Deformation Mechanisms beneath Shallow Foundations Brendan McMahon Trinity Hall University of Cambridge A thesis submitted for the degree of Doctor of Philosophy September 2012 The experiments of 280 years have demonstrated amply for every solid substance examined with sufficient care, that the strain resulting from small applied stress is not a linear function thereof. Bell (1973) Declaration I hereby declare that, except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification at this, or any other, university. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This dissertation contains no more than 65,000 words, inclusive of references, footnotes, tables and equations, and has less than 150 figures. Brendan T. McMahon September 2012 i Acknowledgements I would like to sincerely thank my supervisor, Professor Malcolm Bolton, for his supervision and assistance throughout my time in Cambridge. His encouragement and direction, coupled with his enthusiasm, has provided me with great support and help. Many thanks must go to my advisor Dr Gopal Madabhushi and to Dr Stuart Haigh. They have both been very helpful during my time and advice was always given with a friendly face - no matter what the problem. Many discussions have clarified my understanding of soil mechanics and assisted in my analysis. The Schofield Centre Technical Staff merit a big thank you. The help of John Chandler, Kristian Pether, Chris McGinnie and Richard Adams made my centrifuge testing possible. The production of components for my tests was completed supportively and cooperatively and a helping hand for general tasks has always been available. Thank you also to Anama Lowday who has kept the whole Schofield Centre running smoothly. I would like to acknowledge my fellow researchers within the Schofield Centre for their advice, help and most importantly, their friendship. These include Matthew Kuo, Zheming Li, Ruaidhri Farrell, Philip Walbridge, Paul Vardanega, Paul Shepley, Charlie Heron, Alec Marshall and Takaaki Kobayashi. A special mention must go to Sidney Lam and Mark Stringer for all their help with clay and sand models when I first started. I am extremely grateful to the Cambridge Australia Trust for providing funding through a Poynton Cambridge Australia Scholarship and to the Principals of UK Universities for the Overseas Research Students Awards Scheme which made this research possible at Cambridge University. I am also grateful for the additional financial support given in the final stages of this research through the Lundgren Fund and Engineering Department. I am thankful to Trinity Hall for providing me with a home away from home. I always felt very welcome and was given the opportunity to meet new friends through academic, sporting and social activities. Through my time at Trinity Hall I met Katie, something for which I am very grateful. Finally, thank you to my family and friends back home in Australia. Their unstinting encouragement and support has helped me through difficult times and kept me smiling at the end. iii iv Abstract Shallow foundations can provide the most economical solution for supporting small-scale structures. The design approach is quite simple considering the ultimate bearing capacity and working-load settlement. Research has shown that settlement calculations, determined using a linear-elastic approach, usually govern the design but this approach is inappropriate because soil is highly non-linear, even at small strains. The result is that significant discrepancies are observed between predicted and actual settlements. This uncertainty has seen the development of settlement-based approaches such as Mobilisable Strength Design (MSD). MSD uses an assumed undrained mechanism and accounts for soil non-linearity by scaling a triaxial stress-strain curve to make direct predictions of footing load-settlement behaviour. Centrifuge experiments were conducted to investigate the mechanisms governing the settlement of shallow circular foundations on clay and saturated sand models. Clay model tests were performed on soft or firm kaolin beds, depending on its pre-consolidation. Sand model tests were performed on relatively loose Hostun sand saturated with methyl-cellulose to slow consolidation. One-dimensional actuators were developed to apply footing loads through dead-weight or pneumatic loading. A Perspex window in the centrifuge package allowed digital images to be captured of a central cross-section, during and after footing loading. These were used to deduce soil displacements by Particle Image Velocimetry which were consistent with footing settlements measured directly. Deformation mechanisms are presented for undrained penetration, consolidation due to transient flow, as measured by pore pressure transducers, and creep. A technique was developed for discriminating consolidation settlements from the varying rates of short and long-term creep of clay models. Using MSD, a method for predicting the undrained penetration of a spread foundation on clay was proposed, using database results alone, which then provided estimates of creep and consolidation settlements that follow. The importance of the undrained penetration necessitated further investigation by using the observed undrained mechanism as the basis of an ellipsoidal cavity expansion model. An upper-bound energy approach was used to determine the load-settlement behaviour of circular shallow foundations on linear-elastic and non-linear clays, with yield defined using the von Mises’ yield criterion. Linear-elastic soil results were consistent with those obtained from finite element analyses. The non-linear model, as described by a power-law, showed good agreement with both centrifuge experiment results and some real case histories. The single design curve developed through this model for normalised footing pressure and settlement could be used by practising engineers based on existing soil correlations or site investigations. Keywords: circular shallow foundations, centrifuge, clay, sand, load-settlement, cavity expansion, non-linear v Contents Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures xiii List of Tables xix Nomenclature xxi 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Shallow Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Summary of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Literature Review 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Bearing Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Bearing Capacity Design . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Settlement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 One-Dimensional Consolidation . . . . . . . . . . . . . . . . . . 16 2.3.2 Undrained Settlement . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.3 Consolidation Settlement . . . . . . . . . . . . . . . . . . . . . . 27 2.3.4 Rate of Settlement . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.5 Secondary Settlement . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Allowable Settlement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 Precompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 vii CONTENTS 2.6 Mobilisable Strength Design - A New Approach . . . . . . . . . . . . . 46 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Modelling Techniques 55 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Centrifuge Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.1 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Centrifuge Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Model Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 Footings and Loading Method . . . . . . . . . . . . . . . . . . . 60 3.4.2 Particle Image Velocimetry . . . . . . . . . . . . . . . . . . . . . 63 3.4.2.1 PIV Performance . . . . . . . . . . . . . . . . . . . . . 66 3.5 Instrumentation and Data-Acquisition . . . . . . . . . . . . . . . . . . 68 3.5.1 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5.2 Load Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5.3 MEMS Accelerometer . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.4 Pore Pressure Transducer . . . . . . . . . . . . . . . . . . . . . 69 3.5.5 Standpipe Pore Pressure Transducer . . . . . . . . . . . . . . . 69 3.5.6 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.6 Model Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.1 Clay Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.1.1 Clay Properties . . . . . . . . . . . . . . . . . . . . . . 70 3.6.1.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . 72 3.6.2 Sand Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.6.2.1 Sand Properties . . . . . . . . . . . . . . . . . . . . . . 75 3.6.2.2 Sand Pouring . . . . . . . . . . . . . . . . . . . . . . . 76 3.6.2.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . 76 3.6.3 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6.3.1 Methyl-cellulose Preparation . . . . . . . . . . . . . . . 78 3.6.3.2 Saturation Procedure . . . . . . . . . . . . . . . . . . . 79 3.6.4 Package Completion and Loading . . . . . . . . . . . . . . . . . 83 3.7 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.8 Summary of Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.8.1 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 viii CONTENTS 4 Clay Results 89 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 Clay Unloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 Spin-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Self-Weight Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4.1 Soft Clay Models . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4.2 Firm Clay Models . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5 Footing Settlement - Virgin Loading . . . . . . . . . . . . . . . . . . . 95 4.5.1 Firm Clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.1.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.1.2 “Undrained” Penetration . . . . . . . . . . . . . . . . 98 4.5.1.3 Consolidation and Creep . . . . . . . . . . . . . . . . . 104 4.5.2 Soft Clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5.2.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5.2.2 “Undrained” Penetration . . . . . . . . . . . . . . . . 111 4.5.2.3 Consolidation and Creep . . . . . . . . . . . . . . . . . 113 4.5.3 Back-Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5.3.1 Bearing Capacity Test on Soft Clay . . . . . . . . . . . 115 4.5.3.2 Undrained Penetration . . . . . . . . . . . . . . . . . . 118 4.5.3.3 Time Effects . . . . . . . . . . . . . . . . . . . . . . . 121 4.6 Footing Settlement - Further Loading . . . . . . . . . . . . . . . . . . . 124 4.6.1 Firm Clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.6.1.1 “Undrained” Penetration . . . . . . . . . . . . . . . . 125 4.6.1.2 Consolidation and Creep . . . . . . . . . . . . . . . . . 127 4.7 Experiment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.8 Undrained Mechanism Analysis . . . . . . . . . . . . . . . . . . . . . . 128 4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5 Cavity Expansion Model 135 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.2 Cavity Expansion Literature . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.3.1 Elastic Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.3.2 Plastic Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.4 Validation of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ix CONTENTS 5.5 Cavity Expansion Analysis of Circular Shallow Foundations . . . . . . 143 5.5.1 Assumed Deformation Field . . . . . . . . . . . . . . . . . . . . 144 5.5.1.1 Ellipsoidal Model . . . . . . . . . . . . . . . . . . . . . 144 5.5.1.2 Hemispherical Region . . . . . . . . . . . . . . . . . . 147 5.5.2 Strain Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.5.3 Work Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.5.3.1 Elastic Work beyond the Bounding Radius . . . . . . . 151 5.5.3.2 Load-Settlement Behaviour . . . . . . . . . . . . . . . 152 5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.6.1 Effect of Hemispherical Radius . . . . . . . . . . . . . . . . . . 153 5.6.2 Effect of Mesh Size . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.6.3 Load-Settlement Behaviour . . . . . . . . . . . . . . . . . . . . 154 5.6.4 Implications for Design . . . . . . . . . . . . . . . . . . . . . . . 155 5.7 Mechanism Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6 Non-linear soil 161 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2 Non-Linear Investigations . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.3 Non-linear Soil Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.4 Work Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.4.1 Work beyond Bounding Radius . . . . . . . . . . . . . . . . . . 164 6.4.2 Work within Bounding Radius . . . . . . . . . . . . . . . . . . . 165 6.4.2.1 Plastic Work . . . . . . . . . . . . . . . . . . . . . . . 165 6.4.2.2 Non-Linear Work . . . . . . . . . . . . . . . . . . . . . 165 6.4.2.3 Load-Settlement Behaviour . . . . . . . . . . . . . . . 166 6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.5.1 Load-Settlement Behaviour . . . . . . . . . . . . . . . . . . . . 166 6.5.1.1 Effect of b . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.5.2 Design Implication . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.6 Mechanism Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.7 Centrifuge Results Comparison . . . . . . . . . . . . . . . . . . . . . . 171 6.8 Case Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.8.1 Bothkennar Footing Test . . . . . . . . . . . . . . . . . . . . . . 173 6.8.2 Case History Results Comparison . . . . . . . . . . . . . . . . . 176 x CONTENTS 6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7 Sand Results 181 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.2.1 Sand and Model Properties . . . . . . . . . . . . . . . . . . . . 182 7.2.2 Soil Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.2.3 Observed Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 186 7.2.4 Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.3 Back-Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.4 Experiment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8 Conclusions 197 8.1 Circular Shallow Foundations on Clay . . . . . . . . . . . . . . . . . . . 198 8.2 Cavity Expansion Model . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.3 Circular Shallow Foundations on Sand . . . . . . . . . . . . . . . . . . 200 8.4 Implications for Practice . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.5 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.5.1 Footing Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.5.2 Soil Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.5.3 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.5.4 Further Sand Tests and Investigation . . . . . . . . . . . . . . . 203 References 205 xi CONTENTS xii List of Figures 1.1 Different footing types and at two stages of construction (photographs courtesy of OPA (2006); TxDOT (2003)) . . . . . . . . . . . . . . . . . 3 1.2 Settlement definitions (from Burland and Wroth, 1974) . . . . . . . . . 4 1.3 Palace of Fine Arts in Mexico City has undergone significant total set- tlement (Manrique, 2005) . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Leaning Tower of Pisa (Wilmot, 2007) . . . . . . . . . . . . . . . . . . 5 1.5 Cracked walls due to differential settlements (Connor, 2004) . . . . . . 6 2.1 General shear failure mechanism beneath a shallow foundation . . . . . 11 2.2 Plot of stiffness versus strain and the typical strain ranges for a number of structures (Mair, 1993) . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Notation for a uniformly loaded circle . . . . . . . . . . . . . . . . . . . 18 2.4 Vertical stress below a circle on half-space . . . . . . . . . . . . . . . . 19 2.5 Newmark chart for vertical stress with influence value 0.001 (Newmark, 1942) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 Settlement and plot of influence factors (Poulos and Davis, 1974) . . . 22 2.7 Atkinson (2000) method for determining settlement . . . . . . . . . . . 25 2.8 Plots of stress against settlement and settlement at various depths against applied stress from a full-scale test (Lehane, 2003) . . . . . . . . . . . . 26 2.9 Plot of settlement coefficient against pore pressure coefficient . . . . . . 29 2.10 Pore pressure isochrones . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.11 Degree of consolidation versus dimensionless time factor . . . . . . . . . 38 2.12 Degree of settlement/pore pressure dissipation against time for an im- permeable circular footing but otherwise permeable top soil surface and permeable base below the soil layer . . . . . . . . . . . . . . . . . . . . 39 xiii LIST OF FIGURES 2.13 Method to find coefficient of secondary compression and time at end of primary consolidation (modified Augustesen et al., 2004) . . . . . . . . 42 2.14 Prandtl (1921) mechanism adopted for the displacement pattern (from Osman and Bolton, 2005) . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.15 Load-settlement curve obtained from soil stress-strain data (from Osman and Bolton, 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.16 Measured and predicted load-settlement curves for a footing at Kinnegar site (from Osman and Bolton, 2005) . . . . . . . . . . . . . . . . . . . . 52 3.1 Photograph of centrifuge package . . . . . . . . . . . . . . . . . . . . . 59 3.2 Schematic diagram of centrifuge package . . . . . . . . . . . . . . . . . 60 3.3 Dead-weight system used to hold the footing above the surface and to apply the load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Assembled beam showing the loading method components . . . . . . . 63 3.5 Patch from image 1 being searched for in image 2 (White and Take, 2002) 64 3.6 Mylar card and control markers required for the transformation procedure 65 3.7 Computer controlled consolidometer with computer to monitor pore pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.8 Pore pressure transducer location map and zone of influences . . . . . . 73 3.9 Procedure required for inserting each PPT through the base of the box 74 3.10 Package during the process of sand pouring in the sand pouring room . 77 3.11 PPT placement in the sand for a 100 mm diameter footing . . . . . . . 78 3.12 Package ready for saturation . . . . . . . . . . . . . . . . . . . . . . . . 80 3.13 Mass flux rate of successful saturation . . . . . . . . . . . . . . . . . . 82 3.14 Package during saturation and the completely saturated model . . . . . 82 3.15 Centrifuge package ready for testing . . . . . . . . . . . . . . . . . . . . 84 4.1 Suction and effective stress induced during the final stage of unloading 90 4.2 Spin-up and consolidation data . . . . . . . . . . . . . . . . . . . . . . 91 4.3 Effective consolidation during spin-up of soft clay (SC-1) . . . . . . . . 93 4.4 Pore pressure dissipation during consolidation . . . . . . . . . . . . . . 94 4.5 Pore pressure distribution for entire duration of test (model scale) . . . 95 4.6 Raw laser and load cell data for determining the zero-time of the footing 97 4.7 Matching of laser and PIV data to get average footing settlement (test 3A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 xiv LIST OF FIGURES 4.8 Footing settlement measured by PIV and laser data at model scale . . . 99 4.9 Average footing load, settlement and excess pore pressure for the first 100 s at model scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.10 Raw and interpolated “undrained” mechanisms at time t1 = 10.5 s (30 hours) for 100 mm diameter footing on stiff clay (test 3A) . . . . . . . . 102 4.11 Volumetric strain (%) at t = t1 (test 3A) . . . . . . . . . . . . . . . . . 103 4.12 Engineering shear strain (%) at t = t1 (test 3A) . . . . . . . . . . . . . 103 4.13 Mechanism at time t2 (30.5 s at model scale) for 100 mm diameter footing on stiff clay (test 3A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.14 Volumetric strain (%) at t = t2 (test 3A) . . . . . . . . . . . . . . . . . 105 4.15 Comparison of mechanism between “undrained” penetration (t1) and during the consolidation and creep phase (t2) for test 3A . . . . . . . . 106 4.16 Footing settlement and excess pore pressure for test 3A (model scale) . 106 4.17 Development of the deformation mechanism throughout the 2 hour test of footing 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.18 Consolidation and creep mechanism for test 3A between 140 and 600 seconds at model scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.19 Settlement plot at various depths within the soil for test 3A at model scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.20 Test 3A normalised and cumulative settlement with depth at model scale109 4.21 Footing settlement and excess pore pressure measured along the footing edge at model scale for test 1A . . . . . . . . . . . . . . . . . . . . . . 110 4.22 Average footing load, settlement and excess pore pressure for the first 200 s of test 1A (model scale) . . . . . . . . . . . . . . . . . . . . . . . 111 4.23 Mechanism at time t1 (3.5 seconds at model scale) for test 1A . . . . . 112 4.24 Comparison of mechanism between “undrained” penetration (t1) and during the consolidation and creep phase (t2) for test 1A . . . . . . . . 112 4.25 Consolidation and creep mechanism between t2 and 600 s (test 1A) . . 113 4.26 Contour plot demonstrating some slight heave for test 1A . . . . . . . . 114 4.27 Bearing capacity test information . . . . . . . . . . . . . . . . . . . . . 117 4.28 Profile of in-situ and pre-consolidation stress, overconsolidation ratio, undrained shear strength and mobilisation strain with depth for test 3A 119 4.29 Stress-strain curve at characteristic depth of 0.3D using the kaolin database of Vardanega et al. (2012) . . . . . . . . . . . . . . . . . . . . . . . . . 119 xv LIST OF FIGURES 4.30 Predicted undrained penetration plotted against the undrained settle- ment from the centrifuge experiments . . . . . . . . . . . . . . . . . . . 121 4.31 Creep model showing creep and consolidation settlements for test 3A . 122 4.32 Additional pressure of 70 kPa applied for 2 hours at model scale (test 3A-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.33 “Undrained” penetration mechanism at time t1 for test 3A-1 . . . . . . 126 4.34 “Undrained” settlement plotted against predicted values for the addi- tional pressures applied via compressed air . . . . . . . . . . . . . . . . 126 4.35 Clay particles between the footing and Perspex for the soft clay test in comparison to typical results . . . . . . . . . . . . . . . . . . . . . . . . 129 4.36 Comparison of centrifuge and Osman and Bolton (2005) mechanisms for settlement at centre of footing of 2 mm . . . . . . . . . . . . . . . . . . 130 4.37 Ellipses with relatively normal interpolated displacements . . . . . . . . 131 4.38 Clay mechanism on photograph for test 3C (100 mm footing on stiff clay)133 5.1 Method of characteristics from ABC for the Prandtl mechanism (Martin, 2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2 Energy method used to determine the load-settlement behaviour . . . . 144 5.3 Evolution of ellipsoids to hemispheres for rh = D . . . . . . . . . . . . 146 5.4 Ellipsoidal mechanism for a footing with diameter D and settlement w 149 5.5 Comparison of the cavity expansion mechanism and that observed in the centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.6 Vector difference between mechanisms of Figure 5.5 . . . . . . . . . . . 150 5.7 Effect of increasing the extent of the ellipsoidal mechanism . . . . . . . 154 5.8 Load-settlement behaviour for soil with G/cu = 100 . . . . . . . . . . . 155 5.9 Load settlement behaviour for soil with G/cu = 167 . . . . . . . . . . . 155 5.10 Footing stress plotted against (w/D ×G/cu) with expression for linear range and design example point . . . . . . . . . . . . . . . . . . . . . . 156 5.11 Ellipsoidal cavity expansion and Osman and Bolton (2005) mechanisms 158 5.12 Load-settlement behaviour for Osman and Bolton (2005) mechanism with new work calculation procedure . . . . . . . . . . . . . . . . . . . 158 5.13 Footing stress plotted against (w/D × G/cu) for the ellipsoidal cavity expansion and Osman and Bolton (2005) mechanisms . . . . . . . . . . 159 xvi LIST OF FIGURES 6.1 Constitutive soil models used and the comparison between MSD, EMSD and finite difference methods from Klar and Osman (2008) . . . . . . . 162 6.2 Example stress-strain curve demonstrating the extra strain energy for non-linear analysis of soil . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.3 Load-settlement behaviour using the ellipsoidal mechanism with soils of varying overconsolidation ratio for b calculated using Equation 6.2 . . . 167 6.4 Load-settlement behaviour using the ellipsoidal mechanism for soils with varying overconsolidation ratio for b = 0.6 . . . . . . . . . . . . . . . . 167 6.5 Load-settlement behaviour for normally consolidated soil and overcon- solidated soils (OCR = 20 and OCR = 40) showing the effect of using the calculated value of b and b = 0.6 on the ellipsoidal mechanism . . . 168 6.6 Footing stress plotted against (w/D × 1/γM=2) with linear fitting . . . 169 6.7 Difference between using calculated b and b = 0.6 on Osman and Bolton (2005) mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.8 Load-settlement relationship for varying OCR on Osman and Bolton (2005) mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.9 Single design line for non-linear soils with both mechanisms . . . . . . 171 6.10 Design line with centrifuge experiment results . . . . . . . . . . . . . . 172 6.11 Summary of soil profile to 7 m depth at Bothkennar (from Jardine et al., 1995) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.12 Undrained strength properties at Bothkennar (from Jardine et al., 1995) 175 6.13 Overall load-displacement behaviour (from Jardine et al., 1995) . . . . 176 6.14 Non-linear cavity expansion model with observations from a footing test performed in Bothkennar (Jardine et al., 1995) . . . . . . . . . . . . . . 177 6.15 Cavity expansion model with non-linear soil and some case histories . . 179 7.1 Uneven load distribution due to uneven surface in test S-1 . . . . . . . 183 7.2 Discrepancy in PIV and laser results due to sand surface in test S-2 . . 183 7.3 Observed bearing pressure in test S-2 in model scale time . . . . . . . . 185 7.4 Immediate excess pore pressure dissipation and small movements (model scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.5 Raw mechanism observed for the 5 m prototype scale footing . . . . . . 186 7.6 Interpolated mechanism of undrained penetration and consolidation . . 187 7.7 Engineering shear strain (%) for the 5 m prototype scale footing . . . . 188 7.8 Volumetric strain (%) beneath the 5 m prototype scale footing . . . . . 188 xvii LIST OF FIGURES 7.9 Volumetric strain (%) along the centreline of the footing (compression positive) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.10 Stress determination at the representative depth before and after loading to find σ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.11 Total displacement mechanism (shear plus consolidation) of a footing on saturated sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 xviii List of Tables 2.1 Notation as used in Eurocode 7 . . . . . . . . . . . . . . . . . . . . . . 12 3.1 Scaling laws for centrifuge modelling . . . . . . . . . . . . . . . . . . . 57 3.2 Angular velocity to achieve testing level of 100− g at the soil surface . 58 3.3 Footing load and application method . . . . . . . . . . . . . . . . . . . 62 3.4 Camera, Field of View (FOV) data and theoretical precision . . . . . . 67 3.5 Typical instrument calibration factors . . . . . . . . . . . . . . . . . . . 68 3.6 Polwhite E clay properties (IMERYS, 2008) . . . . . . . . . . . . . . . 71 3.7 Properties of Fraction E sand (Haigh and Madabhushi, 2002) . . . . . . 71 3.8 Properties of Hostun sand (from Chian et al., 2010; Mitrani, 2006) . . . 75 3.9 Photograph capturing times used in centrifuge experiments . . . . . . . 85 3.10 Summary of centrifuge tests . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1 Summary of virgin loaded centrifuge tests on clay showing footing di- ameter, average load and relevant timings . . . . . . . . . . . . . . . . 96 4.2 Nominal soil properties at z = 0.3D and the predicted undrained settle- ment using MSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.3 Footing settlements and implied Poisson’s ratio for the given creep and consolidation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.4 Footing settlements and implied Poisson’s ratio for the given creep and consolidation model for further loading of foundations . . . . . . . . . . 127 6.1 Case histories of some shallow foundation tests and calculated mobilisa- tion strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.1 Principal stresses and equivalent triaxial stress before loading and in the active and passive regions after loading . . . . . . . . . . . . . . . . . . 192 xix LIST OF TABLES 7.2 Back-analysis of two verified footing tests with observed and predicted settlements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 xx Nomenclature Roman Symbols A area b empirical exponent C methyl-cellulose concentration (%) Cα secondary compression index cu undrained shear strength cv one-dimensional coefficient of consolidation D foundation diameter D10, D50, D90 sand grain diameter 10%, 50% and 90% finer respectively E Young’s modulus e void ratio G shear modulus g acceleration due to gravity Gs specific gravity h height I1, I2 influence factors Iρ influence factor xxi NOMENCLATURE Ip plasticity index K earth pressure coefficient k soil permeability L length Mc compatibility factor mv coefficient of volume change N number of gravities Nc bearing capacity factor OCR overconsolidation ratio ps cavity pressure Q volumetric flow rate q foundation pressure qu undrained strength in triaxial compression r radius rc cavity radius sc shape factor t time U rate of settlement parameter u pore pressure Uc uniformity coefficient v20 kinematic viscosity at 20◦C W work w footing settlement xxii NOMENCLATURE x, y Cartesian coordinate system Y uniaxial yield stress z depth Greek Symbols α creep slope ∆ differential settlement γ˙ strain rate ε˙1, ε˙2, ε˙3 major, intermediate and minor strain rates γ shear strain γ′ buoyant unit weight γ∗ equivalent linear shear strain γe, γr characteristic strains γM=2 mobilisation strain ˆ˙ε strain rate invariant σˆ1, σˆ2, σˆ3 major, intermediate and minor deviatoric stresses respectively Λ empirical exponent µ settlement coefficient ν Poisson’s ratio ω angular velocity φ friction angle ρ cavity radial displacement σ stress σ′0 initial mean effective stress xxiii NOMENCLATURE σ1, σ2, σ3 major, intermediate and minor principal stresses respectively σc limiting cavity stress τ shear stress ε1, ε2, ε3 major, intermediate and minor principal strains respectively εr, εθ, εz radial, circumferential and vertical strain respectively εv volumetric strain Subscripts 0 initial c consolidation crit critical d drained e elastic f footing max maximum min minimum mob mobilised n− l non-linear nc normally consolidated oed oedometer p plastic s secondary u undrained ult ultimate w water xxiv Chapter 1 Introduction 1.1 Background While the impact of the global economic downturn on localised construction might have varied between individual countries, overall the industry has weathered the crisis quite well and has recently resumed a healthy state. Architects continue to push the design envelope to create more interesting and innovative structures. As a result, engineers are faced with greater design challenges and the construction sector must adapt to meet new requirements. All structures in contact with the ground require a foundation for support. There are two types of foundations in Geotechnical Engineering - spread and piled. Due to the need to capitalise on the increase in strength and stiffness of soil with depth, piled foundations are generally used for larger structures, such as a high-rise building. However, the majority of structures world-wide are small-scale - for which spread foun- dations are frequently utilised. Where sufficient capacity is available near the ground surface, projects such as low-rise buildings (up to about five storeys in height), houses, tanks and even wind turbines might adopt a shallow foundation system. 1.2 Shallow Foundations Shallow foundations are often far more cost effective than other possible solutions. The relatively quick design and construction process makes it a very popular design choice. Thus, the most common design undertaken by a Geotechnical Engineer in their 1 1. INTRODUCTION working life will be that of a shallow foundation (French, 1999). Shallow foundations spread the load that needs to be resisted over a larger area and are hence also referred to as spread foundations. Shallow foundations can be rectangular, a strip or circular - dependent on the type of load being supported. Figure 1.1(a) shows that rectangular (or square) footings will often be used for columns, strip footings will support load-bearing walls and circular footings can be used to support the load from a tank. Shallow footings are predominantly constructed with reinforced concrete. Concrete is used primarily because of its strength in compression; but also for its resistance to harsh environments, such as those found below ground. Steel bars, or mesh, are frequently used as reinforcement as they are most economical. Shallow footings also provide the most simple construction process. The required excavation takes place and appropriate form work is installed to outline the footing shape. Often a blinding layer of concrete is laid at the base to provide a clean, dry and level working surface (Tomlinson, 1970). The reinforcement is laid and tied as specified ensuring there is sufficient cover to prevent corrosion. The concrete is then poured and the appropriate curing and drying procedure is followed. Figures 1.1(b) and 1.1(c) show a wind turbine shallow foundation during the concrete pour stage, and a completed rectangular footing with column reinforcement respectively. Two issues are considered in the design of shallow foundations - ultimate bearing capacity and settlement. The recognised method for determining the ultimate bearing capacity of a shallow foundation is based on plasticity theory. A typical approach would determine the bearing capacity of the footing and divide it by some factor of safety (typically 3) to produce an allowable working load. However, the large safety factor is not usually justified on the basis of uncertainty in applied loads or soil strength, but rather the necessity to control settlement. The settlement at the allowable work- ing load (and obviously the ultimate capacity) can still be significant, resulting in a serviceability failure of the superstructure. Therefore, it is necessary to estimate the settlement of a shallow foundation under an applied load. In general practice it is usual to calculate settlement using elasticity theory together with soil stiffness measured in a one-dimensional oedometer test. Soil, however, is highly non-linear even at small strains. Structures can experience settlement, tilt and distortion. Figure 1.2 shows defini- 2 1. INTRODUCTION (a) Typical footing loads and shapes (b) Shallow foundation for wind turbine (c) Completed rectangular footing Fig. 1.1 Different footing types and at two stages of construction (photographs courtesy of OPA (2006); TxDOT (2003)) tions for these deformations, as proposed by Burland and Wroth (1974). Figure 1.2(a) shows that a structure can experience two types of settlement - uniform and differ- ential. Significant uniform settlement can be withstood by most structures because no additional stresses are induced in members. The Palacio de Bellas Artes (Palace of Fine Arts), shown in Figure 1.3, is founded on the well known soft clay of Mexico City and since its construction has undergone a uniform settlement of over four metres (Nadgouda, 2006). Although the Palace has settled significantly the building is still structurally adequate and remains open to the public, albeit with the original ground floor entrance now utilising a downward set of stairs. Differential settlement results from an uneven settlement across the horizontal plane of a foundation and structure. Differential settlement causes tilt and angular distortion of a structure, as shown 3 1. INTRODUCTION (a) Settlement w, relative settlement δw and rotation θ (b) Tilt ω and relative rotation (angular distortion) β (c) Relative deflection ∆ over length L Fig. 1.2 Settlement definitions (from Burland and Wroth, 1974) in Figure 1.2(b). A famous example of tilt is the Leaning Tower of Pisa - shown in Figure 1.4. The distortion that results from tilt can cause more serious problems with the performance of a structure. At small levels of differential settlement and distortion damage can be purely aesthetic, such as the cracking of glass facades, or cause functional issues such as doors and windows jamming. Services pipes and their fittings also need to be relatively flexible to allow for some differential movement. At more significant levels of distortion, cracking can occur in load bearing elements - resulting in structural integrity issues and possible collapse - as shown in Figure 1.5. Allowable settlement limits were found to be most adequately described by the relative deflection ratio, ∆/L, as shown in Figure 1.2(c). A structure on isolated spread footings can experience the settlements, tilt and 4 1. INTRODUCTION Fig. 1.3 Palace of Fine Arts in Mexico City has undergone significant total settlement (Manrique, 2005) Fig. 1.4 Leaning Tower of Pisa (Wilmot, 2007) distortion that were shown in Figure 1.2. Typically, codes of practice will provide al- lowable values of ∆/L and wmax to limit the settlement of a structure. Considering the typical spacing of columns, the resulting values of these limits are similar. Unexpected settlements can result from deficient calculation procedures in the design phase or from inadequate site investigation. An inappropriate consideration of the soil-structure in- teraction and estimated building stiffness can result in the actual applied footing stress differing from its design value. The properties of the soil, such as non-linear behaviour, variation in the ground conditions and time-effects can also cause discrepancies in ex- 5 1. INTRODUCTION Fig. 1.5 Cracked walls due to differential settlements (Connor, 2004) pected and actual settlements. A more accurate prediction method for settlement of individual footings can minimise these differential settlement concerns. Research must first be carried out, therefore, to investigate the monolithic settlement and deformation mechanisms of spread foundations. 1.3 Summary of Research Research into the bearing capacity and settlement of shallow foundations has been performed for nearly a century. Time and the development of technology has provided a greater understanding of soil behaviour, but the abundance of research that continues today - particularly in the area of shallow foundations - is indicative of the uncertainty that still persists. In the design of shallow foundations De Beer (1965) and Meyerhof (1965) respec- tively concluded that the settlement limit is the critical parameter on sand and clay soils. Osman and Bolton (2005) also states that “excessive total or differential settle- ments are a main cause of unsatisfactory building performance”. This research was performed to investigate the settlement of shallow foundations with the following ob- jectives: 6 1. INTRODUCTION 1. Perform centrifuge experiments to observe the deformation mechanisms beneath circular shallow foundations on saturated sand and clay soil models. Centrifuge experiments are advantageous because their validity does not depend on an a priori understanding of the constitutive behaviour of the soil. Improvements in camera technology and the development of Particle Image Velocimetry (PIV) by White et al. (2003) allow actual deformations within soil models to be observed and analysed. 2. Utilise the observed deformation mechanism to extend the settlement based ap- proach of Mobilisable Strength Design (MSD) of Osman and Bolton (2005) in the design of shallow foundations on clay. 3. Extend the method, which is currently designed to predict undrained settlements on clay, to long-term settlements allowing for consolidation and creep. Similarly, consider the prediction of settlements on sand. 4. Apply the results of databases of the deformability of sands and clays to estimate settlements in the tests, compare to existing experimental results and to make recommendations accordingly. 