Energy-Based Evaluation and Remediation

Energy-Based Evaluation and Remediation of Liquefiable Soils

Russell A. Green

Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Civil Engineering

Advisory Committee: Dr. James K. Mitchell, Chair Dr. M. Gutierrez Dr. T. Kuppusamy Dr. J.R. Martin Dr. S.F. Obermeier Dr. M.P. Singh

August 6, 2001 Blacksburg, Virginia

Keywords: Liquefaction, remediation, ground improvement, densification, energy, earthquake

Energy-Based Evaluation and Remediation of Liquefiable Soils Russell A. Green Dr. James K. Mitchell, Chair Charles E. Via, Jr. Department of Civil and Environmental Engineering ABSTRACT Remedial ground densification is commonly used to reduce the liquefaction susceptibility of loose, saturated sand deposits, wherein controlled liquefaction is typically induced as the first step in the densification process. Assuming that the extent of induced liquefaction is approximately equal to the extent of ground densification, the purpose of this research is to assess the feasibility of using earthquake liquefaction data in remedial ground densification design via energy-based concepts. The energy dissipated by frictional mechanisms during the relative movement of sand grains is hypothesized to be directly related to the ability of a soil to resist liquefaction (i.e., Capacity). This hypothesis is supported by energy-based pore pressure generation models, which functionally relate dissipated energy to residual excess pore pressures. Assuming a linearized hysteretic model, a “simplified” expression is derived for computing the energy dissipated in the soil during an earthquake (i.e., Demand). Using this expression, the cumulative energy dissipated per unit volume of soil and normalized by the initial mean effective confining stress (i.e., normalized energy demand: NED) is calculated for 126 earthquake case histories for which the occurrence or non-occurrence of liquefaction is known. By plotting the computed NED values as a function of their corresponding SPT penetration resistance, a correlation between the normalized energy capacity of the soil (NEC) and SPT penetration resistance is established by the boundary giving a reasonable separation of the liquefaction / no liquefaction data points. NEC is the cumulative energy dissipated per unit volume of soil up to initial liquefaction, normalized by the initial mean effective confining stress, and the NEC correlation with SPT penetration resistance is referred to as the Capacity curve. Because the motions induced during earthquake shaking and remedial ground densification significantly differ in amplitude, duration, and frequency content, the

dependency of the derived Capacity curve on the nature of the loading needs to be established. Towards this end, the calibration parameters for energy-based pore pressure generation models are examined for their dependence on the amplitude of the applied loading. The premise being that if the relationship between dissipated energy and pore pressure generation is independent of the amplitude of loading, then the energy required to generate excess pore pressures equal to the initial effective confining stress should also be independent of the load amplitude. However, no conclusive statement could be made from results of this review. Next, first order numerical models are developed for computing the spatial distribution of the energy dissipated in the soil during treatment using the vibratory probe method, deep dynamic compaction, and explosive compaction. In conjunction with the earthquake-derived Capacity curves, the models are used to predict the spatial extent of induced liquefaction during soil treatment and compared with the predicted spatial extent of improvement using empirical expressions and guidelines. Although the proposed numerical models require further validation, the predicted extent of liquefaction and improvement are in very good agreement, thus giving credence to the feasibility of using the Capacity curve for remedial ground densification design. Although further work is required to develop energy-based remedial densification design procedures, the potential benefits of such procedures are as follows. By using the Capacity curve, the minimum dissipated energy required for successful treatment of the soil can be determined. Because there are physical limits on the magnitude of the energy that can be imparted by a given technique, such an approach may lead to improved feasibility assessments and initial designs of the densification programs.

Dedication

To the loving memory of my father. To my mother, who had the arduous task of raising six incorrigible children by herself after the untimely death of my father. To my wife, who has supported me in all my endeavors. To my son, who makes life fun; see Figure D-1.

Figure D-1. Four-year-old Owen William Green watching cartoons in Daddy's office while waiting for Daddy to arrive home late from school.

v

Acknowledgements First and foremost, praise and thanks goes to my savior Jesus Christ for the many blessing undeservingly bestowed upon me. More than any others, my interaction with five individuals has shaped the way I think as an engineer: 

Professor James K. Mitchell, University Distinguished Professor, Emeritus, Virginia Tech.



Mr. Asadour H. Hadjian, Senior Technical Staff, US Defense Nuclear Facilities Safety Board.



Professor William J. Hall, Emeritus Professor of Civil Engineering, University of Illinois, Urbana-Champaign.



Dr. William F. Marcuson, III, Emeritus Director, Geotechnical Laboratory, US Army Waterways Experiment Station.



Dr. Robert Ebeling, Information Technology Laboratory, US Army Waterways Experiment Station.

I feel extremely privileged and grateful for their tutelage, mentoring, and friendship. If anything contained in this thesis is deemed meritorious, it most assuredly can be attributed to the lessons of these individuals; I alone take credit for the less than meritorious content. Particularly, my sincere gratitude goes to my advisor, Dr. James K. Mitchell, for his guidance, encouragement, and tremendous patience during the course of this research, without which the completion of this work would not have been possible. I am proud to be number seventy two. The author also extends his gratitude to the other members of his advisory committee who provided advice and guidance: Professors M. Gutierrez, T. Kuppusamy, J.R. Martin, S.F. Obermeier, and M.P. Singh. Although not committee members, the dedication of Professors J.M. Duncan and G.M. Filz to the students and the Department of Civil Engineering is gratefully acknowledged.

vi

Professors Ricardo Dobry and Thomas O’Rourke acted as the MCEER reviewers for geotechnical research and provided valuable comments throughout the commencement of this research. Drs. Carmine P. Polito, Joseph P. Koester, and John A. Bonita generously provided the laboratory data used in this research, while Carmine patiently taught me the finer points of laboratory testing. The following individuals took time to either read and/or discuss portions of my thesis, thus improving the final product: A. Hadjian, R. Ebeling, C.P. Polito, R.O. Davis, P. Byrne, I.M. Idriss, E. Kavazanjian, M. Chapman, W.S. Degen, R.E. Kayen, P.W. Mayne, W.A. Narin van Court, C. Soydemir, A. Gwal, G.R. Martin, W. Joyner, and S. Dickenson. Through the wonders of the internet and e-mail, I was able to correspond with numerous individuals from around the world who gracious provided me with references and/or information used in this thesis: J.L. Figueroa, E.V. Leyendecker, R.D. Borcherdt, J.M. Hagerty, A.E. Holeyman, F. Ostadan, B. Muhunthan, R.J. Fragaszy, F. Fernandez, M.D. Trifunac, J.M. Roesset, J. Kuwano, T. Kagawa, and K.T. Law. My experience at Virginia Tech was greatly enhanced by the interaction with other students. Youngjin Park’s integrity and work ethic was both inspiring and humbling. The numerous discussions with Wanda Cameron, Carmine Polito, Miguel Pando, and Guney Olgun greatly increased my understanding of geotechnical engineering. The friendship of Trish Gallagher, Jeremy Britton, Jeff McGregor, Aaron Muck, Chris and Diane Baxter, and many others in geotech group made the stressful times more bearable. Youngjin Park and William White got my research back on track after being plagued with computer problems. Additionally, William White painstakingly proofread large portions of this thesis. Finally, the love and support of my wife, Chris, and son, Owen, turned any fears of failure into desires to succeed. Thank you.

vii

Grant Information Financial support for this research was provided by the Multi-Disciplinary Center for Earthquake Engineering Research (MCEER), SUNY Buffalo, NY: Geotechnical Rehabilitation: Site and Foundation Remediation #MCEER-01-2033. The author received additional support from the Via Fellowship, Department of Civil and Environmental Engineering, Virginia Tech.

Table of Contents Abstract...................................................................................................................... ii Grant Information...................................................................................................... iv Dedication.................................................................................................................. v Acknowledgements.................................................................................................... vi Table of Contents....................................................................................................... viii List of Tables............................................................................................................. xiv List of Figures............................................................................................................ xvi Chapter 1. Introduction............................................................................................. 1 1.1 Objective of Research................................................................................... 1 1.2 Background and Approach to Solution......................................................... 1 1.3 Organization of the Thesis............................................................................ 5 Chapter 2. Review of Liquefaction Evaluation Procedures..................................... 9 2.1 Introduction................................................................................................... 9 2.2 Overview of the Procedures.......................................................................... 10 2.2.1 Stress-based procedure......................................................................... 10 2.2.2 Strain-based procedure......................................................................... 17 2.2.3 Energy-based procedures..................................................................... 23 2.2.3.1 Procedures developed from earthquake case histories............... 26 2.2.3.1.1 Gutenberg-Richter approaches.......................................... 26 2.2.3.1.1a Davis and Berrill (1982)............................................ 26 2.2.3.1.1b Berrill and Davis (1985)............................................ 28 2.2.3.1.1c Law, Cao, and He (1990)........................................... 31 2.2.3.1.1d Trifunac (1995).......................................................... 33 2.2.3.1.2 Arias intensity approaches................................................. 40 2.2.3.1.2a Kayen and Mitchell (1997)........................................ 40 2.2.3.1.2b Running (1996).......................................................... 45 2.2.3.2 Procedures developed from laboratory data............................... 47 2.2.3.2.1 Alkhatib (1994)................................................................. 47 viii

