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Total stress dynamic analysis of Coquitlam Dam Dharmasetia, Charissa W. 2000

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TOTAL STRESS DYNAMIC ANALYSIS OF COQUITLAM DAM  by  CHARISSA W. DHARMASETIA B.A.Sc, The University of British Columbia, 1995  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA July, 2000 © Charissa Dharmasetia, 2000  In presenting this thesis in partial  fulfilment  of the  requirements  for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his or  her  representatives.  It  is  understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  <?*<f-'A/££Z/Ai&  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  3/  AOfST,  JooO  ABSTRACT  Earthquake-induced liquefaction of loose saturated sands can cause large deformations to occur resulting in flow slides. The near catastrophic failure of the Lower S a n Fernando Dam due to the 1971 earthquake is one of the most well known examples of a flow slide. However, the concern is not limited to flow slide situations in which the driving stresses are greater than the residual strength after liquefaction. condition.  Significant deformations can occur for a non-flow slide  These are caused by a reduction in stiffness of liquefied materials as well as from  inertia forces caused by the earthquake motion. Limit equilibrium type analyses have been used reliably in the prediction of the occurrence of a flow slide. However, difficulty has been encountered in the prediction of earthquake-induced deformation in the event that a flow slide is not predicted to occur. The problems have arisen from the modeling of the behaviour of liquefied soils. Soil exhibits stiff behaviour under cyclic loading and the strains associated up to the point of triggering of liquefaction are small. Upon liquefaction triggering, soil behaves like a liquid and initially strains under very small shear stresses. However, upon further deformation, the soil dilates and regains stiffness and strength. Most deformation analysis procedures overly simplify the effects of earthquake inertia forces and do not adequately model the stress-strain behaviour of liquefied soil. The proposed total stress procedure attempts to take account of both of the above effects. The procedure is separated into three main phases of cyclic-induced liquefaction behaviour of sands:  the pre-  triggering, triggering, and post-triggering response of soils. The triggering of liquefaction in each element of a soil structure is predicted by weighting the cyclic shear stresses induced by a prescribed base motion.  Upon triggering of liquefaction, liquefied stress-strain parameters are  assigned to zones predicted to liquefy as they occur. The pre-triggering and triggering phases of the procedure were verified using S H A K E analyses.  Similarly, the post-triggering phase of the procedure was compared with results  obtained from Bartlett and Youd's empirical equation. In both cases, reasonable agreement was  found. The method was finally applied to the Coquitlam Dam and the results compared with two of the more commonly used deformation analyses such as variations of the Modified Modulus method and Jitno and Byrne's extended Newmark method. The predicted results from all three methods are in reasonable agreement.  TABLE OF CONTENTS  ABSTRACT  »  TABLE OF CONTENTS  iv  LIST OF TABLES  ix  LIST OF FIGURES  x  LIST OF SYMBOLS  xiv  ACKNOWLEDGEMENTS  xvii  CHAPTER 1  1  INTRODUCTION  1  CHAPTER 2  4  BEHAVIOUR OF SATURATED UNDRAINED SOILS  4  2.1  Introduction  4  2.2  Monotonic Loading Behaviour  4  2.3  Cyclic Loading Behaviour  7  2.3.1  Cyclic Resistance  12  2.4  Post-Liquefaction Cyclic Loading Behaviour  15  2.5  Post-Liquefaction Monotonic Loading Behaviour  19  2.5.1  Residual Strength  2.5.2  Limiting Shear Strain  24 26  2.6  Post-Liquefaction Volumetric Deformation  27  2.7  Summary  30  CHAPTER 3  31  iv  EVALUATION OF EARTHQUAKE INDUCED DEFORMATIONS OF EARTH STRUCTURES ..31 3.1  Introduction  31  3.2  Empirical Methods  32  3.2.1  Hamadaetal.  3.2.2  Bartlett and Youd  3.3  Pseudo Static Methods  3.3.1  Newmark's Method  3.3.2  Byrne's Extended Newmark Approach  3.4  Two Dimensional Methods  3.4.1  Modified Modulus Approach  3.4.2  Strain Potential / Dynamic Stress Path Approach  3.4.3  Jitno and Byrne's Approach  32 33 34 35 38 41 42 42 43  3.5  Non-Linear Effective Stress Analysis  44  3.6  Summary  45  CHAPTER 4  46  PROPOSED TOTAL STRESS METHOD  46  4.1  Introduction  46  4.2  Triggering of Liquefaction  47  4.2.1  Cyclic Stress Ratio  47  4.2.2  Cyclic Resistance Ratio  49  4.3  Proposed Method  49  4.3.7  Pre-Liquefaction Trigger  51  4.3.2  Liquefaction Trigger  52  4.3.3  Post-Liquefaction Trigger  54  4.3.4  Mohr-Coulomb Model  59  4.3.4.1  Introduction  59  4.3.4.2  Yield Sudace  60 v  4.3.4.3  4.4  Non-associative  Flow Rule  Required Parameters  60  61  4.4.1  Pre-liquefaction Parameters  61  4.4.2  Post-Liquefaction Parameters  62  4.4.2.1  Residual Strength  62  4.4.2.2  Limiting Shear Strain  62  4.5  Summary  63  CHAPTER 5  64  VERIFICATION OF TOTAL STRESS PROCEDURE  64  5.1  Introduction  64  5.2  Triggering Verification Using SHAKE  65  5.2.1  Description of FLAC  5.2.2  Case Analyzed.  68  5.2.3  Results  69  5.3  Post-Triggering Verification Using Bartlett and Youd Method  65  70  5.3.1  Case Analyzed.  71  5.3.2  Results  71  5.4  5.3.2.1  Effect of G/G  5.3.2.2  Effect of Critical Damping Ratio  74  5.3.2.3  Effect of water table  75  5.3.2.4  Effect of Earthquake Time Histories and Magnitude  75  max  ratio  Summary  74  82  CHAPTER 6  83  PAST ANALYSES OF COQUITLAM DAM  83  6.1  Introduction  83  6.2  Field and Laboratory Testing  85 vi  6.2.1  Field Exploration  85  6.2.1.1  Mud Rotary Drilling  86  6.2.1.2  In-Situ Testing  88  6.2.1.2.1 Dynamic Cone Penetration Testing  88  6.2.1.2.2 Pressuremeter Testing  88  6.2.1.2.3 Electric Cone Penetration Testing  89  6.2.2  Laboratory Testing  91  6.2.2.1  Index Tests  91  6.2.2.2  Resonant Column Tests  92  6.2.2.3  Consolidated  92  6.2.2.4  Cyclic Triaxial Tests  94  6.2.2.5  Post Cyclic Monotonic Tests  94  6.3  Undrained Triaxial Test  Seismic Stability Analyses  95  6.3.1  1979 Analyses  95  6.3.2  1984 Analyses  98  6.4  Other Studies  101  6.5  Summary  101  CHAPTER 7  102  TOTAL STRESS DYNAMIC ANALYSES OF COQUITLAM DAM  102  7.1  Introduction  102  7.2  Pre-Earthquake Analysis  102  7.2.1  Construction Analysis  103  7.2.1.1 Model Parameters  103  7.2.1.2 Results  106  7.2.2  End of Reservoir Filling Analysis  107  7.2.2.1 Flow Model Parameters  109  7.2.2.2 Results  110  vii  7.3  Total Stress Dynamic Analysis  113  7.3.1  Introduction  7.3.2  Liquefaction Triggering Analysis  7.3.3  Flow Slide Analysis  116  7.3.4  Deformation Analysis  117  7.3.4.1  Model Parameters  7.3.4.2 Results 7.4  Proposed Total Stress Procedure  7.4.1  Model Parameters  7.4.2  Results  113 114  117  119 121 121 123  7.5  Deformation Due to Consolidation  127  7.6  Summary  129  CHAPTER 8  131  CONCLUSIONS AND RECOMMENDATIONS  131  BIBLIOGRAPHY  134  APPENDIX 1  140  viii  LIST OF TABLES  Table 3. 1 Post-Liquefaction Stress-Strain Parameters  41  Table 4. 1 Equivalent Number of Cycles  48  Table 5. 1 Summary of Earthquake Histories used in Bartlett-Youd Verification  75  Table 6. 1 Summary of Mud Rotary Drill Holes  87  Table 6. 2 Summary of Laboratory Tests  91  Table 7. 1. Summary of Dam Material Parameters used in Static Analysis  106  Table 7. 2. Summary of Dam Material Properties for Seepage Analysis  110  Table 7. 3 Corrected Cyclic Resistance Ratio for Coquitlam Dam Materials  114  Table 7. 4 S H A K E input parameters  115  Table 7. 5 Results of Liquefaction Induced Displacement of Coquitlam Dam  119  Table 7. 6 Summary of Parameters Used in Proposed Model  122  ix  LIST OF FIGURES  Figure 2.1 Characteristic Undrained Monotonic Response of Saturated Soils (Chern, 1985)  5  Figure 2. 2 Characteristic Undrained Response of Saturated Sands Under Cyclic Loading (Vaid and Chern, 1985)  9  Figure 2. 3 Stress Ratio versus Number of Cycle to Liquefaction (De Alba, 1976)  11  Figure 2. 4 K o versus Effective Overburden Pressure (Pillai and Byrne, 1994)  12  Figure 2. 5 Correction Factor for Static Shear ( N C E E R , 1997)  13  Figure 2. 6 Post-Liquefaction Stress-Strain Curves (Kuerbis, 1989)  17  Figure 2. 7 Surface Acceleration versus Relative Displacement (Byrne and Mclntyre., 1994)  18  Figure 2. 8 Approximate Stress-Strain Relationship during Shaking Table Tests (Sasaki et al., 1992)  18  Figure 2. 9 Post-Liquefaction Monotonic Behaviour of Saturated Sands (Seed, 1979) a. Sacramento River Sand, D = 40%; b. Mine Tailings, D = 9 5 % R  R  20  Figure 2. 10 Effect of Excess Pore Pressure on Post-Liquefaction Behaviour of Saturated Sands (Yasuda, 1994)  22  Figure 2.11 Characterization of Post-Liquefaction Curve (Thomas, 1992)  23  Figure 2. 12 Shear strength of strain-softening Syncrude Sand (Vaid et al. 1998)  25  Figure 2. 13 Relationship Between Corrected "Clean Sand" Blowcount (N^o-os and Undrained Strength (Sr) (Seed and Harder, 1990)  26  Figure 2. 14 Volumetric Strain versus Maximum Amplitude Shear Strain for Different Relative Densities (Ishihara and Yoshimine, 1992)  28  Figure 2 . 1 5 Volumetric Strain versus Factor of Safety Against Liquefaction (Ishihara and Yoshimine, 1992)  29  Figure 3. 1 Idealized Potential Sliding Slope (Newmark, 1965)  35  Figure 3. 2 Work Energy Approach to Newmark Method (Byrne, 1990)  37  Figure 3. 3 Idealized Pre- and Post- Liquefaction Behaviour of Sand (Haile et al., 1996)  38  x  Figure 3. 4 Work Energy Approach to Extended Newmark (Jitno, 1995)  38  Figure 3. 5 Linear and Non-Linear Stress-Strain Curves (Byrne, 1990)  39  Figure 4.1  50  Cyclic Resistance Ratio Based on S P T Blow Count, M 7.5 ( N C E E R , 1997)  Figure 4. 2 Cyclic Induced Liquefaction Behaviour of Sands  51  Figure 4. 3 Cyclic-Induced Shear Stress History  52  Figure 4. 4 Relationship Between Normalized Shear Stress and Number of Cycles to Liquefaction (Byrne, 1990)  53  Figure 4. 5 Modeled Cyclic Induced Behaviour of the Soil Element  55  Figure 4. 6 Stress-Strain Response of Soil  55  Figure 4. 7 Actual and Idealized Post-Liquefaction Behaviour of Saturated Sands Under Cyclic Loading  56  Figure 4. 8 Definition of Loading and Unloading in Proposed Model  57  Figure 4. 9 Comparison Between Actual and Modelled Hysteretic Loops  58  Figure 4. 10 Linear Elastic-Plastic Behaviour of Soils  59  Figure 5. 1 F L A C Calculation Cycle (FLAC 3.3 manual, 1995)  65  Figure 5. 2 Variation of Normalized Critical Damping Ratio with Frequency ( F L A C 3.3 manual, 1995)  67  Figure 5. 3 Shear Moduli and Damping Attenuation Curves  68  Figure 5. 4 Comparison of Dynamic Response Between F L A C and S H A K E  70  Figure 5. 5 Comparison of Displacements Predicted by Proposed Model and Bartlett-Youd  73  Figure 5. 6 Comparison of Response to Different Earthquake Records Scaled to P G A 0.15g ....78 Figure 5. 7 Acceleration Time Histories of Wildlife and Caltechb Earthquakes  79  Figure 5. 8 Acceleration Time Histories of Gilroy and S a n Fernando (a) Earthquakes  80  Figure 5. 9 Comparison of Response to Different Earthquake Records Scaled to 0.27g  81  Figure 6. 1 Plan and Location of Coquitlam Dam  84  xi  Figure 6. 2 Typical Cross Section of Coquitlam Dam (not to scale)  85  Figure 6. 3 Typical Cone Penetration Resistance Profile of Core Material  90  Figure 6. 4 C U Triaxial Test Results of Core Material  93  Figure 6. 5 Cyclic Resistance of Core Material of Coquitlam Dam  94  Figure 6. 6 C U Triaxial Test Results of Liquefied Core Material  96  Figure 6. 7 1979 Computed Cyclic Stress Ratio Contours Based on Hand Calculations  97  Figure 6. 8 1979 Computed Cyclic Stress Ratio Contours Based on Dynamic Analyses  98  Figure 6. 9 1984 Computed Cyclic Stress Contours Based on Hand Calculations  99  Figure 6. 10 S O I L S T R E S S Deformation Analysis Based on Loss in Stiffness  100  Figure 7. 1 Grid Used For Analysis of Coquitlam Dam  105  Figure 7. 2 Material Zones For Analysis of Coquitlam Dam  105  Figure 7. 3 Mohr-Coulomb Parameter Fit to Hyperbolic Parameter  106  Figure 7. 4 End of Construction Displacement Vectors  108  Figure 7. 5 Vertical Displacements Along Centreline of Dam  108  Figure 7. 6 End of Construction Vertical Stress Contours  108  Figure 7. 7 End of Reservoir Filling Flow Vectors  111  Figure 7. 8 End of Reservoir Filling Pore Pressure Contours  111  Figure 7. 9 End of Reservoir Filling Total Head Contours  111  Figure 7. 10 End of Reservoir Filling Displacement Vectors  112  Figure 7. 11 End of Reservoir Filling Vertical Stress Contours  112  Figure 7. 12 Liquefaction Triggering of Coquitlam Dam  118  Figure 7. 13 Limit Equilibrium Flow Slide Analyses  118  Figure 7. 14 Typical Post-Liquefaction Displacement Vector Pattern  120  Figure 7 . 1 5 Comparison of Velocity History of Crest of Dam  121  Figure 7. 16 Zones Predicted to Liquefy by Proposed Method  123  Figure 7. 17 Post-Liquefaction Displacement Pattern Predicted by Proposed Procedure  125  xii  Figure 7 . 1 8 Post-Liquefaction Displacement Pattern Assuming Liquefiable Zones Triggered at the Same Time  125  Figure 7. 19 Cyclic Shear-Strain Plot Predicted in Dam Core Material  126  Figure 7. 20 Displacement Pattern Due to Volumetric Deformations Only  128  Figure 7. 21 Distorted Grid Due to Post-Liquefaction Deformation and Settlement  128  xiii  LIST OF SYMBOLS  (N^  S P T blow counts normalized to an overburden stress of 1 tsf (98 kPa)  (Ni)eo  (Ni) normalized to 60 percent of hammer energy  (Ni)6o-cs  Equivalent clean sand blow count  (tdy)max  Maximum dynamic shear stress  a  Ratio of shear stress on horizontal plane to effective stress, direction of loading with repect to bedding plane  B  Bulk modulus  c  Cohesion  CRR  Cyclic resistance ratio  CSR  Critical stress ratio (Chapter 2) or Cyclic stress ratio (subsequent Chapters)  D  1 0  Effective grain size  D  5 0  Mean grain size  D a m  AN D  X  e q  r  Maximum damping ratio Increment in equivalent number of cycles Relative density  Au  Excess pore water pressure increment  e  a  Axial strain  E  V  Volumetric strain  F  Force  <|)  Friction angle of soil  FL or Fug  Factor of safety against liquefaction  f  Shear yield function  s  g  Acceleration due to gravity  G  Shear modulus  y  Shear Strain  YL  Limiting shear strain  Gliq  Liquefied shear modulus  Ymax  Maximum amplitude of shear strain  xiv  G g  m a x  Maximum shear modulus Shear potential function  s  Ysat  Saturated unit weight of soil  k  Horizontal or vertical seismic coefficient  k  Shear modulus number (dynamic analysis)  2  k2,max  Maximum shear modulus number (dynamic analysis)  Ka  C R R correction factor due to effect of static shear bias  kb  Bulk modulus number  kg  Shear modulus number (Mohr-Coulomb, static analysis)  K  C R R correction factor for confining stress  CT  X  plastic modulus number  m  Bulk modulus exponent  N  Number of cycles  n  Shear modulus exponent  Pa  Atmospheric pressure  PGA  Peak ground acceleration  PT  Phase transformation state  r  Stress reduction factor due to soil depth  d  Pdry  Bulk dry density of soil  a'  3  Minor effective principal stress  a'  c  Effective confining stress  a'  0  Initial effective confining stress  a'vo  Effective overburden stress  G'I  Major effective principal stress  CT  Deviator Stress  SPT  Standard penetration test  S  Residual strength  d  r  SS  Steady state  a  Total overburden stress  v 0  CTxx  Normal stress in x direction  (%  Normal stress in y direction  xv  o-zz  Normal stress in z direction  T  Shear Stress  TIS  Cyclic shear stress causing liquefaction in 15 cycles  T  Dynamic or cyclic shear stress  i  D Y  Static shear stress  s t  txy  Shear stress on horizontal plane  v  Dilation angle  co.  Angular frequency  co  min  Minimum angular frequency  E,j  Critical damping ratio  c^min  Minimum critical damping ratio  xvi  ACKNOWLEDGEMENTS  I would like to express my sincere gratitude to my supervisor, Dr. P . M . Byrne for his support and guidance during the course of my studies and research. I would also like to thank my colleagues in the Geotechnical Engineering Research Group at the University of British Columbia, particularly, Dr. Michael Beaty for reviewing this thesis and Dr. Humberto Puebla for making suggestions for solving problems encountered in F L A C . Financial support provided by B C Hydro and the valuable work experience gained through the Professional Partnership Program was sincerely appreciated. Thank you to Dr. Michael Lee of B C Hydro, who also reviewed this thesis and provided helpful suggestions and edits. Finally, I thank G o d and Our Lady for the strength needed to help me complete this and also to my family for their loving support, understanding, and most importantly patience in dealing with my banterings. I would also like to thank my office-mates for their friendship and support. Last, but not least, I am very grateful to Karen Savage and Saman Vazinkhoo for their cooperation, encouragement and motivation.  xvii  Chapter 1. Introduction  CHAPTER 1  INTRODUCTION Dynamic loading of saturated granular soils causes an increase in porewater pressure. Depending on the duration and intensity of earthquake motion and the permeability of the soil, the pore pressures may rise to equal the initial overburden stress in which case liquefaction is said to occur. This rise in porewater pressure causes a reduction in the strength and stiffness of the soil. The reduced strength is generally referred to as the residual strength.  Liquefaction may cause  large displacements depending on the earthquake intensity and duration and on the initial static shear stress acting on the structure. There are many case histories of catastrophic failures of earth structures which have occurred as a result of earthquake-induced liquefaction. One of the more well known examples of such behaviour is the 1978 failure of two tailings dams associated with the Mochikoshi gold mine in Japan in which a large volume of tailings materials was released. Near catastrophic failures have also occurred as a result of liquefaction due to earthquake loading.  In 1971, an earthquake hit the San Fernando Valley in California. The earthquake  caused major damage to the area, in particular to the Upper and Lower San Fernando Dams. An upstream flow slide which extended to the upper portion of the downstream slope left the Lower San Fernando Dam with only approximately 1.5 m of freeboard.  Although the damage to the  Upper S a n Fernando Dam was less severe, the crest of the dam moved 1.5 m downstream and settled about 0.8 m. The movements of the Lower San Fernando Dam and Mochikoshi Tailings Dams occurred because the driving forces within the structures exceeded the residual strengths of the soils. The movements come to a stop when large changes in geometry have caused a reduction in the difference between the driving stresses and residual strengths. If the changes in geometry  1  Chapter 1. Introduction  cannot cause the stresses and strengths to be balanced, unlimited deformation, such as those observed in the Mochikoshi tailings dams, may occur. Although flow slides may not occur, situations in which the driving stresses are less than the residual strength can cause be a major concern. Large deformations can be induced from the reduction in stiffness of liquefied materials as well as from inertia forces caused by the earthquake base motions. The damage to the Upper San Fernando is an example of such behaviour. Limit equilibrium analyses have long been used to assess the stability of earth structures under static and dynamic conditions. These types of analyses have been shown to be reliable in predicting the flow slide potential of earth structures.  The flow slide potential is assessed by a  factor of safety in which the driving stresses are compared to material strengths.  If a factor of  safety of less than unity is computed then a flow slide may occur. If the factor of safety is greater than unity then a flow slide is not likely to occur.  However, the deformations associated with  liquefaction may still be significant. Newmark was the first to advance the concept of assessing seismic response by computing displacements rather than a factor of safety.  Newmark proposed a single degree of  freedom model in which soil is assumed to exhibit rigid plastic behaviour. The method works well in situations where deformations occur on a distinct surface.  However, the method does not  consider the displacements due to the reduced strength and stiffness of liquefied soils. More sophisticated two-dimensional procedures have been developed as a result of the increase in understanding of the behaviour of liquefied materials. The two more commonly-used methods are Lee's Modified Modulus Method and Jitno and Byrne's extended Newmark Method. Both procedures use finite elements to compute deformations. The Modified Modulus approach assumes that displacements are due to the reduction in strength and stiffness of the liquefied soil and does not consider inertia forces induced by earthquake loading. Jitno and Byrne extended Newmark's approach to a multi degree of freedom system in which the loss in strength and stiffness of liquefied soils is considered. The method has proven to be reliable, however, because the displacements are computed based on energy balance of the system, the  computed  2  Chapter 1. Introduction  displacements are sensitive to the relative stiffness of adjacent layers as well as to the grid geometry. Rigorous non-linear effective stress methods are more fundamental than the simple total stress procedures. The rise in porewater pressure during earthquake motion can be modeled and liquefaction is triggered in different zones at different times.  However, these analyses are  complicated and have not been fully validated. A proposed total stress procedure is presented in this thesis. The procedure considers the two main effects of liquefaction induced deformations: the reduction in strength and stiffness and the inertia forces due to earthquake loading.  The method combines triggering and  deformation analyses together into one procedure. Triggering of liquefaction in each element can be predicted by weighting the cyclic shear stress amplitude during a prescribed base motion. Post-liquefaction parameters are assigned when liquefaction is considered to have been triggered. The triggering of liquefaction was approximately verified against the results using S H A K E . Furthermore, the resulting displacements were validated against the predictions using Bartlett and Youd's empirical approach. The proposed method was then applied to Coquitlam Dam. In this thesis, the results using the proposed procedure on Coquitlam Dam were compared to the results using the more conventionally used methods to predict liquefaction induced displacements.  3  Chapter 2 Behaviour of Saturated Undrained  Soils  CHAPTER 2  BEHAVIOUR OF SATURATED UNDRAINED SOILS 2.1 Introduction Cohesionless soils can be susceptible to excessive deformations under undrained loading conditions. This phenomenon, liquefaction, is due to porewater pressure build up and decrease in effective stresses and can be initiated by static or dynamic loading or by changes in groundwater conditions. Undrained response due to static or monotonic loading as opposed to cyclic loading is usually considered separately. Interest in monotonic loading is related to failure associated with flow slide.  The characteristic behaviour is large deformations due to low shear resistances.  Interest in cyclic loading is related to the accumulation of deformations under repeated loading such as in earthquake shaking and/or a flow slide.  2.2 Monotonic Loading Behaviour Monotonic loading behaviour of saturated sands has been investigated by a number of researchers  [eg.  Castro(1969),  Sivathayalan(1994)].  Lee  and  Seed(1970),  Chern(1985),  Thomas(1992),  and  Typical triaxial compression undrained response can be characterized by  the 3 curves shown in Fig 2.1. Different responses are associated with the relative density and initial confining stress prior to shearing. At a given initial effective confining stress, the behaviour changes from type 1 to 3 with increasing relative density. Type 1 response represents a strain softening or contractive behaviour wherein the sand exhibits a large reduction in shear resistance under large strains. This type of response is termed liquefaction (Castro, 1969; Seed, 1979; Vaid and Chem,1985). Initial development of strains and pore pressure is slow until an effective stress ratio corresponding to a peak deviator stress is  4  Chapter 2 Behaviour of Saturated Undrained Soils  Normal Effective Stress  Figure 2.1 Characteristic Undrained Monotonic Response of Saturated Soils (Chern, 1985) 5  Chapter 2 Behaviour of Saturated Undrained Soils  attained.  This effective  stress ratio, termed  the  critical  stress ratio (CSR) (Vaid  and  Chern,1983,1985), has been found to be unique for a given sand in undrained compression loading (Vaid and Chern, 1985; Vaid et al., 1989). In contrast, the C S R in extension or in simple shear is not unique and is lower than in compression. The C S R increases with an increase in void ratio but is always smaller than that in compression mode. Once the C S R has been attained, the soil experiences a sudden decrease in resistance along with a sudden increase in pore pressure with strain. The soil resistance eventually reaches a minimum constant value, termed the steady state (SS) strength (Castro, 1969). strength is related to initial void ratio (Castro, 1969).  The S S  However, at a given void ratio, the S S  strength in compression is greater than that in extension (Vaid et al., 1989). In type 2 response, strain softening behaviour occurs over a limited strain range then transforms into strain hardening behaviour after a minimum strength is reached.  This type of  behaviour has been termed limited liquefaction (Castro, 1969). Like type 1 response, strain softening is initiated at the C S R and continues until a minimum undrained strength is reached. However, upon further deformation, the soil begins to regain strength as the pore pressure decrease occur.  The state where the soil changes from  contractive to dilative behaviour is termed phase transformation (PT) (Ishihara et al., 1975). In the effective stress diagram, this state is represented by the point of sharp reversal of the effective stress path. The friction angle mobilized in the phase transformation phase is identical to that mobilized in steady state in type 1 response (Vaid and Chern, 1985). Upon further straining past the P T state, the effective stress path follows the ultimate failure line. This line is unique for a given water deposited sand (Vaid and Chern, 1985; Vaid and Thomas, 1994). Type 3 response represents strain hardening behaviour with no loss in shear resistance. The shear resistance increases with increasing strains.  In contrast to type 2 response,  the  reversal in effective stress path is gradual once it has crossed the P T / S S line and approaches the ultimate failure line.  6  Chapter 2 Behaviour of Saturated Undrained Soils  2.3  Cyclic Loading Behaviour Undrained cyclic loading causes a cumulative increase in pore pressure which may lead  to the development of large shear strains. The soil would then be considered as having liquefied. This strain development can be due to either liquefaction, limited liquefaction, or cyclic mobility, depending on the initial state of the sand. A close link exists between the response of sands under monotonic and cyclic loading. It has been shown that liquefaction and limited liquefaction occur in the same manner in cyclic as in monotonic loading (Castro, 1969; Vaid and Chern, 1985). Liquefaction-type behaviour in cyclic loading is shown in Fig. 2.2a and b. A s in monotonic loading, pore pressure and strain development are slow until the effective stress state crosses the C S R line. Upon intersecting the C S R line, the effective stress path reaches the PT or S S line, at which point the soil resistance decreases to a minimum value upon further straining. If the initial static shear stress level is greater than the steady state strength, unlimited deformation may occur. The C S R and P T / S S lines are the same in cyclic as in monotonic loading, implying that they are unique (Vaid and Chern, 1985). Figures 2.2c and d show a limited liquefaction-type behaviour under cyclic loading. Strain softening behaviour is initiated when the effective stress path reaches the C S R line. Large strains occur as the stress path moves from the C S R to PT lines. A s in monotonic behaviour, a sharp turn around is observed when the stress path intersects the P T line. This results in the development of large strains (Fig 2.2c).  Subsequent unloading causes a large increase in pore  pressure leading to a transient state of zero effective stress. Strain recovery in this process is small (Fig 2.2c). Further unloading in the extension region of cyclic loading causes large strains to develop as the sand deforms under virtually zero stiffness. Upon subsequent re-loading in the compression region, the sand exhibits very soft behaviour until it reaches the strain obtained prior to the first zero effective stress condition, after which it regains much of its stiffness. Cyclic mobility due to cyclic loading is shown in Figures 2.2e and f.  Strain and pore  pressure accumulate slowly with increasing number of cycles. Deformation is not due to marked  7  Chapter 2 Behaviour of Saturated Undrained Soils  strain softening at any loading stage. Upon reaching the P T line, the effective stress path sharply turns around, accompanied by the development of significant strains. Further unloading causes large pore pressures to develop, bringing the effective stress close to a state of zero effective stress. Subsequent re-loading finally causes a transient zero effective stress condition and the sample undergoes large deformation in the process. Similar to the limited liquefaction response, the concern with cyclic mobility is with the accumulation of limited deformation with continued cyclic loading. The type of response under cyclic undrained loading depends on factors such as deposition, grain angularity, initial void ratio, effective confining stress, initial shear stress level, induced cyclic stress and number of loading cycles. Because liquefaction can be loosely defined as the development of large strains, it can be caused by any of the above three mechanisms. Cyclic resistance to liquefaction is defined as the cyclic stress required to cause a specified strain in a specific number of cycles. In other words, if the cyclic resistance is exceeded, the response of the soil will change from small strain to large strain.  8  Chapter 2 Behaviour of Saturated Undrained Soils  Figure 2. 2 Characteristic Undrained Response of Saturated Sands Under Cyclic Loading (Vaid and Chern, 1985) 9  Chapter 2 Behaviour of Saturated Undrained Soils  2.3.1 Cyclic Resistance to Triggering of Liquefaction  Cyclic triaxial tests, cyclic simple shear tests, and cyclic torsional tests have been used to evaluate the liquefaction resistance of saturated sands by many researchers.  Liquefaction  resistance has been found to be dependent on initial confining stress, shaking intensity, number of loading cycles, and relative density. It is generally considered that the ratio of cyclic shear stress to initial confining stress, referred to as the cyclic resistance ratio (Seed, 1984) is an important parameter in liquefaction analysis.  For the triaxial loading condition, this ratio is defined as  GD/ZG'O, where CT is a single amplitude of cyclic axial stress and a ' is the initial effective confining d  0  stress. The results of cyclic triaxial tests are usually plotted in terms of cyclic stress ratio ( C S R ) versus number of cycles to liquefaction.  Liquefaction in laboratory tests is achieved when a  specifed double amplitude axial strain of usually 5 percent is obtained. Typical results are shown in Fig 2.3. The cyclic resistance ratio is usually multiplied by 0.65 in order to represent simple shear loading conditions. This correction is considered conservative since it can range from 0.66 to about 0.78 depending on relative density and confining stress (Sivathayalan, 1994).  