A BIOT FORMULATION FOR GEOTECHNICAL EARTHQUAKE ENGINEERING APPLICATIONS by YU BAO B.S., Tianjin University, China, 1998 M.S., Tianjin University, China, 2001 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Doctor of Philosophy Department of Civil, Environmental, and Architectural Engineering 2006 For Tote #: Vol.Issue: 67-05 School Code: 0051 E COL-BOUL-E Copyright: Bound:N LCP: N Size:N Pub No Author DEGREE PAGES See : 3219029 Bao, Yu 240 241 CAO 2006 DATA SHEET USE THESE INSTRUCTIONS FOR BEPRESS SUBMISSIONS. Do not photo lib. pg. do not use acct number on sch list. 1o f 10Page AREA REWORK Author Check-in Publishing Camera 2006 Publication Notes : A Biot@formulation for @?geotechnical earthquake @?engineering applications 174934-831-0 Pagination Note : Standing Order : Format : Geo Class : Scan :USA B SEP-20-2006 elec tpi umi+1 ii-xv 1-225 241 N EXPO EBeam : Y Normal size (8 1/2 X 11) Camera and EBeam This thesis entitled: A Biot Formulation for Geotechnical Earthquake Engineering Applications written by Yu Bao has been approved for the Department of Civil, Environmental, and Architectural Engineering ___________________________________ Stein Sture ___________________________________ Hon-Yim Ko Date_____________________ The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline Bao, Yu (Ph.D., Civil Engineering) A Biot Formulation for Geotechnical Earthquake Engineering Applications Thesis directed by Professor Stein Sture The mechanical behavior of saturated soil is mainly governed by the interaction between the soil skeleton and the pore fluid, and this interaction may lead to significant loss of strength known as liquefaction under seismic loadings. The main objective of this thesis is to develop and implement a cyclic constitutive model capable of modeling soil skeleton dilatancy during earthquake excitation. The constitutive model is based on the fuzzy-set plasticity theory and enhancement is made on the description of dilatancy behavior under cyclic loading. A robust Biot formulation, in which the governing equations of motion of the soil mixture are coupled with the global mass balance equations, is developed to describe the realistic behavior of saturated soil. The finite element discretization is established without neglecting the convective terms. An unconditionally stable implicit time integration scheme, Hilber-Hughes-Taylor α method is used and an iterative algorithm based on Newton-Raphson method is developed to solve the nonlinear time-discretized problem. A numerical study of sand liquefaction is performed and compared with the centrifuge experimental results to show the capabilities of the proposed formulation on pore water pressure generation and strength loss occurred in loose granular soil deposit under cyclic loading. The computed results show good agreement with the experimental data. The capability of the enhanced fuzzy-set model in simulating cyclic soil behaviors including liquefaction is validated. It is concluded that the developed Biot formulation and computational procedure are an effective means to assess liquefaction potential and liquefaction-related deformations. iv Dedication To My Family For your love and encouragement v Acknowledgments I would like to express my most sincere appreciation to my advisor Professor Stein Sture for his invaluable guidance, encouragement and support throughout this study. I would like to thank Professor Hon-Yim Ko for his friendship, and his guidance and support in centrifuge modeling. I am also very grateful to Professor Richard Regueiro for his sharing of knowledge and support. I am grateful to Professor Tad Pfeffer and Professor Carlos Felippa for serving as my defense committee. I would like to thank Professor Ronald Pak, who was my academic advisor during my first year of Ph.D. study. My appreciation goes to Dr. Yu-Ning Ge for his help in understanding the fuzzy- set plasticity constitutive model and Dr. Sung Ryul Kim for his help in the centrifuge experiments. My thanks go to my parents, my husband Miao and my son Dylan for their love, encouragement and support. vi CONTENTS CHAPTER 1 INTRODUCTION……………………………………………………………1 1.1 Motivation……………………………………………………………… 1 1.2 Research Focus………………………………………………………… 4 1.3 Scope of Work and Layout………………………………………………5 2 BACKGROUND AND