of the grid) and can continue to any desired depth inside the grid. A s the cone moves downward in the grid, the stresses below the cone tip w i l l increase. This ensures that the build-up of stresses around the cone tip is correctly accounted for. The ability to perform large strain analysis in this approach can be considered as an advantage over the first approach. To verify the second approach, the measurements of tip resistance in the calibration chamber are compared with the numerical predictions. One o f the applications of the second approach is the analysis o f cone penetration in layered soil. This is an issue that has been addressed inadequately in the literature. The second approach is capable of producing numerical values for interface distance. These values are compared with experimental findings. Most of the correlations of cone tip resistance with soil parameters are derived from experimental measurements in the calibration chamber test. However, chambers have a finite size, and it is argued that the size of the chamber can influence the measured tip resistance. Also the type of boundary condition in the chamber can influence the experimental measurements. Therefore, the result of a penetration test in the field can 16 be quite different from the measurements in the chamber given the soil in the field and in the chamber have the same properties. The difference is called "chamber size effect" in the literature. In this study, the effect o f chamber size and its boundary are investigated. A sensitivity analysis is also performed in this study to investigate which soil properties contribute most to the measurements of cone tip resistance. Finally, a preliminary analysis of pore pressure response during penetration is performed. This is also an issue that is addressed insufficiently in the literature. Chapter 2 provides a description of calibration chamber tests and the different boundary conditions during testing. Methods of sample preparation are briefly discussed. In this study, the computer program F L A C has been used to carry out the numerical analyses. Chapter 3 provides a brief explanation of this computer program. In chapter 4, the constitutive model and the parameters used for sand are described briefly. The two approaches in cone penetration are also presented in this chapter, and the numerical predictions are compared with experimental measurements. The applications of the proposed numerical procedure (soil layering, chamber size effect, etc.) wi l l be discussed in chapter 5. 17 In chapter 6 a summary o f this study is presented together with a discussion of the need for future research in cone penetration analysis. Appendix A contains the experimental data of cone tip resistance in the calibration chamber that was used to compare with the predictions o f the numerical analyses. In Appendix B, the reliability of the computer program F L A C is first verified through numerical analysis of the bearing capacity problem for which theoretical solution is available. The proposed model for analysis o f tip resistance was also compared with other independent solutions in C P T modeling. 18 C H A P T E R 2: CALIBRATION C H A M B E R TESTING 2.1. Introduction Theories or approaches for interpretation of cone penetration test results in the field must be calibrated against experimental data. For a cohesive material, where penetration is essentially undrained, cone tip resistance depends strongly on undrained shear strength (s u) of the material. B y performing in-situ vane shear tests or laboratory tests on undisturbed clay samples taken at locations close to the cone path, a reasonable correlation between cone tip resistance and undrained shear strength of clay can be established. For sandy soils, however, due to the problems associated with sample disturbance, the above approach is practically inadequate. Instead, C P T measurements are performed in large laboratory vessels with known density o f sand subjected to known stresses at the boundaries. These vessels are called calibration chambers. To interpret the properties o f a sandy material, the measurements o f C P T in the field are compared with the measurements in the calibration chambers. 19 2.2. Cal ibra t ion chambers The first advanced calibration chamber where boundary stresses and strains could be measured was built in 1969 at the Country Roads Board (CRB) in Australia (Holden, 1991). Nowadays, the chambers used in research differ in a number o f ways such as dimensions, nature and form of control of boundaries, deposition procedure, and capability to handle saturated specimens. Ghionna and Jamiolkowski (1991) provide an extensive list o f most calibration chambers in use nowadays around the world. Figure 2.1 from Bellotti et al. (1982) shows a general set-up of calibration chamber testing used by E N E L - C R I S (the geotechnical, structural, and hydraulic research laboratory o f the Italian Electricity Company) in Mi lan , Italy. The equipment consists of a flexible wall chamber, a loading frame, a mass sand spreader for sand deposition, and a saturation system. The height of E N E L - C R I S calibration chamber is 1.5 m, and its diameter is 1.2 m. The sand specimen is prepared by the pluvial deposition through air. The sample is then consolidated under the desired K 0 value for the test. This is performed by gradually applying vertical stresses while restraining radial movement at the sample lateral boundary. For over-consolidated samples, the vertical stress is decreased in small increments until the desired O C R ratio is reached. The sand specimen is enclosed at the side and base by rubber membranes; the side membrane is sealed around an aluminium plate which forms the top rigid boundary o f 20 Vertical pressure manometer and transducer • B Piston water I i Chamber water To air regulator Pressure transducer Reaction jack Lid Top plate Centre hollow bush Double wall barrel Side membrane Base membrane Piston Cylinder —i Base Rod to displ. transducer Fig . 2.1. General arrangement for E N E L - C R I S calibration chamber, (from Lunne et al., 1997 based on Bellotti et al., 1988) the specimen and transfers the thrust of the chamber piston from the sand to the lid. A hole in the center o f the lid allows the penetration o f the cone into the sand specimen. It is also possible to saturate the sand samples. However, Bellotti et al. (1988) argue that the saturation effect on the cone tip resistance is not significant; and penetration in sand occurs in virtually drained conditions. Therefore, dry sand has usually been tested at E N E L - C R I S calibration chamber. 21 Knowledge o f the relationship between cone resistance and the relative density and stress state is important in interpreting the cone penetration tests in sand. Each test results in one value of tip resistance for given values of relative density and stress. A large number o f tests, covering the range of densities and stresses of interest, provide the basis for regression analysis of the data and empirical establishment of the relationship between the tip resistance, relative density, and the stress state. 2.3. Limitations of calibration chamber tests Been et al. (1988) and Ghionna and Jamiolkowski (1991) have reviewed the problems associated with calibration chambers. Sample age, types of sand tested, and sample size and boundary effects are among the important issues to be discussed. 2.3.1. Sample age and cementation Calibration chamber tests are performed on specimens of freshly reconstituted sands. The fabric o f these samples of sand may be different from that o f natural soil deposits which may have a highly developed structure. The structure and aging effects have been reported to have a significant effect on measured cone resistance (Schmertmann, 1991). 22 Light cementation can have a moderate impact on penetration resistance. For a lightly cemented soil the magnitude of cohesion intercept (c') is relatively small, and generally does not exceed 20 to 40 kPa (Rad et al. 1986). In contrast, even very light cementation has an important influence on the stiffness, especially at small and intermediate strain levels. Probably, the influence of light cementation on the penetration resistance should be linked to the contribution from increased stiffness of the sand (Ghionna and Jamiolkowski, 1991). 2.3.2. Types of sand tested Most calibration chamber tests have so far been performed on uniform, clean predominantly silica sands, referred to as "academic" soil by Ghionna and Jamiolkowski (1991). Natural sand deposits are seldom as uniform, and generally contain some small amount of fines that may significantly influence the engineering behavior. In addition, many relevant engineering problems are linked to more crushable and compressible materials such as carbonate sands which are usually slightly cemented. Strictly speaking, the correlations are only valid for sands of similar grading and mineralogy to the sands the correlations are based upon. To extend calibration chamber research to more realistic soils such as silty, crushable, and lightly cemented sands is a desirable aim for future work (Ghionna and Jamiolkowski, 1991). 23 2.3.3. Chamber size and boundary effect Depending on whether stresses are kept constant or displacements are zero at the lateral and bottom sample boundaries, there are four different types o f boundary conditions that can be applied in this type of calibration chamber. These are listed in Table 2.1. Table 2.1: Boundary conditions available in Italian calibration chambers. Type of boundary La tera l boundary Bottom boundary condition condition condition B C 1 Horizontal stress = Constant Vertical stress = Constant B C 2 Horizontal strain = 0 Vertical strain = 0 B C 3 Horizontal strain = 0 Vertical stress = Constant B C 4 Horizontal stress = Constant Vertical strain = 0 None of these four different boundary conditions simulate the field condition perfectly. The larger the chamber size, the less significant is the difference between results obtained in the chamber and the results obtained in the field. This means that the boundary conditions in the chamber can influence the results of penetration resistance i f the chamber size is small. Parkin and Lunne (1982), based on penetration test results for two different chamber sizes and two different penetrometer sizes, have concluded that for loose sands, chamber size and boundary conditions do not have a significant effect on cone resistance. For dense sands, on the other hand, the effects are considerable. Lunne and Christophersen (1983), based on chamber test results on Hokksund sand, suggested that for a chamber to cone diameter ratio of 50, the difference in tip resistance obtained in the chamber and the field should be small. Jamiolkowski et al. (1985) proposed a formula to relate the tip resistance obtained in the chamber to the tip resistance in the field. 0 .2 (DR%-30) , % , field = %, cc ( 1 + " } < 2 -In the above formula, q C ; C C is the experimental value of tip resistance observed in the calibration chamber, D R is the relative density, and qc,field is the corrected tip resistance expected to be measured in the field for the same sand with the same relative density and the same in-situ stresses as in the chamber. The above formula implies that for loose sand with a relative density of 30%, the experimental results in the chamber and the field are basically similar; and no calibration chamber size effect should be considered. However, as the relative density o f sand in the chamber increases, the size effect wi l l become larger. The above formula is valid for a cone penetrometer with a projected cone area of 10 cm 2 in a 1.2 m diameter chamber; and it is based on the experimental results obtained on calibration chamber tests under BC1 type boundary condition. 25 Schnaid and Houlsby (1991) have documented experimentally the importance of chamber size effects on the ultimate cavity stress obtained from pressuremeter tests. They suggest that the finite chamber dimensions affect the cone tip resistance to approximately the same extent as the pressuremeter. Mayne and Kulhawy (1991) assumed that, regardless of the relative density and stress state, a chamber diameter to cone diameter ratio of 70 is sufficient to achieve the "free field" condition. However, as Salgado et al. (1998) argue, the choice of a suitable diameter ratio at which size effect is no longer important should also be based on the relative density, stress state, and other pertinent parameters. From the above discussion, it appears that in sands o f low and moderate compressibility the use of calibration chambers having dimensions sufficiently large to avoid the chamber size effects for dense specimens is probably not practical due to its high cost. It has also been suggested that the chamber size effects can probably be solved in a more cost effective manner by combining centrifuge testing of sands with miniature cones and large size calibration tests (Ghionna and Jamiolkowski, 1991). 2.4. Selected database Several groups of database for calibration chamber test results are available in the literature. Most of the calibration chamber tests have been performed on Ticino sand, Hokksund sand, Monterey sand, and Toyoura sand. In this study the experimental data were taken from the results of calibration chamber testing on Ticino sand at E N E L - C R I S calibration chamber in Italy. It is believed that the test results at E N E L -CRIS calibration chamber on Ticino sand are among the most reliable databases reported in the literature [Lunne et al. (1997)]. This database, taken from Lunne et al. (1997), is reproduced in Appendix A . L o Presti et al. (1992) provide the properties of Ticino sand as is shown in Table 2.2. Table 2.2: Properties of Ticino sand. D 5 0 , mm 0.58 Dio, mm 0.36 C u 1.50 G s 2.685 Cmax 0.931 6min 0.574 Yd, max, k N / m 3 16.73 Yd, min, k N / m 3 13.64 <})cv, degree 34.8 Angularity Sub-angular to angular Mineralogy 30% quartz, 5% mica The selected database together with the properties of Ticino sand are used in this study to verify the proposed model for cone penetration analysis. 27 C H A P T E R 3: A N INTRODUCTION TO F L A C 3.1. Introduction In this study, the F L A C (Fast Lagrangian Analysis of Continua) program has been used to simulate the cone penetration process. This code is an "explicit, finite difference program" that performs a "Lagrangian analysis", and is a suitable tool to model non-linear material behavior such as soil ( F L A C 3.4 Manual, 1998). In this chapter, some basic definitions of these words are first briefly introduced. The advantages and disadvantages of this numerical program are discussed next; and finally the suitability of the program to analyze the cone penetration process is presented. 3.2. Finite difference method The finite difference method is a numerical technique used for the solution of sets of differential equations. In the finite difference method, every derivative in the set of governing equations is replaced by an algebraic expression written in terms of the field variables (e.g., stress or displacement) at discrete points in space. 2 8 3.3. Equations of motion Both static and dynamic problems can be handled in F L A C . However, the formulation is based on the dynamic equations of motion. One reason for doing this is to ensure that the numerical scheme is stable even i f the physical system being modeled is unstable. With non-linear materials, there is always the possibility of physical instabilities. One penalty for including the full law of motion is that the user must have some physical feel for what the system is doing; F L A C is not a black box that w i l l give the "solution" at the end of its calculation phase. Since this code models a non-linear system as it evolves in time, the interpretation of results may be more difficult than a conventional finite element program. 