2.2.3.2.2 Liang (1995)...................................................................... 49 2.2.3.3 Other Approaches....................................................................... 50 2.2.3.3.1 Mostaghel and Habibaghi (1978, 1979)............................ 50 2.2.3.3.2 Moroto and Tanoue (1989)................................................ 50 2.2.3.3.3 Ostadan, Deng, and Arango (1996, 1998)......................... 50 2.3 Overview of the Parameter Study................................................................. 50 2.3.1 Objective.............................................................................................. 50 2.3.2 Discussion of expected trends.............................................................. 52 2.3.3 Results of parameter study and summary of observed trends.............. 56 2.3.4 Commentary on procedures................................................................. 67 2.3.4.1 Gutenberg-Richter approaches.................................................... 67 2.3.4.2 Arias intensity approaches.......................................................... 68 2.4 Conclusions................................................................................................... 75 Appendix 2a: Normalization of measured SPT N-values.................................. 76 Chapter 3. Mechanism and Mathematical Representation of Energy Dissipation................................................... 78 3.1 Introduction................................................................................................... 78 3.2 Mechanisms of Energy Dissipation in Sands................................................ 78 3.2.1 Frictional Dissipation Mechanism....................................................... 79 3.2.2 Viscous Dissipation Mechanism.......................................................... 85 3.3 Modeling of Energy Dissipation................................................................... 87 3.3.1 Hysteresis loops................................................................................... 87 3.3.2 Equivalent Lineariztion and Damping Ratios...................................... 89 3.3.3 Final Comments on D.......................................................................... 95 3.4 Use of Dissipated Energy to Quantify Capacity........................................... 96 3.5 Computing Dissipated Energy from Laboratory Tests................................. 97 3.5.1 Cyclic Triaxial Test.............................................................................. 100 3.5.2 Cyclic Simple Shear Test..................................................................... 101 3.5.3 Hollow Cylinder Triaxial - Torsional Shear Test................................ 102 3.5.4 Use of the Derived Equations............................................................. 104

ix

Chapter 4. Energy-Based Excess Pore Pressure Generation Models...................... 105 4.1 Introduction.................................................................................................. 105 4.2 Energy-Based Pore Pressure Models from Published Literature................. 106 4.2.1 N-NS Model......................................................................................... 107 4.2.2 MH Model............................................................................................ 108 4.2.3 DB1 Model........................................................................................... 108 4.2.4 BD Model............................................................................................. 109 4.2.5 DB2 Model........................................................................................... 109 4.2.6 YTI Model............................................................................................ 110 4.2.7 LCH Model.......................................................................................... 112 4.2.8 YS Model.............................................................................................. 113 4.2.9 Hsu Model............................................................................................ 114 4.2.10 Liang Model....................................................................................... 115 4.2.11 OAY Model......................................................................................... 116 4.2.12 FSKL Model....................................................................................... 117 4.2.13 WTK Model........................................................................................ 118 4.3 Commentaries on the Models....................................................................... 119 4.4 GMP Model.................................................................................................. 120 4.5 Load Dependency of Calibration Parameters............................................... 126 4.6 Summary....................................................................................................... 128 Chapter 5. Proposed Energy-Based Liquefaction Evaluation Procedure................ 129 5.1 Introduction................................................................................................... 129 5.2 Mathematical Expression for Computing Dissipated Energy....................... 129 5.2.1 Determination of ................................................................................ 131 5.2.2 Determination of G and D.................................................................... 131 5.2.3 Determination of Neqv........................................................................... 134 5.3 Capacity Curve.............................................................................................. 142 5.4 Liquefaction Evaluation................................................................................. 151 5.5 Parameter Study............................................................................................. 151

x

5.6 Comparison of Energy-Based Capacity Curve Derived from Field Data with Laboratory Test Data................................... 156 Appendix 5a: Earthquake time histories used in the parameter study to develop the correlation relating Richter magnitude (M), epicentral distance (ED) from the site to the source, and number of equivalent cycles (Neqv)................................................................................... 178 Appendix 5b: Profiles used in the parametric study to develop the correlation relating Richter magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv)............................................................. 181 Appendix 5c: Energy-based magnitude scaling factors (MSF)........................... 186 Chapter 6: Determination of Seismologic Parameters for Proposed Liquefaction Evaluation Procedure.......................................... 195 6.1 Introduction.................................................................................................... 195 6.2 Procedure for determining epicentral distance (ED) and magnitude (M) of the design earthquake................................................. 196 6.3 Currently Used Procedures for Determining amax.......................................... 200 6.4 Proposed Procedure for Determining amax..................................................... 206 6.4.1 Approximate Spectral Shapes of USGS Rock Outcrop Motions..........210 6.4.2 Response of Uniform Soil Profiles....................................................... 216 6.5 Comparison of proposed procedure with currently used amplification curves.............................................................................. 222 Appendix 6a: Procedure for computing an equivalent uniform profile............... 230 Chapter 7. Remedial Ground Densification Techniques.......................................... 234 7.1 Introduction.................................................................................................... 234 7.2 Vibro-Compaction......................................................................................... 239 7.2.1 Vibrocompaction...................................................................................241 7.2.2 The Vibratory Probe Method................................................................ 249 7.2.2.1 Terra-Probe.................................................................................. 251 7.2.2.2 Double Tube Rod and Rod with Projectives................................252 7.2.2.3 Vibro Wing.................................................................................. 254

xi

7.2.2.4 Franki TriStar Probe (or Y-Probe)............................................... 255 7.2.2.5 Double-Y or Flexi Probes............................................................ 256 7.2.2.6 Frequency of Vibration................................................................ 260 7.3 Deep Dynamic Compaction........................................................................... 267 7.4 Explosive Compaction................................................................................... 279 7.4.1 Mechanisms of Densification............................................................... 282 7.4.2 Procedures for Selecting Charge Size and Spacing.............................. 290 7.4.2.1 Kok Approach.............................................................................. 290 7.4.2.2 Gohl et al. Approach.................................................................... 296 7.4.2.3 Narin van Court and Mitchell Approach..................................... 298 7.5 Comparison of the Mechanical Energy Input by the Densification Techniques......................................................................... 300 7.5.1 Vibro-Compaction................................................................................ 300 7.5.2 Deep Dynamic Compaction.................................................................. 302 7.5.3 Explosive Compaction.......................................................................... 302 7.5.4 Discussion............................................................................................. 303 7.6 Summary........................................................................................................ 304 Chapter 8. First Order Numerical Models of Remedial Ground Densification Techniques.......................................................... 306 8.1 Introduction.................................................................................................... 306 8.2 Numerical Model Verification and Validation.............................................. 307 8.3 Numerical Modeling of Vibro-Compaction...................................................308 8.3.1 Proposed Model.................................................................................... 308 8.3.2 Attenuation Relationships..................................................................... 313 8.3.3 Comparison of Proposed First Order Model Predictions with Empirical Guidelines................................................. 324 8.4 Numerical Modeling of Deep Dynamic Compaction.................................... 329 8.4.1 Approximate Input Motion for Deep Dynamic Compaction................ 329 8.4.2 Response of the Soil at Depth for Deep Dynamic Compaction............341 8.5 Numerical Modeling of Explosive Compaction............................................ 351 8.6 Higher Order Numerical Modeling for Remedial Ground Densification...... 361 xii

8.7 Summary and Conclusions............................................................................ 362 Chapter 9. Summary and Conclusions....................................................................... 365 9.1 Restatement of Research Objective.............................................................. 365 9.2 Overview of Research................................................................................... 365 9.3 Summary of Major Findings......................................................................... 366 9.4 Recommendations for Future Work..............................................................369 References.................................................................................................................. 370 Vita............................................................................................................................. 393

xiii

List of Figures Figure D-1. Four-year-old Owen William Green watching cartoons in Daddy's office while waiting for Daddy to arrive home late from school............................................................ v Figure 1-1. An example of a sand boil that occurred during the Loma Prieta earthquake.................................................................. 1 Figure 1-2. Photograph of intact structures overturned as a consequence of soil liquefaction during the 1964 Niigata, Japan earthquake..................................................................... 2 Figure 1-3. Organization of Thesis.......................................................................... 8 Figure 2-1. Stress reduction factor (rd) to account for soil column deformability............................................................................ 12 Figure 2-2. Magnitude scaling factors proposed by various investigators.......................................................................................... 13 Figure 2-3. Cyclic resistance ratio (CRR) curve...................................................... 15 Figure 2-4. Recommended values for overburden pressure correction factor K............................................................................... 16 Figure 2-5. Iterative solution of Equation (2-11) to determine the effective shear-strain () at a given depth in a soil profile.................................................................. 18 Figure 2-6. Plots of the Ishibashi and Zhang (1993) shear modulus degradation curves for various initial mean effective confining stresses......................................................... 19 Figure 2-7a. Laboratory test results conducted on samples of varying relative densities. The shear strain at which excess pore pressures are measured for all the samples is slightly greater than 10-2 percent.............................. 21 Figure 2-7b. Laboratory test results conducted on samples of varying initial effective confining stresses and prepared by various methods. The shear strain at which excess pore pressures are measured for all the samples is slightly greater than 10-2 percent................................... 22

xvi

Figure 2-8a. Commonly used measures of site-to-source distance........................... 24 Figure 2-8b. Comparison of the central tendencies of various earthquake magnitude scales, with moment magnitude (MW) currently being the most accepted scale. Although in the past the term “Richter magnitude” was used synonymously with ML, it currently is used to refer to a conglomeration of magnitude scales (e.g., Krinitzski et al. 1993)................................................................................................ 25 Figure 2-9. Liquefaction chart proposed by Davis and Berrill (1982).................... 27 Figure 2-10. Material attenuation factor (A) as a function of the dimensionless distance (a).................................................................... 29 Figure 2-11. Liquefaction chart proposed by Berrill and Davis (1985).................... 30 Figure 2-12. Liquefaction chart proposed by Law, Cao, and He (1990)................... 32 Figure 2-13. Comparison of the boundaries proposed by Davis and Berrill (1982) and Trifunac No.1 to separate the data points representing liquefaction and no-liquefaction............................ 35 Figure 2-14. Graphical illustration of the site-to-source distance r*......................... 41 Figure 2-15. Results of a statistical analysis of the variation of rb as a function of depth in a soil profile...................................................42 Figure 2-16. Liquefaction curve proposed by Kayen and Mitchell (1997).................................................................................................... 44 Figure 2-17. Liquefaction curve proposed by Running (1996)................................. 46 Figure 2-18. Correlation of the dimensionless parameter Normalized Maximum Energy (NME) and maximum acceleration (amax)..................................................................................................... 47 Figure 2-19. Correlation between relative density (Dr) and energy ratio (ER)............................................................................................... 48 Figure 2-20. FS-profiles for the stress-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft............................................... 54