LITERATURE REVIEW………………………… 7 2.1 Laboratory Tests…………………………………………………………8 2.2 Centrifuge Modeling……………………………………………………15 2.3 Constitutive Modeling………………………………………………….16 2.4 The Theory of Mixtures……………………………………………… 24 2.5 Finite Element Implementation…………………………………………26 2.6 Time Integration Scheme……………………………………………….27 3 ENHANCED FUZZY-SET PLASTICITY CONSTITUTIVE MODEL… 29 3.1 Fuzzy Sets………………………………………………………………30 3.2 Classical Plasticity Theory…………………………………………… 31 3.3 Fuzzy-Set Plasticity Theory…………………………………………….35 3.4 Stress Control Formulation in p-q Space……………………………….38 3.5 Stress Control Formulation in Cartesian Stress Space………………….46 3.6 Strain Control Formulation…………………………………………… 50 3.7 2-D Plane Strain Formulation………………………………………… 52 3.8 Kinematic Mechanism of Deviatoric Membership Function d γ ………54 vii 3.9 Kinematic Mechanism of Locking Membership Function l γ …………62 3.10 Model Parameters………………………………………………………67 3.11 Model Capabilities on Cyclic Mobility…………………………………67 3.12 Model Responses……………………………………………………….71 3.13 Dilatancy Parameters………………………………………………… 91 3.14 Model Calibration………………………………………………………93 3.14.1 Unconstrained Numerical Optimization…………………… 93 3.14.2 Numerical Optimization in Constitutive Model Calibration 96 4 THEORY OF MIXTURES FOR FLUID-SATURATED POROUS MEDIA…………………………………………………………………… 98 4.1 Average Quantities…………………………………………………… 99 4.2 Laws of Balance……………………………………………………….103 4.2.1 Balance of Mass…………………………………………….104 4.2.2 Balance of Linear Momentum…………………………… 107 4.2.3 Balance of Angular Momentum…………………………….110 4.2.4 First Law of Thermodynamics (Balance of Energy)……….113 4.2.5 Second Law of Thermodynamics (Entropy Inequality)…….113 4.3 Field Equations for Saturated Soil…………………………………….114 5 FINITE ELEMENT FORMULATION….……………………………… 119 5.1 The Finite Element Method (FEM)………………………………… 119 5.2 Matrix Form of the Field Equations for Saturated Soil……………….122 5.3 FEM Form of the Balance of Linear Momentum in the Solid Phase…125 5.4 FEM Form of the Balance of Linear Momentum in the Fluid Phase…129 5.5 FEM Form of Conservation of Mass for the Mixture…………………132 5.6 Combination of the Discretized Governing Equations……………… 134 viii 6 COUPLED FINITE ELEMENT – INFINITE ELEMENT MODEL…… 137 6.1 Mixed Displacement – Pore Pressure Element……………………… 138 6.1.1 2-D 9-node Lagrangian Isoparametric Quadrilateral Element…… 139 6.1.2 4-node Lagrangian Isoparametric Quadrilateral Element…………145 6.2 2-D 6-Node Infinite Element…………………………………………147 6.2.1 1-D 2-Node Infinite Element………………………………………148 6.2.2 2-D 6-Node Infinite Element…………………………………… 150 6.3 2-D Coupled Finite Element - Infinite Element Numerical Model… 158 7 TIME INTEGRATION AND NONLINEAR ANALYSIS……………… 159 7.1 Direct Time Integration Techniques………………………………… 160 7.1.1 Newmark Method…………………………………………………161 7.1.2 Hilber-Hughes-Taylor α -Method………………………………….163 7.2 Implementation of Hilber-Hughes-Taylor α -Method……………… 165 7.3 Newton-Raphson Method: Nonlinear Analysis……………………….167 7.4 Calculation Procedure……………………………………………… 169 8 VERIFICATION OF NUMERICAL SIMULATION…………………… 172 8.1 Centrifuge Modeling of Soil Liquefaction…………………………….173 8.1.1 Centrifuge Model Test on a Layer of Liquefiable Sand Deposit… 173 8.1.2 Experimental Results………………………………………………176 8.2 Fully Coupled FEM Code “DYNSOILS”…………………………… 187 8.2.1 Pre-Processing Module…………………………………………… 187 8.2.2 Post-Processing Module…………………………………………….188 8.2.3 Analysis Module……………………………………………………188 8.3 Case 1: A Liquefiable Layer of Sand Deposit ix – Comparison of Numerical Simulation with Centrifuge Experiment 190 8.4 Case 2: A Footing Subject to Vertical Sinusoidal Loading………… 206 8.5 Summary………………………………………………………………215 9 SUMMARY, CONCLUSION AND RECOMMENDATION FOR FUTURE WORK…………………………………………………………………… 216 9.1 Summary and Conclusions……………………………………………216 9.2 Recommendation for Future Work……………………………………218 BIBLIOGRAPHY………………………………………………………………219 [...]