3.4. Expl ic i t , t ime-marching scheme Explicit time integration methods permit the response of a dynamic system to be completely determined in terms of the information in previous time intervals. The central concept in this method is that the equations always operate on known values that are fixed for the duration of that time interval. The advantage of this method is that no iteration process is necessary in calculating stresses or strains even i f the constitutive law is nonlinear. Another advantage is the capability of treating very 29 large problems with only modest computer storage requirements, since the global matrices of the system are not formed. However, with explicit methods, stability typically requires that the time step be small enough that information does not propagate across more than one element in each time step. This means that large numbers of steps must be taken, and computational time can become very lengthy. 3.5. Lagrangian analysis The cone penetration process is basically a large strain phenomenon. The soil located on the cone path experiences large deformations as the cone penetrating downward clears its path from the soil below. A realistic analysis o f cone penetration should account for the large deformations in the soil around the cone. F L A C is a program that can handle the large deformations as well. The large strain option implemented in F L A C requires the coordinates to be updated at each time step. The incremental displacements are added to the coordinates so that the grid moves and deforms with the material it represents. This is termed a "Lagrangian" formulation. The constitutive law used in F L A C at each time step is a small strain one. However, it is argued that it is equivalent to a large strain formulation over many steps ( F L A C 3.4 Manual, 1998). 30 Fig . 3.1. Overlayed quadrilateral elements used in F L A C . 3.6. Notes on gr id elements The grid representing the solid body is divided into a finite difference mesh composed of quadrilateral elements. F L A C subdivides each element into two overlayed sets of constant strain triangular elements as shown in Fig . 3.1. Triangles a and b constitute one set (or pair) of constant strain triangular elements, and triangles c 31 and d constitute the other one. The force vector exerted on each node is taken to be the mean of the two force vectors exerted by the two overlayed quadrilaterals. If one pair of triangles becomes badly distorted, e.g., i f the area of one triangle becomes much smaller than the area of its companion, then the corresponding quadrilateral is not used in the analysis; and only the nodal forces from the other more reasonably shaped quadrilateral are used. If both overlayed sets of triangles are badly distorted, F L A C gives the "Bad Geometry" message in the large strain option; and execution of the program is halted. In the small strain option, however, no message is given, and the program execution is continued. 3.7. Disadvantages of F L A C A s mentioned briefly in section 3.4, the main disadvantage o f F L A C is its requirement for choosing a small time-step to guarantee the stability o f the numerical scheme. This means that the computational time can be very long. The solution time with F L A C is determined by the speed of the compression wave between nodal points. Problems that contain large disparities in elastic moduli are very inefficient to model with F L A C . For example, in the analysis o f pore pressure during cone penetration the bulk modulus of water is very much larger than that of the soil structure; and the solution time is inefficient. Also, the disparity in element 32 size can impose inefficiency to the solution time. The use of elements with length to width ratio of around one is recommended. In F L A C , it is discouraged to use elements that have a length to width ratio of more than five ( F L A C 3.4 Manual, 1998). This means that it is not practical to use small size elements in the neighborhood o f larger ones in the model; and the number of elements in the grid model generally becomes large, which makes the solution even slower. 3.8. Why FLAC ? A s it wi l l be seen later in the next chapter, the analysis of the cone penetration process involves imposing a displacement pattern to the soil elements around the cone, so that the pattern of deformation induced by the penetration analysis resembles that during actual penetration testing. In the analysis to be discussed in detail later, the soil elements undergo a prescribed deformation pattern. Therefore, in this type of analysis, emphasis is on producing a pattern of deformation around the cone. Analyses of this type are basically routine in F L A C , i.e., it is a routine procedure to apply a prescribed deformation pattern to a solid body, and calculate the response of the system, e.g. stresses, due to the imposed deformation. This capability of the program can be regarded as a favorable factor in facilitating analysis o f these classes of problems. Another advantage of this program is its scheme of calculation procedure that requires a step by step analysis. Since each step corresponds to time, the solution to 33 the cone penetration problem can be found in time as well . A reasonable solution to the penetration problem should include the effect of penetration stresses on the other elements along the cone path as the cone is going downward. In other words, the solution should be able to simulate the penetration process as it evolves in time and in space. This important requirement of realistic cone penetration analysis can be easily accommodated with F L A C . There are several in-built constitutive models in F L A C . The choice of the constitutive model depends on the type of material in the problem to be solved. Through "fish" functions introduced in F L A C , the in-built constitutive models can be changed to comply with the demands of the problem. This broadens the capability o f the program to be a suitable tool for the penetration analysis. This is because model parameters for elements located in the vicinity of the cone tip, where stresses are very high, can be changed through a fish function. This capability of F L A C enables the analysis of cone penetration to be carried out in a more realistic manner. 34 C H A P T E R 4: SIMULATION OF C P T T I P RESISTANCE IN S A N D 4.1. Introduction The cone penetration process can be modeled by a variety of approaches. Two different approaches for cone penetration modeling have been pursued during this research program. Emphasis is on the second approach; and it w i l l be discussed at greater length later in this chapter. The results presented in this thesis are based on the second approach. However, it is useful to describe briefly the modeling procedure and the results obtained in the first approach. In this chapter, the constitutive law used for analysis is presented first. The two different approaches for analysis of cone penetration in sand are presented subsequently. The second approach is verified by comparing the numerical predictions with the experimental results from calibration chamber tests. Finally, a review of the reliability o f the experimental data itself is also presented. The computer code F L A C has been used for all o f these approaches. Also, in the results to follow in this chapter and the next, it is assumed that the penetration is fully drained unless it is specifically noted. 3 5 4.2. Constitutive law for sand The Mohr-Coulomb elasto-plastic model was chosen for this problem. The parameters for this model are: bulk modulus, shear modulus, friction angle, and dilation angle. The values of stresses in close proximity to the cone tip are very much higher than those in the far field, and it is argued that the model parameters w i l l therefore be different in the near and far field. Hence, in simulating the calibration chamber tests, the Mohr-Coulomb soil parameters are considered to be stress dependent. The stress dependent relations for shear and bulk modulus used in the Mohr-Coulomb soil model are: G = K G P A ( ^ ) N ( 4 - 1 ) B = K B P A ( ^ ) M ( 4 - 2 ) In the above relations, c'm is the mean effective stress, P A is the atmospheric pressure, equal to 1 kg/cm 2 =98.1 kPa. The exponents m and n can range between 0 .2 to 0.7; and are taken to be 0 .6 . K G and K B are shear and bulk stiffness numbers that are assumed to depend only on the relative density of the sand in the calibration chamber. Values of constrained modulus (M) are measured for each test in the calibration chamber test; and these values are given in Appendix A . Knowing values of constrained modulus (M), it is possible to determine the constrained modulus number through a relation similar to equations 4-1 or 4 - 2 , i.e. 36 M = K M P A ( ^ ) N (4-3) "A Figure 4-1 shows the relation between values of K M and relative density for the series of normally consolidated Ticino sand tested under B C 1 boundary condition. Each point in this figure represents the K M value associated with the measurement of constrained modulus for each test, and the dotted line in the figure is the best fit line for all the data points considered. In other words, the dotted line describes the relation between the constrained modulus number and the relative density o f the sand in the calibration chamber tests. Having established a relation between K M and the relative density, it is now possible to find the values for K G and K B through elasticity formulas relating the constrained modulus to bulk or shear modulus provided a Poisson's ratio value is assumed. In this study, Poisson's ratio is taken to be 0.25 for sand. However, Poisson's ratios of 0.2 and 0.3 were also used in a few o f the numerical analyses. The outcome of these analyses shows that Poisson's ratio does affect the predicted values of tip resistance, but its influence is not significant. Though the values for K M were calculated from the experimental results on normally consolidated sand under B C 1 boundary condition, the same K M values were also used in the numerical analyses for over-consolidated sand and other types of boundary condition. Also, the data points in Fig. 4-1 are rather scattered. This is partially due to the value of modulus exponent (n) used in relation 4-3. However, it can be argued that for the 37 1400 1200 —\ 1000 — 800 H 600 —i 400 T — 1 — T 70 80 Relative Density, DR % 100 Fig. 4.1. Variation of constrained modulus number with sand relative density. Data from calibration chamber tests on Ticino sand. 38 simple Mohr-Coulomb model used in the numerical analysis, the relation between K M and the sand relative density represented by the best fit line in Fig . 4-1 can be considered sufficient for choosing the modulus numbers required in the model. It is also noted that these values of K G and K B are in the range of values reported by Byrne et al. (1987). This is especially true for lower relative densities. However, improvements in the model and a better choice of the parameters are warranted for further studies. The modulus numbers that are used in the constitutive model are given in Table 4.1 As mentioned previously, these values depend only on the relative density. The Mohr-Coulomb model used also needs to be defined in terms o f plastic parameters; namely friction angle and dilation angle. Baligh (1975) argues that since the tip resistance obtained in granular material often exceeds the level o f stresses ordinarily encountered in other soil mechanics application, a realistic analysis o f tip resistance in sand must therefore be based on the response of the soil at elevated stresses. The response differs from the common behavior in two important aspects: a. the decrease of the angle of internal friction with the mean normal stress, i.e. the Mohr-Coulomb failure envelope, is not straight but is actually convex, and b. the significant decrease in volume which takes place upon shearing even for dense granular media (Baligh, 1975). In order to determine the shear strength parameters of Ticino sands used in the calibration chamber tests, a series of triaxial tests were carried out by E N E L / I S M E S 39 in Italy. Baldi et al. (1986) have summarized the results of these tests in terms of the curvilinear formula proposed by Baligh (1975): % - c r f f tanc^'0+tana 2 l ° S l 0 l ^ \Z-3 * A J (4-4) Where iff = shear stress on the failure surface at failure, a'ff = effective normal stress on the failure surface at failure, a = angle which describes the curvature of the failure envelope, and <p'o = secant angle of friction at a'ff = 2.72 P A . Table 4.1 also shows the values of (j)'o and a as obtained from specimens at three different values of relative density. Table 4.1. Parameters used for deformation and shear strength of Ticino sand. D R K G K B 4>'o a % (deg) (deg) 45 195 325 38.2 4.2 65 230 385 40.2 6.5 85 290 480 42.9 8.1 D R = average relative density of the tested specimens, at the end of consolidation. The dilational characteristics o f the sand in the model were introduced by the following relationship which relates the dilation angle to the friction angle at failure and constant volume friction angle: 40 sin cp = sin <j)'f - sin <t>cv (4-5) Parameters in the above relation are defined as: cp = dilation angle, (}>'f = friction angle at failure, and $ c v = constant volume friction angle for Ticino sand, measured to be 34.8 degrees as described by Salgado et al. (1997b). 4.3. Size of the numerical grid In this study, predicted values of tip resistance are compared with the experimental measurements obtained from the E N E L - C R I S calibration chamber. The dimensions of the numerical grid are chosen to be the same as those of the ENEL-CRIS calibration chamber. The grid has a height of 1.5 m, and a diameter of 1.2 m. Because the axisymmetric option is used in the analysis, the figures representing the grid show a section of the grid having a radius of 0.6 meters. 4.4. First approach in cone penetration The method used in the first approach is basically a generalization of the method that has successfully been used for predicting the failure load of shallow foundations. Griffiths (1982) and de Borst and Vermeer (1984) have presented the result of their finite element analysis o f cone penetration in clay. Their methodology is basically similar to this approach. In this approach, the variation of stresses versus displacements at points close to the cone tip is monitored. The collapse (or failure) 41 load is reached when the stresses remain constant with continued increase in displacement. In the first approach the cone is placed at mid depth of an axisymmetric grid. The mesh close to the cone tip is more congested with elements so that the variation of response around the cone could be monitored with higher accuracy. Because the problem has symmetry about the vertical axis, the axisymmetric option can be used for this three dimensional problem to reduce the number of elements in the solution procedure. In this model, the cone is separated from the surrounding soil by interface elements. Two types of interface elements were used: conical interface elements separating the conical tip from the surrounding soil, and the cylindrical interface elements separating the cylinder above the cone tip from the surrounding soil. Figure 4-2 shows the grid in the vicinity of the cone together with interface elements. Only small strain analysis could be investigated in this approach. The implementation of the large strain option resulted in discontinuation of program execution due to the "Bad Geometry" message in F L A C . This message simply means that the execution cannot proceed due to the large geometric distortion of the elements. To simulate the penetration process, the points associated with the cone in the grid are given a downward movement. This results in an increase in the stresses below and around the cone tip. Figure 4-3 shows the variation of cone factor, N c = (q c-o v)/s u, with respect to the penetration depth for an analysis in clayey material. In the above relation, q c is the tip resistance, a v is the total vertical in-situ stress, and su is the 42 . 