1.4 Dissertation Outline An introduction to shallow foundations has been presented and the importance of settlement considerations in shallow foundation design has been demonstrated. This dissertation contains 8 chapters with the following layout: Chapter 2 provides a review of literature on shallow foundation design. The theory developed for bearing capacity design and the design method according to Eurocode 7 is presented. The components of total settlement are introduced and some different design methodologies for each, including recommendations from Eurocode 7, are given. Finally, the relatively new design approach of MSD is introduced and the need for a real deformation mechanism from centrifuge testing is discussed. Chapter 3 outlines the modelling techniques used for this research. The benefits of centrifuge testing and associated scaling laws are introduced. The centrifuge package, 1-D actuators developed to apply the footing loads and the instrumentation used are discussed. The image processing technique of PIV was used to produce deformation mechanisms, and so particular attention is placed on explaining the process and its use 7 1. INTRODUCTION in the analysis of results. Outlines of the preparation methods for model soil bodies of both clay and saturated sand are also given before the centrifuge test procedure and a summary of tests are provided. Chapter 4 presents footing test results on clay soil bodies. Deformation mechanisms, footing pressure and settlement, and subsequent excess pore pressure results are given for both soft and stiff clays. An existing database of results from 115 triaxial tests is used to validate the results of a bearing capacity test that was performed on the soft clay model. Following this, estimated values of settlement are calculated and compared to experimental results. A summary of this chapter is presented in McMahon and Bolton (2012). Chapter 5 uses the observed undrained mechanisms to provide the basis for the ellipsoidal cavity expansion mechanism utilised in an energy approach to find the load- settlement behaviour of circular foundations on linear-elastic perfectly-plastic soil. Re- sults are shown to compare well with those published from finite element analyses. Parts of this chapter are presented in McMahon et al. (2013) Chapter 6 extends the approach of Chapter 5 by considering the non-linear be- haviour of soil. The power-law stress-strain relationship developed from a database of triaxial tests is used to describe this non-linearity. The load-settlement relationship developed is compared to the experimental results and some real case histories. Chapter 7 presents a set of experimental results from the footing tests performed on saturated sand. A database of sand tests is used to compare theoretical values of settlement with the centrifuge results. Some of these results are published in McMahon and Bolton (2011). Chapter 8 summarises the most significant results obtained in this research and suggests some possible areas for future work. 8 Chapter 2 Literature Review 2.1 Introduction The concept of a shallow foundation is simple - to provide resistance by spreading the required supporting load over a suitable area. The design and construction is relatively simple, and the materials used are common and relatively inexpensive, making shallow foundations an efficient foundation system for most small-scale structures. Shallow foundations may be defined as having a depth to breadth ratio of less than 1 (Atkinson, 1993) and can be rectangular, strip or circular. Extensive research over nearly a century has been performed into shallow founda- tions but with no significant variation between the results presented. Safety factors are used in practice without much knowledge of their origin and purpose. The uncer- tainty associated with geotechnical engineering has ensured that much research is still performed today. A geotechnical site investigation is usually one of the first tasks to be performed on a construction project. The load required to be supported is determined nearer the end of the design process; given structural designs are performed top-down. However, foundations are constructed first, which signifies the need for an efficient and more accurate design process. This requires a better understanding of what actually happens upon loading a soil with a foundation - the aim of this research. This chapter introduces the two design criterion of ultimate bearing capacity and settlement for shallow foundations. The ultimate bearing capacity for undrained and 9 2. LITERATURE REVIEW drained conditions is discussed along with the particular design method of Eurocode 7. Generally the settlement governs the design, and therefore greater emphasis on settlement literature is provided. A brief overview of allowable settlement and the use of precompression for minimising settlements is then presented. Finally the new design approach of Mobilisable Strength Design (MSD) is introduced and the need for validation through physical modelling is discussed. 2.2 Bearing Capacity There are two approaches which can be adopted in determining the ultimate bearing capacity of foundations. The classical approach uses a rigid-plastic bearing mechanism of indentation, shear and heaving. The second approach uses a cavity expansion ide- alisation beneath the foundation. The current design procedure for determining the ultimate bearing capacity of shallow foundations on clays and sands, based on the classical approach, is now described. 2.2.1 Bearing Capacity Design Prandtl (1921) used the theory of plasticity to analyse the penetration of hard metal bodies in to softer materials. Reissner (1924) continued this research by investigating weightless materials with internal friction. Terzaghi (1943) extended the research to apply to the bearing capacity of shallow foundations. A bearing capacity failure mech- anism for a general shear failure is shown in Figure 2.1. To this day, Terzaghi’s theory remains the basis for design, but some refinements have been applied by a number of other researchers through the years - examples include Skempton (1951), Meyerhof (1963), Hansen (1970) and Vesic (1973). The procedure given in Section 6 of Eurocode 7 outlines the requirements for the design of spread foundations. It is required that a commonly recognised procedure to determine the bearing resistance be adopted, and Annex D of Eurocode 7 provides one such example. This design method is based on the work and research of a number of authors. The approach given in Eurocode 7 would be used extensively by practising engineers in Europe, and possibly even other nations. Adopting a different approach would generally result in a similar value being obtained, with slight differences resulting 10 2. LITERATURE REVIEW Fig. 2.1 General shear failure mechanism beneath a shallow foundation from the factors within each method. The method of Annex D in Eurocode 7 is now given as a demonstration. It is said that the short-term, undrained, and long-term, drained, capacities must be determined. For clarity, the procedure of determining the undrained and drained bearing capacities is given using the notation used within the code - as shown in Table 2.1. Undrained Conditions The design bearing resistance, R, on a soil with undrained shear strength cu may be calculated from: R/A′ = (pi + 2)cubcscic + q (2.1) With the dimensionless factors for: ˆ the inclination of the foundation base: bc = 1− 2α/(pi + 2) ˆ the shape of the foundation: sc = 1 + 0.2(B′/L′) for a rectangular shape and sc = 1.2 for a square or circular shape ˆ the inclination of the load, caused by a horizontal loadH: ic = 12 ( 1 + √ 1− HA′cu ) with H ′ ≤ Acu 11 2. LITERATURE REVIEW Table 2.1 Notation as used in Eurocode 7 A′ = B′ × L′ Design effective foundation area b Design values of the factors for the inclination of the base, with subscripts cohesion c, surcharge q and weight density γ B Foundation width B′ Effective foundation width D Embedment depth e Eccentricity of the resultant action, with subscripts B and L H Horizontal load i Design values of the factors for the inclination of the base, with subscripts c, q and γ L Foundation width L′ Effective foundation width m Exponent in formulae for the inclination factor i N Bearing capacity factors, with subscripts c, q and γ q Overburden or surcharge pressure at the level of the foundation base q′ Design effective overburden pressure at the level of the foundation base R Resistance of soil s Shape factors of the foundation base, with subscripts c, q and γ V Vertical load α Inclination of the foundation base to the horizontal γ′ Design effective weight density of the soil below the foundation level θ Direction angle of H ϕ′ Angle of shearing resistance in terms of effective stress 12 2. LITERATURE REVIEW Drained Conditions The design bearing resistance on a soil with apparent cohesion c′ may be calculated from: R/A′ = c′Ncbcscic + q ′Nqbqsqiq + 0.5γ ′B′Nγbγsγiγ (2.2) With the design values of dimensionless factors for: ˆ the bearing resistance: Nq = epi tanϕ ′ tan2(45 + ϕ′/2); Nc = (Nq − 1) cotϕ′; Nγ = 2(Nq − 1) tanϕ′ where δ ≥ ϕ′/2 (rough base); ˆ the inclination of the foundation base: bc = bq − (1− bq)/(Nc tanϕ′) bq = bγ = (1− α tanϕ′)2 ˆ the shape of foundation: sq = 1 + (B′/L′) sinϕ′ for a rectangular shape; sq = 1 + sinϕ′ for a square or circular shape; sγ = 1− 0.3(B′/L′) for a rectangular shape; sγ = 0.7 for a square or circular shape; sc = (sqNq − 1)/(Nq − 1) for rectangular, square or circular shape ˆ the inclination of the load, caused by a horizontal load H: ic = iq − (1− iq)/(Nc tanϕ′); iq = [1−H/(V + A′c′ cotϕ′)]m; iγ = [1−H/(V + A′c′ cotϕ′)]m+1; Where m = mB = [2 + (B′/L′)]/[1 + (B′/L′)] when H acts in the direction of B′; m = mL = [2 + (L′/B′)]/[1 + (L′/B′)] when H acts in the direction of L′; In cases where the horizontal load component acts in a direction forming an angle θ with the direction of L′, m may be calculated by: m = mθ = mL cos2 θ +mB sin2 θ For the particular case of a shallow foundation on the surface of a purely cohesive soil with undrained shear strength cu, Equation 2.1 simplifies to R/A′ = (pi + 2) cu sc. The value of (pi+2) in Equation 2.1 is the bearing capacity factor of a strip on the surface, as determined by Prandtl (1921). The bearing capacity factor Nc is the ultimate bearing pressure normalised by the undrained soil shear strength and therefore represents the 13 2. LITERATURE REVIEW global factor accounting for the effects of footing shape and roughness, and hence Nc = (pi + 2) sc. For example, investigations by Cox et al. (1961) and Eason and Shield (1960) using plasticity theory found bearing capacity factors of smooth and rough circular shallow foundations of Nc = 5.69 and Nc = 6.05 respectively. Similar research is less frequent in recent times, which could be ascribed to the theory being well understood and field results correlating reasonably well with calculated values. The UK National Annex to Eurocode 7 (Annex D) provides no provisions on Section 6 of Eurocode 7. The sample method given in Annex D, however, fails to consider the depth and ground inclination factors shown in Equations 2.1 and 2.2. The omission of the depth factor provides a conservative estimate, but ignoring the ground inclination factor errs on the wrong side of safety. It is said that a suitable alternative method can be adopted - and should be, where required. Therefore, for the design of most shallow foundations in the United Kingdom the same process will be used as for those in Europe, as dictated by Eurocode 7. In Australia there is no particular standard for the design of shallow foundations. Small-scale projects, such as houses, can find guidance in AS 2870: Residential Slabs and Footings. The primary purpose of AS 2870 is to classify a site based on the ground movement - in reference to the expected shrink and swell movement - and then provide a standard design for the footings. This is a function of the soil type (sand or clay), ground moisture, change in moisture related to the seasons and also the ground profile. Large-scale projects would often adopt Eurocode 7 or possibly even American State Highway Codes. Concerns about the use of a rigid-plastic material to represent soil produced the second approach for determining foundation stiffness and ultimate bearing capacity - a cavity expansion idealisation. This method has primarily been used for deep founda- tions such as piles, where capacity is provided by the resistance that the soil offers to the expansion of a cavity corresponding to the volume indented by the pile. The undrained deformation field observed in this research was shown to be better represented by a cav- ity expansion mechanism. For consistency, the cavity expansion literature is introduced with the model developed in Chapter 5. It is widely published that if a building foundation is loaded up to its ultimate bearing capacity then it will have undergone considerable deformation resulting in pos- sible damage to the superstructure. Some examples include Meyerhof (1965), De Beer 14 2. LITERATURE REVIEW Fig. 2.2 Plot of stiffness versus strain and the typical strain ranges for a number of structures (Mair, 1993) (1965), Atkinson (1993) and Osman and Bolton (2005). To limit the settlement the ultimate bearing capacity is reduced to an allowable design load by applying a factor of safety. Recommendations are given for the value of the factor of safety, but it is governed by the soil conditions and generally between 2 and 3 (Atkinson, 1993; Peck et al., 1974). Skempton (1951) indicated that the use a of small factor of safety can result in sig- nificant settlement. This movement will almost certainly result in the structure failing its serviceability criteria - with the possibility of collapse if structural elements are dam- aged. It is generally more satisfactory and economical to adopt a higher factor of safety in order to prevent damage to even sensitive components, such as architectural features which frequent modern design. Typical working levels of strain that structures sustain were suggested by Mair (1993) and are shown in Figure 2.2. However, no database was provided, and the suggested upper limits might be regarded as conservative. Settlement criteria dictates the design of shallow foundations. The most common cause of structural damage is differential settlement between elements of a structure, 15 2. LITERATURE REVIEW and therefore a settlement-based approach to design may be more appropriate. Pro- fessor Bolton’s Rankine Lecture (2012) on ‘Performance-based design in geotechnical engineering’ presented such an argument. Similar to bearing capacity research, there has been extensive research in to the settlement of shallow foundations. The progression of structural designing and devel- opments in technology and instrumentation has seen a renewed interest in the research of settlement of late. Some significant and interesting contributions are now presented. 2.3 Settlement The conventional calculation of the total settlement of a footing, wt, regards it as the sum of primary and secondary components. The primary settlement is composed of the immediate undrained component, wu, and the consolidation component, wc, while the secondary settlement, ws, is attributed to creep. The total settlement is: wt = wu + wc + ws (2.3) The components of settlement are now introduced, with example design methods given, and appropriate soil testing methods for determining the necessary parameters discussed. Some limits of working load settlement - as governed by the structure being supported (Hansen, 1967) - are then presented. The use of precompression to limit a structures exposure to settlement is introduced, and then the settlement based approach of MSD is explained. 2.3.1 One-Dimensional Consolidation Early analyses were performed using conventional one-dimensional analysis, as devel- oped by Terzaghi, for determining the total settlement, w, as a result of a stress change, ∆σz. It is termed one-dimensional because only vertical strains, and hence settlements, occur due to appreciable lateral restraints. It was assumed that the thickness of the soil layer is small compared to the width of the footing, implying that horizontal strains can be neglected (Atkinson, 1993). This assumption is similar to the strain conditions 16 2. LITERATURE REVIEW in the oedometer test. Therefore, the oedometer is used to measure the coefficient of volume change, mv, for determining settlement. Appropriately, this settlement is also termed the oedometer settlement, woed. Also developed by Terzaghi was an expression for the rate of settlement so the degree of settlement could be determined at a time t1. This was related by the parameter U . w = woed = ∫ h 0 mv.∆σz.dz wt=t1 = U.woed (2.4) The coefficient of volume change, mv, is not a soil constant and must be deter- mined through an oedometer test performed at conditions corresponding to those in- situ (Atkinson, 1993). It is a function of the void ratio, e, and is given as: mv = − 1 1 + e de dσ′ (2.5) The remaining parameter for the determination of the settlement is the stress in the soil as a result of loading at the surface. There are numerous methods of determining the change in stress as proposed by researchers. A very simple approach adopts a 2:1 slope from the surface down to the bedrock (Bowles, 1996). The earliest and most common method, however, is that of Boussinesq (1885). This solution was for a point load on the surface of a semi-infinite, homogeneous, isotropic elastic, weightless half-space. Expressions for the vertical (σz) and horizontal (σr) stresses beneath the centreline of a uniformly loaded circle (Poulos and Davis, 1974) with notation shown in Figure 2.3 are: σz = q [ 1− { 1 1 + (D/2z)2 }3/2 ] σr = σθ = q 2 [ (1 + 2ν)− 2(1 + ν)z (D2/4 + z2)1/2 + z3 (D2/42 + z2)3/2 ] (2.6) where ν is the Poisson’s ratio of the soil. For any point beneath a circular load Poulos and Davis (1974) provided charts for the vertical, horizontal and shear stresses, as well as deflections. The plot of vertical stress beneath a circular load is given in Figure 2.4. 17 2. LITERATURE REVIEW Fig. 2.3 Notation for a uniformly loaded circle Newmark (1942) developed a graphical procedure for determining the stresses within a profile loaded with a uniform pressure. The benefit of this procedure is that the stress at any depth can be determined for any shape of uniform pressure. Newmark charts use an influence value which is a function of the number of units resulting from the subdivision of the chart. The procedure involves equating the depth of interest to the standard scale given on the chart. Using the determined scale the load shape is drawn with the point of interest at the origin of the chart. The sum of enclosed units is mul- tiplied by the surface pressure and the chart influence value to give the stress at that point. Newmark (1942) dictates that the accuracy is sufficient for practical purposes. A Newmark chart for vertical stresses with an influence value of 0.001 is given in Figure 2.5. Other numerical methods exist for solving the Boussinesq equation. Poulos and Davis (1974) give a number of variations of load shapes with different load distributions such as linear and triangular loads. It must be noted that, although the use of these methods is common practice, the approximation of a real soil profile by that of a semi-infinite, homogeneous, isotropic, elastic mass is unsatisfactory (Davis and Poulos, 1968). A common method for solving the one-dimensional consolidation equation is to divide the soil into a number of layers. This is because the void ratio of the soil and the stress vary with depth. A more accurate answer is achieved with a greater number of layers, as a result of using smaller thicknesses. The benefit of this procedure is that only a summation is required, rather than an integration. If the soil is uniform, however, 18 2. LITERATURE REVIEW Fig. 2.4 Vertical stress below a circle on half-space then there is no need to divide the soil into layers as a single average calculation can be performed. In true one-dimensional conditions, due to the lateral strain restraint in the oe- dometer, only vertical strains occur and hence immediate undrained settlement cannot occur (Davis and Poulos, 1968; Skempton and Bjerrum, 1957). Examples of in-situ one-dimensional conditions were provided by Skempton and Bjerrum (1957). These include a loaded area of horizontal extent which is far greater than the thickness of the clay layer, resulting in negligible lateral strains; or when a thin clay layer is between beds of sand or rock. These conditions only occur occasionally, and often the first case is assumed. Therefore, the theory of one-dimensional consolidation was extended for layers of thick clay - where immediate settlement would occur. Even today, some de- sign engineers will take the oedometer settlement as the consolidation settlement while others will use it as the total final settlement. Skempton et al. (1955) performed an investigation in to the settlement of twenty structures and found that the oedometer settlement had a stronger correlation with the total final settlement. 19 2. LITERATURE REVIEW Fig. 2.5 Newmark chart for vertical stress with influence value 0.001 (Newmark, 1942) 2.3.2 Undrained Settlement Loading of a soil, through a foundation, is immediately resisted by an increase in the pore pressure. The deformation is undrained, resulting in an immediate settlement that occurs at constant volume. It is clear, therefore, that the undrained immediate settlement can only be calculated using a three-dimensional approach. Biot (1941) presented a theory for three-dimensional consolidation, which when reduced to one dimension was shown to be equivalent to the theory of Terzaghi (Cryer, 1963). Modelling the soil as linear-elastic, with properties: Poisson’s ratio, ν, and Young’s modulus, E, Skempton and Bjerrum (1957) proposed a method for determining the settlement using the expression: w = qD 1− ν2 E Iρ (2.7) where w is the immediate settlement due to a net foundation pressure, q, applied over a foundation of width D. The influence factor Iρ is a function of the depth of the soil layer and the shape of the loaded area. Poulos and Davis (1974) produced a number of charts for the influence factor based on different footing shapes and soil boundary conditions. The chart of influence factors for a loaded rigid circle and a simplified 20 2. LITERATURE REVIEW equation for use within it is given in Figure 2.6. An expression for the particular case of settlement of a rigid circular punch on a deep elastic bed with shear modulus G was developed by Davis and Selvadurai (1996) and is given by: w = pi 8 qD (1− ν) G (2.8) Poisson’s ratio, ν, for immediate settlement calculations, such as those in Equations 2.7 and 2.8, is taken as 0.5. This value corresponds to an incompressible material, which is satisfactory given both soil and water particles are incompressible. Thus, this corresponds to undrained, constant volume loading with the load being resisted by an increase in pore pressure (Osman et al., 2007). The value of Young’s modulus for immediate settlement in Equation 2.7 should be the undrained modulus, Eu. Skempton and Bjerrum (1957) argue that the soil properties, for clay in this case, can be determined from an undrained triaxial test. The most important feature of the triaxial test is that the lateral strain can be measured and controlled (Davis and Poulos, 1968). Also of particular note is that the influence of circular loads, due to axial symmetry, is easily described by triaxial parameters (Muir Wood, 1990). Skempton et al. (1955) claim that the stress applied by a structure is small in comparison to the ultimate failure load and therefore the stress-strain data within this range is linear. Jardine et al. (1984) determined through experimental work that the purely elastic response of low plasticity clays rarely extended beyond a shear strain of 0.01%. Soil is highly non-linear even at small strains - as demonstrated in Figure 2.2. The value of Young’s modulus is affected by a number of parameters such as stress history, shear strain amplitude and sample disturbance. The calculation of immediate settlement is a strong function of the selected value of undrained modulus (Schnaid et al., 1993) and discrepancies in the predicted and observed immediate settlements are frequent and can be significant. This can be attributed not only to the use of elastic theory for a non-linear material, but also to an inappropriate value of elastic modulus. Triaxial tests remain the common tool for determining mechanical behaviour for design. Recent researchers, however, had the benefit of seeing the development of in- situ testing apparatus. Scott (1980) suggests that results from laboratory tests can be erroneous and therefore recommends that in-situ tests be used. Aubeny et al. (2000) 21 2. LITERATURE REVIEW Fig. 2.6 Settlement and plot of influence factors (Poulos and Davis, 1974) 22 2. LITERATURE REVIEW researched the difference in strength obtained between pressuremeter and triaxial tests. The major discrepancy was attributed to sample disturbance. An alternative method to determine Young’s modulus is to use, if possible, observed and recorded results of adjacent structures in back-analyses (Scott, 1980). Similar to its recommendation about the bearing capacity of a spread footing, Eu- rocode 7 requires that a commonly recognised method for evaluating settlements be used. Annex F of Eurocode 7 provides sample methods for evaluating the immedi- ate and consolidation settlement. It dictates that the undrained settlement can be determined using either the stress-strain method or the adjusted elasticity method. Naturally, it is recommended that the appropriate undrained parameters be used. The requirement of codes of practice to use a commonly recognised method raises the ques- tion of making progress in design. An engineer is entitled to use their engineering judgment but codes of practice are relatively slow in keeping up with the new methods that have been developed for a better estimate of settlement. The stress-strain method of Eurocode 7 can be used for cohesive and non-cohesive soils. It is essentially the same procedure as the one-dimensional settlement theory described in Section 2.3.1, with some additional recommendations. Stresses due to surface loading are computed using elasticity theory, generally assuming a homoge- neous isotropic soil as used in the Boussinesq solution. The strains are calculated using appropriate stiffness moduli determined either from laboratory tests, with the recommendation that results be calibrated with field tests, or from field tests alone. The strains are then integrated to find the settlement. For a summation approach it is recommended that a sufficient number of layers be used for thinner stratum and a resulting better prediction. The adjusted elasticity method can also be used for cohesive and non-cohesive soils and is basically the three-dimensional approach described. The expression given has the usual parameters of bearing pressure, foundation diameter and elastic modulus. The influence factor is termed the settlement coefficient and is denoted f . The settle- ment coefficient is a function of the shape and dimensions of the foundation, variation of stiffness with depth, thickness of the soil layer, Poisson’s ratio, the distribution of bearing pressure and the point at which settlement is being calculated. The Young’s modulus is recommended to be determined, if possible, from structures in similar con- ditions - otherwise laboratory or field tests may be used. Eurocode 7 only allows this 23 2. LITERATURE REVIEW method to be used if no significant yielding occurs, as the soil can no longer be con- sidered elastic, and that the stress-strain behaviour of the ground can be considered to be linear. Great caution is also required in the case of non-homogeneous ground. Thus, even Eurocode 7 recognises the uncertainty and inadequacy of current settlement models. The effect of different footing conditions have been investigated, with resulting ef- fects accounted for by the inclusion of more influence factors in the design process of Equation 2.7. Fox (1948), for example, developed an influence factor, IF , to suggest that the settlement is reduced when the footing is placed at some depth within the ground. Bowles (1996) uses elasticity theory to provide another expression for deter- mining the immediate settlement of soil, which utilises the influence factor from Fox (1948). Although it claims that the principal components of settlement are inelastic, such as particle rolling, sliding and grain crushing, it is convenient to treat the soil as pseudo-elastic material. The expression of Bowles (1996) for immediate settlement is: w = qD 1− ν2 E ( I1 + 1− 2ν 1− ν I2 ) IF (2.9) where the influence factors I1 and I2 are functions of footing dimension and the depth of the soil bed. Although the equation is used widely the predictions are even argued by the author to not agree well with measured settlements. The problem is directed towards the use of the equation. Bowles (1996) recommends a number of steps to utilise this procedure. For example, a weighted average is justifiably obtained for the Young’s modulus, as engineers may adopt a value just below the footing which is not representative of the influence depth. Also, round foundations are converted into an equivalent square, which Davis and Poulos (1972) have shown that for elastic materials this is appropriate for bearing capacity and settlement calculations. An example of a structure with known settlement is provided by Bowles (1996) using a Poisson’s ratio of 0.35 in a calculation for immediate settlement. This is not common practice and technically not correct. This value of ν = 0.35 may simply have been adopted to produce settlements similar to the measured in support of the proposed method. Atkinson (2000) investigated non-linear soil stiffness in routine design, based on elasticity in reference to a vertically loaded circular foundation. The method for deter- mining the settlement incorporates the result that soil stiffness parameters decay with 24 2. LITERATURE REVIEW settlement in both model and full-scale foundations as in a triaxial test sample. This form of self-similarity allows a plot of stiffness against strain to be scaled to normalised footing settlement (Osman et al., 2007). Atkinson (2000) found that the normalised settlement ratio (settlement to footing width) was two to three times larger than the corresponding axial strains in a triaxial test. Good correlation was found on centrifuge and model plate loading tests when a factor of three was utilised. The procedure de- veloped by Atkinson (2000) is shown in Figure 2.7. Essentially, a triaxial stress-strain curve is constructed from small strain stiffness, strength and failure strain data. The relationship between triaxial and in-situ conditions is developed by scaling the strain axis by a factor of three. Fig. 2.7 Atkinson (2000) method for determining settlement The design process iterates around a loop, also shown in Figure 2.7, until the load, dimensions, stiffnesses and settlements are all compatible. If, however, the limiting settlement is known then no iteration is required and the maximum load is simply determined (Atkinson, 2000). Lehane (2003) conducted a full-scale test on a square shallow foundation in Kin- negar. The footing was 2 m × 2 m × 1.7 m thick with its base 0.2 m below the 25 2. LITERATURE REVIEW water table. It was found that the relationship between applied stress and mean pad settlement was strongly non-linear from the earliest stages of loading. Figure 2.8(a) demonstrates the applied stress against the mean pad settlement. Load was applied with concrete blocks and so the time-dependent settlement was able to be observed in between the addition of each block. A range of instrumentation was utilised in the test to allow the monitoring of a number of parameters, for example settlement plates were placed beneath the footing centreline. Results of the settlement at each plate for a given applied load were demonstrated together with a mean settlement for the pad. It is observed that nearly half of the footing settlement is due to compression of the soil between depths of 1 m and 2 m. This plot is shown in Figure 2.8(b). (a) Applied stress against mean pad set- tlement (b) Settlement against applied stress Fig. 2.8 Plots of stress against settlement and settlement at various depths against applied stress from a full-scale test (Lehane, 2003) A back-analysis was performed by Lehane (2003) to determine the Young’s modulus for given settlements. A plot of Young’s modulus against normalised settlement was produced. The result was very similar to that obtained in a triaxial test, even with the factor of three to account for the difference in strains between triaxial and in-situ strains - as proposed by Atkinson (2000). Thus the method of Atkinson (2000) was confirmed for the quick evaluation of undrained settlement. As shown, a good deal of research into the immediate settlement of shallow footings 26 2. LITERATURE REVIEW has been performed. Only the most relevant and noteworthy contributions have been discussed in this review. Although some research claims to have good correlation between predicted and measured settlements, most of the theory is based on properties which soil does not possess. Davis and Poulos (1968) suppose that results will generally show the significant discrepancies between predicted and observed settlement, and are therefore less likely to be reported than good agreements. The biggest discrepancy generally occurs in the calculation of the immediate settlement. Mobilisable Strength Design (MSD) has been developed at Cambridge University and, like Atkinson (2000), uses self-similarity; but, more appropriately, it is based on plasticity. As MSD reflects true soil behaviour it is discussed in more detail in Section 2.6. Following the immediate undrained settlement pore pressures begin to dissipate, resulting in consolidation settlement. 2.3.3 Consolidation Settlement The differential equation governing the consolidation as a result of the dissipation of excess pore pressures in one-dimensional flow was formulated by Terzaghi. The equa- tion is based on the assumption that the pore water flows according to Darcy’s law (Seed, 1965). Other assumptions include that the soil is semi-infinite in extent, homo- geneous, isotropic and elastic, completely saturated and that both the soil particles and pore water are incompressible. The basic equation for one-dimensional consolidation is given as: ∂u ∂t = cv ∂2u ∂z2 + ∂σz ∂t (2.10) where the one dimensional coefficient of consolidation is equal to cv = k/(mvγw) where k and γw are the soil permeability and unit weight of water respectively. In the three-dimensional problem of shallow foundations there are two ranges of stress increase. The first, as discussed, is the undrained condition and utilises ν=0.5. The second stress range corresponds to the drained phase and uses the elastic expres- sions given in Equations 2.7 and 2.8 with ν = ν ′. Davis and Poulos (1968) suggest that ideally the stresses should be altered gradually from the undrained to the drained phase because the stress distribution within the soil changes and the deviator stress 27 2. LITERATURE REVIEW beneath the centre of the footing increases. It is recommended, however, that as the calculation procedure is only really approximate, performing two distinct calculations is satisfactory. Davis and Poulos (1968) also found that for increasing values of Poisson’s ratio the immediate settlement contributes to ultimate settlement more significantly. The plot showing the influence factor for a number of Poisson’s ratio values was given in Figure 2.6. The relationship between Young’s modulus and the elastic shear modulus, G, is given as: G = E 2(1 + ν) (2.11) For an elastic material the shear and volumetric effects are decoupled, giving equiv- alent undrained and drained elastic shear moduli, thus G′ = Gu. Based on this equality the undrained and drained Young’s moduli are related using: Eu = 3E ′ 2(1 + ν ′) (2.12) Parameters E ′ or G for use in Equations 2.7 and 2.8 can be obtained from the oedometer modulus Eoed if some value for ν ′ can be assumed. A linear elastic analysis of one-dimensional compression provides that: Eoed = E(1− ν ′) (1 + ν ′)(1− 2ν ′) = 2G(1− ν ′) (1− 2ν ′) (2.13) Values of ν ′ for clays at small strains typically fall between 0.25 and 0.35 (Wroth, 1975). The procedures given can now be used to determine both the immediate settle- ments, wu, and consolidation settlements, wc of shallow foundations. The unique value of elastic shear modulus G, drained or undrained, can also be used in Equation 2.8 to deduce the ratio of the final consolidation settlement to the undrained penetration, confirming Davis and Poulos (1968), as given by: wc wu = (1− 2ν ′) (2.14) 28 2. LITERATURE REVIEW Fig. 2.9 Plot of settlement coefficient against pore pressure coefficient Eurocode 7 recommends that the settlement caused by consolidation be calculated by assuming a confined one-dimensional deformation of the soil. Thus, results from an oedometer test can be used to evaluate the settlement. Research has found and Eurocode 7 states that the addition of the immediate and consolidation settlement often leads to an overestimate of the settlement. It is therefore allowed that empirical corrections may be applied. Skempton and Bjerrum (1957) noted the discrepancy and as a result refined the oedometer settlement calculation to make the total final settlement a sum of the imme- diate settlement, as calculated using the discussed elastic method, and the oedometer settlement multiplied by a settlement coefficient, µ. The settlement coefficient is a semi- empirical factor dependent upon the geometry of the problem and the pore pressure coefficient A - as determined experimentally in an undrained triaxial test (Skempton, 1954). The settlement coefficient is given in Figure 2.9 for use in the expressions of settlement and the rate of settlement, given as: w = wu + µ.woed (2.15) wt=t1 = wu + U.