10  Chapter 2 Behaviour of Saturated Undrained Soils  N u m b e r of  Cycles,  N  c  O00  Figure 2. 3 Stress Ratio versus Number of Cycle to Liquefaction (De Alba, 1976) 2.3.1.1 Effect of Confining Stress The effects of confining stress on the cyclic resistance of sands have been investigated by means of a cyclic triaxial device on reconstituted Fraser River sands (Thomas, 1992) and on undisturbed samples from frozen Duncan Dam foundation material under high confining stresses (Pillai and Byrne, 1994). The results were plotted in terms of a correction factor, K o , defined as the cyclic resistance ratio of soil at a ' divided by the ratio at a ' = 100 k P a , versus effective 0  0  confining stress, a ' . 0  As shown in Fig. 2.4 (Pillai and Byrne, 1994), the values for K decreases with increasing CT  effective confining stress, implying that the cyclic resistance ratio decreases with increase in effective confining stress.  However, the increase in confining stress is associated with an  increase in relative density. A s a result, the K„ values simulate an increase in confining stress as well as in relative density. The results from both tests plot above the values proposed by Seed and Harder(1990). This indicates that the values previously used may have been conservative. 11  Chapter 2 Behaviour of Saturated Undrained Soils  1.2  o  A — o  ©  :  •  X N ^ \ 0.8  :  \.  ~=^y^ ^ m  \  \ "v.  _  A  -=ft  ,,  '—— '  "\  •—-  Dr=60%  a 1-  Dr=65%  s. Seed and Harder (1990) ^  —- --ai-r-j-rt  o  Duncan Dam X \  .8 '  „  \ 0.6  : .  lings  1  ~Dr=60%  ^  o-  —  -Df=66%-  0.4  O  -  Dr=70% _  «4  ~Dr=20%  \ ~°  0.2  Dr=40% Dr=60%  i 500  0  .  i  ' 1000  i 1500  Effective Confining Stress (kPa)  Figure 2. 4 Ko versus Effective Overburden Pressure (Pillai and Byrne, 1994) 2.3.1.2 Effect of Relative  Density  The effect of relative density on the cyclic resistance of sands has been well documented. Liquefaction resistance increases with increase in relative density at all levels of confining stress although it is more pronounced at lower confining stresses.  2.3.1.3 Effect of Static Shear Bias The effects of static shear on the liquefaction potential of sands have been investigated by a number of researchers (Vaid and Finn, 1979; Vaid and Chern, 1983; Vaid and Chern, 1985; Pillai and Stewart, 1992). It has been found that the effects of static shear depends on the relative density of sands as well a s on the confining stress. A s a result, a correction factor, K , defined as H  the ratio of the C R R at any initial static shear and the C R R at zero static shear has been  12  Chapter 2 Behaviour of Saturated Undrained Soils  developed, is usually plotted versus a,  the ratio of shear stress on a horizontal plane to the  effective normal stress. A s shown, in Fig. 2.5, for dense sands, the cyclic resistance increases with increase in initial static shear.  In contrast, the cyclic resistance decreases with increase in static shear for  fairly loose sands.  2.0  T  T  a ' < 3 tsf vo  D « 55-70% (N,)6o= 14-22 r  1.5  Ka i.o  D « 45-50% (N,)«o«8-12 r  0.5  h  D «35% (N,)6o«4-6 r  A. 0.1  0.2  0.3  0.4  a=(T /a 5) s  Vf  Figure 2. 5 Correction Factor for Static Shear (NCEER, 1997) 2.3.1.4 Effect of Fines Content Based on field observations of liquefaction and non-liquefaction in sands, it has been considered conservative to ignore the presence of fines contents greater than 5% in silty sands (Seed et al., 1985).  A s a result,  research has concentrated on liquefaction in clean sands.  However, recent studies in sand and silt mixures by means of cyclic triaxial device (Kuerbis et al.,  13  Chapter 2 Behaviour of Saturated Undrained Soils  1988; Ishihara and Kosecki, 1989; Koester, 1992) have shown that the presence of non-cohesive or non-plastic fines may show liquefaction potential as great or greater than that in clean sands.  Liquefaction in silty sands and silts is usually defined by the development of 5 % or 10% double amplitude strains rather than the attainment of 100% pore pressure in a cyclic triaxial device. This is because it has been found that significant deformations develop prior to significant development of pore pressure.  This is especially true in undisturbed silts, in which, with the  exception of loose silts, the development of pore pressure is similar to that in medium dense sands (Zhou et al., 1995).  However, in general, over 8 0 % pore pressure is developed in  reconstituted silty specimens when liquefaction is considered to have been triggered. The fines content and the plasticity of fines have been found to most strongly affect the cyclic resistance of silty sands.  For a given relative density, cyclic strength decreases with  increase in the content of low plasticity fines of up to 20 to 30 percent (Kuerbis et al., 1988; Koester, 1992; Singh, 1994; Erten and Maher, 1995).  However, if the relative density is  expressed in terms of sand skeleton, the cyclic strength increases only slightly with increasing fines content to 20 percent. This suggests that the silts in silty sands only occupy the void spaces in the sand skeleton and the behaviour is essentially controlled by the sand skeleton void ratio (Kuerbis et al., 1988).  In contrast, the cyclic strength increases with increase in silt content  greater than about 30 percent although never exceeds that of the clean parent sand (Koester, 1992; Singh, 1994). Most researchers have found that the cyclic strength increases with increase in plasticity (Ishihara and Koseki, 1989; Prakash and Sandoval, 1992).  However, Zhou et al. (1995) have  found that the cyclic resistance decreases with increase in plasticity in reconstituted specimen although the reverse is true in undisturbed specimens.  This may be due to the increase in  interparticle cementation between fines particles with time, resulting in more difficult particle separation during cyclic loading. To  a  lesser extent,  the  cyclic  resistance of  silty  sands  is  influenced  by  the  overconsolidation ratio and time. Ishihara et al. (1989) compiled a data base of case histories of 14  Chapter 2 Behaviour of Saturated Undrained Soils  liquefaction of silty sands. They found that cyclic resistance increases with time and with increase in overconsolidation ratio.  2.4 Post-Liquefaction Cyclic Loading Behaviour The cyclic loading behaviour past the liquefaction state has been investigated in the laboratory by a number of researchers by means of a triaxial device (Seed and Lee, 1966; Vaid and Chern, 1985; Kuerbis, 1989) and a triaxial torsion shear device (Towhata and Ishihara, 1985). Typical stress-strain response under isotropic consolidated triaxial loading conditions are shown in Fig. 2.6. Limited liquefaction behaviour is shown in Fig 2.6a. Axial strains prior to liquefaction are small compared to those upon liquefaction.  Once liquefaction has occurred in the extension  mode, large strains develop. Loading in the compression region causes the sample to deform under low stiffness over a limited amount of strain after which strain hardening behaviour is exhibited. Subsequent loading in the extension region causes the sample to deform under almost zero stiffness for a greater amount of strain. Cyclic mobility behaviour is shown in Fig 2.6b.  Similar response as that in limited  liquefaction is observed, although stiffness is regained over a smaller strain range.  The post-  liquefaction strain increment initially increases with each cycle until a certain number of cycles is reached after which the strain increment decrease slightly. There seems to be a maximum cyclic strain which additional load cycles do not significantly alter. This is consistent with the findings of De Alba et al. (1976) in the large simple shear device. Laboratory test results show that, after initial deformation under zero stiffness, stiffness of liquefied soils increases with increase in strains under cyclic loading.  In fact, even loose soils  under low static shear can exhibit dilative behaviour. This implies that liquefied soils can transmit earthquake induced shear waves to overlying layers.  Field evidence in terms of recorded  acceleration data from the Wildlife site in the 1987 Superstition Hills earthquake (Byrne and Mclntyre, 1994) and results from shaking table tests (Sasaki et al., 1992) confirm this.  15  Chapter 2 Behaviour of Saturated Undrained Soils  The Wildlife site, California,  is a gently sloping area located near the Alamo river  consisting of loose saturated silty sands which has liquefied a number of times in past earthquakes. Prior to the 1987 Superstition Hills earthquake, it had been heavily instrumented to monitor the possible re-occurrence of liquefaction.  Holzer et al. (1989) presented the field  behaviour. Byrne and Mclntyre (1994) analyzed the recorded earthquake accelerations at ground surface and at the top of the unliquefied layer.  The accelerations at both locations were  integrated to obtain displacements. The surface acceleration was then plotted versus the relative displacements between the two layers. The resulting plot, shown in Fig. 2.7, is similar to a stress versus strain plot since shear stress and strain are proportional to acceleration and relative displacement respectively. The plot shows a substantial decrease in stiffness at some point during earthquake loading indicating the triggering of liquefaction in some layers between the surface and nonliquefied base. Similar to the plot shown in Fig 2.6, significant deformation is developed upon liquefaction although stiffness increases upon further deformation. A series of shaking table tests were conducted by Sasaki et al. ( 1992) in order to investigate the deformation mechanism of liquefied sands. Acceleration, excess pore pressure, and lateral displacement were recorded and plotted during cyclic loading. In addition, similar to Byrne and Mclntyre, acceleration versus lateral displacement at the surface relative to the shaking table was plotted as shown in Fig. 2.8.  16  Chapter 2 Behaviour of Saturated Undrained Soils  AXIAL STRAIN  (PERCENT)  Figure 2. 6 Post-Liquefaction Stress-Strain Curves (Kuerbis, 1989)  Chapter 2 Behaviour of Saturated Undrained Soils  200  Relative Displacement (cm)  Figure 2. 7 Surface Acceleration versus Relative Displacement (Byrne and Mclntyre., 1994)  Figure 2. 8 Approximate Stress-Strain Relationship during Shaking Table Tests (Sasaki et al., 1992)  18  Chapter 2 Behaviour of Saturated Undrained Soils  T h e plot shows that after a number of cycles, the soil stiffness decreases noticeably causing significant deformations to develop in the sand. further straining.  However, the soil regains stiffness upon  In addition, consistent with the observations of De Alba et al. (1976), there do  not s e e m to be any significant difference between the strain increments in the 20th and 40th cycles.  2.5 Post-Liquefaction Monotonic Loading Behaviour Failures of the Lower San Fernando Dam and Dam No. 2 of the Mochikoshi tailings dam indicate that large deformations can occur after cessation of earthquake shaking.  Monotonic  loading tests performed on cyclically liquefied material are used to investigate the behaviour of soils under such conditions. A comparison between the pre- and post-liquefaction monotonic stress-strain behaviour of two types of sand is shown in Fig. 2.9 (Seed, 1979).  In both sands, a pore pressure ratio of  100% is developed during cyclic loading prior to monotonic loading. A s shown in Fig. 2.9a., the pre-liquefied sand gains strength steadily to attain a peak deviator stress of about 7 k g / c m at 2  approximately 20% axial strain.  In contrast, the liquefied sand loses most of its stiffness over a  large strain before the sample dilates at about 20% axial strain and exhibits strain hardening behaviour. The sample regains most of its strength when an axial strain of about 40% has been developed.  19  Chapter 2 Behaviour of Saturated Undrained Soils  Figure 2. 9 Post-Liquefaction Monotonic Behaviour of Saturated Sands (Seed, 1979) a. Sacramento River Sand, D = 40%; b. Mine Tailings, D = 95% R  R  As shown in Fig. 2.9b, similar stress-strain behaviour is observed in dense mine tailings sand.  In contrast to the looser sample, the sample deforms under zero stiffness over a smaller  strain range of nearly 10% and regains most of its strength when an axial strain of 30% has been developed.  This implies that the rate of stiffness build-up increases with increasing relative  density which is consistent with the findings of Thomas (1992). Simple shear post-liquefaction tests on undisturbed samples of Duncan Dam foundation material have shown that samples with initial static shear stress are stiffer than those without static shear stress (Salgado and Pillai, 1993).  Post-liquefaction stiffnesses were reduced by  about 2 to 50 times the pre-liquefaction stiffness.  It is important to note that  samples were  cyclically loaded to a maximum shear strain of 4% and not necessarily until a pore pressure of 100% has been developed.  20  Chapter 2 Behaviour of Saturated Undrained Soils  T h e effects of pore pressure ratio developed during cyclic loading on post-liquefaction monotonic behaviour have been investigated by means of triaxial and torsional shear devices (Yasuda et al., 1991,1994; Thomas, 1992). In addition, the effects of additional cyclic loading past the liquefaction stage on the post-liquefaction response and expressed as a factor of safety has been studied (Yasuda et al., 1994). A s shown in Fig. 2.10, the liquefied stiffness of soil decreases with increasing excess pore pressure ratio. Stiffnesses were reduced by as much as 500 to 1000 times the original stiffnesses when 100% pore pressure is developed. Stiffnesses are decreased further when samples are cycled past the initial liquefaction stage. Recent studies on the undrained response of loose Syncrude sand under various loading conditions by Vaid et al. (1998) have shown that the liquefied strength and stiffness of the soil is dependent on the direction of loading relative to the direction of deposition. T h e liquefied sand is strongest when the load is applied in the same direction as the direction of deposition and weakest when applied perpendicular to the direction of deposition. Strengths were reduced by as much as a factor of 5. Thomas (1992) showed that the post-liquefaction undrained stress-strain curves on Fraser River Sand can be characterized into three distinct regions as shown in Fig. 2.11. 1 is the region with near zero stiffness.  Region  Region 2 is a region of increasing stiffness and can be  approximated by a parabolic curve. T h e stress-strain curve in region 3 is nearly linear. The stress-strain response has been found to become stiffer with increasing density. The length of region 1 increases with decrease in relative density and the slope of region 3 increases with increase in relative density.  For loose sands, a similar trend is observed as the stiffness of  the response increases with increase in confining stress. However, the effect of confining stress is not as apparent at higher relative densities.  21  Chapter 2 Behaviour of Saturated Undrained  Soils  (a)  20  30  40  50  60  80  Shear strain , r (96)  (b) Dr=46.4~49.5% (kgf/cm ) o 5I 1  1  o 4 CO* t o <J CD •>  l_  1  [——i  '  Triaxial compression test  7/* /  •du/o '=0.38 / /  1  1  . Toyoura Sand ' o '=0.5kgf/m  2  c  ^u/Oe=0.26^///  .  /  ^u/o v0.94 e  55 2  /  t_  o  o 1 > o ° 0  —i  .4u/o 'e1.0 c  l  4 6 Axial Strain, c (%)  i  8  i  10  Figure 2.10 Effect of Excess Pore Pressure on Post-Liquefaction Behaviour of Saturated Sands(Yasuda, 1994)  22  Chapter 2 Behaviour of Saturated Undrained Soils  Figure 2.11 Characterization of Post-Liquefaction Curve (Thomas, 1992)  23  Chapter 2 Behaviour of Saturated Undrained Soils  2.5.1 Residual Strength T h e residual strength and limiting shear strain are the most sought after parameters when testing soils under post-liquefaction monotonic loading. T h e residual strength value for soils exhibiting contractive behaviour is the steady state strength.  However, for soils exhibiting limited  liquefaction behaviour, the post liquefaction strength depends on the strain level. suggested that the residual strength  is the strength  achieved after the dilation  It has been process is  complete.  This residual strength value may be very high if the soil exhibits strong dilative  response.  However, it is commonly assumed that the residual strength is the strength at the  phase transformation state. This value is considered to be conservative. Based on laboratory testing, the residual strength has been found to be influenced by relative density, fines content, and direction of loading. dependent on vertical consolidation pressure, a' , vo  Because the residual strength is also  it is often expressed as a fraction of a' vo  In  general, the residual strength decreases with decrease in relative density and increase in fines content. Triaxial compression and extension, simple shear, and hollow cylinder tests on Syncrude sand (Vaid et al., 1998) have shown that the residual strength is anisotropic and that the residual strength  ratio,  s /a' , r  v0  decreases as much as by a factor of 5 as loading changes  compression to extension.  from  Compression loading occurs when the major principal stress is  perpendicular to the plane of deposition and is defined as a  = 0°.  Simple shear loading  corresponds to a = 4 5 ° and extension loading to a = 9 0 ° . A s shown in Fig. 2.12, for a void ratio of about 0.70, the values range from s/a'vo ~ 0.3 to 0.06. The residual strength can be determined directly from laboratory testing or indirectly by correlating equivalent clean sand S P T ( N ^ o values to a residual strength using the empirical chart developed by S e e d and Harder (1990) and shown in Fig. 2.13. T h e residual strength can also be obtained by using empirically derived equations such as the ones developed by Stark and Mesri (1992) and Byrne (1990) as shown below: a) Byrne (1990)  24  Chapter 2 Behaviour of Saturated Undrained Soils  S =0.0284 Pa- (°- < ' »-) 173  r  JV )  e  > 0.087 o r , , /  E q . 2. 1a  b) Stark and Mesri (1992)  = 0.0055 - ( J V1, ^60-e.v )  E q . 2. 2b  400 a'^400  kPa, b=0.5,  e.=0.790 a  a b and a held constant during shear mt  0  r  = 0  30 — -90 8 ^1  -  £3  4 5  10  ( 55)  Figure 2.12 Shear strength of strain-softening Syncrude Sand (Vaid et al. 1998)  However, the residual strength ratio of Duncan Dam foundation material has been found to be 0.21, which is almost 3 times greater than that predicted by Stark a n d Mesri's empirical relationship (Salgado and Pillai, 1993).  Lab test results on Tia Juana silty sand and Lagunillas  sandy silt yielded residual strength ratios varying from about 0.08 to 0.18 (Ishihara, 1993). Backcalculated residual strength ratios from c a s e histories of silty sands which have liquefied yielded values in the same range.  25  Chapter 2 Behaviour of Saturated Undrained Soils  2000  1600 o  z  ui cr  •  E A R T H Q U A K E - IN0UCE0 LtOUEFACTION ANO SLIOINC C A S E HISTORIES WHERE S P T OATA A N D R E S I D U A L S T R E N G T H P A R A M E T E R S HAVE S E E N M E A S U R E D .  o  E A R T H Q U A K E - INOUCEO LIQUEFACTION A N O SLIOING C A S E HISTORIES WHERE S P T OATA A N D R E S I D U A L S T R E N G T H P A R A M E T E R S HAVE B E E N ESTIMATED.  •  C O N S T R U C T I O N - I N O U C E O L I Q U E F A C T I O N A N D SLIOING  CASE  HISTORIES.  1200  cc < X  io Q Ul  800  z  < cr Q  z 3  400 LOWER S A N F E R N A N D O DAM  _l <  g ui CE  O  4  8  EQUIVALENT  C L E A N  12 SAND  16  S P T BLOWCOUNT.  20 (N,)  24 6  0  .  c  s  Figure 2.13 Relationship Between Corrected "Clean Sand" Blowcount (N^eo-cs and Undrained Strength (Sr) (Seed and Harder, 1990). 2.5.2 Limiting Shear Strain The  limiting shear strain is the limited amount of shear strain that could be developed  during cyclic loading regardless of the cyclic stress ratio and number of cycles.  It is often  assumed to be the strain corresponding to the point at which the residual strength has been developed. T h e limiting shear strain can be determined from laboratory testing or by correlating (Ni)6o-cs values with limiting strain values as proposed by S e e d et al. (1985).  Alternatively, the  limiting strains can be obtained from the empirical relationship developed by Byrne (1990):  ^  =  1 0  (22-0.05.</V,  W  E  q  2  2  The limiting shear strain depends on the relative density of the soil. Loose soils exhibiting contractive behaviour may have limiting shear strain values ranging from 10 to possibly over 30%.  26  Chapter 2 Behaviour of Saturated Undrained Soils  O n the other hand, the limiting shear strain of medium dense to dense soils may only range from 2 to 5%.  2.6 Post-Liquefaction Volumetric Deformation Excess pore pressures are generated when undrained saturated soils are subjected to cyclic loading caused by earthquake cessation of earthquake  shaking.  Pore pressures dissipate during and at the  loading causing volumetric strains.  surface as ground settlement.  T h e s e strains manifest on the  Although post-liquefaction settlement may not be as damaging as  lateral ground displacements, field evidence have shown that settlement should not be ignored. Lee and Albaisa (1974) investigated the cyclic induced settlement behaviour of saturated sands by means of cyclic triaxial tests. They found that the reconsolidation volumetric strains due to dissipation of excess pore pressure increases with increasing grain size of the soil, decreasing relative density, and increasing excess pore pressure generated during cyclic loading. In addition, they found that the volumetric change increases with additional dynamic loads past the condition of development of 100% pore pressure ratio.  Tatsuoka et al. (1984) studied reconsolidation volumetric strains after 100% excess pore pressure has been generated by cyclic undrained simple shear loading and found that the amount of settlement strongly depends on relative density and maximum shear strain developed in the soil.  This agrees well with the findings of Lee and Albaisa. They also found that settlement is  insensitive to effective pressure prior to cyclic loading.  Tokimatsu and Seed (1987) plotted available data in terms of volumetric strain versus relative density.  Their plot shows that volumetric strains are strongly dependent on relative  density and cyclic shear strain.  However, their data show that there is a maximum cyclic shear  strain above which cyclic shear strain does not affect the reconsolidation volumetric strains. Ishihara and Yoshimine (1992) plotted available data in a series of curves in terms of volumetric strain versus maximum shear strain for several values of relative density.  T h e plot,  shown in Fig. 2.14, supports the idea that a maximum volumetric strain due to consolidation  27  Chapter 2 Behaviour of Saturated Undrained Soils  exists. Interestingly, the value is attained when the maximum amplitude of shear strain is at least 8% for all relative densities.  The data can also be plotted in terms of factor of safety against  liquefaction. A s a result, Ishihara and Yoshimine prepared a chart to approximate the volumetric strains if both the factor of safety against liquefaction and the relative density are known.  This  chart is shown in Fig. 2.15.  Figure 2.14 Volumetric Strain versus Maximum Amplitude Shear Strain for Different Relative Densities (Ishihara and Yoshimine, 1992)  28  Chapter 2 Behaviour of Saturated Undrained Soils  Figure 2.15 Volumetric Strain versus Factor of Safety Against Liquefaction (Ishihara and Yoshimine, 1992)  29  Chapter 2 Behaviour of Saturated Undrained Soils  2.7 Summary T h e understanding gained from laboratory tests have allowed engineers to develop methods to model liquefaction behaviour and predict response.  This chapter reviewed the  behaviour of soils under saturated undrained monotonic and cyclic loading based on extensive laboratory testing by numerous researchers.  Deformations associated with liquefaction can be  the result of a loss in strength and stiffness upon liquefaction or the dissipation of the excess porewater pressures generated during liquefaction. Deformations obtained prior to triggering of liquefaction are small compared to those that occur after liquefaction.  Upon liquefaction, the soil temporarily behaves like a liquid and may  deform significantly under a driving stress.  T h e amount of strain the soil must undergo before  recovering strength and stiffness depends on its relative density. Despite the liquid-like behaviour over a relatively large strain range, it has been shown that liquefied soil can transmit earthquake induced shear stresses.  30  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  CHAPTER 3  EVALUATION OF EARTHQUAKE INDUCED  DEFORMATIONS OF EARTH  STRUCTURES 3.1 Introduction There are three basic concerns when dealing with seismic analyses of earth structures. The first is whether or not cyclic loading due to earthquake shaking will induce liquefaction. This procedure is well documented and usually involves a triggering analysis wherein the the factor of safety against triggering of liquefaction is a s s e s s e d by comparing the cyclic resistance ratio with the cyclic stress ratio caused by the design earthquake.  A factor of safety of less than unity  means that liquefaction is likely to occur. The second concern is whether the  residual strength of the liquefied materials is  adequate to prevent the occurrence of a flow slide.  This involves a limit equilibrium analysis  where the stability of a potential sliding mass is a s s e s s e d by a factor of safety.  The factor of  safety is the ratio of the shear strength of the soil to the driving shear stress. A residual strength is assigned to those zones predicted to have liquefied. A factor of safety of less than unity implies that the earth structure is not stable and that large displacements may occur.  If the factor of  safety is greater than unity, flow failure may not occur, although the displacements may not be acceptable. The magnitude of these displacements is the third concern in liquefaction analysis. Prediction of earthquake-induced displacements of earth structures is a more difficult problem. A reliable estimate should take account of displacements due to inertia forces as well as those due to the reduced liquefied soil stiffness under gravity loads. In the past 30 years, various methods have been developed to estimate seismic deformations of earth dams, embankments and natural slopes.  Methods vary from mechanics-based simple one-dimensional methods or  rigorous effective stress methods to empirical based formulas.  31  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  Sophisticated effective stress methods are more fundamental and complicated than the more simple one-dimensional methods.  However, in some c a s e s , displacements predicted from  simpler methods are more appropriate.  The various methods developed are summarized along  with their respective advantages and disadvantages in the following paragraphs from the most simple to the more complicated.  3.2  Empirical Methods There are mainly two empirical equations used to predict potential seismic-induced  deformations.  Both  equations  were  developed  using  field  data  of  liquefaction-induced  displacements from past earthquakes and have considered soil conditions and topography.  3.2.1 Hamadaetal. Hamada et al. (1987) compiled data of ground displacements observed from sites liquefied during the 1964 Niigata, 1971 San Fernando, and 1983 Nihonkai-Chubu earthquakes, which have earthquake magnitudes of greater than M7.0. T h e database involved mainly of gently sloping sites consisting of loose, clean, uniform sand of medium grain size. The  data  showed  that  the  liquefaction-induced  displacements  influenced by the ground slope and thickness of liquefied layer.  are  most  strongly  A s a result, Hamada et al.  proposed the following empirical relationship to estimate displacements: i D = 0.75//2#  where  \_ 3  Eq.3. 1  D = displacement, meters H = thickness of liquefied layer, meters 9 = the larger of the ground slope or slope of base of liquefied layer, percent  32  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  A  comparison between  displacements estimated  from  the  empirical equation  and  displacements observed from the field data on which the equation was based found that most of the displacements ranged from about half to twice the predicted value.  This large scatter is  probably because many key factors such as soil type, soil density, and level of shaking intensity were not accounted for in the equation. T h e equation implies that soils with S P T ( N ^ o values of 4 and 20 would give the same displacement assuming they have both liquefied.  3.2.2 Bartlett and Youd Bartlett and Youd (1995) used a database of liquefaction induced lateral spread case histories from eight different major earthquakes in Japan and the United States to develop their more comprehensive empirical method. The earthquakes 9.2.  ranged in magnitude from M 6.4 to M  They used regression analysis in order to determine which factors such as earthquake  magnitude, soil conditions, and topography, most strongly influence lateral spread displacement. Bartlett and Youd found that two models with different parameters were required to predict lateral spread displacement for free face and ground slope conditions. Their models are as follows: a. Free Face Model log(D +0.01) = -16.366 + 1.178M - 0.927 log R - 0.013R + 0.657 logW + 0.348 l o g T H  + 4.527 log(100-Fi ) - 0.922 D 5 0 5  15  15  E q . 3. 2a  b. Ground Slope Model log(D +0.01) = -15.787 + 1.178M - 0.927 log R - 0.013R + 0.429 log S + 0.348 l o g T H  +4.527 log(100-F ) - 0.922 D 5 0 15  where  D  H  15  15  E q . 3. 2b  = horizontal displacement, meters  M = moment magnitude of earthquake R = horizontal distance to nearest seismic source, kilometers W = 100 * Height of free face/L, distance from channel S = ground slope, percent  33  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  T  1 5  = cummulative thickness of liquefied layer with ( N ^ o < 15, meters  F15 = average fines content in layer T , percent 1 5  D50  15  = average mean grain size in layer T15, millimeters  Although Bartlett and Youd's model is superior to Hamada's empirical equation, the results are very sensitive to earthquake magnitude, distance to seismic source, and to the average fines content.  Because the model was developed from primarily western U.S. and  Japanese data, it is most applicable to sites with similar characteristics. For example, the model yields reasonable predictions for regions with earthquake magnitude ranging from 6.0 to 8.0, underlain by shallow layers of sandy material ( F  15  < 50%) with ( N ^ o values less than 15.  In  addition, the depth to the liquefied layer should be less than 15 meters. Bartlett and Youd compared the displacements measured from their data set and compared them with those predicted by their equation.  About ninety percent of their estimated  displacements are within half to twice the observed displacements.  3.3  Pseudo-Static Methods The seismic stability of earth structures were initially assessed by a pseudo-static limit  equilibrium analysis. This type of analysis is similar to that used to evaluate flow slide potential. However, in contrast, the effects of earthquake shaking are accounted for and represented by constant horizontal and/or vertical accelerations. The pseudostatic accelerations produce inertial forces which act through the centroid of the failure mass. A factor of safety of less than unity may be acceptable because the actual vibrational earthquake loads act only for a finite time. This type of analysis gives an index of stability but does not give any estimation of displacements. Newmark (1965) was the first to develop a procedure wherein displacements can be predicted.  34  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  3.3.1 Newmark's Newmark  Method proposed a simple method to predict earthquake-induced displacements of  earth structures. A potential sliding soil wedge is modelled a s a rigid block resting on an inclined plane as shown in Fig. 3.1.  