0.820 . 0.810 _ 0.800 m . 0.790 . 0.780 / / / / / / ! / / / / / • 3 / / C r I f / { ) i fl / / . 0.770 / / / / / . 0.760 . 0 750 0 740 / / / / / / s / _ 0.730 1 1 1 -0.150 -0.050 o.c ( ' T 60 0. 50 i 0. 1 250 r 10"-0 ) 1 350 I 0.4 50 1 0 1 550 I ( m i )650 0 1 750 Fig . 4-2: The shape of the grid in the vicinity of the cone tip (used in the numerical analysis for the first approach). The thick black lines are interfaces separating the cone from the surrounding soil. The cone is placed at mid-depth o f the grid. 43 15 o LL O c o o 10 - \ Hi 5-^ s u = 30 kPa "I 1 1-0.10 Penetration of Tip, m 0.00 0.20 Fig. 4.3. A reasonable variation of cone factor with penetration of tip in clay soil. (first approach). 44 undrained shear strength of clay. A simple elasto-plastic Mohr-Coulomb law is used to model the clay behavior. The shear modulus is assumed to be 9000 kPa, and the bulk modulus to be 300000 kPa. These values correspond to a Poisson ratio close to 0.5, and are assumed to remain constant throughout the model. The undrained shear strength is taken to be 30 kPa. In the analysis, the total vertical in-situ stress is assumed to be 100 kPa, and Ko is assumed to be 0.5. A s is shown in Figure 4-3, the cone factor, N c , seems to reach a constant value of about 13.1 with continued penetration. This value of cone factor is in the range reported in the literature and obtained during practical cone penetration testing in clay. It should be noted that although the slope of the curve diminishes with increase in penetration of tip, the curve does not become completely flat. However, it has been argued that this rather flat curve is acceptable for practical purposes. The situation is not so for penetration in sand. Figure 4-4 shows the variation of cone tip resistance versus the penetration of the tip in sand where it may be seen that the tip resistance is still increasing after a penetration of 0.2 meters, and no failure load can be distinguished. This means that with continued penetration the stresses do not reach a constant value. This general trend for sand is in contrast with the collapse (or failure) load definition used for this approach, and it is unacceptable. 45 (0 CD O c (0 40 30 jS 20 H .15 co CD a : 10 H 0 0.00 0.10 Penetration of tip, m 0.20 Figure 4-4. Unacceptable variation of tip resistance with penetration of tip in sandy soil, (first approach). 46 To overcome the difficulties associated with this approach, a different procedure for modeling was pursued, which wi l l be discussed next. 4.5. Second approach in cone penetration In order to physically simulate penetration in the second approach, the soil nodal points located along the cone path are pushed away in a systematic process that starts from the top of the grid, and can continue to any desired depth into the grid. The axisymmetric configuration is also used for this approach. F ig 4-5 shows the inner boundary nodal points at an arbitrary depth in the grid. In this approach, all these inner boundary points are displaced horizontally as well as vertically. In reality, the inner boundary points are initially located on the axis o f symmetry. However, in the numerical analysis for this approach, the points located on this axis do not accept any horizontal displacement; and cannot be pushed horizontally. In order to bypass this difficulty, the grid points located on the axis of symmetry were removed from the grid; and the inner boundary was relocated at a distance equal to a quarter of the cone radius. Therefore, in the numerical analysis, the inner boundary points are displaced a horizontal distance equal to three quarters of cone radius. For a standard cone with an area of 10 cm 2 , the cone radius is 17.9 mm; and the distance that the inner boundary points are displaced horizontally is 13.4 mm. The nodal points are also displaced a vertical downward distance o f 15 mm. 47 r./i -. 0.850 -. 0.830 - _ 0.810 " m . 0.780 -. 0.770 -D, _ 0.750 E F 0.730 . 0.710 i • i < i ' i i I i 0.000 0.020 0.040 0.060 0.080 0. 1 100 T L 0.1 I I 20 I 0.140 i Fig. 4.5. The deformation pattern around the cone tip obtained in the analysis. (second approach) 48 This downward displacement is necessary to produce a deformation pattern in the analysis that is similar to the experimental observations. The figure shows that during the modeling procedure, the nodal point A has already been pushed away from its initial location at A ' by giving a horizontal as well as a vertical downward displacement. The vector A ' A in the figure shows the displacement vector for this point. The nodal points B , C, and D are now being pushed away from the inner boundary. The nodal point B has just been displaced sufficiently, and wi l l no longer be pushed. As the process of imposing displacement on point B is halted, point E starts to be displaced; and this process continues systematically until all the inner boundary nodal points are displaced a horizontal and a vertical distance of 13.4 mm and 15 mm respectively. This procedure basically produces a vertical cylindrical hole with a radius equal to the cone radius in the grid. Baligh (1975) argues that penetration process belongs, to a large extent, to the class of displacement or strain controlled problems where primary consideration should be given to displacements. Baligh (1975) also argues that this is in contrast to a majority o f geotechnical problems which are stress controlled, and where primary consideration is given to satisfy equilibrium at the price of often neglecting strain compatibility. Based on experimental observations, Baligh (1975) also argues that the deformation pattern, obtained during penetration in two different soil types is more or less the same. Hence, the main criterion for this approach is to produce a deformation pattern in the analysis that is similar to the experimental observations. Fig. 4-6 shows the pattern of deformation around a cone tip obtained 49 Fig 4.6. The deformation pattern around the cone tip obtained in experiments. (after van den Berg, 1994). 50 from experiments (van den Berg, 1994). It is seen that the deformation pattern obtained in the analysis is relatively similar to that obtained during experiments. The second approach models the penetration process in a realistic way. The penetration modeling starts at the top of the grid, and progresses into the grid, and can end at any desired depth in the grid, meaning that the modeling process is simulating the cone moving downward in the ground. Figures 4.7 and 4.8 show the location of cone tip at different depths in the grid. Since cone penetration is basically a large strain phenomenon, the soil under the cone tip undergoes a severe deformation pattern; and it is necessary to use the large strain option in the analysis to better model the process. The ability to use the large strain analysis in the second approach is regarded as another advantage of the present approach compared to the first approach. A s mentioned previously, the grid points located on the axis o f symmetry were removed from the grid; and the inner boundary was relocated at a distance equal to a quarter of cone radius. Since the area ignored is only about 5% of the cone area, it can be argued that the error introduced is not significant. Figures 4-9 and 4-10 show typical patterns of vertical and horizontal stresses around the cone tip respectively. The figures correspond to a typical penetration analysis in sand with a relative density of 70% and vertical in-situ stress of 300 kPa. The K 0 value in the analysis is taken to be 0.5, and the boundary condition type used for this typical analysis is B C 1 . The comparison between these two figures shows that the magnitude of the horizontal stresses around the cone is larger than the vertical 51 L 1.600 L 1.200 L 0.800 L 0.400 L o.ooo m Fig . 4.7. The location o f cone tip after a penetration o f about 0.35 m. Fig . 4.8. The location o f cone tip at mid grid depth, after a penetration of 0.75 m. 53 Fig 4.9. Contours of vertical stresses around the cone tip. (contour A= 20 MPa, contour D= 5 MPa, contour interval= 5 MPa) 54 5 MPa ~r~ ' T i i \ i 1 1 1 1 1 1 — T 1 1 ,——| .— 0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 m 0.475 0.767 0.762 0.757 L 0.752 m |_ 0.748 0.743 0.738 0.732 0.727 0.722 P O M ) Fig 4.10. Contours of horizontal stresses around the cone tip. (contour A= 30 MPa, contour F= 5 MPa, contour interval= 5 MPa) 55 stresses. This response is typical for the results obtained with this approach. While the stresses in the far field have not increased significantly, a very large increase in stresses has occurred in close proximity to the cone tip. The largest stresses occur close to the cone; and they decrease rapidly at points farther from the cone. The magnitude of horizontal stresses around the cone tip is 30 M P a decreasing to 5 M P a along a horizontal radius of about 45 mm. The vertical stresses also decreased from 20 M P a to 5 M P a along a shorter distance of 30 mm. This shows that there are high stress gradients around the cone tip. More importantly, it is also noted that the stresses behind the cone tip decrease suddenly as the cone tip penetrates downward in the grid. This reveals an even higher gradient of stresses between the elements close to the cone tip and the elements above the cone shoulder. 4.6. Var ia t ion of horizontal stress ahead and behind the tip Fig . 4-11 shows the variation of the horizontal stresses in the soil elements located 0.15 m ahead as well as behind the tip. In this figure, the location of the cone shoulder is used as a reference, and it is associated with the zero value in the y axis. The length of the friction sleeve is about 0.13 m; and the variation of horizontal stresses shown along the 0.15 m length behind the tip is sufficient to demonstrate the variation along the whole length of the friction sleeve. The figure shows that for the elements located at about 0.15 m ahead o f the tip, the increase in the horizontal stress is not significant. For the elements located closer to 56 Fig 4.11. Variation o f horizontal stress ahead and behind the cone tip. 57 the cone tip, the horizontal stress gradually increases. It is also seen that for the elements in contact with the cone tip, the horizontal stress increases significantly. The figure also shows that for the elements behind the cone tip, located at about 30 to 40 mm above the cone shoulder, the magnitude of the horizontal stress sharply decreases. However, the horizontal stress gradually increases for the elements located further behind the tip. The maximum horizontal stress occurs at a distance of about 60 mm behind the shoulder. Above this point, the horizontal stress decreases very slightly, and remains relatively constant thereafter. The second approach predicts seemingly reasonable magnitudes of the horizontal stresses above the cone shoulder. For the typical penetration analysis considered above, the average magnitude of the horizontal stress on the elements above the cone shoulder is about 925 kPa. The values of active and passive earth pressure coefficients for this typical analysis are 0.23 and 4.4 respectively. For an in-situ vertical stress of 300 kPa, the active earth pressure is estimated to be about 70 kPa, and the passive earth pressure to be about 1300 kPa. It is seen that the magnitude of the horizontal stress above the shoulder that is predicted in the analysis is within the range of active and passive pressures. However, it is noted that the prediction o f the horizontal stresses above the cone shoulder obtained in the numerical analysis should be regarded as approximate. In the second approach, no interface element was modeled between the soil and the cone; and attempts to model the shearing response of the soil along the friction sleeve located above the cone shoulder have not been successful. The magnitudes of 58 shearing stresses that are directly predicted with this approach do not seem to be reliable. However, it is possible to estimate the magnitude of the shearing stresses along the friction sleeve by considering the horizontal stresses in the elements behind the cone tip. Assuming an interface angle of, for example, 10 degrees between sand and the friction sleeve, and for an average horizontal stress of 925 kPa in the elements above the cone shoulder, the magnitude of friction resistance is estimated to be about 165 kPa. For this typical example considered, the numerical analysis results in a cone tip resistance o f 19 M P a . This gives a friction ratio of 0.85%. This value of friction ratio is within the typical range of friction ratios (0.5% to 1.2%) obtained in experimental observations (Campanella, 1995). In order to predict the soil response above the cone shoulder with a higher confidence, a more rigorous and fundamental approach should be pursued; and improvements in the modeling procedure are warranted. It is believed that for improved modeling of response of the soil along the friction sleeve, it would be necessary to incorporate interface elements in the analysis. 4.7. Comparison of numerical results with experimental values in calibration chamber The numerical results for sands are compared with the experimental values obtained from penetration tests in the E N E L - C R I S calibration chamber in Italy. The test results together with the properties of sand used and type of boundary condition for each test are given in Lunne et al. (1997). This is reproduced in Appendix A . 59 Numerical results for all four different types of boundary conditions are compared with experimental values. F ig 4-12 shows a typical result of the numerical analysis obtained with the second approach. In the analysis, the tip force is calculated through the integration of vertical and shearing stresses at elements in contact with the cone. The tip force is then divided by cone area to give the tip resistance. In the figure the variation of tip resistance with the penetration of the tip is presented. The height o f the grid in the numerical analysis is 1.5 m, which is the same height as the E N E L - C R I S calibration chamber. Figure 4-12 shows that the cone tip resistance decreases when the cone approaches the bottom boundary. This is consistent with the experimental observations in calibration chambers, and is discussed in detail in chapter 5. The experimental values of cone tip resistance presented in Appendix A are the values of tip resistance measured at mid-height during calibration chamber tests. In order to compare the results of the analysis with the experiments, the predicted values of cone tip resistance were also taken at mid-height of the numerical grid, i.e. the values of cone tip resistance at point A in Fig. 4-12 were compared with experimental measurements. In the following sections, the results for each boundary type are presented separately. 4.7.1. Compar ison for BC1 type boundary condition Figure 4-13 shows the predicted values of tip resistance versus the experimental values for a series of tests with B C 1 (see Table 2.1) type boundary condition for 60 Cone Tip Res i s tance, MPa 5 10 15 20 25 Fig. 4.12. A typical profile of tip resistance versus depth obtained in the analysis. 