xvii

Figure 2-21. FS-profiles for the strain-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft............................................... 55 Figure 2-22. FS-profiles for Davis and Berrill (1982): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft............................................... 57 Figure 2-23. FS-profiles for Berrill and Davis (1985): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft............................................... 58 Figure 2-24. FS-profiles for Law, Cao, and He (1990): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft............................................... 59 Figure 2-25. FS-profiles for Trifunac No.1: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft....................................60 Figure 2-26. FS-profiles for Trifunac No.3: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft....................................61 Figure 2-27. FS-profiles for Trifunac No.4: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft....................................62 Figure 2-28. FS-profiles for Trifunac No.5: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft....................................63 Figure 2-29. FS-profiles for Kayen and Mitchell (1997): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft..................................................................................................64

xviii

Figure 2-30. FS-profiles for Running (1996): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft....................................65 Figure 2-31. FS-profiles for Alkhatib (1994): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft....................................66 Figure 2-32. a) Soil profile having a liquefiable layer lying below a stiff desiccated crust. b) Duplication of in-situ stresses using simple shear device on a soil sample from the liquefiable layer................................................ 69 Figure 2-33. Comparison of expressions for duration of strong ground motion as a function of magnitude........................................... 72 Figure 2-34. Comparison of the “depth reduction factors” rb for the Arias Intensity approach and (rd)2 from the stressbased approach...................................................................................... 73 Figure 2-35. Comparison of Arias intensity liquefaction curve and CRR................................................................................................ 74 Figure 3-1. Conceptualization of the energy imparted to the soil by an earthquake and the portions dissipated by friction and viscous drag.................................................................. 79 Figure 3-2. Contact forces and stresses between two equal sized spheres of radius R....................................................................... 81 Figure 3-3. Radius of the contact area (a) and the normal stress (c) across the contact area of the two spheres...................................................................................................82 Figure 3-4. Relative slippage of the spheres: no slippage at T = 0; gross slippage at T = fN. At intermediate values 0 < T < fN, there is an annulus of slippage surrounding a zone of no slippage.........................................................83 Figure 3-5. Photograph of an actual sphere, which in contact with another sphere was subjected to an oscillating tangential force 0 < T < fN. Wear marks formed by the annulus of slippage between the spheres can be clearly identified........................................................................ 84

xix

Figure 3-6. Typical acceleration time history with the arrival of the large amplitude shear waves occurring early in the record.......................................................................................... 84 Figure 3-7. Comparison of the variation of logarithmic decrement with amplitude for dry and saturated Ottawa sand in torsional oscillation...................................................... 86 Figure 3-8. Comparison of the variation of logarithmic decrement with amplitude for dry and saturated glass beads in torsional oscillation........................................................ 87 Figure 3-9. Hysteresis loop resulting from: a) the application and removal of a tangential force T. b) the application and removal of a shear stress ........................................... 88 Figure 3-10. Hysteresis loops from a stress-controlled cyclic triaxial test............................................................................................. 89 Figure 3-11. Graphical definitions of stored energy and total dissipated energy................................................................................... 89 Figure 3-12. Rheological models and corresponding hysteresis loops for hysteretic and equivalent linear materials............................. 91 Figure 3-13. Quantities used in defining damping ratio (D)..................................... 92 Figure 3-14. Development of shear modulus degradation curves............................ 93 Figure 3-15. Plots of the Ishibashi and Zhang (1993) shear modulus degradation curves for various initial mean effective confining stresses......................................................... 95 Figure 3-16. Stresses acting on a differential element a) tensorsuffix notation b) engineering notation................................................. 98 Figure 3-17. The dissipated energy per unit volume for a soil sample in cyclic triaxial loading is defined as the area bound by the deviator stress - axial strain hysteresis loops, Equation (3-18)......................................................... 101 Figure 3-18. Stress conditions for a hollow cylinder triaxialtorsional shear test................................................................................ 102

xx

Figure 4-1. Procedure for determining calibration parameters for YTI Model from laboratory test data............................................... 111 Figure 4-2. Graphic illustration of how PEC is determined from cyclic test data. The data shown in this figure is from a cyclic triaxial test conducted on Yatesville clean sand.............................................................................................. 121 Figure 4-3a. Comparison of measured and computed residual excess pore pressures in various silt-sand mixtures having varying densities. Symbols are values computed using the GMP Model and the lines are the measured values. All samples were run in stress controlled cyclic triaxial tests............................................................... 122 Figure 4-3b. Comparison of measured and computed residual excess pore pressures in samples tested in various configurations. Symbols are values computed using the GMP Model and the lines are the measured values.................................................................................................... 123 Figure 4-4. Correlations among CSR, PEC, and Dr for Monterey silt-sand mixtures (FC  30%).............................................................. 125 Figure 4-5. Correlations among CSR, PEC, and Dr for Yatesville silt-sand mixtures (FC  30%)............................................ 125 Figure 4-6a. Alternate plot of the correlation relating PEC, Dr, and CSR for Yatesville fine grained sand-silt mixtures.................................................................................................127 Figure 4-6b. Alternate plot of the correlation relating PEC, Dr, and CSR for Monterey medium grained sandsilt mixtures........................................................................................... 128 Figure 5-1. Quantities used in defining damping ratio (D)..................................... 130 Figure 5-2. Iterative solution of Equation (5-6) to determine the effective shear-strain () at a given depth in a soil profile.......................................................................................... 132 Figure 5-3. The determination of shear modulus and damping ratios from the respective degradation curves...................................... 133 Figure 5-4. Graphical representation of the dissipated energy per unit volume for an equivalent cycle of loading.............................. 133

xxi

Figure 5-5. A correlation relating earthquake magnitude and equivalent number of cycles. The data points labeled S-I and A-1 are assumed to be from a different study than the rest of the data. Also shown in this figure are several M-Neqv pairs that are commonly presented in tabular form in published literature................................................................................135 Figure 5-6. Illustration of the procedure used to develop the Neqv correlation. In this procedure, the dissipated in a layer of soil, as computed from a site response analysis, is equated to the energy dissipated in an equivalent cycle of loading multiplied by Neqv.................................................................................. 136 Figure 5-7. Shear-stress shear-strain hysteresis loops, output by SHAKEVT, for a given depth in a soil profile subjected to an earthquake acceleration time history........................................................................................... 137 Figure 5-8. A comparison of the NED values computed by SHAKEVT and Equation (5-7) multiplied by Neqv. Figures a) and b) correspond to the two horizontal components of motion at the same site. The sum of the Neqv for each component represents the Neqv for total motions experienced at the site............................................... 138 Figure 5-9. A fit surface to the computed Neqv for one of the profiles shown in Appendix 5b. The black dots shown in this plot are the computed Neqv values................................... 139 Figure 5-10. Contour plot of the average fit surface for each of the 12 profiles used in the parameter study. The contours are of constant Neqv as a function of epicentral distance and Richter magnitude. The near field - far field boundary superimposed on the contour plot is that proposed by Krinitzsky et al. (1993); note this boundary is only given up to M7.5.................................................................................................. 141 Figure 5-11a.Energy-based Capacity curve developed from 126 liquefaction field case histories............................................................. 144 Figure 5-11b.Energy-based Capacity curve developed from 126 liquefaction field case histories............................................................. 145

xxii

Figure 5-12. FS-profiles computed using the proposed energybased procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft............................................................................ 153 Figure 5-13. FS-profiles for the stress-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft............................................... 154 Figure 5-14. FS-profiles for the strain-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at a approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft..................................................................................................155 Figure 5-15. Stress-strain hysteresis loops for a) total stress numerical analysis, b) stress controlled cyclic triaxial test and c) strain controlled cyclic triaxial test.. ....................................................................................................... 156 Figure 5-16. Two idealized hysteresis loops for a stresscontrolled, undrained cyclic test, where loop 1 occurs earlier in the test than loop 2..................................................... 158 Figure 5-17. The ratio of the energies dissipated during two cycles of loading in an undrained stress controlled cyclic test.............................................................................................. 160 Figure 5-18. Normalized dissipated energies computed from cyclic triaxial test data using Equations (5-15a) and (5-15b). The data shown in this figure is that listed as m0e76c28 in Table 5-2a......................................................... 163 Figure 5-19. NEC values computed from stress controlled cyclic triaxial tests for samples having silt contents below the limiting value. Also shown is the capacity curve derived from field case histories. The negative relative densities were generated by moist tamping the laboratory samples, thus creating specimens having larger void ratios than the maximum predicted by the index tests using dry soil......................................................................................... 166

xxiii

Figure 5-20. NEC values computed from stress controlled cyclic triaxial tests for samples having silt contents above the limiting value. Also shown is the capacity curve derived from field case histories................................................................................................. 167 Figure 5b-1. Profiles 1, 5, and 9 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60..................................................................................... 182 Figure 5b-2. Profiles 2, 6, and 10 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60..................................................................................... 183 Figure 5b-3. Profiles 3, 7, and 11 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60..................................................................................... 184 Figure 5b-4. Profiles 4, 8, and 12 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60................................................................... 185 Figure 5c-1. Capacity curve developed from cyclic triaxial laboratory tests. Each point on the plot represents a separate test. Each test is conducted on similar samples subjected to varying amplitude CSR. For isotropically consolidated cyclic triaxial tests, the CSR = d/2’o, where d is the applied deviator stress, and ’o is the initial effective confining stress..................................................................................................... 186 Figure 5c-2. Illustration of how MSF were developed from laboratory liquefaction curves...............................................................187