m.g.A(t)  water table  m.g.N  7~77  r  earthquake motion  a. Potential sliding surfaces of an earth dam.  b. Idealized potential sliding mass.  Figure 3.1 Idealized Potential Sliding Slope (Newmark, 1965)  T h e soil is assumed to exhibit rigid plastic behaviour where deformations occur when earthquake driving forces exceed the yield resistance of the block.  T h e yield resistance is the  force which causes the factor of safety against sliding to equal unity and has a value of mgN where m is the mass of the block, g is the gravity acceleration, and N is the yield acceleration in terms of fraction of gravity. Displacements of the block for any time history of base motion can be computed by double integrating the difference between the base acceleration and yield acceleration at times where the base acceleration is greater than the yield acceleration to obtain relative velocity.  The  relative velocity is then integrated to obtain displacements. Newmark applied his procedure to estimate the displacements of a sliding block due to four western U S earthquakes which were normalized to the same peak acceleration and velocity. He considered cases where the soil resistance is similar uphill as downhill (symmetrical) and  35  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  where the yield acceleration in one direction is significantly higher that the other (unsymmetrical). The following equations were proposed as a result of his analyses: a. Symmetrical Resistance  V  N  2gN  A  2  D = - — •(!--)  Eq.3.2  b. Unsymmetrical Resistance  V  2  D =  N/A > 0.5  (1  2gN  A  )•—  E q . 3. 3a.  A N  V  A  2gN  N  2  D =  0.13 < N/A < 0.5  A \  Eq. 3.3b.  D = ^— 2gN  N/A < 0.13  where  Eq. 3.3c.  V = velocity of the mass, taken as peak ground velocity A = maximum earthquake acceleration in terms of fraction of g.  Newmark's equation can also be developed from energy principles where the work done by the external forces (W ) minus the work done by the stress field (W ) is equal to the change ext  in)  in kinetic energy. This principle is expressed as :  W ,-W =±M-(V -V ) 2  e  int  f  2  = -±M-V  2  E q . 3. 4  where the final resting velocity, V , is equal to zero, and V is the specified initial velocity of the f  block which comes from earthquake forces greater than the yield resistance of the potential sliding block.  36  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  The external force on the block, as shown in Fig. 3.2.  is the gravity driving force which is  constant with displacement. T h e work done by the external force is the area beneath the driving force line. T h e work done by the internal force is the area beneath the soil resistance line. If only one earthquake pulse is considered, E q . 3.5. can manipulated and reduced to  V  2  D=  E q . 3. 5  2gN  If six velocity pulses were considered this equation would be identical to Newmark's formula in E q . 3.4c for unsymmetrical case N/A < 0.13.  V = Velocity M = Mass of the block D = Seismic displacement  ^  Force oil Resistance V  Driving force, Mg sin Ot (b)  Displacement, D  Figure 3. 2 Work Energy Approach to Newmark Method (Byrne, 1990)  This  method  displacements.  has  proven  to  be  very  useful  for  predicting  earthquake-induced  However, its assumption that soils behave in a rigid-perfectly plastic manner is  too simplistic and may cause the displacements to be underestimated.  Also, because yield  acceleration depends on effective stress, hence on pore pressure, it would be too difficult to obtain a value for yield acceleration  because pore pressures change considerably during  earthquake loading (Seed, 1966). Finally, the method only gives one displacement value along a potential sliding surface rather than an overall displacement pattern.  37  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  3.3.2 Byrne's Extended Newmark  Approach  In contrast to the assumption made in Newmark's approach, soil does not exhibit rigid perfectly plastic behaviour when liquefied but rather a large reduction in stiffness is observed when the pore pressure rises to cause the effective stresses to drop down to zero. strain curves more ideally resemble the curves shown in Fig. 3.3.  A s a result,  The stressByrne (1990)  extended Newmark's approach to incorporate the essentials of the stress-strain characteristics of loose saturated sands into a work-energy approach, as shown in Fig. 3.4.  b  Pre-liquefaction  H O  S„/a'„  DZ  m  ID  I  •p  Shear Strain, y  Figure 3. 3 Idealized Pre- and Post- Liquefaction Behaviour of Sand (Haile et al., 1996)  Strain Figure 3. 4 Work Energy Approach to Extended Newmark (Jitno, 1995)  38  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  Point P represents the pre-earthquake stress state of a soil element in an earth structure. Upon liquefaction, the stress state drops from point P to point Q, which occurs at a low strain. The resistance increases with strain until it reaches a residual strength value of S . Because the r  driving stresses from the slope are constant, the system accelerates as it deforms. A s a result, the system has a velocity when it reaches point R, where the resistance is equal to the driving stress. The system continues to deform until the external work done by the driving force is equal to the internal work done by the system.  If, in addition, the system has a velocity due to  earthquake shaking it will continue to deform to point T. A s shown in Fig. 3.4. Newmark's method does not consider the displacements from point P to S. Byrne developed equations to estimate liquefaction induced displacements by modelling a slope as crust lying on a layer of liquefied soil.  The liquefied layer is represented as a block  resting on an inclined plane. The liquefied layer can be assumed to exhibit linear elastic plastic or non-linear elastic plastic behaviour a s shown in Fig 3.5.  4-*  Linear  Soil Resistance  x,  •a  Non-linear  Shear Strain, y  Figure 3. 5 Linear and Non-Linear Stress-Strain Curves (Byrne, 1990)  The developed equations are the following: 39  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  a. Linear Elastic Plastic  D<D  D>D  L*  D=DB+  L  *?L  +  Eq.3.6a  1  K ^ -  D = D„ + - ( £ > , - £ > „ ) +  L  E q . 3. 7b  b. Non-Linear Elastic Plastic  D<D  D = — ( ^ - ^ - - M V  U  (\MV -\K D 2  2  L  D > D  where  D =  L  2  Eq.3.7a  )  +S D ) R  L  ^  E q . 3. 8b  D = liquefaction induced displacement D  st  = static displacement corresponding to point R in Fig. 3.5  D = displacement corresponding to mobilisation of residual strength, S L  K = stiffness of liquefied soil, equal to T / y s t  L  r  L  M = soil mass V = velocity of soil mass T t = initial static shear stress s  YL = limiting shear strain T h e above equations were developed considering one velocity pulse. This was justified because the displacements due to pulses prior to liquefaction are small compared to those that occur upon liquefaction.  T h e displacements due to velocity pulses after liquefaction are  accounted for by the introduction of a factor of safety against liquefaction (Jitno, 1995). Table 3.1 was proposed by Byrne (1996) in which the limiting shear strain and residual strength ratio can be determined if the S P T (N^eo value and the factor of safety against liquefaction are known. T h e  40  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  effect of the accumulation of strain with velocity pulses is accounted for in the residual strain. For example, a strain increment of loose sand may be 2 or 3% per cycle. T e n cycles would cause 20 to 30% shear strain.  Table 3.1 Post-Liquefaction Stress-Strain Parameters s,/a'o  (NOeo  7L F RIG«1.0  FTRIG ®  T  0-4  0.05-0.10  25-50  0.10-0.20  10-25  0.15-0.40  8 -15  20-35  15-20  0.30 - 0.50  5 -10  15-25  >20  >0.50  <5  < 15  4-  10  10-  The  0.5  15  displacements  predicted  from  this  method  > 100 30->  were  100  compared  to  displacements  observed from North American case histories (Jitno, 1995). The predicted displacements using a linear stress-strain assumption were close to the observed displacements.  In contrast, most of  the predicted displacements using a non-linear stress-strain assumption were within half to twice the measured values.  This suggests that a linear stress-strain assumption may be  more  appropriate for North American case histories. Although the methods described above consider inertia forces and the loss in stiffness due to liquefaction, they predict displacements on only a distinct failure surface. A s a result, they are not able to predict overall deformation of earth structures.  3.4 Two Dimensional Methods One-dimensional methods are only able to predict liquefaction-induced displacements on a potential failure surface.  In the field, however, deformations may result from the accumulation  of strain increments throughout the earth structure. A s a result, more realistic displacements can be predicted by analyses performed on a more general field rather than on a specific surface.  41  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  Simple two-dimensional methods using the finite element approach have been proposed by a number of researchers, a few of which will be summarized below.  3.4.1 Modified Modulus Approach Lee (1974) and Yasuda et al. (1992) among others have proposed a simple procedure for predicting liquefaction-induced displacements of earth structures.  Permanent  displacements  occur under the pre-earthquake static stresses in the structure as a result of a loss of strength and decrease in stiffness of the liquefied soil. A hyperbolic stress-strain relationship is assumed for the soil.  The procedure requires that a simple gravity turn on finite element analysis be  performed twice:  once using the pre-earthquake stiffnesses to establish reference deformations  and compute pre-earthquake stresses; the second time using the reduced soil stiffness under the stresses computed previously kept constant.  T h e difference  in deformation  between  both  analyses is computed and assumed to represent the deformation due to earthquake. Although this method is able to give an overall deformation pattern of an earth structure and accounts for reduction in stiffness due to liquefaction, it may underestimate the earthquakeinduced displacements.  This is because it does not consider the effect of inertia forces or  deformation due to dissipation of excess pore pressures. However, there are field examples such as the failure of the Lower San Fernando Dam and Dam No. 2 of the Mochikoshi tailings dam, where deformations occur after cessation of earthquake  shaking and inertia forces due to  earthquake loading need not be considered.  3.4.2 Strain Potential / Dynamic Stress Path Approach T o account for the effects of inertia forces, S e e d and his coworkers proposed a procedure for predicting earthquake-induced displacements using a strain potential concept. T h e procedure involves a series of steps which must be more or less rigorously observed and approached with caution in order to obtain reasonable results (Seed, 1979).  T h e steps can be summarized as  follows:  42  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  a) Compute pre-earthquake stresses in embankment using finite element method. b) Compute the stresses and dynamic response of the structure induced by a selected design earthquake using a dynamic analysis procedure. c) Subject representative samples of the embankment materials to the combined effects of the initial static stresses and the superimposed dynamic stresses and determine their effects in terms of the generation of pore water pressures and the development of strains. Perform a sufficient number of these tests to permit similar evaluations to be made, by interpolation, for all elements comprising the embankment. d) Use strains obtained from laboratory induced by the combined effects of dynamic and static stresses to estimate the overall deformation of the dam. In this method, effects of the earthquake are represented by equivalent static stresses which can be converted to nodal point forces in a finite element mesh.  The equivalent static  stresses are read off from static stress-strain curves of soils comprising each element of the embankment and represent the stresses required to cause the strain potentials as determined in step b. Although this method accounts for inertia forces in the embankment, it has been found to underestimate displacements in Upper San Fernando Dam.  This may be due to the fact that it  assumes that the post-earthquake curves can be represented by the pre-cyclic stress-strain curve. This procedure may be applicable to elements which are not predicted to liquefy but is not valid for those elements which have. A s shown in Chapter 2, the initial shear modulus of postliquefaction stress-strain curves are significantly less than that of the pre-cyclic stress-strain curve. A s a result, this method only partly accounts for the effect of a reduction in stiffness.  3.4.3 Jitno and Byrne's Approach In order to obtain an overall deformation pattern, Jitno(1995) extended Byrne's onedimensional method as described above to a two-dimensional system where a pseudo-dynamic finite element approach is used.  T h e procedure is similar to the modified modulus approach in  43  Chapter 3 Evaluation of Earthquake Induced Deformations of Earth Structures  that two finite element analyses are performed:  the first using pre-earthquake parameters to  establish reference deformations; the second using post-earthquake stress-strain parameters. However, in this procedure, the displacements in the second analysis must satisfy the energy principles as described in section 3.3.1. T h e displacements can be computed from the solution of the following equation:  Eq. 3. 8  where  [K] = global stiffness matrix of system {A} = nodal displacement {F} = static load vector {AF} = additional load vector to produce energy balance of E q . 3.5  T h e internal work done in a multi-degree-of-freedom system is the work done by the element stresses and strains and the external work corresponds to the work done by a static load vector, {F}-{A}. The additional force, AF, is equal to the product of a horizontal or vertical seismic T  coefficient, k, and the weight of a soil element, w.  However, k does not correspond to the peak  ground acceleration but is rather the result of an iterative procedure to balance the energy equation.  3.5 Non-Linear Effective Stress Analysis The best approach to accurately predict earthquake-induced displacement is a non-linear effective stress analysis wherein the soil response during an earthquake is simulated by means of a constitutive model developed from experimental observation.  Many constitutive models have  been developed which can more or less accurately predict the cyclic behaviour of soils although no one model can predict the behaviour of all types of soils. Most of the  proposed models are  incorporate different hardening properties.  based on complicated plasticity theories  which  A non-associative flow rule can be used which fully  couples shear stress and volumetric strain.  If the volume is constrained, shear-induced  44  Chapter 3 Evaluation of Earthquake Induced Deformations  of Earth  Structures  contractive volumetric strains cause an increase in pore water pressures thus a decrease in effective stresses. Displacements can be computed by solving the equations of motion using a step-by-step integration procedure. Changes in effective stress can be considered at every step. Unlike the simpler methods, in which liquefiable zones are assumed to trigger at the same time, both the triggering of liquefaction in different elements and the post-liquefaction displacement can be predicted. Although the procedure is more fundamental than other methods, it is complex, time consuming, and as a result costly. earthquake induced displacements.  In some cases the simpler methods can equally predict the An effective stress analysis is preferable to the simpler  methods particularly when drainage effects are important (Byrne, 1996).  3.6  Summary The various available methods for predicting earthquake-induced displacements have  been summarized. Most of the methods either use simplifying assumptions which neglect key parameters to a reliable prediction of liquefaction-induced displacements or are so complex that they become time-consuming, expensive, and inefficient for most applications. There is room for improvement in the procedures for predicting earthquake-induced displacements. The following chapter presents a proposed procedure which attempts to capture the essential features of soil behaviour under cyclic loading in a total stress approach.  45  Chapter 4 Proposed Two-Dimensional Method  CHAPTER 4  PROPOSED TOTAL STRESS METHOD  4.1 I n t r o d u c t i o n As discussed in the previous chapter, a reliable simple procedure to predict liquefactioninduced displacements should identify liquefied zones and account for displacements due to inertia forces and those due to the reduced stiffness of the liquefied soil. T h e present available simple methods use assumptions which overly simplify the effects of earthquake inertia forces and fail to adequately model the stress-strain behaviour of liquefied soil. A s a result, a total stress dynamic analysis procedure is proposed. In the proposed procedure, stiff pre-liquefaction stress-strain parameters are used initially. The time history of shear stress for a prescribed base input motion is computed for each element. Each pulse is weighted and when or if sufficient pulses occur, liquefaction is triggered for that element.  Soft post-liquefaction stress-strain parameters are assigned to the element and the  analysis is continued. In this manner, the various elements of the soil structure liquefy and soften at different times and allow both the amount and pattern of displacement to be predicted as shaking proceeds. In this chapter, the proposed two-dimensional method is presented.  The procedure by  which liquefaction is triggered in each element as well as the constitutive model used to represent the stress-strain behaviour of the liquefied soil are discussed.  Finally, the key parameters to be  used in the analysis are presented along with recommended values.  First, however,  the  traditional triggering analysis procedure is summarized and presented.  46  Chapter 4 Proposed  Two-Dimensional  Method  4.2 Triggering of Liquefaction The determination of liquefied zones has traditionally been a s s e s s e d by using the method proposed by Seed and his coworkers (1990).  A factor of safety against liquefaction, F , which L  compares the cyclic resistance ratio, C R R , with the cyclic stress ratio, C S R , is computed: F = CRR/CSR  E q . 4. 1  L  A factor of safety of less than unity implies that liquefaction would likely occur.  4.2.1 Cyclic Stress Ratio T h e C S R is usually evaluated using a total stress equivalent linear dynamic analysis program such as S H A K E (Schnabel et al., 1972) or F L U S H (Lysmer et al., 1974). Both programs use a frequency domain approach in which a base motion is represented by its harmonics.  The  response in terms of shear stresses or accelerations is the sum of the responses from each harmonic. T h e response is based on the solution of the wave equation. T h e nonlinearity of soil behaviour in terms of shear modulus and damping is accounted for by the use of equivalent linear soil properties.  An iterative procedure to obtain shear modulus and damping values compatible  with the effective strains in each layer is used. S H A K E performs a one-dimensional dynamic response analysis based on a continuous solution of the wave equation.  In contrast, F L U S H performs a two or an approximate  dimensional analysis by assuming a lumped mass system.  three-  Two-dimensional analyses can  generally provide sufficient accuracy except in cases where the structure is subject to significant three-dimensional effects such as high dams in narrow canyons wherein 3-D analyses may be preferable (Seed and Harder, 1990). One-dimensional dynamic analyses are generally not recommended for dams although they can provide reasonably accurate estimates of cyclic stresses within embankments, especially ones with flat slopes and high crest length to dam height ratio. dimensional analyses of the  Lower San Fernando Dam  However,  one- and two-  have shown that one-dimensional  47  Chapter 4 Proposed Two-Dimensional  Method  analyses underestimate both accelerations and cyclic shear stresses near the crest and upper faces of the embankment (Seed and Harder, 1990). T h e C S R is evaluated from the maximum induced shear stress computed by the dynamic analysis from the following relationship:  C S * =  where  (T ) d y  CT' O V  m a x  f  t  6  '  5  (  r  *  )  n  "  Eq.4.2  = maximum shear stress computed from the dynamic analysis,  = the initial effective overburden stress  T h e factor, 0.65, is an equivalent loading factor which converts the random earthquake induced shear stresses into an equivalent uniform cyclic shear stress acting over a number of cycles.  T h e equivalent number of cycles depend on the earthquake magnitude as depicted in  Table 4.1 (Seed et al., 1975), based on a study by Seed and Idriss (1967). equivalent uniform shear stress equal to 0 . 6 5 ( T ) dy  For example, an  would act over 15 cycles for an earthquake  max  magnitude of 7.5.  Table 4.1  Equivalent Number of Cycles  Magnitude  Number of  Earthquake  Cycles  5'U  2-3  6  5  6 /  10  3  4  A 8 /  15 26  1  2  Alternatively, the C S R can be evaluated from an empirical relationship developed in conjunction with the simplified liquefaction evaluation procedure proposed by S e e d and Idriss (1971) based on the results of S H A K E analyses, as follows:  CSR = 0 . 6 5 - - ^ r where  d  Eq. 4. 3  a = peak ground acceleration  48  Chapter 4 Proposed Two-Dimensional Method  g = acceleration due to gravity a'vo, a'vo = total and effective overburden stress respectively r = stress reduction factor due to soil depth d  4.2.2 Cyclic Resistance Ratio O n e of the best methods of evaluating C R R is by testing undisturbed samples of representative materials.  The number of cycles to cause liquefaction is determined for different  cyclic stress ratios and the results are plotted on a semi-logarithmic scale as shown earlier in Fig. 2.3.  T h e C R R is usually taken as the cyclic shear stress required to cause liquefaction in 15  cycles.  However this method is limited to relatively large and important projects due to the  difficulty and high cost associated with obtaining high quality undisturbed samples. The  CRR  is generally  determined  by using S P T testing  and  Seed's  liquefaction  assessment chart, shown in Fig. 4.1, which empirically correlates the C R R with ( N ^ o values based on past liquefaction experience.  T h e C R R from Seed's chart is for an  magnitude of 7.5 corresponding to 15 cycles.  earthquake  A s discussed in Chapter 2, the C R R must be  corrected for overburden stress, static shear bias, and earthquake magnitude. Alternatively, the C R R can also be evaluated from empirical charts correlating C R R and cone penetration (CPT)  tip resistance. Stark and Olsen (1995) have compiled cone penetration  liquefaction assessment charts for different types of soils ranging from clean sands, silts to gravelly soils.  4.3 Proposed Method As shown in Chapter 2, there are three main phases in the cyclic-induced liquefaction behaviour of sands which are depicted in Fig. 4.2.  In the pre-liquefaction triggering phase the soil  exhibits stiff behaviour as it is subjected to cyclic loading. Because the strains developed at this phase are relatively small, the soil shear moduli are usually determined from small strain tests such as resonant column tests.  Upon the triggering of liquefaction, the second phase, the soil  49  Chapter 4 Proposed Two-Dimensional Method  behaves like a liquid and the shear stresses drop down to zero a s the shear stresses try to reverse directions.  Further straining in the post-liquefaction phase, the third phase, causes the  soil to deform significantly under nearly zero shear stiffness before finally regaining strength upon even further deformation.  T h e shear stiffnesses in the post-liquefaction phase are reduced by  factors of 25 up to 500 of the pre-liquefaction stiffness (Haile et al., 1996). 0.6  Percent Fines = 35  15  <5  0.5  A|0  o 0.4 I -CRR curves for 5,15, and 35 percent fines, respectively I  >  ca 0.3  GO o  U  0.2  FINES CONTENT ^ 5%  ijf  0  0.1  Modified Chinese Code Proposal (clay content = 5%) ®  ^30  Marginal Liquefaction Liquefaction  Adjustment Pan - American data Recommended Japanese data By Workshop Chinese data 10  20  30  No Liquefaction • o A  40  50  Corrected Blow Count, (Ni)^ Figure 4.1  Cyclic Resistance Ratio Based on SPT Blow Count, M 7.5 (NCEER, 1997)  50  Chapter 4 Proposed Two-Dimensional Method  Liquefaction Trigger  x,  f  y Post-Liquefaction Trigger  Figure 4. 2 Cyclic Induced Liquefaction Behaviour of Sands  T h e above three phases are captured simply in the proposed procedure. T h e manner in which the three phases are modeled are presented and discussed.  4.3.1 Pre-Liquefaction Trigger T h e pre-liquefaction analysis of the proposed method is similar to that of the current methods in that the soil is assumed to exhibit linear behaviour and the cyclic induced shear stress is compared to the cyclic resistance. Unlike the current methods, however, the proposed method does not consider the nonlinear soil behaviour with respect to shear modulus and damping through an iterative procedure.  Parameters  correspond to the uniform  Rayleigh stiffness-  proportional viscous damping and constant shear moduli which are estimated as a fraction of the maximum shear modulus, G  m a x  , parameters are selected prior to analysis. T h e s e pre-liquefaction  parameters are maintained in each element during the dynamic shear stress history until the element is predicted to liquefy.  51  Chapter 4 Proposed Two-Dimensional Method  4.3.2 Liquefaction Trigger The triggering of liquefaction is evaluated by tracking the dynamic shear stress history for each element.  An example of a dynamic shear stress history for a prescribed base motion is  shown in Fig. 4.3.  The cyclic shear stress, t , dy  is computed at each time step and is defined as  the difference between the current shear stress, x , and the static bias, t . xy  st  The static bias, T „ is s  equal to the shear stress calculated from static analysis. A maximum value of T  D Y  defines a half-  cycle or a cyclic pulse, so that the dynamic shear stress history for an element can be represented as a series of half-cycles or pulses which cumulatively contribute to the triggering of liquefaction.  Liquefaction is predicted to occur when a number of pulses of sufficient amplitude,  has been accumulated.  A  \ /*  /  ^<Jy  '  }  :  ^ / ^ ^ W r r r r i  ! ?  •  :  Half eyck  time  j  Figure 4. 3 Cyclic-Induced Shear Stress History  A s shown in Fig. 4.3, the dynamic shear stress history for an element is non-uniform and must be converted to an equivalent uniform cyclic history in order for liquefaction to be adequately predicted.  The amplitude of the uniform cyctic history is taken to be T , the cyclic shear stress 1 5  required to cause liquefaction in 15 cycles.  Each pulse is weighted to an equivalent number of  cycles, A N , at T . Liquefaction is predicted when the summation of A N eq  1 5  eq  is equal to 15.  Weighting of each cyclic pulse is carried out using the relationship between x number of cycles required to cause liquefaction.  dy  and  The cyclic shear stress is normalized with  52  Chapter 4 Proposed Two-Dimensional Method  respect to T . 1 5  This relationship, as shown in Fig. 4.4, is based on laboratory test data.  Application of the relationship is as follows. weighted such that A N  e q  For T  equal to x , the corresponding half cycle is  D Y  15  is equal to 0.5 and liquefaction is triggered in 30 half-cycles or 15 full  cycles in that element. O n the otherhand, if r  dy  is equal to 1.5T , then A N 15  e q  is approximately equal  to 2.5 and liquefaction is triggered in 6 half-cycles or 3 full cycles of x = 1.5T , as schematically dy  shown in Fig. 4.4. Similarly, if i  d y  is equal to about 0.75x , then A N 15  e q  15  is equal to 0.075 liquefaction  is triggered in 100 full cycles of the above amplitude in that element. Because the dynamic shear stress histories in all elements are not equal, liquefaction in each element is triggered at different times. 2.0 1.8 1.6  I  I  I  I I I I  I  I  i  l  l  !  !  J  I  \ •\ \  Approximate  F i t Curve  1.4 1.2  •A  in  ~>  1.0 0.8 0.6 0.4 0.2 0.0  I I I II I  T  1  I I I II I  10  I I I I II  100  1000  Number of C y c l e s to Liquefaction  Figure 4. 4 Relationship Between Normalized Shear Stress and Number of Cycles to Liquefaction (Byrne, 1990)  The relationship shown in Fig. 4.5 is mathematically expressed as follows:  (—1.0)  T  D Y  /T  1 5  >  1  0  AA^=0.5-10  (0.5-1*) ' r  5  Eq. 4. 4a  53  Chapter 4 Proposed Two-Dimensional  Method  (I*-1.0) (0.40-^2-)  T /x d y  where  T T  D Y  N 5  AN N  1 5  1 5  < 1.0  AN  eq  =0.5-10  Eq. 4.4b  r , s  = half-cycle stress amplitude = cyclic shear stress required to cause liquefaction in 15 cycles e q  = equivalent number of cycles at T  1 5  = 15 cycles  0.5 = half cycle. It has been stated that in order for the equivalent number of cycle weighting process to be consistent, Fig. 4.4 should be represented by a single linear relationship on a log-log plot (Idriss, 1998). However, a study, as shown in the next chapter, showed that this procedure is consistent. Upon liquefaction, the soil temporarily behaves as a liquid and has very low resistance to shear stresses.  In this procedure, this transition phase into a liquid state was conservatively  ignored and a condition of 100% pore pressure was immediately imposed upon the element as soon as liquefaction is triggered, as shown in Fig. 4.5.  This effect is simulated in the numerical  model by setting the horizontal stresses equal to the vertical stress stresses equal to zero (T^ = 0) after the summation of A N  e q  (CTXX  = a  z z  = a ) and the shear  exceeds 15.  yy  In addition, post-  liquefaction stress-strain parameters are assigned to those zones which are estimated to have liquefied.  4.3.3 Post-Liquefaction Trigger As discussed earlier, loading after liquefaction has been triggered causes the soil to deform considerably under very low shear stresses. However, after a significant amount of strain has been developed, the liquefied soil eventually regains strength and stiffness. The stiffness of the liquefied soil has been found to be reduced by a factor of 25 to 500 times that of pre-liquefied soil as shown in Fig. 4.6.  54  Chapter 4 Proposed Two-Dimensional Method  Figure 4. 5 Modeled Cyclic Induced Behaviour of the Soil Element  Pre-liquefaction b H  CJ V>  S,/o'„  IS)  e in  Shear Strain, y  Figure 4. 6 Stress-Strain Response of Soil In the proposed procedure, the liquefied soil is assumed to exhibit bi-linear elastic plastic behaviour.  T h e behaviour observed in the laboratory as well as that idealized by the model are  shown in Fig. 4.8.  Although the non-linear behaviour of the liquefied soil in loading mode appears  55  Chapter 4 Proposed Two-Dimensional Method  to be overly simplified by using a linear stress-strain model, analyses on case histories using both linear and non-linear assumptions have shown that a linear assumption predicts response as well as or better than a non-linear assumption for monotonic post-liquefaction loading (Jitno, 1995).  A aviour observed soil ber !  model approximation  I  •  .— —  /  /  1  •  /  j  ^ Y  j Ylim  Y = 1/10 y,  im  Figure 4. 7 Actual and Idealized Post-Liquefaction Behaviour of Saturated Sands Under Cyclic Loading T h e plot shown in Fig. 4.2 as well as the plots shown in Chapter 2 show that liquefied soil exhibits stiffer behaviour in unloading than in loading mode.  A s a result, different loading and  unloading bi-linear shear moduli are required to capture the response of the idealized soil behaviour under cyclic loading. A n examination of the laboratory data indicates that the ratio of the unloading and loading modulus can vary from about 7 to over 12.  