61 Relative densities 53.2% to 92.8% Vertical stress 61.8 kPato 715.1 kPa Over-consolidation ratio 1 to 14.67 Kg values 0.370 to 1.296 0 10 20 30 40 50 Measured Tip Resistance, MPa Fig . 4.13. Agreement of predicted and measured tip resistance for B C 1 type boundary condition (second approach). 62 normally consolidated as well as over-consolidated Ticino sand. In this type of boundary condition, constant stresses are maintained in the horizontal as well as the vertical directions in the calibration chamber. These stresses are equal to the stresses after completion of Ko consolidation. For these series of tests, the relative density ranged from 53.2% to 92.8%, and the vertical stress in the chamber ranged from 61.8 kPa to 715.1 kPa. The K 0 values were in the range 0.370 to 1.296, and the O C R values ranged from 1 for normally consolidated sand to 14.67 for over-consolidated sand. The points in this figure are close to the line with a slope of 45 degrees, indicating that the predicted values obtained from numerical analysis are in agreement with the experimental values obtained in calibration chamber testing. It is also noted that for values o f tip resistance more than 35 M P a , the numerical procedure systematically under-predicts the tip resistance values. These points correspond to experiments in which confinement stresses were high. It may be possible to improve the under-prediction by fine adjustment of the parameters that are used in the model. 4.7.2. Compar ison for B C 3 type boundary condition In this type of boundary condition (BC3), a constant stress is imposed on the bottom boundary during penetration testing, but the strain in the lateral boundary is kept zero. This type o f test has also been performed frequently in calibration chamber testing. For these series of tests, the relative density ranged from 40.2 % to 94.4 %, and the vertical stress in the chamber ranged from 62.8 kPa to 716.1 kPa. The O C R value is 63 between 1 for normally consolidated sand to 14.41 for over-consolidated sand. The Ko value ranged between 0.39 to 1.356. Figure 4-14 shows the comparison between the experimental results and numerical values obtained in the second approach. Most of the points are slightly above the 45 degree line. This means that the second approach over-predicts the test results to some extent. However, generally speaking, the difference between the test results and the numerical values is not more than 30%. In the sections to follow, a critical examination of the database itself w i l l be discussed. In his numerical modeling of cone penetration, Salgado (1993) also reports an over-prediction for B C 2 and B C 3 type boundary conditions. Salgado et al. (1998) associate this over-prediction to experimental difficulties in achieving perfect B C 3 conditions. They argue that it is probably not possible to maintain, in a flexible wall calibration chamber, a radial displacement equal to zero everywhere at the sample boundary; and because of this compliance at the sample boundaries, the resulting experimental values of penetration resistance are smaller than predicted by numerical analysis. 4.7.3. Compar ison for B C 4 type boundary condition In this type of boundary condition (BC4), a constant stress is applied at the radial boundary. However, the bottom boundary is restrained, i.e., the vertical displacement is kept zero during the testing procedure. Only one test, test no. 148, is performed with the above boundary condition at the E N E L - C R I S calibration chamber with Ticino sand. Table A3 in Appendix A provides the test information together with the 64 Relative densities 40.2% to 94.4% Vertical stress 62.8 kPato 716.1 kPa Over-consolidation ratio 1 to 14.41 Kg values 0.390 to 1.356 0 10 20 30 40 Measured Tip Resistance, MPa Fig . 4.14. Comparison between predicted and measured tip resistance for BC3 type boundary condition (second approach). 65 measured value of tip resistance (25.9 MPa) during testing. For this test, the predicted tip resistance is 26.2 M P a . The difference between measured and predicted value of tip resistance is only 1.2%; showing an agreement that is very good. 4.7.4. Comparison for BC2 type boundary condition In this type of boundary condition (BC2), displacement at all the boundaries (radial and bottom) is kept zero during testing. Test no. 150 is the only test in the selected database that is performed with this type of boundary condition. The measured tip resistance is 29.0 M P a (Appendix A, Table A4), and the predicted tip resistance is 37.0 M P a . For this test the numerical analysis over-predicts the measured value by 25%. This trend was also seen in B C 3 type boundary condition tests in which numerical predictions were generally higher than the measured values. This gives more evidence that for these two types of boundary conditions (BC2 and B C 3 ) the chamber walls were not completely restrained. The inevitable compliance introduced during testing has resulted in a decrease in the measured values of tip resistance. To support this point further, a numerical analysis was performed for this test with the distinction that the boundary conditions were chosen to be stress controlled, i.e. a numerical analysis with B C 1 type boundary condition. The numerical analysis resulted in a tip resistance of 24.8 M P a . The fact that the measured tip resistance (29.0 MPa) lies between the two numerical values obtained under two extreme boundary conditions is an indication that for B C 2 and B C 3 type boundary conditions 66 there has been some compliance on the calibration chamber walls during actual testing procedure. 4.8. A review of the experimental results The database chosen for this research work is given in Lunne et al. (1997), and is mainly derived from chamber tests on Ticino sand and Hokksund sand. It is believed that the test results are of high quality, and that they can be used for deriving new correlations or checking new theories or interpretation methods (Lunne et al. 1997). Nevertheless, some uncertainty should be expected for any experimental work. Sand samples may not be prepared uniformly; and the relative density of the sample may be in error by ±5% (Bolton, 1986). The irregular penetration resistance versus depth curves resulting from possible heterogeneity of the sample will increase the degree of uncertainty on what value to choose for the penetration resistance. Salgado (1993) reports six different types of penetration profiles versus depth obtained during tests in the calibration chamber. Based on these different curves, he argues that it is not always easy to interpret with absolute certainty the value of tip resistance. As a consequence of these combined uncertainties, Salgado (1993) argues that no theoretical procedure can be expected to yield results better than ±15%, and that a procedure that predicts most experimental penetration resistance values within ±30% is strongly supported by experimental results. 67 To investigate the degree of uncertainty in the experimental results reported in the database for the E N E L - C R I S calibration chamber, a few of the test results are reviewed in the following: Tests 172 and 173 are tested under B C 3 boundary type condition. Relative density, vertical stress, Ko value, O C R , and constrained modulus for both these tests are very close (Appendix A , Table A2). It is expected that the value of tip resistance measured for both these tests to be also close. Nevertheless, the tip resistance measured for test no. 172 is 19.9 M P a and for test no. 173 is 13.3 M P a . This means that the uncertainty in the measured values can be as much as 20%. Tests 020 and 028 are also examples that are worth mentioning. These tests were performed at the same measured (93.1%) relative density. The effective vertical and horizontal stresses are very similar. The measured constrained modulus for both tests is also very similar. The boundary condition type is B C 3 for both tests (Appendix A , Table A2). Nevertheless, the tip resistance for test no. 020 is 39.1 M P a and for test no. 028 is 36.2 M P a . The difference is about 8%. This indicates that even at essentially identical conditions, an uncertainty in the range of about 5% can easily be encountered during experiments. The experimental results for test no. 169 and test no. 171 seem to have a higher uncertainty. Both tests are type B C 3 boundary condition. The calibration chamber test information for both tests are given in Table A 2 in Appendix A . Data indicates that the sand in test no. 169 is slightly denser than the sand in test no. 171. The effective vertical stress for test no. 169 is 507.2 kPa and for test no. 171 is 114.8 kPa. 68 Also, the Ko value for test no. 169 is 0.504 whereas for test no. 171 is 0.458. This means that test no. 169 is under very much larger stresses than test no. 171. Assuming that the water table is close to the ground surface, test no. 169 is simulating a cone penetration test at a depth of approximately 50 meters whereas test no. 171 is simulating a penetration at a depth of approximately only 11 meters. The constrained modulus for test no. 169 is also slightly higher. Comparing these values, test no. 169 should result in values of tip resistance larger than test no. 171. The database shows that the tip resistance for test no. 169 is 7.3 M P a and for test no. 171 is 14.2 M P a ; result that basically contradicts our general understanding of C P T test interpretation. The same situation is also encountered in test no. 067 and 068. These are B C 1 type boundary condition test (Appendix A , Table A l ) . Test no. 067 is performed under higher vertical and horizontal stresses, and the constrained modulus at this test is also larger. However, the measured tip resistance for test no. 067 is 20.6 M P a and for test no. 068 is 23.6 M P a . This result cannot be interpreted with any existing C P T test. In fact for test no. 067, it was expected that the experimental value o f cone tip resistance would be larger than that for test no. 068. To investigate these anomalies, the same database reported by other investigators was checked. Baldi et al. (1986) presented a large database of calibration chamber tests. Salgado (1993) also presented a comprehensive database for calibration chamber tests of different types of sand at different locations around the world. It is noted that tests 169 and 171 are not listed in their database. Also not listed are tests 165 and 166 (Table A 2 , Appendix A ) . These are among tests in which the measured and predicted 69 values of tip resistance are very much different. In fact, the source of data for these specific tests was not found in the literature. Figure 4-15 is similar to Fig. 4-14 with the distinction that suspicious points in the database are removed. The general agreement between measured and predicted tip resistance improves to some extent. 4.9. Summary of all data Fig. 4-16 contains all the data points for the four different boundary conditions. This figure basically summarizes the general behavior of the proposed model. The total number of data points involved in this figure is 59. Most of the data points are congested around the 45 degree line, indicating that the model can predict the experimental values of calibration chamber tests reasonably well. The error band for the majority of the data points is about ±25%. However, among the data points, there are also few points that have a relatively larger error band. 70 Relative densities 46.2% to 94.4% Vertical stress 62.8 kPa to 716.1 kPa Over-consolidation ratio 1 to 14.41 Kg values 0.390 to 1.356 0 10 20 30 40 Measured Tip Resistance, MPa Fig. 4.15. Comparison between predicted and measured tip resistance for BC3 type boundary condition after removing the suspicious measurements (second approach). 71 Relative densities 46.2% to 94.4% Vertical stress 62.8 kPato 716.1 kPa Over-consolidation ratio 1 to 14.67 Kg values 0.370 to 1.356 0 10 20 30 40 50 Measured Tip Resistance, MPa Fig . 4.16. Comparison of measured and predicted values of tip resistance for all types of boundary conditions. Dashed lines indicate difference between predicted and measured tip resistance (q c) value of 25%. 72 CHAPTER 5: M O D E L APPLICATIONS 5.1. Introduction In the previous chapter, the numerical model was proposed; and the reliability o f the model was investigated by comparing the predicted values of tip resistance with the measured values in the experimental chambers. In this chapter, several applications of the proposed model for C P T testing w i l l be presented. In the analyses to follow, the same constitutive relation that has been adopted in Chapter 4 is used, and the model parameters are the same as well . The relative importance of horizontal and vertical in-situ stress on the magnitude of tip resistance w i l l be discussed first. The effect of different boundaries as well as chamber diameter wi l l be addressed next. The implications of the proposed model in the analysis o f tip resistance in layered soil are presented later in this chapter. Finally, the result of pore pressure analysis in saturated soil is discussed. A l l the numerical analyses were carried out for a standard cone; i.e. a cone tip with an area of 10 cm 2 , and an apex angle of 60 degrees. 73 5.2. Which stress affects the tip resistance: horizontal or vertical? There has been a debate in the literature, (e.g. Baldi et al. 1986, Houlsby and Hitchman 1988, Parkin 1988, Houlsby and Wroth 1989), regarding which stress (horizontal or vertical) is more influential on the values of tip resistance obtained in cone penetration testing. Based on their experimental observations using the cone penetrometer in sand in the Oxford calibration chamber, Houlsby and Hitchman (1988) argue that, for a given density of sand, the cone tip resistance depends on the in-situ effective horizontal stress, and not on the effective vertical stress. To investigate this issue, two different series of numerical analyses were performed. The sand used for both these series has the same relative density, assumed to be 70%. The analyses are carried out for B C 1 type boundary condition, the height of the numerical grid is 1.5 m, and its diameter is 1.2 m. In one series of numerical analyses, the effective vertical stress is constant, and assumed to be equal to 300 kPa. However, the effective horizontal stress varied from 120 kPa to 480 kPa; i.e. K 0 values ranged from 0.4 to 1.6. A total of five analyses were carried out in this series. Figure 5-1 shows the variation of tip resistance versus the variation of effective horizontal stress for this series of numerical analysis. The increase in tip resistance with increase in horizontal stress is noted in the figure. 74 40 DR=70% CT'v = 300 kPa 30 — 20 H 10 1 1 1 1 i 200 300 400 Effective In-Situ Horizontal Stress, kPa 100 500 Fig 5.1. Effect of effective in-situ horizontal stress on tip resistance. The second series also consists of five numerical analyses. For these analyses, the effective horizontal stress is constant, and assumed to be equal to 300 kPa. However, the vertical stress varied from 120 kPa to 480 kPa. The variation o f tip resistance versus the effective vertical stress is shown in F ig 5-2. It is seen that this wide range of increase in the vertical stress has resulted in no appreciable increase in tip resistance. Basically, the magnitude of tip resistance remains virtually unchanged for 40 0. 