... granular soils and Biot field equations used in overall analysis Traditional concepts of critical state soil mechanics, state parameter and phase transformation surface are introduced in the enhanced model The new development of a strongly dilative/contractive phase beneath the failure surface is added and new analytical developments for the dilatancy parameter are presented The numerical optimization... numerical techniques, material model formulation and so on, reliable and accurate predictive methods have yet to be developed 2 Since 1960’s, significant progress has been made in understanding the liquefaction phenomena There are three main approaches: (1) field observations before, during and/or after earthquakes, (2) laboratory experiments, and (3) numerical simulations Among the theoretical studies,... acceleration records show a strong influence of soil dilation or contraction at large cyclic shear strain excursions Dilation phases can cause significant regain in shear stiffness and strength and lead to a strong restraining effect on the magnitude of cyclic and accumulated permanent shear strains, while a contractive soil skeleton typically leads to immediate liquefaction Under cyclic loading, a. .. with relative density ranging from 68% to 79% (Koga and Matsuo, 1990) Fig 2.5 shows a representative of the recorded accelerations and pore water pressures Fig 2.6 shows the time histories of 12 acceleration and lateral displacement in a shake table test conducted by Ishihara et al (1991) Sasaki et al (1991, 1992) reported a series of tests on a liquefiable layer of loose sand prepared by an underwater... Pioneering and traditional liquefaction studies related to the overall phenomenon and cyclic mobility include the work by Seed and Lee (1966), Casagrande (1975), Castro (1975), Castro and Poulos (1977) and Seed (1979) 8 2.1 Laboratory Tests A thorough review of laboratory tests and case studies of sand liquefaction was conducted by Ishihara (1993) The triaxial test has been widely used for laboratory testing... experiments to validate the numerical procedure and compare physical and simulated phenomena This thesis consists of 9 chapters A literature review of liquefaction research related to cyclic mobility is included in Chapter 2 Chapter 3 presents the formulation and improvement of the fuzzy-set plasticity model, along with the necessary material parameters calibration and model responses Chapter 4 addresses... Stress-strain Curve and Stress-path for Nevada Sand of Dr = 60% Obtained from Cyclic Undrained Simple Shear Test (Arulmoli et al., 1992) 13 Fig 2.5 Recorded Time Histories of Acceleration and Pore Water Pressure (Koga and Matsuo, 1990) 14 Fig 2.6 Time Records of Acceleration and Lateral Displacement Responses (Ishihara et al 1991) The laboratory testing results show that: 1) At low confinement, sands undergo... procedures for model calibration are investigated In this thesis, the governing equations of motion of the soil mixture are coupled with the global mass balance equations, and necessary assumptions are made to obtain the equivalent Biot s equations from the general balance equations The x ( s ) − u − x ( f ) formulation is used in the finite element spatial discretization, where x ( s ) , u and x ( f... granular soils under both monotonic and cyclic loading conditions to obtain constitutive parameters The cyclic triaxial test requires that the apparatus should be capable of applying extensional as well as compression loads to the soil specimen so that the cyclic stresses can reverse between triaxial compression and extension state Fig 2.1 Shows the result of a typical triaxial test when a cyclically... analysis The basic idea is to place elements with a special shape function for the geometry at the infinite boundary The accuracy of numerical analysis needs to be validated However, due to the difficulty of predicting when and where a major earthquake will occur and the general random nature of these events, most field liquefaction failures have occurred at sites which were not instrumented The advent of . EBeam : Y Normal size (8 1/2 X 11) Camera and EBeam This thesis entitled: A Biot Formulation for Geotechnical Earthquake Engineering Applications written by Yu Bao has been approved for. Camera 2006 Publication Notes : A Biot@ formulation for @ ?geotechnical earthquake @ ?engineering applications 174934-831-0 Pagination Note : Standing Order : Format : Geo Class : Scan :USA B SEP-20-2006 elec tpi umi+1 ii-xv 1-225 241 N EXPO . Bao, Yu (Ph.D., Civil Engineering) A Biot Formulation for Geotechnical Earthquake Engineering Applications Thesis directed by Professor Stein Sture The mechanical behavior of saturated