Hence, an average  constant value of 10 is assumed. This is also shown in Fig. 4.8. Loading and unloading modes need to be defined in order to differentiate the shear moduli assigned to a zone.  This is done by storing, updating, and comparing  earthquake-induced shear stress, ( T ) d y  m a x  the maximum  , in each zone to the 'current' zone shear stress, T , at d y  56  Chapter 4 Proposed Two-Dimensional Method  each time step.  If t  d y  is greater than ( T ) x , then the loading shear modulus is assigned to the dy  ma  zone and the ( T ) x is then updated to equal T . d y  m a  dy  In contrast, unloading mode is defined by the case where i is depicted in Fig. 4.8.  If x  d y  stays the same sign as ( T ) d y  the unloading modulus is still used in the zone.  When t  m a x  d y  is less than 0.99 ( T ) d y  , and  , but starts to increase in value, then changes sign relative to ( T )  d y  m a x  d y  m a x  , then  the loading modulus is used and the maximum shear stress is reset to the current shear stress value and updated at each time step.  tk T  pre-liquefaction curve  ^^\\dea\\T.ed  ~  i * ^  r (tj„  w  0.99(T„)4  /  .  0.99{t0„  (^)_  j  1  L  Idealized post- liquefaction curve  Figure 4. 8 Definition of Loading and Unloading in Proposed Model  T h e use of loading and unloading shear moduli creates shear stress versus strain loops when modeling soil under dynamic loading conditions. These loops, known as hysteresis loops, dissipate energy and act to damp the soil system. Although damping is somewhat accounted for in the system, it is not accurate because the modeled loops only approximate the hysteresis loops observed from laboratory tests and case histories.  A comparison between hysteresis loops  observed from the model and those observed from laboratory data is shown in Fig. 4.9 and indicates that the damping applied in the model may be greater than that experienced under field and laboratory conditions, especially when hysteretic damping is combined with viscous damping.  57  Chapter 4 Proposed Two-Dimensional Method  Different loading and unloading shear moduli were used only after liquefaction has been triggered. This is because equivalent linear properties, which have been obtained in S H A K E and F L U S H , are used in the pre-liquefaction analysis. T h e loading liquefied shear modulus used is computed from the ratio of the residual shear strength to limiting shear strain. This is depicted in Fig. 4.10. 40 Actual Hysteresis Loop Modeled HysteresisLoop  20  CO CO  o  H  co O  -20  -40  -15  -10  0 S h e a r Strain, y  10  15  Figure 4. 9 Comparison Between Actual and Modeled Hysteretic Loops  58  Chapter 4 Proposed Two-Dimensional Method  Figure 4.10 Linear Elastic-Plastic Behaviour of Soils A s shown in Chapter 2, soil residual strength is anisotropic and can display up to five times as much strength in compression loading than in extension loading.  This was not  considered in this version of the proposed model because it does not significanlty affect the model's application presented in this thesis. However, it is recommended that strength anisotropy be incorporated in a more refined version of this proposed model.  4.3.4 Mohr-Coulomb Model  4.3.4.1 Introduction The total stress analysis described here cannot be performed using programs based on a frequency domain approach. Because the cyclic shear stresses must be assessed for at least each half cycle, a computer program based on a time domain approach is needed. The computer code  F L A C (Fast Lagrangian Analysis of Continua) which is a two-dimensional explicit finite  difference program was selected for the analyses. A subroutine which performs the proposed procedure was written for use in F L A C .  A more detailed description of F L A C is presented in  Chapter 5. F L A C has several built-in constitutive models, one of which is the simple and verified Mohr-Coulomb model. The procedure uses a modified Mohr-Coulomb model to simulate the soil behaviour under static and dynamic loading conditions.  59  Chapter 4 Proposed Two-Dimensional Method  T h e Mohr-Coulomb model assumes soil exhibits simple linear-elastic plastic behaviour, as shown in Fig. 4.8.  In terms of plasticity, the model features a yield surface, a flow rule, and has  been modified to include a definition for loading and unloading for dynamic problems.  4.3.4.2 Yield Sudace The yield surface is defined by the Mohr-Coulomb shear yield function:  f,=&-o' N -r2cJW;<0 3  where  Eq.4.5  4  f = shear yield function s  CT'I = major effective principal stress  <> j = friction angle c = cohesion  1 + sin d> a ' = minor effective principal stress 3  =  1 - sin (j> T o determine loading states, trial incremental principal effective stresses are assumed to be completely due to elastic principal strain increments and related by Hooke's law.  Principal  effective stresses are updated at each step by summing the incremental principal effective stresses and the previous stresses.  If the new stresses exceed the failure criterion, they are  modified using plasticity theory. Plastic strain increments are related to the principal stresses by a flow rule.  4.3.4.3 Non-associative Flow Rule T h e flow rule has the following form:  As? = where  E  q  4  6  Aej = plastic principal strain increment P  5g = shear potential function corresponding to a non-associative flow rule s  5o'i = principal effective stress increment The shear potential function has the form:  60  Chapter 4 Proposed Two-Dimensional Method  E q . 4. 7  8, = ° " i - o - W v 3  where  Af,, =  1 + sin v  , v = dilation angle  1 - sin v  4.4 Required Parameters T h e key parameters  in the total stress procedure are the common Mohr-Coulomb  variables such as cohesion, c, friction angle,  shear modulus, G , and bulk modulus, B. For pre-  liquefaction dynamic analyses, the key parameters are the shear modulus and damping.  For  post-liquefaction analyses, the key parameters are the residual strength, expressed as a cohesion value, and the loading and unloading shear moduli.  4.4.1 Pre-liquefaction Parameters The main parameter in the pre-liquefaction portion of the analysis are the equivalent elastic shear modulus, G , which is taken to be a fraction of G modulus at low shear strains, G  m a x  m a x  , and damping.  The shear  , can be determined from laboratory tests such as resonant  column tests, from empirical correlations with cone penetration data, in-situ shear wave velocities, or from a data base of values. The fraction of shear modulus value to use is more difficult to determine and depends mainly on the input dynamic motion and the nonlinear stress-strain properties of the soil.  The  value was taken based on the average of the values output in S H A K E or F L U S H analyses. Similarly, the damping value is very difficult to determine.  F L A C has the option of using  Rayleigh- or "local"-type damping. Unfortunately, there is no database of values for Rayleigh-type damping parameters.  A s shown in Chapter 5, the value was taken as the average of the values  output in S H A K E or F L U S H analyses.  61  Chapter 4 Proposed Two-Dimensional  Method  4.4.2 Post-Liquefaction Parameters  4.4.2.1 Residual  Strength  A s presented in Chapter 2, the  residual strength can be determined directly from  laboratory testing or indirectly by correlating equivalent clean sand S P T ( N ^ o values to a residual strength using the empirical chart developed by S e e d and Harder (1990). T h e residual strength can also be obtained by using empirically derived equations such as the ones developed by Stark and Mesri (1992) and Byrne (1990). It was shown in Chapter 2 that the residual strength depends significantly on the pore pressure developed during liquefaction, relative density, fines content, and direction of loading, i.e. direction of major principal stress.  For example, based on Vaid et al.(1998), a change in the  direction of loading can change the residual strength by a factor of up to 5. Because the direction of major principal stress can range in the field condition from extension to simple shear to compression (Byrne et al., 1998), it is recommended that the residual strength be varied with direction of loading or that the lowest, most conservative, residual strength be used.  4.4.2.2 Limiting Shear Strain T h e "limiting" shear strain, as defined in this procedure, is the strain required to mobilize the residual strength. modulus,  Giiq,  This limiting shear strain is mainly required to define the liquefied shear  along with the residual strength. T h e limiting shear strain value can be determined  from laboratory testing and can vary significantly depending on the value of the relative density of the soil as well as on the liquefaction behaviour. Most empirical shear strain values, such as those proposed by S e e d et al. (1985), refer to a maximum shear strain value which can not be exceeded rather than as the strain required to mobilize the residual strength although they have been used in that context.  In addition, these  empirical values were based on pseudo-dynamic post-liquefaction procedures. Because of this, the range of values to be used in the proposed procedure should be determined by comparing the results of analyses using the proposed procedure with field experience.  62  Chapter 4 Proposed Two-Dimensional Method  4.5 S u m m a r y A relatively simple total stress procedure for predicting earthquake-induced behaviour has been presented.  In this procedure, liquefaction is triggered in each zone or element individually  and post-liquefaction parameters are assigned to liquefied zones as they occur. T h e triggering of liquefaction is predicted by weighting the cyclic stress pulses as they occur to obtain an equivalent number of cycles and accumulating cycles for a prescribed base motion. Unlike other total stress methods, earthquake  time histories are directly incorporated in the displacement analysis.  Displacements due to the effects of earthquake inertia forces and reduction in soil stiffness are accounted for.  T h e analysis can be performed using a computer program based on a time  domain approach such as F L A C .  A subroutine incorporating the procedure has been written for  use in F L A C .  63  Chapter 5. Verification of Total Stress Procedure  CHAPTER 5 VERIFICATION OF TOTAL STRESS PROCEDURE 5.1 Introduction As in any other new method for predicting liquefaction induced displacements, the procedure must be validated against c a s e histories in order to be credible. T h e proposed procedure was verified against Bartlett and Youd's relationship for ground slope conditions. A s summarized in Chapter 3, Bartlett and Youd's equation is based on observations of lateral displacement case histories. The procedure is also compared to two state-of-practice methods in predicting liquefaction induced displacements.  T h e three different methods were applied to  analyses of Coquitlam Dam. T h e triggering process is comparable to the equivalent linear method.  T h e equivalent  linear method, which is embodied in the program S H A K E , has been shown to give reasonably accurate results. T h e method essentially captures the variation of shear modulus and damping ratio with strain. The method approximates the nonlinear behaviour of soil by using an equivalent elastic shear modulus, G , that is compatible with the average strain level for the particular input motion. S H A K E uses a different ratio of G / G  m a x  in F L A C , but for simplicity a constant G / G  value is assigned to all zones. In addition, the type  m a x  for each layer. A similar procedure can be used  of damping available in the computer code F L A C is different from that used in S H A K E . This chapter presents and compares the results of dynamic analyses performed using F L A C and S H A K E to validate the pre-triggering procedure. verified against Bartlett and Youd's empirical equation. discussed.  T h e post-triggering procedure is  These results are also presented and  T h e comparison between the proposed procedure and other methods applied to  Coquitlam Dam will be presented in Chapter 7.  Available data and past analyses of Coquitlam  Dam will be summarized in Chapter 6.  64  Chapter 5. Verification of Total Stress Procedure 5.2 Triggering Verification Against SHAKE  5.2.1 Description FLAC  of FLAC  (Fast  Lagrangian  Analysis of  Continua)  is a  two-dimensional  explicit  finite  difference program initially developed by Cundall in 1986 for use in rock mechanics problems.  It  has since been extended for use in soil mechanics. Static, seepage, and dynamic analyses can be performed using F L A C . F L A C uses the full dynamic equations of motion in order to model both static and dynamic systems.  The equations of motion are used to compute new velocities and  displacements from stresses and forces.  Strains are computed from displacments and new  stresses are derived from strains through a constitutive or stress-strain relation.  This process, as  shown in Fig. 5.1, performed in one timestep in each element, is repeated using the new stresses and forces. A static problem is solved by damping the equations of motion. applied to each node which is proportional to the unbalanced force.  A damping force is  A s the system reaches  equilibrium, the unbalanced force decreases.  New velocities and displacements  New stresses or forces  Stress/Strain Relation (Constitutive Equation)  Figure 5.1 FLAC Calculation Cycle (FLAC 3.3 manual, 1995)  65  Chapter 5. Verification of Total Stress  Procedure  F L A C is more often used to perform static while either S H A K E or F L U S H is commonly used in dynamic analyses.  S H A K E is a one-dimensional dynamic analysis program in which  strain dependent damping and shear moduli are represented by equivalent constant values which are obtained by iteration.  F L U S H is the two-to-three dimensional version of S H A K E .  F L A C is  able to handle the nonlinearity of the shear modulus better than S H A K E but the computation time in F L A C is significantly slower than that in S H A K E . In the proposed procedure, a constant equivalent shear modulus ratio, G / G  m a x  , is used to  account for the nonlinearity, for simplicity and to decrease computation time. This constant shear modulus ratio, G / G The fraction of G  m a x  m a x  , can be used in the F L A C model to produce the similar results to S H A K E .  in S H A K E varies with depth in order to obtain moduli compatible with effective  shear strain at all points of the system. In contrast, the fraction of G  m a x  used in F L A C is assumed  constant with depth. T h e type of damping used in F L A C is different from that used in S H A K E . essentially two types of damping available in dynamic F L A C :  local damping and  damping. Unlike the damping in S H A K E , Rayleigh damping is frequency dependent. damping ratio,  There are Rayleigh  The critical  ^ , at any angular frequency, m„ of the system can be found from the mass and  stiffness proportional damping constants ( F L A C 3.3 manual, 1995).  From equations relating ^  with Mi, it can be shown that mass-proportional damping is dominant at lower angular frequency ranges and stiffness-proportional damping is dominant at higher angular frequency ranges.  The  damping ratio is a minimum at a frequency where mass and stiffness damping each supply half of the total damping force. In F L A C , the user defines this minimum or critical damping ratio £  m i n  , and  frequency (o . T h e damping ratio is approximately constant over a 3:1 frequency range centering min  about the co , as depicted in Fig. 5.2. min  be approximated by choosing co  min  Frequency independent damping, as used in S H A K E , can  to be close to the center of the dominant frequency range of the  model.  66  Chapter 5. Verification of Total Stress Procedure  u5> 3 2 1  -|  10  15  20  1  1  r  25  30  CO — — — — — -  Mass Proportional Only Stiffness Proportional Only total  Figure 5. 2 Variation of Normalized Critical D a m p i n g Ratio w i t h Frequency (FLAC 3.3 manual, 1995)  Local damping was originally used to equilibrate static simulations in F L A C .  It is attractive  to dynamic analysis because unlike Rayleigh damping it is frequency independent. damping operates by subtracting and adding mass at a gridpoint at velocity extremes.  Local This is  performed at every cycle of calculation. T h e proportion of energy removed can be related to a fraction of critical damping. The local damping coefficient used in F L A C is equal to the product of pi and critical damping ratio. Local damping, mass-, or stiffness-proportional Rayleigh damping, or a combination of the latter two types of damping can be used n F L A C .  Past detailed analyses to study the effect of  damping in F L A C , included in the Appendix, have shown that the use of combined mass- and stiffness- proportional Rayleigh damping gives more accurate results.  Mass- or stiffness-  proportional Rayleigh damping only can give reasonable results if the critical damping ratio is doubled and c o  miri  is centered about the dominant frequency of the system. Local damping gives  67  Chapter 5. Verification of Total Stress Procedure  acceptable results in terms of displacement although the resulting acceleration time histories are not comparable to S H A K E .  5.2.2  Case  Analyzed  A 100-ft (30.48-m) soil column was analyzed using both S H A K E and F L A C .  A 30-  sublayer or zone column was used in both S H A K E and F L A C . The column is homogeneous and elastic and its stiffness properties are stress dependent.  The shear modulus and damping  attenuation relationships corresponding to sand and gravel, which are available in S H A K E , were used, as shown in Fig. 5.3. The value of the maximum shear modulus number, k  2ma  x , was 49.1.  The maximum damping ratio used in S H A K E was 26 based on the equations developed by Hardin and Drnevich (1972). The water table was assumed to be at ground surface. The soil density was assumed to be 0.129 kips/ft or 2066 kg/m . The Caltechb earthquake record scaled to 0.32 3  3  g was used in the analysis. The vertical sides in the F L A C column were "attached" to simulate a continuous column, which is assumed in S H A K E . The base of the column is fixed in F L A C but is an elastic half-space in S H A K E .  a) Shear Moduli Attenuation Curves  —  —  —  —  b) Damping Attenuation Curves  (3) ROCK and (6) TILL (SHAKE Manual, Dmax=4.6) (7) SAND (D/Dmax=1-G/Gmax, Byrne (1990))  Figure 5. 3 Shear Moduli and Damping Attenuation Curves  68  Chapter 5. Verification of Total Stress Procedure  5.2.3 Results T h e resulting fraction of k ax computed in the S H A K E analysis ranged from 0.211  to  2m  0.358 at the top of the column. With the exception of the top of the column, the average G / G  m a x  ratio is 0.215. T h e average critical damping ranged from 0.133 to 0.177, with an average critical damping ratio of 0.175, not including the top of the column.  F L A C tends to underpredict the  response for corresponding input parameters. This is because F L A C damping is a minimum at 4 Hz and all other frequencies will have higher damping. Comparable results are obtained when a fraction of k ax of 0.215 and critical damping ratio of about 10 % about a central frequency of 4 2m  Hz are used in the F L A C model. This central frequency corresponds to the dominant frequency of the Caltechb earthquake  record.  T h e maximum shear stress ratio, cyclic stress ratio, and  acceleration time history calculated for the top of the column using S H A K E and F L A C  are  compared and shown in Fig. 5.4. Although it has been shown that F L A C results are comparable to those computed by S H A K E , it is recommended that S H A K E analyses be performed to verify the results using F L A C . Analyses have shown that under a similar magnitude earthquake, acceptable results can be computed in F L A C for the fraction of k m 2  a x  of 0.215 and critical damping ratio of 10 % . However, it  should be noted that S H A K E uses cyclic or pre-liquefaction parameters for the entire column throughout the duration of the input motion.  In reality, the response may be significantly different  since liquefaction in layers or zones may be triggered at different times.  Because F L A C uses a  time domain approach, it gives the user the ability to assign post-liquefaction parameters in zones when liquefaction is predicted to occur in a particular zone. T h e present comparison of the L F A C model with S H A K E aimed to verify the triggering procedure.  T o validate the post-triggering  response of the model, the results can be compared with the empirical equations developed by Bartlett and Youd.  69  Chapter 5. Verification of Total Stress  Procedure  Comparison Between F L A C and S H A K E Acceleration Histories at Top of Column  SHAKE  FLAC  S o 2  i  1  CD  -2  H -  (.1 • •  -  -4  I 10  15  20 Time (s)  Cyclic Stress Ratio 0.5  30  35  S h e a r Stress Profile  Cyclic Stress Ratio Profile 0  25  Shear Modulus Profile  Shear Stress (kPa) 1  0  20  40  60  80  Shear M odulus (kPa) 100  0  20000  _^  10  -4  10  10  20  H-  20  20  30  30  30  40  |  40000  i  I  Figure 5. 4 Comparison of Dynamic Response Between FLAC and SHAKE 5.3 Post-Triggering Verification Using Bartlett and Youd Method The  earthquake  induced deformations  predicted  by the proposed procedure are  compared to those predicted by Bartlett and Youd's (1995) relationship for ground displacements. T h e equation requires a number of topographical and geotechnical parameters and is based on case histories from a number of earthquakes.  70  Chapter 5. Verification of Total Stress Procedure  5.3.1  Case  Analyzed  A 15-zone, 15-m high column was analyzed in F L A C . A smaller column than that used in the trigger check was used to reduce computation time.  Liquefaction triggering was compared to  S H A K E for this column as well. T h e geometry and soil parameters used in the analysis are within the limits for which the Bartlett and Youd's equation are valid. T h e water table was assumed to be 1 m below ground surface and 14 m of the column was allowed to liquefy. An S P T ( N ^ o value of 10 was assumed for the entire column.  A G/G  m a x  and damping ratios of 0.22  and  0.08,  respectively, were used based on comparisons with S H A K E analyses. T h e Caltechb earthquake record, scaled to a peak ground acceleration value of 0.32 g, was used. Three values of residual shear strength ratio varying from 0.1 to 0.3 were used in order to assess the effect of the strength on the results. Similarly, the residual or limiting strain was varied, ranging from 0.025 to 0.2. T h e slope of the column was varied from 0.0° to 5.0°. T h e corresponding input parameters  for Bartlett and Youd's model are  cumulative  thickness of liquefied layer, T , of 14 m and average fines percentage of 0 % . Rather than using 1 5  peak ground acceleration, the  input earthquake  parameters  for the empirical equation  are  magnitude and epicentral distance. Using the attenuation relationship developed by Idriss (1991) for an earthquake  magnitude of 6.6 and peak ground acceleration of 0.32  epicentral distance is about 17.5 km.  g, the computed  This is approximately equal to that computed based on  attenuation relationships developed by Joyner and Boore (1988), Campbell (1990), and Geomatrix (1992). Like the F L A C analysis, the average mean grain size in the liquefied layers and the slope were varied.  5.3.2  Results T h e results were plotted in terms of displacement versus slope for residual strength ratios  of 0.1, 0.2, and 0.3.  T h e effect of varying limiting shear strains as well as a comparison to the  findings of Bartlett and Youd are also plotted and shown in Fig. 5.5.  71  Chapter 5. Verification of Total Stress Procedure  For all three residual strength values, the results are comparable to those computed by Bartlett and Youd's relationship. strength  values.  approximately  For a  In general, the displacements decrease with increasing residual  given value  of  residual  strain, the  1/3 as the residual strength increases by 0.1.  displacements  decrease  by  Similarly, for a given residual  strength ratio, the displacements increase with increasing residual strain.  This is reasonable  because increasing the residual strains have the effect of decreasing the post-liquefaction shear modulus or stiffness. For a residual strength value of 0.1 and slope of 2 ° , the displacements increase by a factor of 3, from over 1 m to over 3 m, as the residual strains increase from 2.5 % to 20  %.  Similarly, the  displacements  increase from  about 0.7  m to over 2  m and  from  approximately 0.5 m to 1.5 m for residual strength values of 0.2 and 0.3 respectively. T h e results computed for a residual strength value of 2.5% compare most favourable with those predicted by Bartlett and Youd for all residual strength ratios. This is in agreement with the observations of Byrne (1996) and Dobry (1998).  In addition, based on Fig. 5.5, the residual  strength ratios appear to be associated with average grain size. For example, a residual strength ratio of 0.1 corresponds to a mean grain size of 0.07 mm which infers a fine to silty sand. This is in agreement with the findings of Koester (1992) who found that sands containing fines have lower residual strengths than clean sands.  Similarly, sands of mean grain size of about 0.25 mm and  0.5 mm correspond to residual strengths of 0.2 and 0.3 respectively. T h e model results are comparable to those predicted by Bartlett and Youd for this specific set of input parameters and earthquake history.  T o ensure that the model yields reliable results  under different circumstances, the sensitivity of the model was tested with respect to G / G critical damping  ratio, water level, and most importantly  different earthquake  m a x  ratio,  records and  magnitude.  72  Chapter 5. Verification of Total Stress Procedure  b. su/s'vo = 0.2 +  Bartlett-Youd, D 5 0 = 0.07mm  £  Bartlett-Youd, D 5 0 = 0.5mm  Ylim = QJ2+  -im = 0£5i~ y  lim  = 0.025  T 2  T 3  Slope (degrees) c. su/s'vo = 0.3 +  Bartlett-Youd, D 5 0 = 0.07mm  #  Bartlett-Youd, D 5 0 = 0.5mm  Ylim =  0.2 -¥lim-=  T 0  1  0.1.  T 2  3  4  5  Figure 5. 5 Comparison of Displacements Predicted by Proposed Model and Bartlett-Youd  73  Chapter 5. Verification of Total Stress Procedure 5.3.2.1 Effect of G/G  max  ratio  T h e effect of fraction of maximum shear modulus on model prediction was analyzed by varying the G / G  m a x  and maintaining all other parameters constant. It was assumed that the  residual strength ratio is 0.1, limiting shear strain is 0.2, and the ground slope is 0.5°. The water table was maintained at 1 m below ground surface and the ( N ^ o value for the entire column is 10. The G / G  m a x  was varied from 0.2 to 0.4 to 0.8.  Lateral displacements of 1.92 m, 2.01 m, and 2.15  m were predicted for each of the above G / G  m a x  prediction is not significantly sensitive to the G / G The effect of increasing G / G  m a x  values. m a x  Based on these results, the model  value, for the Caltechb earthquake record.  is to decrease the time to triggering of liquefaction in specific  zones. Hence, the predicted displacements increase slightly.  5.3.2.2 Effect of Critical Damping Ratio In the previous analyses, a critical damping ratio of 0.08 centered about 4 Hz was used throughout the analyses.  Despite the fact that an average critical damping ratio of 0.15  was  determined from S H A K E analyses, a value of 0.08 was used because it approximates an average of the pre-liquefaction and post-liquefaction viscous damping.  A s discussed in Chapter 4, once  liquefaction is triggered, a portion of damping is accounted for by the stress-strain loops created by the dynamic motion in which a soil element loads and unloads under different stiffnesses. T h e most desireable scenario is one in which the viscous damping can be reduced once liquefaction is triggered.  Unfortunately, this option was not available in F L A C 3.3 since the damping cannot be  varied by element or zone. The sensitivity of the model to the critical damping ratio was tested by varying the critical damping values from 0.08 to 0.12 to 0.16 under the same conditions as specified above.  The  lateral displacements were computed to be 0.92 m, 0.85 m, and 0.63 m for damping ratios of 0.08, 0.12, and 0.16 respectively.  Based on these results, the model is moderately sensitive to  the chosen damping values. For damping ratios of 0.12 and 0.16, subsequent analyses on a column under steeper slope of 2 ° showed a difference In results varying from 2.3 m to 1.46 m. This is likely because as the damping increases, the variation in damping with frequency also  74  Chapter 5. Verification of Total Stress  increases.  Procedure  A s a result, damping at frequencies other than the central frequency are likely to be  considerably larger. Based on the present study, a critical damping ratio of 0.08 is recommended at this time. This value may vary for different earthquake records.  5.3.2.3  Effect of water table The water table was varied to a s s e s s the effect of the level of the water table on the  model prediction. A s the water table was lowered, the estimated displacements decreased. This is consistent with the fact that as the water table was lowered, the number of zones which liquefied was also reduced.  5.3.2.4  Effect of Earthquake  Time Histories and  Magnitude  Including the Caltechb record, the column was analyzed using a total of 6 different earthquake time histories of different magnitudes and peak ground accelerations, P G A . Most of the records were scaled to 0.15 g and 0.27 g and the response of the column to the different records were assessed.  T h e epicentral distances were also computed accordingly using Idriss'  method for use in the Bartlett and Youd approach. T h e main characteristics of the earthquake histories are summarized in Table 5.1.  Table 5.1 Summary of Earthquake Histories used in Bartlett-Youd Verification PGA  Vmax  Duration  Mag  (g)  (cm/s)  (s)  Caltechb (modified)  6.6  0.32  19.6  40  Wildlife (shortened)  6.6  0.15  7.2  40  31  36.4  San Fernando  (a)  6.6  0.27  30.5  54.4  32.1  21.2  San Fernando  (b)  6.6  0.12  17.2  45.7  42.4  43.5  Loma Prieta - Gilroygc  7.1  0.35  28.9  40  28.7  22.8  Loma Prieta - Corralitos  7.1  0.48  47.5  40  6.9  17.2  Earthquake Record  Epicentral Distance (km) actual  Idriss (est) 17.8  Identical soil parameters were used for all analyses to assess the effects of using different earthquake base input records. The water table was assumed to be 1 m below ground surface, (1^)60 value of 10 was used, and the critical damping ratio was 0.08.  T h e only difference between  75  Chapter 5. Verification of Total Stress Procedure  analyses is the central frequency for Rayleigh Damping.  T h e central frequency was set at the  dominant frequency of the earthquake record, which in turn was determined from fast fourier transforms.  The residual strength ratio was kept constant at 0.2, while the residual strain and  slope were varied to generate plots similar to Fig. 5.5.  Predictions computed from Bartlett and  Youd's relationship were also plotted for comparison. It should be noted that the comparison may not be a fair one because the pre-liquefaction parameters may not be appropriate. Both the Caltechb and Gilroy records were scaled down to 0.15 responses with those of the Wildlife site and San Fernando record (b).  g to compare the  In all cases where the  earthquake histories are scaled, the distance to fault rupture, using Idriss' relationship, was recalculated for use in Bartlett and Youd's equation. A s shown in Fig. 5.6, the response predicted using the Gilroy record is comparable to that predicted by Bartlett and Youd. Although not shown, the response using the Caltechb record is similar.  In contrast, the model overpredicts the  response with respect to Bartlett and Youd for the Wildlife and San Fernando records. In general, liquefaction was predicted to have occurred at about 5 to 6 seconds.  For the Wildlife record in  particular, as shown in Fig. 5.7 and 5.8, the peak ground acceleration, P G A , occurred at a later time.  