0) U £ CQ +J (0 "55 ct Q. i— T3 Oi +-> o a. Q. DR=70% a ' h = 300 kPa 30 20 H 10 1 ' 1 1 1 200 300 400 Effective In-Situ Vertical Stress, kPa 100 500 Fig 5.2. Effect of effective in-situ vertical stress on tip resistance. this series of analysis in which the horizontal stress remains constant. This is an indication that the tip resistance is affected by in-situ horizontal stress, and not by in-situ vertical stress. This is supported by experimental observations of Houlsby et al. (1988) and others. The above results also mean that the over-consolidation ratio (OCR) does not directly influence the values of cone tip resistance for sand. This means that, at a given 76 relative density, two sandy soils with different ovei -consolidation ratios may have the same tip resistance i f their horizontal stresses are the same. This is supported by experimental observations of Fioravante et al. (1991) and Salgado et al. (1997a). 5.3. Effects of boundaries The effect of different types of calibration chamber boundary conditions on the experimental results of C P T has long been recognized. In this work, verification has been made to investigate whether the numerical analyses produce results in support of the experimental findings. To this end, several analyses were performed. In these analyses, the diameter of the numerical grid has been increased from 1.2 m to 18 m, 2.4 m, 3.0 m and 4.0 meters. These values correspond to chamber diameter to cone diameter ratio of 33.6, 50.4, 67.2, 84, and 112 respectively. The numerical grid with a diameter of 1.2 m corresponds to the rather largest diameter calibration chambers in use today. For each grid, the four categories of boundary conditions (BC1, B C 2 , B C 3 , and BC4) have been analyzed. Fig . 5-3 shows the variation of tip resistance versus chamber radius for different boundary conditions. As the figure shows, for B C 1 and B C 4 boundary conditions, an increase in chamber radius results in an increase in tip resistance. The rate of increase reduces as the chamber radius increases. 77 1.2 m d i a . c h a m b e r 32 i , 4 m d i a . c h a m b e r 28 24 20 H 16 12 o o o 1 A Boundary Condition + BC1 # B C 2 O B C 3 A BC4 T T 1 r 40 60 80 100 Ratio of Chamber Diameter to Cone Diameter 20 T 120 Fig. 5.3. Effect of chamber size and boundary condition on tip resistance(dense sand). [DR=90%, rj'v = 70 kPa, Ko=0.5] 78 For boundary conditions B C 2 and B C 3 , where the lateral boundary is strain controlled, the tip resistance decreases as the chamber radius increases. A boundary condition with strain controlled condition represents a boundary that is restrained against deformation, i.e., the boundary is acting as a stiff wall. If a boundary simulates a stiff wall, the closer the stiff wall is to the cone tip, the higher the tip resistance should be. This trend is opposite to the numerical results obtained for B C 1 and B C 4 type boundary condition. However, all these trends are physically reasonable. For a B C 1 or B C 4 type boundary condition, the calibration chamber walls are stress controlled, representing a flexible wall at the boundary. If a flexible wall is located closer to the cone tip, the tip resistance becomes smaller. The trend of variation in tip resistance due to the type and location of the boundary conditions observed in Fig. 5-3 is in agreement with the above physical interpretation. In all o f the analyses carried out in Fig. 5-3, the relative density of sand is equal to 90%, and the in-situ vertical stress is assumed to be 70 kPa, and Ko to be equal to 0.5. These values of relative density and in-situ stresses correspond to sand in a dense state. Fig. 5-4 shows the effect of chamber size and different boundary conditions on tip resistance for sand in a loose state. This figure is similar to Fig. 5-3. The only difference between these figures is that the results of tip resistance are obtained for 1.2 m dia. chamber 4 m dia chamber W Q. CD O c ra •*-> w *w <D Q. T 20 40 60 80 100 Ratio of Chamber Diameter to Cone Diameter 120 Fig. 5.4. Effect of chamber size and boundary condition on tip resistance (loose sand). [DR=50%, o 'v = 700 kPa, K 0=0.5] 80 sand in a loose state. The relative density for the sand in Fig. 5-4 is 50%, effective vertical stress is assumed to be 700 kPa, and K 0 to be equal to 0.5. Comparing figures 5-3 and 5-4, it can be concluded that for loose sand, a chamber that has a diameter to cone diameter ratio of 33.6, which corresponds to the rather largest chambers in use today, can simulate the field condition practically well. This means that the difference in penetration resistance obtained in the chamber will not be much different from that in the field. However, for dense sand, the diameter of the chamber should be larger; i.e. a chamber diameter to cone diameter ratio more than 100 is necessary i f the calibration chamber test results are going to be compared directly with field results. It is noted that for the size of the calibration chambers in existence nowadays, the measured tip resistance in dense sand in the calibration chamber should be modified to reflect the magnitude of the tip resistance in the field. 5.4. Effect of vertical (bottom) boundary condition In this section, the profile of tip resistance versus depth of penetration is compared for two different bottom boundary conditions in the chamber. Results of calibration chamber testing shows that as the cone approaches a bottom boundary having a B C 1 type boundary, the tip resistance drops to reflect the flexible bottom boundary. Fig. 5-5 shows the analytical results for the variation of tip resistance versus depth as the cone penetrates downward. The sand modeled is loose with a relative density of 50%, o sz •*-> a. © Q 0 0.0 -I Tip Resistance, MPa 10 20 30 0.3 H c o •*3 0.6 —| ro L _ CD c 0) D_ 0.9 —\ 1.2 H 1.5 1 1 BC1 * 4 \ • A • A • A ABC4 Loose sand DR=50% a ' v = 700 kPa K n = 0.5 Fig. 5.5. Profile of tip resistance with penetration depth for B C 1 and B C 4 boundary condition (loose sand). 82 effective vertical stress of 700 kPa, and the Ko value is 0.5. To investigate exclusively the effect of bottom boundaries, and to eliminate the effects that the lateral boundaries might have, the chamber walls in the model are located at two meters from the axis of symmetry; i.e., a four meter diameter chamber. The height of the chamber is 1.5 m. Figure 5-5 shows that for BC1 type boundary condition, the initial decrease in tip resistance starts at a depth of approximately 1.2 m from the top of the grid. This is to say that the cone penetrating in loose sand "feels" the effect of the bottom stress boundary at a distance of 0.3 m from the bottom boundary, or about 8.5 times cone diameter. To see i f this drop in tip resistance is uniquely due to the presence of an approaching boundary, a numerical analysis with the same sand and the same chamber size but with B C 4 type boundary condition was also performed. The figure also shows the profile of tip resistance versus depth of penetration in a chamber with B C 4 type boundary condition, i.e. with the bottom boundary being restrained against displacement. The tip resistance is seen to increase as the cone approaches the bottom boundary. This result is physically expected. The distance at which the cone "feels" the effect of B C 4 bottom boundary is the same; i.e. about 0.3 m from the bottom boundary. The experimental studies show that the interface distance at which the cone "feels" the effect of the bottom boundary is dependent on relative density of the sand, and its 8 3 stress state. To verify this experimental observation, the same analysis with the same chamber size was carried out, but the sand has a relative density of 90%, and the effective vertical stress is 70 kPa. The K 0 value is unchanged, and it is assumed to be 0.5. The high relative density and the low vertical (and consequently confining stress) represents a sandy soil in a dense state. Figure 5-6 shows the variation of the tip resistance versus depth of penetration for a sandy soil in a dense state. For BC1 type boundary condition, the tip resistance starts to decrease at a distance of about 0.75 m from the bottom boundary. For B C 4 type boundary condition, the tip resistance starts to increase at the same distance. This means that the effect that is seen is solely due to the effect of the bottom boundary. For the sand in a dense state, the interface distance is about 21 times cone diameter. This distance is significantly larger than that obtained for the loose sand. 5.5. Analysis for layered soils The analysis of tip resistance in layered soil has been addressed inadequately in the literature. Vreugdenhil et al. (1994) presented a method for analyzing the cone tip resistance in layered soil. However, their analysis is based on elastic theory. In the sections to follow, the results of numerical analyses showing the effects of soil layering on penetration resistance are discussed. The results of tip resistance for loose 84 Tip Resistance, MPa 10 20 30 40 c o 2 o c Q. <D Q Dense sand DR=90% o ' v = 70 kPa Kn = 0.5 Fig. 5.6. Profile of tip resistance with penetration depth for B C 1 and B C 4 boundary condition (dense sand). 85 sand over dense sand and for dense sand over loose sand as well as the results of tip resistance for a medium with two different soils, i.e., sand and clay are presented. For these series of analyses of tip resistance in layered soil, the height of the numerical grid is 2.3m, and its diameter is 1.2 m. The in-situ effective vertical stress is assumed to be 100 kPa, and Ko is 0.5. The analyses are carried out for B C 1 type boundary condition. 5.5.1. Loose sand over dense sand Figure 5-7 shows the result of the numerical analysis of tip resistance for cone penetration in sand with two different relative densities. The loose sand has a relative density of 50% and the dense sand below has a relative density of 90%. The thickness of the loose top layer is 0.75 m. The figure shows that as the cone tip approaches the dense layer the tip resistance increases. This increase in tip resistance occurs at a distance of 0.25 m above the dense layer. In other words, the tip senses the effect of the approaching layer 0.25 m ahead. This interface distance is approximately 7 times the cone diameter. The result of experimental measurements of Treadwell (1976) for interface distance in a layered system composed of alternating dense and loose layers in a calibration chamber are shown in Fig. 5-8. The experimental measurement of Treadwell shows that the tip senses an interface distance of 3 to 10 cone diameters ahead and behind 86 Tip Resistance, MPa 0 5 10 15 20 25 2.1 Fig. 5.7. Result of drained cone penetration analysis for loose sand overlying dense sand ( a ' v = 100 kPa, K 0 = 0.5). 87 UNIT PENETRATION RESISTANCE, q f (kN/m ) 1000 2000 10 20 x H 0. u a 3 a. 30 40 1 r - i » a. Monterey No. 0 Sand 3. Pluvial Compaction Q . 60 degree apex angle 0> • H/B*8 o • Loose 0» • D r = 9% o> • e = 0.80 •o» • -o • o • • • o • - • 0 • -• o • • o • • o • • 0 • — • o < • o • • o • Dense • • 0 o • • D r = 84% — • o • e = 0.60 • o • • o • • o • • 0 • — • o • -• ° _ Layered * • 0 • • o • • o * - • o • -• • o o j • • 0 Dense Lot 3se • 0 D r =78% D r = 11% — • • o 0 e = 0.62 e = D.80 • o • o Loose = 7% r e = 0.81 i 7.8 a Bu W Q M 8 +- 15.5 i - 2 3 . 3 Fig . 5.8. Experimental measurement of penetration resistance in a layered soil (from Mitchell et al., 1998 based on Treadwell, 1976) 88 the tip. Lunne et al. (1997) note that the distance over which the cone senses an interface increases with material stiffness. In soft materials the diameter of the sphere of influence can be as small as two or three cone diameters, whereas in stiff materials the sphere of influence can be up to 10 to 20 cone diameters. It is seen that the predicted value of interface distance based on the numerical analysis is well within the experimental range. While the cone is inside the dense sand, but close to the loose layer above, its tip resistance is affected by the presence of the layer above. It takes a distance of 0.18 m for the cone tip to be influenced solely by the dense sand. The distance that the cone senses the top layer behind is about 5 cone diameters, which is again in agreement with experimental findings of Treadwell (1976). The figure also shows that as the cone is approaching the bottom stress boundary, the tip resistance starts to decrease. This indicates that the cone in dense sand is beginning to pick up the effects of the bottom boundary, which is located at a depth of 2.3 meters, i.e., it is sensing the bottom boundary at a distance of about 0.5 m. This sensing distance corresponds to about 14 cone diameters. 5.5.2. Dense sand over loose sand The result of the analysis for a dense sand layer over loose sand is shown in Figure 5-9. The same values of in-situ vertical and horizontal stresses are used for this Tip Resistance, MPa 0 5 10 15 20 25 2.1 Fig. 5.9. Result of drained cone penetration analysis for dense sand overlying loose sand (G'V = 100 kPa, K 0 = 0.5). 90 analysis. The thickness of the dense sand is 1.56 m. The larger thickness was chosen to allow the cone in the dense layer to reach its tip resistance without the influence of the soil layer below. Once the cone approaches the bottom loose layer, as the figure shows, its tip resistance gradually decreases. The cone in the dense sand senses the effect of the approaching loose sand at a distance of 0.56 m, which is about 16 times the cone diameter. This value of interface distance is larger than the value obtained for the previous case. This may be because the dense sand, due to its higher stiffness, can project forward its influence over a wider zone [Mitchell and Brandon (1998)]. Thus, although the relative stiffness is the same for both cases, the sensing distance is much greater when a dense sand overlies a looser sand. As the cone penetrates further into the loose sand, the effect of the overlying dense layer decreases; at a distance of 0.17 m the effect of top dense layer vanishes, and the tip resistance is only affected by the layer in which the cone is penetrating. Comparing this figure with Fig. 5-7, it is seen that the cone resistance is not yet affected by the presence of the bottom boundary. For this analysis the location of the bottom boundary is the same as for Fig. 5-7. This further indicates that the interface distance depends on soil stiffness. Figure 5-9 shows that the tip resistance near the start of penetration (top of the grid) is higher, and then it gradually drops. The larger values obtained in the analysis occur as a result of restraining the top boundary in order to simulate the rigid top platen used 91 in most experimental calibration chambers today. This effect is specially seen in penetration analysis of very dense sand. 5.5.3. Sand on clay Figure 5-10 shows the numerical analysis of penetration in a sand layer above a clay. The sand has a relative density of 70%, and the clay has an undrained shear strength of 30 kPa. The shear modulus for clay is assumed to be 9000 kPa, and the bulk modulus to be 300000 kPa. This corresponds to a Poisson ratio close to 0.5. Model parameters for the clay layer are assumed to be constant, and not to be stress dependent. The constitutive law used for clay is a very simplified one since the main interest in this research is on sand. The values of in-situ effective vertical and horizontal stresses are 100 kPa and 50 kPa respectively, which are the same values chosen for the previous analysis. The sand layer covers the top 1.74 m of the grid. While the cone is in sand, its tip resistance is not affected by the clay layer until it reaches a distance of 0.45 m from the clay layer below. This distance is about 12.5 times the cone diameter. However, the tip resistance in the clay layer is only slightly affected by the sand layer above. It needs only a penetration of less than 0.05 m for the cone in clay, or about 1.7 cone diameters, to be solely dependent on the clay layer itself, and not the sand layer above. This demonstrates that the radius of the zone that affects the cone is much less for clay than for sand. 0.0 0.3 — 0.6 —\ c o 2 0.9 —I t> c o D_ "S 1-2 H _c a o Q 1.5-1.8 H 2.1 Tip Resistance, MPa 5 10 15 I I * Sand, DR = 70% y J | Clay, s u = 30 kPa Fig. 5.10. Result of drained cone penetration analysis in sand into undrained clay. ( a ' v = 1 0 0 k P a , K 0 = 0.5) 93 In the analysis, the unchained shear strength of clay is assumed to be 30 kPa, and the in-situ vertical stress is taken to be 100 kPa. Fig. 5-10 shows that the magnitude of tip resistance in the clay layer is about 500 kPa. This gives a cone factor (Nc) o f 13.3 for the clay layer. This value of cone factor for clay is in the acceptable range reported in experimental observations (Campanella, 1995). 