µ.woed (2.16) 29 2. LITERATURE REVIEW Research has also been performed into non-homogeneous soils. Values of the influ- ence factor for soils possessing a varying Young’s modulus with depth was developed and used in the same process. Shahin et al. (2002) used an artificial neural network (ANN) with 189 case records of known settlements of shallow foundations on cohesionless soils. The purpose of an ANN is to develop a relationship between a set of inputs and corresponding outputs using the data alone - thus no assumptions or simplifications are required, contrary to the given methods. Obviously, the greater number of examples provides a better correlation and thus it is often termed a ‘data driven approach’. As new data becomes available it is added and the model must then be re-trained to determine new model coefficients to minimise any error. The result is a tool that can provide quick estimates of settlement without the need for calculations. The validation set of data obtained a coefficient of determination, r2 of 0.819, a root-mean-square error of 11.04 mm and a mean absolute error of 8.78 mm. These magnitudes of error may be considered satisfactory to avoid significant differential settlement but this process needs more case records to be included and further analysis. The greatest problem associated with using an ANN is that the developed rela- tionship which is used to arrive at a settlement is not easily explained by the model. Similar to other empirical models the data must be calibrated and inputs should be consistent, and thus obtained from a common testing procedure or methodology. The use of an ANN for the settlement prediction of shallow foundations is a powerful tool but would not easily be introduced to practising engineers given the inability to see the model. A result in support of a hand calculation may, however, provide a greater degree of confidence. As discussed, laboratory tests for determining soil properties are performed in ei- ther an oedometer or triaxial apparatus. Parameters measured in laboratory tests are strongly dependent upon the quality of the sample. Sample set-up and the testing procedure, such as loading path and rate; affect the quality of the results. Trained personnel are also required to ensure correct performance of the apparatus through to analysis and interpretation of raw data (Atkinson, 2000). Human error can easily cause results to be erroneous. Disturbance of soil has been considered in research for a significant period of time. 30 2. LITERATURE REVIEW Disturbance to clay samples can occur as a result of tube sampling, sample transport and in preparation of the laboratory test (Hight et al., 1992a). The destructuring that results causes the observations of stiffnesses to be lower than those back analysed from field measurements (Jardine et al., 1986). Questions are also raised given such small samples are used to represent a much larger area - thus neglecting the possibility of inhomogeneous soil conditions. Jardine et al. (1986) also performed research on the triaxial test and found that discrepancies between stiffnesses determined in a laboratory and in the field were more likely to be caused by inadequacies in strain measuring techniques. This is because strains below shallow foundations are small, requiring very precise data of an even smaller scale. As demonstrated, Atkinson (2000) applied a correction factor to equate triaxial and in-situ strains. Thus, no conclusive evidence has ever been provided as to the validity of laboratory testing. Given discrepancies, however, reasonable agreement between settlements of footings determined from the triaxial test was observed (Jardine et al., 1986). Results in an analysis by Mair (1993) found that data obtained from a self-boring pressuremeter and an anisotropic consolidated triaxial test on a high quality thin-walled sample agreed very strongly. Aubeny et al. (2000) claim that the pressuremeter is a unique in-situ test because of the existence of a sound theoretical basis for deriving the complete shear stress-strain-strength properties of the surrounding soil directly from the measured expansion (and/or contraction) curve. In practice, however, problems have been experienced in obtaining reliable strength properties. Evaluation of experimental results has allowed correlations to be determined. For example, Ladd et al. (1977) performed direct simple shear tests to determine the effect of the overconsolidation ratio on the shear strength of clay. Kulhawy and Mayne (1990) provides a comprehensive set of correlations for estimating soil properties, each with their historical evolution and appropriate statistical and reliability parameters. Correlations taken from Kulhawy and Mayne (1990) for use in this research include anisotropy and rate effects. With time, more results from experiments have become available and technology development has aided the production of databases. Using a large number of test results can provide very strong statistical correlations. Vardanega et al. (2012) used triaxial test results to develop an expression to describe the non-linear stress-strain 31 2. LITERATURE REVIEW behaviour of kaolin and linked it to the stress history. Oztoprak and Bolton (2012) used results from a number of laboratory tests on sand to develop a hyperbolic stiffness relationship. For clarity, the correlations are introduced when they are utilised in the analysis of Chapters 4, 5 and 7. Performing laboratory experiments on clays, although it has some issues, is still used for design projects. Obtaining samples of sands for laboratory tests, however, is significantly more difficult. Settlements of shallow foundations on sands, therefore, are generally based on in-situ tests and elasticity theory (Nova and Montrasio, 1991). Determining parameters in laboratory tests for sand samples involves considerable time and expense. Values obtained are questionable due to sample and handling dis- turbance (Schmertmann, 1970). Methods required just for obtaining a sample include freezing or grouting of the sand. The time and cost has made it common practice to use in-situ methods. The main parameter which dictates the settlement of sands is the relative density (Atkinson, 1993). Examples of instruments used to determine the engineering properties of sand include penetration tests, plate load tests and pres- suremeters. The most common are the standard penetration test (SPT) and cone penetration test (CPT) as they are relatively cheap (Hughes et al., 1977). Given the relatively high permeability of sands any settlement will be drained and will occur quite rapidly. Atkinson (1993) suggests that a simple design procedure is to relate the allowable bearing pressure to the relative density. The number of blows required to move the tip of the SPT through the soil is correlated with the relative density and then the allowable bearing pressure. The CPT is used to determine a deformation modulus based on the cone resistance. A number of methods for correlat- ing the test results with soil properties have been proposed over the years. Viana da Fonseca (2001) performed full-scale tests in Portugal and found poor correlation with results and some very conservative predictions. Anderson et al. (2007) also performed a full-scale test and found that good correlation was obtained with very high values of stress. This is to be expected because the test methods deform the soil to high levels of strain and are therefore better suited to ultimate capacity parameters rather than compressibility for settlement predictions (Anderson et al., 2007; Hughes et al., 1977). Hughes et al. (1977) demonstrate that although plate loading tests and pressureme- ter tests are more expensive the data can be analysed more rigorously in order to di- rectly produce design parameters. It has been observed that they are very good for 32 2. LITERATURE REVIEW determining the non-linear behaviour of soils. Self-boring pressuremeters were developed with the intention of removing soil at the same rate as the rate of penetration whilst offering full support to the surrounding soil in order to minimise disturbance (Hughes et al., 1977). The bearing capacity as determined from pressuremeter data provided higher values than other testing methods (Chiang and Ho, 1980). This demonstrates that the pressuremeter is not only good at low levels of strain, but also possibly better at higher levels of strain. The associated higher cost of using a pressuremeter may, therefore, be justified. Briaud (2007) observed during full-scale testing that footing deformation is a result of lateral expansion. As the pressuremeter test imitates this deformation it was recommended the pressuremeter be used for both sands and clays. The dilatometer also provides direct measurements of compressibility. In the full- scale test performed by Anderson et al. (2007) it was found that both the dilatometer and pressuremeter provided good predictions of settlement at small loads. This is more suitable as actual loads resisted by footings are significantly lower than the actual capacity - hence why settlement design generally governs. Again, proposed methods have been refined by researchers through development and more experimenting. De Beer (1965) concluded that oedometer tests should be replaced by triaxial test- ing for determining compressibility parameters of sand. Viana da Fonseca (2001) found, however, that settlement predictions determined using parameters obtained from triax- ial test results did not compare well with the observed settlements from footing tests. The primary reason for the poor prediction was attributed to the underestimation of stiffness - possibly due to disturbance. A number of tests must also be performed, given the parameters are strongly dependent upon the stress level. This supports the need for in-situ testing methods for cohesionless soils. Finite element analyses performed by Anderson et al. (2007) found that results, as expected, are a strong function of the input parameters. A number of testing methods were used, and it was possible that a method which gave good in-situ results provided a poor correlation in the finite element analysis. In general, in-situ methods over-predict the settlement of foundations on cohesion- less soils such as sands. However, it was shown that plate loading tests, dilatometer and pressuremeter tests provide reasonable predictions at small applied loads (Ander- 33 2. LITERATURE REVIEW son et al., 2007). This is the relevant data for this analysis and is discussed further in the method of MSD. Briaud (2007) performed an investigation into spread footings on sand using a load- settlement curve approach with data obtained from pressuremeter tests. The proposed process requires pre-boring pressuremeter tests to be performed to obtain curves of cavity wall pressure versus relative increase in cavity radius. A mean curve is developed using an averaging technique where data at particular depths is considered in greater proportions, down to a depth of approximately three footing diameters. Naturally, the soil at shallow depths has a greater weighting. Correction factors were developed using finite element analyses for footing shape, eccentric load, footing inclination and proximity to a slope. An investigation into the combined effects was not performed, but, as is common in design procedures, individual influence factors are multiplied to account for a combination. The appropriate influence factors are applied to produce a footing curve with the ultimate result being a short term load-settlement curve. Long- term settlement is determined by multiplying settlements by (t/t1)n where t is the design life of the structure, t1 is one hour and n is obtained from creep pressuremeter tests. Briaud (2007) used data from twenty-four full-scale and centrifuge footing tests and twenty finite element models to produce a reasonably promising correlation. This procedure is similar to that of MSD but contains some flaws. MSD utilises a deforma- tion mechanism, and therefore stress-strain data can be obtained from any number of testing procedures. MSD is useful for any medium, therefore, as an appropriate testing method can be adopted - whereas Briaud (2007) only uses the pressuremeter. Although the pressuremeter is a popular design tool, no conclusive proof has been provided that it is the most suitable method. It could also be argued that a pre-boring approach is more likely to cause some disturbance to the soil than the self-boring technique. Briaud (2007) is generally concerned with square and rectangular footings founded primarily on sand. The limited number of results on clay, however, were reasonably good. This method needs to be extended for varying soil types and is, in general, more expensive in both testing and calculation. In determining the settlement of shallow foundations it is also important to de- termine the profile of settlement across and adjacent to the footing - not just at a particular point such as the footing centre. Predictions must at least match the pro- 34 2. LITERATURE REVIEW file, if not being equal in value. This is critical in assessing differential settlement and distortion of structures (Mair, 1993). It is possible that, given settlement, a redistribution of stresses in a structure is possible. Significant redistribution generally does not occur at working loads of struc- tures but can occur under extreme load cases or as a result of differential settlement. More accurate prediction of differential settlement can be achieved by considering the soil-structure interaction. Total settlement is primarily a function of the soil properties but differential settlement is often governed by the structural stiffness (Scott, 1980). Rigid foundations will have a uniform settlement with varying bearing pressure beneath the footing, generally greater at the edges and smaller in the centre - similar to the Boussinesq distribution (Meyerhof, 1965). Conversely, flexible footings will have a uniform pressure but varying settlement (Atkinson, 1993). The effect of footing rigidity was not explored in this research. The interface friction angle between the footing and soil surface will affect the deformation mechanism that is observed. Direct shear tests by Bolton et al. (1989) determined an interface friction angle between aluminium and kaolin of approximately 21◦. This can be considered a relatively smooth interface, but the use of cling film to prevent the clay surface drying during centrifuging will be shown to have made the soil-footing interface more rough. A number of the assumptions in Terzaghi’s consolidation equation (Equation 2.10) are rarely seen in-situ. It is assumed that the permeability and compressibility of the soil is constant - not only with depth but also during consolidation. This is incorrect because the increased effective stress and settlement will cause a reduction in both parameters. Real footings will generally not be founded on homogeneous and isotropic soils, however, these experiments were performed to investigate this perfect case. There is, of course, scope for further experimenting on layered soils to investigate the effect of non-homogeneous and anisotropic soils. It is also assumed that the soil layer is semi-infinite in depth. For foundations on elastic material a depth of five diameters approximately approaches the asymptote for infinite depth (Jardine et al., 1986; Poulos and Davis, 1974). Full scale tests by Jardine et al. (1995) found that the settlement recorded at points approximately two diameters below the surface were negligible. This is because most of the strains and settlement occur in a small depth below the footing, as supported by the field tests of Lehane (2003) that were shown in Figure 2.8(b). 35 2. LITERATURE REVIEW Also, the strength of soils increase with depth due to the increased effective stress. Jardine et al. (1986) performed research into the influence of non-linear soil stress- strain properties on soil-structure interaction. A significant number of discrepancies between non-linear and elastic behaviour were discussed. At the time of the research the use of non-linear behaviour in design was too time consuming and expensive. More recent research has showed that a theory based on strain-hardening plasticity would describe with reasonable accuracy the observed behaviour of shallow foundations (Nova and Montrasio, 1991). MSD is an example of one such method and it is discussed in Section 2.6. The theory of elasticity for stresses and settlements is used in the theory of set- tlement of shallow foundations. The use of this is known to be incorrect due to the non-linear behaviour of soils. It is still used today because of the absence of a more suitable method. The result of assumptions in the theory not corresponding to in-situ conditions is that errors of 50% or more may be found (Scott, 1980). Current engineers often expect this kind of error, and accept them because of the relatively simple design process. In addition, it is easy and not much more expensive to be more conservative in the design of superstructures. Nevertheless, a serious issue in the present day is sus- tainability and this sort of approach and attitude can have a significant environmental cost. This again supports the need for a performance-based design approach. In assessing the settlement of structures it is important to not only predict the extent of the settlement but also to analyse the rate at which it will occur. 2.3.4 Rate of Settlement The differential equation governing consolidation, shown in Equation 2.10, is based on the assumption that the pore water flows according to Darcy’s law (Seed, 1965). Darcy’s law describes the flow in a porous medium and a basic form of the equation is given as: Q = A.k. ∆h L (2.17) where Q is the volumetric flow rate, A is the flow area and ∆h/L is the change in head ∆h over a length L. To normalise solutions Terzaghi developed a settlement ratio, U 36 2. LITERATURE REVIEW Fig. 2.10 Pore pressure isochrones and a dimensionless time factor, Tv, given as: Ut = wt w∞ Tv = cvt H2 (2.18) where wt is the settlement at time t, w∞ is the final settlement (as calculated using the presented methods) and H is drainage path length. The relationship between Ut and Tv is a function of the soil geometry, the associated drainage conditions and the initial distribution of excess pore pressure (Atkinson, 1993). Assuming an initial uniform excess pore pressure, parabolic isochrones were developed to show the sequence of dissipation at each value of Tv. If there is only one-way drainage then only half of the plot is utilised. These isochrones are shown in Figure 2.10. Curves were also produced for the degree of consolidation for given values of Tv and are shown in Figure 2.11. Research has found that there is a significant discrepancy between the rate of set- 37 2. LITERATURE REVIEW Fig. 2.11 Degree of consolidation versus dimensionless time factor tlement determined through the above one-dimensional procedure and that observed in the field. The rate of settlement in-situ is often greater than that predicted (Davis and Poulos, 1972; McNamee and Gibson, 1960; Scott, 1980) because the assumption of one-dimensional flow is not appropriate. Some lateral pore pressure dissipation will occur (Davis and Poulos, 1972) and this is not represented in the oedometer test. This again supports the theory that shallow foundation settlement is a three-dimensional problem. Biot (1941) gave a three-dimensional theory of consolidation which also utilises Darcy’s law for the flow of water. However, due to the complex nature of the equations, there have only been a few solutions evaluated by researchers (Davis and Poulos, 1972). Through a numerical finite difference solution, Davis and Poulos (1968) solved a basic diffusion equation. This produced a series of curves of average degree of pore pressure dissipation, Up, over time and described by: Up = 1− ∫ H 0 ut.dz ∫ H 0 u0.dz (2.19) where ut is the excess pore pressure at time t and u0 is the initial excess pore pres- 38 2. LITERATURE REVIEW Fig. 2.12 Degree of settlement/pore pressure dissipation against time for an imper- meable circular footing but otherwise permeable top soil surface and permeable base below the soil layer sure. For a homogeneous soil the degree of consolidation settlement, Us, can be shown to be equal to Up if there is no redistribution of total stresses during consolidation. This is true for the one-dimensional theory of Terzaghi and thus U represents both Us and Up. In the real problem of shallow foundations, however, stress redistribution can, and will generally, occur, and therefore it is only assumed that Us ≈ Up (Davis and Poulos, 1972). Plots were produced for four combinations of frequently used hy- draulic boundary conditions (impermeable or permeable). The particular example of an impermeable circular footing with an otherwise permeable surface and permeable base below the soil layer is provided in Figure 2.12, and represents the experiments performed for this research. The rate of consolidation for clay can take a significant time due to its low perme- ability. The centrifuge scales time, so that long term tests can be performed in a much shorter period of time. The rate can, therefore, be investigated through centrifuge experiments. Often circular footings are designed based on an equivalent area square. Davis and Poulos (1972) found similar values of ultimate settlement but there were considerable 39 2. LITERATURE REVIEW differences in the rate of settlement. The rate of consolidation can also be affected by the degree of saturation. For example, a partially saturated soil (one containing air bubbles) will undergo greater immediate settlement due to the comparably greater compressibility of the air. Thus, the rate will not match the already greater settlement of the soil. Another example is that a soil containing gas bubbles will have a slower rate than the predicted. This is because gas bubbles can impede flow, resulting in a slower dissipation of pore pressures (Lowe et al., 1964). This highlights the need to simulate in-situ conditions when performing experiments. Soils below the water table will generally be fully saturated, and therefore all care was taken when preparing samples for centrifuge testing. Sandy soils are known to have a greater permeability. Often the rate of settlement has not been considered because most settlements occur quickly. However, Schmert- mann (1970) demonstrates that settlement does continue to occur - suggesting a creep type of deformation. The division between primary and secondary settlements, how- ever, is an arbitrary separation of a continuous process. One suggested method for defining the end of consolidation time is to extrapolate the approximately linear data of pore pressure versus settlement to a value of zero pore pressure (Crawford, 1964). 2.3.5 Secondary Settlement Once consolidation is complete, settlement continues to occur in the form of secondary settlement - also known as creep. As the pore pressures have dissipated during con- solidation, creep results from the time-dependent rearrangement of soil particles under constant effective stress (Fox, 2003). Research into secondary settlement has only been extensive in about the last twenty years. Some of the progress is discussed. Secondary consolidation is generally insignificant for clays and sands compared to the immediate and consolidation settlements (Skempton and Bjerrum, 1957). It will, however, be shown that this is not necessarily the case for footings on clay. Peats and organic soils generally undergo rapid consolidation, and often more significant and extensive, long-term secondary compression (Bowles, 1996; Fox, 2003). 40 2. LITERATURE REVIEW The common approach to determine the creep of a soil is to utilise the secondary compression index, Cα. This is constant with time and given as: Cα = ∆e ∆ log t (2.20) Jardine et al. (1995) performed tests on Bothkennar clay and determined a value of the creep coefficient as Cα = 0.02 after approximately 420 days. By performing research into precompression, Alonso et al. (2000) found that sec- ondary compression was a function of the overconsolidation ratio. This point is sup- ported by the use of precompression, which is discussed in Section 2.5. As expected a soil with a high overconsolidation ratio is less prone to creep and therefore has a lower index. The expression derived for the secondary compression coefficient was: Cα = 0.008[0.1 + 0.9 exp(−13(OCR− 1))] (2.21) which gives Cα = 0.008 for normally consolidated soil (OCR = 1), and for soils with high overconsolidation ratios tends towards Cα = 0.0008. Mesri and Godlewski (1977) developed the Cα/Cc concept for the analysis of sec- ondary compression. It was observed that the magnitude and behaviour of Cα with time is directly related to the magnitude and behaviour of Cc with consolidation pres- sure (Mesri and Castro, 1987). Values for soils that will undergo creep were given as: Cα Cc =    inorganic soft clays x = 0.04± 0.01 highly organic plastic clays x = 0.05± 0.01 (2.22) where Cα is shown in Equation 2.20 and Cc is given as: Cc = ∆e ∆ log σ′v (2.23) 41 2. LITERATURE REVIEW The creep settlement (GeoRG, 2004) is then calculated using: Ss = Cα 1 + ep H log( ts tpf ) (2.24) where ep is the void ratio at the time of complete primary consolidation, tpf , and ts is the time over which secondary compression is to be calculated. The development of Equation 2.24 introduced a strain variation of the secondary compression index, Cα ε (Augustesen et al., 2004), which is also commonly referred to and given by: Cα ε = ∆e (1 + ep)∆ log t (2.25) One of the difficulties associated with the creep model is knowing when the creep deformation begins, and hence determining ep and tpf in Equations 2.24 and 2.25. Naturally, the end of primary consolidation occurs when excess pore water pressures are zero. Without instrumentation, however, it is possible to use oedometer test data to determine the time that primary consolidation ends by fitting two straight lines and finding the intersection (Augustesen et al., 2004). The secondary creep coefficient, Cα or Cα ε, is simply the slope of the second line depending on what parameter is plotted on the y-axis. This process is demonstrated in Figure 2.13. Fig. 2.13 Method to find coefficient of secondary compression and time at end of pri- mary consolidation (modified Augustesen et al., 2004) It can be argued that the test duration in which the secondary compression index is determined is not long enough. Any prediction for in-situ conditions is simply an 42 2. LITERATURE REVIEW extrapolation of the fitted data. It is, for example, recommended that oedometer tests be left as long as possible with a one week minimum (Yin, 1999). The creep model could also be considered flawed given that at infinite time the settlement will be infinite. It is more likely that there will be a plateau in settlement at a given significant period of time. Yin (1999) argued such, and produced a new creep function which could describe the non-linear behaviour of creep, with a limiting creep parameter. It could, however, be argued that the model is sufficient for the time periods that are usually being used in the design process. Structures are often designed for a life of between 50 and 100 years, and therefore the model may be adequate (Mesri and Vardhanabhuti, 2005). The components of settlement and their importance in the design process has now been introduced. It must be reaffirmed that the calculation of settlement is only approximate - as supported by the uncertainty in Eurocode 7. The settlement allowed in design is primarily governed by the superstructure. 2.4 Allowable Settlement The allowable settlement is a function of a number of parameters. Each mode of defor- mation is primarily dependent upon the type of superstructure and its functionality. This includes properties such as the construction material, the stiffness of the structure during and after construction, and also the services utilised in the building. Other con- siderations include the confidence in which the allowable settlement can be specified and rate of ground movement (Eurocode 7). Much research has been performed into the allowable settlement of structures and some of the results are discussed herein. Early research by Terzaghi and Peck (1948) concluded that most ordinary structures can withstand a differential settlement between adjacent columns of 3/4 inch (19 mm). An allowable total settlement for shallow foundations was suggested at 1−2 inches (25− 50 mm) - the lower limit applying to footings and the upper limit to rafts. However, a single value cannot be utilised given the number of factors associated with the allowable settlement. Skempton and MacDonald (1956) contributed significantly to the tolerable settle- ment criteria for structures - with many values very similar to those adopted in design 43 2. LITERATURE REVIEW today. Research was based on the observations of settlement and damage on both steel and reinforced concrete frame buildings, and buildings with load-bearing walls. This included buildings founded with shallow footings. Primary observed damage was the result of angular distortion, as was shown in Figure 1.2. This was, therefore, chosen as the critical parameter for settlement of structures. Cracking of load-bearing walls or panel walls in frame structures was deemed likely to occur when the ratio of set- tlement to length of member, ∆/L, exceeded 1/300. Structural damage is probable when ∆/L exceeds 1/150. As a design criterion that provides a factor of safety against cracking, a value of ∆/L = 1/500 was suggested (Skempton and MacDonald, 1956). Eurocode 7 adopts these values as its recommendation for what is termed normal, routine structures. Examples of more complex structures include buildings with cranes or a number of services. Crane rails require more strict imposed limits to simply allow the crane to be serviceable. Buildings with a number of services can not undergo significant movement because of the possibility of pipes cracking. This could result in potentially hazardous substances leaking. More detailed analyses are considered necessary for structures which are considered to be out of the ordinary. Some researchers suggest that the allowable settlement is best expressed as a func- tion of the footing diameter. De´court (1992) used an investigation of SPT tests to propose that the allowable settlement should be taken as 0.75% of the footing diame- ter. Nova and Montrasio (1991) investigated shallow foundations on sand and suggested that the allowable settlement is typically 1% of the footing diameter. However, adopt- ing the allowable settlement as a function of the footing diameter is not necessarily appropriate, and so the authors suggest that these limits are only applicable to smaller footings. These values can then be seen to correspond with the common limits. Olson and Lai (2002) utilises the Skempton and MacDonald (1956) value for limiting distortion to determine the maximum settlement of an isolated foundation on clay as 4 inches (101 mm) and 2 inches (50 mm) for sand. Eurocode 7 takes the conservative view, and prescribes a maximum total settlement of 50 mm. The rate of settlement is an important parameter because a high proportion of the settlement may occur during construction, and will therefore not be a problem. Also the type of structure and material is important because stresses can be redistributed, allowing greater settlements to occur without structural integrity issues. Peck et al. 44 2. LITERATURE REVIEW (1974) states that steel structures can withstand more differential settlement than con- crete frames, while brick and masonry structures can withstand approximately three times more than concrete. McKinley (1964) considers that the remote chance of struc- tural collapse is due to the elasticity of materials, safety factors inherent in the design process and the ability of a structure to redistribute stresses. Surveying methods, such as precise levelling, allow the movements of structures to be monitored. This allows the early arrest of significant settlements to be performed, resulting in a more cost efficient repair. Given the amount of research into shallow foundations, monitoring should only be required for particularly important or sensitive structures as simple reassurance for the client. Olson and Lai (2002) also indicate that no recent research has been performed on allowable settlement. This is attributed to the fact that calculations are so conservative that foundation problems have diminished significantly. A new method for the design of shallow foundations could reduce the conservatism currently involved without negating safety. There are a number of methods to reduce the settlement a structure experiences - of which precompression is one common method. 2.5 Precompression A number of methods are used to compress the soil in an effort to reduce the settlement that the superstructure experiences. However, the time required for this process is often unavailable, and the cost can be too significant. Alonso et al. (2000) describes precompression as being widely utilised in geotechnical engineering to improve the foundation characteristics of soft, fine-grained soils. Aldrich (1965) claims that it is normal practice for light buildings, oil storage tanks, and highway embankments and bridges. Generally earth fill is used as the precompression load. Dependent on soil properties, a significant proportion of consolidation settlement can be removed by applying a preload over a certain period of time. This process essentially removes plastic strains within the soil with a resulting increase in stiffness. In cases like this, the secondary settlement may now become significant, in particular for sensitive structures. One such example is given in Koutsoftas et al. (1987) for the development of an airport in Hong Kong. The research found that there is a significant 45 2. LITERATURE REVIEW reduction in secondary compression deformation if the soil is overconsolidated - even to a modest degree. Thus, a precompression load greater than that of the structure load provides an effective means of reducing the secondary settlement to which the structure is exposed. Naturally the load is removed before construction commences. Precompression can also be achieved by lowering the water table to increase the effective stresses. The advantage of this process is that effectively a pre-load is applied without causing considerable shear stresses in the foundation soil (Aldrich, 1965). By using vertical sand drains the rate of settlement can be increased, and therefore precompression is accomplished. Sand drains are utilised to reduce the drainage dis- tance for pore fluid upon loading. Handy (2002) demonstrates that sand drains were used to hasten primary consolidation for Kansai International Airport in Japan. A more common application of sand drains is highway embankments. Pore water can also removed using the technique of vacuum preloading. This method has been well developed because it is cheaper then applying fill. A signifi- cant amount of land has been reclaimed in China using this method. It has also been used extensively in remediation works and in the stabilising of structures (Chu et al., 2008). Precompression is a good method for reducing the settlement a structure undergoes. However, the time for precompression is normally between one and five months, and owners rarely allow longer due to economic reasons (Aldrich, 1965). If settlements could be determined accurately then the need for precompression may subside. As discussed, research in to shallow foundation settlement has been extensive in recent years because of the unsuitable methods currently adopted and also the time, economic and environmental savings which could be made with a better design process. A new method for the design of shallow footings has been developed at Cambridge University and is based on the mobilised strength of the soil. 2.6 Mobilisable Strength Design - A New Approach Excessive total or differential settlements are a main cause of unsatisfactory building performance. Occasionally, consolidation settlement is unexpected, but the primary discrepancy is caused by the use of linear-elasticity in determining the immediate 46 2. LITERATURE REVIEW Fig. 2.14 Prandtl (1921) mechanism adopted for the displacement pattern (from Osman and Bolton, 2005) undrained settlement (Osman and Bolton, 2005). Jardine et al. (1986) demonstrates that although linear-elasticity is convenient for soil-structure interaction calculations, it can be misleading, unless the non-linear behaviour of soil is considered. Based on the theory of plasticity and the mobilisable soil strength concept, Osman and Bolton (2004) proposed a new design method for retaining walls. A stress path in a representative soil zone was treated as a curve of plastic soil strength mobilised as strains develop. These strains are inserted into the plastic deformation mechanism to predict displacements. Thus, both ultimate strength and serviceability are considered in the process of Mobilisable Strength Design (MSD). Osman and Bolton (2005) inves- tigate the load-settlement behaviour of shallow foundations using the MSD approach. The MSD approach utilises plasticity theory, but instead of a rigid perfectly-plastic material, strain-hardening is incorporated. An undrained soil deformation mechanism was assumed within the boundaries of the classical plane-strain Prandtl (1921) bearing capacity mechanism. The mechanism consists of three distributed shear zones (active, fan and passive) which deform compatibly and continuously with no relative sliding at the boundaries. Strains and compatible displacements are established using the stress increment and condition of equilibrium. The Prandtl mechanism is shown in Figure 2.14. By applying axisymmetric conditions, an expression for the strains can be deter- mined from the first derivative of displacement. The average shear strain mobilised, 47 2. LITERATURE REVIEW γmob, in the deforming soil is determined as the spatial average of the shear strain in the whole volume of the deformation zone. The footing settlement, wu, is normalised by the footing diameter, D. A relationship is then established between the average shear strain and normalised footing settlement using the displacement field. This gives: γmob = ∫ vol γ.dvol∫ vol dvol = Mc wu D (2.26) The compatibility factor, Mc, was shown in Osman (2005) to be 1.35. Houlsby and Wroth (1983) argue that the geometry of the deformation mechanism should be optimised according to the rate of increase of shear strength with depth, resulting in a varying compatibility factor. However, MSD uses the soil properties from a represen- tative location. Although the solution avoided the use of any slip lines it was shown through an overall energy balance that the average shear stress mobilized within the mechanism cmob was related to the average applied pressure, σmob, by the same “bearing capacity factor” Nc currently employed at failure: σmob = Nccmob (2.27) Equations 2.26 and 2.27 provide the scaling from a simple soil test curve of (τ ,γ) to a loading curve (q, wu). The soil test data can be determined from undrained triaxial tests or even in-situ tests such as the pressuremeter. Jardine et al. (1986) shows that conventional laboratory tests often produce stiffnesses far lower than those back anal- ysed from field measurements. This discrepancy has resulted in a strong move towards in-situ testing. It is also argued that the differences between laboratory and field stiff- nesses of London Clay result more from inadequacies in conventional laboratory strain measuring techniques than from sampling disturbance or time-dependent threshold ef- fects. It may be more expensive to utilise a pressuremeter, but the additional cost may be justified in the design and construction phase. The compromise of MSD is that it couples together an equilibrium solution based on the mobilisation of a constant shear stress, with a kinematic solution based on the creation of an average mobilised shear strain. The load-settlement curve obtained 48 2. LITERATURE REVIEW Fig. 2.15 Load-settlement curve obtained from soil stress-strain data (from Osman and Bolton, 2005) from a soil stress-strain curve is demonstrated in Figure 2.15. It can be seen that the non-linear behaviour is considered with the compatibility factor and bearing capacity factor both determined from plasticity analyses. The MSD methodology has been introduced, but designers need to know the lo- cation from which the stress-strain data should be obtained. Soil elements can differ in properties due to the stress history and hence can follow a different stress path upon loading. Thus, elements can have different stress-strain responses and would mobilise different shear stresses. The approach taken is to perform a weighted aver- age approach to select a representative shear strain that mobilises the required shear strength. Stress-strain data and the value of cu is obtained from the representative location and utilised in MSD. The representative location beneath a pad of diameter D is taken as 0.3D. This is very near to the centroid of the deformation mechanism - corresponding to 0.273D. If a single point is to represent the settlement properties of a small area then effects like disturbance must be minimised to ensure an accurate representation. The utilisation of a representative soil element simplifies the design of shallow foun- dations. As shown by Lehane (2003) in Figure 2.8, the majority of settlement for the full-scale footing was between depths of 1m and 2m. It is well known that the shear 49 2. LITERATURE REVIEW modulus, G, and cohesion, cu exhibit non-linear behaviour with depth. Full geotech- nical site investigations still need to be performed for construction projects to locate any possible geotechnical problems within the soil. Given satisfactory observations, the data from the representative location can be utilised. The MSD approach needs to be extended to cover soils which have layers of differing properties within the deformation mechanism. This is quite likely in most soils, given decreasing permeability and com- pressibility naturally occurs with depth. There is also the possibility of two material types within the settlement region or even thin sand layers. A possible approach to supplement these problems is to again take a weighted average of the soil properties to determine a new representative location. To verify the MSD approach, Osman and Bolton (2005) performed a limit analysis, finite element analyses and back-calculations from full-scale experiment results. A limit analysis calculation was performed to verify the consistency associated with the use of the Prandtl mechanism. The value of the bearing capacity coefficient Nc = 5.86 is only 3% higher than the classical value of Cox et al. (1961) for a smooth circular footing. This does not confirm the accuracy of using the Prandtl mechanism to describe the displacement field beneath a shallow foundation, but it is claimed to be encouraging. Although the significant soil movements in the developed deformation field do approx- imately fall within the Prandtl mechanism, there are still some movements outside this region. This implies that the soil must remain elastic beyond this purely plastic region, and this can cause problems with equilibrium. In order to provide equilibrium within the mechanism, Osman and Bolton (2005) induce significant heave adjacent to the footing. This may not be realistic with most soils and it will be shown in Chapter 4 that the actual mechanism beneath a shallow circular footing more closely resembles a cavity-expansion type mechanism. London clay was simulated in a finite element analysis by inputting a variation of shear modulus and undrained shear strength with depth. The results of the analysis showed that the significant soil movement occurred mainly within the boundary of the Prandtl mechanism - and therefore it could be used to predict displacements. The finite element analysis results had a maximum discrepancy of 10% from those predicted by MSD. This difference was a slight overestimation by MSD, which is obviously the better side on which to err. The greatest discrepancy occurred at a settlement between 30− 40 mm. This is quite a significant settlement, and it could be presumed that any structure which has undergone this magnitude of movement has already experienced 50 2. LITERATURE REVIEW problems - possibly a serviceability failure. An important point to note is that parameters of interest for analysis adopted in the finite element analysis are very similar to the experimental conditions of this research. This includes the footing diameter, the height and edge space of the soil. The mesh domain was simply chosen to eliminate boundary effects. Results of the settlement from a square footing on Bothkennar clay was analysed with an equivalent circular diameter, determined by equating the areas. Equating areas is a common practice, but Osman and Bolton (2005) claim that there is no theoretical reason for doing so. The Hansen (1970) correction factor for footing embedment, fd, was utilised and applied to the bearing capacity coefficient in order to accommodate for the embedded footing. No stress-strain data was available at the representative location, but engineering judgement was used, with the data provided at various heights to determine the parameters for the MSD analysis. Agreement with results was good up to a settlement of approximately 20 mm. A back-analysis was also performed on a footing at the Kinnegar site. Figure 2.16 shows the comparison between measured and predicted load-settlement curves. There is good agreement between results and predictions in the domain of working settlements. Osman and Bolton (2005) reason that as MSD deformation is controlled by the average soil stiffness, both triaxial compression and extension data should be taken into account. As soil deformation does not conform to either exactly, it is at the discretion of the designer to use which data and in what proportion. Soil anisotropy influences the settlement of shallow foundations on soft clay. This is due to the inclination of the major principal stress to the vertical in the zones beneath the footing. Figure 2.16 shows that the average Kinnegar site data of the compression and extension conforms well to the measured data. This would indicate that anisotropy can be considered in calculations by simply taking an average of the soil stiffness. MSD is a simple approach for the design of shallow foundations - but it has some limitations. A particular example is the analysis of double-layered soils. Also, although shallow footings are not the most efficient system for combined loads, an approach for the combination of these actions has not yet been developed. Osman et al. (2007) introduces the separate load cases of vertical, horizontal and moment loading. It is 51 2. LITERATURE REVIEW Fig. 2.16 Measured and predicted load-settlement curves for a footing at Kinnegar site (from Osman and Bolton, 2005) possible that a combined actions equation could be developed for combined load cases. The MSD approach considers the effects of non-linear non-homogeneous soil be- haviour on the settlement of shallow foundations. Osman and Bolton (2005) showed that good predictions were made in the serviceability range of full-scale test structures and finite element analyses. Importantly, the satisfaction of serviceability limits lead generally to sufficient ultimate capacity. Osman et al. (2007) compared both MSD and Atkinson’s method, and found that both methods provide results accurate to within 20% for typical working loads - determined as a fraction of the ultimate bearing ca- pacity. The concept and theoretical approach of MSD is simple and it could be adopted for the design of shallow foundations. Although it proved possible to superimpose theoret- ical load-settlement curves from MSD and non-linear finite element analyses based on identical rate-independent stress-strain relations, a check on actual deformation mech- anisms and load-settlement-time data would be valuable. The centrifuge experiments for this research were performed, accordingly, with the aim of enhancing foundation 52 2. LITERATURE REVIEW calculation methods. This validation could provide design engineers with an approach that has the support of physical modelling. 