xxiv

Figure 5c-3. Illustration of liquefaction curve plotted on log-log scales........................................................................................ 188 Figure 5c-4. Contour plot of Neqv as a function of epicentral distance and Richter magnitude............................................................ 190 Figure 5c-5. A comparison of Ambraseys’ MSF curve and the set of curves developed in this study.............................................. 192 Figure 5c-6. Liquefaction data computed using Ambraseys’ MSF curve............................................................................................. 193 Figure 5c-7. Liquefaction data computed using the set of MSF curves developed in this study.............................................................. 194 Figure 6-1. Example de-aggregation matrix for a 2500 year return period event obtained from the USGS web site......................................................................................................... 197 Figure 6-2a. Early curves quantifying the amplification ratios as function of site conditions and amplitude of rock acceleration................................................................................... 201 Figure 6-2b. Curve quantifying the amplification ratios for soft soil sites................................................................................................. 201 Figure 6-3. Comparison of average 5% damped response spectral shapes computed from strong-motion data recorded at rock sites in CEUS (solid line) and WUS (dashed line). CEUS average shape is from recordings of the mb = 6.4 Nahanni aftershock (Canada). The WUS average shape is from recordings for the San Fernando ML = 6.4 and Imperial Valley ML = 6.6 events (California)....................................... 202 Figure 6-4. Example of the seismic data retrieved from the USGS web site...................................................................................... 205 Figure 6-5. Curves quantifying the amplification ratios used in the NEHRP provisions...................................................................... 206 Figure 6-6. Illustration of proposed approach for determining amax........................................................................................................ 209 Figure 6-7. Characteristics and analytical geometry of the parabolic representation of UHS........................................................... 211

xxv

Figure 6-8a. Parabolic approximation of uniform hazard spectrum (UHS) for Salt Lake City, UT for seismic hazard having a 500 year return period (i.e., 10% probability of exceedance in 50 years). The spectrum is plotted on three scales (i.e., log-log, semi-log, and arithmetic) to facilitate the comparison of the actual spectral acceleration values and the parabolic approximation. The curves shown in the above plots were computed using Equation (6-8), and the black dots are the pga, Ss, and S1 values obtained from the USGS web site. The open circles are additional spectral acceleration values obtained from the USGS....................................................................... 213 Figure 6-8b. Parabolic approximation of uniform hazard spectrum (UHS) for Memphis, TN for seismic hazard having a 2500 year return period (i.e., 2% probability of exceedance in 50 years). The spectrum is plotted on three scales (i.e., log-log, semi-log, and arithmetic) to facilitate the comparison of the actual spectral acceleration values and the parabolic approximation. The curves shown in the above plots were computed using Equation (6-8), and the black dots are the pga, Ss, and S1 values obtained from the USGS web site. The open circles are additional spectral acceleration values obtained from the USGS....................................... 213 Figure 6-9. Response ratios and parabolic approximations for 4 different impedance ratios (IR). Each point in the above plot represents a separate site response analysis.................................................................................................. 219 Figure 6-10. amax spectra for San Francisco, CA, corresponding to IR = 2, 5, 10, 15, and 20. The largest amplification occurs for sites having a fundamental period close to Tmax of the rock outcrop motion (i.e., T  0.222sec)............................. 222 Figure 6-11a.Estimated characteristic periods for rock motions as a function of distance from the causative fault................................. 223

xxvi

Figure 6-11b.Comparison of earthquake period – distance relations for M7 events. The correlations of Seed et al. (1969) and Idriss (1991) define the characteristic period of the ground motion as the period corresponding to the largest spectral acceleration value for the actual earthquake response spectrum. The correlation proposed by Rathje et al. (1998) defines the characteristic period as the period corresponding to the largest spectral acceleration for the smoothed response spectrum of the ground motion................................................................................................... 223 Figure 6-12a.Skyline time history recorded on a rock outcrop at approximately 70km from the epicenter of the Loma Prieta earthquake................................................................................... 225 Figure 6-12b.Response spectrum of the Skyline time history and its parabolic approximation. Also shown is the 500 year UHS for San Francisco, scaled to the pga of the Skyline motion............................................................... 225 Figure 6-13. Procedure for approximation of the amax spectrum for the Skyline motion (Tpga = 0.133sec) by scaling the amax spectrum computed for a parabolic response spectrum having a similar shape to the Skyline spectrum with Tpga = 0.044sec.............................................................. 227 Figure 6-14. amax spectra for representative long and short epicentral distance motions for the 1989 Loma Prieta earthquake................................................................................... 228 Figure 6-15. Comparison of amplification curve and amax values computed using the proposed procedure................................... 229 Figure 6a-1. Chart used in determining the fundamental period of soil profile......................................................................................... 232 Figure 7-1. General trends of how the physical processes for reducing liquefaction susceptibility affect the stress-based liquefaction curve............................................................. 236 Figure 7-2. Applicable grain-size ranges for liquefiable soil improvement methods........................................................................... 237

xxvii

Figure 7-3. Commonly used grid patterns for compaction points for implementing vibro-compaction techniques to large areas...................................................................................................... 240 Figure 7-4. A 1937 photograph of an early vibrocompaction system or “Pfeilerruettler” (vibrating pile)........................................... 241 Figure 7-5. Schematic diagram of the equipment and process of soil densification using vibrocompaction technique........................ 242 Figure 7-6. Densified zones resulting from vibrocompaction................................. 243 Figure 7-7. Common vibrators used in vibrocompaction. The relative sizes of the vibrators are not to scale....................................... 244 Figure 7-8. Conical movement of vibrator unit....................................................... 245 Figure 7-9. Illustration of the horizontal impacting forces and torsional shear induced by the vibroflot................................................247 Figure 7-10. Approximate variation of post-compaction relative density and tributary area per compaction point................................... 248 Figure 7-11. Variation of post-treated CPT tip resistance with distance from compaction point for two different vibrators................................................................................................ 249 Figure 7-12. Conceptual illustration of the orientation of the exciting force of the vibratory probe.................................................... 250 Figure 7-13. Conceptualization of the vibratory probe-soil interaction............................................................................................. 251 Figure 7-14. Terra-Probe, not to scale....................................................................... 252 Figure 7-15. a) Double rod probe, and b) Probe with projectives............................. 253 Figure 7-16. Penetration resistance of pre- and post-treated soil as a function of fines content................................................................ 254 Figure 7-17. Swedish vibro-wing.............................................................................. 255 Figure 7-18. Franki TriStar probe or Y-probe........................................................... 256 Figure 7-19. K-factor as a function of pile impedance.............................................. 258

xxviii

Figure 7-20. Double-Y or Flexi probe. Not to scale; no dimensions given.................................................................................. 259 Figure 7-21. Results from study comparing the influence of vibratory frequency on densification.................................................... 261 Figure 7-22. Velocity time history recorded at 3.5m from the vibratory probe during switch on.......................................................... 262 Figure 7-23. Frequency spectra of vertical peak particle velocities for the Y- and double-Y probes............................................................ 263 Figure 7-24. Relationship between optimal vibration frequency and the distance between velocity transducer and probe..................................................................................................... 264 Figure 7-25. Compactability of soils for the vibratory probe technique, based on the electric cone penetration test (CPT) with friction sleeve measurements...................................... 265 Figure 7-26. Correlation relating initial penetration resistance, layer thickness, and the required vertical peak ground acceleration to densify the soil................................................. 266 Figure 7-27. Correlation relating the peak ground acceleration, initial penetration resistance, and average induced settlement of treated layer..................................................................... 267 Figure 7-28. Conceptual illustration of deep dynamic compaction........................... 268 Figure 7-29. Grouping of soils for dynamic compaction. Zone 1 soils are most suitable for deep dynamic compaction.......................... 269 Figure 7-30. Stages of soil conditions as a result of successive passes of deep dynamic compaction..................................................... 270 Figure 7-31. The effects of the high and low energy phases of deep dynamic compaction and the effects of aging.............................. 272 Figure 7-32. Trends between apparent maximum depth of influence and WH................................................................................. 273 Figure 7-33. Lateral movements 3m from the centerlines of the drop points...................................................................................... 275 Figure 7-34. Relationship between tamper mass and drop height............................. 277

xxix

Figure 7-35. Partition of energy among the three waves for vertical vibration of a disk on an elastic half-space.............................. 279 Figure 7-36. Schematic illustration of the times histories from the cyclic expansion and contraction of the gas bubble: a) radius of gas bubble, b) gas pressure, c) displacement of the bubble wall, and d) velocity of the bubble wall............................................................... 284 Figure 7-37. Shear strain resulting from the conical shaped wave front............................................................................................. 285 Figure 7-38. Proposed mechanism of blast induced liquefaction resulting from the differences in the bulk compressibilities of the pore water and soil skeleton and the plastic volume change of the soil skeleton.......................................................................................... 287 Figure 7-39. Illustration of undrained, high pressure, isotropic consolidation test that induces liquefaction. Not to scale. Such tests were performed on sands to verify the proposed blast-induced liquefaction mechanism............................................................................................ 288 Figure 7-40a.Settlements of the ground surface due to blasting at a depth of 7m.................................................................................... 289 Figure 7-40b.Results of pre- and post-blast screw plate load test settlements. Ratios less than 1.0 indicate decreased strength and stiffness, while ratios greater than 1.0 show increases............................................................ 290 Figure 7-41. Functional relationship between pore pressure ratio (usd/’) and HN. The above figure was taken from a hand drawn figure, and the scales are not exact................................. 291 Figure 7-42. Optimal R-W combination required for densification is shown as the dark shaded zone......................................................... 293 Figure 7-43. Correlation relating HN and surface settlement expressed in terms of vertical strain......................................................293 Figure 7-44. Comparison of the vertical strain corresponding to usd/’ = 0.8 for blast data to the vertical strain from earthquake data.............................................................................295