A s a result, large pulses are applied to the system after liquefaction has been predicted to  have been triggered. In addition, the acceleration amplitudes following the occurrence of the P G A are comparable to the P G A . This is not the case for the Caltechb or Gilroy records. For these records, the P G A is reached at a relatively early point in time and, in general, the subsequent acceleration amplitudes are significantly lower than the P G A . T h e effect of relatively high pulses after liquefaction has been predicted to occur may have contributed to the overprediction of the response with respect to the Bartlett and Youd approach for the Wildlife and San Fernando earthquake records. However, the ability of the model to respond to high acceleration amplitudes after the P G A is the reason for which this approach is prefered to other approaches in which the entire earthquake history is not considered in the calculation of the post-liquefaction deformations. T h e Gilroy acceleration record was scaled down to 0.27  g and the response was  compared to those predicted using the Caltechb and San Fernando (a) records. A s shown in Fig. 5.9, the predicted response to the Caltechb and Gilroy records is comparable to Bartlett and 76  Chapter 5. Verification of Total Stress Procedure  Youd.  In contrast, like the previous example, the San Fernando record tends to overpredict the  response.  A close examination of Table 5.1  between the actual epicentral distance and  indicates that there is a significant  difference  that computed by Idriss' attenuation relationship.  If  the epicentral distance computed from Idriss' relationship is used in Bartlett-Youd's equation, then the model response and that predicted by Bartlett and Youd are comparable.  This was also  observed in the proposed model's response to the Loma Prieta Corralitos record.  77  Chapter 5. Verification of Total Stress Procedure  a. Wildlife R e c o r d , pga = 0.15g  2  1  3 Slope (degrees)  b. S a n Fernando R e c o r d , pga = 0.12 g +  Bartlett-Youd, D50 = 0.07mm  #  Bartlett-Youd, D50 = 0.5mm  Ylim = 0.2  2  3 Slope (degrees)  c. Gilroy R e c o r d , scaled to pga = 0.15 g +  Bartlett-Youd, D50 = 0.07mm  #  Bartlett-Youd, D50 = 0.5mm  Ylim = 0.2  Ylim = 0.1  -""  + Ylim = 0.05 " Tii'm = U.U25"  2  3 Slope (degrees)  Figure 5. 6 C o m p a r i s o n of Response t o Different Earthquake Records Scaled t o PGA 0.15g  78  Chapter 5. Verification of Total Stress Procedure  a) Wildlife E a r t h q u a k e , P G A = 0.15 g  20  Time (s)  0.5 0.4  b) Caltechb E a r t h q u a k e , P G A = 0.32 g —\  0.3 0.2  —\  3  0.1  o 2  0.0  o  -0.1  —\  -0.2  —  -0.3  —  -0.4  —  -0.5 10  20  30  40  Time (s)  Figure 5. 7 Acceleration Time Histories of Wildlife a n d Caltechb Earthquakes  79  Chapter 5. Verification of Total Stress Procedure  -0.4  —\  0  10  20  30  40  Time (s)  d) S a n Fernando Earthquake (a), P G A = 0.27 g  ~l 10  20  30  40  Time (s)  Figure 5. 8 Acceleration Time Histories of Gilroy and San Fernando (a) Earthquakes 80  Chapter 5. Verification of Total Stress Procedure  a. Caltechb Record, pga = 0.32g -|-  Bartlett-Youd, D50 = 0.07mm  A  Bartlett-Youd, DSO = 0.5mm  ,im = YMrn. :  0,2  Jlim = - 0 . 1 '  .+  Yi  2  i m  = 0.025  3  Slope (degrees) b. San Fernando Record, pga = 0.27g -|-  Barlletl-Youd, D50 = 0.07mm  4k  Sarllett-Youd, D50 = 0.5mm  Ylim = 0 . 2  •^Tta— 0T4  —...szS^m^&^r^Y,  2  .  i m  = 0.025  3  Slope (degrees) c. Gilroy Record, pga = 0.27g -|-  Bartlett-Youd, D50 = 0.07mm  A  Bartlett-Youd, D50 = 0.5mm  Ylim = 0 . 2  Ylim = 0 . 1 _  Ylim = 0 . 0 5 Ylim = 0 . 0 2 5  T 2  3  Slope (degrees)  Figure 5. 9 Comparison of Response to Different Earthquake Records Scaled to 0.27g  81  Chapter 5. Verification of Total Stress Procedure  5.4 S u m m a r y The proposed F L A C model has been verified, with respect the pre-liquefaction  and  liquefaction triggering procedure, by comparing the results with those obtained by the computer program  SHAKE.  The  model can predict  a similar  response to S H A K E ,  although  it is  recommended that S H A K E analyses be performed and compared to that predicted from the F L A C model to ensure that liquefaction is triggered at the right time if at all. The model was also compared to the predictions using Bartlett and Youd's equations in order to validate the post-liquefaction deformation procedure.  Reasonable comparisons of the  computed displacements were found. Sensitivity analyses were also performed and it was found that the model is most sensitive to earthquake input record. Comparable results were observed when the actual epicentral distance for a particular earthquake  history is the same as that  computed from Idriss' relationship and input into the Bartlett and Youd equation.  In additon, the  analyses indicate that the model is relatively insensitive to the fraction of maximum shear modulus for the cases studied. It is suspected that these parameters could become very significant if the traditional factor of safety against liquefaction is close to 1.0. T h e fact that the earthquake  record plays a significant part in the prediction of the  earthquake induced displacements indicate that other simple methods may be deficient under certain circumstances. T h e proposed total stress approach is compared to the more commonly used methods by application to dynamic analyses of Coquitlam Dam in Chapter 7.  Available information on  Coquitlam Dam is first summarized in Chapter 6.  82  Chapter 6. Past Analyses of Coquitlam Dam  CHAPTER 6  PAST ANALYSES OF COQUITLAM DAM  6.1  Introduction Coquitlam Dam is one of 61 dams at 43 sites in British Columbia maintained by B . C .  Hydro. A s shown in Fig. 6.1, it is located in the Port Moody Conservation Reserve approximately 20 km east of Vancouver.  The dam retains Coquitlam Lake which has a maximum operating  volume capacity of 230 x 10 m . Water in the reservoir is directed via a tunnel to Buntzen Lake, 6  for power generation. District.  3  In addition, it is a water supply source for the Greater Vancouver Regional  The municipalities of Coquitlam and Port Coquitlam are approximately 8 to 15 km  immediately downstream of the dam. Coquitlam Dam is a hydraulic fill embankment dam. T h e existing dam is 30 m high, 290 m long, and 200 m wide at the base. T h e dam has an upstream composite slope of 1V:2H with berms and downstream composite slopes ranging from 1V:2.5H, idealized section is shown in Fig. 6.2.  1V:3H, to 1V:5V, and an  The dam consists of upstream and downstream rockfill  berms, sand and gravel shells, and a sandy silt to silty sand core with interlayered  sand.  Subsurface explorations to 30 m depth below the base of the dam revealed that it is underlain by glacio-fluvial deposits comprising of layers of very stiff silt, dense sand and gravel, and dense sand underlain by bedrock. A bedrock ridge outcrops downstream of the left abutment. T h e dam was built between 1909 and 1913 by the Vancouver Power Company.  It was  constructed by building two rockfill toes first and then hydraulically placing sand and gravel against the rockfill. T h e sandy silt core was sluiced into the centre portion of the dam through two sets of double flumes.  Since construction, the dam has been rehabilitated twice.  T h e cross-section  shown in Fig. 6.2. is the present one.  83  Chapter 6. Past Analyses of Coquitlam Dam  Chapter 6. Past Analyses of Coquitlam Dam  The dam was rehabilited as a result of two seismic stability studies. T h e first study was implemented in 1979 when the Water Management Study recommended that the stability of Coquitlam Dam be reviewed under the then current design earthquake. study indicated that the dam must be rehabilited.  The results of the 1979  Subsequent studies revealed that the dam  required further reinforcement. This led to the implementation of the 1984 seismic stability study and rehabilitation.  In addition to the dynamic analyses, the studies included field and laboratory  testing programs.  1 HYDRAULIC FILL CORE  6 RAISED CREST (1955)  2 HYDRAULICALLY PLACED SHELLS  7 SILT FOUNDATION  3 ROCKFILL TOES  3 SAND & GRAVEL STRATUM  4 ROCKFILL & GRANULAR REINFORCEMENT  9 BEDROCK  5 ADDITIONAL REINFORCEMENT (1934-1905)  Figure 6. 2 Typical Cross Section of Coquitlam Dam (not  to scale)  6.2 Field and Laboratory Testing  6.2.1  Field  Exploration  Field exploration programs were carried out at Coquitlam Dam in 1979 and 1984 as part of the seismic stability studies.  In 1979, the program was carried out to obtain undisturbed  samples of the dam material, perform in-situ tests, and install monitoring devices. T h e program  85  Chapter 6. Past Analyses of Coquitlam Dam  consisted of mud rotary drilling, dynamic cone penetration testing, and pressuremeter testing. Two field programs were carried out in 1984 in order to a s s e s s the potential slides in the spillway channel and further refine the rehabilitation program.  Mud rotary drilling and electric cone  penetration tests were performed.  6.2.1.1 Mud Rotary Drilling A total of 14 mud rotary holes were drilled through the dam and/or foundation in order to obtain soil stratigraphy, samples for laboratory testing, and permeability tests.  More than 15  instruments were installed in the process. Standard penetration tests were performed in the upstream and downstream shell material. A summary of the drill holes is shown in Table 6.1. Drilling in the hydraulic-filled portion of the dam was said to be easy and rapid indicating that the core material is soft. The core material was found to comprise of very soft sandy silt to silty sand with occasional fine to medium sand s e a m s . Full recovery of undisturbed samples was difficult with an average recovery of only 69 % for 50 thin-walled Shelby tubes. Four holes were drilled through the upstream shell.  T h e upstream shell consists of  relatively dense sands and gravels which progressively become finer and less dense toward the hydraulic fill core of the dam. The S P T blow counts corrected to a standard overburden pressure of 1 ton/ft , decrease toward the core of the dam from an average value of 40 to 29 to finally 13. If 2  a safety hammer is assumed for energy corrections then the average ( N ^ o values would reduce to 36 to 26 to 11 for drill holes DH84-8, DH84-9, and DH84-10 respectively.  Using Skempton's  relationship (1986), which correlates relative density with blow count, the corresponding relative density of the upstream shell would vary from about 75 to 60 to about 45 to 50 % .  86  Chapter 6. Past Analyses of Coquitlam Dam  Table 6.1 Summary of Mud Rotary Drill Holes DRILL  GROUND  HOLE  ELEV.  ZONE  FIELD TESTING or  LABORATORY TESTING  SAMPLING  (m) DH79-1A  160.10  core  Shelby sample  index tests  DH79-1B  160.10  core  Shelby sample  resonant column, cyclic triaxial  DH 79-2  158.48  core  Shelby sample  resonant column, cyclic triaxial  DH 79-3  160.11  Shelby sample  resonant column, cyclic triaxial  DH84-1  160.11  DH 84-2  160.06  DH 84-3  159.89  core, foundation left abutment left abutment core  Shelby sample, S P T  C U triaxial  DH 84-4  155.81  core  Shelby sample  C U triaxial, cyclic triaxial  DH 84-5  155.83  DH 84-6  133.90  DH 84-7  151.35  DH 84-8  154.46  DH 84-9  154.44  DH 84-10  154.49  upstream shell downstream foundation downstream shell upstream shell upstream shell upstream shell  SPT SPT  index tests  SPT  index tests  SPT SPT SPT  Only one hole was drilled through the downstream shell. Drilling was difficult and had to be terminated above the foundation level when circulation of drilling mud could not be maintained. This suggests that the gradation of the downstream shell is coarser than that in the upstream shell, thus has higher permeability. discounting extremely  The average S P T ( N ^ o value of the downstream shell,  high values, is about 36.  Based on Skempton's relationship, this  corresponds to a relative density of 75 % . Similarly, only one hole was drilled through the downstream foundation. Drilling was also difficult due to the dense nature of the material and continuous mud loss, particularly at the higher elevations. Uncorrected standard penetration values were generally greater than 100.  87  Chapter 6. Past Analyses of Coquitlam Dam  6.2.1.2  In-Situ Testing  6.2.1.2.1  Dynamic Cone Penetration Testing  In 1979, six dynamic cone penetration tests were conducted mainly in the core material but included portions of the upstream shell. T h e tests, carried out by Becker Drilling Ltd., consists of driving a "Becker sleeved cone" into the ground using a hydraulically operated 63.6-lb. trip hammer dropped from a height of 762 mm (30 in.). The "Becker sleeved cone" has a maximum diameter of 50.8 mm and uses a 200-mm long, 50.8 mm diameter sleeve mounted on 38 mm diameter rods. In all cases, the cone was driven to the base of the dam but could not penetrate the stiff silt foundation. The penetration resistances increase with depth, varying from about 2 to 10 blows per foot with an average of approximately 4 blows per foot in the core material. it was interpreted to correspond to a C R R value of only 0.05. resistances were not corrected for depth.  In 1979,  It should be noted that the  T h e difference in penetration resistance between the  core and part of the upstream shell material which is closest to the core is not distinct, with an average value of just less than 10. It is evident that the penetration resistance in the downstream shell material is greater.  6.2.1.2.2  Pressuremeter Testing In 1979, 11 self-boring pressuremeter tests through drillhole D H 79-1B were performed  in the core material.  T h e purpose of the testing program was to determine  the stiffness  characteristics of the hydraulic core material under slow cyclic loading and to examine the dilation characteristics of the core material.  Eight of the tests were simple expansion tests and the  remaining 3 tests were slow cyclic tests. It was found that the core is locally heterogeneous with zones of compact and loose soils. T h e behaviour of the soil varied from slightly dilative to slightly contractive.  No increase in pore pressure was observed albeit the tests were performed above  the phreatic surface. Based on the results in 1979 and on studies by Vaid et al. (1981), the C R R was interpreted to be about 0.11.  88  Chapter 6.  Past Analyses of Coquitlam Dam  A number of difficulties were encountered during the field testing program. It was difficult to maintain water in the casing which indicates the core material has a high permeability. unexpected of silty material.  This is  In addition, towards the end of the testing program, the hole  collapsed when the pressuremeter was pulled out.  6.2.1.2.3  Electric Cone Penetration Testing  Four electric cone penetration tests were performed in the hydraulic fill core material in 1984.  T h e pore water pressures were measured behind the cone tip in holes C P T 84-1 and C P T  84-3.  In contrast, the pore pressures were measured on the cone face for holes C P T 84-2 and  C P T 84-4. T h e typical cone penetration resistance profile is shown in Fig. 6.3. Interpretation of the cone shows that the core consists of two types of materials.  Material  A has a cone tip resistance of about 1 M P a , friction ratio of over 4 % and does not generate significant pore pressure. This is interpreted as silty clay to silty sand.  Material B has cone tip  resistance of about 2.5 M P a , friction ratio averaging 2 % , and generates pore pressures between 300 to 500 kPa, corresponding to a silty sand type material.  Both materials are interlayered with  sand to silty sand and are considered susceptible to liquefaction.  Based on the empirical charts  developed by Stark and Olson (1995), the C R R is about 0.10 for both of these materials.  The  pore pressure profile indicates a heterogeneous material which exhibits dilatant to contractant behaviour.  89  Chapter 6. Past Analyses of Coquitlam Dam  90  Chapter 6. Past Analyses of Coquitlam Dam  6.2.2 Laboratory  Testing  A summary of laboratory tests performed on Coquitlam Dam material is shown in Table 6.2.  Table 6. 2 Summary of Laboratory Tests TEST  MATERIAL  YEAR  NO. TESTS  Index Tests  Core  1979, 1984  Downstream Shell  1984  38 7  Downstream Fndn  1979, 1984  9  Resonant Column  Core  1980  6  C U Triaxial on undisturbed samples  Core  1984  3  C U Triaxial on remoulded samples  Core  1984  4  C U Triaxial on liquefied undisturbed samples  Core  1984  4  C U Triaxial on liquefied remoulded samples  Core  1984  2  Cyclic Triaxial on undisturbed samples  Core  1979, 1984  14  6.2.2.1 Index Tests Index tests were performed on hydraulic fill material, downstream shell and foundation material.  They were carried out by all the laboratories that tested Coquitlam Dam samples and  consisted of moisture content, grain size, Atterberg limits, wet and dry densities, void ratio, and specific gravity. In general, the recovered hydraulic fill core material consists of soft, saturated, uniform, non-plastic, gray silt with some fine to medium sand lenses and trace organics. T h e material is easily disturbed. T h e moisture content is about 30 % and the dry and wet densities are about 1500 k g / m and 2000 k g / m respectively. T h e plasticity index averages about 5 % and the void 3  3  ratio is 0.95. Tests on the downstream shells showed that it consists of well graded sand and gravel with some to trace silt.  The moisture content is about 14 % and the dry and wet densities are  1800 and 2070 k g / m respectively. 3  91  Chapter 6. Past Analyses of Coquitlam Dam  6.2.2.2 Resonant Column Tests In 1980,  6 resonant column tests were performed on the core material in order to  determine the variation of shear modulus and damping with shear strains. T h e average value of the shear modulus at low strain, G  m a x  , is about 90000 kPa.  T h e corresponding dimensionless  shear modulus number, K a x . has been calculated to be about 27. 2m  6.2.2.3 Consolidated  Undrained Triaxial Test  Three and four consolidated undrained (CU) triaxial tests were perfomed on undisturbed and reconstituted core samples respectively. T h e tests indicate that the silty core exhibits dilative behaviour under compression for both undisturbed and reconstituted samples.  However,  reconstituted samples assumes a relatively more contractive behaviour than the  the  undisturbed  samples which can be categorized as a limited liquefaction type of behaviour. The strength at the phase transformation state as a fraction of the confining stress is conservatively estimated to be 0.3.  This strength is developed at strains of about 1 % to 1.5 % .  material dilates  rapidly upon further straining  and  For undisturbed samples, the  increases in strength.  In contrast,  the  reconstituted samples dilate slightly upon further straining with a corresponding small increase in strength. Typical test results are shown in Fig. 6.4. A s mentioned previously, the recovery of the Shelby tube samples was relatively poor at about 69 % . There is a possibility that the core material sampled correspond to the stiffer layers in the soil. Therefore, it would be reasonable and conservative to assume that the core material behaves more like the weaker reconstituted sample than the undisturbed sample.  92  Chapter 6.  Past Analyses of Coquitlam Dam  Figure 6 . 4 CU Triaxial Test Results of Core Material  93 /  Chapter 6. Past Analyses of Coquitlam Dam 6.2.2.4 Cyclic Triaxial Tests Fourteen cyclic triaxial tests were performed in 1979 and 1984 to determine the cyclic resistance of the hydraulic fill core material. T h e tests indicate that the core material will liquefy in 15 cycles when the cyclic stress ratio is about 0.15 under cyclic triaxial conditions. Under simple shear conditions, the corresponding cyclic stress ratio is 0.10.  It is important to note that  liquefaction was considered to have triggered when a double amplitude shear strain of 5 % rather than 100 % pore pressure ratio has been attained. A plot of the cyclic stress ratio versus number of cycles to liquefaction is shown in Fig. 6.5.  I  2  3  4  6  6 7 89I0  NUMBER  20  30  40 5060708090KX)  200  300  OF C Y C L E S , N  Figure 6. 5 Cyclic Resistance of Core Material of Coquitlam Dam 6.2.2.5 Post Cyclic Monotonic Tests In 1984, monotonic triaxial tests were performed in compression on samples after cyclic loading to determine the strength properties of the core material after it had been liquefied. general, the  material  exhibited  limited  liquefaction  behaviour with the  exhibiting less dilative behaviour than the undisturbed samples.  In  remoulded samples  T h e tests showed that the  samples gained strength to the value before liquefaction. This corresponds to a residual strength ratio of 0.3. However, the strains required to reach the strength was 25 % rather than only 1 % . It  94  Chapter 6.  Past Analyses of Coquitlam Dam  is important to note that in most c a s e s , the samples were cyclically loaded until double amplitude strains of 5 % have been attained developed.  rather than when 100 % pore pressure ratio has been  It has been shown that the post-liquefaction strength and stiffness decrease with  increasing pore pressure ratio.  In addition, as summarized in Chapter 2, recent studies have  shown that the liquefied strength of material is strongest in compression loading. In the field, soil elements are subjected to simple shear, and extension loading as well, which have weaker undrained strengths. The typical results are shown in Fig. 6.6.  6.3 Seismic Stability Analyses  6.3.1 1979 Analyses Preliminary seismic analyses were carried out on Coquitlam Dam in 1979/1980 in which the triggering of liquefaction only was assessed. Liquefaction potential was evaluated, in terms of factor of safety, by comparing the cyclic resistance ratio (CRR) with the cyclic stress ratio (CSR) induced by the design earthquake.  T h e assessment was performed using a simplifed procedure  as well as a comprehensive one. In the simplified procedure, the liquefaction resistance was obtained empirically from blow count from Becker Cone Penetration testing, and was equal to 0.05.  T h e cyclic stress ratio was  obtained empirically from the equation (Eq. 4.2) developed by S e e d and Idriss. The peak ground acceleration (PGA) was assumed to be 0.12  g, corresponding to the  Maximum  Credible  Earthquake (MCE) evaluated in 1979/1980. T h e results of the analysis indicate that the upstream shell and upper portion of the core will liquefy. The computed cyclic stress ratio contours on the 1979 Coquitlam Dam cross section is shown in Fig. 6.7.  95  Chapter 6. Past Analyses of Coquitlam Dam  CE iii  8 AXIAL  12  8  16  12  AXIAL  STRAIN %  16  STRAIN %  200 Ul  cn z>  900  cc  Q.  2  600  TEST ru  -400  \^ TESTS  -TEST  ,  L3.  r  TL  TL4,1 UP  o X  a  TL 2R  -600 300  600  900 P  1200  1500 1800  (kPa)  Figure 6. 6 CU Triaxial Test Results of Liquefied Core Material  96  Chapter 6. Past Analyses of Coquitlam Dam  In the more comprehensive procedure, the liquefaction resistance was obtained from laboratory testing.  A s discussed earlier, the C R R was determined to be 0.10.  The cyclic stress  ratio was determined from a two-dimensional analysis of the deepest cross-section of the dam. Static stresses were determined using a finite element program called ISBILD, construction sequence of the dam was simulated.  in which the  The dynamic stresses were computed using  the program Q U A D 4 which consideres both variable stiffness and damping.  The San Fernando  record scaled to a peak acceleration of 0.12 g was used as input base motion. The results indicate that the upper upstream shell and core material would liquefy under the M C E . The cyclic stress ratio profile is shown in Fig. 6.8. Because liquefaction may cause slumping in the upper part of the dam, rockfill berms were added to the slopes. T h e rehabilitation was carried out in 1980.  SIMPLIFIED SEED - HAND CALCULATIONS  Figure 6. 7 1979 Computed Cyclic Stress Ratio Contours Based on Hand Calculations  97  Chapter 6. Past Analyses of Coquitlam Dam  QUAP 4  Note: Liquefaction predicted to have triggered in shaded  zones  Figure 6. 8 1979 Computed Cyclic Stress Ratio Contours Based on Dynamic Analyses 6.3.2 1984  Analyses  Between 1980 and 1984, further studies indicated that the dam needed to be further reinforced. In result, in 1984, the liquefaction potential of proposed remediated dam sections were evaluated under the operating base earthquake, O B E , which has a P G A of 0.20 g and the M C E which has a P G A of 0.35 g.  In addition, the flow slide and deformation potential considering the  loss in stiffness when the core liquefies were assessed. The simplified approach was used in the triggering analysis. The cyclic resistance of the core material and upstream shell material were obtained using data from cyclic triaxial tests and in-situ testing.  The C R R of the upstream shell was interpreted to be 0.285, based on S P T blow  count information corrected from overburden stress. A s shown in Fig. 6.9, the results indicated that under the M C E , all of the core material and a portion of the upstream shell will liquefy.  98  Chapter 6.  Past Analyses of Coquitlam Dam  Uote: Liquefaction predicted in shaded zone  Figure 6. 9 1984 Computed Cyclic Stress Contours Based on Hand Calculations  A limit equilibrium analysis was conducted to a s s e s s the flow slide potential of the dam. A number of failure surfaces were generated and a factor of safety against flow slide was computed. Friction angles of 2 ° and 2 0 ° for the liquefied core and upstream shell respectively were assumed. T h e s e strengths were considered to be equivalent to using full effective strength angle of 3 5 ° and pore pressure ratios, r , of 95% in the core and 50 % in the shell. T h e results indicated that a flow u  slide above the upstream rockfill berm may occur under the M C E . However, a flow slide was not predicted to occur under the O B E . Earthquake deformations due to inertia force and to loss in stiffness when the core liquefies were analyzed using Newmark analysis respectively.  analysis and finite element  post-liquefaction stress  T h e yield acceleration under the O B E , determined from limit equilibrium  analysis, was computed to be 0.14g  and 0.17g  for the upstream and downstream slopes  respectively. The displacements due to inertia force is predicted to about 7 cm in the upstream direction and 1 cm in the downstream direction. For the M C E , the critical acceleration is zero and therefore a flow slide is predicted.  Under the proposed remediated design, it was considered that  the top of the dam would unlikely drop below El. 155.5 due to the geometry of the d a m .  99  Chapter 6. Past Analyses of Coquitlam Dam  Finite element program S O I L S T R E S S .  post earthquake  static analyses were performed using the computer  The post-cyclic stiffness parameters, obtained from laboratory tests,  were reduced to 1/25 of the pre-cyclic parameters.  T h e results indicate that under the O B E , the  dam crest may settle vertically about 0.6 m and the upper portion of the downstream slope may move outward 0.3 m. Under the M C E , only an additional crest settlement of 0.3m was predicted. T h e dam deformation pattern is shown in Fig. 6.10.  Figure 6.10 SOILSTRESS Deformation Analysis Based on Loss in Stiffness  In addition, the deformation due to dissipation of excess pore water pressures was predicted.  Based on a bulk modulus of compressiblity of 4 x 1 0  s  m /kN and an estimated 2  average excess pore pressure generated during liquefaction of 300 kPa, the volumetric strains were computed to be about 1.2 % .  Assuming that volumetric strains are due to vertical strains  only, the crest settlement was conservatively estimated to be 0.4 m. The total estimated displacements of the reinforced dam was computed to be the sum of the displacements due to inertia force, loss in stiffness, and dissipation of pore pressure.  Under  the O B E , the total vertical displacement was calculated to be 1 m. A vertical displacement of 6.5 m, corresponding to a loss in freeboard, was estimated under the M C E .  Because the normal  operating reservoir level is less than the level of the rockfill berm, this amount of deformation was considered acceptable. In 1986, the dam was rehabilitated to the section shown in Figure 6.9.  100  Chapter 6.  Past Analyses of Coquitlam Dam  6.4 Other Studies Dam breach analyses have been performed for Coquitlam Dam.  T h e studies showed  that should the unlikely failure of the dam occur under any circumstance, very little warning time is available to the downstream residents and the number of people potentially at risk is estimated to be about 32,000.  6.5 Summary Coquitlam Dam has been rehabilited twice since construction as a result of seismic stability analyses in 1979 and 1984.  T h e 1979 seismic analyses comprised of a liquefaction  triggering assessment only. T h e core and upper portion of the upstream shell were predicted to liquefy under the 1979 M C E . slumping of the upper dam.  Rockfill berms were added to the slopes due to the possibility of Subsequent analyses in 1984 indicated that additional rehabilitation  was necessary. Liquefaction triggering, flow slide, and deformation analyses were performed on the proposed remediated section.  Under the M C E , which had P G A of 0.35 g, the core and a  portion of the upstream shell were predicted to liquefy, resulting in the possibility of a flow slide. However, it was considered unlikely that overtopping would occur under the normal operating reservoir level if the dam were remediated to the proposed geometry.  Consequently, the dam  was remediated to the proposed geometry. Since progressed.  1984,  the  state-of-practice  in earthquake  induced deformation  analysis  has  Although failure by overtopping under the normal operating reservoir level was not  predicted in 1984, the most current analysis procedures were applied to ensure that the seismic performance of the dam is still acceptable.  101  Chapter 7. Total Stress Analyses of Coquitlam Dam  CHAPTER 7  TOTAL STRESS DYNAMIC ANALYSES OF COQUITLAM DAM 7.1  Introduction Since the last (1984) seismic analysis of Coquitlam Dam, the state-of-practice procedure  has changed due to a better understanding of materials under cyclic loading conditions. Although the earthquake-induced deformations that were estimated previously accounted for the effects of inertia forces and loss in stiffness, better procedures are now available. A s discussed in Chapter 3, the previously calculated deformation due to the inertia forces is only one-dimensional and does not account for the deformation due to two-dimensional deformation of the structure.  For a dam  with high consequences of failure, the seismic stability should be re-evaluated using the most current procedures. Three different total stress deformation procedures were performed.  T h e proposed  method is a total stress dynamic analysis. T h e other two methods include the calculations of the deformations due to loss in stiffness only and those due both to loss in stiffness and earthquake velocity pulse, similar to Jitno's method. compared.  T h e results obtained by these three methods were  Static and seepage analyses, in which construction sequence and reservoir filling  were simulated, were also performed to obtain the pre-earthquake initial stresses. All static and dynamic analyses of the dam were performed using F L A C version 3.3.  7.2 Pre-Earthquake Analysis Prior to performing the dynamic analyses on the dam, the pre-earthquake initial stresses are calculated. This involved simulating construction of the dam and performing an analysis due to the loading of the reservoir water.  The pore water pressures in the dam and hence the  effective stresses can be calculated.  102  Chapter  7. Total Stress Analyses  7.2.1 Construction  of Coquitlam  Dam  Analysis  A two-dimensional plane strain static analysis was performed on the idealized crosssection of the Coquitlam Dam using the computer code F L A C .  