2.7 Summary This chapter has introduced some of the literature associated with shallow foundations. The two design parameters of ultimate bearing capacity and settlement have been presented. The most common method for determining the ultimate bearing capacity adopts the classical rigid-plastic mechanism. As an example, the design procedure of Eurocode 7 was given in Section 2.2 for the undrained and drained bearing capacities. A less common method for the bearing capacity utilises a cavity expansion idealisation and this approach was briefly introduced. Deformation fields observed in this research were found to resemble a cavity expansion mechanism and hence this is discussed in more detail later. As research has shown, settlement generally governs the design of shallow founda- tions and therefore a greater emphasis on this literature was provided. The components of settlement - undrained, consolidation and secondary settlement - were introduced in Section 2.3. A number of design approaches were discussed, which all assume that the soil is linear elastic. Soil, however, is non-linear even at very small strains, and therefore discrepancies between theoretical and actual results is common. Relevant laboratory and in-situ testing methods for the parameters required in the design process were also discussed. The allowable settlement of structures, which dictates the limits for shallow foun- dations, was introduced in Section 2.4. The method of precompression to remove set- tlements before the superstructure via additional load, sand drains or vacuum loading was introduced in Section 2.5. The relatively new approach of MSD, which utilises plasticity theory incorporating strain hardening, was introduced in Section 2.6. The method is relatively straight- forward and is supported by finite element analysis results. Although a sound tech- nique, finite element analysis should be coupled with a physical modelling approach. Improvements in instrumentation and developments in technology and analysis tech- niques such as Particle Image Velocimetry (PIV) (White et al., 2003) have enhanced 53 2. LITERATURE REVIEW the power of physical modelling in demonstrating the actual behaviour of soil-structure systems. Centrifuge testing is used in this research to elucidate one of the simplest such systems, a circular foundation applying load to the surface of clay and saturated sand beds. Chapter 3 now presents some of this centrifuge technology, and the experimental methods and instruments used in conducting this research. 54 Chapter 3 Modelling Techniques 3.1 Introduction There are a number of techniques which can be used to test geotechnical models. Centrifuge testing, together with Particle Image Velocimetry (PIV), provided the most appropriate method for investigating the deformation mechanisms beneath shallow foundations. This chapter presents centrifuge modelling as a technique and introduces the rele- vant scaling laws. The testing package, modelling techniques of footing loading method and PIV, and then the instrumentation used in this research are discussed. The prepa- ration methods for consolidated clay and saturated sand models is provided before the chapter is concluded with a summary of the testing programme. 3.2 Centrifuge Modelling The purpose of centrifuge modelling is to produce equivalent stresses and strains be- tween model and prototype. Testing geotechnical models at 1 − g is a quick method, but the real soil behaviour is not modelled accurately due to its stress dependent non- linearity. Therefore, this provides a qualitative, rather than quantitative, set of results. Full-scale tests offer an accurate set of results at one location, although the time re- quired, associated cost and the fact that deformation mechanisms can not be fully 55 3. MODELLING TECHNIQUES observed, make it an occasionally used method. Performing finite element analyses is currently very popular, both in academia and industry. The validity and accuracy of the results, however, depends on the use of an appropriate soil constitutive model and the selection of values for its parameters. Often over 10 soil parameters are required to be input into the constitutive model. Obtaining samples for testing in the laboratory can be affected by disturbance and therefore may not provide a true indication of the soil behaviour. In addition, finite element analyses can be time consuming and expen- sive. Although a sound technique, it should, where possible, be coupled with a physical modelling method. Centrifuge testing provided the most appropriate method for the objective of this research - to observe the deformation mechanisms beneath shallow foundations. The 10 m diameter Turner Beam Centrifuge at Cambridge University (Schofield, 1980) was utilised for this research. 3.2.1 Scaling Laws In order to scale down the dimensions, to produce a model, the gravity must be scaled up to produce equivalent stresses and strains in the prototype. An increase in gravity, g, is achieved by rotating a model at radius, r, at an angular velocity, ω in a cen- trifuge. The resultant increase in gravity, represented by the factor N , is given by the expression: N × g = r × ω2 (3.1) Schofield (1980) and Schofield (1981) performed a dimensional analysis to provide relevant scaling laws for centrifuge testing. Results relevant for this research are given in Table 3.1. Ovesen (1979) performed a number of centrifuge experiments to investigate the par- ticle size effects of circular foundations on sand. Different model sizes and accelerations were investigated which all corresponded to the same prototype size. It was found that some deviations from common results were only recorded when the foundation diam- eter to grain size ratio was less than about 15. The ratio of foundation diameter to average grain size in this research was a minimum of 350, and therefore the clay and sand beds should have behaved like a continuum - as at prototype scale. 56 3. MODELLING TECHNIQUES Table 3.1 Scaling laws for centrifuge modelling Parameter Model/Prototype Gravity N Length (physical dimensions) 1/N Area 1/N2 Volume 1/N3 Stress 1 Strain 1 Seepage Velocity N Time (consolidation) 1/N2 As shown in Table 3.1 the seepage velocity, and therefore pore pressure dissipation, occurs N times faster in the centrifuge. Dynamic centrifuge experimenters adjust the hydraulic conductivity of the soil models to solve a discrepancy in time scaling laws between dynamic events and seepage events (Stringer, 2012). This approach was used in this research to slow the process of seepage velocity, and attempt to observe consolidation due to footing loading in sand beds. Flow of fluid through the pores is governed by Darcy’s Law, as discussed in Section 2.3.4. However, the form of the equation, shown in Equation 2.17, does not include a term for the viscosity of the fluid. This is because generally the fluid is water, which has a viscosity at 20◦C of 1.0 cSt = 10−6 m2/s. The permeability term of the soil, however, is a function of both the unit weight and viscosity of the pore fluid. The velocity of flow through a soil is inversely proportional to the fluid viscosity (Scott, 1963). Therefore for a centrifuge test conducted at N − g the hydraulic conductivity can be reduced by a factor N by increasing the viscosity of the pore fluid by a factor of N . A frequently used pore fluid is hydroxyl-propyl methyl cellulose, the benefit of which is the flexibility of choosing, and ease associated with adjusting the viscosity. In addition, no safety precautions are required with use. The undrained and consolidation mechanisms beneath a footing on saturated sand was not successfully observed in this research programme, however, and this is discussed with the results from the experiments in Chapter 7. The variation in the gravity field through the depth of the model was not relevant in this research because the stress, strain and settlement were assumed and found to occur at a relatively small depth below the footing and soil surface. It was appropriate, 57 3. MODELLING TECHNIQUES therefore, to set the testing level at the surface of the soil. Calculations were performed and are shown in Table 3.2. It was determined, and used in testing, that to achieve 100− g at the surface of the soil an angular velocity of 156.7 rpm should be adopted. Table 3.2 Angular velocity to achieve testing level of 100− g at the soil surface Parameter Notation Equation Value Centrifuge Radius to Base Plate R Known 4.125 m Offset to Base of Sand d Measured 0.10 m Height of Model (Depth of Soil) Hm Measured 0.38 m Desired g-level at Surface N Designated 100-g Radius to Top Surface of Soil rt R-d-Hm 3.64 m Angular Velocity ω √ Ng/rt 16.41 rad/s Angular Velocity ω 156.7rpm The radial acceleration provided by the centrifuge will vary within a model, and is primarily a function of the centrifuge radius. A larger centrifuge radius produces a smaller radial error within a given model. The relatively large centrifuge used in this research, together with the small offset of the outer footings from the model centre, provided a negligible radial error (less than 0.2%). A slight lack of symmetry in the mechanisms for the outer footings was observed and this is discussed further later. 3.3 Centrifuge Package The significant pressures imparted during centrifuge testing require a strong test pack- age to be used (Schofield, 1980). The plane strain box adopted for this research was comprised of an aluminium U-frame, steel back plate and a Perspex window at the front. The package consisted of a detachable octagon base, to fit on the centrifuge swing, and the rectangular box. The internal dimensions of the box were 790 mm× 200 mm× 560 mm deep. Aluminium was used to manufacture a U-frame with steel used for the back and front plates during consolidation of clay. As smaller pressures are imparted on the side walls, both during consolidation and centrifuge testing, aluminium could 58 3. MODELLING TECHNIQUES be used and thus the weight of the box was reduced whilst still possessing sufficient strength. The Perspex window was mounted and sealed within a steel frame which was then bolted onto the side walls. The internal faces of the box were plated with finished and polished hard chrome for minimal friction surfaces. O-rings were used to seal the joints between all components to provide a good seal, including the capacity to hold vacuum. The base of the aluminium U-frame, and hence the box, had a grid of 81 holes which could be used for drainage, the insertion of instruments or to act as inputs for model saturation. Unused holes were simply capped with plugs. All components were cleaned thoroughly and greased before assembling the package. A photograph of the assembled package is shown in Figure 3.1 and a schematic diagram, with testing apparatus, is shown in Figure 3.2. Fig. 3.1 Photograph of centrifuge package 59 3. MODELLING TECHNIQUES Fig. 3.2 Schematic diagram of centrifuge package 3.4 Model Techniques 3.4.1 Footings and Loading Method To observe the deformation mechanisms through the Perspex window and facilitate analysis in axial symmetry, semi-circular footings were adopted. Pneumatic cylinders and solenoids were adopted as 1-D actuators for each of the footings. The solenoids and pneumatic cylinders were manufactured by Festo, a supplier of automation technology, with the cylinders having a 50 mm piston diameter and a 25 mm stroke. An aluminium rod was used to bridge the footing and the pneumatic cylinder at the centre of gravity. The large plane-strain strong box utilised in this research allowed more than one footing test, considering boundary effects and disturbance, to be performed within each centrifuge experiment. The approximate footing spacing was first found using the outline of the Prandtl mechanism - as shown in Figure 3.8. Although there was a slight overlap in these boundaries, it was confirmed during testing that there was 60 3. MODELLING TECHNIQUES no actual overlapping of the zone of influences. A consistent load of 100 kPa was chosen as the design load for two of the footings through all sand and clay tests. This load of 100 kPa was applied through the dead-weight of aluminium footings, of density 2700 kg/m3, requiring a thickness of approximately 40 mm. The benefit of using a dead-load application of pressure was that the load could be accurately determined through weighing the footing. The third footing used compressed air to apply load to the footing. Before testing, with the piston retracted, the footing was positioned just above the soil surface. Compressed air was then applied through the solenoid to the bottom of the cylinder during spin-up and consolidation to hold the footing above the soil. When ready, the air pressure was slowly reduced to the solenoid, and hence pneumatic cylinder, resulting in the footing falling under its own weight to the soil surface. The reduction in pressure was relatively slow, so as to minimise significant inertial effects sufficiently while ensuring the load was not applied gradually. The fall-height of each footing was dependent upon the amount of soil consolidation. Minimal consolidation was observed in sand samples, while up to 5 mm was seen in the soft clay models. The mechanism used to hold the footing was designed to ensure that no load was applied to the soil apart from the dead weight of the footing. A 3.5 mm diameter hole was drilled through the centre of gravity of the footing. A 5.5 mm diameter hole was then countersunk in to the bottom of the footing for a depth of 30 mm. As the countersunk depth was greater than the piston stroke it was ensured that the aluminium connecting rod could not reach the soil. An M3 bolt was then inserted through the footing and screwed in to this rod which connected to the pneumatic cylinder. The footing rested on the head of the bolt during spin-up and consolidation. Figure 3.3 demonstrates the system before loading and during the loading phase. Footing 1, approximately 11 mm thick, was rigidly fixed to the aluminium connect- ing rod via a bolt. The initial load applied was, therefore, the weight of the footing, connecting rod and piston within the pneumatic cylinder. This corresponded to 72 kPa at the centrifuge testing level of 100 − g. At equal time intervals compressed air was applied to the top of the pneumatic cylinder to increase the footing load applied. Foot- ing loads were, initially, applied to the soil independently - about 15 minutes apart to ensure no combined movements were observed. Later tests were conducted, however, by loading the outside footings (footings 1 and 3) simultaneously, with the middle 61 3. MODELLING TECHNIQUES Fig. 3.3 Dead-weight system used to hold the footing above the surface and to apply the load footing (footing 2) loaded 15 minutes later. A summary of the footing sizes, loading methods and magnitude of load are shown in Table 3.3. Table 3.3 Footing load and application method Footing 1 2 3 Model D 100 mm 50 mm 100 mm Prototype D 10 m 5 m 10 m Load Type Pneumatic Dead Load Dead Load Load / kPa 70 → 140 → 280 100 100 To ensure and verify that no load was applied from the connecting rod or M3 bolt during testing a load cell was placed in the connecting rod. This could also be used to quantify any friction occurring on the Perspex-footing interface. Although the footings were placed relatively tightly against the Perspex window, both footing and Perspex surfaces were cleaned and greased before package preparation. Channel section beams were utilised as footing gantries for mounting the pneumatic cylinder and solenoid. A photograph is shown in Figure 3.4. 62 3. MODELLING TECHNIQUES Fig. 3.4 Assembled beam showing the loading method components 3.4.2 Particle Image Velocimetry Particle Image Velocimetry (PIV) is a velocity-measuring technique with original appli- cations in the field of experimental fluid mechanics (Adrian, 1991). White et al. (2003) describes the modification implemented to apply the theory to geotechnical testing. PIV operates by tracking the texture of a soil within a patch through a series of images - and hence tracking the soil movement. PIV requires control markers on the Perspex window to turn the image space into an object space, and thus physical dimensions. The performance achieved by PIV exceeds previously listed methods and the precision and accuracy is comparable to local instrumentation (White et al., 2003). The theory is explained in White et al. (2003) but a brief explanation is now provided. The initial stage of the PIV process assigns a mesh of patches to the first digital image. A search patch in the second image is examined to find the displacement of the test patch. A cross-correlation is then performed from which the highest correlation, also termed degree of match, corresponds to the displaced position. Computational time is reduced by conducting the correlation process in the frequency domain by taking the Fast Fourier Transform of each patch and then following the convolution 63 3. MODELLING TECHNIQUES theorem. The location of the correlation peak is established to sub-pixel precision through the fitting of a bicubic interpolation around the highest integer peak. This process is repeated for all test patches and all images specified by the user. This procedure is demonstrated in Figure 3.5. Fig. 3.5 Patch from image 1 being searched for in image 2 (White and Take, 2002) Control markers, with known object-space coordinates (x, y) are used to transform image-space results in to real physical space. As the control markers on the Perspex window do not move, any distortions are quantifiable. Each source of image distortion discussed in White et al. (2003) is explicitly modelled to yield a 14-parameter trans- formation, examples of which include refraction through the Perspex and movement of the lens due to elevated g-levels. To determine the location of control markers in object-space requires the use of a certified photogrammetric reference field - a Mylar card, in this case. Multiple- threshold centroiding is used to locate the centroid of the dots, both on the card and the control markers. This, in turn, provides the image-space coordinates of the control markers. The Mylar card was not of sufficient size to capture the entire Perspex win- dow in one photograph and therefore an overlapping procedure had to be performed. Photographs were taken ensuring that a common row or column of control markers 64 3. MODELLING TECHNIQUES Fig. 3.6 Mylar card and control markers required for the transformation procedure was in each photograph. Adjacent photographs capturing a coincident row are demon- strated in Figure 3.6. Using the rotation equation shown in Equation 3.2, the common control markers were used to ensure the angle between markers was the same between each image. x′ = x.cos(θ) + y.sin(θ) y′ = y.cos(θ)− x.sin(θ) (3.2) 65 3. MODELLING TECHNIQUES 3.4.2.1 PIV Performance The performance of PIV can be assessed by precision, accuracy (both positional and movement) and resolution. The random difference between multiple measurements of the same quantity represents the precision. Accuracy is defined as the difference between a measured quantity and the true value. The smallest interval obtainable in a reading is the resolution (White et al., 2003). The precision of PIV is a strong function of the patch size, L, and through an investigation an expression was derived empirically, as shown in Equation 3.3. A decrease in the precision was also evident with decreasing texture. White et al. (2003) indicate that sand has its own sufficient texture for PIV - given the appropriate lighting. Clay, however, has a uniform brightness and therefore no natural texture, requiring the use of coloured sand on the front face. The face of each footing was also textured with paint. ρpixel = 0.6 L + 150000 L8 (3.3) The accuracy of PIV is governed by the photogrammetric calibration procedure. Therefore, if the centroiding process yields incorrect true coordinates of the control markers, the transformation from image-space to object-space will not be valid. The use of the Mylar card was examined by Take (2003) and given due care will provide accurate results. White et al. (2003) report that the accuracy, precision and resolution of PIV are all an order of magnitude higher than previous-image based deformation methods and are comparable to local instrumentation. For verification and validation, lasers were also used to measure footing displacement. These were mounted above the footing as shown in Figure 3.4. The cameras utilised initially in testing were Canon Powershot S80 8-megapixel units. These were replaced with Canon Powershot G10 14.7-megapixel cameras for the later tests. One camera was used for each individual footing which prevented the need for meshing photographs together for analysis of the results. The S80 cameras had an image space field of view (FOV) of 3264 × 2448 pixels while the object space FOV was designated to 290 × 218 mm for all three footings. The G10 cameras had 66 3. MODELLING TECHNIQUES better resolution, providing an image space FOV of 4416 × 3312 pixels and an object space FOV of 350× 262 mm for 100 mm diameter footings, and 290× 218 mm for the 50 mm footing. The object space dimensions, and hence camera location, were chosen to include all regions considered in the Prandtl mechanism shown in Figure 2.14. The standard patch size utilised in this research was L = 50 pixels for the 8- megapixel cameras and L = 128 pixels for the 14.7-megapixel units for which the errors, according to Equation 3.3, are 0.012 and 0.005 pixels respectively. Given the image and object space fields of view this corresponds to the theoretically attainable preci- sion of 0.001 mm (290 mm/3264 pixels× 0.012 pixels) for the S80 8-megapixel units. A summary for both the S80 8-megapixel and the G10 14.7-megapixel cameras used, field of view for particular footings and the corresponding theoretical precision are given in Table 3.4. Table 3.4 Camera, Field of View (FOV) data and theoretical precision Camera Footings FOV (pixels) FOV (mm) L (pixels) Precision (mm) S80 All 3264× 2448 290× 218 50 0.001 G10 D = 50mm 4416× 3312 290× 218 128 0.0003 G10 D = 100mm 4416× 3312 350× 262 128 0.0004 Table 3.4 portrays a very precise process based on the small values of theoretical precision. This value, however, is for a single displacement and hence the error accumu- lates as the process is performed through a series of displacements. The actual precision will also be affected by the soil texture, quality of lighting during photograph captur- ing and control marker location. White et al. (2003) dictates that multiple-threshold centroiding determines the location to be found to accuracy better than 0.1 pixels re- sulting in a precision of 0.01 mm. In general movements smaller than this value were ignored. Lighting was provided by way of two 240 V, 75 Watt bulbs adjacent to the camera gantry. Slight glare was observed in photographs but not within the area of interest. Texture was added to the front face of each footing by coating them with white paint and then, once dry, spraying from a distance with black spray paint. This provided sufficient texture for the footings to be tracked by PIV. 67 3. MODELLING TECHNIQUES 3.5 Instrumentation and Data-Acquisition All instruments used in this research were calibrated appropriately before and after each experiment. Typical calibration factors for instruments used are shown in Table 3.5, followed by a brief explanation of each. Table 3.5 Typical instrument calibration factors Instrument Calibration Factor Laser 3.1 mm/V Load Cell 900 N/V MEMS Accelerometer 55 g/V Pore Pressure Transducer 85 kPa/V Standpipe Pore Pressure Transducer 140 kPa/V 3.5.1 Lasers Lasers were used to measure the displacement of each footing for the testing pro- gramme. Baumer Electric distance sensors (OADM 12I6430/S35A) are class 2 lasers with a range of 16 − 26 mm, a resolution of 0.002 − 0.005 mm and a linearity error of 0.006 − 0.015 mm. Due to the small working range, mounts had to be manufactured to position the lasers just above each footing, as seen in Figure 3.4. Laser data was used together with footing PIV results to determine the exact loading time and also to verify the magnitude of displacement. 3.5.2 Load Cell Load cells were adopted to verify the load being applied to the soil. Friction on the footing-Perspex interface could cause discrepancies between theoretical and measured applied loads. Miniature diaphragm load cells (F259) manufactured by Novatech, with a 1 kN capacity, were utilised for all three footings. The load cells have a linearity error and hysteresis of ±5 N, a repeatability error of ±1 N and a creep change (after 20 minutes) of ±0.2% of the applied load. The load cells had threaded ends and could thus be screwed in to the connecting rod, these are also shown in Figure 3.4. 68 3. MODELLING TECHNIQUES 3.5.3 MEMS Accelerometer A MEMS Accelerometer was attached to the exterior face of the back plate at the level of the soil surface. The surface was chosen to validate and verify the desired centrifuge velocity based on the 100 − g gravity level as demonstrated in Table 3.2. The accelerometer (ADXL193), manufactured by Analog Devices, is for a single axis with a capacity of 120− g and a non-linearity of 0.2%. The accelerometer was coated with Araldite® and attached to the package using superglue. 3.5.4 Pore Pressure Transducer Pore pressure transducers were used within soil models to observe the static and excess pore pressure as a result of the footing loading. High performance miniature pressure transducers from Druck (PDCR81) were utilised with 1 bar capacity at shallower depths and 7 bar at greater depths. Protective porous stones were used in all tests. These are described along with the insertion methods and locations in Sections 3.6.1.2 and 3.6.2.3. 3.5.5 Standpipe Pore Pressure Transducer A pore pressure transducer was used at the base of the standpipe for clay tests to monitor the water table for the duration of the experiment. The general purpose pressure transducer manufactured by Druck (PDCR810) has a capacity of 7 bar with a combined non-linearity, hysteresis and repeatability of ±0.1%. The transducer had a male threaded end, so it was simply screwed in to the base of the standpipe. 3.5.6 Data Acquisition Instruments used in experiments were plugged in to a junction box on the package which was then connected to a central computer located on the beam centrifuge. An internal data acquisition card connected the beam centrifuge computer to the control room acquisition system. DASYLab© software was used for data acquisition and recording during testing. 69 3. MODELLING TECHNIQUES Data was recorded from all instruments at a frequency of 50 Hz to ensure the immediate undrained behaviour of the footing load was recorded. The cameras were each attached to separate computers on the beam centrifuge via a USB connection. Access to these computers was performed by remote access from the control room. PSRemote© was utilised to control the operation of the cameras in the centrifuge control room during testing. The time lapse feature within the soft- ware was utilised to take photographs automatically at desired time intervals. The smallest achievable interval between capturing of photographs using this feature was approximately 5 seconds. The techniques, instruments and equipment utilised in this research have now all been introduced. Preparation of the soil body, both sand and clay, required careful and repeatable methods. These are now described. 3.6 Model Preparation Soil body preparation was performed using easily repeatable methods to ensure that consistent results were produced. The properties of the sand grains and clay particles are given along with associated methods. 3.6.1 Clay Tests 3.6.1.1 Clay Properties Polwhite E clay was used in this research for clay tests. It is a high quality, medium particle size kaolin which is produced from deposits found in the South West of England. It has the properties shown in Table 3.6. Vyon sheets and filter paper were placed at the base of the package to prevent base plug blockages. A drainage layer was provided through a dense layer of Fraction E sand - poured for a depth of 30 mm using the automatic sand pourer (described in greater detail in Section 3.6.2.2). Double sieving, a small nozzle and a drop height of 630 mm ensured that a minimum relative density of 88% was achieved. This corresponds to a 70 3. MODELLING TECHNIQUES Table 3.6 Polwhite E clay properties (IMERYS, 2008) Property Value 300 mesh residue 0.05% maximum ≥ 10µm 35% maximum ≤ 2µm 25% minimum Specific gravity 2.6 pH 5.0 Surface area 8 m2/g Oil absorption 33g/100g Water soluble salt content 0.15% SiO2 50% Al2O3 35% unit weight of γ = 15.6 kN/m3. Fraction E sand was supplied by David Ball Group plc and the properties are given in Table 3.7 (Haigh and Madabhushi, 2002). Table 3.7 Properties of Fraction E sand (Haigh and Madabhushi, 2002) Property Value emin 1.010 emax 0.555 Gs 2.65 D10 0.095mm D50 0.14mm D90 0.15mm A layer of filter paper was placed on top of the sand drainage layer to prevent clay and sand mixing. Before pouring in the clay slurry the sand was saturated slowly by means of a standpipe and inlets in the base of the box. For workability, grease was applied to all internal walls. The Polwhite E clay powder was added to a mixing tank with water at a 100% moisture content level (approximately twice the liquid limit). Vacuum was applied to the tank and mixing only took place while under maximum attainable vacuum. The clay slurry was then carefully transferred into the centrifuge package, ensuring no air bubbles were trapped in the clay. The clay slurry was poured to a height 71 3. MODELLING TECHNIQUES of approximately 750 mm, through the use of a package extension. To prevent clay particles escaping around the piston a layer of filter paper and vyon was again used. The piston provided a self weight of 2.2 kPa to the soil. After being left with this load applied for a few days, the centrifuge package was placed in a computer controlled consolidometer and load applied to the clay - approximately doubling the load every three days up to the required level - as shown in Figure 3.7. Fig. 3.7 Computer controlled consolidometer with computer to monitor pore pressures 3.6.1.2 Instrumentation The insertion of pore pressure transducers in the clay models required the use of the grid of holes in the package base. At a basic level the procedure required a small hole to be drilled, the PPT inserted and then the hole back filled with clay slurry. Ensuring that the clay returned to equilibrium, and no weak spot was present behind each PPT, required the insertion to be performed before the consolidation load was doubled for the last time - thus at 70 kPa for the 140 kPa consolidation pressure and at 250 kPa for the 500 kPa test. A prediction, therefore, had to be made as to the 72 3. MODELLING TECHNIQUES final height of the clay based on the previous consolidation data, thus the λ-line. The first line of holes in the base of the box was 35 mm from the front face. Therefore, to be able to obtain a reasonable pore pressure reading and the desired location of 15 mm from the Perspex face (to ensure no effect was observed from the Perspex), the PPTs had to be installed at an angle. The design of this centrifuge package was performed with the task of PPT installation considered. For this, a stand with a gear winding handle was fabricated. Another gear was then simply bolted on to the side of the centrifuge package. This allowed work to be carried out on the package at any angle. Ceramic stones were fitted in front of the sensing face of each PPT for protection from clay particles. To ensure that a quick response was observed to changes in pore pressure, the ceramic stones were saturated. This was achieved by inserting the PPT and stone into a chamber, applying a vacuum and then submerging them in water and applying a high positive pressure. Each step was left for a few hours and the whole procedure repeated to ensure no air was trapped in the stone, and hence providing a quick response time. Pore pressure transducers were placed at a depth corresponding to one half of the footing diameter of the footing. This depth also matched with the base of the active and passive triangles in the Prandtl mechanism, as shown in Figure 2.14. A map demonstrating the desired locations is shown in Figure 3.8. In addition a PPT was placed at sample mid-depth to assist in monitoring the static pore pressure within the model. Fig. 3.8 Pore pressure transducer location map and zone of influences Using the known PPT depth, a length from the package base and the required insertion angle into the package were able to be determined. The package was placed 73 3. MODELLING TECHNIQUES at the appropriate angle using a combination square and protractor, approximately 4◦. A horizontal hole was then carefully drilled and excavated using a hand drill - all confirmed with a spirit level. The PPT was placed in the hole and then back filled by way of injection with a clay slurry. The package base was sealed around the PPT cord using a rubber washer. This whole process is demonstrated in Figure 3.9. Once all transducers were inserted the clay was returned to the consolidometer pressure that was loaded before removal, and left to equilibriate before increasing the load to the final level. Fig. 3.9 Procedure required for inserting each PPT through the base of the box Once the maximum pre-consolidation pressure was reached and all excess pore pressures dissipated, the unloading proceeded in increments of no more than 80 kPa. 74 3. MODELLING TECHNIQUES After each load reduction the clay was allowed to swell. The final unloading step, however, from 60 kPa to 0 kPa was performed without allowing any water to enter the model. Completing the final step of unloading was performed knowing that no air would enter the clay. The induced effective stress on the model helped reduce centrifuge consolidation time. The centrifuge package was removed from the consolidometer and final preparations were made. The front and back plates were removed in order to cut and level the clay surface to the required depth from the top of the package. Coloured sand was gently blown on to the front surface of the clay to provide the required texture for PIV analysis. The back plate and Perspex front were then greased before being bolted back onto the package. To prevent drying of the surface, cling film was placed on the clay. Where footing load occurred semi-circles were cut out, of a size approximate to the footing diameter. A larger semi-circle of cling film was greased and placed at these locations. This ensured that the clay surface did not dry out and that no resistance was present through friction between the cling film layers upon footing loading. The clay sample was now ready for the remaining package components to be in- stalled and for the centrifuge test to be conducted. 