xxx

Figure 7-45. Comparison of observed strains to maximum values predicted by Equation (7-15d) and from Figure 7-4............................. 298 Figure 7-46. Comparison of measured and predicted final normalized tip resistances..................................................................... 299 Figure 7-47. Current log recorded during vibrocompaction...................................... 301 Figure 8-1. First order model of the vibratory probe method.................................. 310 Figure 8-2. Iterative solution of Equation (8-3) to determine the shear-strain () at a given distance from the vibratory probe...................................................................................... 311 Figure 8-3. The determination of shear modulus and damping ratios from the respective degradation curves...................................... 312 Figure 8-4. Comparison of attenuation relations given by Equation (8-5) (i.e., geometrical spreading only) and given by Equation (8-6) (i.e., combined geometrical spreading and material damping). The data points shown in the plot are peak particle velocities resulting from the dropping of a 6ton tamper from 27m................................................................................... 318 Figure 8-5. Attenuation of peak ground acceleration with distance from the vertically vibrating probe excited at 16 and 25hz........................................................................... 319 Figure 8-6. Comparison of attenuation expressions to measured field data for three excitation frequencies............................ 322 Figure 8-7. Measured vibration levels at a depth of 14ft during the placement of stone columns............................................................ 323 Figure 8-8. For a triangular grid pattern, the distance from a compaction point to the center of the compaction points is approximately 60% of the distance between the compaction points............................................................. 325 Figure 8-9. Simple model predictions using peak particle velocities: a) The maximum extent of liquefaction for 20 and 50hz excitation frequencies, b) Liquefaction fronts for 0.5, 1, 5, 10, and 15sec of vibrations at 20hz.............................................................................. 326

xxxi

Figure 8-10. Simple model predictions using steady state particle velocities: a) The maximum extent of liquefaction for 20 and 50hz excitation frequencies, b) Liquefaction fronts for 1, 5, and 25sec of vibrations at 20hz............................................................. 327 Figure 8-11. Conical surface depression created by vibrocompacting without backfill......................................................... 329 Figure 8-12. Viscously damped, single degree of freedom oscillator used to represent the soil profile........................................... 330 Figure 8-13. The influence on boundary conditions on E’. a) Condition of zero lateral stress, b) Condition of zero lateral strain......................................................... 334 Figure 8-14. Comparison of Ey, Ec, and ELa as a function of Poisson’s ratio....................................................................................... 335 Figure 8-15. An infinitely long rod may be modeled using an appropriately selected damper.............................................................. 336 Figure 8-16. A comparison of computed and recorded acceleration time histories. The recorded time histories are from Mayne and Jones (1983).......................................... 339 Figure 8-17. Predicted and measured peak acceleration values as a function of drop height of the tamper............................................ 340 Figure 8-18. Truncated cone used to model induced stresses from impacting tamper on the surface of a profile. The soil modulus is assumed to vary parabolically from ELa at a depth of 10ft to Ec at a depth of 40ft, as shown in Figure 8-19b. Above a depth of 10ft, the soil modulus is assumed equal to ELa..............................................343 Figure 8-19. Variation of: a) propagation velocity of a compression-extension wave, and b) soil modulus, as a function of depth in the profile...................................................... 345 Figure 8-20. Variation of the constant of proportionality relating deviatoric and axial strains as a function of depth................................ 346 Figure 8-21. General axial stress-strain behavior of soil subjected to impact loading.................................................................. 348

xxxii

Figure 8-22a.Modified shear modulus degradation curve to incorporate strain hardening effects...................................................... 348 Figure 8-22b.Axial stress-strain relation corresponding to modified Ishibashi and Zhang shear modulus degradation curve shown in Figure 8-22a. For comparison, the commonly used hyperbolic stress-strain relation is shown, which does not account for strain hardening effects...................................................... 349 Figure 8-23. Comparison of the predicted depth of induced liquefaction using the numerical model proposed by the author to the predicted depth of improvement using empirical expressions.................................................................. 351 Figure 8-24. Relative contribution of the rr and  as for two distances from the cavity wall...............................................................355 Figure 8-25. Empirical attenuation relations for deep, concentrated charges; refer to Table 8-3............................................... 357 Figure 8-26. Comparison of the predicted radial extent of induced liquefaction using the numerical model proposed by the author to the predicted radial extent of improvement using empirical guidelines proposed by Ivanov (1967) and Kok (1981)......................................... 360 Figure V-1. Just South of the DMZ in Korea, nineteen-year-old Lance Corporal Green stands guard of the General’s latrine, ensuring no enlisted Marines use the facility. (February 1986, 2:00am, -10F)........................................................... 394

xxxiii

List of Tables Table 2a-1. SPT Normalization and Correction Factors.......................................... 77 Table 5-1.

Liquefaction case histories.................................................................... 146

Table 5-2a. Cyclic Triaxial Laboratory Test Data: Monterey Sand – Silt Mixtures............................................................. 169 Table 5-2b. Cyclic Triaxial Laboratory Test Data: Yatesville Sand – Silt Mixtures.............................................................172 Table 6-1.

Amplification Ratios (Fa) as a Function of the Site Class and Mapped Short Period Maximum Considered Earthquake Spectral Acceleration (Ss).................................................. 204

Table 6-2a. Tmax and P values for UHS for WUS cities...........................................215 Table 6-2b. Tmax and P values for UHS for CEUS cities......................................... 215 Table 6-3.

Combinations of P and Tmax selected to represent the UHS for the CEUS and WUS............................................................... 216

Table 6-4a. , , and  coefficients for the Response Ratio for WUS...................................................................................................... 221 Table 6-4b. , , and  coefficients for the Response Ratio for CEUS.................................................................................................... 221 Table 7-1.

Approaches to increasing Capacity and decreasing Demand................................................................................................. 234

Table 7-2.

Specifications of commonly used vibrators.......................................... 245

Table 7-3.

Published values for n........................................................................... 274

Table 7-4.

Applied energy guidelines for densifying various soils. See Figure 7-29 for the definitions of the soil Zones........................... 276

Table 7-5.

Equipment requirements for different size tampers.............................. 278

Table 8-1.

Geometric damping coefficients........................................................... 315

Table 8-2.

Proposed Classification of Earth Materials by Attenuation Coefficient......................................................................... 316

xiv

Table 8-3.

Coefficients for blast attenuation expression........................................ 356

Table 8-4.

Higher order models for remedial ground densification........................362

xv

Chapter 1. Introduction 1.1 Objective of Research The states-of-practice for performing earthquake liquefaction analyses and remedial ground densification designs have evolved relatively independent of each other. This is in spite of the fact that liquefaction is typically induced in saturated sands as part of the remedial ground densification process. The goal of this research is to assess the feasibility of using the vast amount of data collected over the years on earthquake induced liquefaction for remedial ground densification design via energy-based concepts. 1.2 Background and Approach to Solution When subjected to vibrations, loose sand tends to compact and decrease in volume. If the sand is saturated and drainage is prevented, or if the vibrations are rapid enough that drainage is unable to occur, this tendency for volume decrease results in the transfer of the effective overburden stress to the pore water (Martin et al. 1975). Liquefaction, as referred to in this thesis, is condition when the excess pore pressure equals the initial effective overburden stress. This phenomenon has occurred in almost all large earthquakes, with the most commonly observed manifestation being “sand boils” on the ground surface. An example of a sand boil that occurred during the 1989 Loma Prieta earthquake on Bay Farm Island, Alameda, California is shown Figure 1-1. Not Youngjin Park

Figure 1-1. An example of a sand boil that occurred during the Loma Prieta earthquake. (Slide No. LPGeotech69, Loma Prieta Collection, Earthquake Engineering Research Center, University of California, Berkeley).

1

The destructive effects of liquefaction were fully realized in the aftermath of the 1964 Anchorage, Alaska and Niigata, Japan earthquakes. The most dramatic consequence of liquefaction during the Niigata earthquake was the overturning of intact structures, such as shown in the photograph by Professor Joseph Penzien presented in Figure 1-2. More frequent than the bearing capacity failures shown in this figure is damage resulting from the differential settlement of buildings upon the dissipation of excess pore pressures subsequent to liquefaction (e.g., Ishihara and Yoshimine 1992).

Figure 1-2. Photograph of intact structures overturned as a consequence of soil liquefaction during the 1964 Niigata, Japan earthquake. (Slide No. S3160, Steinbrugge Collection, Earthquake Engineering Research Center, University of California, Berkeley). In the 1960’s numerous critical structures, such as nuclear power plants and large earth dams, were being designed and constructed in the United States. The seriousness of potential failures of these facilities due to liquefaction led to massive research efforts aimed at understanding the liquefaction phenomena and developing procedures for evaluating its potential in the field. The most widely used liquefaction evaluation method is the stress-based procedure first proposed by Seed and Idriss (1971) and Whitman (1971). This procedure is mostly empirical and based on laboratory and field observations. It has been continually refined as a result of newer studies and the increase in the number of liquefaction case histories (e.g., NRC 1985, NCEER 1997, Youd et al. 2001).

2

Less commonly used is the strain-based liquefaction evaluation procedure proposed Dobry et al. (1982), which was derived from the mechanics of two interacting idealized sand grains and then generalized for natural soil deposits. This approach is based on the premise that a threshold shear strain needs to be exceeded before the initiation of the transfer of the effective overburden stresses from the soil skeleton to the pore water. One reason for its limited uses is that the procedure only predicts the exceedance of the threshold strain, which is required for liquefaction to occur, but does not imply that liquefaction will occur. Starting in the late 1970’s and early 1980’s, numerous energy-based liquefaction evaluation procedures have been proposed (e.g., Davis and Berrill 1982, Liang 1995, and Kayen and Mitchell 1997). Even though the first energy-based procedures were proposed almost twenty years prior, just recently these procedures are being viewed as “an important new direction in analysis of liquefaction…” (NCEER 1997). The use of energy to quantify the Capacity of systems subjected to earthquake loads is not unique to soils, but also has been applied to structural systems (e.g., Zahrah and Hall 1982, McCabe and Hall 1987, and Uang and Bertero 1988). One impetus for use of energybased methods in earthquake design and analyses is that seismologists have long been quantifying the energy released during earthquakes, i.e., Demand, (e.g., Gutenberg and Richter 1956). Accordingly, quantifying the Capacity of soil and structural systems in terms of energy was a logical step in the evolution of earthquake studies. Various techniques have been developed to mitigate the risk of liquefaction, including remedial ground densification by deep dynamic compaction, explosive compaction, and vibro-compaction. All of these techniques involve imparting energy into the soil to breakdown the structure as a first step in the densification process. When applied to saturated sands, a controlled liquefaction is induced, thus allowing the particles to rearrange to a denser packing concurrent with the dissipation of the excess pore pressures (Mitchell 1981).