T h e computer program allows  multiple stage loading in which the construction sequence can be simulated by adding the soil materials in layers. Equilibrium is obtained by "stepping" through the calculation sequence prior to the addition of a new layer. O n c e equilibrium has been established, the gridpoint displacements of the top layer are set to zero, since displacements are incurred during the stepping process. The cross section of the grid used for the analysis is shown in Fig. 7.1. This cross-section is the same as that used in previous seismic analyses and is considered to represent the deepest cross section of the dam. T h e grid is composed of about 3610 zones. The base of the crosssection is fixed in the horizontal and vertical directions and the upstream and downstream end boundaries are fixed in the horizontal direction only. T h e dam was constructed in 10 layers each of over 2 m thick. Although it is likely that pore pressures were developed during construction due to the hydraulic fill placement method, the analysis was performed using drained stiffness parameters.  It is judged that by now the pore pressures would have had dissipated. Because the  stresses of interest are the "current" stresses and not those representing the stresses immediately after construction, this procedure is considered acceptable.  7.2.1.1  Model  Parameters  T h e Mohr Coulomb model, which is a built-in feature of the F L A C computer program, was used in the analysis. This model was modified so that the shear and bulk modulus parameters are stress dependent.  The main input parameters are the shear modulus number, kg, bulk  modulus number, kb, shear modulus exponent, n, bulk modulus exponent, m, friction angle, <(>, dilation angle, u, and cohesion, c.  The parameters used to represent the dam materials were  based on results of laboratory testing, in-situ testing, and published values available  from  literature. As summarized in Chapter 6, the dam and its foundation are made up of 6 different materials. The dam is comprised of rockfill toes, rockfill berms, upstream and downstream sand 103  Chapter 7. Total Stress Analyses of Coquitlam Dam  and gravel shells, and a hydraulic fill sandy silt to silty sand core. relatively thin stiff silt layer which overlies a dense sand layer. parameters were assigned to represent the dam materials.  T h e dam is underlain by a In these analyses, 8 sets of  T h e upstream sand and gravel shells  were further subdivided into 3 different zones based on the in-situ test results.  The relative  density of the upstream shell material decreases as it approaches the hydraulic fill core.  The  different material zones are shown in Fig. 7.2. The soil parameters for the hydraulic fill core were determined based on laboratory test results. The bulk modulus value was computed directly from laboratory test results.  Numerical  single element tests were performed in which the shear modulus, friction angle, and dilation angle were varied to produce a best fit with the laboratory results.  T h e parameters obtained were  compared with the recommended values proposed by Stark et al.  (1994) to assess whether or  not they are reasonable. The upstream and downstream shell parameters were based on empirical correlation with relative density values developed by Byrne et al. (1987) for sands.  Using the relative density  which was estimated from in-situ test results, parameters are selected for the hyperbolic model as suggested by Byrne et al. (1987).  T h e parameters were converted to correspond to the Mohr-  Coulomb model by simulating a single element test using both the Mohr-Coulomb and the hyperbolic models. Best-fit Mohr-Coulomb parameters were determined by trial and error. A plot comparing the Mohr-Coulomb response with the hyperbolic response is shown in Fig. 7.3.  Similar  plots were generated for each of the sand and gravel and dense sand foundation material properties.  Properties for the stiff silt were determined in a similar manner from a data base of  silty material developed by Stark et al. (1994). T h e upstream and downstream rockfill parameters were obtained from values recommended by Saboya and Byrne (1991) based on laboratory testing and back-analysis of rockfill dams.  The Mohr-Coulomb parameters  used for static  analyses are shown in Table 7.1.  104  Chapter 7. Total Stress Analyses of Coquitlam Dam  Figure 7.1 FLAC Grid Used For the Analysis of Coquitlam Dam  (a) (b) (c) (d)  Dense Sand Foundation Stiff Silt Foundation Upstream and Downstream Rockfill Upstream Sand and Gravel Shell (a)  (e) (f) (g) (h)  Upstream Sand and Gravel Shell (b) Upstream Sand and Gravel Shell (c) Downtream Sand and Gravel Shell Hydraulic Fill Core  Figure 7. 2 Material Zones Used For the Analysis of Coquitlam Dam  105  Chapter 7. Total Stress Analyses of Coquitlam Dam  Table 7.1 Summary of Dam Material Parameters used in Static Analysis Dam Material  kg  n  m  kb  (j) de  u de  g  g  Pdry  kg/m  Dense Sand Foundation  425  0.5  650  0.25  60  1800  Stiff Silt Foundation  80  1.0  115  1.0  45  1750  Upstream and Downstream Rockfill 125  0.25  250  0.35  48  2000  Upstream Sand and Gravel Shell (a) 350  0.5  450  0.25  47  1800  (b)  200  0.5  400  0.25  43  1800  (c)  125  0.5  300  0.25  38  1800  Downstream Sand and Gravel Shell 350  0.5  450  0.25  47  1800  Hydraulic Fill Sandy Silt Core  0.8  45  0.8  37  33  8  3  1500  400 Mohr-Coulomb  300  Hyperbolic  GO CO  £ co  200  100  !  o.o2  ,  !  ,  0.04  !  0.06  ,  !  0.1  0.08  Strain  Figure 7. 3 Mohr-Coulomb Parameter Fit to Hyperbolic Parameter 7.2.1.2 Results T h e results of the construction were plotted in terms of displacement vectors and vertical or principal stress contours.  The displacement vectors, shown in Fig. 7.4, are oriented almost  exclusively vertically downward with some horizontal components along the  upstream  and  downstream boundaries of the dam. T h e largest magnitude vectors appear to be in the middle of the dam.  A plot of the vertical displacement profile along the centerline of the dam, depicted in  Fig. 7.5., confirms that the largest vertical displacements occur in the middle of the dam.  These  106  Chapter 7. Total Stress Analyses of Coquitlam Dam  observations are in agreement with both case histories and expected for a dam with a relatively stiff foundation.  T h e stiffness of dam material decreases with increase in dam height but is  countered by the decrease in overburden stress with increase in dam height so that the maximum deformation occurs at about midheight of the dam. T h e vertical stress contours shown in Fig. 7.6 show low stresses in the core and slightly higher stresses in the stiffer shells, indicative of arching. This is not unreasonable because the difference in stiffness between the shell and core material is significant. Although the 1979 static analysis indicated that there are no areas of arching it should be noted that the relative stiffness between  the  comparable.  core  and  upstream  and  downstream  shells assumed  in that  analysis  was  However, subsequent laboratory and in-situ test results indicate that the core  material is very soft.  7.2.2 End of Reservoir Filling Analysis T h e end of reservoir filling state can be modeled by performing either a coupled or an uncoupled stress-flow analysis.  A coupled stress-flow analysis is performed by solving for  stresses, displacements, and pore pressures together.  An uncoupled analysis is performed by  solving the flow separately from the stress to obtain steady state pore pressures or seepage forces.  Having computed the pore pressures, the resulting stresses and deformations can be  computed from two methods as follows: Method A 1.  Specify total unit weights - saturated where appropriate.  2.  Specify boundary water pressures.  3.  Specify internal pore water pressures.  Method B 1.  Specify submerged unit weights below the phreatic surface.  2.  Specify seepage forces below the phreatic surface.  Method A was selected for Coquitlam Dam.  107  Chapter 7. Total Stress Analyses of Coquitlam Dam  150 cr o  CO > CD  100-1 J  0  I  I  _1  !_  50  100  I  ,  I  I  L_  150  200  I  I  I  I  250  l_  _i  u  i  300  350  Figure 7. 4 E n d of C o n s t r u c t i o n D i s p l a c e m e n t V e c t o r s  150-  o ca > JS LU  100 0.4  0 J  0  50  100  I  150  .  I  200  L  250  300  350  Figure 7. 5 Vertical D i s p l a c e m e n t s Due to C o n s t r u c t i o n A l o n g Centreline of D a m in metres  1 50—  03  > CD  100-  I  50  I  I  I  ,  100  I  I  I  J  ,  150  I  1  1  L_  200  250  350  300  Figure 7. 6 E n d of C o n s t r u c t i o n Vertical S t r e s s C o n t o u r s in k P a  108  Chapter 7. Total Stress Analyses of Coquitlam Dam  Both stress and seepage analyses were performed using F L A C .  The boundary water  pressure under full reservoir elevation of 155.0 m was applied on the upstream boundary.  The  pore pressures and saturation were fixed for the upstream boundary and the pore pressures only were fixed on the top and downstream boundary of the dam. The steady state pore pressures are computed by performing a seepage analysis in which the mechanical computation option in F L A C is turned off and the flow computation option is turned on. T h e analysis was carried out until flow equilibrium is reached in which the inflow is approximately equal to the outflow. Having reached flow equilibrium, the flow computation option in F L A C is turned off and the mechanical computation option is turned on. W h e n the flow computation option is turned off, F L A C only subtracts porewater pressures from the total stresses to obtain effective stresses and does not consider porewater pressures as loads. T o overcome this problem, the pore pressures were added to the total stresses for at least one time step. T h e boundary water pressures were applied incrementally to avoid "shocking" the system.  Computation was performed until force  equilibrium of the system was reached.  7.2.2.1 Flow Model Parameters T h e parameters required in seepage analysis in F L A C are the permeability and void ratio or porosity. The seepage parameters for the dam materials were based on laboratory testing and published typical values (Craig, 1992).  The vertical permeability for the hydraulic fill core was  determined from the consolidation portion of triaxial testing.  T h e horizontal permeability was  based on Hazen's empirical formula (Craig, 1992) relating permeability with effective grain size as follows:  k = lO' -Df  Eq. 7. 1  2  0  where  k = permeability in m/s D-io = effective grain size in mm.  The  permeability  for the  upstream  and  downstream  sand and  gravel  were  also  determined using Hazen's formula. T h e permeability parameters for the remaining materials were  109  Chapter 7. Total Stress Analyses of Coquitlam Dam  based on typical values.  Table 7.2 summarizes the seepage properties for Coquitlam D a m  materials.  Table 7. 2 Summary of Dam Material Properties for Seepage Analysis Dam Material Dense Sand Foundation Stiff Silt Foundation Upstream and Downstream Rockfill Upstream Sand and Gravel Downstream Sand and Gravel Hydraulic Fill Core  k (m/s)  porosity  Se 5e 5e 10" 10" k - 5e" k - 1.5e'  0.3 0.4 0.43 0.35 0.3 0.39  3  6  2  4  3  5  x  6  v  Since the permeability constant, K, required in F L A C is the proportionality constant in Darcy's Law and is expressed in terms of pressure rather than head, the above permeability values were divided by the unit weight of water for input into F L A C .  7.2.2.2  Results The results of the seepage analysis were plotted in terms of flow vectors, pore pressure  and total head contours and are shown in Figs. 7.7, 7.8, and 7.9 respectively. A s shown in Fig. 7.7, the largest flow vectors are in the downstream rockfill toe. Although the inflow and outflow are equal, the output flow is concentrated over a smaller area than the input flow, hence the outflow velocity vectors are greater. reasonable.  Both the pore pressure and total head contours appear to  A s expected, the pore pressures increase with depth below the phreatic surface.  T h e total head contours are similar to those obtained from SEEPA/V analyses performed in 1995. The results of the end of reservoir filling stress analysis, which is intended to account for the boundary loads, are shown in Fig. 7.10 and 7.11 in terms of vertical effective stress contours and displacements. T h e vertical effective stresses appear to be reasonable. T h e displacement vectors, shown in Fig. 7.10, indicate that the crest moves upwards and downstream and are reasonable. However, the magnitude of the displacements are likely too large.  110  Chapter 7. Total Stress Analyses of Coquitlam Dam  Chapter 7. Total Stress Analyses of Coquitlam Dam  This is due to the assumption that the water level rises from the bottom of the grid to the reservoir water level. A more accurate assumption would be to model the water level to rise from Elevation 150 m to the reservoir water level.  Also, the reload moduli should be used in order to be  compativle with the stress path of the type of soil element.  However, the stresses remain more or  less the same if the unloading moduli are used.  200-  150 >  > 41 |  4111,  i  100-  5 0  100  150  200  T  250  300  350  Figure 7.10 End of Reservoir Filling Displacement Vectors  150-  ro > UJ  100-  1000  50  Figure 7.11  100  150  200  250  300  350  End of Reservoir Filling Vertical Stress Contours in kPa  112  Chapter 7. Total Stress Analyses of Coquitlam Dam  7.3 Total Stress Dynamic Analysis  7.3.1 Introduction Three total stress deformation analyses were carried out, as follows: 1.  Gravity Only  2.  Gravity Plus Velocity Pulse  3.  Proposed Total Stress Dynamic Analysis (Self Triggering and Gravity Plus B a s e Acceleration)  T h e first two methods are considered to be traditional procedures in which liquefaction triggering, flow slide, and deformation analyses are performed separately. T h e proposed method considers liquefaction triggering and the liquefaction induced displacements in the same process and will be presented in the next section. The traditional methods were performed to compare with the proposed method. The first deformation analysis procedure (Gravity Only) is similar to Lee's Modified Modulus approach in that the deformations are caused by gravity loading only as a result of the loss of strength and stiffness in the liquefied soil.  The liquefaction effect, in which the liquefied  soil behaves like a liquid, is handled by adjusting the stress distribution so that the horizontal stresses in the liquefied zones are equal to the vertical stresses and the shear stresses are equal to zero. In addition, the liquefied zones are assigned reduced strength and stiffness parameters. This stress state and reduced stiffness result in an imbalance in forces within the structure and cause deformation to occur until equilibrium is achieved by stress redistribution.  Although the  inertia forces due to seismic loading are not considered, analyses using the dynamic option in F L A C allow consideration of inertia forces induced by the initial acceleration of the soil elements under the out of balance forces. T h e second deformation analysis (Gravity Plus Velocity Pulse) is comparable to the extended Newmark approach proposed by Jitno and Byrne (1995). The deformations are caused by gravity loading and a velocity pulse equal to the maximum ground velocity pulse.  Only one  113  Chapter 7. Total Stress Analyses of Coquitlam Dam  velocity pulse is considered because the deformations prior to triggering are negligible.  The  earthquake pulses which occur after liquefaction are accounted for using the residual or limiting strain values which have been correlated to factor of safety against liquefaction.  T h e post-  liquefaction parameters suggested by Byrne (1991) was shown in Table 4.1. In the above method the entire dam was considered to have a velocity at the time of liquefaction equal to the maximum ground velocity at the time of liquefaction. T h e liquefied zones were then assigned their reduced stiffness and strength parameters and the initial stresses were redistributed. All liquefied elements are assumed to liquefy at the same time.  7.3.2 Liquefaction Triggering Analysis The liquefaction potential was assessed by comparing the cyclic resistance ratio with the developed cyclic stress ratio.  The cyclic resistance ratio, C R R , for the various dam materials  were evaluated from laboratory test results and various in-situ test results such as cone penetration tip resistance and standard penetration blow count. overburden stress using the chart shown in Fig. 2.4.  T h e C R R was corrected for  T h e corrected cyclic resistance ratios  assigned for the dam materials are summarized in Table 7.3.  Table 7. 3 Corrected Cyclic Resistance Ratio f o r Coquitlam Dam Materials Dam Material  Source  CRRcORR  Dense Sand Foundation  0.5  S P T blow count  Stiff Silt Foundation  0.5  S P T blow count  Upstream and Downstream Rockfill Upstream Sand and Gravel Shell (a)0.4  will not liquefy (free draining) (avg)  (b)  0.3 (avg)  (c)  0.15 (avg)  Downstream Sand and Gravel Shell Hydraulic Fill Sandy Silt Core  will not liquefy (not saturated) 0.1  C P T tip, cyclic triaxial tests  The cyclic stress ratios induced by the earthquake base motion were computed using a one-dimensional analysis using the computer code S H A K E .  Because the Coquitlam Dam slopes  are relatively flat and wide, boundary effects are judged to be minimal.  In addition, as mentioned  114  Chapter 7. Total Stress Analyses of Coquitlam Dam  in the preceding chapters, it has been shown that no significant difference in results were observed between one- and two-dimensional analyses. Several soil columns representing the section between the upstream to the downstream sand and gravel shells were analyzed in S H A K E .  Columns representing the rockfill zones of the  dam were not analyzed because the rockfill berms are not considered to liquefy. The soil parameters used in the analysis are the unit weights, maximum shear modulus number and the maximum damping ratio in percent. T h e maximum damping ratios for both the sand and gravel shell and core material were evaluated from relationships developed by Hardin and Drnevich (1972). T h e maximum shear modulus numbers for the sand and gravel shell and hydraulic fill core were determined from correlations with S P T blow count and resonant column tests respectively.  T h e shear modulus number, (k2) , is determined based on the S P T blow max  count value through the following relationship, developed by Tokimatsu and S e e d (1987):  (*2L =20-(Ar,)J  Eq.7.2  x  The S H A K E input parameters are summarized in Table 7.4  Table 7. 4 SHAKE input parameters Dam Material  y  (kips/ft )  3 1  sat  (k2)max  D  max  Upstream Sand and Gravel Shell (a) 0.129 (2.067)  65  26.2  (b)  0.129 (2.067)  55  26.2  (c)  0.129 (2.067)  45  26.2  Downstream Sand and Gravel Shell 0.129 (2.067)  65  31.2  Hydraulic Fill Core  27  20  0.1217 (1.95)  (%)  1. Note: Values in parentheses in metric units of kg/m x 1000. The base input motion used in the S H A K E analysis corresponds to the Caltechb earthquake record scaled to 0.32 g. As shown in Fig.7.12, for an earthquake with pga of 0.32 g, almost the entire hydraulic fill core and a portion of the upstream sand and gravel shell will liquefy. The factor of safety against liquefaction triggering in the core material is relatively low ranging from 0.5 to almost 1.0.  115  Chapter 7. Total Stress Analyses of Coquitlam Dam  7.3.3 Flow Slide  Analysis  T h e rise in porewater pressure associated with liquefaction causes a loss in strength and stiffness in the liquefiable zones. Although the dam slopes are stable under static conditions, the development of residual strength of the weakened liquefied core and portion of the upstream shell material may lead to the occurrence of a flow slide. The flow slide potential of Coquitlam Dam was a s s e s s e d by using a limit equilibrium analysis computer program X S T A B L version 5.1. A s presented in the preceding section, nearly the entire dam core and a portion of the upstream sand and gravel shell will liquefy under the Caltechb earthquake scaled to a P G A of 0.32g. Undrained strength ratios of 0.1 and 0.2 were assigned to the liquefied core and upstream sand and gravel shells respectively.  Although laboratory test results indicate that the residual  strength ratio of the core material is about 0.3, a value of 0.1 was assigned to the core material because the laboratory samples were tested in compression, which tends to give a higher residual strength.  In addition, the tested samples were not liquefied to 100% pore pressure ratio, which  also results in higher strengths than cases where sample are liquefied to 100 % pore pressure. Back-calculated values from cases histories of flow failure of silty sand to sandy silt materials have been shown to range from 0.1 to 0.2 (Ishihara, 1993).  T h e strengths assigned to the  liquefied zones are considered to be conservative. T h e results, depicted in Fig. 7.13, show that a flow slide is not likely to occur. The factor of safety in the upstream slope is 1.49, and 1.87 in the downstream slope.  The difference in  result from the 1984 analyses, which predicted that a flow slide in the upstream direction is likely to occur is due to the strength values used.  In 1984, friction angles of 2° and 2 0 ° were assumed  for the liquefied core and upstream sand and gravel shell. T o produce the same factors of safety in X S T A B L , these friction angles correspond to undrained residual strengths of 5.7° and 11.3°, respectively.  116  Chapter 7. Total Stress Analyses of Coquitlam Dam  7.3.4 Deformation  Analysis  7.3.4.1 Model Parameters Although a flow slide is not predicted to occur, the deformations liquefaction  may still be significant and were assessed.  Deformation  associated with  analyses  performed  presently should consider the effects of liquefaction as well as the inertia forces induced by the base motion. Two commonly used deformation analyses methods were used. T h e post-liquefaction stress strain curve for both analyses was assumed to be bilinear. Similar to the limit equilibrium analysis, the residual strength ratio was assumed to be 0.1 and 0.2 for the liquefied core and the upstream sand and gravel materials, respectively. strain values were based on Byrne's recommended values as shown in Table 4.1.  T h e residual T h e limiting  strain values were assumed to be 15% for the upstream shell material and 25% for the core material. A  maximum  ground velocity of  19.6  cm/s was  used for the  second  deformation  procedure. This velocity corresponds to the Caltechb earthquake record scaled to 0.32 g.  117  Chapter 7. Total Stress Analyses of Coquitlam Dam  Note: Factor of safety d e c r e a s e s with lighter fill tone  I  i  i  100  120  140  1  1  1  200  220  1  160  180  '  1  '  240  1  '  260  , 280  Figure 7.12 Liquefaction Triggering of Coquitlam Dam Post-Liquefaction Limit Equilibrium 10 most critical s u r f a c e s , Minimum Bishop F O S = 1.867  170  145  r  h  120 I 8p  1  1  1  1  105  1  '  1  130  '  1  155  '  1  180  '  1  205  '  1  230  1  255  280  Post-Liquefaction Limit Equilibrium 10 most critical s u r f a c e s , Minimum Bishop F O S = 1.486 170  L  145  h  120  I 80  '  1  105  '  1  130  '  ' 155  '  1  180  '  1  '  205  1  230  '  1  1  255  1  280  Figure 7.13 Limit Equilibrium Flow Slide Analyses  118  Chapter 7. Total Stress Analyses of Coquitlam Dam  T h e analyses were performed using the dynamic mode in F L A C in order to account for the inertia forces due to the motion of the deforming mass. Rayleigh damping of about 10% about a central frequency of 4 Hz was used. T h e frequency of 4 Hz corresponds approximately to the predominant undamped frequency, based on analysis of the acceleration history of the crest of the dam,  7.3.4.2 Results T h e velocity pulse was applied in both the upstream and downstream horizontal directions as well as in a downward vertical direction to assess the effect of direction of velocity pulse on the response of the structure. T h e results for both approaches are compared in Table 7.5.  Table 7. 5 Results of Liquefaction Induced Displacment of Coquitlam Dam Crest Displacement (m) PROCEDURE  Horizontal (+ve d/s)  Vertical (+ve  Gravity Only  -0.0836  -1.09  Gravity Plus Horizontal Velocity Pulse (d/s)  -0.0103  -1.102  Gravity Plus Horizontal Velocity Pulse (u/s)  -0.0341  -1.1075  Gravity Plus Vertical Velocity Pulse (down)  -0.0348  -1.075  up)  As shown in the above table, the displacements due to inertial loads represented by single velocity pulses result in an increase in crest deformation by about 0.1 m larger than those due the loss in stiffness only.  In fact, for the case where a velocity pulse is applied in a vertical  downward direction, the predicted displacements are actually less than those computed based on gravity loading or loss in stiffness only.  However, the difference is less than 0.015  m can be  considered more or less negligible. T h e computed vertical crest displacement due exclusively to a loss in stiffness of 1.09 m is comparable to that found in 1984 of about 0.9 m under the M C E . pattern, shown in Fig. 7.14,  is also similar.  The deformation vector  It is interesting to note that the patterns in the  119  Chapter 7. Total Stress Analyses of Coquitlam Dam  upstream and downstream shells are similar in shape to the failure surfaces predicted by X S T A B L , shown earlier in Fig. 7.10.  0  50  100  150  200  250  300  Figure 7.14 Typical Post-Liquefaction Displacement Vector Pattern  In contrast, the displacements due to inertia force cannot be compared to those found in 1984.  In 1984, the displacements due to inertia force were computed for a one-dimensional  potential failure surface as opposed to a two-dimensional system as performed in the present analyses.  In addition, the peak horizontal velocities for both the O B E and M C E of 0.36 m/s and  0.40 m/s respectively, estimated in 1984, are greater than that a s s u m e d in the present analyses which is only 0.19 m/s. those predicted in 1984  Despite this fact, the computed displacements are slightly greater than under the O B E of about 0.06  m.  Displacements of over 6.5  m,  corresponding to a flow slide, were predicted under the M C E in 1984. Based on these results, a velocity pulse of 0.19  m/s does not appear to affect the  predicted displacements significantly. A plot comparing the horizontal velocity history of the crest of the dam under the cases of gravity loading only and gravity loading plus horizontal downstream velocity pulse shows that horizontal velocities calculated using the first method (gravity only) are slightly greater than that calculated using the second method (Fig. 7.15).  If a velocity pulse of  about 0.3 m/s were applied to the dam, it is likely that a greater response will be observed. However, because the Caltechb input motion corresponding to P G A of 0.32 g was chosen, a  120  Chapter 7. Total Stress Analyses of Coquitlam Dam  velocity pulse corresponding to the peak horizontal velocity should be used for comparative purposes. 0.40  - i  L o s s in S t i f f n e s s P l u s Inertia F o r c e ( H o r i z o n t a l V e l o c i t y P u l s e ) — —  0.20  L o s s in S t i f f n e s s O n l y  o CD >  0.00  H  -0.20  I  1  115000  1  120000  I  1  125000  I  I  1  130000 No. Steps  135000  1  I  1  140000  I 145000  Figure 7.15 Comparison of Velocity History of Crest of Dam  7.4  Proposed Total Stress Dynamic Analysis Procedure  7.4.1 Model  Parameters  The same material parameters used in the above triggering analysis were used in the pre-liquefaction of the dam.  T h e shear modulus values were based on S P T blow count values  where available. A Fish subroutine function, written in the F L A C data file, can be used to compute the shear modulus values based on S P T blow count input.  T h e shear modulus values assigned  for the rockfill material was slightly stiffer than those computed for the dense portion of the sand and gravel shells. Comparison with the results of the S H A K E analyses showed that the average fraction of maximum shear modulus, G / G  m a x  , should be about 0.3 for all zones in the dam.  Post-liquefaction parameters for all material zones in the dam were assigned in the input data file.  All materials were assigned a residual strength ratio of 0.3 with the exception of the  121  Chapter 7. Total Stress Analyses of Coquitlam Dam  softer portion of the upstream sand and gravel shell and the hydraulic fill core material.  Residual  strength ratios of 0.1 and 0.2 were assumed for the hydraulic fill and soft portion of the upstream shell material respectively. A liquefaction strain value of 2.5 % , which was in agreement with the findings of the verification analyses, was used for all dam materials.  A summary of the input  parameters can be found in Table 7.6.  Table 7 . 6 Summary of Parameters Used in Proposed Model Dam Material  ( N i )  6  Sf/a'vo  0  Dense Sand Foundation  120  0.3  Stiff Silt Foundation  100  0.3  Upstream and Downstream Rockfill Upstream Sand and Gravel Shell (a)  70  0.3  35  0.3  (b)  25  0.3  (c)  12  0.2  35  0.3  Downstream Sand and Gravel Shell Hydraulic Fill Sandy Silt Core  27  0.1  Similar to the deformation analyses, the present analyses were performed using the dynamic mode option in F L A C .  Combined stiffness- and mass-proportional Rayleigh damping  assuming a minimum critical damping ratio of 4 % about a central frequency of about 4 Hz was used. The central frequency corresponds to the predominant frequency of the input base motion. A minimum damping ratio of only 4 % was assumed because in the event that the core liquefies, additional damping due to the cyclic or 'ratchet' effect of the post-liquefaction model will occur. Because the core is subjected to low or negligible static shear stresses, cyclic loading may cause large cyclic shear stress-strain loops. A s a result, damping will take place due to these loops. A s presented in Chapter 4, this cyclic 'ratchet' effect is caused by the different loading and unloading moduli. Free field boundaries were applied at the upstream and downstream boundaries in order to minimize wave reflections and achieve free field conditions. The Caltechb earthquake history was input as a velocity history at the base of the geometry because, as described in the Appendix, F L A C version 3.3 does not integrate acceleration histories accurately. 122  Chapter 7. Total Stress Analyses of Coquitlam Dam  7.4.2 Results T h e analysis predicts that the entire core and a large portion of the upstream sand and gravel shell will liquefy.  A plot showing the zones predicted to liquefy by this analysis indicates  that the results are somewhat comparable to those found from the S H A K E analyses (Fig. 7.12). However, a larger portion of the upstream shell is predicted to liquefy in the proposed procedure than that predicted by S H A K E .  This discrepancy in results may be due to either two-dimensional  effects or to the damping values used.  It is likely that the difference is due to the latter because  the fraction of critical damping used in the S H A K E analyses were approximately 17 % , which is considerably greater than the 4 % used in these analyses.  In result,  the system is significantly  underdamped in the pre-liquefaction phase of the response. In this case, the the error is on the conservative side.  However, it should be noted that 4 % damping is applied to frequencies of  about 4 Hz. For frequencies other that 4 Hz, the damping ratio is greater than 4 % .  0  50  100  150  200  250  300  Figure 7.16 Zones Predicted to Liquefy by Proposed Method  A plot of the shear stress versus shear strain curve in the core material, although not shown here, indicated that liquefaction  occurred at approximately  6 s.  This corresponds  approximately to the time when the computed maximum shear stress has already occurred in the S H A K E analyses.  In S H A K E , the maximum shear stress in the core was estimated to occur at  about 5.7 s to 6.1 s.  123  Chapter 7. Total Stress Analyses of Coquitlam Dam  Using the proposed procedure, the crest is predicted to deform 0.6 m and 1.1 m in the horizontally upstream and vertically downward directions, respectively.  T h e displacement in the  vertical direction is comparable to those predicted to occur due to the loss in stiffness only as well as those due to loss in stiffness and inertia force.  