3.6.2 Sand Tests 3.6.2.1 Sand Properties The sand used for tests on saturated sand was Hostun sand which comes from a quarry in Droˆme, France. The properties of Hostun sand are shown in Table 3.8 (Chian et al., 2010; Mitrani, 2006). Table 3.8 Properties of Hostun sand (from Chian et al., 2010; Mitrani, 2006) Property Value emin 0.555 emax 1.010 Gs 2.65 D50 0.34mm 75 3. MODELLING TECHNIQUES 3.6.2.2 Sand Pouring Sand test models were prepared using the automatic sand pourer (Zhao et al., 2006). The automatic sand pourer has the ability to achieve a uniform density with the benefit of repeatability. Relative density calibration tests were performed to ensure appropriate values were achieved in the centrifuge package. The magnitude of movement due to footing load was small on the sand tests, and so tests were conducted on a model with the reasonably low relative density of 49%. In order for saturation to occur in a uniform manner up through the model, flow was promoted initially in a horizontal direction rather than upwards. This was performed by adding a layer of gravel, poured by hand, to ensure a loose layer, for a depth of approximately 30 mm. Sheets of filter paper were placed either side of this gravel layer. The package was then moved to the sand pourer room and made level before starting. Angle sections, made from thin aluminium sheet, were placed around the perimeter of the box opening to ensure that only the sand which was being poured within the model area entered the box. To provide additional texture for the PIV process some Hostun sand was dyed blue using fabric dye. This blue sand was mixed with ordinary Hostun sand in a ratio of approximately 1:10. The coloured Hostun sand mix was only used in the area analysed in the process of PIV - the top 250 mm of the package. The sand used was weighed before and after filling the hopper for accurate property determination. Appropriate health and safety precautions were taken during all sand pouring activities. Figure 3.10 shows the package during the process of sand pouring. 3.6.2.3 Instrumentation Similar to clay tests, the same layout of pore pressure transducers was adopted for sand tests, as shown in Figure 3.8. In addition to the eight below the footings, two transducers were placed at the base of the box to measure and monitor the static head. Sintered bronze stones were fitted to each PPT to prevent the entry of sand in to the transducer chamber. Owing to the more porous nature of these stones, no saturation was required. The pore pressure transducers were placed at the appropriate height 76 3. MODELLING TECHNIQUES Fig. 3.10 Package during the process of sand pouring in the sand pouring room and location during the sand pouring process - presenting a far less rigorous procedure than clay tests. Again, a nominal distance of 15 mm was used between the Perspex and PPT sensing face. Figure 3.11 shows the placement of three such transducers. Upon completion of sand pouring, the surface was levelled off to ensure a horizontal surface. Preparations for saturation then proceeded. 3.6.3 Saturation Saturation of sand models was achieved by controlling a pressure difference between the sand model and the methyl-cellulose holding tank. The methyl-cellulose solution was prepared during the process of sand pouring. 77 3. MODELLING TECHNIQUES Fig. 3.11 PPT placement in the sand for a 100 mm diameter footing 3.6.3.1 Methyl-cellulose Preparation The use of methyl-cellulose as a pore fluid has been discussed in Stewart et al. (1998). The viscosity is determined using: v20 = 6.92C 2.54 (3.4) where v20 is the kinematic viscosity at a temperature of 20◦C and C is the concentration of the methyl-cellulose solution in percent. As previously introduced, the viscosity of pore fluid for testing should be a factor N greater than the viscosity of water. The viscosity of water at 20◦C is 1.0 cSt = 10−6m2/s and thus a viscosity of 100 cSt was utilised. Methyl-cellulose solution was produced by adding the appropriate weight of methyl- cellulose powder to de-aired water. A cap full of disinfectant was also added to the mixture to prevent the growth of any bacteria. In addition, about 50 mL of a surfactant, in this case rinse aid, was added to reduce the surface tension associated with such a viscous fluid. It was confirmed through testing, that the addition of this liquid did 78 3. MODELLING TECHNIQUES not affect the viscosity of the methyl-cellulose solution. The mixture was stirred under vacuum for about 2 days, or until it was evident no lumps of powder were present. Approximately 25 L of methyl-cellulose solution was required for each sand test. 3.6.3.2 Saturation Procedure Saturation was performed using the CAM-Sat system outlined in Stringer and Mad- abhushi (2009) and Stringer et al. (2009). At a basic level, saturation was performed under vacuum with a driving head for the fluid provided through a pressure difference applied between the centrifuge package and the methyl-cellulose tank. To prepare the model for saturation the instrument cables were taped to the inside package faces above the sand. The top of the package was cleaned before using a greased o-ring to again provide a good seal. The package lid was then bolted on. The package lid had a bulkhead connector with valves for both vacuum and carbon dioxide lines, a pressure gauge (capable of measuring both positive and negative pressures) and a safety valve. Four methyl-cellulose inlets from the tank were connected at equal intervals in the base of the box. Carbon dioxide (CO2) was used to flush the sand models. The benefit of using carbon dioxide is that it is more soluble in water than air, and hence the degree of saturation can be increased (Lacasse and Berre, 1988). The vacuum pump was switched on, the valve on the package lid opened and by incrementally adjusting the vacuum regulator, the model was allowed to reach the highest attainable vacuum pressure, at least −90 kPa, and then left for approximately an hour. The pressure was then returned to atmospheric, or slightly above, through opening the carbon dioxide valve. This was performed at a steady pressure so as to not produce a depression in the sand surface beneath the inlet point. For safety purposes a 0.5 bar gauge safety valve was fixed to the lid to ensure no significant positive pressure could be applied to the package. The process of vacuum pressure followed by carbon dioxide was repeated to ensure a thorough flushing of the package. Once vacuum pressure was again reached the saturation was ready to commence. The valve at the base of the methyl-cellulose tank was opened allowing methyl-cellulose to flow in to the base of the package. Figure 3.12 shows the package in the saturation room ready for the process to commence. The aim of saturating a soil sample is to achieve full and perfect saturation, Sr = 1, 79 3. MODELLING TECHNIQUES Fig. 3.12 Package ready for saturation 80 3. MODELLING TECHNIQUES while causing minimal disturbance to the soil. Vacuum systems are used because it is possible to achieve a higher degree of saturation. The introduction of a pressure difference, by adjusting the vacuum level on the methyl-cellulose tank, subjects the box to a hydraulic gradient and hence flow results. The CAM-Sat system monitors the mass flux, and adjusts the pressure difference accordingly, in order to maintain the designated flow-rate range. If the flow rate is too significant, as a result of a high hydraulic gradient, the soil can undergo disturbance - including liquefaction. Liquefaction occurs when the soil effective stress, σ′, falls below zero. Using Terza- ghi’s theory of effective stress, σ′ = σ − u, the pore pressure, u, must be less than or equal to the total vertical stress, σ. Ignoring head losses along the methyl-cellulose pipes, liquefaction can be prevented if the maximum head is restricted to γdryH. Both these parameters were known through the sand pouring operation. A relative density of 49% was achieved, resulting in γdry = 14.5 kN/m 3 for a depth of sand H = 0.35 m, permitting a maximum head of 5 kPa. As CAM-Sat had not yet been successful in achieving saturation of such a high viscosity model, trial saturations were performed. The observed flow rate was well below typical values for lower viscosity models, achieving approximately 120 g/hour. The system, therefore, continued to increase the pressure difference between the tank and package, with pressures recorded far greater than the calculated safe value of 5 kPa. To this end, a pressure difference limiter was placed within the CAM-Sat system. Friction losses along the pipes were not quantified but a pressure difference limit of 8 kPa was never exceeded. As a visual monitor within the package, pipes were placed in the back corners with the bottom open in the gravel layer and the top sitting above the sand surface. This portrayed the pressure difference actually applied at the package and acted as backup for the maximum adopted pressure difference. No methyl-cellulose solution flowed out of the top of the pipe and hence the maximum safe head was not surpassed. Figure 3.13 shows the successful saturation flux rate of test S-2. The saturation was ceased by closing the valve at the base of the tank when approx- imately 10 mm of methyl-cellulose was on top of the soil surface. The methyl-cellulose was left on the surface until shortly before the test, when a small amount was removed - leaving a level of approximately 8 mm above the surface. This ensured that, due to the curvature which results from centrifuging, the methyl-cellulose, and therefore the water table, was coincident with, if not slightly above, the sand surface at the centre 81 3. MODELLING TECHNIQUES Fig. 3.13 Mass flux rate of successful saturation of the box. Figure 3.14 shows a partially saturated package and the final saturated model. The saturated sand model was now ready for the experiment apparatus to be in- stalled and the centrifuge test conducted. Fig. 3.14 Package during saturation and the completely saturated model 82 3. MODELLING TECHNIQUES 3.6.4 Package Completion and Loading All discussed instruments and components were bolted on to the footing beams and front and back plates. All loose cables were cable tied to ensure there were no loose components during the test. Balance calculations were completed for the centrifuge with an allowable tolerance of ±2 kg. A photograph of the package just before loading is shown in Figure 3.15. 3.7 Test Procedure Spin up of the centrifuge to the testing level of 100 − g was performed in 20 − g increments. Photographs and readings were recorded at each stage. Clay models were left for approximately 3 hours to allow the clay to consolidate. A constant water supply was provided to the clay through a standpipe with an overflow coinciding with the clay surface, thus maintaining the water table at the clay level. Progress was monitored through both the pore pressure transducers in the model and the standpipe. Saturated sand tests were left for roughly 30 minutes to allow any consolidation to take place. Before reducing the compressed air pressure to the appropriate footing solenoid, a time lapse was initiated in PSRemote©. Table 3.9 shows the parameters used in the time lapse setup, including the interval, duration and number of photos. During clay tests the pneumatic footings had each load applied and observed for about 2 hours. Time lapse settings were simply repeated for each increase in load increment. This corresponded to a total of about 6 hours of loading and data for both the pneumatic and dead-load footings. In sand tests, each pneumatic footing load was applied for 20 minutes, resulting in 60 minutes of data for both footing load-types. Resultant total centrifuge test durations were approximately 2 hours for sand and 10 hours for clay. Upon completion of the loading sequence the air pressure was returned to all solenoids, to raise the footings, and the centrifuge was spun down. The package was removed from the centrifuge and the soil excavated. 83 3. MODELLING TECHNIQUES Fig. 3.15 Centrifuge package ready for testing 84 3. MODELLING TECHNIQUES Table 3.9 Photograph capturing times used in centrifuge experiments Soil Body Footing Load Interval (seconds) Duration (minutes) Number of Photos Total Time (minutes) Clay Dead 5 15 180 15 10 15 90 30 20 30 90 60 30 30 60 90 300 270 54 360 Pneumatic1 5 5 60 5 10 5 30 10 60 50 50 60 300 60 12 120 Sand Dead 5 15 180 15 20 45 135 60 Pneumatic1 5 5 60 5 20 15 45 20 3.8 Summary of Tests As part of this research, 5 successful centrifuge tests were performed. Table 3.10 portrays a summary of the tests, giving the footing diameter and the length of data that was recorded. The ID assigned for each test, SC, FC and S, represent soft clay, firm clay and sand respectively. 3.8.1 Test Problems As with any centrifuge testing programme, problems were incurred during this research. Some of these were: ˆ The Canon Powershot S80 8-megapixel cameras have been used extensively in centrifuge tests by researchers at the Schofield Centre. As a result, issues devel- oped with the cameras during this research programme resulting in some footing 1The quantities shown were used for each loading increment. The time lapse sequence was re- started with each load increase. 85 3. MODELLING TECHNIQUES Table 3.10 Summary of centrifuge tests Sample ID Soil Type Footing Diameter - Test ID (Duration of Data) 1 (left) 2 (centre) 3 (right) SC-1 Soft Clay 50 mm - 1B (Bearing test) 50 mm - 1A (5 hours) 100 mm - 1C (5 hours) FC-1 Firm Clay Camera fault 50 mm - 2A (1 hour) 100 mm - 2B (1 hour) FC-2 Firm Clay 100 mm - 3A (6.5 hours) 50 mm - 3B (1 hour) 100 mm - 3C (6.5 hours) S-1 Sand 100 mm (1.5 hours) 100 mm (High-speed camera) 50 mm (1 hour) S-2 Sand 100 mm (1.5 hours) 100 mm (High-speed camera) 50 mm (1.5 hours) data not being captured - as shown in Table 3.10. These units were eventu- ally replaced with Canon Powershot G10 14.7-megapixel cameras which provided higher quality photographs and, in turn, better PIV results. ˆ An attempt was made to capture the undrained mechanism for shallow founda- tions on sand through the use of a high speed camera - the MotionBLITZ® Cube from Mikrotron. This camera has a 3-megapixel capacity at up to 525 frames per second. The low resolution, however, resulted in the small movements observed not being detectable by the PIV process. ˆ Some instruments, particularly pore pressure transducers, would occasionally cal- ibrate adequately but then drop-out during centrifuge testing. Connections could be checked, but ultimately, the instrument data was not recorded. ˆ The centrifuge itself experienced some problems during testing. The most serious occurred in test FC-1 where problems with the bearings resulted in the test being stopped prematurely. Only undrained and short-term consolidation data was able to be observed in this test. ˆ The most problematic aspect of this research was in the process of saturating sand models. As discussed, the CAM-Sat system had not yet been successful with such high viscous saturation. Trial runs saw models destroyed through liquefaction. Test S-1 experienced a few points of local liquefaction and so saturation was completed using a top-down method. The surface of the sand was inundated with methyl-cellulose solution and once vacuum was removed the solution was 86 3. MODELLING TECHNIQUES drawn down through the model. As the small problem areas occurred well below the surface and area of interest for each footing, the centrifuge test was carried out and results analysed for this research. All the knowledge accumulated through this research was implemented for test S-2, culminating in a successful model saturation. The procedure used has been presented. 3.9 Summary This chapter has presented the apparatus developed and the instruments and modelling techniques used to perform this research. The technique of centrifuge modelling was introduced and its appropriateness for the investigation of deformation mechanisms beneath shallow foundations was presented. The centrifuge strong package used was introduced in Section 3.3 and the development of 1-D actuators to control footings and apply the loads was described in Section 3.4.1. The technique of PIV used to observe deformation mechanisms was described in Section 3.4.2, and instruments utilised to measure and validate relevent parameters in Section 3.5. Preparation of both clay and saturated sand models were outlined in Section 3.6. Finally, the centrifuge test procedure was outlined in Section 3.7 and a summary of the testing programme with some of the problems incurred was given in Section 3.8. Analysis of observations recorded from the described instruments and PIV results from the captured photographs was performed following each centrifuge test. The results from tests on clay and saturated sand bodies have been presented separately, in Chapters 4 and 7 respectively. 87 3. MODELLING TECHNIQUES 88 Chapter 4 Clay Results 4.1 Introduction The development of technology and techniques such as Particle Image Velocimetry (PIV) has provided the opportunity to observe actual deformation mechanisms beneath shallow foundations. These methods were utilised in a series of centrifuge experiments performed on the 10 metre diameter beam centrifuge to investigate shallow circular foundations. This chapter presents results from the tests performed at a nominal 100− g imposed at the surface of clay soil bodies. A sample set of results is given for a pneumatically loaded footing on firm clay, and is supported by some selected results from a dead-weight footing on soft clay. The processes of unloading the clay from the consolidometer, and spin-up and self-weight consolidation in the centrifuge are initially presented. Following this, the approach used to determine the undrained penetration and the corresponding observed mechanism are given. Consolidation and creep settlements are also investigated and presented. Using the results from a bearing capacity test that was performed as valida- tion, a back-analysis is performed to demonstrate good correlation between actual and predicted settlements. A model for creep is introduced, which allows the consolida- tion and creep settlements to be quantified. These results are shown to be dependent on the undrained penetration, and therefore this chapter is concluded with a further Parts of this chapter are currently under review for Ge´otechnique as McMahon, B. T. and Bolton, M. D. (2012) Circular foundations on clay 89 4. CLAY RESULTS investigation into the undrained settlement, which is then used as the basis for a cavity- expansion type model. 4.2 Clay Unloading After consolidation was complete up to 140 or 500 kPa the soil was unloaded in incre- ments of no more than 80 kPa, during which the soil was permitted to swell. The final unloading stage from 60 kPa to zero was performed without allowing water to enter the clay. This final step is demonstrated in Figure 4.1. It can be seen that the suction is effectively 60 kPa for about the first twenty minutes, before some suction is lost. Not all water can be removed from in and around the piston, and obviously the filter paper and porous plastic (vyon) hold some water. This results in some loss of the suction near the soil surface. In addition, the process of cutting and smoothing the surface of the clay requires the use of some water, which also reduces the induced effective stress. This did not cause any problems for the clay - it simply increased the required self-weight consolidation time in the centrifuge slightly. Fig. 4.1 Suction and effective stress induced during the final stage of unloading 90 4. CLAY RESULTS 4.3 Spin-up The centrifuge was spun-up in approximately 20 − g increments. Once the centrifuge reached the testing level of 100−g the model was allowed to spin for the clay to undergo self-weight consolidation. This process continued for about 3 hours, or until the pore pressures could be seen to have significantly dissipated. Figure 4.2 portrays the spin- up process with the standpipe pressure head. Once the testing level was reached the flow rate of water into the standpipe was adjusted appropriately to ensure that it was constantly overflowing. As portrayed, after the consolidation time the standpipe head is constant and the soil is ready for testing. Fig. 4.2 Spin-up and consolidation data 4.4 Self-Weight Consolidation Results of the self-weight consolidation are presented for both soft and firm clay models for completeness. 4.4.1 Soft Clay Models A PIV analysis was performed on images taken during self-weight consolidation and is shown together with a plot showing the consolidation with depth in Figure 4.3. As 91 4. CLAY RESULTS expected, the clay consolidation at the surface is greater than that at depth. Consol- idation results from excess pore pressures generated due to the soil weight increasing as a result of the elevated g-level. The surface settlements are the greatest due to the cumulative nature of the consolidation. It is also evident in Figure 4.3 that there is a significant change in slope at a model scale depth of approximately of 75 mm. The buoyant unit weight of the clay was γ′ = 7.8 kN/m3 which corresponds to a depth of 75 mm for an effective stress of 60 kPa - the value induced after unloading. Thus the soil between the surface and a depth of 75 mm is effectively swelling during centrifug- ing, whilst all soil below is consolidating down the κ-line up until the pre-consolidation pressure of 140 kPa. Beyond this point it is normally consolidating, and hence moving along the λ-line. The radial effect of centrifuging is also evident in the edge soil profile of Figure 4.3. The PPTs were used to monitor dissipation of pore pressures due to self-weight consolidation. Figure 4.4 portrays the pore pressure from 3 PPTs positioned across the model, with the locations indicated, for the 3 hour period after spin-up which was used to consolidate the soil. A significant amount of pore pressure dissipation was observed before the footing tests were performed. Complete dissipation was not observed until about 8 hours into the test, effectively near the end of centrifuging, and therefore footing loads had to be applied with some remaining excess pore pressure still present. 4.4.2 Firm Clay Models Figure 4.5 shows the pore pressures recorded by a PPT at a depth of 55 mm below the loading area of a 100 mm diameter footing for a firm clay model. The initially negative value (u = −30 kPa) corresponds to the initial state of effective stress in the soil prior to centrifuging. As discussed, the process of unloading and working on the clay reduced the initial suction of u = −60 kPa. The stepped increase to u = 85 kPa corresponds to centrifuge spin-up, and the subsequent reduction over the next 3.5 hours corresponds to self-weight consolidation. The test was suspended briefly in order to correct a camera fault (with this time frame removed). The centrifuge was spun-up again and allowed to consolidate for 1 hour before the footing was gently lowered, with the pore pressure responding accordingly, and then dissipating once again. This was repeated twice more in subsequent loading stages effected by the compressed air piston. It could 92 4. CLAY RESULTS Fig. 4.3 Effective consolidation during spin-up of soft clay (SC-1) 93 4. CLAY RESULTS Fig. 4.4 Pore pressure dissipation during consolidation 94 4. CLAY RESULTS again be argued that the footing load was performed too early, but Figure 4.5 shows that full self-weight consolidation was only achieved after about 8 hours. Any further soil movement due to self-weight consolidation during footing loading was accounted for by performing a PIV analysis in an area unaffected by the footings (the far-field). Fig. 4.5 Pore pressure distribution for entire duration of test (model scale) Once self-weight consolidation was substantially complete the footing loads were applied. 4.5 Footing Settlement - Virgin Loading As shown in Table 3.10, three centrifuge tests were conducted on clay, with a total of 7 footing tests that provided load-settlement-time data for the virgin loading of the soil. Table 4.1 offers a summary of these tests, indicating the pre-consolidation pressure, σv,max, footing diameter, the time taken to apply the footing load to the soil and the load-test duration. The footing tests that encountered camera problems are not listed. A camera with higher resolution was used for footing test 3A (in Table 4.1) which provided more detailed images and better results from PIV. These higher resolution images, for a pneumatically loaded 100 mm diameter footing on firm clay, are therefore used as an exemplar below. These are then briefly followed by supporting results of a 50 mm footing on soft clay (test 1A). 95 4. CLAY RESULTS Table 4.1 Summary of virgin loaded centrifuge tests on clay showing footing diameter, average load and relevant timings Test σv,max (kPa) D (mm) Load (kPa) Footing Time to Apply Load (s) Total Load Duration (hrs) Test Label 1 140 50 100 Centre 0.5 5 1A 50 117 Left 2 0.01 1B 2 500 50 100 Centre 2 1 2A 100 100 Right 6 1 2B 3 500 100 72 Left 7 2 3A 50 100 Centre 5 6 3B 100 100 Right 11 6 3C 4.5.1 Firm Clay Applying load through the dead-weight method is analogous to oil or water storage tank loading. One such example is the Doris tank in the Ekofisk oil field which, when flooded, applied a significant load immediately (Bjerrum, 1973; Gerwick and Hognestad, 1973). This method provides a worst-case for foundation settlement, as the soil can not stiffen through consolidation with each loading increment. The advantage is that clear mechanisms were observable in the centrifuge over the whole timescale, from 1 second (3 hours at prototype scale) up to 6 hours (7 years at prototype scale). The procedure used for determining the undrained penetration is now presented, followed by the illustration of deformation mechanisms and a discussion of consolidation and creep. 4.5.1.1 Consistency The laser measurement of footing settlement and the load cell data were used to de- termine the zero-time of each footing. This process is shown in Figure 4.6. At approx- imately 4649 s the laser and load cell start to register that the footing has begun to fall towards the surface. As expected, each instrument simultaneously demonstrates a slight bump, and this was interpreted as the point at which the footing landed on the surface of the clay. The load cell records a bump as resistance is offered by the clay, and this pushes the load cell into compression. The laser records the bump because it is effectively falling under its own weight, so it slows and stops when hitting the surface. A zero-time was taken roughly in the middle of this bump. As readings were 96 4. CLAY RESULTS obtained at a frequency of 50 Hz it was possible to determine that the zero-time for this footing was 4649.10 s. This method was used to zero the laser, load cell and PPT data and hence determine the undrained penetration of each footing. Fig. 4.6 Raw laser and load cell data for determining the zero-time of the footing The footing faces were textured with paint to allow PIV analyses to be carried out. Analysis commenced with a run performed on the footing only. As the footing was falling from above the surface the PIV data alone could not be used to verify the undrained penetration. Following the zeroing of the laser data it was possible to compare the raw PIV results with the zeroed laser data. This is shown in Figure 4.7. It is evident that the curves match very well. Therefore, the data was matched at a specified time and hence the average footing settlement and the timings of the PIV results was determined. The settlement is considered an average as the PIV is measuring the settlement at the front while the laser is measuring near the back of the footing. Rotations should have been fairly restricted because of the Perspex window, but because footings were not pressed hard against the Perspex (for minimal friction, and to ensure that the footing could fall under its own weight) it is possible that some rotations were observed. This was observed for the 100 mm footing on soft clay and for this reason the results were ignored. This is discussed further later. Figure 4.7 also shows that after the third load increase the laser could no longer measure the movement because the footing was now beyond its 10 mm capability. 97 4. CLAY RESULTS Fig. 4.7 Matching of laser and PIV data to get average footing settlement (test 3A) Once the settlements were matched it was evident that good correlation was ob- served. This suggests that either there was no footing rotation, or the footing settled uniformly but with a slight rotation. Figure 4.8(a) shows the comparison for the entire duration of the footing loading. Figure 4.8(b) demonstrates that for the initial load- ing, over approximately 2 hours, there is a good comparison between the observations. From here, therefore, only movements recorded by the lasers are presented. 4.5.1.2 “Undrained” Penetration The immediate response of a saturated soil to a load is to resist volume change, with settlement occurring through shear. Excess pore pressures beneath the footing can be used to verify when drainage has occurred. Figure 4.9 portrays footing measurements for the first 100 seconds of testing. The load cell data shows that the average pressure to be applied to the soil was approximately 72 kPa. As a consequence of reducing the compressed air pressure that was holding the footing suspended, the footing was gen- tly lowered so that its full load took approximately 7 seconds to apply (time t0). By weighing the loading components before the experiment, this load magnitude was inde- pendently estimated, indicating that any friction between the footing and the Perspex 98 4. CLAY RESULTS (a) Footing settlement for centrifuge test duration (b) Footing settlement for the first loading phase of the centrifuge test (test 3A) Fig. 4.8 Footing settlement measured by PIV and laser data at model scale 99 4. CLAY RESULTS interface was negligible. Figure 4.9 also portrays the footing movement and associated excess pore pressure measured nominally on the footing centreline at a depth of 0.55D. An “undrained” penetration of 0.82 mm occurred once the full load was applied at time t0 and this generated an excess pore pressure of about 25 kPa. The excess pore pressure on the centreline can then be seen to rise to approximately 27 kPa at time t1. An increase of this sort was seen in all tests and can be attributed to load redistribution due to peripheral drainage (Mandel, 1953). Because transient drainage occurs more rapidly at the edge of the load, the soil beneath the edge would settle faster if the load were flexible. In the case of a rigid footing, or in Mandel’s analysis of a stiff embankment on a soft soil, the effect is to redistribute load from the edges towards the centre. This causes excess pore pressure reduction beneath the edge and an increase beneath the centre. Schiffman et al. (1969) referred to this general phenomenon as the Mandel-Cryer effect. This interpretation is supported by results from PPTs along the footing edge line, which demonstrated a far less pronounced increase in excess pore pressure. The load redistribution appears to be taking place as soon as the load has been fully applied, at time t0 = 7 s in Figure 4.9, but more regional dissipation of pore pressure affecting the PPT located at a depth 0.55D below the footing centre only became obvious after time t1 = 10.5 s. Fig. 4.9 Average footing load, settlement and excess pore pressure for the first 100 s at model scale A PIV analysis was also performed on a selected area of soil beneath the footing for the entire duration of the test. Wild vectors resulting from the position of control markers or loss of texture were removed from these results. The mechanism produced 100 4. CLAY RESULTS from PIV once the full load had been applied is shown in Figure 4.10(a). This cor- responded to a time of 10.5 seconds; 3.5 seconds after the full footing load had been applied to the soil - denoted t1 in Figure 4.9. A script developed for Matlab which interpolates vectors of movement to replace the lost wild vectors was utilised in all analyses. Also, soil movements smaller than the capabilities of PIV, as discussed in Section 3.4.2, were removed. The result is given in Figure 4.10(b). The soil movement beneath the footing is about 0.86 mm - as also observed by the laser data in Figure 4.9. Figure 4.10(b) shows that the footing is moderately rough because of the relatively small horizontal movements immediately below the footing. There is not much adjacent heave invoked, and the mechanism resembles cavity expansion rather than rigid-plastic indentation. The volumetric strain within the soil was numerically formulated from the dis- placement mechanisms using another application of PIV - geoSTRAIN8. The grid of displacements is divided into a network of constant strain triangles, each defined at the centroid. The volumetric strain is simply calculated as the sum of principal strains. Further information of the formulation procedure can be found in White (2002). Figure 4.11 shows the volumetric strain at time t1. Figure 4.11 shows there is some volumetric strain at the surface of the clay with slightly more below the right of the footing. The greater volumetric strain on this side shows that the footing did not land on the surface perfectly horizontally which may have resulted from centrifuge radial effects. As discussed, drainage at the edge of the footing occurs immediately. The interface between the footing and Perspex also made it possible for some water to move between the footing and Perspex, draining the plane of symmetry. It is also possible that some additional strain may have resulted from the inertial effect of the footing falling to the surface. It was intended that the footing loads should be applied slowly to minimise this effect. Strain information is also lost at the soil surface as no triangle can be performed for the top deformations. Irrespective of these arguments, the small and localised volumetric strain beneath the footing does confirm the mechanisms in Figure 4.10 are broadly undrained. The engineering shear strain at time t1 is given in Figure 4.12. It can be seen that there is a relative concentration of shear strains beneath the edge of the footing. Shear strains are also observed down to a depth of about 0.5D beneath the footing. 101 4. CLAY RESULTS (a) Raw data “undrained” mechanism (b) Interpolated “undrained” mechanism Fig. 4.10 Raw and interpolated “undrained” mechanisms at time t1 = 10.5 s (30 hours) for 100 mm diameter footing on stiff clay (test 3A) 102 4. CLAY RESULTS Fig. 4.11 Volumetric strain (%) at t = t1 (test 3A) Fig. 4.12 Engineering shear strain (%) at t = t1 (test 3A) 103 4. CLAY RESULTS 4.5.1.3 Consolidation and Creep The footing deformation between t1 and t2 (about 30 seconds after loading), indicated on Figure 4.9, occurs with the excess pore pressure at 0.55D beneath the centre drop- ping 4% from its peak. A PIV analysis was also performed at time t2 = 30.5 seconds and is given in Figure 4.13 along with the volumetric strain plot in Figure 4.14. Figure 4.13 shows a very similar mechanism to that of the immediate mechanism in Figure 4.10. The volumetric strains confirm that drainage has occurred at the surface adjacent to the footing and immediately beneath the footing. There is some volumetric strain at a depth of about 0.3D but it is not symmetrical. This may suggest that there was a small gap between the clay and Perspex window. Therefore, the settlement during this period may be attributed to the following mechanisms: 1. Local drainage of the soil under the edge of the footing, coupled with added total stress being carried under the centre, leading to additional undrained penetration. 2. General creep following the shear mechanism induced by footing penetration. The similarity of the mechanisms at times t1 and t2 warranted further investigation. A comparison of soil movement vector magnitude was completed and it was confirmed that each soil patch vector was simply amplified by a common factor. Figure 4.15 demonstrates that the mechanism movements were multiplied by a factor of 1.12 on average. So the combination of factors discussed above contributed an additional set- tlement equal to 12% of the undrained penetration during this time period, a rate of 26% for a factor of 10 on time. The footing settlement and excess pore pressure for the whole 2 hour loading stage is shown in Figure 4.16 and the deformation mechanism at four times throughout are shown in Figure 4.17. It is now even clearer that a continuous increase in settlement with the passage of time on a log-scale, from t1 to t2, occurred with negligible pore pressure dissipation occurring at a depth of 0.55D. It can also be seen that excess pore pressures had substantially dissipated after approximately 2000 seconds (time t3). The pore pressure then continues to fall to a value of about −2 kPa. This is attributable to the initial excess pore pressure after self-weight consolidation, shown in Figure 4.5, which had nevertheless been taken as datum during the loading event. 104 4. CLAY RESULTS Fig. 4.13 Mechanism at time t2 (30.5 s at model scale) for 100 mm diameter footing on stiff clay (test 3A) Fig. 4.14 Volumetric strain (%) at t = t2 (test 3A) 105 4. CLAY RESULTS Fig. 4.15 Comparison of mechanism between “undrained” penetration (t1) and during the consolidation and creep phase (t2) for test 3A A deformation mechanism for consolidation and creep occurring between times of 140 and 600 seconds is presented in Figure 4.18. The magnitude of movement beneath the footing is approximately 0.25 mm. The mechanism demonstrates a quasi one- dimensional compression of soil beneath the footing between these times. Fig. 4.16 Footing settlement and excess pore pressure for test 3A (model scale) 106 4. CLAY RESULTS (a ) M ec ha ni sm at 10 0 s (1 2 da ys ) (b ) M ec ha ni sm at 44 0 s (5 0 da ys ) (c ) M ec ha ni sm at 17 50 s (2 00 da ys ) (d ) M ec ha ni sm at 70 00 s (8 00 da ys ) F ig . 4. 17 D ev el op m en t of th e de fo rm at io n m ec ha ni sm th ro ug ho ut th e 2 ho ur te st of fo ot in g 3A 107 4. CLAY RESULTS Fig. 4.18 Consolidation and creep mechanism for test 3A between 140 and 600 seconds at model scale Continued footing settlement after excess pore pressures have substantially dissi- pated at t3 = 2000 s, seen in Figure 4.16, indicates drained creep of the soil due to the footing load. No drained creep deformation mechanism could be observed in the PIV analysis, however, due to the relatively small magnitude of movement, and “noise” in the data. PIV can also be used to plot the settlement with time at any location beneath the footing. Figure 4.19 shows the footing settlement together with the footing centreline settlements at depths of 0.1D, 0.25D, 0.5D, 0.75D and 1.0D. In addition, a depth of 0.3D is shown, as it represents the characteristic depth in the MSD method. Figure 4.19 demonstrates that almost all of the footing settlement occurs within a depth of one footing diameter. Using these results it was possible to normalise the centreline settlements by the footing settlement. This, therefore, represents the cumulative set- tlement of the soil with depth, and is shown in Figure 4.20. Figure 4.20 shows that approximately 40% of the undrained penetration occurs within a depth of 0.1D, but as consolidation occurs this reduces to 30% of the settlement. Approximately half of the footing settlement occurs within a depth of 0.5D and 90% of the settlement has occurred by a depth of 1.0D. These plots can be compared with those of Lehane (2003) for a 2 m × 2 m square footing shown in Figure 2.8(b). Lehane (2003) showed that slightly more than 50% of the settlement had occurred at a depth of half the 108 4. CLAY RESULTS footing width and approximately 75% of the settlement had occurred by a depth equal to the width. Caution must be observed between the comparison of square and circular footings, but a general observation indicates reasonably similar results. Fig. 4.19 Settlement plot at various depths within the soil for test 3A at model scale Fig. 4.20 Test 3A normalised and cumulative settlement with depth at model scale 109 4. CLAY RESULTS 4.5.2 Soft Clay A brief set of results is provided for test 1A (as shown in Table 4.1) for support of the results shown previously, and to show long-term results. 4.5.2.1 Consistency The same process of consistency was performed for the laser, load cell and PIV data - again showing good results. The results for the full test corresponding to 19000 s (6 years at prototype scale) are shown in Figure 4.21 for the laser movements and a PPT nominally beneath the footing edge at a depth of 0.4D Fig. 4.21 Footing settlement and excess pore pressure measured along the footing edge at model scale for test 1A Figure 4.21 shows that the load was applied far quicker then it was for the discussed pneumatic footing. The load was applied in less then 1 second, resulting in a spike in excess pore pressure which can be attributed to dynamic effects. The excess pore pressure quickly returns to the static reading. An increase in excess pore pressure is again evident and was discussed in Section 4.5.1.2. The excess pore pressure eventually falls to a value of about −2 kPa corresponding to the excess remaining due to self- weight consolidation. Early data of the average bearing pressure, footing settlement 110 4. CLAY RESULTS and excess pore pressure along the footing centreline is given in Figure 4.22. The mass of the dead-weight footings corresponded to a bearing pressure of 100 kPa and this was observed in the experiment - again indicating a negligible effect from friction. Fig. 4.22 Average footing load, settlement and excess pore pressure for the first 200 s of test 1A (model scale) 4.5.2.2 “Undrained” Penetration A PIV analysis was performed on the soil, and using the same process of cleaning vectors and interpolating the results the undrained penetration, at time t1, was found and is given in Figure 4.23. An “undrained” penetration of about 2 mm was observed - far greater than for the stiff clay. This mechanism is very similar to that observed in the stiff clay, which also shows the footing was relatively rough. A comparison of mechanisms between time t1 and t2 showed similar results. In this period, the mechanism displacements showed an increase in magnitude by 5%. This result is shown in Figure 4.24. 