5.6. Sensitivity analysis In real field penetration testing, the cone tip resistance is affected by a combination of soil properties; and the effects of the soil properties cannot be separated. Also, in reality, soil properties are inter-related, e.g., sandy soils with higher friction angles may have higher dilatancy properties, or may exhibit higher modulus. However, it is instructive to investigate the separate effect of each soil property on the cone tip resistance to see i f any of the soil properties has a more important contribution in the magnitude of the cone tip resistance. In the following, a series of analyses is carried out in which only one soil property is changed, and the rest are assumed to be constant, and not dependent on the property that is changed. For these series sensitivity analyses, the height of the numerical grid is 1.5m, and its diameter is 4 m. The in-situ effective vertical stress is taken to be 300 kPa, and Ko is 0.5. The analyses are carried out for B C 1 type boundary condition. 94 5.6.1. Effect of soil friction angle on tip resistance Figure 5-11 shows the variation of tip resistance with the variation of soil friction angle. A s the figure shows, the tip resistance increases from about 7 M P a to 9 M P a for an increase in friction angle of 32 degrees to 44 degrees. A l l other parameters are considered constant. For these analyses, dilation is assumed to be zero, and constrained modulus to be 100000 kPa. Assuming a Poisson's ratio of 0.25, this corresponds to a bulk modulus of about 50000 kPa, and a shear modulus of about 30000 kPa. These values are well within the range of soil moduli that can be chosen for a typical sandy soil. Furthermore, the soil properties are assumed to be constant throughout the whole domain, and the properties are not stress dependent. Though in reality, the soil properties change around the cone tip, and dilation is not equal to zero in the whole domain, these values are merely taken to exclusively investigate the effect of only one soil property. It is seen that for a wide range of friction angles between 32 to 44, which is the range that most sandy soils fall into, the change in tip resistance is about ± 1 5 % . It is concluded that the effect of soil friction on the cone tip resistance is not large. It is noted that the small influence of friction angle on tip resistance that is seen in Fig. 5-11 is in reality the effect of constant volume friction angle on tip resistance. This is because in these series of analyses, the effect of dilation has not been 14 12 H 10 —\ B 8 H 6 - ^ Constrained Modulus = 100000 kPa Dilation Angle = 0 deg CT'v = 300 kPa K n = 0.5 4 H 30 32 34 36 38 40 42 Friction Angle, deg 44 46 Fig. 5.11. Effect of variation of friction angle on tip resistance. considered, and the dilation angle in this series of analyses is taken to be a constant value equal to zero. The range of values of constant volume friction angle for a sandy soil is between 32 to 36 degrees, which is smaller than the selected range shown in Fig. 5-11. Therefore, it can be argued that the effect of constant volume friction angle on cone tip resistance is even less than 15% that is obtained for a larger range of values of friction angles between 32 and 44 degrees. Constrained Modulus = 100000 kPa Friction Angle = 40 deg a' v = 300 kPa K 0 = 0.5 -4 -2 0 2 4 6 8 10 12 Dilation Angle, deg Fig. 5.12. Effect of variations in dilation angle on tip resistance. 5.6.2. Effect of dilation on tip resistance Figure 5-12 shows the variation of tip resistance with the soil dilation angle while other soil parameters are held constant. For these series of analyses, friction angle is 40 degrees, and the sand constrained modulus is 100000 kPa. The soil dilation angle is changed from - 2 degrees to 10 degrees. This range basically covers the dilatancy characteristics of most sandy soils. In the figure, a negative dilation angle corresponds to a contractive (loose) sand; and a positive dilation angle corresponds to a dilative (dense) sand. The larger dilation angle is associated with a more dilative sand. As the figure shows, the tip resistance changes from a low value of about 7.5 M P a for a soil dilation of - 2 degrees to a high value of about 13 M P a for a soil dilation of 10 degrees. Taking a mid value between these two lower and upper values as a reference point, it can be concluded that for the range of dilation angles considered, the tip resistance is influenced by as much as 30%. Comparing the two figures of 5-11 and 5-12, it can be seen that the cone tip resistance is more sensitive to dilation characteristics of a sandy soil than its constant volume friction angle. 5.6.3. Effect of modulus on tip resistance Figure 5-13 shows the variation of tip resistance with the variation in soil constrained modulus. The range of constrained modulus is 25000 kPa to 125000 kPa. Assuming Poisson's ratio to be 0.25, this gives a range of bulk modulus of about 12000 to 60000 kPa, and a range of shear modulus of about 8000 to 40000 kPa. These ranges generally cover the stiffness characteristics of most sandy soils. 98 14 12 H 0 25000 50000 75000 100000 125000 150000 Constrained Modulus, kPa Fig. 5.13. Effect of variation of soil modulus on tip resistance. For all these series of analyses, the soil friction angle is assumed to be 40 degrees, and dilation angle to be zero. For this range of modulus, the tip resistance changes from about 3 M P a to about 10 M P a . This shows that the soil modulus has the 99 strongest effect on the tip resistance. Taking a mid value between these two values as a reference point, it is noted that for the range of the modulus considered, the influence of the change in modulus is as much as 75%. Based on the above analyses, it is concluded that the soil modulus has the most influence on cone tip resistance. Dilatancy characteristics of sand also can change the tip resistance to a large extent. However, the sand constant volume friction angle does not play a significant role in the value of resistance at the cone tip. 5.7. Pore pressure analysis In this research work, a preliminary analysis is carried out to investigate the pore pressure response during cone penetration. The coupled stress flow option implemented in F L A C has been used in the analysis. Fig 5-14 shows a typical result for pore pressure response at the middle of cone face ( U l ) and behind the cone tip (U2) for different values of soil permeability. In these analyses, it is assumed that the sand has a relative density of 50%. The in-situ effective vertical stress is taken to be 300 kPa, and K 0 to be 0.5. The equilibrium pore pressure is assumed to be 150 kPa. For this series of numerical analysis, the value of the bulk modulus of water is assumed to be 2 x l 0 5 kPa, and the tension limit for water to be zero kPa. The rate of penetration for these series of analyses is taken to be the same as the standard rate of penetration of the cone, i.e. 20 mm/s. In these series of CO CL <D" l _ _3 CO CO CD i_ Q_ ci> o Q_ CO CO CD O X HI 300 150 H -150 —\ - 3 0 0 —1 1—I I 111l l | 1—I I H I M 1E-8 1E-7 1E-6 Permeability, m/s l — I I I 11 i i | 1—I I 111 I I | 1E-5 1E-4 Fig. 5-14. Numerical prediction of variation of excess pore pressure during penetration on the mid cone face ( U l ) and behind the tip (U2) with soil permeability. analyses, the height of the numerical grid is 1.5 m, and its diameter is 1.2 m. The analyses are carried out for a B C 1 type boundary condition. As the figure shows, for low values of permeability, (in the range of 5x l0" 8 m/s), the pore pressure at the cone face is higher than the equilibrium pore pressure; and the 101 excess pore pressure is positive. These low values of permeability of the soil inhibit the water flow freely around the cone, and the situation is close to the undrained case. The numerical results show that as permeability of the soil increases ( l x l O " 7 m/s), the excess pore pressure generated on the cone face decreases. However, as the figure shows, its magnitude is still positive, but smaller than the case with lower permeability. It is seen that with more increase in permeability, the excess pore pressure on the cone face drops substantially. This means that with higher permeability values, the situation is close to a drained case. This is intuitively expected. The figure also shows that for soil permeability values in the range of 5x10" g m/s to l x l O " 7 m/s, the excess pore pressure generated at the cone shoulder (U2) is negative. With continued penetration, the elements already in contact with the cone tip are eventually pushed away, and the large stresses on these elements are released in a short period of time. This generates a highly negative pore pressure in the elements at the cone shoulder. For a non-plastic soil with the permeability in the range of 5x10" m/s to l x l O " 6 m/s, this highly negative excess pore pressure cannot be drained quickly; therefore, negative excess pore pressures are predicted at U 2 location. As the permeability of the soil increases, the excess pore pressure can be drained in a shorter time, and the excess pore pressure predicted at U 2 location is less negative. As the figure shows, for a permeability value of 5xl0" 5 m/s, the soil drainage characteristics 102 improves, and the excess pore pressure can be drained faster, and therefore, no appreciable amount of pore pressure is detected at the cone shoulder. Fig. 5-14 shows that for the low values of permeability (in the range of 5x l0" 8 m/s to l x l O " 7 m/s), the excess pore pressure at U 2 location is about minus 150 kPa. In the analysis the equilibrium pore pressure is taken to be 150 kPa. Therefore, the magnitude of pore pressure during penetration is close to zero. This means that the magnitude of pore pressure during penetration has reached to the value of tension limit specified for water for this series of analyses. Fig. 5-15 shows the C P T U sounding profiles from Alex Fraser Bridge site. This site is a U . B . C . research site which is located on Annacis Island in N e w Westminster, B . C . Fig. 5-15 shows that the site is covered with sand fill to a depth of 3 meters near the water table depth. Beneath the sand fill, there is approximately 1 meter of clayey silt deposits which are underlain by sand to a depth of 5 meters where there is a one meter silt layer. Beneath 8 meters, the sand is fairly clean to a depth of about 19 meters. From 19 to about 23 meters weaker silt lenses exist. The silty lenses are marked by sharp decreases in penetration pore pressures and increases in friction ratio in combination with very reduced cone bearing. Only pore pressure behind the tip (U2) was measured in this sounding. Details of the stratigraphy of the site are given by Campanella and Kokan (1993). At a depth of 23 meters, the characteristics of the in-situ soil are similar, but not exactly, to what have been assumed in the numerical analysis. The sand in-situ is 103 INTERPRETED Cone Tip Resistance (bar) Friction Ratio (%) U2 (m H20) PROFILE Fig . 5.15. C P T U sounding from Alex Fraser Bridge-research site. 104 estimated to have a relative density of 50% (Campanella, 1995). The effective in-situ vertical stress is about 300 kPa at this depth. For this normally consolidated site, the Ko value is estimated to be between 0.45 to 0.5. The magnitude of equilibrium pore pressure is about 200 kPa, which is slightly more than what is assumed in the analysis. The permeability at this location is estimated to be between l x l O " 7 m/s to l x l O " 6 m/s (Lunne et al. 1997). The field measurement of penetration pore pressure at U 2 location for a depth of 23 meters shows that the excess pore pressure is about minus 200 kPa, and the magnitude of pore pressure during penetration has decreased sharply to nearly zero value. The value of pore pressure predicted in the numerical analysis for a permeability range of l x l O " 7 m/s to l x l O " 6 m/s also shows a sharp decrease in the magnitude of pore pressure. The predicted excess pore pressure is about minus 150 kPa, which is lower than the field measurement. However, the magnitude of pore pressure during penetration predicted in the numerical analysis is close to zero, which is in agreement with the pore pressure measurement at this depth in the field. The numerical prediction of excess pore pressure (Fig. 5-14) shows that, in a soil with permeability in the range of 5xl0" 5 m/s, no excess pore pressure is generated during penetration. It is also seen that in the clean sand layer, located at a depth between 8 to 19 meters, no excess pore pressure is generated during penetration. This is a satisfactory agreement between field measurements and numerical predictions. 105 Fig 5-16 shows the contours of excess pore pressure around the cone tip. For this figure, the soil has a permeability of 5xl0" 7 m/s. It is noted that the excess pore pressure around the cone tip is close to zero, meaning that for this value of permeability, no significant excess pore pressure is detected at U l location. It is also seen that at the cone shoulder the value of excess pore pressure is negative. Fig 5-17 shows the contours of excess pore pressure around the cone tip for a soil with a permeability of 5xl0" 5 m/s. A soil with higher permeability can drain the excess pore pressure in a shorter time. As the figure shows, the excess pore pressure values on the cone face and behind the tip are very close to zero. The physical value of bulk modulus is 2 x l 0 6 kPa for pure water at room temperature. However, the bulk modulus of water used in the analyses, as mentioned previously, is 2 x l 0 5 kPa. The reason for choosing a smaller value for the bulk modulus of water is that the use of very high values of bulk modulus for water is discouraged in F L A C analyses ( F L A C 3.4 Manual, 1998). This is because the solution convergence becomes very slow for high values of water bulk modulus resulting in long execution times for each pore pressure analysis. A series of pore pressure analyses was also carried out by using a value of bulk modulus of water equal to l x l O 6 kPa. In this series of analysis, for a permeability value of 5xl0~ 7 m/s, the prediction resulted in negative values of excess pore pressure on the mid cone face ( U l location). This seems to be unusual. For this series of numerical analyses a smaller time step has been used, and program execution time 106 -0.020 0.020 0.060 0.100 0.140 0.180 Fig. 5.16. Contours of excess pore pressure around the cone tip. (Contour B: -125 kPa, Contour H: 25 kPa, Contour interval: 25 kPa) (DR=50%, soil permeability=5e-7 m/s) 107 Fig. 5.17. Contours of excess pore pressure around the cone tip. (Contour C: -1 kPa, Contour G: 1 kPa, Contour interval: 0.5 kPa) (DR=50%, soil permeability=5e-5 m/s) 108 has increased significantly. Nevertheless, it seems that for this high value of water bulk modulus the solution has not yet converged. This typical analysis of pore pressure shows that the general performance of the presented model is satisfactory, and in agreement with the experimental measurements of pore pressure during penetration. 