In the proposed procedure, however, the  horizontal displacement is significantly greater than those predicted by the previous methods. This is likely due to the larger upstream sand and gravel zone predicted to liquefy in this method than in the other method.  In fact, as shown in Fig. 7.17, the largest displacements occur in the  upstream sand and gravel berm.  In this area, the predicted maximum displacements are about  1.51 m in the upstream and upwards direction. Although the displacements are significant, they are considered unlikely to cause overtopping because in fact the highest elevation of the berm and the core never drop below the reservoir water level. In an additional run, the critical damping ratio was increased to 8 % in order to assess the effect of the damping on the response. Although not shown, the upstream liquefied zone is only very slightly reduced.  T h e horizontal and vertical displacements of the crest are 0.35 m in the  upstream direction and 0.65 m downwards respectively.  T h e s e values are nearly half of the  displacements predicted when using a critical damping ratio of 4 % . Another different run was carried out to study the effect of time of liquefaction triggering. The entire liquefiable upstream sand and gravel shell and hydraulic fill core material were predicted to liquefy at a dynamic time of 6 s. T h e predicted displacements of the crest are 0.083 m upstream and 0.935 m downwards in the horizontal and vertical directions.  T h e vertical  displacement increases by nearly 0.3 m from that predicted when liquefaction in different zones are triggered at different times. less than in the previous run.  In contrast, the horizontal displacement is approximately 0.3 m A plot of the displacement vectors, shown in Fig. 7.18, indicates  that the displacement pattern is similar to those obtained in the gravity only and gravity plus velocity pulse deformations analyses. This is expected because liquefaction in liquefiable zones in those analyses are assumed to all occur at the same time.  124  Chapter 7. Total Stress Analyses of Coquitlam Dam  m  10(H  0  50  100  150  200  250  300  Figure 7.17 Post-Liquefaction Displacement Pattern Predicted by Proposed Procedure  Figure 7.18 Post-Liquefaction Displacement Pattern Assuming Liquefiable Zones Triggered at the Same Time  125  Chapter 7. Total Stress Analyses of Coquitlam Dam  T h e cyclic shear stress-strain plot of a zone in the core material is shown in Fig. 7.15. T h e curves indicate that large loops are traced during post liquefaction response of the core material. A s a result, damping is accounted for directly in model for this material. This indicates that the use of a critical damping ratio of only 4 % is reasonable, although the preliquefaction triggering may be higher.  However, the error would be on the conservative side and would be  acceptable. Assuming a direct relationship between damping ratio and displacement for a critical damping ratio of 4 % , horizontal and vertical displacements of  about 0.16 m and 1.8 m are  estimated if all liquefiable materials are assumed to liquefy at the same time. By comparing these results with those of the previous run of 0.58  m and 1.07  m in the horizontal and vertical  directions, whereby liquefaction in different zones is triggered at different times, it appears that the time of liquefaction triggering is very important. 100  — |  CO  -100  1  Figure 7.19 Cyclic Shear-Strain Plot Predicted in Dam Core Material  126  Chapter 7. Total Stress Analyses of Coquitlam Dam  7.5  Deformation Due to Consolidation T h e deformation caused by the dissipation of excess porewater pressures will add to the  total displacement due to liquefaction.  A semi-empirical approach was used to evaluate these  deformations induced by consolidation. T h e volumetric deformations can be estimated from a relationship developed by Ishihara and Yoshimine (1992). As summarized in Chapter 2, materials liquefied to 100 % pore pressure ratio behave like a liquid and cannot resist shear stresses.  Upon drainage, the material consolidates under the  applied stresses causing volumetric strains. This results in surface settlement. As described previously, the in-plane (G ), out-of-plane (a ) horizontal stresses, and the x  z  vertical stresses (a ) are set equal to the pore water pressure to simulate soil liquefaction. y  The  bulk and shear moduli in the liquefied zones are adjusted such that volumetric strains only will occur due to the change in effective stresses. The magnitude of the strains was estimated from Ishihara and Yoshimine's chart. T h e change in stresses is equal to the effective stresses under gravity loading. This method was verified in F L A C ; using both a single element and a column geometry, prior to its application on the dam. According to Ishihara and Yoshimine's relationship, a volumetric strain of about 3 % is predicted.  It should be noted, however, that laboratory tests have shown that the volumetric  strains associated with silty sands are generally about 1.2 % to 1.5 % , somewhat less than that obtained from the chart for clean sands. In result, the computed deformations are likely to be very conservative. For a volumetric strain of 3 % in the liquefied elements due to consolidation, the crest setllement was computed to be approximately 0.6 m. T h e displacement pattern due to volumetric deformation only is shown in Fig. 7.16.  T h e resulting total crest movement including that due to  inertia forces is about 1.6 m down in the vertical direction and 0.7 m in the upstream horizontal direction. A plot of the magnified deformed grid for comparison with the original grid is shown in Fig. 7.17.  127  Chapter 7. Total Stress Analyses of Coquitlam Dam  128  Chapter 7. Total Stress Analyses of Coquitlam Dam  7.6 Summary T h e proposed Total Stress Dynamic Analysis procedure was applied to Coquitlam Dam. T h e proposed method captures both liquefaction triggering and post liquefaction displacement prediction in a relatively simple procedure.  This method allows liquefaction of elements to be  triggered at different times, which can signifantly affect the predicted deformation for a twodimensional geometry  with different material properties.  The  results were discussed  and  compared with those obtained using traditional triggering methods and deformation analyses considering loss in stiffness only and loss in stiffness plus velocity pulse. The results of the proposed procedure were compared to those using S H A K E and predicted a larger liquefied zone in the upstream shell material. The reason for this is likely the small value of critical damping ratio.  A low value of critical damping ratio is required because in the event that  the core liquefies, damping is accounted for in the stress-strain loops. This value has a significant effect on the predicted deformations. For the Coquitlam Dam applied under a Caltechb earthquake motion, crest displacement results estimated  using the proposed procedure are comparable to those using the  traditional deformation methods although the pattern of displacement is different.  more  Significant  deformation is predicted in the upstream sand and gravel shell berm, which can be attributed to the larger zone of liquefaction in the upstream shell material.  However, overall, the proposed  procedure compares favourably with the traditional methods and most importantly illustrates the importance of trigger time in the liquefaction-induced deformation analyses. The results of all of the present analyses, in terms of maximum displacement and deformation  pattern,  are comparable to those obtained  in 1984  under the O B E .  This is  reasonable considering that the peak ground earthquake acceleration used in these analyses is similar to the 1984 O B E . A larger magnitude design earthquake may cause significantly larger displacements in the crest, however,  due to the width and stiffness of the rockfill berms,  overtopping of the dam may not occur.  This should be verified if the design earthquake is  significantly larger than that used in the present analyses and if more up to date laboratory or field  129  Chapter 7. Total Stress Analyses of Coquitlam Dam  testing indicates that the portion of the upstream shell closest to the rockfill berms is less dense than assumed in these analyses.  130  Chapter 8. Conclusions and Recommendations  CHAPTER 8  CONCLUSIONS AND RECOMMENDATIONS  Earthquake induced liquefaction of loose saturated soils can cause severe damage to earth structures. O n e well-known example is the failure of the Lower San Fernando Dam in which the driving stresses exceed the residual strengths of the liquefied soil. However, the development of significant deformations is not restricted to situations in which the driving stresses exceed the residual strengths.  Laboratory tests and field examples indicate that liquefied soils can still  transmit some finite amount of cyclic shear stresses. A s a result, inertia forces are generated even in the post liquefaction phase. This can cause significant displacements particularly if there is an initial static shear stress acting on the structure. A total stress procedure was presented in which the deformations due to the reduced stiffness and strength of liquefied soil as well as those due to the earthquake induced inertia forces are computed.  Unlike other total stress procedures, such as Jitno and Byrne's in which  inertia forces after liquefaction are accounted for indirectly in the residual strains based on a factor of safety, the present method includes the inertia forces in the analysis. In addition, liquefaction is triggered in each element at different times.  Triggering of liquefaction is predicted by weighting  the cyclic shear stresses induced by a prescribed motion and converting each half cycle into an equivalent number of cycles. been exceeded.  Liquefaction is triggered when a specified number of cycles has  Upon liquefaction, post liquefaction properties are assigned to the liquefied  element and the stresses are reset.  A FISH subroutine which performs the procedure has been  written for use in the computer code F L A C . The ability of the procedure to predict liquefaction triggering was verified against the computer code S H A K E .  The results indicate that the method yields results comparable to  S H A K E . However, the results are sensitive to the model parameters chosen for the analysis. For  131  Chapter 8. Conclusions and Recommendations  the present it is recommended that S H A K E analyses be performed in order to compare the results with the proposed procedure. The predicted liquefaction induced displacements were also compared to those predicted by Bartlett and Youd's empirical equation for sloping ground.  T h e results indicate that the  displacements are comparable. Sensitivity analyses on the input parameters of the model show that the pre-liquefaction parameters do not significantly affect the results. This may be because the high acceleration amplitudes in the earthquake record used in these analyses only occur early on in the time history.  A s a result, the predicted post liquefaction response would not be  significantly affected by the time of liquefaction triggering. However, if relatively high acceleration amplitudes are maintained throughout the duration of the earthquake, then triggering time may become important. T h e procedure was applied to seismic analyses of Coquitlam Dam. Coquitlam Dam is a 30 m high hydraulic fill dam with high consequences of failure, the dam has been classified as an extrerne hazard.  The seismic stability of the dam has not been investigated since 1984.  Since  then the state of practice in seismic assessment of earth structures has progressed. The results of the proposed procedure were compared to displacements predicted using variations of the more common methods such as the modified modulus method and Jitno and Byrne's extended Newmark  method.  The results are comparable also illustrate the importance of taking into  account that liquefaction of soil elements could occur at different times.  Based  on the  above  results obtained  using the  proposed method, the  following  recommendations are made for future research: 1.  Perform a sufficient number of analyses comparing the results of liquefaction triggering with those of S H A K E in order to develop a data base of recommended fraction of shear modulus, G/G  2.  m a x  and damping values accounting for earthquake base motion.  Perform additional one-dimensional analyses using different earthquake base imput motions to better understand the effect of different earthquakes on the proposed procedure.  132  Chapter 8. Conclusions and Recommendations  3.  Compare the results of the liquefaction triggering with the results using a two-dimensional method such as F L U S H  4.  Verify the procedure with a number of actual case histories.  5.  Fine-tune the liquefaction triggering procedure so that the non-linear stress strain behaviour of soil is directly incorporated by using different shear moduli compatible with strain level at a particular time during dynamic motion. A s a result, damping is directly incorporated into the procedure and will not adversely affect the post-liquefaction triggering phase of the procedure.  6.  Develop a non-linear shear stress-strain relationship for use in F L A C , which is not possible in S H A K E or F L U S H , and verify with case histories and compare with other procedures.  133  Bibliography  BIBLIOGRAPHY  1.  Bartlett, S . F . and Youd, T . L . , 1992. "Empirical Prediction of Lateral Spread Displacement," P r o c , 4th U S - J a p a n Workshop on Earthquake Resistant Design, Lifeline Facilities and Countermeasures for Soil Liquefaction, Honolulu, Hawaii, N C E E R , State University of New York, Buffalo.  2.  B C Hydro, 1980. "Coquitlam Dam and Reservoir - Memorandum on Seismic Stability," Report No. H1171, March 1980.  3.  B C Hydro, 1984. "Coquitlam Dam Rehabilitation - Memorandum on Design," Report No. H1757, September 1984.  4.  Byrne, P.M., 1990. 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"Soil Liquefaction and Cyclic Mobility Evaluation for Level Ground during Earthquakes," J . Geotech. Engrg., A S C E , 105(2), pp. 201-255.  45. S e e d , R.B. and Harder, L.F., 1990. " S P T - B a s e d Analysis of Cyclic Pore Pressure Generation and Undrained Residual Strength," Proc. H. Bolton Seed Memorial Symposium, Vancouver, B C , 2, 351-377.  46. S e e d , H.B. and Idriss, I.M., 1967. "Analysis of Soil Liquefaction: Niigata Earthquake," J . Soil Mech. Found. Div., A S C E , 93(SM3), pp. 83-108.  47. S e e d , H.B. and Idriss, I.M., 1971. "Simplified Procedures for Evaluating Soil Liquefaction Potential," J . Soil Mech. Found. Div., A S C E , 97(SM9), pp. 1249-1273.  48. S e e d , H.B. and Lee, K.L., 1966. "Liquefaction of Saturated Sands during Cyclic Loading," J . of the Soil Mech. and Found. Div., A S C E , 92(SM6), pp. 105-134.  49. S e e d , H.B., Lee, K.L., Idriss, I.M., and Makdisi, F.I., 1975. "The Slides in the San Fernando Dams During the Earthquake of February 9, 1971," J . Geotech. Engrg., A S C E , 101(GT7), pp. 651-688.  50. S e e d , H.B., Tokimatsu, K., Harder, L.F., and Cjung, R., 1985. "Influence of S P T Procedures in Soil Liquefaction Resistance Evaluations," J . Geotech. Engrg., A S C E , 111(12), pp.  51. Singh, S., 1994. "Liquefaction Characteristics of Silts," Ground Failures Under Seismic Conditions, Geotechnical Special Report No. 44, A S C E , pp. 105-115.  52. Stark, T . D . , Ebeling, R.M., and Vettel, J . J . , 1994. "Hyperbolic Stress-Strain Parameters for Silts," J . Geotech. Engrg., A S C E , 120(2), pp. 420-441.  137  Bibliography  53. Stark, T . D . and Mesh, G . , 1992. "Undrained Shear Strength of Sands for Stability Analysis," J . Geotech. Engrg., A S C E , 118(11), pp. 1727-1747.  54. Stark, T . D . and Olson, S . M . , 1995. "Liquefaction Resistance Using C P T and Field C a s e Histories," J . Geotech. Engrg., 121(12), A S C E , 856-1216.  55. Taboada-Urtuzuastegui, V . M . and Dobry, R., 1998. "Centrifuge Modeling of EarthquakeInduced Lateral Spreading in Sand," J . Geotech. Engrg., A S C E , 124(12), pp. 1195-1206.  56. Tatsuoka, F., Sasaki, T., and Yamada, S., 1984. "Settlement in Saturated Sand Induced by Cyclic Undrained Simple Shear," P r o c , 8th World Conference on Earthquake Engineering, San Fransisco, Vol. 3, pp. 255-262.  57. Thevanayagam, S., Ravishankar, K., Mohan, S., 1994. "Steady-State Strength, Relative Density, and Fines Content Relationships for Sands," Transportation Research Record 1547, 61-67.  58. Thomas, J . , 1992. "Static, Cyclic and Post Liquefaction Undrained Behaviour of Fraser River Sand," M.A.Sc. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, B C .  59. Tokimatsu, K. and S e e d , H.B., 1987. "Evaluation of settlements in Sand due to Earthquake Shaking," J . Geotech. Engrg, A S C E , 113(8), pp. 861-878.  60. Towhata, I. and Ishihara, K., 1985. "Undrained Strength of Sand Undergoing Cyclic Rotation of Principal Stress Axes," Soils and Foundations, Japanese S o c . S M & F E , 25(2), pp. 135-147.  61. Vaid, Y . P . , Byrne, P.M., and Hughes, J . M . , 1981. "Dilation Angle and Liquefaction Potential," J . Geotech. Engrg., A S C E , 107(GT7), pp. 1003-1008.  62. Vaid, Y . P . and Chern, J . C , 1983. "Mechanism of Deformation During Cyclic Undrained Loading of Saturated Sands," International Journal of Soil dynamics and Earthquake Engineering, 2(3), pp. 171-177.  63. Vaid, Y . P . and C h e m , J . C , 1985. "Cyclic and Monotonic Undrained Response of Saturated Sands," Advances in the art of testing soils under cyclic conditions, A S C E Convention, Detroit, Michigan.  64. Vaid, Y . P . and Finn, W.D.L., 1979. Engrg., A S C E , 105(10).  "Static Shear and Liquefaction Potential," J . Geotech.  65. Vaid, Y . P . and Thomas, K., 1995. "Liquefaction and Postliquefaction Behaviour of Sand," J . Geotech. Engrg., A S C E , 121(2), pp. 103-173.  138  Bibliography  66. Vaid, Y . P . , . "Liquefaction of Silty Soils," Ground Failures Under Seismic Conditions, Geotechnical Special Report No. 44, A S C E , pp. 1-15.  67. Vaid, Y . P . , Chung, E.K.F., and Kuerbis, R., 1989. "Preshearing and Undrained Response of Sand," Soils and Foundations, Tokyo, Japan, 29(4), pp. 49-61.  68. Vaid, Y . P . , Sivathayalan, S., Uthayakumar, M., Eliadorani, A., Konrad, J . M . , Hofman, B.A., and Robertson, P.K., 1998. "Static and Cyclic Undrained Behaviour of Soils at the C A N L E X Sites," Draft Version of C A N L E X report.  69. Yasuda, Y . P . , Masada, T., Yoshida, N., Nagase, H., Kiku, H., Itafuji, S., Mine, K., and Sato, K., 1994. "Torsional Shear and Triaxial Compression Tests on Deformation Characters of Sands Before and After Liquefaction," Proc. 5th U S - J a p a n Worshop on Liquefaction Large Ground Deformation, N C E E R , State University of New York at Buffalo.  70. Yasuda, N., Matsuomo, N., 1993. "Dynamic Deformation characteristices of Sands and Rockfill Materials," Canadian Geotechnical Journal, 30, pp. 747-757  71. Yasuda, Y . P . , Nagase, H., Kiku, H., and Uchida, Y., 1991. "A Simplified Procedure for the Analysis of the Permanent Ground Displacement," Proc. 3rd J a p a n - U S Workshop on Earthquake Resistant Design of Lifeline Facilities and Countermeasures for Soil Liquefaction, edited by T . D . O'Rourke and M.Hamada, Report No. N C E E R - 9 1 - 0 0 0 1 , N C E E R , State University of New York at Buffalo, pp. 225-236.  72. Yasuda, Y . P . , Nagase, H., Kiku, H., and Uchida, Y., 1992. "A Mechanism and a Simplified Procedure for the Analysis of Permanent Ground Displacement due to liquefaction," Soils and Foundations, 32(1), pp. 149-160.  139  Appendix I  APPENDIX I  140  Appendix I  TO:  Dr. Peter Byrne  FROM:  Charissa Dharmasetia  SUBJECT:  DATE:  13 November, 1997  Damping and Integration in Dynamic F L A C  A s requested a package was prepared summarizing the results of a dynamic analysis on a column using the programs F L A C and S H A K E in order to assess the effect of damping in F L A C on the system. A total of 20 simple cases in F L A C were examined using two different input base accelerations, four types of damping for each input  motion, and three values of  central  frequencies for cases where Rayleigh damping was used. Two cases were analyzed in S H A K E using the same two input base accelerations as in F L A C . Analyses in F L A C and S H A K E were carried out to at least four seconds after the end of dynamic motion. In general, it was found that F L A C and S H A K E results compare favorably, with the same damping factor, when Rayleigh damping is used.  In addition, the resulting base displacement histories obtained from F L A C were examined and compared to base displacements manually calculated by numerically double integrating  the  acceleration histories. The formulations developed by Wilson and Clough (1962) were used to integrate the acceleration histories. A total of eight cases in F L A C , varying material properties and damping were performed using two different input base acceleration. The results show that F L A C does not calculate the displacements correctly and that the base displacement histories, which should only be dependent on input accelerations, vary with damping factors and material properties.  DAMPING CASES ANALYZED A 100-ft (or 30.48-m) simple column was analyzed in both S H A K E and F L A C . The column is homogeneous and elastic and its properties are stress independent and have no modulus or damping attenuation. T h e damping factor used was 0.1 and the shear modulus input was 1840  141  Appendix I  kips/ft in S H A K E corresponding to 88100 k P a in F L A C . The water table was assumed to be at 2  ground surface. T h e soil density was assumed to be 0.1185 kips/ft in S H A K E and 1900 k g / m in 3  3  FLAC.  T h e column was analyzed under two different input motions: a simple harmonic input with 0.2g amplitude and a frequency of 4 Hz lasting 10 seconds, and the Caltechb earthquake record reduced to last 37 seconds. T h e analysis was carried out to a time of at least 4 seconds after the end of earthquake motion in order to observe the effects of damping.  In addition, in F L A C , the column was analyzed using the four different types of damping available:  1.  Rayleigh Damping (both mass and stiffness proportional); •  Central Frequency, f  •  f  •  fmin > dominant input frequency;  min  2.  , = dominant input frequency;  < dominant input frequency;  Rayleigh Damping (mass proportional damping only); •  Central Frequency, f in, = dominant input frequency;  •  f  min  •  f  min  m  3.  4.  min  < dominant input frequency; > dominant input frequency; Rayleigh Damping (stiffness proportional damping only);  •  Central Frequency, f i , = dominant input frequency;  •  f  min  •  f  min  m  n  < dominant input frequency; > dominant input frequency;  Local Damping.  Rayleigh Damping was originally used in structural analysis to damp the natural oscillation modes of the system. A s a result, it is frequency dependent. Its equations are expressed in matrix form  142  Appendix I  with components proportional to mass and stiffness. Bathe and Wilson developed an expression whereby the critical damping ratio, X, at any angular frequency, CQJ, of the system can be found from the mass and stiffness proportional damping constants. From the equation, it can be shown that mass-proportional damping is dominant at lower angular frequency ranges and stiffnessproportional damping is dominant at higher angular frequency ranges. The critical damping ratio reaches a minimum at a frequency where mass and stiffness damping each supplies half of the total damping force. In F L A C , the user defines this minimum critical damping ratio, \  m  m  , and  frequency, f n. T h e damping ratio is constant over approximately a 3:1 frequency range about the mi  minimum frequency, f  min  . A s a result, frequency independent damping can be approximated by  choosing 2~ to lie in the centre of the dominant range of frequencies present in the numerical model. This 'minimum frequency', f  min  is termed central frequency in F L A C .  In F L A C , combined mass- and stiffness-proportional damping, mass-proportional damping only, and stiffness-proportional damping only can be specified. Theoretically, if combined mass and stiffness proportional damping is used, F L A C model response to frequencies outside the 3:1 frequency range about f  min  , should be over damped. If mass-proportional damping only is used,  model response to frequencies, f, less than f~ should be 'overdamped' and response to f greater than f should be 'underdamped'. Conversely, if stiffness-proportional damping only is used, model response to f less than f  min  should be 'underdamped' and response to f greater than f i should be m  n  'overdamped'.  Local damping was originally used to equilibrate static simulations in F L A C . It is reported to be attractive to dynamic analysis because it is frequency independent and as a result, unlike the Rayleigh damping option, frequency does not have to be specified. Local damping operates by subtracting and adding mass at a gridpoint at velocity extremes, thus is performed twice per oscillation cycle. T h e proportion of energy removed can be related to fraction of critical damping. T h e local damping coefficient used in F L A C is equal to the product of pi and critical damping ratio.  143  Appendix I  According to the manual, local damping appears to give good results for a simple case but is untested for more complicated situations. It was tested here, on a supposedly simple case. Another one of the more attractive features of local damping is that it uses larger time steps than Rayleigh damping. Unfortunately, it was recommended that in order to be able to compare these results to those with Rayleigh damping, the same time step as that used in Rayleigh damping should be used.  The maximum shear stress profile and the acceleration history of the top of the column were used to compare the results of S H A K E and F L A C . T h e acceleration history of the base of the column was also plotted in order to check whether the input motion is the same as that in S H A K E .  RESULTS Simple Harmonic Wave Rayleigh Damping (both mass and stiffness-proportional) Central Frequency, f  min  = 4 Hz  T h e acceleration histories of the top of the column, analyzed in F L A C and S H A K E , compare favorably. In both S H A K E and F L A C , the initial response is greater than the 'steady state' acceleration amplitude. However, these maximum initial amplitudes are greater in S H A K E m/s) than in F L A C (2.4  (2.62  m/s). The 'steady state' amplitudes are identical. After shaking, the  residual oscillations in both F L A C and S H A K E match although the amplitudes in S H A K E are slightly greater than in F L A C . The acceleration at the top of the column comes to rest at the same time in S H A K E as in F L A C .  T h e shear stress profiles determined from F L A C and from S H A K E also match up nearly perfectly. This is shown on the attached plots.  Central Frequency,  f  min  = 1.5 Hz  144  Appendix I  The initial acceleration history response is significantly greater in the S H A K E model than in the F L A C model. Similarly, the 'steady state' and the post earthquake oscillation responses are also greater in magnitude in the S H A K E model than in F L A C . This indicates that the F L A C system is more damped than the S H A K E system. This response corresponds to what is expected from theory. However, uncharacteristic of overdamped systems, the F L A C model comes to rest at a later time than the S H A K E model.  The F L A C shear stress profile is similar in shape to the S H A K E profile although seems shifted down. T h e magnitude of the F L A C profile is slightly less than that of the S H A K E profile.  Central Frequency, f  min  = 8 Hz  The magnitudes of the initial as well as the 'steady state' acceleration responses of the top of the column are greater in the S H A K E model than in the F L A C model. Similarly, the response after base motion has ended is greater in magnitude in S H A K E  than in F L A C . This response  corresponds to theory. Unlike the previous case, however, the residual oscillations come to a rest earlier in F L A C than in S H A K E .  The shear stress profiles predicted by S H A K E and F L A C models are similar in shape. However, the magnitude of the profile in F L A C is greater than in S H A K E . O n c e again, this shows that the F L A C system is more damped than the S H A K E system.  Rayleigh Damping (mass-proportional damping only) Central Frequency, f  min  = 4 Hz  The steady state acceleration history responses of the top of the column from F L A C and S H A K E are similar. However, unlike the both mass- and stiffness-proportional damping case, the initial peak in acceleration in F L A C (3 m/s ) is greater than in S H A K E (2.62 m/s). In addition, after 2  shaking, the residual oscillations do not match in phase or magnitude. T h e amplitudes of the oscillations in F L A C are greater than those in S H A K E and come to rest at a later time. This  145  Appendix I  indicates that the F L A C system is less damped than the S H A K E system. This is consistent with theory because for X \  = 0.1, f  m n  min  = 4 Hz, mass-proportional damping only supplies half of the  specified damping at 4 Hz (in other words, damping ratio is actually only about 0.05).  T h e shear stress profile in F L A C and S H A K E match in shape but not in amplitude. T h e amplitude in F L A C is greater than in S H A K E .  If the critical damping ratio operating at 4 Hz is doubled to 0.2, the acceleration history response in F L A C matches to that in S H A K E . Oscillations after the end of dynamic motion come to a rest at the same time. This corresponds to theory since the actual damping ratio supplied by mass proportional damping operating at 4 Hz is 0.1.  The shear stress profile in F L A C and S H A K E match in both shape and amplitude.  Central Frequency, f  min  = 1. 5 Hz  T h e acceleration history response in F L A C is more transient that in previous runs. A 'steady state' response takes longer to achieve than previously. T h e initial response is significantly greater in amplitude in the F L A C model than in the S H A K E model. The amplitude gradually decreases to a steady state amplitude slightly greater than in S H A K E . After dynamic motions has stopped, residual accelerations continue to oscillate but decrease at a slower rate than the  SHAKE  response. T e n seconds after the end of dynamic motions, the accelerations in F L A C still have not come to a rest.  This behaviour shows that the F L A C model is less damped than the S H A K E  model. T h e response is even less damped than the previous run. T h e s e observations are consistent with theory, since even less damping is applied at the system frequency of 4 Hz than at 1.5 Hz.  T h e shear stress profile response in the F L A C model is similar in shape but greater in magnitude than the S H A K E response.  146  Appendix I  Central Frequency, f  min  = 8 Hz  T h e acceleration history response of the F L A C model compares very well with that obtained from S H A K E . This response is also consistent with theory since for Nin = 8 Hz, damping for system frequencies less than 8 Hz is greater than those for 8 Hz. It appears that the critical damping ratio of the system at 4 Hz is about 0.1 as opposed to 0.05 at frequency 8 Hz.  The maximum shear stress profile response of F L A C also compares favorably with that of the S H A K E model.  Rayleigh Damping (stiffness-proportional damping only) Central frequency, f,, = 4 Hz The acceleration history response of the top of the column in F L A C is slightly greater than the response in the S H A K E model. The F L A C model also comes to a rest at a later time after dynamic motion than the S H A K E model. This indicates that the F L A C model is less damped than the S H A K E model and is consistent with theory. According to theory, a critical damping ratio of 0.05 is applied at frequency of 4 Hz because stiffness damping contributes to only half of the specified critical damping ratio at the central frequency.  