111 4. CLAY RESULTS Fig. 4.23 Mechanism at time t1 (3.5 seconds at model scale) for test 1A Fig. 4.24 Comparison of mechanism between “undrained” penetration (t1) and during the consolidation and creep phase (t2) for test 1A 112 4. CLAY RESULTS 4.5.2.3 Consolidation and Creep A mechanism for consolidation and creep was also determined and is given in Figure 4.25 which occurs between t2 and a time of 600 s. The movements are much less than the undrained penetration. The results are not as clear as those shown in Figure 4.18 due to the lower resolution camera, but suggest a similar mechanism of quasi one-dimensional compression. Fig. 4.25 Consolidation and creep mechanism between t2 and 600 s (test 1A) Figure 4.21 shows that the long term behaviour is for the settlement to slow down significantly. The creep rate of the soil must decrease with time and this is supported by these results. A contour plot can also be used to show the ‘effected depth’ of soil beneath a footing loading. Figure 4.26 shows a contour plot of settlement beneath this footing. It is evident that most settlement has occurred by a depth equal to one footing diameter - which was also demonstrated in Figure 4.19. 113 4. CLAY RESULTS Fig. 4.26 Contour plot demonstrating some slight heave for test 1A 4.5.3 Back-Analysis As introduced in Chapter 2, there now exists a number of correlations for soil pa- rameters formed from databases containing a number of soil test results. Previously published data on the strength and stiffness of clays, and the settlement and bearing capacity of circular footings, is first used to estimate the bearing capacity of a 50 mm diameter footing on the soft clay (test 1B), for comparison with the measured value. Having demonstrated an acceptable match, the same relationships are used to back- analyse the undrained load-penetration data of all tests - with test 3A used as the example. A model for consolidation and creep is then proposed. 114 4. CLAY RESULTS 4.5.3.1 Bearing Capacity Test on Soft Clay Ladd et al. (1977) investigated the effect of the degree of overconsolidation on the undrained shear strength through direct simple shear tests and found: cu σ′v,0 = ( cu σ′v,0 ) nc (OCR)Λ (4.1) where cu is the undrained shear strength, σ′v,0 is the vertical effective stress, nc denotes normally consolidated, OCR is the overconsolidation ratio and Λ is an empirical ex- ponent which can be taken as 0.8 but a better fit is obtained if reduced from 0.85 to 0.75 with increasing overconsolidation. For the kaolin used in these tests, Vardanega et al. (2012) quote a value of (cu/σ′v,0)nc = 0.23 pertinent to triaxial compression tests carried out at an axial strain rate of 1.2% per hour, which corresponds to a shear strain rate of 5.0× 10−6 s−1. Osman and Bolton (2005) adopted a characteristic depth z = 0.3D to provide a representative stress-strain relation for use in the prediction of undrained footing settlements following the MSD framework. A non-linear elastic finite element analysis was performed by Osman and Bolton (2005) to demonstrate that the behaviour of a circular footing on stiff clay was closely matched by MSD calculations based on a uniform soil with its properties selected for the characteristic depth. The same representative depth of 0.3D has been used in the current back-analyses. The clay in test 1 was consolidated to 140 kPa before centrifuging. Therefore, for the 50 mm diameter footing in test 1B the characteristic depth is 15 mm, offering an in-situ stress σ′v,0 = 11.7 kPa, with OCR = 12, and cu = 20 kPa from Equation 4.1. Comparative data for all the tests is given in Table 4.2. The broadly undrained mechanisms in Figures 4.10 and 4.23 show that the footing is relatively rough, and therefore a bearing capacity factor of Nc = 6.05 is adopted. This produces a theoretical bearing capacity of qult = 121 kPa, based on triaxial compression data. Figure 4.27(a) demonstrates the configuration of the load cell used during the exper- iment and shows how the load cell readings were interpreted. Figure 4.27(b) presents the results from the bearing capacity test 1B and indicates an average bearing capacity of qult = 117 kPa, which compares extremely well with the predicted value. This may be fortuitous, however, because the effects of embedment, anisotropy and strain rate 115 4. CLAY RESULTS have so far been neglected. The bearing pressure of 117 kPa is seen in Figure 4.27(b) to be acting after only 2 s, when the settlement is about 12 mm. At this depth of embedment the overburden pressure is 20 kPa, so the net bearing pressure is about 97 kPa. This corresponds to an average shear strain estimated from Equation 2.26 of 1.35 × 12/50 = 32.4%, which is far beyond the point of peak strength recorded in the triaxial tests on the same soil reported by Vardanega et al. (2012). These tests typically exhibited 30% softening at a gross overall shear strain of 15%, although it must be recognised that the actual magnitude of strain and the degree of softening are a function of localisation, whose severity depends on deformation constraints. If, notwithstanding, the shear strain rate during the bearing failure is taken as 0.162 s−1, that is about 3 × 104 times faster than the triaxial tests that were used to estimate cu. Kulhawy and Mayne (1990) demonstrated a correlation between undrained strength and strain rate for 209 undrained triaxial tests on a total of 26 clays: cu cu,0 = [ 1 + 0.1log10 ( γ˙ γ˙0 )] (4.2) This suggests that the clay strength in test 1B should be increased by a factor of 1.45, giving a theoretical net bearing capacity of 175 kPa. However, some allowance should also be made for anisotropy. Kulhawy and Mayne (1990) demonstrated that for kaolin with a plasticity index of 33% the ratio of undrained strength in triaxial tests in extension and compression typically falls in the range 0.55± 0.15. Furthermore, Osman and Bolton (2005) demonstrated that the field load- ing tests of a 2 m square footing on soft silt reported by Lehane (2003) were approxi- mately consistent with the average of the strengths measured for cores in compression and extension. It might therefore be expected that the ratio of the operational strength in a bearing capacity problem and the shear strength in triaxial compression would be about 0.77. Applying this to the previous estimate of 175 kPa for the fast penetration of the footing, the estimated bearing capacity reduces to 135 kPa. This is 39% higher than the observed net bearing pressure applied in the test, which is taken to signify 39% post-peak softening, similar to that observed in the triaxial tests reported by Var- danega et al. (2012). This back-analysis of bearing capacity test 1B must be regarded as broadly satisfactory. 116 4. CLAY RESULTS (a) Load cell readings during footing suspension and footing loading (b) Footing settlement and bearing pressure of test 1B at model scale Fig. 4.27 Bearing capacity test information 117 4. CLAY RESULTS 4.5.3.2 Undrained Penetration Using triaxial compression test data, Vardanega et al. (2012) developed the simple stress-strain model: τmob/cu = 0.5(γ/γM=2) b in the range 0.2 < τmob/cu < 0.8 (4.3) with empirical expressions for the exponent b (Equation 4.4) and for the mobilisation strain γM=2 (Equation 4.5) as functions of the overconsolidation ratio in the range 1 to 20. b = 0.011(OCR) + 0.371 (4.4) γM=2 = 0.0040(OCR) 0.680 (4.5) For 115 tests it was found, however, that b was in the range of 0.3 to 1.2 with an average value of 0.6. For ease of calculation, a value of 0.6 has been used for all subsequent calculations, corresponding to OCR ≈ 21. Values of nominal undrained strength and non-linear stiffness for the kaolin clay tested in triaxial compression, from a consolidation history equivalent to any desired depth in the centrifuge model, can then be determined using Equations 4.1, 4.3 and 4.5. These values can subsequently be corrected for embedment, rate and anisotropy effects, as explained earlier. The firm clay of test 3 was consolidated to a pressure of σ′v,max = 500 kPa. Figure 4.28 shows the corresponding profiles of stress, overconsolidation ratio and nominal strength, together with the location of the characteristic depth of 30 mm. At this depth OCR = 21, cu = 63 kPa and γM=2 = 0.031. The nominal stress-strain curve, deduced from Equation 4.3, is given in Figure 4.29, and this will be modified for use in MSD calculations of settlement in the tests. The mobilised shear stress in the soil beneath footing 3A is determined as τmob = q/Nc = 72/6.05 = 12 kPa. This was the least heavily loaded of the model founda- tions. The nominal degree of mobilization (τmob/cu) is just below the validated range for Equation 4.3. This is indicated on the stress-strain curve of Figure 4.29 and corre- sponds to a nominal shear strain γ = 6.3× 10−3. The average shear strain-rate during 118 4. CLAY RESULTS Fig. 4.28 Profile of in-situ and pre-consolidation stress, overconsolidation ratio, undrained shear strength and mobilisation strain with depth for test 3A Fig. 4.29 Stress-strain curve at characteristic depth of 0.3D using the kaolin database of Vardanega et al. (2012) 119 4. CLAY RESULTS undrained loading can now be computed as 9× 10−4 s−1 using the information that it took 7 s for the load to be fully applied. This is 180 times faster than the triaxial tests, so Equation 4.2 implies that the undrained strength would have been enhanced by a factor of (1 + 0.1log10180) = 1.23 had it been fully mobilized. The operational strength accounting for anisotropy should be reduced by factor 0.77, as discussed above. Embedment is negligible in this case, so no allowance for overburden or softening is appropriate. The net reduction in the estimated shear strength in bearing is, therefore, found by applying a factor of 0.94. Introducing this correction into Equation 4.1 the mobilization increases to τmob/cu = 0.20 and the estimated shear strain rises to 6.9× 10−3. The predicted undrained settlement in test 3A, therefore, is 0.51 mm compared to the observed settlement of 0.82 mm, 7 seconds after the load was applied. Considering the multiplicity of influences - a non-linear strength profile with depth, non-linear stiffness, rate and anisotropy effects - the error might be regarded as tolerable. Table 4.2 shows the nominal soil properties for each test and its predicted undrained settlement, wu,pred. Figure 4.30 shows the predicted undrained settlement plotted against the experimental settlement, wu,exp, at the time of first application of the full load. The measurements are typically 0.2 mm larger than the predictions, which might relate to a lack of perfect fit at the clay-footing interface. Table 4.2 Nominal soil properties at z = 0.3D and the predicted undrained settlement using MSD Test Label Soil Properties at z = 0.3D wu,pred (mm) z (mm) σ′v,0 (kPa) OCR cu (kPa) γM=2 γ˙ (s−1) cu,mod (kPa) τmob cu,mod γ 1A 15 11.7 12 20 0.022 0.1226 22 0.76 0.044 1.62 1B 15 11.7 12 20 0.022 0.1620 22 2A 15 12.2 42 55 0.050 0.0136 57 0.29 0.020 0.76 2B 30 24.3 21 63 0.031 0.0018 61 0.27 0.011 0.84 3A 30 24.0 21 63 0.031 0.0009 59 0.20 0.007 0.51 3B 15 12.0 42 55 0.050 0.0055 55 0.30 0.022 0.81 3C 30 24.0 21 63 0.031 0.0010 59 0.28 0.012 0.88 120 4. CLAY RESULTS Fig. 4.30 Predicted undrained penetration plotted against the undrained settlement from the centrifuge experiments 4.5.3.3 Time Effects The secondary settlement due to foundation loading could be described by: ∆ws = αw log10 ( t+ ∆t t ) (4.6) where α varies between an undrained creep slope αu and a drained creep slope αd. Although there is a lack of directly supportive evidence, corrections to the oper- ational undrained strength cu will be permitted to apply at all intermediate strains prior to failure simply by adjusting cu in Equation 4.3 and Figure 4.29, without alter- ing either of parameters b and γM=2. There is some evidence, for example in Lo Presti 121 4. CLAY RESULTS et al. (1997), that the stiffness at about 1% strain enjoys a similar rate-enhancement to undrained strength. It is easy to show from Equation 4.3 that, with b = 0.6, a 10% reduction in cu causes a 19% increase in shear strain for a maintained shear stress inducing moderate strains (i.e. for 0.2 < τmob/cu < 0.8). Therefore, it is taken that αu = 0.19. The drained creep slope αd was determined by fitting a line to the experimental data following excess pore pressure dissipation. For test 3A, this was determined to be αd = 0.11. The transition between 3-D axially symmetric undrained creep and 1-D drained creep is uncertain. The creep slope α, in this case, was taken to change in proportion to the degree of consolidation settlement. Figure 4.31 plots the data of settlement versus the logarithm of time for test 3A, with salient points and trend-lines marked to demonstrate the back-analysis using Equation 4.2 for the rate correction of strength, 4.3 for the stress-strain relation, and 4.6 for creep, as discussed. Fig. 4.31 Creep model showing creep and consolidation settlements for test 3A At time t0 in Figure 4.31 the load has just been fully applied and the settlement wu = 0.82 mm. By time t1 a clear trend has emerged of settlement increasing linearly at a rate of 0.26 mm or 31% per factor 10 on time, which extrapolates back to a settlement of 0.84 mm at time t0, perhaps suggesting that the elimination of 0.02 mm 122 4. CLAY RESULTS of clay unevenness at the foundation interface took 30 seconds to accomplish. The settlement increment of 31% per factor 10 on time between t1 and t2 in Figure 4.31 should be compared with 19% as predicted by the undrained creep relation of Equation 4.6. The increment of 26% per factor 10 on time, deduced from Figure 4.15 by averaging over the whole PIV region, rests between these values, suggesting that the region furthest from the soil surface is simply creeping whereas that closer to the footing edges is also draining. Note that an accompanying drop of 4% in excess pore pressures occurred even at a depth of 0.55D during this interval. As discussed, the drained creep rate was less than the predicted undrained creep rate. Three cycles of log10t after the foundation first touched the clay, the excess pore pressures shown in Figure 4.16 had effectively dissipated, and the rate of settlement had dropped. This reduction must be expected, since the consolidated soil is further from failure. The solution of Senjuntichai and Sapsathiarn (2006) for transient flow below a rigid impermeable circular footing, loading a deep poro-elastic bed, also required 3 cycles of log10t for its effective completion. In accordance with Equation 2.3 therefore, the components wu = 0.82 mm of immediate undrained settlement, ws = 0.46 mm of creep settlement, and wc = 0.37 mm of consolidation settlement, have been marked on Figure 4.31 at the end of loading. Beyond time t3, only drained creep continues to take place, and at the slightly slower rate. This construction on the load-response data of test 3A permits the back-calculation of an effective Poisson’s ratio ν ′ = 0.27 from Equation 2.14, which is consistent with the data of Wroth (1975). Table 4.3 records similar quantities arising from the back-analysis of the other tests. The variation in the implied value of ν ′ is seen to be acceptable. A prediction for the consolidation settlement was determined using the method of Poulos and Davis (1974) for comparison with the experimental value. A secant value of shear modulus Gsec was established from Figure 4.29 and then a drained Young’s modulus was determined through the use of Equations 2.11 and 2.12. This determined a predicted consolidation settlement wc,pred = 0.28 mm, which compares well with the observed value of wc = 0.37 mm found using the creep model. This result is also shown in Figure 4.31. The similarity shows some support for the developed creep model, and also for the use of elastic parameters in the determination of consolidation settlements. 123 4. CLAY RESULTS Table 4.3 Footing settlements and implied Poisson’s ratio for the given creep and consolidation model Test Label wt (mm) wu (mm) wc (mm) ws (mm) αu αd ν ′imp 1A 3.20 1.94 0.61 0.65 0.19 0.05 0.34 2A 1.65 1.05 0.28 0.32 0.19 0.07 0.37 2B 1.64 0.91 0.32 0.41 0.19 0.09 0.32 3A 1.65 0.82 0.37 0.46 0.19 0.11 0.27 3B 1.64 1.11 0.27 0.26 0.19 0.07 0.38 3C 1.98 1.02 0.36 0.60 0.19 0.11 0.32 4.6 Footing Settlement - Further Loading Incremental loading of the soil, as applied through the use of compressed air, could simulate a typical building construction project. As the building increases in height the footing load increases. The result is reduced settlement as the soil is able to stiffen through consolidation following each loading stage. A data set from one additional loading, from 72 kPa to nominally 140 kPa for the firm clay is presented along with a summary of results from all of the tests. 4.6.1 Firm Clay Through the use of compressed air an additional pressure of about 70 kPa was applied to the top of the pneumatic cylinder. The load cell, however, did not register that 140 kPa was applied to the soil. This is attributed to slight rotations of the footing which consequently affected the load cell readings. Load cells are very sensitive to rotations and the resulting bending moments. It can be concluded that about an additional 70 kPa was applied to the soil because of the similar value of excess pore pressure that was generated beneath the footing. Figure 4.32 shows the footing settlement and excess pore pressure measured on the centreline at a nominal depth of 0.53D (reduced due to the settlement in the virgin loading phase). Figure 4.32 shows similar characteristics to that discussed in Section 4.5. The excess pore pressure of about 30 kPa supports the argument that an additional load of 124 4. CLAY RESULTS Fig. 4.32 Additional pressure of 70 kPa applied for 2 hours at model scale (test 3A-1) 70 kPa was applied. The Mandel-Cryer effect is again evident, although slightly less pronounced and the footing settlement behaviour with time is also very similar. 4.6.1.1 “Undrained” Penetration The “undrained” penetration at time t1 in Figure 4.32 is 0.68 mm - 17% less than the initial virgin loading of the soil under a similar average pressure. A PIV analysis at this time produced the mechanism shown in Figure 4.33. The mechanism appears to be a combination of the virgin undrained mechanism and the consolidation and creep mechanism. This suggests that as the clay is permitted to stiffen, the undrained mechanism is tending towards that of quasi one-dimensional compression. The most simple approach for the foundation design of a structure which under- goes additional loading after an extended period would be to determine the settlement using the final value of applied pressure, but on the initial soil characteristics. The loading intervals in the centrifuge corresponded to increases in pressure every 2 years at prototype scale and so this is analogous to vertical extensions applied to a super- structure, albeit very slowly. This approach is conservative and would probably be adopted by practising engineers. As discussed, the effects of consolidation and creep 125 4. CLAY RESULTS Fig. 4.33 “Undrained” penetration mechanism at time t1 for test 3A-1 Fig. 4.34 “Undrained” settlement plotted against predicted values for the additional pressures applied via compressed air 126 4. CLAY RESULTS result in the soil becoming stiffer: hence the conservative result. The approach also requires no further soil testing and therefore a cost-saving is made. Naturally, more accurate results could be determined if an in-situ investigation was performed (if prac- tically possible). Using the approach discussed in Section 4.5.3 a load-settlement curve was produced with the three total pressures and their cumulative broadly undrained settlement assuming an average loading time of 5 s; as given in Figure 4.34. Figure 4.34 shows good prediction of settlement at lower pressures. The centrifuge test results fall on the non-conservative side but this was also evident in the virgin- loading results of the soil. The final pressure of 405 kPa is approximately 10% more than the original bearing capacity of the soil - well beyond actual design pressures and settlements. Hence, this pressure was only applied for a few minutes at model scale. Further investigation of the strengthening and stiffening of the soil due to footing loading is warranted but is not performed here. 4.6.1.2 Consolidation and Creep A similar analysis of consolidation and creep was performed for the additional footing loadings. Table 4.4 shows the results for total applied pressures of 140 kPa (3A-1) and 280 kPa (3A-2) together with the original virgin loading. It is evident that the drained creep ratio decreases significantly and is almost zero at the end of 3A-2. Values of ν ′imp are still consistent with those from the literature. Table 4.4 Footing settlements and implied Poisson’s ratio for the given creep and consolidation model for further loading of foundations Test Label wt (mm) wu (mm) wc (mm) ws (mm) αu αd ν ′imp 3A 1.65 0.82 0.37 0.46 0.19 0.11 0.27 3A-1 1.43 0.68 0.35 0.40 0.19 0.09 0.24 3A-2 5.62 3.19 1.17 1.26 0.19 0.01 0.32 3A-3 5.17 No pneumatically loaded footings were tested on soft clay as the discussed bearing capacity test was performed in its place. As usually occurs with experimental testing, some problems occurred. These are now briefly discussed. 127 4. CLAY RESULTS 4.7 Experiment Problems The placing of the semi-circular foundations against the Perspex window caused some problems. If the footing was placed too firmly there was potential for friction to effect the loading, or for the footing not to fall to the surface. If the footing was not placed firmly enough against the Perspex, then there was the possibility that clay might enter the footing-Perspex window interface - as shown in Figure 4.35(a) for a 100 mm footing on soft clay. As could be expected, this problem was much more evident for the soft clay model. Figure 4.35(b) shows a photograph from a firm clay test where relatively little clay moved between the footing and Perspex. Due to the significant amount of clay that entered the space in Figure 4.35(a) the results were discarded as the load cell, laser and PIV results were all affected. The load cell recorded inconsistent bearing pressures due to the rotation and associated bending moment. Laser and PIV measurements of footing settlement were different because of the footing rotation. The results presented in this chapter have demonstrated the importance of the undrained penetration on the total settlement of shallow circular foundations. The pre- dictions using MSD were reasonable, but this method adopted an assumed deformation field. Given that actual mechanisms have now been observed, further investigation was performed and is now presented. 4.8 Undrained Mechanism Analysis Osman and Bolton (2005) defined a displacement field based on the Prandtl mechanism to be used in MSD. The mechanism was for a smooth footing, and results observed in the centrifuge indicate a relatively rough footing. Nonetheless, a comparison between the mechanisms can be performed. The mechanism with comparable centre settlement of the footing is shown with the experimental results (from Figure 4.23) in Figure 4.36. It can be seen that there is very little heave invoked and that the soil movement continues for a greater depth into the soil. The difference in the theoretical and experimental mechanism warranted a new theoretical mechanism to be developed. Initial investigation and thought yielded the idea that each vector looked normal to ellipses drawn beneath the footing. A script was developed in Matlab to investigate further. A series of ellipses were drawn on the 128 4. CLAY RESULTS (a) Clay between footing and Perspex window on soft clay test (b) Relatively little clay between the footing and Perspex on stiff clay Fig. 4.35 Clay particles between the footing and Perspex for the soft clay test in com- parison to typical results 129 4. CLAY RESULTS Fig. 4.36 Comparison of centrifuge and Osman and Bolton (2005) mechanisms for settlement at centre of footing of 2 mm mechanism of Figure 4.23. At each point on the ellipse a displacement was determined by taking an interpolated value of the actual displacements nearest the point. This displacement was then drawn on the ellipse. Figure 4.37(a) shows the interpolated displacements in red with the centrifuge data. Figure 4.37(b) shows the interpolated data and ellipses only. It can be seen that the displacement along each ellipse is approximately equal and that the displacements are relatively normal to each ellipse. The ellipses shown in Figure 4.37 were estimated initially but showed great promise for a mechanism. A relationship for the location of the ellipse focus with depth was developed and was used in a cavity expansion model for the undrained deformation mechanism. This is presented in Chapter 5. 130 4. CLAY RESULTS (a) Interpolated displacements with centrifuge data (b) Interpolated displacements Fig. 4.37 Ellipses with relatively normal interpolated displacements 131 4. CLAY RESULTS 4.9 Summary A series of centrifuge experiments was performed to investigate the settlement and de- formation mechanisms associated with circular shallow foundations on overconsolidated clay beds. The following points were presented and discussed: ˆ Observations associated with the unloading of the soil and centrifuge spin-up and then self-weight consolidation were presented. The deep clay bed of the model meant that footing tests had to be performed with some self-weight consolidation excess pore pressure still present. Any subsequent clay body settlement was removed from the footing settlement by performing a PIV analysis in the far- field. ˆ Laser data of footing movement and load cell data of pressure were used to de- termine the zero-time of loading and undrained penetration for each foundation. PIV and laser results were then compared to ensure uniform settlement of the foundation. This comparison and calibration of data ensured consistency within the results. ˆ Footing test durations ranged from 3 hours to 7 years at prototype scale. Set- tlement time data and mechanisms for the initial loading of a 100 mm diameter footing on firm clay were presented in detail together with some supporting results from a 50 mm diameter footing on soft clay. Early analysis showed a redistribu- tion of stresses as drainage occurred adjacent to the footings - as confirmed by volumetric strain plots. Pore pressure responses showed the corresponding soil- foundation response ranged from fast undrained shearing with significant rate effects, through transient drainage with contemporaneous creep, to fully drained creep. Where possible, mechanisms were provided of each of these phases. ˆ The majority of settlements were shown to occur within a depth of about one footing diameter - deeper than that in the idealised Prandtl mechanism. ˆ Correlations developed through extensive experimental testing and the formation of databases were used to perform a back-analysis on a bearing capacity test performed on soft clay. The success of this analysis depended on making three adjustments to the measured soil strengths: a rate effect, an anisotropy effect and by accounting for the corresponding overburden pressure in calculating net 132 4. CLAY RESULTS bearing pressures. Although a simplification of reality, the reasonably accurate prediction of bearing capacity allowed the method to be adopted for predicting the undrained penetration of the footing tests. A reasonable accuracy was again achieved. ˆ Creep settlements before and during consolidation were predicted simply by in- terpolating between undrained settlements which were taken to increase by creep at 19% per factor 10 on time from combining the power law for stress versus strain with the proposition that strength should reduce by 10% per factor 10 on time, and the drained creep found from the experimental data. This assessment of creep allowed the consolidation settlement to be determined and an implied value of ν ′ to be calculated, which was consistent with published results. ˆ The essential step to predicting footing behaviour by the presented method is a good prediction of its undrained penetration. An ellipsoidal cavity expansion mechanism was introduced and is now used to investigate the load-settlement behaviour of a linear-elastic perfectly-plastic clay bed. For a better visual repre- sentation this chapter is concluded with the broadly undrained mechanism shown on a captured photograph in Figure 4.38. Fig. 4.38 Clay mechanism on photograph for test 3C (100 mm footing on stiff clay) 133 4. CLAY RESULTS 134 Chapter 5 Cavity Expansion Model for Bearing Capacity and Settlement of Circular Shallow Foundations on Clay 5.1 Introduction The design of a shallow foundation requires consideration of both the ultimate bearing capacity and the settlement at the working load. Settlements are often critical due to the vulnerability of typical structures to differential settlements between the various elements of the foundation system. This chapter presents a cavity expansion model for linear-elastic perfectly-plastic soil. The use of cavity expansion for the bearing capacity of shallow foundations is not the classical approach and therefore a review of some of the research that has been performed is presented. The yielding behaviour follows the von Mises’ criteria and the method for determining the energy is provided. This is then used for the ellipsoidal model and results are presented. Finally, a comparison of the cavity expansion model and the mechanism of Osman and Bolton (2005) for linear-elastic perfectly-plastic soil is provided. Parts of this chapter have been accepted for publication by Ge´otechnique as McMahon, B. T., Haigh, S. K. and Bolton, M. D. (2012) Cavity expansion model for the bearing capacity and settlement of circular shallow foundations on clay. doi:10.1680/geot.12.P.61 135 5. CAVITY EXPANSION MODEL 5.2 Cavity Expansion Literature There are two approaches which can be adopted in determining the bearing capacity of foundations. The classical approach which adopts a local mechanism of indentation, shear and heaving was introduced in Chapter 2, and the design approach according to Eurocode 7 was given. The ultimate bearing capacity, qult, for the simple case of a shallow foundation on the surface of a purely cohesive soil with undrained shear strength, cu, is determined using: qult = cuNc sc (5.1) where Nc and sc are the bearing capacity and shape factors respectively. Concerns about the use of a rigid-plastic material to represent soil, however, produced the second approach for determining foundation stiffness and ultimate bearing capacity - a cavity expansion idealisation. This method has primarily been used for deep foundations such as piles where capacity is provided by the resistance that the soil offers to the expansion of a cavity corresponding to the volume indented by the pile. Using spherical cavity expansion theory combined with plasticity, the bearing capacity factor has been determined to be a function of both the strength and stiffness of the soil. This arises from the recognition that soil in the far field must remain elastic if the indentation of the foundation remains finite. Bishop et al. (1945) investigated the theory of indentation and determined the pressure required to enlarge a spherical cavity infinitely through plastic flow of the soil, ps, as: ps = 2Y 3 ( 1 + 3 ln rp rc ) (5.2) where Y is the uniaxial yield stress of the material, and: rp rc = ( E (1 + ν)Y )1/3 (5.3) Assuming that for soil Y is equivalent to the triaxial deviatoric stress, qu, and assuming undrained behaviour, for which ν = 0.5 and E = 3G, this produces the limiting stress for expansion of a spherical cavity as: 136 5. CAVITY EXPANSION MODEL ps = 2 3 qu ( 1 + ln ( 2 G qu )) (5.4) For the particular case of a pile, Gibson (1950) recognised that Equation 5.2 cor- responds to the ultimate capacity, but with no cohesion being mobilised on the base of the pile. In reality, full cohesion on this surface would be mobilised, and combining this assumption with the finite-strain theory of Swainger (1947), produced: Nc = 4 3 ( 1 + ln E cu ) + cotα (5.5) where the base of the pile is taken to be a cone with semi-angle α, and E is the linear equivalent of Young’s Modulus, taken for convenience as the secant modulus to one- half of the ultimate compressive stress. Skempton (1951) utilised this expression and adopted a semi-angle of α = 45◦ for shallow foundations, giving: Nc = 4 3 ( 1 + ln E cu ) + 1 (5.6) Meyerhof (1951) also considered full mobilisation along the pile surface but defined Young’s Modulus as the initial tangent to triaxial stress-strain curves (Konrad and Law, 1987). For the particular case of a rough shallow circular foundation, it was determined that Nc = 6.18. Vesic (1972) analysed the expansion of a spherical cavity in ideal soils, but also considered the effects of volume change. For an undrained spherical cavity expansion of a purely cohesive soil the result was: Nc = 4 3 ( 1 + ln G cu ) (5.7) Carter et al. (1986) investigated cylindrical and spherical cavities in cohesive fric- tional soils. The relationship between cavity pressure and expansion was found for small deformations. The particular case of a purely cohesive material was shown to have an undrained spherical cavity limit pressure factor identical to that in Equation 5.7. 137 5. CAVITY EXPANSION MODEL Finite element analyses have been widely used in the consideration of the ultimate bearing capacity of shallow foundations. Taiebat and Carter (2000) performed an analysis, producing Nc = 5.7 for a rough footing on clay with G/cu = 100. A refined analysis in Taiebat and Carter (2010), through a finer mesh and selection of a flow rule closer to the Tresca criterion, produced a more accurate value of Nc = 6.17 for the same soil. Gourvenec and Randolph (2002) investigated stiffer clay, with G/cu = 167, determining the ultimate capacity factor as being Nc = 5.91. These values are 5.8% less, 2.0% more and 2.4% less respectively, than the classical value of Nc = 6.05. The broadly undrained deformation mechanisms observed in the centrifuge were shown in Figures 4.10 and 4.23 of Chapter 4. It can be seen that the displacement field resembles a cavity-expansion more closely than the rigid-plastic solution by method of characteristics shown in Figure 5.1. Fig. 5.1 Method of characteristics from ABC for the Prandtl mechanism (Martin, 2003) A method for determining the load-settlement behaviour of a circular surface foun- dation on linear-elastic perfectly-plastic soil using an energy approach based on a kine- matically admissible deformation mechanism for cavity expansion is now presented. In order to evolve smoothly from a plane punch at the foundation to a hemispherical cavity expansion at depth, the cavity expansion is taken to be ellipsoidal rather than spherical, as an approximation to the behaviour seen in the centrifuge test data. 138 5. CAVITY EXPANSION MODEL 5.3 Analysis Procedure In order to carry out an analysis based on energy conservation, it is necessary to calculate the work done in deforming the soil within the axisymmetric mechanism. Rigid, perfectly-plastic analyses in plane strain can calculate the work done per unit volume, following Shield and Drucker (1953), as: W = cuε1 (5.8) However, for an elastic perfectly-plastic material in axial symmetry, it is important to calculate both the elastic and the plastic work due to all of the stress components acting on the soil. 5.3.1 Elastic Work The energy associated with elastic work can be determined using Hooke’s Law. For the undrained case, with Poisson’s ratio ν = 0.5, the three-dimensional expression can be reduced to: E [ ε1 ε2 ] = [ 1 −1/2 −1/2 1 ][ σ1 − σ3 σ2 − σ3 ] (5.9) As the undrained Young’s modulus, Eu, is related to the shear modulus, G, by Eu = 3G, this can be inverted to give: [ σ1 − σ3 σ2 − σ3 ] = G [ 4 2 2 4 ][ ε1 ε2 ] (5.10) The rate of elastic work per unit volume is given by: ∂We/∂w = σ1ε˙1 + σ2ε˙2 + σ3ε˙3 (5.11) If a mechanism that is geometrically similar for all foundation displacements is assumed, then it can be shown that: 139 5. CAVITY EXPANSION MODEL εn = ∂εn ∂w = ε˙nw [n = 1, 2, 3] (5.12) Given that there is no volumetric strain, ε3 = −ε1 − ε2 can be used with Equa- tions 5.10 and 5.11 to produce the elastic work rate per unit volume: ∂We ∂w = 4G ˆ˙ε 2 w (5.13) where the appropriate strain invariant function is written: ˆ˙ε 2 = [ε˙21 + ε˙ 2 2 + ε˙1ε˙2] (5.14) 5.3.2 Plastic Work Yield was determined using the isotropic von Mises’ yield criterion: (σ1 − σ3) 2 + (σ2 − σ3) 2 + (σ1 − σ2) 2 = 2q2u (5.15) where qu is the undrained strength in triaxial compression. Substitution of the expres- sions in Equations 5.10 and 5.14 into Equation 5.15 shows that at yield: ˆ˙ε = 1 2 √ 3 qu G 1 w (5.16) It can be shown from Equation 5.9 that for an undrained elastic material, the total strain vector is parallel to the deviatoric stress, σˆ = σ − p. As the von Mises’ yield criterion is circular in the pi-plane, for a material exhibiting associated flow, the direction of the incremental plastic strain vector, ε˙p, is also parallel to the deviatoric stress. If a mechanism that remains geometrically similar for all footing displacements is assumed, the direction of the total strain vector will remain constant under both elastic and plastic deformations. As the plastic and total strain rates are parallel, it is 140 5. CAVITY EXPANSION MODEL thus implied that no further elastic strain can occur post-yield, with the plastic strain rate being equal to the total strain rate. It thus follows that in the zone of plastic deformation, no incremental elastic work is done. As the total strain rate in the plastic zone is equal to the plastic strain rate, it follows that: σˆ2 σˆ1 = ε˙2 ε˙1 and σˆ3 σˆ1 = ε˙3 ε˙1 (5.17) This can be substituted along with the condition of no volumetric strain, into the yield criterion of Equation 5.15. This produces an expression for the major principal deviatoric stress: σˆ1 = qu ε˙1√ 3 ˆ˙ε (5.18) The rate of plastic work due to the deviatoric stress per unit volume is given by: ∂Wp/∂w = σˆ1ε˙1 + σˆ2ε˙2 + σˆ3ε˙3 (5.19) Substitution of the expressions in Equations 5.17 and 5.18 determines the plastic work rate per unit volume to be: ∂Wp ∂w = 2qu ˆ˙ε√ 3 (5.20) The validity of the discussed approach is verified by analysing an elastic perfectly- plastic spherical cavity expansion and comparing with the results presented by Bishop et al. (1945). 5.4 Validation of Model Using the energy method discussed, a solution can be determined for the expansion of a spherical cavity. Equations 5.11, 5.13, 5.14, 5.19 and 5.20 are utilised, but the derivatives are now taken with respect to cavity expansion, ρ, rather than footing 141 5. CAVITY EXPANSION MODEL displacement. The radial displacement of a spherical cavity is given as: ρ = dV 4pir2 (5.21) where dV is the change in cavity volume. The radial and circumferential strains, also the principal strains, are: εr = 2ρ r = dV 2pir3 , εθ = − ρ r = − dV 4pir3 (5.22) which can be substituted into Equation 5.14 to find that: ˆ˙ε = √ 3r2c r3 (5.23) This can now be compared with the value at yield given by Equation 5.16 to show that: r3p = r 3 c dV Vc G cu (5.24) Equation 5.23 is utilised in calculating the rates of elastic and plastic work per unit volume, as portrayed in Equations 5.13 and 5.20 respectively. These work rates are then integrated over the volume to give a limiting stress, σc, of: σc = 2 3 qu ( 1 + ln ( 2 G qu )) (5.25) which is identical to the lower bound solution of Bishop et al. (1945) shown in Equation 5.4, which was based on equilibrium stresses that conform to the yield criterion. The assumption of a deformation mechanism and the balancing of work and en- ergy must, in principle, lead to an upper-bound estimate of collapse loads for perfectly rigid-plastic materials. Similarly, the assumption of an arbitrary but kinematically admissible deformation mechanism for elastic materials leads in principle to an over- estimation of strain energy under an applied load, and to an underestimation of the displacement of its point of application. Accordingly, the energy method proposed here should provide an upper-bound to the true load-displacement relationship. The 142 5. CAVITY EXPANSION MODEL equivalence of Equations 5.25 and 5.4 simply confirms the exact nature of both the lower-bound and upper-bound solutions. In spherical symmetry, only one deformation mechanism is possible, so the upper and lower bounds naturally coincide at the correct solution. Equally, this confirms the accuracy of the upper-bound energy calculation. Klar and Osman (2008) investigated the load-displacement solutions for shallow foundations using deformation fields and energy methods. It confirms that the de- formation field that corresponds to the exact solution should provide accurate values. Other deformation fields, however, should overestimate the accurate value because the internal energy cannot be minimised to the exact value. 