3

The evolutionary process in the advancement of remedial ground densification techniques was concisely given by Greenwood (1991), from which the following is paraphrased. Soil improvement techniques have evolved in response to the need to solve a problem. The unknown, and often unsuspected, influences on and by the improvement techniques at the outset are too numerous to allow accurate theoretical prediction of the results in advance of full-scale trials. Consequently, the techniques evolve empirically, often without a full understanding the physics of the process. Equipment, once developed, may be applied in unsuitable circumstances, and this may lead to a reappraisal of its design and so on to further development. Performance data and environmental influences are not always acquired systematically, and data, which are obtained, are often guarded as commercial secrets. Accordingly, a well-developed theoretical understanding of a process rarely precedes its initial practical development. Further improvement of remediation techniques and equipment often results from a more complete understanding of the physical processes, facilitated by knowledge gained from other areas of engineering and construction. In line with Greenwood’s statements, the goal of this thesis is to assess the applicability of knowledge gained from earthquake induced liquefaction and settlement to remedial ground densification design. As stated above, several energy-based liquefaction evaluation procedures have been proposed, many of which use empirical correlations derived from earthquake case histories relating the energy required to induce liquefaction to the resistance of the soil, as measured by SPT penetration resistance, shear wave velocity, etc. Such correlations are referred to herein as Capacity curves, and more specifically energy-based Capacity curves. In applying the Capacity curves to remedial ground densification design, the energy required to induce liquefaction, as determined from the Capacity curve, is minimum energy required to be imparted to the soil by the various remediation techniques. Because there are physical limits on the magnitude of the energy that can be imparted by a given technique, such an approach may lead to improved feasibility assessments and initial designs of the densification programs.

4

Because the frequency content, duration, and amplitude of earthquake motions differs from those induced by the remediation techniques, central to the feasibility of using Capacity curves in the design of ground densification programs is whether the Capacity of the soil, as quantified by some measure of energy, is dependent or independent of the characteristics of the applied loading. And as a corollary, if the energy Capacity of the soil is load dependent, how sensitive is the dependency? For example, is the dependency such that the Capacity curves can be used for designing vibro-compaction programs, but for not deep dynamic compaction and blast densification? These questions are addressed herein. 1.3 Organization of the Thesis This thesis consists of two parts: earthquake liquefaction evaluation and ground improvement, as depicted in Figure 1-3. Chapters 2-6 make up Part I of the thesis and cover earthquake liquefaction evaluations. Because the focus of the thesis is the application of knowledge gained from earthquakes to ground densification and not the development of a new energy-based liquefaction evaluation procedure, existing liquefaction evaluation procedures were first critically reviewed for possible use. This review is presented in Chapter 2. Early in the literature review, it was realized that different measures of energy were used to quantify the Capacity of the soil (e.g., dissipated energy, Arias intensity). Additionally, even for a consistent measure of energy, the expressions used to compute the Capacity differed from study to study. As a result, the fundamentals and mathematical representation of “energy” are reviewed in Chapter 3. Complementing energy-based liquefaction evaluation procedures are energy-based pore pressure generation models. These models provide a basis for relating energy to soil Capacity, defined as the energy required to generate excess pore pressures equal to the initial effective overburden pressure. Existing energy-based pore pressure models are reviewed in Chapter 4, and a new model is proposed. In the review of the existing and proposed pore pressure generation models, attention is given to whether or not the

5

models’ calibration parameters are load dependent. If the various models’ calibration parameters are independent of the loading, then it is likely that Capacity curve developed from earthquake case histories is also load independent, and therefore, applicable to the loads induced during remedial ground densification. Quantifying Capacity in terms of the cumulative energy dissipated in a unit volume of soil, normalized by the initial mean effective confining stress, up to the point of initial liquefaction, a new energy-based liquefaction evaluation procedure is proposed in Chapter 5. From analyzing earthquake case histories, a Capacity curve is derived. Additionally, the field based Capacity curve is compared with normalized laboratory test data. Part II of the thesis covers ground improvement. Chapter 7 reviews three remedial ground densification techniques: vibro-compaction, deep dynamic compaction, and explosive compaction. Particular emphasis is placed on the mechanisms associated with the techniques that break down the soil structure, and therefore, induce liquefaction. Also, currently used empirical expressions and guidelines for designing densification programs are reviewed. In Chapter 8, first order numerical models for computing the spatial distribution of the energy dissipated in the soil during remedial densification are proposed. In conjunction with the energy-based Capacity curve presented in Chapter 5, the first order models are used to predict the spatial extent of induced liquefaction by the three remediation techniques. To assess the feasibility of using the energy-based earthquake Capacity curve in remedial densification design, the model predictions are compared with the currently used empirical expressions and guidelines. Finally, a summary of research findings and conclusions are given in Chapter 9. Although slightly deviating from the focus of the thesis, two peripheral topics are covered in Appendix 5c and Chapter 6. Energy-based magnitude scaling factors (MSF) for use

6

with the existing stress-based liquefaction evaluation procedure are proposed in Appendix 5c. The proposed MSF are a natural extension to a correlation developed as part of the proposed energy-based liquefaction evaluation procedure. In Chapter 6, approaches are outlined for determining the seismological parameters required for using the proposed energy-based liquefaction evaluation for earthquakes.

7

Part I Liquefaction Evaluation Existing EnergyBased Liquefaction Evaluation Procedures

Chapter 2

Peripheral to the focus of the thesis Central to the focus of the thesis

Fundamentals of Energy Dissipation

Chapter 3

Part II

The focus of the thesis

Ground Improvement

Energy-Based Pore Pressure Generation Models

Chapter 4

Procedures for Determining: amax, M, ED

Current Design of Remedial Ground Densification Techniques

Chapter 7

Proposed Energy-Based Liquefaction Evaluation Procedure

Chapter 5

Energy-Based Liquefaction Curve and Remedial Ground Densification

Chapter 8

Chapter 6

Energy-Based Magnitude Scaling Factors

Appendix 5c Figure 1-3. Organization of Thesis. 8

Chapter 2. Review of Liquefaction Evaluation Procedures 2.1 Introduction Numerous methods have been proposed for evaluating the liquefaction potential of soil deposits. The purpose of this chapter is to review selected approaches, with emphasis placed on energy-based procedures. The intent of the review is to assess the procedures for potential use in the design of remedial densification programs for liquefiable soils. With the exception of Liang (1995), all the procedures discussed have simplified forms: their implementation does not require site response analyses. The procedures are presented in terms of Demand, Capacity, and Factor of Safety, where Demand is the load imparted to the soil by the earthquake (both amplitude and duration), Capacity is the Demand required to induce liquefaction, and Factor of Safety is defined as the ratio of Capacity and Demand. The procedures are presented in this way for consistency and to facilitate comparisons. To do this, several of the procedures are presented in alternate forms from those published in the referenced literature. Furthermore, to facilitate computations, curve-fitting techniques were employed to develop equations corresponding to charts and graphs relied on by several of the procedures. However, it is emphasized that the alternate forms of presentation and the use of curve-fit equations in no way change the basic formulations and assumptions of the procedures. For the procedures that correlate soil Capacity to field tests, only correlations with SPT N-values are presented. The reason for this is that the majority of the procedures only have such correlations. Also, correlations with SPT N-values serve the intent of the review, which is to assess the potential of the procedures to be used in the design of remedial ground densification programs, not to assess the merits of the various field tests. The appendix at the end of this chapter outlines the Youd et al. (2001) recommended factors for normalizing measured SPT N-values for overburden pressure, hammer energy, borehole diameter, rod length, and sampling method, with the result being designated as N1,60. The liquefaction evaluation procedures proposed by Davis and Berrill (1982),

9

Berrill and Davis (1985), and Trifunac (1995) use correlations with SPT N-values normalized only for overburden pressure. It is the opinion of the author that in implementing these procedures, SPT N-values using all the normalizations given in the appendix should be used. The normalizations standardize the SPT N-values and reduce the variability in the measured blow counts by various drill rigs and operators. Furthermore, several of the procedures require correction of the SPT N-values for fines content. However, no two procedures use the same correction factors. Accordingly, fines content correction factors are presented with the corresponding procedures. Rather than delving into diversified theories of the various procedures, a “cookbook” presentation is given for each. The procedures are presented in the following order: the stress-based procedure, the strain-based procedure, and energy-based procedures. The energy-based procedures are grouped into approaches developed using earthquake case histories, approaches developed from laboratory data, and other approaches. This last group includes studies that warrant mention but are either similar to other approaches discussed more in depth or are not fully developed liquefaction evaluation procedures. After the “cookbook” presentation, the results and observed trends of a parameter study are presented. These results help to assess the procedures. Finally, general and specific commentaries are given on the assumptions and formulations of several of the procedures. 2.2 Overview of the Procedures 2.2.1 Stress-based procedure The most widely used method for evaluating liquefaction is the stress-based procedure first proposed by Seed and Idriss (1971) and Whitman (1971). This procedure is largely based on empirical observations of laboratory and field data and has been continually refined as a result of newer studies and the increase in the number of liquefaction case histories (e.g., NRC 1985, NCEER 1997, Youd et al. 2001).