T h e magnitude of the maximum shear stress profile is also greater in the F L A C model than in the S H A K E model.  If the fraction of critical damping operating at the center of frequency 4 Hz is set to 0.2, then the stiffness-proportional contribution is 0.1 at 4 Hz. Theoretically, the F L A C response should be comparable to the S H A K E response. This response is somewhat observed, however, in general, the magnitude of the top acceleration history response is slightly less in F L A C than in S H A K E . However, the F L A C model comes to a rest at a later time than the S H A K E model. This may be because not enough damping is applied at the other system frequencies.  147  Appendix I  T h e shear stress profile in the F L A C . m o d e l is comparable to that in S H A K E model. However, the response is F L A C is slightly less damped than that in the S H A K E model at depths greater than 15 m.  Another significant drawback to using the stiffness-proportional damping only option in Rayleigh damping, besides the poor results is that the time step used is significantly smaller than that used in both mass- and stiffness-proportional damping or mass-proportional damping only. T h e time step used in the stiffness-proportional damping only option is on the order of 10:5 rather than 10-4 used in the previous options. A s a result, the time it takes to run analyses using this option is significantly longer than in other options.  Central frequency, f = 1. 5 Hz min  T h e initial, steady state, and post earthquake magnitude of the acceleration response of the top of the column is slightly greater in S H A K E than in the F L A C model. In general, the F L A C model is more damped than the S H A K E model. However, the F L A C model comes to a rest at a later time than the S H A K E model. This is likely because not enough damping is applied at the other system frequencies. This response is reasonable because the damping ratio at a frequency of 1.5 Hz should be greater than about 0.05. Because there is a linear relationship between damping ratio and frequency in stiffness-proportional damping, the damping at 3 Hz should be 0.1. Therefore, at a frequency of 4 Hz, the damping is greater than 0.1.  The magnitude of the shear stress profile for depths less than about 16 m is greater in S H A K E than in F L A C . However, for greater depths, the reverse is true.  Central frequency, f  min  = 8 Hz  148  Appendix I  T h e acceleration history response in the F L A C model is somewhat erratic but is in general greater in magnitude than the S H A K E response. T h e post-dynamic response is also greater in magnitude in F L A C than in S H A K E . In F L A C , the model does not come to a rest even 10 s after the end of dynamic motion. This indicates that the F L A C model is less damped and is consistent with theory.  The shear stress profiles of the F L A C and S H A K E models are similar in shape. However, the magnitude in the F L A C model is greater than in the S H A K E model.  Local  Damping  T h e damping coefficient was set equal to 0.314, which is equal to the product of the fraction of critical damping and pi. T h e time step was also set to be equal to that used in Rayleigh damping of 8.274e-4 in order to be able to compare the response.  T h e top acceleration history response in F L A C does not compare in amplitude or shape with the response in S H A K E . T h e steady harmonic shape of constant amplitude is somewhat observed in F L A C , however, there are periodic waves of amplitudes of about 7.0 m/s2. In addition, the amplitude of the response drops suddenly in comparison to S H A K E but continues to oscillate at a very small amplitude even 10 s after dynamic motion has stopped.  Similarly, the shear stress profiles in F L A C and S H A K E do not match in shape. However, the magnitudes of the shear stresses in F L A C are somewhat comparable those in S H A K E . This is shown on the attached plots.  Caltechb earthquake motion Prior to analyzing the column in F L A C under different types of damping, the column was analyzed with the Caltechb input motion undamped in order to determine the frequency components that contained the most energy. T h e s e frequency components are required to specify the damping in Rayleigh damping to ensure damping is centered about the most dominant frequencies. The  149  Appendix I  significant frequencies depend on the natural as well as on the forcing frequencies. In order to determine the significant frequencies, a fast fourier transform was performed on the acceleration history of the top of the undamped column.  A s shown on the attached spectral density plots, the dominant frequencies of the column are about 2 Hz followed by about 5 Hz followed by 9 Hz. T h e centre of the most significant frequencies is about 4 Hz, or 3.72 Hz to be more precise. This central frequency of about 4 Hz is coincidental with the dominant frequency of the Caltechb acceleration record. Incidentally, plots of the variation of critical damping ratio with frequency at central frequencies of 3.72 Hz, 1 Hz, and 9 Hz were also plotted on the Fourier spectrum plots in order to show the damping ratios used at the significant frequencies.  A fast fourier transform was also performed on the Caltechb record in order to check for the frequency content to ensure that it will not affect the numerical accuracy of wave transmission. According to  Kuhiemeyer and  Lysmer, the following  condition  must  be  met  for  accurate  representation of wave transmission through a model:  A/  10  C  where Al is the spatial element size, X is the wavelength associated with the highest frequency component that contains appreciable energy, f is highest frequency component in hertz, C is the speed of propagation giving the lowest natural mode C p (p-wave) or C , (s-wave)).  150  Appendix I  where G is the shear modulus, K is the bulk modulus, and p is the mass density.  For G = 88100 kPa, K = 234933 kPa, p = 1900 k g / m C = 215 m/s, C = 431 m/s, therefore, C = 3  s  C . s  p  From the spectral density plot of the Caltechb acceleration record, the highest appreciable  frequency component is less than 10 Hz. T h e associated wavelength is 22 m. From the above equation, Al = 2.2 m. All of the grid elements are 1.01  m. Therefore, the Caltechb acceleration  input does not have to be filtered for high frequencies to ensure numerical accuracy for wave transmission.  Rayleigh  Damping (Mass- and stiffness-proportional  damping)  Central Frequency, f , = 3.72 Hz min  The acceleration histories of the top of the column of F L A C and S H A K E are comparable. However, for dynamic time less than about 10 s, when the acceleration amplitude is the most significant, the response in S H A K E is slightly greater in amplitude than the response in F L A C . This response is consistent with theory because the significant frequencies during this time is likely about 2.0 Hz, in which case, the F L A C response would be more damped. However, after the time of 10 s, the responses of F L A C and S H A K E are almost identical, including the time at which the top acceleration comes to a rest.  T h e shear stress profile of F L A C matches with that of S H A K E in magnitude and shape down to a depth of 15 m. Beyond a depth of 15 m, the shear stress magnitude of F L A C is slightly greater than that of S H A K E . It is reasonable that the response in F L A C and S H A K E are not identical because the maximum shear stresses with depth may not necessarily occur under the same mode. However, it is expected that the F L A C response be less than the S H A K E response in magnitude because at all frequencies other than the central frequency, the damping ratio is greater than 0.1.  151  Appendix I  Central Frequency, f  mirt  = 1 Hz  The acceleration history response at the top of the column in F L A C is lesser in magnitude than in S H A K E at all times. T h e F L A C response also comes to a rest earlier than S H A K E after the end of dynamic motion. This response is consistent with theory because the F L A C model is more damped than the S H A K E model.  Similarly, the shear stress profile is lesser in magnitude in the F L A C model than in the S H A K E model. However, at a depth greater than about 20 m, the F L A C response becomes greater than the S H A K E response.  Central Frequency, f  min  = .9 Hz  For dynamic time less than 15 s, the acceleration response in F L A C is less in magnitude than in S H A K E . The response is in fact more damped than in the previous case. However, after a time of 15 s, the response in F L A C is comparable to the response in S H A K E . This response is reasonable because for the system frequency of 2 Hz, the critical damping ratio is significantly greaterthanOA. However, for frequencies greater than about 5 Hz, the critical damping ratio is about 0.1.  The shear stress profiles of the F L A C and S H A K E models are comparable. For depths less than about 17 m, the S H A K E profile is slightly greater than the F L A C profile. However, below 17 m, the F L A C profile is greater than the S H A K E profile.  Rayleigh Damping (Mass-proportional damping only) Central Frequency, fmin = 3.72 Hz It is difficult to compare the acceleration history response of the top of the column in F L A C with that in S H A K E because they are out of phase. In general, however, the F L A C response is slightly greater in amplitude than the S H A K E response. This indicates that the F L A C response is less damped than the S H A K E response. This response agrees with theory.  152  Appendix I  T h e shear stress profile in F L A C is also greater in magnitude than that in the S H A K E model. In addition, the shape of the F L A C profile is slightly different than that of the S H A K E model.  If the critical damping ratio is doubled to 0.2, the acceleration response of the F L A C model becomes more comparable to the S H A K E model. For dynamic times less than 10 s, there are only occasional times where the F L A C response is less than the S H A K E response. For times greater than 10 s, the F L A C and S H A K E  responses are almost identical. This response is  reasonable.  Similarly, shear stress profile of F L A C is comparable to that of S H A K E . However, the F L A C profile is slightly greater in magnitude than that of S H A K E .  Central Frequency, f  min  = 1 Hz  At all times, the acceleration response in F L A C is significantly greater in amplitude than in S H A K E . At a time of 40 s, 3 s after the end of dynamic motion, the F L A C model has not yet come to a rest and continues to oscillate at a high amplitude relative to the S H A K E model. This response is consistent with theory because at all frequencies greater than 1 Hz, the damping ratio is less than 0.05.  T h e magnitude of the shear stress profile in F L A C is significantly greater than in S H A K E . This supports the theory that the F L A C model is appreciably less damped than the S H A K E model.  Central Frequency,  f in = m  9 Hz  For dynamic times less than 15 s, the acceleration response in the F L A C model is slightly less in amplitude than the response in the S H A K E model. However, for times greater than 15 s, the response in F L A C and S H A K E are nearly identical. This response is reasonable because for frequencies less than 9 Hz, the damping ratio is greater than 0.05. For frequencies less about 5  153  Appendix I  Hz, the damping ratio is less than 0.1. Because the most significant frequencies are about 5 Hz or less, the F L A C model is more damped than the S H A K E model.  T h e shear stress profile of the F L A C model is comparable to that of the S H A K E model. However, the magnitude of the F L A C profile is slightly greater than that of the S H A K E model. Although the most significant frequencies are more damped in F L A C than in S H A K E , it is possible that the frequencies causing the greatest shear stresses are less damped than in S H A K E .  Rayleigh Damping (stiffness-proportional damping only) Central Frequency, f  min  = 3.72 Hz  T h e acceleration response in the F L A C model is significantly greater in amplitude than in the S H A K E model. For times less than 10 s, the F L A C response is in phase with the  SHAKE  response. However, for times greater than 10 s, the F L A C response becomes out of phase. The magnitude of the F L A C response is only slightly greater than the S H A K E response after a time of 20 s. At time 40 s, when the S H A K E model has already come to a rest, the F L A C model continues to oscillate at a very small amplitude. This response is reasonable because for frequencies less than about 8 Hz (~2*f ), the damping ratio is less than 0.1. min  T h e shear stress profile in F L A C is significantly greater than the profile in S H A K E .  When the critical damping ratio of 0.1 operating at central frequency of 3.72 Hz is doubled to 0.2, the acceleration history response in the F L A C model is only slightly greater in amplitude than in the S H A K E model. For times greater than 20 s, the F L A C response is more comparable in magnitude. Although the critical damping ratio is doubled, the F L A C response is still less damped than the S H A K E response, this is probably because at the most significant frequency of the system of about 2 Hz, F L A C damping is less than 0.1.  154  Appendix I  T h e shear stress profile in the F L A C model is greater in magnitude than in the S H A K E model although to a lesser extent than in the case where ^  Central Frequency, f  min  m i n  = 0.1.  = 1 Hz  With the exception of the response between times 10 and 15 s, the acceleration response in F L A C is less than the response in S H A K E . Between the times of 10 and 15 s, the F L A C response is comparable to the S H A K E  response. This response is reasonable because for system  frequencies of greater than 2 Hz, the damping ration is equal to or greater than 0.1. Since most of the significant frequencies are greater than 2 Hz, F L A C response theoretically should be equally damped or only slightly more damped than the S H A K E response.  T h e shear stress profiles in the F L A C and S H A K E models are comparable.  For depths less tan  15 m, the S H A K E response is slightly greater than the F L A C response. For depths greater than 15 m, the reverse is true.  Central Frequency, f  min  = 9 Hz  T h e acceleration history response in F L A C is significantly greater than the S H A K E response at all times. T h e F L A C response gradually decreases in magnitude with time, since the amplitude of the forcing motion is relatively small at times greater than 10 s. At a time of 40 s, the F L A C model has not yet come to a rest. This response is consistent with theory since at all frequencies less than 9 Hz, the critical damping ratio is less than or equal to only 0.05.  Similarly, the shear stress profile in the F L A C model is significantly less than in the S H A K E model.  Local  Damping  Similar to the harmonic case, the dynamic time step was set to match the step used in the Rayleigh analysis. In this case, the time step used was 1.058e-4. 155  Appendix I  For a damping factor of 0.314, the top acceleration history of the F L A C model is greater in magnitude than the response in S H A K E . In fact, the F L A C accelerations show no signs of coming to rest towards the end of the history.  T h e shear stress profile obtained in F L A C is greater in magnitude than that obtained from S H A K E . In addition, as opposed to the S H A K E results, the F L A C profile is jagged not smooth.  BASE DISPLACEMENT ANALYSIS Recently, concerns have been expressed over whether F L A C calculated base displacements correctly in dynamic F L A C . Previously, it had been found that the calculated base displacement histories in F L A C varied with material properties. In order to address these concerns, the following cases were analyzed:  1.  Simple Harmonic Base Motion - Acceleration Input: •  Rayleigh Damping (mass and stiffness proportional), f j = 4 Hz;  •  Rayleigh Damping (mass-proportional damping only), f  min  •  Rayleigh Damping (mass-proportional damping only), f  min  •  G = 22,000 kPa, B = 60,000 kPa.  m  n  , = 4 Hz; = 8 Hz;  2. Caltechb Earthquake Record - Acceleration Input: •  Rayleigh Damping (mass and stiffness proportional), f  min  = 3.72 Hz;  •  Rayleigh Damping (mass and stiffness proportional), f  min  = 1 Hz;  •  Rayleigh Damping (mass-proportional damping only), f  •  G = 22,000 kPa, B = 60,000 kPa.  min  = 3.72 Hz;  156  Appendix I  In all cases, the base displacement and velocity histories were plotted and compared to the displacements and velocities integrated from the input acceleration records. Both acceleration records were integrated using formulations developed by Wilson and Clough(l 962). All cases using the simple harmonic base motion were plotted on one plot and all cases using the Caltechb Earthquake Record were plotted on another.  INTEGRATION Wilson and Clough presented a step-by-step integration procedure to calculate the response of lumped-mass systems to arbitrary dynamic loads. This approach is an alternative to the modesuperposition method of solving the differential equations representing the equilibrium of the lumped-mass system. T h e step-by-step method involves the direct numerical integration or the equilibrium equations. It can be assumed that the acceleration varies linearly (or higher order) within a time increment. This method was applied to both the simple harmonic acceleration record and the Caltechb acceleration record.  It was assumed that the acceleration varies linearly with time. T h e integration assuming linear variation was compared to the results of the direct integration of the simple harmonic acceleration record and was found to match. T h e following equations were used:  {*}, ={a} + y - { J t } ,  {x} ={b} + t  o  ~{x},  where  At  2  157  Appendix I  RESULTS Simple Harmonic Motion Acceleration Input As shown on the attached plot, the base displacement histories computed in F L A C for any of the above cases do not match the history integrated from the acceleration input motion. Although the amplitude of the displacements match, none of the histories in F L A C match each other. There seems to be a tendency for all histories to move towards one direction to different degrees varying with material properties and damping. This tendency is carried out even after earthquake motion has ceased with the exception of cases where mass-proportional only damping is used. For these cases, the relative displacements with time remain constant. This indicates that F L A C is not integrating properly. Because displacements are integrated directly from velocities, it is likely that F L A C is not integrating the velocities properly.  A plot comparing the base velocity histories computed for the various cases examined is also attached. Contrary to previous beliefs, the plot shows that for all cases, the F L A C results match the velocity history integrated from the acceleration input motion. Strangely, it appears that F L A C does calculate the base velocities properly.  T h e shear strain history response in F L A C for the case where Rayleigh Damping, with central frequency of 4 Hz, at a depth of 55 ft or 16.8 m was compared to that in S H A K E . A s shown in the attached plot, the response in F L A C and S H A K E match nearly perfectly. A plot of the maximum shear strain profile is also attached. It shows that the shear strain profile also match perfectly. T h e s e results indicate that F L A C computes the relative displacements correctly which would explain the reason why in the damping analyses the results in F L A C compare well with those in S H A K E under certain conditions. This is reasonable because the stresses are computed from strain rates which in turn are computed from velocities. The resulting displacements, however, are wrong.  158  Appendix I  Velocity Input All of the above four cases were re-run using the first integral of the acceleration history, the velocity history, a s the input motion. T h e base displacement and base acceleration histories of all cases were plotted and compared with the histories determined by integration. T h e shear strain history at a depth of 55 ft as well as the maximum shear strain profile response in F L A C were plotted and compared to the results in S H A K E .  As shown on the attached plot, the base displacement histories of all cases match perfectly with the displacement input history integrated from velocity. Similarly, the base acceleration history also matches perfectly. Like the case where the acceleration base input was used, the shear strain history at depth 55 ft and the maximum shear strain profile in F L A C matches  nearly  perfectly with that in S H A K E .  The acceleration histories of the base as well as the top of the column in the F L A C model using combined mass and stiffness Rayleigh damping, f  min  = 4 Hz, were plotted with the corresponding  histories in S H A K E . T h e s e plots show that the responses, like the case where an acceleration input was used, are comparable to the response in S H A K E . The maximum shear stress profile in F L A C is also comparable to the profile in S H A K E .  Caltechb Input Motion Acceleration Input Similar to the results using simple harmonic wave input, the attached plot shows that the base displacements are not computed correctly in F L A C . In these runs, however, the differences in displacement histories between the various cases are more subtle. T h e reason is likely because  159  Appendix I  the differences are small compared to the total magnitude of displacements. In general, F L A C tends to overpredict the displacements, in this case it is approximately by 10 percent.  The plot of the comparison of the velocity histories with the integrated velocity history shows that the velocity is computed correctly in F L A C . The shear strain history response at a depth of 55 ft in F L A C also compares favorably with the response in S H A K E . The maximum shear strain profiles in F L A C and S H A K E also compare at a depth above 15 m. Below 15 m, the F L A C response is slightly greater in magnitude than the S H A K E response.  Velocity Input If the velocity history, integrated from the Caltechb acceleration history, is used as the base input motion, the base displacement history used in F L A C matches perfectly with the displacements obtained from direct numerical integration of the velocity history. Similarly, the base acceleration histories of all the cases match with the original Caltechb acceleration history.  The shear strain history maximum shear strain profile in F L A C and S H A K E also match comparably. The response is the same as when an acceleration input is used.  The acceleration histories of the base and top of the column in the F L A C model using a velocity base motion are compared with the S H A K E responses. The responses are the same as when an acceleration base input is used. Comments Although F L A C appears to calculate the velocities and not the displacements correctly, R is likely the reverse is true because it is unreasonable that the displacements are improperly calculated when they are only simply integrated from velocities only. This was examined by integrating the velocity history output by F L A C by using the Wilson and Clough method to obtain displacement history and comparing with the displacement history directly output by F L A C . It was found that the integrated displacements match perfectly with F L A C . Thus F L A C does integrate displacements  160  Appendix I  correctly. The problem is likely because a negligible error is introduced in the integration of the velocities from the acceleration histories which is magnified in the displacement histories when they are integrated from the velocity history. Although F L A C does not integrate the velocities correctly when an acceleration base input motion is used, the errors are negligible. A s a result, the stresses and strains are not appreciably affected. T h e errors become apparent  only when  displacements are required.  CONCLUSIONS Maximum shear stress profile and the surface or top acceleration histories were used to compare the dynamic results of a 100-ft homogeneous and elastic column modeled in F L A C and S H A K E in order to assess the effects of the different types of F L A C damping on the system. T h e types of damping compared in the analyses are: Rayleigh damping with combined mass and stiffness contribution, Rayleigh damping with mass-proportional damping only, Rayleigh damping with stiffness-proportional damping only, and local damping. For cases where Rayleigh damping was used, the central frequency was varied about the significant frequencies of the system. All cases were analyzed by applying a simple harmonic base motion and the Caltechb earthquake motion.  The results indicate the following:  •  For identical critical damping ratios in F L A C and S H A K E , the F L A C response is comparable  to the  SHAKE  response provided that Rayleigh  damping with  combined mass- and stiffness- proportional damping contribution, centered about the significant frequencies of the system, is used. The significant frequencies depends on the natural frequencies of the system as well as the input motion and can be determined  by performing a fast fourier transform on the  undamped  history response of interest.  161  Appendix I  •  If the central frequency is specified outside the 3:1  frequency range of the  significant frequencies of the system, the response is 'overdamped'. •  Comparable results in F L A C  using Rayleigh damping with mass-proportional  damping only or stiffness-proportional damping only can be achieved by varying the central frequency as well as the damping ratios operating at the  central  frequency, provided that the system has only one significant frequency, which is unlikely. •  In Rayleigh damping, mass-proportional damping is dominant over the  lower  frequency ranges, whereas stiffness-proportional damping is dominant over the higher frequency ranges. •  Although local damping has the advantage of having a larger time-step than Rayleigh  Damping and  is frequency  independent,  it has been found to  be  completely useless.  Base displacement histories were also compared in order to a s s e s s whether they are computed correctly. T h e displacements were compared for cases where damping and material properties were varied. All cases were analyzed by applying a simple harmonic acceleration and velocity history inputs as well as the Caltechb acceleration and velocity history dynamic inputs. T h e results indicate the following:  •  When an acceleration history input is applied, negligible errors are introduced in the integration of the history to obtain the velocity history. T h e errors, however, are magnified when the velocities are further integrated to obtain displacements. Displacements are, thus, overestimated.  •  Because the errors appear to be negligible, the stresses and strains computed are not significantly affected since they are computed from velocities.  162  Appendix I  •  It is recommended that acceleration histories are integrated and dynamic input is applied as a velocity history.  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I  X * *  | *« •12.  « 0  S. cu  a £  1  c w  ~co ?-s 3=  C  II  2.  <I UJ ^  I§ s a:  UJ  Cn  s  co o I 5 |5 u. I i  E i Li *  5  S«  »5 CO Q.  E  .c Ct c I co a '5  O j a g  C O  §  <  O o =  « g.  < Ia  X co | . e  IL  1 °>  g  . 82 *-8s| "  eg  o  §1  CO  w  o.  E o o  E o o  (zs/ui) uo!iej3|aoov  CQ  5  (ZS/U|) U0!)eJ3|333V  171-  172  173  Appendix I  in i--  o in  _  o  CM  (ui) mdaa  c E 3 o  c E o  o -a «O  <-> 5 o  ll  st CU •jr  11  Dl  CL S O 9-  §.  m ui  55 ° 8 a. 5S  >< 1 s  g*  Sit9- n  O  O -Q  X  -8 a  X  c o  CD  If 111 9.  <  c  9  °  E  X — CO 6 TJ m CC  CO '5. S TJ  £  c  s  CB  O S-  211 U. E  " I S ^ a•  O C  "-6  3  "S  K  0  •*- o ' c  m  '3  x ^ tt  <  C ^ CO o  1.?  i= 5.  o »6 o - a  O  9  si  o *  ««  CO U l CL  a  E o O  E o  o  (ZS/UI) U0!ieJ9|30DV  (zs/ui) UO|lBia|0D0V  174  Appendix  I  LO  E 3  O  O o 0)  o QCO U) 0) i_  55 .c CO  E 3  o  o  a ^_  " 5  o  O II CD g>  o Q. S 11  £ &  Co p m | 0 g  6 5  i -  >>? :  «  x ff-  oj;  . a l l  •2 •§ " S  0)  — (D O U  S  s  CD  CD ~  S  < If  <^ 8 <t < «£  < ^  •§ 1  C  u  o  H—  C  Lu  LU  o 8  in 5  o =  o * tn  a.  co * a  LU  5 33  ou. E O  ^3  O Scj  < 33 LL  3  i  E CL E  CO  s  O 3  o  a) Ct  CO . y ' s »  1 S 4 CO | . -  J  c  t «) 3  CD « c — (DC 5 Q  E  TO  W Q  <  •a  II  S |  E o  o  E o  o  (ZS/Ul) UOI1BJ9I300V  (ZS/Ui) uoueJ3|a33v  175'  Appendix I 10  c  E 3  o  o _ co  o  o CD  cu  c  CO  o  O ^_ o O CD  O  ll  ?  CL  S 6  o o  1 R! c c  • II  cn cn X  «  |  1 ^ O -8 °  5  ®  o  2  c  CD  ca  11  0>  OJ  o o  w>  CO  WJ to  •it  < If  2 cc < .g\£ x  Ul  CO TJ C CO  O <  I*  o  I*  3 <J B3  3  e c  o c o  o c o at  CO CL  CO CL  •£5  E  E o o  E o o  (ZS/ui) u o j i e j a i a o o v  (zs/ui) u o g i e i a i c o o v  176'  Appendix!  E 3  O  o CD  n  o CO  10  CO CD  CO CO CD  CO  o co (ui) utctea  c E  E  3  o  £  »-  ° II II c  O  S  B- -  Cu  ;:. m  _ 2 ° 0  a  s l  £  CQ  o O ^  S .  I-  •-  *-  CO  0) CD  o ^  n S  Zi  CD O O  D)  S  < X  < fjj CO |g  co •o c  1 !s  O < _l Ii.  Lu E i  i  8# £ ui  CO  «  Q. E o o  p  »  ID c" »  ft S- a  E o  CO  o |  O  CD  < I s ui a . E  S S " E  IS  c  c5 ~  j j »~ -3  O <  o s i  P  c  f 8 CD m S  0)  I  5  S |i .2  .sM=  O  o  i  -  o  Eg  6 i  O CO  3  CO  O)  <  C o o  (zs/ui) u o i i e i a i s s o v  (ZS/Ul) U O J I B J 3 I 3 3 3 V  178?"  Appendix  I  co  (ui) u-dsa  I  E O 0 •I-  °  O n e CD at  « ^  tt  Cu  ° tt °  ° - -S g O g-' H E =J *5 .-9 9  to  - 2 §  O fa  §  FBI  x | | CUV)  1  ? « <D W  O  C  ct.  CO  _  a s 2 a » • If g « | O O  -  0)  - rtj § Q  £ f  CO tt cc TJ f § C CO E £  II  L.  O  < TO * E  g  « »s  o  <.  O  3 8  •§ o E o f  -J IL  Q.  " I <I2 < •X•• 9, s>  •a •§ o C  3 -S  4= w  < o> cc 35  (f n -S ^-  !« s tt  o  III  ° s  " t r u e  5. a  t l  3  "  E  3  o  5  § s i at ^ co *- X • O 100  > o S  CL E o  C u, E o  <t  o  o  (zs/ui) uogiejatsoov  (zs/ui) u o i i e j o i a o o v  17m  Appendix I  to r-  (iu) gjdaa  c E  C  E  3  o  O  fi  ° *«  o  o ° £  CL -S £  O  i  i  X  c  §  c CO  O " 'a O "»  8 si  C  < I S ui § t  ^ s?  < s.? x CO  t | B. S •D § °  CO  !s  o  O l s  <  U. E S c  0  Q-  ^  UJ  .2  a E  ~  us S £ So c o  S g>  ai 8 I a> § I  « ?!  O  * ^  5 | §  C a. S ° « 8 • - c o ,  ?  g- ~-  o o  o*S  s  .c; » F  0  S 5- CQ TO O »- 2 » ° 8S  i  g  | -S>  <2J *o • CZ V) %  1  u.  E o  E o  o  5  o  CM  >(jS/Ul)  U0|IBJ3|300V  (zs/iu) uoitejaiaoav  480  !  Append/*/  LO  (ui) wdaa  c  E o  « ; O  a> CQ «•—  1  2 ^  0  5  II -p  g-E  •»  II  • -.  O <" * > . -S o CO  o  Si  Xo  o -8 S. 0)  s_.5  I  a> o o <  <  9  ^ e  .9  E  1=  LU  < X S CA fc u c ca  < X  a  co l l  to  CO  O «> a <  O <  3 Ct  H  LU  g  _l LU  o o tn co Q.  E o o  j  5a ui LU  |^  ™ a: u CO (o  «S 5  E o  *  o  (zs/ui) uooejaiaosv  (ZS/Ul)  U0!)BJ3|300V  181  Appendix I  E O  o  _  o  LO  o  0)  o  0.  CO CO CD  w  o co (ui)  c E -o  E  -  Mldaa  *1 o ^ °>  £  «  CD CO  Z.  ca  Q.  cn £ S •£  H  | cs  w  0  O  ^  tco  o  V>  X  x c o  c  o  i s  co  *k_ S -g  < UJ  < Ul  <  a> 8 6 U O  s  ca O <  o  tn c" S 9 €  1 0  1 t  o  < s.g>  a  to  JJQ C  <  IS  .91  6  9.1  •° 12  s - §>  tn *  CD  If  o q,  Si S t  o 8 .a§!«"  co o CL  E o  CO  a.  p  E o  CO  6  5  o  o  0  0  (zs/ui)  9  T T  uojiejaiaoov  (ZS/Ul) U 0 ! l 8 J 3 | 3 O 3 V  163?  Appendix I  Appendix I  TD O  s§ oi O  c  O  o  c  o o o to 2 § o o> 'c to3 o >» f£  o >. E s E"  55  E  E  o  S  Itl  5  <>•  !E  O  'o o CD  c <D  E  » s  >CD  s-l  m  cn CO  a> < cn - i ra  CO  E o  f  y^ <  I  I  o  .9 o o  <J  —I Li-  > T3  m  X "> CO To  f to  i  CU C CC CO V) & o to  §  m  CO  <  <U  - s  0)  c  c  CO "D <1) 2 cu Ol oo 2c  < I  <n a> cn  CO X  o  CD  O  O  CO « X c a g a.  E 55 8o  CD Ir  H-  O  CL >. i_  O  a.  TT  Xi c to  x:  ra  >. "o o  CL  E o o  >  O  <  Uc CD  1 CD  u-  C  oU) CO CL  E o O E o O  (iu) luduiaaeidsja  (S/Ul) /UpoiOA  (%) "IBJ'S  185  Appendix I  -186  Appendix I  . 187  

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