5.5 Cavity Expansion Analysis of Circular Shallow Foundations In this investigation, an energy method is used to determine an upper-bound on the bearing capacity of a circular shallow foundation using cavity expansion methods. Con- ventional cavity expansion methods assume expansion of a hemispherical cavity below the foundation which expands outwards with spherical symmetry. These methods ignore the work done in the hemispherical zone below the foundation. In this inves- tigation an ellipsoidal cavity expansion approach is analysed, as this allows a smooth transition between a flat punch at the soil surface and spherical cavity expansion in the far-field, allowing an upper-bound solution to be attempted. Figure 5.2 illustrates notation and the global mechanism. In the near-field, ellipsoids smoothly transition from a flat punch at the ground surface to a hemisphere of radius rh. Beyond this radius, conventional spherical cavity expansion occurs. The plastic radius, rp, divides the plastic and elastic zones of the soil and can lie in either the ellipsoidal or the spherical zone. In order to make the calculation domain finite, a bounding radius, rb, in the spherical zone is chosen with the work done outside this radius being calculated on the basis of purely elastic spherical cavity expansion. This approach is valid provided that rp < rb for the foundation settlement considered. An upper-bound on the bearing capacity of a shallow foundation can be found by equating the work done in moving the foundation to the energy stored or dissipated 143 5. CAVITY EXPANSION MODEL Fig. 5.2 Energy method used to determine the load-settlement behaviour within the soil using an assumed mechanism. In the case of the mechanism described here, this work can be subdivided into elastic work beyond the bounding radius, and both elastic and plastic work within the bounding radius. For simplicity, the effect of induced anisotropy has been ignored, however it could be considered negligible at the typical small working settlement of a footing. 5.5.1 Assumed Deformation Field 5.5.1.1 Ellipsoidal Model A new method is now used to describe the movement within the hemisphere of radius rh in terms of a series of ellipsoids with a resultant soil displacement normal to the ellipsoid at each point. The first ellipsoid occurs at the footing base and hence is the special case of a circle (the footing) with normal displacement w. This corresponds to a perfectly rough footing base. Ellipsoids then grow in size and gradually transition towards being a hemisphere at rh. Figure 5.3 portrays the ellipsoids tending towards 144 5. CAVITY EXPANSION MODEL hemispheres at the hemispherical radius - in this case one footing diameter. The transition is accomplished through moving the foci of the ellipsoids, allowing their eccentricity to change. Compatibility is maintained using: δ = Af Ae × w (5.26) where δ is the displacement normal to the ellipsoid, and Ae and Af are the surface areas of that ellipsoid and the footing respectively. The axisymmetric nature of the problem results in the ellipsoids being oblate spheroids. The general equation of an oblate spheroid in cylindrical coordinates is given by: r2 a2 + z2 b2 = 1 where a > b (5.27) The focal radius of an ellipse can be expressed as: f = √ a2 − b2 (5.28) At the soil surface the ellipse is a straight line, thus f = a = rf and b = 0 corresponding to a focus at the footing edge. Circles are a special case of an ellipse, where a = b corresponding to a focus at the centre. The analysis presented allows the focal radius to vary from rf to zero as a linear function of the major-axis radius a. The expression for the focal radius is, thus, given by: f = ( rf rf − rh ) a− rfrh rf − rh = rf (a− rh) rf − rh (5.29) The defined focus relationship in Equation 5.29 can be equated to the general focus definition to produce an expression for the minor-axis radius, b, as: b2 = a2 − r2f (a− rh) 2 (rf − rh)2 (5.30) 145 5. CAVITY EXPANSION MODEL Fig. 5.3 Evolution of ellipsoids to hemispheres for rh = D 146 5. CAVITY EXPANSION MODEL Substituting Equation 5.30 into the general ellipse equation (Equation 5.27) pro- duces a quartic equation in terms of the major axis a. This is given by: (r2h − 2rfrh)a 4 + (2r2frh)a 3 +(2r2rfrh − z 2r2f + 2z 2rfrh − z 2r2h − r 2r2h − r 2 fr 2 h)a 2 +(−2r2r2frh)a+ (r 2r2fr 2 h) = 0 (5.31) A grid of points (r, z) was created and utilised in solving Equation 5.31 using the roots function within Matlab. Thus, the major-axis radius, a, was determined for each point in the soil. The minor-axis radius was then found using Equation 5.30. The surface area of the ellipsoid, Ae, passing through each grid point could hence be calculated and consequently the magnitude of displacement at each point found. The soil displacement is defined as being normal to the surface of the ellipsoid. The magnitude of displacement at each point is determined using Equation 5.26, in which the surface area of half an oblate spheroid is given by: Ae = pia 2 + piab2 2f ln ( a+ f a− f ) (5.32) For ease of strain calculation, the displacement, δ, was separated into radial and vertical components u and v respectively, as shown in Figure 5.2. The slope of the displacement vector, denoted dz/dr, was combined with the known length of each vector to produce: dz dr = a2 b2 (z r ) , u = δ √ 1 + ( dz dr )2 , v = u dz dr (5.33) 5.5.1.2 Hemispherical Region Beyond the designated hemispherical boundary, the soil displacements are normal to hemispheres. Hemispheres are a particular case of an ellipsoid, where a = b = r, and thus a similar approach can be adopted. The deformation mechanism produced using this model is shown in Figure 5.4. It must be noted that the mechanism suffers continuity issues at the footing edge due to a 147 5. CAVITY EXPANSION MODEL gross change in geometry. Below the foundation soil displacements are purely vertical, whereas at the soil surface they are purely horizontal. Investigation demonstrated that changing the geometry at the footing edge had negligible effect on the value of footing pressure and settlement. The strains within the displaced footing area have not been removed for this anal- ysis. Movements are relatively small compared to footing diameter and therefore any error is negligible. Figure 5.5 shows a comparison between the experimental results of Chapter 4 and the cavity expansion model, and Figure 5.6 shows the vector difference between the mechanisms. The difference is relatively small, demonstrating a good comparison, with the most significant difference occurring in the horizontal direction in the shear fan zone of the Prandtl mechanism. It may be possible to modify the relationship changing the ellipsoids focal radii for a closer match. It is again evident in Figure 5.5 that the footing-soil interface in the centrifuge tests was relatively rough. 5.5.2 Strain Calculation Strains in each direction can be calculated, following Osman and Bolton (2005), by taking the first derivative of displacement. These are given by: εr = − ∂u ∂r γrθ = 0 εθ = − u r γθz = 0 εz = − ∂v ∂z γzr = − ∂v ∂r − ∂u ∂z (5.34) The major and minor principal strains, 1 and 3, and the intermediate principal strain, 2 can be determined by: ε1 = 1 2 ( −εθ + √ ε2θ + γ 2 rz − 4εrεz ) ε2 = εθ ε3 = 1 2 ( −εθ − √ ε2θ + γ 2 rz − 4εrεz ) (5.35) 148 5. CAVITY EXPANSION MODEL Fig. 5.4 Ellipsoidal mechanism for a footing with diameter D and settlement w 149 5. CAVITY EXPANSION MODEL Fig. 5.5 Comparison of the cavity expansion mechanism and that observed in the centrifuge Fig. 5.6 Vector difference between mechanisms of Figure 5.5 150 5. CAVITY EXPANSION MODEL 5.5.3 Work Calculation 5.5.3.1 Elastic Work beyond the Bounding Radius Using equilibrium and compatibility, an expression for the work done outside the radius rb can be determined. Equilibrium in this zone dictates that: ∂σr ∂r + 2 σr − σθ r = 0 (5.36) for radii greater than the plastic radius. The elastic strains are: εr = 1 E (σr − 2νσθ) εθ = 1 E (σθ − νσr − νσθ) (5.37) Rearranging the constitutive equations provides an expression relating the devia- toric components of stress and strain: (σr − σθ) = E 1 + ν (εr − εθ) (5.38) The surface area of a hemisphere, radius r, is Ah = 2pir2. As discussed, compati- bility holds, and in this zone is given by: pir2fw = 2pir 2δ (5.39) Expressions for circumferential and radial strain were given in Equation 5.22. Com- bining these with compatibility from Equation 5.39 gives: εr − εθ = 3 2 r2f w r3 (5.40) Equations 5.38 and 5.40 can now be substituted into the equilibrium condition (Equation 5.36) and integrated between an infinite radius, corresponding to an in-situ stress of p0, and the boundary radius, rb with the plastic radius stress σb: 151 5. CAVITY EXPANSION MODEL ∫ σb p0 dσr = −3Er2f w 1 + ν ∫ rb ∞ dr r4 (5.41) Ignoring the in-situ stress, thus assuming weightless soil and setting p0 = 0, and using the relationship G = E/[2(1 + ν)], Equation 5.41 was solved to find: σb = E 1 + ν r2f w r3b = 2G r2f w r3b (5.42) which is also verified by Yu (2000). Work is determined by W = ∫ F.dδ which for an elastic material is simply given as W = 12 (σbA) δb. Using compatibility (Equation 5.39) and the surface area of the bounding hemisphere, the elastic work outside the bounding radius is given by: Wb = piG r4f r3b w2 (5.43) To ensure consistency, the derivative with respect to footing displacement is required to allow an incremental analysis to be performed. The elastic work rate beyond the bounding radius, therefore, is: ∂Wb ∂w = 2piG r4f r3b w (5.44) 5.5.3.2 Load-Settlement Behaviour The rate of elastic work outside the boundary radius was added to the rates of plastic and elastic work and then integrated over the appropriate regions within the bounding radius. This was then equated to the footing work. The relationship between footing stress and displacement is hence: q = 1 pir2f   ∂Wb ∂w + ∫ elastic ∂We ∂w dV + ∫ plastic ∂Wp ∂w dV   (5.45) 152 5. CAVITY EXPANSION MODEL The process of solving Equation 5.45, for all foundation settlements, was computa- tionally inexpensive because the mechanism does not change with settlement - as was shown in the experiment results and in Figure 4.17. The work derivative terms are thus constant throughout the process. As the integration of the work terms is carried out numerically rather than analytically, the plastic radius need not be explicitly calcu- lated. By comparing the strain invariant, ˆ˙ε, with its value at yield, given by Equation 5.16, it can be determined whether any soil element is undergoing plastic or elastic deformation and hence which expression of work is appropriate. The preceding analysis has been formulated in terms of the triaxial compressive strength of the soil qu. In order to facilitate comparison with previously published finite element analyses, it will be assumed that: qu = 2cu (5.46) 5.6 Results 5.6.1 Effect of Hemispherical Radius The normalised footing pressure q/cu reduced as the ratio of hemispherical radius to footing diameter rh/D increased. For a soil with G/cu = 100 at a settlement of 5% of the footing diameter the load-settlement behaviour is shown in Figure 5.7. Beyond approximately two footing diameters there is minimal change in the calculated bearing stress with increasing hemispherical radius. Results for the load-settlement behaviour, therefore, were determined using a hemispherical radius of two footing diameters. 5.6.2 Effect of Mesh Size A mesh size of 0.2%D was adopted for the analysis. A finer mesh, corresponding to a size 0.04%D, resulted in a reduction of the footing load by only 0.7%. The significant increase in computation time was therefore not justified. 153 5. CAVITY EXPANSION MODEL Fig. 5.7 Effect of increasing the extent of the ellipsoidal mechanism 5.6.3 Load-Settlement Behaviour In order to make comparisons with previously published finite element results, soils with rigidity, G/cu of 100 and 167 were considered. Figures 5.8 and 5.9 portray the load-settlement behaviour for these soils at varying values of settlement ratio. Figure 5.8 also shows data from the finite element analysis of Taiebat and Carter (2000, 2010) for a soil with G/cu = 100, and Figure 5.9 that of Gourvenec and Randolph (2002) for a soil with G/cu = 167. The figures also show a line to represent the classical value of Nc = 6.05 (Eason and Shield, 1960). Results agree well between the present study and the finite element analyses, especially in the low settlement region in which conventional design would be likely to take place. Analyses by Taiebat and Carter (2000, 2010) demonstrate approximately linear load-settlement behaviour in the small settlement region. This is not reflected in the results of the present study, nor in those of Gourvenec and Randolph (2002), as the soil beneath the edge of the footing was observed to become plastic, even at very small settlements. It is evident from Figures 5.8 and 5.9 that the footing pressure increases with the allowable footing settlement. Taiebat and Carter (2000) found that with a purely vertical load on the foundation no peak load was observed. Gourvenec and Randolph (2002) and Taiebat and Carter (2010), however, found convergence to limiting values of Nc = 5.91 and 6.17 for soils with G/cu = 167 and 100 respectively. While the load- settlement behaviour compares very well in the small settlement range, the results of this analysis continue to increase beyond the limiting value of Nc = 6.05 at large settlements. 154 5. CAVITY EXPANSION MODEL Fig. 5.8 Load-settlement behaviour for soil with G/cu = 100 Fig. 5.9 Load settlement behaviour for soil with G/cu = 167 The analysis described in this investigation gives an upper-bound on foundation loading. Whilst it appears accurate when compared to the finite element solutions at small settlements, it overestimates the strength of the foundation at large settlements. It is probable that once the footing settlement becomes large, the Prandtl mechanism will give a lower upper-bound on applied stress. 5.6.4 Implications for Design Analyses were performed on soils with G/cu values between 10 and 10000. The load- settlement behaviour was found to fall on a single line when plotted as a function of soil rigidity, G/cu, multiplied by the normalised settlement w/D. Figure 5.10 shows the relationship between footing load and soil rigidity multiplied by normalised settlement. The linear range of Figure 5.10 represents the region of typical footing designs for clays. The expression for this range is: 155 5. CAVITY EXPANSION MODEL Fig. 5.10 Footing stress plotted against (w/D ×G/cu) with expression for linear range and design example point q cu = 4.45 + 1.34ln ( w D G cu ) (5.47) Equation 5.47 could be used in design in order to determine the allowable footing pressure based on an allowable settlement and the soils rigidity. As an example, consider a 2 m diameter footing with an allowable immediate undrained settlement of 2 mm founded on London clay. Additional settlements due to consolidation would, of course, also need to be accounted for. It was suggested by Jefferies (1995) that a typical value of G/cu for London clay might be 180. Using these values in Equation 5.47 produces q/cu = 2.15, as indicated on Figure 5.10. In conventional design terms this is equivalent to a factor of safety of 2.8 on the classical bearing capacity factor of Nc = 6.05. This single calculation has accounted for both bearing capacity and settlement. 5.7 Mechanism Comparison Osman and Bolton (2005) utilised the work equation from Shield and Drucker (1953) for an ideally plastic Tresca material for a deformation field developed within the Prandtl mechanism to determine an upper-bound for the theoretical collapse load. It 156 5. CAVITY EXPANSION MODEL was determined that Nc = 5.86 for the soil. However, as a rigid plastic material was used, this approach can only be used for determining the ultimate, or collapse, load and not in the production of a load-settlement curve. The load-settlement curve was produced by finding the mobilised strain on the stress-strain curve (using the mobilised stress) and multiplying by the compatibility factor. This approach could be considered problematic because the soil was taken to increase in stiffness and strength with depth so that a characteristic location could be found to match the triaxial test results. The energy approach given allows the load-settlement behaviour for a linear-elastic perfectly-plastic material following the von Mises’ yield criterion to be determined for any mechanism which remains geometrically similar. It is necessary for a particular soil test to be found to match the new calculation procedure. The deformation field of Osman and Bolton (2005) was formed independently for a comparison. The mechanism was verified when it was confirmed that Mc = 1.35. A comparison of the deformation mechanisms is shown in Figure 5.11. Using the energy method presented, an investigation on the Osman and Bolton (2005) mechanism was performed for soils of varying rigidity, with G/cu equal to 10, 50, 100 and 1000. This is given in Figure 5.12. It is evident that as the soil stiffness increases the soil reaches its maximum capacity sooner. For all values of soil rigidity index, a convergence to an ultimate bearing capacity of Nc = 6.11 is observed. This is different to the value Nc = 5.86 that was calculated in Osman and Bolton (2005). The approach of Osman and Bolton (2005) was to use the work expression from Shield and Drucker (1953) for a material obeying the Tresca yield criterion. The stresses at yield for this criterion must move to a vertex to allow the plastic strain increment to be parallel to the total strain increment. At this point, however, there is no work performed in the intermediate direction. The circular shape of the von Mises’ yield criteria on the pi-plane permits any strain direction, which means that work in the intermediate direction is considered. This was demonstrated in the expression for the developed strain invariant function that was shown in Equation 5.14. The mechanism developed by Osman and Bolton (2005) is for a smooth footing, and the bearing capacity according to the presented energy method with the von Mises’ yield criteria differs from the classical value of 5.69 (Cox et al., 1961) by about 7%. The ellipsoidal cavity expansion method gave a lower upper-bound at small levels of displacement and good correlation with published finite element analysis results. It was suggested that beyond small displacements the Prandtl mechanism is more 157 5. CAVITY EXPANSION MODEL Fig. 5.11 Ellipsoidal cavity expansion and Osman and Bolton (2005) mechanisms Fig. 5.12 Load-settlement behaviour for Osman and Bolton (2005) mechanism with new work calculation procedure appropriate. A similar analysis was performed to investigate the soil rigidity multiplied by normalised footing settlement. Again it was found that all results fell on a single line, and this is shown in Figure 5.13 with the ellipsoidal cavity expansion method for comparison. Although comparing smooth and rough footings, considering footing roughness increases the ultimate bearing capacity of a footing, it is evident that the ellipsoidal cavity expansion mechanism gives a lower upper-bound solution - up to a value of (w/D ×G/cu) ≈ 3. The mechanism of Osman and Bolton (2005) provides a bearing capacity very similar to the classical value for a rough footing of Nc = 6.05 from Eason and Shield (1960). It may be possible to slightly adjust the mechanism so that it provides vertical movements immediately beneath the footing - corresponding to a rough footing. 158 5. CAVITY EXPANSION MODEL Fig. 5.13 Footing stress plotted against (w/D×G/cu) for the ellipsoidal cavity expan- sion and Osman and Bolton (2005) mechanisms 5.8 Summary Centrifuge experiment results suggested a cavity expansion type mechanism beneath circular shallow foundations. This approach has been used for determining the ulti- mate bearing capacity of shallow foundations, but is less common than the classical approach. Some cavity expansion and shallow foundation cavity expansion literature was introduced. An energy approach was introduced for a linear-elastic perfectly- plastic soil following the von Mises’ yield criterion and associated flow which produced expressions for elastic and plastic work based on axisymmetry. This approach was validated by the particular case of spherical cavity expansion. The axisymmetric mechanism with displacements normal to ellipsoids was devel- oped and found to describe the centrifuge experiment results well. This mechanism was utilised in the energy approach to determine the load-settlement behaviour of rough circular shallow foundations. The results were compared with those produced from finite element analyses published in the literature, and were shown to match very well in the small settlement region. A parametric investigation was performed which demonstrated that when the foot- ing load was plotted against the rigidity index multiplied by normalised settlement, a single line was produced. The linear region of this plot represents the design region for shallow foundations and a design example was given for a typical rigidity index of London Clay. 159 5. CAVITY EXPANSION MODEL The cavity expansion mechanism developed was compared with that of Osman and Bolton (2005) and was found to give a lower upper-bound solution for a rough footing up to a value of q/cu ≈ 6 - basically the classical value. The results show that the cavity expansion mechanism could be used in the typical working range, and thereafter, the Prandtl mechanism could be adopted. Soil, however, is highly non-linear even at very small strains. Therefore, this ap- proach is now extended to non-linear soils using an expression for the non-linear be- haviour as developed from a database of triaxial test results. 160 Chapter 6 Extension of Cavity Expansion Model for Non-Linear Soil 6.1 Introduction For simplicity, elastic methods are conventionally used for determining the settlement of shallow foundations. Soil is not elastic, however, and exhibits non-linear behaviour, even at small strains. The cavity expansion model for the bearing capacity and settle- ment of circular shallow foundations on clay, presented in Chapter 5, is now extended to consider the non-linear behaviour of soil. The non-linear behaviour of clay has been characterised using a power law, as shown by Vardanega et al. (2012). New expres- sions for work are developed and the results are then presented. This is followed by a comparison of the design curve with centrifuge results and some case histories that were examined. 6.2 Non-Linear Investigations Klar and Osman (2008) investigated shallow foundations using a process of energy min- imisation to allow the displacement field to change its pattern throughout the loading sequence. This allowed the early stages of loading to use the elastic solution and the later stages to use the plastic solution. This extension of the MSD approach was termed 161 6. NON-LINEAR SOIL (a) Stress-strain curve models used (b) Elastic perfectly plastic model (c) Hyperbolic stress-strain curve (d) Truncated power law Fig. 6.1 Constitutive soil models used and the comparison between MSD, EMSD and finite difference methods from Klar and Osman (2008) extended MSD (EMSD) and was compared with finite difference solutions, found using FLAC, with three different soil constitutive models (elastic perfectly-plastic, hyperbolic stress-strain curve and truncated power law). Levin (1955) developed a deformation field based on the plane strain mechanism of Hill (1950) and this mechanism was also adopted by Klar and Osman (2008) in the analysis. Figure 6.1 demonstrates the con- stitutive models adopted and the results of each model. 6.3 Non-linear Soil Behaviour The analysis presented for a linear-elastic perfectly-plastic soil in Chapter 5 was com- pleted to ensure the process was correct through verification with past research - as was demonstrated. The truncated power law adopted by Klar and Osman (2008), as shown in Figure 6.1(a), had the equation τmob = √ Gcu γ. Vardanega et al. (2012) formed a database from a number of triaxial test results and was able to develop a 162 6. NON-LINEAR SOIL Fig. 6.2 Example stress-strain curve demonstrating the extra strain energy for non- linear analysis of soil non-linear relationship for kaolin clay. The relationship (introduced in Section 4.5.3.2 but presented again here for clarity) is: τmob/cu = 0.5(γ/γM=2) b in the range 0.2 < τmob/cu < 0.8 (6.1) with empirical expressions for the exponent, b, and mobilisation strain, γM=2, as func- tions of the overconsolidation ratio: b = 0.011(OCR) + 0.371 (6.2) γM=2 = 0.0040(OCR) 0.680 (6.3) Equation 6.1 is limited to the range 0.2 < τmob/cu < 0.8, but without further information it is deemed suitable to also use this expression outside of this range. A stress-strain curve is given in Figure 6.2 which demonstrates the extra strain energy that is exhibited by non-linear soil. For 115 tests on a variety of natural clays it was found that b was in the range of 0.3 to 1.2 with an average of 0.6. Results for b as 0.6 and the calculated value using Equation 6.2 are presented for comparison. 163 6. NON-LINEAR SOIL 6.4 Work Calculation The cavity expansion model presented in Chapter 5 is now extended for non-linear soil behaviour, modelled according to Equation 6.1. 6.4.1 Work beyond Bounding Radius The principal strains for a spherical cavity, given in Equation 5.22, can be used to find the shear strain γ: γ = |ε1 − ε3| = 3 δb rb (6.4) Compatibility (shown in Equation 5.39) is combined with Equation 6.4 and then substituted into Equation 6.1 to determine the mobilised shear stress, τmob as: τmob = cu 2 ( 3 2 r2f r3b w γM=2 )b (6.5) Substitution of Equation 6.5 into the expression for equilibrium (Equation 5.36) and integrating between an infinite radius, r = ∞, where σ = p0 and the bounding radius r = rb where σ = σb produces: ∫ σb p0 dσr = −2cu ( 3 2 r2f w γM=2 )b ∫ rb ∞ dr r3b+1 (6.6) The in-situ stress is again ignored by setting p0 = 0. Equation 6.6 can then be solved to find the stress at the bounding radius: σb = 2cu 3b ( 3 2 r2f r3b w γM=2 )b (6.7) Calculating the work is performed by multiplying together the stress at the bound- ing radius, the displacement at the bounding radius and the surface area of the bound- ing hemisphere. This produces the work beyond the bounding radius for non-linear soils, ∂Wb,n−l /∂w as: 164 6. NON-LINEAR SOIL ∂Wb,n−l ∂w = 2pi 3 cu r2f b ( 3 2 r2f r3b w γM=2 )b (6.8) 6.4.2 Work within Bounding Radius 6.4.2.1 Plastic Work The yield strain can be found by setting τmob = cu in Equation 6.1. Yield, therefore occurs when (γ/γM=2) b = 2. The yield strain is γy = b √ 2 γM=2, and this was demon- strated in Figure 6.2. For strains greater than this the soil is plastic, and identical in energy to that described in Section 5.3.2. The incremental work per unit volume, therefore, is given by: ∂Wp ∂δf = 4cu ˆ˙ε√ 3 (6.9) 6.4.2.2 Non-Linear Work The non-linear plastic work expression now used for the stress-strain curve requires the von Mises’ yield criterion to be adapted. Substituting Equations 5.46 and 6.1 into the von Mises’ yield criterion shown in Equation 5.15 produces: (σ1 − σ3) 2 + (σ2 − σ3) 2 + (σ1 − σ2) 2 = 2c2u ( γ γM=2 )b (6.10) As explained in Section 5.3.2 for a material exhibiting associated flow, the direction of the incremental strain vector, ε˙, is parallel to the deviatoric stress, σˆ = σ − p. This result, shown in Equation 5.17, can be substituted along with the condition of no volumetric strain, into the yield criterion of Equation 6.10 to find the major principal deviatoric stress: σˆ1 = cu ε˙1√ 3 ˆ˙ε ( γ γM=2 )b (6.11) 165 6. NON-LINEAR SOIL Substitution of Equations 5.17 and 6.11 into the general expression for plastic work (as shown in Equation 5.19), produces the non-linear plastic work, ∂Wn−l/ ∂w, as: ∂Wn−l ∂δf = 2cu ˆ˙ε√ 3 ( γ γM=2 )b (6.12) 6.4.2.3 Load-Settlement Behaviour The rate of work outside the bounding radius can be added to the rates of non-linear plastic and plastic work, and integrated over the appropriate regions within the bound- ing radius. This is then equated to the footing work. The relationship between footing stress and displacement for non-linear soil is hence: q = 1 pir2f   ∂Wb,n−l ∂w + ∫ non−linear ∂Wn−l ∂w dV + ∫ plastic ∂Wp ∂w dV   (6.13) Similar to the linear-elastic approach, the approach of solving Equation 6.13 was computationally inexpensive because the mechanism does not change with settlement. Also, the plastic radius was not explicitly calculated, but simply the shear strain com- pared to the computed value of yield. The linear-elastic, perfectly-plastic Matlab script that was developed was simply modified for the new expressions of work and yield strain. 6.5 Results The effect of changing the hemispherical radius and reducing the mesh size were similar to those demonstrated in Sections 5.6.1 and 5.6.2 respectively. Therefore no additional figures and discussion are presented. 6.5.1 Load-Settlement Behaviour The load-settlement behaviour was found based on values for b and γM=2 as determined in Equations 6.2 and 6.3 respectively. The results for a normally consolidated soil and soils with varying overconsolidation ratio is given in Figure 6.3. 166 6. NON-LINEAR SOIL Fig. 6.3 Load-settlement behaviour using the ellipsoidal mechanism with soils of varying overconsolidation ratio for b calculated using Equation 6.2 Fig. 6.4 Load-settlement behaviour using the ellipsoidal mechanism for soils with vary- ing overconsolidation ratio for b = 0.6 The database produced a range of values of b between 0.3 and 1.2. An average value of 0.6 was determined, however, and was also used to determine the load-settlement behaviour. The result is shown in Figure 6.4. 6.5.1.1 Effect of b The effect of using the b value calculated according to Equation 6.2 and a mean value of 0.6 is shown in Figure 6.5. The OCR corresponding to b = 0.6 is 20.8. It can be seen that for values less than 20 the average value of b = 0.6 underestimates the footing pressure for a given normalised footing settlement. At an OCR = 40 the footing pressure is overestimated by using the average value b = 0.6. Figure 6.5 also shows that 167 6. NON-LINEAR SOIL the difference in footing pressure is more significant at very small displacements. The error reduces with increasing footing settlement, and at the final normalised settlement value of w/D = 0.05 the discrepancy for both normally consolidated soil and soil with OCR = 40 is approximately 5%. Fig. 6.5 Load-settlement behaviour for normally consolidated soil and overconsolidated soils (OCR = 20 and OCR = 40) showing the effect of using the calculated value of b and b = 0.6 on the ellipsoidal mechanism 6.5.2 Design Implication The small discrepancy between using an average b value and the expression given in Equation 6.2 meant that subsequent analyses were performed using b = 0.6. A parametric study was also performed for the non-linear soil and it was found that the primary benefit of using b = 0.6 is that a single design line can be produced when q/cu is plotted against (w/D × 1/γM=2). This result is shown in Figure 6.6. As expected there is much less of the linear range in this curve (the non-linear part continues for greater values of settlement). Therefore, it is probably more appropriate to use the plot to find the footing pressure for a given settlement and mobilisation strain. For more significant settlements the linear fitting shown in Figure 6.6 can be used, given by: q cu = 3.80 + 1.34ln ( w D 1 γM=2 ) for ( w D 1 γM=2 > 0.5 ) (6.14) 168 6. NON-LINEAR SOIL It is apparent that the slope of the linear section of the design curve is the same between the linear-elastic perfectly-plastic model and the power law model used to describe the non-linear behaviour of the soil. This is expected, as at larger settlements the plastic component of work dominates and so the small strain, be it linear or curved, does not affect the results. Fig. 6.6 Footing stress plotted against (w/D × 1/γM=2) with linear fitting 6.6 Mechanism Comparison The mechanism of Osman and Bolton (2005) was again investigated, but now the non- linear soil behaviour model was used. The effect of using the calculated value of b and average value of b on this mechanism was investigated, and Figure 6.7 shows the results. It can be seen that the curves actually cross at different values of normalised settlement. In general, the effect of b is much smaller than that observed for the cavity expansion mechanism and so, herein, b is taken as 0.6. Figure 6.8 shows the load settlement curves for b = 0.6 for normally consolidated clay and clays with different overconsolidation ratio. Naturally the ultimate collapse value of q/cu = 6.11 is still observed, as the plastic behaviour dominates and the elastic or non-linear portion does not contribute. It was also possible to determine a single design line by performing a parametric study with the mobilisation strain γM=2. Figure 6.9 shows the design lines for both the Osman and Bolton (2005) and ellipsoidal cavity expansion mechanisms for non-linear 169 6. NON-LINEAR SOIL Fig. 6.7 Difference between using calculated b and b = 0.6 on Osman and Bolton (2005) mechanism Fig. 6.8 Load-settlement relationship for varying OCR on Osman and Bolton (2005) mechanism 170 6. NON-LINEAR SOIL Fig. 6.9 Single design line for non-linear soils with both mechanisms soil. In comparison to the linear-elastic model, the two curves are closer together. Overall, the cavity expansion mechanism provides a lower upper-bound and therefore is more appropriate. The mechanism of Osman and Bolton (2005) could be multiplied by a ‘roughness factor’ equal to 6.05/5.69 = 1.06, but given this factor is greater than 1, the curve would become an even higher upper-bound. 6.7 Centrifuge Results Comparison The observed undrained settlements in the centrifuge, as shown in Tables 4.3 and 4.4, can be used with the undrained shear strength, modified for anisotropy and rate effects cu,mod (Table 4.2), to compare with the cavity expansion model with non-linear soil - as shown in Figure 6.10. It is evident that the results are very good, particularly in the small stress and strain region. The results are also reasonably good at greater footing pressures. In this region the cavity expansion model now underestimates the footing pressure. This can be considered acceptable, however, because this region of the plot approximately represents the classical value of the bearing capacity factor for a circular foundation on clay - well beyond the typical working load of a footing. The centrifuge results, therefore, verify the cavity-expansion mechanism and the non-linear power-law soil model. 171 6. NON-LINEAR SOIL Fig. 6.10 Design line with centrifuge experiment results 6.8 Case Histories Literature can provide examples of full-scale testing and settlement monitoring of real structures on shallow foundations. For example, Lehane (2003) performed a full-scale bearing capacity test by monitoring the load and settlement of a square foundation in Belfast - as introduced in Section 2.3.2. Nordlund and Deere (1970) presents load- settlement data from the collapse of a grain elevator in Fargo, North Dakota. A number of researchers have collected similar data to form databases in order to investigate different methods for the estimation of shallow foundation settlement. Elhakim (2005) developed a database of load-case histories to investigate the use of small-strain stiffness in evaluating shallow foundation displacements. Strahler (2012) formed a database to compare the results to those predicted using a hyperbolic load-settlement model. The case histories used as part of this investigation were required to have both good load- settlement and soil property data. The soil properties required for the non-linear cavity expansion model are the undrained shear strength cu and mobilisation strain γM=2. Undrained strength data was presented in most case histories and where possible triaxial test values were used. Some research presented the undrained strength determined from vane tests, plate load- ing tests or the pressuremeter. The value γM=2 is a relatively new concept, however, and so without actual stress-strain curves this value could not be obtained. Vardanega and Bolton (2011) used a database of results to find an expression for the mobilisation strain as: 172 6. NON-LINEAR SOIL γM=2 = 0.0109 (IP ) 0.45 ( cu σ′0 )0.59( σ′0 101.3 )0.28 (6.15) where σ′0, the initial mean effective stress, and cu are measured in kPa. This relation- ship, however, is not particularly strong with r2 = 0.44, and should be used with due care. Data from a square foundation test performed in Bothkennar is now presented as an example procedure for the case histories. 6.8.1 Bothkennar Footing Test Jardine et al. (1995) performed a field test on a 2.2 m square pad embedded 0.8 m. An equivalent circular diameter Dequiv was found by equating the footing areas, providing Dequiv = 2.48 m. The original site investigation of Bothkennar was performed and reported in Hight et al. (1992b). Results for the shallower depths are more appropriate for spread foundations and were presented in Jardine et al. (1995). Figure 6.11 shows a summary of the site soil profile up to a depth of 7 m. Fig. 6.11 Summary of soil profile to 7 m depth at Bothkennar (from Jardine et al., 1995) Using a characteristic depth of 0.3D, and adding the embedded depth of the footing, this depth is 0.8+0.3×2.48 = 1.54 m. Figure 6.11 shows that at this depth the plasticity 173 6. NON-LINEAR SOIL index is Ip = 0.3. Extensive strength testing was performed on the Bothkennar soil, both in-situ and in a laboratory, and some results are shown in Figure 6.12. Figure 6.12(a) shows the number of different methods used as part of the site inves- tigation, and the corresponding mean undrained shear strength for the first 7 metres of depth. Figure 6.12(b) shows the peak strength profile with depth for three labora- tory tests. Anisotropy was considered in Chapter 4 by taking the mean of the triaxial compression and extension strengths. Figure 6.12(a) shows that for tubed Laval sam- ples the triaxial compression strength is cu,c = 22.5 kPa and the triaxial extension strength is cu,e = 8.8 kPa. These can be seen to be similar to the peak values which are shown in Figure 6.12(b). This provides a mean value of undrained shear strength cu = (22.5 + 8.8)/2 = 15.65 kPa. The effect of loading rate was also considered using a similar approach to that in Chapter 4. The footing test was conducted by loading the footing to failure in about 80 hours, which resulted in a mean settlement of wu ≈ 150 mm. This corresponds to a shear strain rate of γ˙ = 1.02× 10−3 hr−1. Hight et al. (1992a) indicates that triaxial tests were performed at an average axial strain rate ε˙a = 6% per day. This corresponds to a shear strain rate of γ˙ = 3.75×10−3 hr−1. Applying the correction of Kulhawy and Mayne (1990) for the effect of loading rate, shown in Equation 4.2, the shear strength should be multiplied by 0.94. The effects of rate and anisotropy are considered by determining the modified undrained shear strength, which here is cu,mod = 15.65 × 0.94 = 14.7 kPa. The ef- fect of footing embedment was simply considered by removing the stress due to the overburden of 0.8 m of soil from the applied footing pressure. The modified undrained shear strength could then be used in Equation 6.15 to determine the mobilisation strain. As an approximation, it is assumed that the initial mean effective stress of the laboratory test σ′0 in Equation 6.15 is equal to the in-situ stress σ ′ v. Figure 6.11 shows that the unit weight of the soil is γ = 17kN/m3 which gives σ′0 = 11.1 kPa. Substitution of this value with the known values of plasticity index and modified undrained shear strength at the characteristic depth into Equation 6.15 provides γM=2 = 0.0041. These soil properties are provided in Table 6.1. The load-settlement data from the footing test (test A) is given in Figure 6.13. The footing was loaded to failure, occurring at 138 kPa. Four points of load and 174 6. NON-LINEAR SOIL (a) Mean undrained shear strengths; 2-6 m depth range (b) Peak undrained shear strengths with depth Fig. 6.12 Undrained strength properties at Bothkennar (from Jardine et al., 1995) 175 6. NON-LINEAR SOIL displacement were chosen from Figure 6.13 and digitised to enable the data to be plotted with the non-linear soil cavity expansion model. The pressure due to footing embedment was removed, as discussed, and the result is shown in Figure 6.14. Fig. 6.13 Overall load-displacement behaviour (from Jardine et al., 1995) Figure 6.14 shows reasonable predictions of undrained penetration from the ellip- soidal cavity expansion model given a correlation was used to determine γM=2. The results may also fall below the model at the smaller displacements due to the footing being relatively smooth compared to the soil. 6.8.2 Case History Results Comparison A similar analysis was performed with a number of footing test results presented in the literature. The footing size, embedded depth zf , soil properties and the corresponding references for all cases are shown in Table 6.1. Where sufficient information was pro- vided, the undrained shear strength was modified to consider the effect of loading rate. If compression and extension triaxial strengths were provided, then a mean value was taken to account for anisotropy, otherwise the compressive strength was multiplied by 0.77 - from the correlation of Kulhawy and Mayne (1990). 