109 C H A P T E R 6: SUMMARY, DISCUSSION, AND F U T U R E RESEARCH 6.1. Summary of the present work The objective of this research has been the development of a numerical analysis procedure that is capable of doing the following: 1. To predict numerically with an acceptable accuracy the tip resistance that is experimentally measured for different types of boundary conditions in calibration chambers. 2. To determine the effects of chamber size on the penetration resistance so that the penetration resistance in the field can be compared with chamber tests for the same sand with the same relative density. 3. To determine numerically the interface distance in layered soil deposits. 4. To investigate which soil property has a more important contribution in the magnitude of the cone tip resistance. 5. To analyze the pore pressure response during cone penetration in sandy soil. 110 The computer program F L A C has been used to carry out all the analyses involved in this study. Two different approaches for cone penetration modeling were investigated. The basic criterion in the first approach is to investigate the stresses that reach a constant value with continued increase in cone tip displacement. The first approach did not result in an acceptable response, and was not pursued any longer during the course of this research program. In the second approach, control is on displacements. This means that the soil particles are given a specified amount of vertical and horizontal displacement. This is in sharp contrast with the methodology adopted for the first approach. The second approach did result in acceptable agreement with experimental values measured in the chamber. In this second approach, during the execution of the program, the soil nodal points located on the cone path are pushed aside systematically from the top of the grid towards the bottom. This produces a cavity similar to the cone shape that is moving downward in the grid. This modeling procedure simulates the penetration process in the field. The sand parameters used in the model are shear modulus (G), bulk modulus (B), friction angle ((()), constant volume friction angle ((j)Cv), and the stress state of the sand (effective vertical and horizontal stress in the chamber or in the field). The parameter values for the sand moduli and friction angle are based on the relative density of the sand before penetration. The numerical results obtained by the proposed penetration modeling were compared with the cone tip resistance measured in calibration chambers for different types of boundary conditions, notably BC1 and B C 3 . The agreement is, in general, in the range of ±25%. In this research, the effect of calibration chamber size is also investigated. The analyses are carried out for chambers of different sizes under all four boundary conditions. Both loose and dense sand are analyzed. The numerical results clearly show that for loose sand, the calibration chamber size effect is not significant. For dense sand, however, the effect can be important. The numerical analysis to predict the interface distance in layered soils was also pursued in this study. This is a unique contribution of this research program. The analyses are carried out for soil layers of two different relative densities, i.e. loose sand on dense sand, and dense sand on loose sand. Analyses are also carried out for soil layers with two different soil types (sand on clay). It is found that the interface distance in layered soils is about 6 to 20 times cone diameter. The distance depends on the relative density of the sand as well as its state. The sand in a denser state "feels" the effects of the approaching layer at a greater distance from the layer boundary. A sensitivity analysis was carried out to investigate which soil property contributes most to the magnitude of the cone tip resistance. Though in reality, soil properties are inter-related, in each of these series of analyses, only one soil property is changed. This was necessary to investigate numerically the separate effect of each soil property on the magnitude of the cone tip resistance. It is numerically shown that the 112 soil modulus has the most influence on the magnitude of the cone tip resistance. Also, the dilatancy characteristics of sand can change the tip resistance to a large extent. The analysis of pore pressure during penetration was also pursued during this research program. This is an issue that has been addressed insufficiently in the literature. The general response of the proposed model and the agreement between numerical predictions of pore pressure and the field measurements are promising. 6.2. Discussion 6.2.1. M o d e l used The soil model used is a Mohr-Coulomb model with stress dependent parameters. In this model, shear modulus, bulk modulus, friction angle, and dilation angle change with the level of stress. This model is one of the simplest models that can be implemented in geotechnical problems. Nevertheless, it can capture the important characteristics of a sandy soil. In this model, the friction angle decreases as the effective mean normal stress becomes large. The dilation angle is related to the friction angle at failure. A drop in friction angle also results in a drop in the dilation angle. As the mean normal stress increases, and the friction angle drops to the constant volume friction angle, the dilation angle becomes zero. This model behavior is widely supported by experiments. 113 The model has some shortcomings. Sand in the field is generally anisotropic. This means that the parameters in the model should be direction dependent. However, the model does not take into account such directional behavior of sand. Also, experimental observations show that the sand behaves nonlinearly even in the early stages of its elastic response. The nonlinear response in the elastic range for sand has not been accommodated in the model presented for this study. 6.2.2. M o d e l parameters for over-consolidated sand Table A l and A2 in Appendix A contain measured values of constrained modulus for normally consolidated as well as over-consolidated sand. As can be seen in these tables, the measured values of constrained modulus for over-consolidated sand are higher than those for normally consolidated sand. It may be argued that the modulus numbers to be used in the model (KG and KB ) for over-consolidated sand should also be higher. Attempts to use higher modulus numbers to model over-consolidated sand has resulted in over-prediction of tip resistance. It should be noted that for over-consolidated sands, the measured constrained modulus during calibration chamber testing corresponds to the unloading phase of their preparation. However, during cone penetration, the soil under the cone is loaded significantly; and therefore, the model parameters should be based on the measurements of constrained modulus during the loading stage of preparation of chamber tests. 114 In this study, the modulus numbers used for modeling over-consolidated sand are chosen to be the same as those for normally consolidated sand, and they depend only on the relative density of the sand in the chamber before penetration. 6.2.3. Limitations of the large strain analysis In the large strain option implemented in the second approach, it is possible to get a premature stoppage of the program execution due to the "Bad Geometry" message in F L A C . This fact has limited the model applications to some extent. In this study, there are 8 elements associated with the cone diameter. This number of elements seems to be sufficient to result in an acceptable response. However, in the large strain option, it may not be possible to refine the mesh around cone in the second approach and investigate its effect on the numerical values of tip resistance with the present analysis procedure. This is because the program execution is halted prematurely due to the "Bad Geometry" message in F L A C . This limits the capability of the program to investigate the element size effect. 6.2.4. A note on imposed displacement boundary As discussed in Section 4.5, the inner boundary points are displaced horizontally as well as vertically. The magnitude of the horizontal distance is based on producing a cylindrical hole with a radius equal to the cone radius in the grid. The magnitude of the vertical distance is chosen to produce a deformation pattern in the analysis similar to the experimental observations shown in Fig. 4.6, and it is taken to be 15 mm. The magnitude of this vertical displacement has been found to result in a reasonable agreement between numerical predictions of cone tip resistance and the experimental measurements in the selected database. To investigate the effect of vertical displacement boundary on the magnitude of the cone tip resistance, several other values than 15 mm were also selected during this research program. It is found that the cone tip resistance differs by as much as ± 1 0 % for values of vertical inner boundary displacements between 12 mm to 18 mm. This demonstrates that the predicted cone tip resistance is affected by the choice of the vertical displacement. However, its effect in this range is not too significant. It should be noted that the vertical distance of 15 mm chosen for displacing the inner boundary nodes in this study is arbitrary. It is possible that for other databases or other model parameters this magnitude needs to be changed slightly to produce a reasonable prediction of tip resistance for the database that is considered. 6.2.5. Sand type in the field versus sand type in the chamber The sand tested in the calibration chamber is clean, non-cemented, unaged sand. However, in the field, the sand deposit may have some amount of fines content. The sand deposit may be aged and also be cemented to some degree. Aging of sand or a light amount of cementation may increase the tip resistance to an appreciable amount. This increase may be due to an increase in strength or stiffness of the cemented material [Ghionna and Jamiliokowski (1991)]. It would be reasonable to question i f the calibration chamber testing results on clean sand can give enough information to the geotechnical engineer to interpret the sand in-situ which may have some fines content or be cemented. It is believed that the features and attributes of naturally occurring sand such as cementation, aging, fabric, or particle size should be reviewed more extensively in both experimental as well as numerical research in cone penetration testing. 6.3. Future studies 6.3.1. Friction resistance along sleeve In the second approach, it was not possible to find, in a rigorous manner, the friction resistance along the sleeve. This is because no interface elements were introduced in the analysis to simulate the interaction between friction sleeve and the surrounding soil. In the second approach, the cone itself is not directly modeled; i.e. no element in the grid is associated with the cone. Therefore, in the present study, sliding of the cone with respect to the surrounding soil is not modeled, and hence the friction resistance along the sleeve cannot be predicted reasonably. 117 6.3.2. Other sand types The database selected for the verification of the modeling procedure is taken from the published results of tests performed on Ticino sand at E N E L - C R I S calibration chamber. Ticino sand is a sand that has been widely used by Italian researchers. Nevertheless, other researchers around the world have used other types of sand in calibration chambers. For example Hokksund sand is also widely used at Norwegian Geotechnical Institute (NGI). Monterey and Toyoura sand are also used at the calibration chamber setup at the University of California at Berkeley. The reason for choosing a smaller database than that available in the literature is the limitation on the time available for the simulation of each test in F L A C . Since F L A C is an explicit code, the time step should be very small; this means that the execution time in analyzing a problem in F L A C can be rather lengthy. This problem mainly limited this research work to a smaller database. It is necessary to extend the verification of the model by selecting other databases on calibration chamber tests that have been published in the literature. This is to investigate whether the procedure introduced in this study can result in numerical values of tip resistance having the same degree of error band and predictability as other databases available in the literature. 118 6.3.3. Pore pressure analysis The analysis of pore pressure introduced in this study should be regarded as a preliminary analysis to tackle this topic in C P T work. The relative density of sand that was used in these analyses is taken to be 50%. However, it is necessary to investigate the response of the model to other smaller or larger relative densities for sand. 119 BIBLIOGRAPHY Acar, Y . B . and Tumay, M T . (1986). "Strain field around cones in steady penetration." Journal of Geotechnical Engineering, ASCE, 112(2), 207-213. Baldi, G . , Bellotti, R., Ghionna, V . , Jamiolkowski, M . , and Pasqualini, E . (1986). "Interpretations of CPT ' s and C P T U ' s , 2 n d Part: Drained penetration of sands." 4th Int. Conf. On Field Instrumentation and In-situ Measurements, Singapore, 143-156. Baligh, M . M . (1975). "Theory of deep site static cone penetration resistance" Publication No . R75-56, Dept. of Civi l Eng., Massachusetts Institute O f Technology. Baligh, M . M . (1985). "Strain path method." A S C E Journal of Geotechnical Engineering, vol. I l l , No . 9, 1108-1136. Been, K . , Crooks, J .H.A. , and Rothenburg, L . (1988). " A critical appraisal of C P T calibration chamber tests." Proc. 1st Int. Symposium on Penetration Testing (ISOPT), V o l . 2, 651-660. Bellotti, R., Crippa V . , Pedroni, S., Ghionna, V . N . (1988). "Saturation of sand specimen for calibration chamber tests" Proc. 1st Int. Symposium on Penetration Testing (ISOPT), V o l . 2, 661-671. Bellotti, R., B izz i , G . , Ghionna, V . (1982). "Design, construction, and use of a calibration chamber." Proc. 2nd European Symposium on Penetration Testing (ESOPTII), Amsterdam, V o l . 2, 439-446. Berezantzev, V . G . , Khristoforov V S . , and Golubkov, V . N . (1961). "Load bearing capacity and deformation of piled foundations." Proc. 5th Int. Conf. on Soil Mechanics and Foundation Engineering, Paris, V o l . 1, 11-15. van den Berg, P. (1994) "Analysis of soil penetration." PhD thesis, Delft University of Technology, Netherlands. van den Berg, P., de Borst R., and Huetink H . (1996). " A n Eulerean finite element model for penetration in layered soil." Int. Journal for Numerical and Analytical Methods in Geomechanics, 20, 865-886. Bishop, R.F . , H i l l , R., and Mott, N . F . (1945). "The theory of indentation and hardness tests" Proc. Phys. Society, 57, 147-159. Bolton, M . D . (1986). "The strength and dilatancy of sand." Geotechnique, 36(1), 65-78. 