10

Demand: The amplitude of the earthquake-induced Demand is quantified by the cyclic stress ratio (CSR). CSR can be determined for any desired depth in a soil profile from site response analyses or by using the “simplified” equation, Equation (2-1).

CSR  where:

 ave a   0.65 max vo rd  ' vo g  ' vo

(2-1)

CSR = Cyclic stress ratio. amax

=

Soil surface of acceleration.

g

= Acceleration due to gravity.

’vo

= Initial effective vertical stress at depth z.

vo

= Total vertical stress at depth z.

rd

= Dimensionless parameter that accounts for the stress reduction due to soil column deformability.

A consistent set of units should be used so that CSR is dimensionless. The range of rd, as a function of depth, is shown in Figure 2-1. The average of this range can be computed using Equation (2-2) (NCEER 1997).

rd =

1.0  0.00765 z

for

z  9.15m

1.174  0.0267 z

for

9.15m  z  23m

0.744  0.008 z

for

23m  z  30m

0.5

for

z  30m

(2-2)

where z is depth in m. Equation (2-3) yields essentially the same results as Equation (22), but may be easier to program for use in spread sheet calculations (NCEER 1997):

rd 

(1.000  0.4113z 0.5  0.04052 z  0.001753z 1.5 ) (1.000)  0.4177 z 0.5  0.05729 z  0.006205z 1.5  0.001210 z 2

11

(2-3)

rd 0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

3 Average values

6

Depth (m)

9 12 15 18 21

Mean values of rd calculated from Equation (2-2) Range for different soil profiles Simplified procedure not verified with case history data in this region

24 27 30 Figure 2-1. Stress reduction factor (rd) to account for soil column deformability. (Adapted from NCEER 1997 and Seed and Idriss 1982). To account for the duration of the earthquake motions, magnitude scaling factors (MSF) are applied to the CSR:

CSRM 7.5 

 ave  CSR   M 7.5 MSF  ' vo MSF  ' vo

(2-4)

This expression defines the Demand imparted to the soil by the earthquake (i.e., Demand = CSRM7.5). Several different correlations for MSF have been proposed, as shown in Figure 2-2. The bases for these relationships are given and discussed in NCEER (1997) and Youd et al. (2001).

12

Magnitude Scaling Factor, MSF

4.5 4.0

Seed and Idriss, (1982) Idriss Ambraseys (1988) Arango (1996) Arango (1996) Andrus and Stokoe Youd and Noble, PL 1.0 implies that liquefaction will not occur, FS  1.0 does not imply that liquefaction will occur. Rather FS  1.0 predicts that there will be gross sliding of the grain-to-grain contact surfaces, which is a requisite for excess pore pressure generation and therefore a requisite for liquefaction. This is in contrast to the stress based procedure for which FS > 1.0 implies that liquefaction will not occur, not necessarily that excess pore pressures will not be generated. Accordingly, the ratio of th and  may be an overly

22

conservative method for computing the factor of safety against liquefaction. However, FS  1.0 provides insight into the expected behavior of the soil in that the residual excess pore pressures will be greater than zero. 2.2.3 Energy-based procedures As opposed to using stress or strain as the base parameters to quantify Capacity and Demand, the procedures presented in this section use various measures of energy. The use of energy was a logical step in the evolution of liquefaction evaluations. The reason for this is twofold. First, seismologists have long been quantifying the energy released during earthquakes and have established simple correlations with common seismological parameters (e.g., Gutenberg and Richter 1956). The second reason is the pioneering study by Nemat-Nasser and Shokooh (1979) showing a functional relationship between the dissipated energy in laboratory samples and generated pore pressures. Chapter 4 gives a brief presentation of Nemat-Nasser and Shokooh’s energy-based pore pressure model, as well as other models. In the presentation given below, the energy-based liquefaction evaluation procedures are categorized into two main groups: 1) procedures developed from earthquake case histories, and 2) procedures developed from laboratory data. A third section entitled “Other Approaches” presents studies warranting mention but are either similar to other procedures presented or are not fully developed liquefaction evaluation procedures. The procedures developed from earthquake case histories are further subdivided into ones quantifying energy using the Gutenberg-Richter energy relation and ones using Arias intensity. The exception to this categorization is in the presentation of the procedures proposed by Trifunac (1995). Trifunac (1995) proposes 5 different energy-based liquefaction evaluation procedures, only one of which uses the Gutenberg-Richter relation. However, for continuity of presentation, all 5 of them are presented in the Gutenberg-Richter section. For purposes of assessing the procedures for potential use in the design of remedial densification programs, preference is automatically given to those developed from

23

earthquake case histories. This is because no study that the author has seen shows that the Capacity of the soil, quantified by any measure of energy, is independent of soil fabric, and none of the reviewed procedures developed from laboratory data accounts for the influence of soil fabric in a way that can be used to evaluate the of Capacity actual field deposits. Finally, inherent to many of the energy-based procedures presented below are parameters quantifying site-to-source distance and earthquake magnitude. However, not all the procedures use the same definition of site-to-source distance or use the same magnitude scale. To facilitate the presentation of the procedures, commonly used measures of siteto-source distance and a comparison of various magnitude scales are shown in Figures 28a and 2-8b, respectively.

Station D5

D4

Surface D2

D3 D1

Epicenter

High Stress zone h Surface of fault slippage Hypocenter D1: D2: D3: D4: D5:

Hypocentral distance, where h is the focal depth Epicentral distance Distance to center of high-energy release (or high localized stress drop) Closest distance to the slipped fault Closest distance to the surface projection of fault rupture.

Figure 2-8a. Commonly used measures of site-to-source distance. (Adapted from Shakal and Bernreuter 1981).

24

9 MS MJMA

MW 8

mB ML

Magnitude

7

mb

6 5 4 ML

MS

3 2

2

MJMA: ML: MS: mb: mB: MW: M:

3

4

5 6 7 Moment Magnitude MW

8

9

10

Magnitude scale used by the Japanese Meteorological Agency Local Magnitude Surface-wave Magnitude Body-wave Magnitude, (short period body waves) Body-wave Magnitude, (long period body waves) Moment Magnitude ML for M< 5.9 Richter Magnitude = MS for 5.9 < M < 8.0 MW for 8.0 < M < 8.3

Figure 2-8b. Comparison of the central tendencies of various earthquake magnitude scales, with moment magnitude (MW) currently being the most accepted scale. Although in the past the term “Richter magnitude” was used synonymously with ML, it currently is used to refer to a conglomeration of magnitude scales (e.g., Krinitzski et al. 1993). (Adapted from Heaton et al. 1986).

25

2.2.3.1 Procedures developed from earthquake case histories 2.2.3.1.1 Gutenberg-Richter approaches Most of the procedures presented in this section use the Gutenberg-Richter energy correlation as the basis for computing the Demand imparted to the soil. The GutenbergRichter energy relation is given as (Gutenberg and Richter 1956):

Eo 101.5M 1.8 where:

(2-18)

Eo =

Total radiated energy from the source (kJ).

M =

Magnitude of earthquake (Richter scale).

2.2.3.1.1a Davis and Berrill (1982) Demand: In addition to the Gutenberg-Richter relation, Davis and Berrill (1982) make three other assumptions in deriving their expression for Demand. First, it is assumed that the magnitude of the energy decreases at a rate proportional to 1/r2, where r is distance from site to the center of energy release. This attenuation model does not include energy lost due to material damping, rather it only accounts for the geometrical spreading of a spherically expanding wave front. Second, it is assumed that the increase in pore pressure is a linear function of dissipated energy. Finally, it is assumed that energy dissipation in the soil due to material damping is proportional to 1/(’vo)0.5, where ’vo is the initial effective overburden stress. Their derived expression for Demand is given as:  r 2  'vo1.5  Demand   1.5 M   10 

where:

r

=

1

(2-19)

Distance from site to the center of energy release (m).

M =

Magnitude of earthquake (Richter scale).

’vo =

Initial effective vertical stress at depth z ( kPa).

Capacity: Similar to the procedure used to determine the CRR curve shown in Figure 2-3, Davis and Berrill grouped earthquake case histories according to the occurrence or non-occurrence of liquefaction. Using Equation (2-19), the Demand imparted to the soil was estimated 26

for each site. Because the distance from the site to the center of energy release was not known for all the cases, hypocentral and epicentral distances were often used. In Davis and Berrill’s presentation, they plotted the inverse of Demand (i.e., 1/Demand) as a function of the N1, as shown in Figure 2-9. low 100 Demand

No Liq. Liq. 10

r 2  'vo 101.5 M

1.5

1.0

high 0.1 Demand 1.0

10 100 Corrected Standard Penetration Value, N1

Figure 2-9. Liquefaction chart proposed by Davis and Berrill (1982).

N1 is the measured SPT N-value adjusted to 1tsf; no hammer-energy correction factors or fines correction factors were applied. Because Davis and Berrill (1982) plotted the inverse of Demand, the boundary giving a reasonable separation of the liquefied and nonliquefied data points defines the inverse of the Capacity of the soil. The Capacity is given by Equation (2-20).  450  Capacity   2   N1 

1

(2-20)

27

where:

N1 =

Measured SPT N-value normalized to 1tsf.