176 6. NON-LINEAR SOIL Fig. 6.14 Non-linear cavity expansion model with observations from a footing test performed in Bothkennar (Jardine et al., 1995) The particular example of Osman et al. (2004) was for a shallow foundation on Lon- don Clay. Significant research has been undertaken into London Clay and a correlation for the mobilisation strain as a function of depth was found by Vardanega and Bolton (2011), such that γM=2 = (−2.84ln(z) + 15.42)/1000. Therefore, this expression was adopted by simply using z = zf + 0.3D. Some load-settlement points were selected from each of the load-settlement results and plotted with the ellipsoidal cavity expansion mechanism using a non-linear soil model. Results for all tests are shown in Figure 6.15. The results show reasonable promise given the correlation for γM=2 is argued by Vardanega and Bolton (2011) to not be particularly strong. The settlement for the footing on London Clay is particularly good, with only a 10% over-prediction of undrained settlement. This small discrepancy could result from the use of correlations to account for anisotropy and to determine the mobilisation strain γM=2. As more tests are added to the database, and a stronger correlation for the mobilisation strain is developed, these could be included in Figure 6.15. It is recommended that for engineers calculating the undrained settlement of real footings that a site investigation determine both the compression and extension strengths in triaxial tests to account for anisotropy. These tests should be conducted at strain rates approximately similar to that which will be incurred by the footing. If these requirements are satisfied, then a reasonably good prediction of undrained penetration should be possible. 177 6. NON-LINEAR SOIL T ab le 6. 1 C as e hi st or ie s of so m e sh al lo w fo un da ti on te st s an d ca lc ul at ed m ob ili sa ti on st ra in L oc at io n Fo ot in g Si ze (D eq u iv ) (m ) z f (m ) I P c u ,m od (k P a) γ M = 2 R ef er en ce B an gk ok 1. 05 × 1. 05 (1 .1 8) 1. 6 0. 48 23 0. 00 58 B ra nd et al . (1 97 2) an d M oh et al . (1 96 9) 0. 75 × 0. 75 (0 .8 5) 1. 6 0. 48 23 0. 00 59 0. 6 × 0. 6 (0 .6 8) 1. 6 0. 48 23 0. 00 59 B el fa st 2 × 2 (2 .2 6) 1. 6 0. 35 15 0. 00 38 L eh an e (2 00 3) B ot hk en na r 2. 2 × 2. 2 (2 .4 8) 0. 8 0. 3 15 0. 00 41 Ja rd in e et al . (1 99 5) Fa rg o 66 .5 × 15 .9 (3 6. 67 ) 1. 8 0. 65 47 0. 00 57 N or dl un d an d D ee re (1 97 0) H ag a 1 × 1 (1 .1 3) 2. 15 0. 15 81 0. 00 67 A nd er se n an d St en ha m er (1 98 2) L on do n D = 35 4. 5 11 6 0. 00 87 O sm an et al . (2 00 4) O tt aw a 3. 1 × 3. 1 (3 .5 0) 1. 3 0. 3 92 0. 01 01 B au er et al . (1 97 6) Sh el lh av en 5 × 14 (9 .4 4) 0. 2 0. 6 13 0. 00 40 Sc hn ai d et al . (1 99 3) T ex as -1 D = 0. 76 0. 61 0. 2 62 0. 00 92 St ue dl ei n an d H ol tz (2 01 0) 2. 74 × 2. 74 (3 .0 9) 0 0. 2 62 0. 00 89 an d St ra hl er (2 01 2) T ex as -2 D = 0. 58 1. 5 0. 3 62 0. 00 89 T an d et al . (1 98 6) 178 6. NON-LINEAR SOIL F ig . 6. 15 C av it y ex pa ns io n m od el w it h no n- lin ea r so il an d so m e ca se hi st or ie s 179 6. NON-LINEAR SOIL 6.9 Summary The linear-elastic perfectly-plastic soil with an ellipsoidal cavity expansion mechanism developed in Chapter 5 was extended to consider the non-linear behaviour of soil. A power law was used to describe the non-linear stress-strain behaviour of the soil in this region. Expressions for the non-linear work both within and beyond the bounding radius were provided. The effect of the power law exponent b was investigated for a derived expression that is a function of the overconsolidation ratio and the average value of b = 0.6. It was determined that the difference in results was negligible. A parametric analysis was performed using b = 0.6, yielding a single design line when the normalised footing pressure q/cu was plotted against the normalised settlement divided by the mobilisation strain (w/D × 1/γM=2). The design curve was shown in Figure 6.6. The non-linear soil model and expressions for work were applied to the mechanism of Osman and Bolton (2005). The result was compared with the current cavity expansion model and it was again determined that the cavity expansion model provides a lower upper-bound solution. The non-linear soil results were compared with the broadly undrained penetrations observed in the centrifuge tests, and showed good correlation. This chapter finished with a comparison of the non-linear soil model with some case histories of footing load tests and monitoring of real structures that are presented in the literature. Only research which presented reliable soil strength data and a plasticity index of the soil were used. A correlation for the mobilisation strain γM=2 was used with properties at the characteristic depth of 0.3D. This comparison was performed for demonstrative purposes and should be reconsidered when stronger correlations are developed using databases with more, and varying data. At this point, the case history results compared reasonably well, within a factor of 3, with the cavity expansion model using non-linear soil. 180 Chapter 7 Sand Results 7.1 Introduction Two centrifuge tests were performed on relatively loose sand models saturated with methyl-cellulose solution. The properties of the sand are given, and the problems observed placing footings on relatively stiff soil are described, in particular for the pneumatically loaded footing which was less able to conform to slopes. Movements observed were much smaller than on the clay and excess pore pressure dissipation occurred almost immediately. Therefore, the mechanisms presented correspond to total displacements encompassing both the undrained penetration and consolidation due to transient flow. Volumetric strain plots are used to distinguish these settlements. The undrained penetrations that were verified by PIV analyses of the soil, are used in a back-analysis by utilising a database of sand laboratory test results for stiffness. The results are shown to compare reasonably well with a new calculation procedure based on the approach of Atkinson (2000). Parts of this chapter were published and presented at the 15th European Conference on Soil Mechanics and Geotechnical Engineering in Athens as McMahon, B. T. and Bolton, M. D. (2011) Experimentally observed settlements beneath shallow foundations on sand 181 7. SAND RESULTS 7.2 Results 7.2.1 Sand and Model Properties Hostun sand was used as the soil body for this research. Grain size distribution tests using Single Particle Optical Sizing analysis (White, 2003) produced a uniformity co- efficient of Uc = 1.6. Using the computer-controlled sand pourer an average relative density of ID = 49% (e = 0.789) was achieved in the model. An attempt to observe consolidation due to footing loading was made by increasing the viscosity of the pore fluid, to slow excess pore pressure dissipation. The sand was saturated with methyl cellulose at a viscosity of 100 cSt - 100 times that of water. As discussed in Section 3.6.3, saturation was performed under vacuum. The resulting prototype saturated unit weight was 18.9 kN/m3 giving a buoyant unit weight of γ′ = 9.1 kN/m3. The sand body was allowed to settle in the centrifuge before the soil was loaded with the footings. 7.2.2 Soil Surface The pneumatically loaded footings were rigidly fastened to the connecting rod and were, therefore, unable to conform to uneven or sloping sand surfaces. Although the sand surface was levelled following both the sand pouring and saturation procedures, a perfectly level surface following centrifuging, evidently, could not be obtained. Figure 7.1 portrays the footing immediately after loading of the soil had occurred in test S-1. It can be seen that most of the footing load is applied on the right hand side, resulting in localised displacements. The radial effect from the centrifuge will have also contributed to this problem. Subsequent load increases tended to spread the contact surface towards the left hand side. McMahon and Bolton (2011) considered that when constructing shallow foundations on hard soils and weak rocks, it is important to ensure that there is a level surface for pre-cast foundations. Design calculations are performed assuming ideal conditions (a level surface) but in-situ this may not occur, potentially resulting in differential settlements and possibly local structural failures. For shallow foundations poured in-situ the importance of the blinding layer is also apparent. The relative size of the sand grains to the footing diameter in the centrifuge model could also provide issues. The average grain size of D50 = 0.34 mm (Table 3.8) presents the possibility of asperities in the sand surface - even if it could be considered level. 182 7. SAND RESULTS Fig. 7.1 Uneven load distribution due to uneven surface in test S-1 Fig. 7.2 Discrepancy in PIV and laser results due to sand surface in test S-2 183 7. SAND RESULTS The effect of a sloping or uneven soil surface could also be observed in the contrast between footing movements measured by the laser and through PIV analysis, and the load cell data. Figure 7.2 presents the footing settlement data using both the laser observations and a PIV analysis of the footing face once the footing load was applied in test S-2. It can be seen that the footing has settled further at the back - as measured by the laser. This could be attributed to a slight rotation (∼ 10−3) of the connecting rod and footing away from the Perspex window. Levelling of soil up to the window was problematic due to the need to protect the PIV control markers. Furthermore, friction against the window would tend to suppress settlement during centrifuging. The rotations of the footing could also be inferred from the load cell observations. Figure 7.3 shows the load cell registered an average footing pressure of 65 kPa. The expected footing pressure was q = 72 kPa, and this discrepancy can be attributed to a rotation of the footing. The load cells are very sensitive to bending moments and this rotation has resulted in a lower bearing pressure being observed. Chapter 4 showed that there was negligible friction between the footings and Perspex for the clay tests and so it is assumed that the full load of q = 72 kPa was applied to the sand, particularly in the early stage of loading, even if friction may have increased with further rotation. Results for the pneumatically loaded footing in test S-1 were discarded due to the localised displacements. The observed settlements from PIV in test S-2 were verified after the magnitude of movement was confirmed by a PIV analysis on the soil immediately beneath the footing. The PIV observations of settlement are also more appropriate to use because this represents the plane of interest. Figure 7.4 shows the excess pore pressure and footing settlement for the first 80 seconds after loading of the soil. It can be seen that the excess pore pressures dissipate almost immediately. The settlement is approximately 0.38 mm at a time of 16 seconds. Observed mechanisms, therefore, represent the sum of undrained penetration and con- solidation. The loading setup of the dead-weight footings, as outlined in Section 3.4.1, meant they were able to conform to slightly sloping surfaces. These results, therefore, are now used as the example set for mechanisms and strain plots. 184 7. SAND RESULTS Fig. 7.3 Observed bearing pressure in test S-2 in model scale time Fig. 7.4 Immediate excess pore pressure dissipation and small movements (model scale) 185 7. SAND RESULTS 7.2.3 Observed Mechanism The 50 mm diameter dead-weight footing, 5 m at prototype scale, applied a dead- weight load of 100 kPa to the soil surface in test S-2. The PIV analysis was performed on photographs taken before and about 3 seconds after loading of the soil. Figure 7.5 shows the raw mechanism for these times. As discussed, due to the high permeability of the sand the mechanism portrays the fully drained mechanism (undrained penetration plus consolidation settlement). It can be seen that heave adjacent to the footing has occurred - something that was not observed in the footing tests on clay. The relatively greater stiffness of sand beds provides the heave that has occurred, with a settlement of 0.68 mm in this case. Fig. 7.5 Raw mechanism observed for the 5 m prototype scale footing The interpolation script was again used to ensure that lost data was recognised and replaced. The result of running this program is shown in Figure 7.6. It can be seen that in the interpolation process the magnitude and direction of some of the heave is lost. This is caused by the zero movement which is obviously recorded above these points (beyond the sand surface). Figure 7.6 confirms the vertical compression below the centre of the footing, with 186 7. SAND RESULTS Fig. 7.6 Interpolated mechanism of undrained penetration and consolidation symmetric shear zones leading to heave adjacent to the footing edge. It is also evi- dent that horizontal movement has occurred directly beneath the footing due to the relatively smooth interface between the footing and sand surface. The raw mechanism, shown in Figure 7.5, appears to be more similar to that de- veloped by Osman and Bolton (2005), as was shown in Figure 4.36. Therefore, this mechanism may best describe those soil beds with greater stiffness. Strain plots can be used to find the depth of influence due to footing loading and also to distinguish the components of settlement. 7.2.4 Strains Engineering shear strain, γ, and volumetric strain, εv, are determined by forming a triangle using the three nearest known displacements. A mesh size of 1 mm was used to perform this strain calculation. Figure 7.7 depicts the engineering shear strain (%) for the interpolated deformation mechanism shown in Figure 7.6, and the boundary of the Prandtl mechanism. It can be seen that the strain field occurs within this boundary, with strains being observed to a depth of about 0.5D. The “noise” that occurs in the process of PIV is also evident. 187 7. SAND RESULTS Fig. 7.7 Engineering shear strain (%) for the 5 m prototype scale footing Fig. 7.8 Volumetric strain (%) beneath the 5 m prototype scale footing 188 7. SAND RESULTS Figure 7.8 shows the volumetric strain (%) beneath the footing, with compression designated as positive. There is some volumetric strain directly beneath the footing and the adjacent heave is highlighted by the negative values of strain. The greater part of the volumetric strain is within the top 3 mm (∼ 8D50) of the surface. This might indicate an uneven or sloping surface due to the sand grains. In order to investigate the settlement due to volume change the volumetric strain (%) along the centreline of the footing was plotted, and is shown in Figure 7.9. Fig. 7.9 Volumetric strain (%) along the centreline of the footing (compression positive) The volumetric strains in Figure 7.9 were separated into the distinctively high strain region of the top 3 mm and the remaining region down to a depth of about 0.5D. Lines of best fit were applied to these regions allowing integrations to be performed and the settlement due to each component to be quantified. The interface zone settlement was determined to be 0.09 mm and the consolidation settlement due to transient flow was wc = 0.16 mm. These can be subtracted from the total observed settlement to obtain an estimate of the undrained penetration wu = 0.43 mm that might have occurred if the load had been applied sufficiently quickly. 7.3 Back-Analysis A back-analysis was performed using the hyperbolic stress-strain relationship of Ozto- prak and Bolton (2012). The equivalent linear shear strain, γ∗, due to footing settle- 189 7. SAND RESULTS ment can be determined using Atkinsons Method (Atkinson, 2000): γ∗ = wu 2D = 0.43% (7.1) The hyperbolic shear stress-shear strain relationship from Oztoprak and Bolton (2012) was developed through a database of 454 sand tests presented in the literature. The expression is given as: G∗ G0 = 1 1 + ( γ∗−γe γr )a (7.2) where the curvature parameter a = U−0.075c . The maximum elastic shear modulus, G0, and characteristic strains, γr and γe, are functions of the soil properties, including voids ratio e, uniformity coefficient Uc, relative density ID and the mean effective stress, σ∗. The expressions for these parameters are: G0 = 57600 (1 + e)3 √ σ∗ γr(%) = 0.0001U −0.3 c σ ∗ + 0.08 e ID γe(%) = 0.0002 + 0.012 γr (7.3) Equation 2.8 showed an expression for the settlement of a rigid circular punch on an elastic bed (Davis and Selvadurai, 1996). From this, it was possible to show in Equation 2.14 that wc/wu = 1 − 2ν ′. Using the known values of these settlements it was found that ν ′ = 0.31, which is typical of conventional design values. Using this value of drained Poisson’s ratio and the centrifuge test results, Equations 7.2, 7.3 and 2.8 with G = G∗ were solved to find σ∗ = 17 kPa. The analysis technique of MSD uses a representative location at a depth of 0.3D beneath a pad footing of diameter D. The classical rigid-plastic mechanism for bearing capacity and the displacement field developed in Osman and Bolton (2005) have an active zone, shear fan and passive zone adjacent to the footing. Figure 7.10 shows a possible simple method for determining the in-situ stresses before and during footing loading. A state of triaxial stress is assumed at these locations and hence a value for the mean effective stress, σ′, can also be determined using the formulae shown within. 190 7. SAND RESULTS Fig. 7.10 Stress determination at the representative depth before and after loading to find σ′ Soil stresses before loading are determined using the at rest earth pressure coef- ficient, K0 = 1 − sin(φmax). The maximum friction angle, φmax is calculated using Equation 7.4 with a critical state friction angle φcrit = 33◦ (Bolton, 1986). φmax = φcrit + 3 ID (ln(20000/σ ′)− 1) (7.4) Utilising the bearing capacity formulation in Equation 2.2, a mobilised friction angle can be determined for the applied load of q = 100 kPa. Given there is no overburden at the footing base it is only an Nγ problem. The mobilised friction angle was determined to be φmob = 23◦. This demonstrates that the soil was not at a loading level near to its bearing capacity. At large strains, sand exhibits dilatancy under shear and its stress state is generally assumed to approach the Mohr-Coulomb strength envelope. In that case, solutions by the method of characteristics show “slip lines” separated not by 90◦ as in the case of Tresca solutions for undrained clay but by (90◦−φ). At small to moderate strains sand 191 7. SAND RESULTS tends to contract slightly, not dilate. Similarly, a Mohr circle of effective stress falls well inside its Mohr-Coulomb strength envelope. A value φmob = 23◦ was determined above and in these circumstances it appears not to be unreasonable to base estimates of mobilised stress on the Prandtl mechanism for shearing at constant volume. This can then be compared with test results, as was shown in Figure 7.7. During loading the stresses in the active and passive zones are determined using the active and passive earth pressure coefficients, Ka and Kp respectively. These are determined by: Ka = 1 Kp = 1− sinφmob 1 + sinφmob (7.5) The representative depth for this footing occurs at z = 1.5 m at prototype scale. Results of calculations performed at the representative depth before and after footing loading are shown in Table 7.1. Table 7.1 Principal stresses and equivalent triaxial stress before loading and in the active and passive regions after loading Parameter Before Loading Active Zone Passive Zone σ′v 14 kPa 114 kPa 14 kPa K 0.33 0.44 2.28 σ′h 4.5 kPa 50 kPa 31 kPa σ′ 8 kPa 71 kPa 19 kPa An operational value for σ′ should represent the average value of mean effective stress during loading. Thus, it is taken as a mean of the values before and after loading, σ′ = 1/2(σ′0 + σ ′ loaded). The loaded value, σ ′ loaded, must represent an average value in space, and is taken as the geometric mean of σ′ below (inside), and surrounding (outside) the loaded footing. Naturally, these are taken to correspond with the active and passive zones in the soil. Therefore, a possible calculated operational mean effective stress, σ′operational, could be: σ′operational = 1 2 ( σ′0 + √ σ′insideσ ′ outside ) = 22.5kPa (7.6) 192 7. SAND RESULTS This operational mean stress is similar to the ideal value σ∗ = 17 kPa calculated earlier. This stress corresponds to a maximum shear modulus G0 = 48000 kPa and a predited settlement of 0.30 mm, in comparison with the observed value of 0.43 mm. It is evident that both the settlement and mean effective stress compare reasonably well between observed results and the predicted values. Two footing tests with laser settlement data being observed and correlating rea- sonably well with PIV analysis, in particular immediately after the footing load, were conducted on sand models. The demonstrated back-analysis procedure was completed for these footings and a summary is shown in Table 7.2. Table 7.2 Back-analysis of two verified footing tests with observed and predicted set- tlements Test Parameters D = 50 mm D = 100 mm wu 0.43 mm 0.21 mm σ∗ 17 kPa 23 kPa σ′operational 22.5 kPa 32 kPa wpred 0.30 mm 0.13 mm Table 7.2 shows reasonable agreement between the centrifuge observed settlements and the predicted values. The method proposed has made simplifications and could be improved with further work. The volumetric strain and subsequent settlement calculations were relatively simple and could have fortuitously provided typical design values of ν ′. The operational stress, σ′operational, could have been reduced by considering the Boussinesq stress distribution with depth - as described in Equation 2.6. This expression can be used to find that the stress at the characteristic depth of z = 0.3D for an elastic material is about 86% of the applied footing pressure, rather than the full bearing pressure that was used. A range of values, lower-bound, mean and upper- bound, for the characteristic strains of Equation 7.3 and a mean value for the curvature parameter were given in Oztoprak and Bolton (2012). Adopting a different set of values, rather than the calculated values may also provide more accurate predictions. The settlement and mechanism of the 100 mm footing may have been affected by rotation and friction, as discussed in detail, and this may have caused the greater 193 7. SAND RESULTS discrepancy in observed and predicted settlement. Also, due to these reasons the actual applied load can not be confirmed for this footing. 7.4 Experiment Problems Chapter 3 presented the details of an attempt to use a high speed camera to capture the undrained mechanism of a footing load on saturated sand. Owing to the relatively small movements that were observed and the low resolution of the camera, this was not able to be captured. The apparatus for the high speed camera was developed and could be used in a new test with greater footing pressure. This may provide an even greater insight, or further verification of the method proposed above. 7.5 Summary Centrifuge experiments were performed to observe the settlements and deformation mechanisms beneath circular foundations on a saturated bed of Hostun sand. The properties of the sand were presented and problems associated with non-conformity at the soil-foundation interface, relatively larger average grain size and the rigidly connected pneumatically loaded footing were discussed. The permeability of sand beds corresponded to almost immediate excess pore pressure dissipation and the relative stiffness resulted in very small footing settlements being observed. A total displacement mechanism for a dead-weight footing applying 100 kPa was provided in Figure 7.6. The footing was seen to be relatively smooth compared to the sand. The stiffness of the sand bed resulted in heave adjacent to the footing being observed. No centrifuge experiments were performed on sand beds with a higher relative density because of the small magnitude of settlements that were observed. Shear strain and volumetric strain plots were also provided. A volumetric strain plot along the footing centreline was used to distinguish an interface zone of one-dimensional compression about 3 mm in depth, overlying a mechanism encompassing both shear and volume change to a depth of about 0.5D. The resulting calculated value of ν ′ = 0.31 was similar to what would typically be used in design. A back-analysis was performed using a database of laboratory sand test results to 194 7. SAND RESULTS determine the operational mean effective stress. An operational stress at the assumed characteristic depth of 0.3D was developed considering the stress before and during footing loading. The stress during loading was, therefore, taken as the geometric mean of the mobilised stresses beneath the footing and beneath the free surface. Similar values of the operational stress and mean effective stress provided reasonable predic- tions of undrained settlement. This method was relatively simple and possibilities for greater accuracy were discussed. To conclude this chapter, the mechanism of undrained penetration and consolidation shown on a captured photograph is given in Figure 7.11. Fig. 7.11 Total displacement mechanism (shear plus consolidation) of a footing on saturated sand 195 7. SAND RESULTS 196 Chapter 8 Conclusions Shallow foundations are one of the most simple design tasks for geotechnical engi- neers. The ultimate bearing capacity is determined based on concepts using plasticity theory, and the results observed in practice and research show reasonably good correla- tion. Settlement calculations, usually the governing design criteria, are calculated using linear-elastic properties of soil despite it being highly non-linear, even at small strains. Uncertainty about methods used for settlement prediction, and discrepancies between predicted and actual settlements, has ensured that research on shallow foundations - after nearly a century - continues today. Structures can experience settlement, tilt and distortion. A uniform settlement can be tolerated by some structures, but can still comprise a serviceability failure. Differential settlements cause tilt and angular distortion of structures - the primary cause of structural damage. This can cause functional issues or possibly even collapse. Examples of structures which have experienced these deformations were introduced and provided motivation for this research. Differential settlement can be minimised if the monolithic settlement of shallow foundations can be predicted more accurately. Some current design methods for shallow foundations and some of their associated de- ficiencies were presented. The relatively new approach of Mobilisable Strength Design (MSD) was developed as a simple design methodology to consider both the bearing capacity and settlement in a single calculation. It idealises a deformation field within the Prandtl mechanism and uses plasticity theory, but also considers strain hardening. MSD, however, needs verification through physical modelling and this provided the aim of this research: to observe deformation mechanisms beneath shallow foundations. 197 8. CONCLUSIONS One-dimensional actuators were developed in order to simulate two types of con- struction. The first used the dead-weight of aluminium footings to apply a load of 100 kPa instantly to the soil, which could simulate a tank loading scenario. The other footings could simulate building projects by applying a load initially through the dead-weight of the footing, and then increasing this load by applying compressed air to pneumatic cylinders. The development of the analysis technique Particle Image Velocimetry (PIV) provided the means for observing the axisymmetric deformation mechanism beneath semi-circular foundations from photographs captured during the experiments. Centrifuge model tests of circular foundations on the surface of over- consolidated clay beds and saturated sand beds provided data of settlements and ac- companying soil deformation mechanisms over times ranging from 3 hours to 7 years at prototype scale. Pore pressure responses confirmed that the corresponding soil- foundation response ranged from fast undrained shearing with significant rate effects, through transient drainage with contemporaneous creep, to fully drained creep. 8.1 Circular Shallow Foundations on Clay Three centrifuge tests were performed on both soft and stiff clay beds, as dependent upon the pre-consolidation pressure. Footing settlement-time data was obtained to- gether with mechanisms in the undrained phase and during consolidation and creep - as verified by pore pressure measurements and volumetric strain plots. The method of preparation and testing of the clay models meant that their effective stress history was known. Furthermore, a suite of undrained triaxial compression tests had been conducted to relate the shear strength profile to stress history. These validated a power curve as the stress-strain relation together with values of its key scaling parameter, the shear strain γM=2 required to mobilize half the shear strength. In a field application this could be replaced by a conventional programme of coring and laboratory testing. The successful back-analysis of undrained footing penetration depended on making three adjustments to the measured soil strengths: a rate effect of 10% per factor 10 on strain-rate or test duration; an anisotropy effect by which it was proposed that the operational strength in bearing was 0.77 times the strength in a triaxial compression 198 8. CONCLUSIONS test; and a finite deformation correction that allowed for soil softening by at least 30% when proportional footing settlements w/D reached 15%, and which accounted for the corresponding overburden pressure in calculating net bearing pressures. Each of these assumptions was recognised to be a simplification of reality, but bearing capacity and footing settlement were predicted with reasonable accuracy. The successful prediction of ultimate consolidation settlements wc following tran- sient flow was based first on the ability to predict undrained settlement wu, and then on the use of elasticity theory to give wc/wu = 1−2ν ′, where Poisson’s ratio was found to be ν ′ ≈ 0.3. The significant times during which transient flow takes place vary over a factor 1000, much longer than with an oedometer. This is because of the influence of local drainage beneath the edge of a footing, which creates additional settlement and load redistribution at an early stage. Creep settlements before and during consolidation, ws, were predicted simply by interpolating between undrained settlements being said to increase by creep at 19% per factor 10 on time from combining the power law for stress versus strain with the proposition that strength should reduce by 10% per factor 10 on time, and the drained creep found from the experimental data. Fully drained creep occurs at a slower rate since the soil is further from failure. The essential step to predicting footing behaviour by this route is a good prediction of its undrained penetration. This mechanism was shown to be better represented by cavity expansion than by classical rigid-plastic bearing mechanisms. 8.2 Cavity Expansion Model The observed broadly undrained mechanism in the centrifuge tests was used as the ba- sis of a cavity expansion model. The analysis initially determined the load-settlement behaviour of a linear-elastic perfectly-plastic soil. An upper-bound energy approach was adopted utilising an axisymmetric mechanism with displacements normal to ellip- soids, as shown in Figure 5.4. The von Mises’ yield criterion with associated flow was utilised to determine the plastic work. The results were found to match very well in the small settlement region with those produced from finite element analyses. 199 8. CONCLUSIONS A parametric investigation was performed which demonstrated that when the foot- ing load was plotted against the rigidity index multiplied by normalised settlement, a single line was produced. The linear-elastic perfectly-plastic soil with the ellipsoidal cavity expansion mech- anism was then extended to consider the non-linear behaviour of soil in Chapter 6. A power law, developed using a database of triaxial test results, was used to describe the stress-strain behaviour of the soil in this region. A similar parametric analysis was performed and a single design curve was found - being a function of normalised footing pressure and normalised footing settlement divided by the mobilisation strain γM=2 - as shown in Figure 6.6. The centrifuge experiment results and some case histories showed good correlation with this curve. This curve, therefore, could be used as part of the design process. 8.3 Circular Shallow Foundations on Sand Hostun sand models at a relative density of 49% were successfully saturated using a methyl-cellulose solution with a viscosity 100 times greater than water. Two successful centrifuge experiments were performed on these models. The permeability of sands re- sulted in excess pore pressures dissipating almost immediately. An attempt was made to capture the undrained mechanism using a high speed camera, but due to the rel- atively high stiffness of sand beds the settlements were too small to be observed. A mechanism of undrained penetration and consolidation was presented and volumetric strain plots used to distinguish the undrained penetration from consolidation settle- ments. The mechanisms demonstrated heave adjacent to the footings. An expression for stiffness developed from a database of laboratory tests was used as part of a back analysis. An operational mean effective stress σ′ bounded within the mechanism and considering the zone of heave was developed, and showed good correlation between predicted and observed settlements. 200 8. CONCLUSIONS 8.4 Implications for Practice This research programme was conducted with a design-based focus in which the out- come has resulted in a number of implications for design. These include: ˆ Observing actual mechanisms that occur beneath shallow circular foundations motivated a new design approach for the undrained bearing capacity and settle- ment of circular footings on clay. Assuming a linear-elastic perfectly-plastic soil provided a design curve which only required one soil parameter - the soil rigidity index - to satisfy both bearing capacity and settlement criteria. Using the more appropriate non-linear soil model, a single design curve was also developed which only requires the mobilisation strain (strain mobilised at half the shear strength) to again satisfy both design criteria. These curves can easily be used by any practising engineer for the design of circular shallow foundations. A correlation was used for the mobilisation strain but it is recommended that an actual site investigation be performed to avoid the use of too many correlations. ˆ There has been a significant amount of investigation into London Clay and a great deal of the results are well documented. By using the design curves that were developed in this research, and some soil strength and stiffness correlations that have been developed through databases of soil tests, it is possible to calculate a reasonable estimate of the undrained penetration of a footing on London Clay, without the need to perform a site investigation. For verification, a site investi- gation should still be performed, but the ability to quickly predict the undrained settlement in practice is invaluable. ˆ This research used centrifuge experiments to observe real mechanisms beneath circular shallow foundations. The subsequent analysis that produced the design curves provide practice with a design process that is based on actual physical modelling results, and can therefore be used with reasonable confidence and un- derstanding. 201 8. CONCLUSIONS 8.5 Directions for Future Work The centrifuge experiments results and the analysis performed for this research have presented further possible areas of research. 8.5.1 Footing Effects The aluminium footings used in the centrifuge experiments were observed to be rel- atively rough. A rougher, and possibly perfectly rough, footing base could easily be achieved by bonding sand grains to the bottom face of the footings. Conversely, foot- ings could be smoothed through further machining to observe the mechanism beneath smooth circular footings. The effect of the footing stiffness could also be investigated by conducting exper- iments with varying model footing thicknesses. The dead-weight method which has been used required thick aluminium to be used, and so different materials may need to be adopted. The thickness of the pneumatically loaded footings could easily be changed to observe the effect of footing stiffness on deformation mechanisms and settlements. 8.5.2 Soil Investigation The method of preparation and testing of the clay models meant that their effective stress history was known. A database of undrained triaxial compression tests had been conducted to relate the shear strength profile to stress history. In a field application this could be replaced by a conventional programme of coring and laboratory test- ing. Although good correlation was observed, an investigation into some in-situ testing methods, such as the pressuremeter and plate loading tests, should be performed be- cause parameters obtained from these tests may be more appropriate. They may also provide the required soil quantities more quickly and cheaply. This research, and in particular the bearing capacity investigation, has effectively simulated plate loading tests and so this may provide better information. Further research could provide a set of guidelines which clearly states the required soil testing method and design approach. As an on-going process, it is recommended that as more results from both laboratory 202 8. CONCLUSIONS testing of clay soils and settlements of shallow foundations become available these, either be added to existing databases or used to create new databases which may provide different and more accurate correlations. This would provide more confidence in calculations performed without performing site investigations. 8.5.3 Creep Mechanisms for creep and consolidation were observed during the centrifuge testing programme. No distinction could be made between these components of settlement and so a new approach to creep prediction was developed. The adopted approach attributed a significant proportion of total settlement due to foundation loading to creep. This suggests that the creep settlements cannot simply be ignored as research suggests. This approach, however, needs confirmation and so further centrifuge testing could be performed. For ease, and to save time, tests should be conducted on models with a smaller depth of clay. This could be achieved by using an aluminium frame within the package so that shallow clay bed models can be created. This would reduce the required self-weight consolidation time and thus allow observations to be recorded for a longer period of time. Centrifuge tests could also be performed for a longer period of time for further drained creep investigation. 8.5.4 Further Sand Tests and Investigation Saturation of sand models proved the most difficult element of this research. Without these problems more sand tests would have been performed. 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