120 de Borst, R., and Vermeer, P .A . (1984). "Possibilities and limitations of finite elements for limit analysis." Geotechnique, 34(2), 199-210. Budhu, M . , and Wu, C S . (1992). "Numerical analysis of sampling disturbance in clay soil." Int. Journal for Numerical and Analytical Methods in Geomechanics, 16, 467-492. Byrne, P . M . , Cheung, H . , and Yan, L . (1987). "Soil parameters for deformation analysis of sand masses." Canadian Geotechnical Journal, 24(3), 366-376. Campanella R . G . (1995). "Interpretation of piezocone test data for geotechnical design." Soil Mechanics Series No. 157, Dept. of Civi l Eng, The University of British Columbia. Campanella, R . G . and Kokan, M . J . (1993). " A new approach to measuring dilatancy in saturated sands" Geotechnical Testing Journal, 16(4) 485-495. Carter, J.P., Booker J.R., and Yeung S.K. (1986). "Cavity expansion in cohesive frictional soils." Geotechnique, 36(3), 349-358. Cividini, A . and Gioda, G. (1988). " A simplified analysis of pile penetration." Proc. 6th Int. Conference on Numerical Methods in Geomechanics, 1043-1049. Collins, I F . , Pender, M . J . , and Wang, Y . (1992). "Cavity expansion in sands under drained loading conditions." Int. Journal for Numerical and Analytical Methods in Geomechanics, 16, 3-23. Durgunoglu, H.T. and Mitchell, J.K. (1975). "Static penetration resistance of soils. I: Analysis." Proc. ASCE Specialty Conference on In-Situ Measurement of Soil Properties, A S C E , New York, V o l . 1, 151-171. Elsworth, D . (1993). "Analysis of piezocone dissipation data using dislocation methods." Journal ofGeotechnical Engineering, ASCE, 119(10), 1601-1623. Fioravante, V . , Jamiolkowski, M . , Tanizawa, F., and Tatsuoka, F. (1991). "Results of C P T ' s in Toyoura quartz sand." Proc. 1st Int. Symposium on Calibration Chamber Testing, Potsdam, New York, 135-146. F L A C User Manual, version 3.4, (1998), Itasca Consulting Group Inc., U S A . Ghionna, V . and Jamiolkowski, M . (1991). " A critical apraisal of calibration chamber testing of sands." Proc. 1st Int. Symposium on Calibration Chamber Testing, Potsdam, New York, 13-39. 121 Griffiths, D . V . (1982). "Elasto-plastic analysis of deep foundations in cohesive soil." Int. Journal for Numerical and Analytical Methods in Geomechanics, 6,211-218. Holden, J.C. (1991). "History of the first six C R B calibration chambers." Proc. 1st Int. Symposium on Calibration Chamber Testing, Potsdam, New York, 1-11. Houlsby, G.T. and Hitchman, R. (1988). "Calibration chamber tests of a cone penetrometer in sand." Geotechnique, 38(1), 39-44. Houlsby, G.T. and Wroth, C P . (1989). "The influence of soil stiffness and lateral stress on the results of in-situ soil tests." Proc. 12th Int. Conf on Soil Mechanics and Foundation Engineering, Rio de Janeiro, V o l . 1, 227-232. H u G . C . Y . (1965). "Bearing capacity of foundations with overburden shear." Sols Soils, 13, 11-18. Huang, A . B . and M a , M . Y . (1994). " A n analytical study of cone penetration tests in granular material." Canadian Geotechnical Journal, 31(1), 91-103. Jamiolkowski, M . , Ladd, C C , Germaine, J.T., and Lancellotta, R. .(1985). "New developments in field and laboratory testing of soils" Proc. 11th Int. Conf. on Soil Mechanics and Foundation Engineering, San Francisco. Kiousis, P .D. , Voyiadjis, G.Z. , and Tumay, M T . (1988). " A large strain theory and its application in the analysis of the cone penetration mechanism." Int. Journal for Numerical and Analytical Methods in Geomechanics, 12, 45-60. Ladanyi, B . (1963). "Expansion of a cavity in a saturated clay medium." Journal of Soil Mechanics and Foundation Div., ASCE, 89(SM4), 127-161. Levadoux, J .N. , and Baligh, M . M . (1986). "Consolidation after undrained piezocone penetration. I: Prediction." Journal of Geotechnical Engineering, ASCE, 112(7), 707-726. L o Presti, D .C .F . , Pedroni, S., and Crippa, V . (1992). "Maximum dry density of cohesionless soils by pluviation and by A S T M D 4253-83: A comparative study." Geotechnical Testing Journal, 15(2), 180-189. Lunne, T., and Christophersen, H.P. (1983). "Interpretation of cone penetration data for offshore sands." Proc. 15th Annual Offshore Technology Conf, Houston, Texas, 181-192. Lunne, T., Robertson, P .K . , and Powell J . M . (1997). uCone Penetration Testing in Geotechnical Practice", Blackie Academic & Professional. 122 Mayne, P .W., and Chen, B . S . Y . (1994). "Preliminary calibration of P C P T - O C R model for clays." Proc. 13th Int. Conf. on Soil Mechanics and Foundation Engineering, The Netherlands, 283-286. Mayne, P .W., and Kulhawy F . H . (1991). "Calibration chamber data base and boundary effects correction for C P T data." Proc. 1st Int. Symposium on Calibration Chamber Testing, Potsdam, New York, 257-264. Meyerhoff, G . G . (1951). "The ultimate bearing capacity of foundations." Geotechnique, 2(4), 301-332. Mitchell, J .K. and Brandon T .L . (1998). "Analysis and use of C P T in earthquake and environmental engineering." In Geotechnical Site Characterization, Proc. 1st Int'l. Conf. on Site Characterization-ISC'98, Atlanta, V o l . 1, 69-97. Parkin, A . and Lunne, T. (1982). "Boundary effects in the laboratory calibration of a cone penetrometer in sand." Proc. 2nd European Symposium on Penetration Testing (ESOPTII), Amsterdam, V o l . 2, 761-768. Rad, N . S . and Tummay, M . T . (1986). "Effect of cementation on the cone penetration resistance of sand: A model study." Geotechnical Testing Journal, 9(3) 117-125. Raschke S.A. and Hryciw R . D . (1997). "Vision cone penetrometer for direct subsurface soil observation." Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(11), 1074-1076. Salgado, R. (1993). "Analysis of penetration resistance in sand." PhD thesis, University of California, Berkeley. Salgado, R., Boulanger, R .W. , and Mitchell, J .K., (1997a). "Lateral stress effects on C P T liquefaction resistance correlations." Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(8), 726-73 5. Salgado, R., Mitchell, J .K., and Jamiolkowski, M . (1997b). "Cavity expansion and penetration resistance in sand." Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(4), 344-354. Salgado, R., Mitchell, J .K., and Jamiolkowski, M . (1998). "Calibration chamber size effects on penetration resistance in sand." Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 124(9), 878-888. Schnaid, F. and Houlsby, G.T. (1991). " A n assessment of chamber size effects in the calibration of in-situ tests in sand." Geotechnique, 41(3), 437-445. 123 Schmertmann, J .H. (1991). "The mechanical ageing of soils." Journal of Geotechnical Engineering, ASCE, 117(12), 1288-1330. Shuttle, D . and Jefferies, M . (1998). "Dimensionless and unbiased C P T interpretation in sand." Int. Journal for Numerical and Analytical Methods in Geomechanics, 22, 351-391. Sloan, S.W., and Randolph, M . F . (1982). "Numerical prediction of collapse loads using finite element methods." Int. Journal for Numerical and Analytical Methods in Geomechanics, 6, 47-76. Teh, C.I. and Houlsby, G.T. (1991). " A n analytical study of the cone penetration test in clay." Geotechnique, 41(1), 17-34. Terzaghi, K . (1943). "Theoretical Soil Mechanics.''' John Wiley and Sons, N e w York. ~ Terzaghi, K . , and Peck, R . B . (1967). "Soil Mechanics in Engineering Practice.'" 2 n d edition, John Wiley and Sons, New York. Treadwell, D . D . (1976). "The influence of gravity, prestress, compressibility, and layering on soil resistance to static penetration." PhD thesis, University of California, Berkeley. Vesic, A . S . (1963). "Bearing capacity of deep foundations in sand." In Stresses in soils and layered system, Highway Research Record, 39, 112-153. Vesic, A . S . (1972). "Expansion of cavities in infinite soil mass." Journal of Soil Mechanics and Foundation Div., ASCE, 98(3), 265-290. Verugdenhil, R., Davis R., and Berrill J. (1994). "Interpretation of cone penetration results in multilayered soils." Int. Journal for Numerical and Analytical Methods in Geomechanics, 18, 585-599. Y u , H.S . and Houlsby, G.T. (1991). "Finite cavity expansion in dilatant soil: loading analysis." Geotechnique, 41(2), 173-183. Y u , H.S. and Mitchell J .K. (1998). "Analysis of cone resistance: review of methods." Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 124(2), 140-147. 124 A P P E N D I X A Cal ibra t ion chamber test results In this appendix, the test results that were used in the present research work are summarized in the following tables. The database was taken from Lunne et al. (1996). The database was also compared with Baldi et al. (1986) and Salgado (1993) to see i f any differences exist. Basically, for almost all of the tests, Baldi et al., Lunne et al. and Salgado have reported the same experimental results. A review of experimental data is discussed in section 4.7. The database is the result of tests on Ticino sand at E N E L - C R I S calibration chamber. In the tables to follow, the experimental results from cone penetration testing for all different boundary condition types are tabulated. Most calibration chamber tests were performed under B C 1 (Table A l ) and B C 3 (Table A2) type boundary conditions. Table A3 and A 4 are for B C 2 and B C 4 type boundary condition. The number of tests carried out with B C 2 and B C 4 boundary condition is not as much as the tests with B C 1 and B C 3 boundary condition type. The database is for a standard cone with a diameter of 35.7 mm, which is equivalent to a projected cone area of 10 cm 2 . 125 The columns in the tables are the following: Test no. = The test number that is referred in the experiments. Yd = D R = CT'V = o' h = O C R K„ = M = tjc,msrd fs = Dry density after consolidation, K N / m 3 . Relative density of the sand in the chamber before penetration, and after consolidation, %. Effective vertical stress before penetration, kPa. Effective horizontal stress before penetration, kPa. Over-consolidation ratio. Maximum applied vertical stress divided by final vertical consolidation stress. Coefficient of lateral stress before penetration. Tangent value of constrained modulus. Measured during final load increment for N C sample. For O C sample, it represents the average secant constrained modulus considering the changes of vertical stress and vertical strain during entire unloading curve. Tip resistance measured at the mid-depth in the chamber, M P a . Tip resistance predicted from numerical analysis at the mid-depth of the grid, M P a . sleeve friction measured at the mid-depth in the chamber, kPa. 126 RJ 0. mi i n CM CD -<t CO co co r-~ •<*• CO Tl- CM co 0 0 ) ( M I - M T - O N T T T - T - I M M W N T - C O O T ' < - - - r - T : r C N " > - c O c O c " c O C N C N ' i -& c o QJ CO CL CM CO CD CM i f o N n eo t o co o i O J CO o> I O CO C N C N O ^ C D C O C N i O r ^ O O C N J ^ C O C O C O I O T -O U O CD CO o i n o m N N C O V CO uo CM O C O C O O C N C O C D C O I C O T— T— co E 0. 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Introduction Since all of the analyses in this study are carried out with the computer code F L A C , it would be desirable to verify the reliability of this code with other boundary value problems with known theoretical solutions. In this section, at first, the bearing capacity problem of a shallow foundation on a cohesive soil is analyzed with F L A C , and the analytical results are compared with the theoretical solution of Prandtl. Thereafter, the numerical results of tip resistance based on the second approach is compared with other solutions available in the literature. B.2. Verification of FLAC program The prediction of collapse loads under steady plastic flow conditions can be difficult for a numerical model to simulate accurately (Sloan and Randolph, 1982). The calculation of bearing capacity of a strip footing on a cohesive frictionless soil is an example for to the class of problems involving steady plastic flow. The objective o f this verification exercise is to see whether F L A C program can reliably predict the collapse load accurately. Fig. B . l shows the numerical grid used for F L A C analysis. A semi-infinite soil medium is modeled by a rectangular grid with a width of 4 m and a height of 1.5 m. i9o no»i) _ 1.600 _ 1.200 . 0 800 _ 0.400 _ 0.000 i , - | i | r- 1 1 ! P 1 0200 0.600 1.000 1.400 1.800 no»i) Fig . B . l . F L A C zone geometry for bearing capacity o f a strip footing. Axis of Symmetry I Fig. B.2. Model boundary conditions used in F L A C . Symmetry is invoked in this plane strain analysis, and the half width of the footing is assumed to be 0.5 m. Fig. B.2 shows the boundary conditions used in the analysis. The cohesive frictionless soil is assumed to be weightless and obeys a simple Mohr-Coulomb elasto-plastic model. The elastic properties are assumed to be 9000 kPa for shear modulus, and 300000 kPa for bulk modulus. These elastic properties correspond to a Poisson's ratio close to 0.5 to model the undrained response of the system. The undrained shear strength of the cohesive material is also taken to be 30 kPa. 132 6 5 - f 4 3 2 — 1 — A A — — - Prandt l S o l u t i o n A F L A C Resu l t T T T 0.00 0.05 0.10 Displacement, m 0.15 0.20 Fig. B .3 . Comparison of theoretical solution of Prandtl with F L A C response. The closed form solution to this bearing capacity problem has been shown by Prandtl to be (Terzaghi and Peck, 1967): q f = (2+TC) su where qf is the bearing capacity load at failure. Fig B.3 shows the normalized load displacement curve obtained from numerical analysis of this problem with F L A C . Also included in the figure is the closed form solution of Prandtl. It is seen that the agreement between predicted F L A C result and 133 Prandtl's solution is very good. This example demonstrates that F L A C can reasonably predict the collapse loads under plastic flow conditions. B.3. Verification of the proposed model for analysis of tip resistance In Chapter 4, the proposed model for analysis of cone tip resistance was verified with published experimental measurements from controlled calibration chamber tests. It may be argued that the proposed model should also be compared with other numerical or theoretical solutions available in the literature. Table B . l shows the comparison between the numerical analysis of tip resistance carried out by van den Berg (1994) and the second approach carried out with F L A C program in this study. The comparison is for both clayey and sandy soils. Table B . 1. Comparison of F L A C solution with van den Berg solution. Soil Type Tip resistance, (kPa) Tip resistance, (kPa) van den Berg solution, FLAC solution Clay 266 270 Sand 280 232 134 Model parameters for clay used by van den Berg are: elastic modulus=6000 kPa, Poisson's ratio=0.49, and su=20 kPa. The in-situ vertical stress is 50 kPa, and Ko is assumed to be 1. For comparison, the same values of model parameters were used in F L A C analysis. As is shown in the Table, the agreement between both studies is very good. Table B . l also shows the comparison between predictions of tip resistance obtained by van den Berg and that obtained in this study. Model parameters used by van den Berg for sand are: elastic modulus=1000 kPa, Poisson's ratio=0.2, friction angle=30 deg, dilation angle=0, and su=5 kPa. The in-situ vertical stress is 35 kPa, and Ko is taken to be 1. The same model parameters were also used in F L A C analysis for comparison. As can be seen, for a sandy material the numerical prediction of van den Berg is slightly larger than that obtained with F L A C . However, the values of predictions do not differ very much, and the agreement is more or less satisfactory. There are few independent solutions available for C P T modeling for sand in the literature with which the proposed method in this study could be compared. Therefore, the main goal in this study was to compare the numerical predictions of cone tip resistance with experimental measurements. It should be recognized that the reliability of any C P T modeling procedure depends on its ability to predict the values of tip resistance obtained in controlled calibration chamber tests or field experiments. 135 

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