Factor of Safety Against Liquefaction: Using the definition of factor of safety as the ratio of Capacity to Demand, substitution of Equations (2-20) and (2-19) results in the following expression for the factor of safety against liquefaction:

N12  r 2  ( ' vo )1.5 FS  450  101.5 M

(2-21)

2.2.3.1.1b Berrill and Davis (1985) Demand: Berrill and Davis (1985) proposed a revised model of their earlier work, Davis and Berrill (1982). Two main revisions were included: 1) a revised pore pressure generation model (i.e, pore pressure is proportional to the square root of dissipated energy), and 2) the inclusion of an expression to account for the inelastic attenuation of seismic energy as it travels from the source to the site. The resulting expression for Demand is given by Equation (2-22).  r 2  ' vo1.5 Demand   1. 5 M  A 10

where:

A

A

=

4



  

0.5

(2-22)

Material attenuation factor, given by Equation (2-23a). 

  x 2 F 2 ( x) e  ax dx 0

F ( x)  a a

1 (1  x 2 )

(2-23a) (2-23b)

= Dimensionless distance, given by Equation (2-23c). kr Qd

where:

(2-23c) r and d have same dimensions. k = dimensionless constant that varies as a function of the source rupture model 28

(= 2.80 for the Brune model: Brune 1970, 1971). d = Source dimension, radius of equivalent circular source, given by Equation (2-23d). Q = Quality factor function of material attenuation (= 280, assumed). d 1.14  10 3  exp 1.35M 

(2-23d)

km

The material attenuation factor, A, is plotted in Figure 2-10 as a function of dimensionless distance, a. For typical cases, it can be expected that A  0.8, which implies that little energy is lost due to anelastic attenuation as the seismic waves travel from the source to the site. Accordingly, the main difference between the Demand expressions proposed in Davis and Berrill (1982) and Berrill and Davis (1985) is that in the 1985 model the pore pressure is assumed proportional to the square root of dissipated energy, as opposed to

Material attenuation factor, A

the linear relationship assumed in the earlier model.

1.0 0.8 0.6 0.4 0.2 0.0 0.001

0.01

0.1

1.0

10

Dimensionless distance, a = kr/Qd Figure 2-10. Material attenuation factor (A) as a function of the dimensionless distance (a). (Adapted from Berrill and Davis 1985).

29

Capacity: Again, categorizing sites according to the occurrence or non-occurrence of liquefaction, Berrill and Davis (1985) plotted the inverse of energy demand (i.e., 1/Demand) as a function of the N1, as shown in Figure 2-11. 50 Liquefaction No Liquefaction

10

 r 2  'vo1.5   1.5 M   A 10 

0.5

5

1.0 0.5

0.2

1

5 10 50 100 Corrected standard penetration value, N1

Figure 2-11. Liquefaction chart proposed by Berrill and Davis (1985).

The boundary giving a reasonable separation of the liquefied and non-liquefied data points defines the inverse of the energy capacity of the soil. The Capacity is given as:  120  Capacity   1.5   N1 

1

(2-24)

30

Factor of Safety: Using the definition of factor of safety as the ratio of Capacity to Demand, substitution of Equations (2-24) and (2-22) yields:

FS 

N11.5  r  ( ' vo ) 0.75 120  A 0.5  10 0.75M

(2-25)

2.2.3.1.1c Law, Cao, and He (1990) Demand: Similar to Davis and Berrill (1982) and Berrill and Davis (1985), Law et al. (1990) derived an expression for energy demand based on the Gutenberg-Richter relation. However, Law et al. made several different assumptions regarding the functional relationship between dissipated energy and excess pore pressure generation, anelastic attenuation of seismic waves along the travel path, and material damping at the site. The resulting expression for Demand, referred to as the seismic energy intensity function (T), is given by Equation (2-26).

T

101.5 M r 4.3

where:

(2-26) T

=

The seismic energy intensity function (i.e., Demand).

r

=

Hypocentral distance from site to source (km).

M =

Magnitude of earthquake (Richter scale).

Capacity: Using the same procedure of grouping sites according to the observance and nonobservance of liquefaction, Law et al. (1990) computed the energy demand imparted to the sites using Equation (2-26) and plotted the results versus N1,60, as shown in Figure 212.

31

SEISMIC ENERGY INTENSITY FUNCTION, T

107 LIQUEFIED SITE NONLIQUEFIED SITE 106

105

104

103

102

SAND

SILTY SAND 101 0.1

10

1

100

CORRECTED SPT RESISTANCE, N1,60 Figure 2-12. Liquefaction chart proposed by Law, Cao, and He (1990). In plotting their data, Law et al. distinguished between the soil types at the site (i.e., sand and silty-sand). As shown in Figure 2-12, boundaries were drawn separating the points representing liquefaction from those of no-liquefaction for each soil type. These boundaries define the Capacities (i.e., Lsand and Lsilt) of the soils as a function of N1,60 and are expressed by Equation (2-27).

L sand  2.28 ( N1,60 )11.5  1010

(2-27a)

L silt 1.14 ( N1,60 )11.5  109

(2-27b)

32

Factor of Safety: Using the definition of factor of safety as the ratio of Capacity to Demand, substitution of Equations (2-27) and (2-26) results in the following expressions for the factor of safety against liquefaction for sand and silty-sand:

2.28  10 10  N1,60 

11.5

FS sand 

(2-28a)

101.5 M

1.14  10 9  N1,60 

11.5

FS silt 

 r 4.3

 r 4.3

(2-28b)

101.5 M

2.2.3.1.1d Trifunac (1995) Trifunac (1995) proposed five separate pairs of expressions for energy Demand and Capacity, only one of which is based on the Gutenberg-Richter energy relation. However, all five are presented in this section and are referred to as Trifunac No.1, Trifunac No.2, Trifunac No.3, etc. In deriving all of the energy demand expressions, Trifunac (1995) retained two of the central assumptions made by Davis and Berrill (1982), namely that the increase in pore pressure is a linear function of dissipated energy and that energy dissipation in the soil due to material damping is proportional to 1/(’vo)0.5. Also, as with Davis and Berrill (1982) and Berrill and Davis (1985), all of the capacity expressions proposed by Trifunac (1995) are expressed as functions of N1, where N1 is the measured SPT N-value adjusted to 1tsf; no hammer-energy correction factors or fines correction factors were applied. More so than the other procedures presented above, the expressions for Demand and Capacity are presented herein in alternate forms from those given in Trifunac (1995). However, with the possible exception of differences resulting from the reduction in the number of significant digits of various coefficients, the expressions presented below give identical results to those given in Trifunac (1995).

33

Trifunac No.1 Demand: The expression for Demand for Trifunac No.1 is based on the Gutenberg-Richter energy relation and is identical to that derived by Davis and Berrill (1982). This expression was presented previously as Equation (2-19) and is repeated below.  r 2  ' vo1.5  Demand   1.5 M   10 

where:

r

=

1

(2-19)

Epicentral distance from site to source (m).

M =

Magnitude of earthquake (Richter scale).

’vo =

Initial effective vertical stress at depth z (kPa).

Trifunac No.1 uses the epicentral distance from the site to the source as an approximation for the distance from site to the center of energy release, the definition of r used by Davis and Berrill (1982). This approximation is coupled with the imposed limitation on Equation (2-19) for 50km  r  (100 to 150km), which is quite restricting. The basis of this limitation is the validity of the assumed 1/r2 relation for geometrical spreading of the seismic energy used in deriving Equation (2-19). Capacity: Although Trifunac No.1 uses the same expression for Demand as Davis and Berrill (1982), a different expression is given for Capacity, presented as Equation (2-29). This expression is based on a different regression of the same data used by Davis and Berrill (1982).  9.92  Capacity     N 1  3.15 

where:

N1 =

3.8

(2-29)

Measured SPT N-value normalized to 1tsf.

A comparison of the Capacity expressions proposed by Davis and Berrill (1982) and Trifunac No.1 is shown in Figure 2-13.

34

Liquefaction No Liquefaction Trifunac No.1 Davis & Berrill (1982)

3

2  r 2  'vo1.5  log  1 .5 M   10 

1

0

-1

0

1 Log(N1)

2

Figure 2-13. Comparison of the boundaries proposed by Davis and Berrill (1982) and Trifunac No.1 to separate the data points representing liquefaction and no-liquefaction. (Adapted from Trifunac 1995).

Factor of Safety: Substituting Equations (2-29) and (2-19) into the definition of factor of safety as the ratio of Capacity to Demand, yields:

FS 

N1  3.153.8  r 2  ( 'vo )1.5

(2-30)

6120  101.5 M

Trifunac No.2 Demand: Alternative to using the Gutenberg-Richter energy relation, Trifunac No.2 computes the energy of the earthquake motions at the site from the Fourier amplitude spectrum of the

35

strong motion accelerations. Empirical regression models may be employed to estimate the Fourier amplitude spectrum from the seismological parameters of the design earthquake and local site conditions (e.g., Trifunac 1976b). Because the energy is computed from the motions at the site (as represented by the Fourier amplitude spectrum), the geometrical spreading and inelastic attenuation of the seismic waves occurring along the travel path from the source to the site are inherently taken into account. Accordingly, this approach alleviates the distance restriction placed on Trifunac No.1. The expression for Demand for Trifunac No.2 is given as Equation (2-31).   ' 1.5  Demand   vo   en  200

en  

F ( )



50

1

(2-31a)

2

d

(2-31b)

where: F() = Fourier amplitude spectrum of strong motion acceleration at the site (units not specified: m/sec??).



= Frequency (rad/sec).

’vo

= Initial effective vertical stress at depth z.

Capacity: By performing a regression analysis of Demands computed using Equation (2-31a) to obtain a reasonable separation of data corresponding to liquefaction and no liquefaction, the following expression for Capacity was obtained for Trifunac No.2:  572  Capacity     N 1  1.95 

2 . 5

(2-32)

Factor of Safety: From Equations (2-32) and (2-31a), the factor of safety against liquefaction for Trifunac No.2 is given as: FS 

N1  1.952.5  ( 'vo )1.5

(2-33)

7.825  10 6  en

36

Trifunac No.3 Demand: In Trifunac No.3, the energy of the earthquake motions at the site is computed from the peak ground velocity (vmax) and the duration of the strong motions (dur). This expression is given as Equation (2-34).   ' vo1.5  Demand    2  v max dur 

where:

1

(2-34)

dur =

Duration of strong ground motion at the site (sec).

vmax =

Peak ground velocity at the site (m/sec).

’vo =

Initial effective vertical stress at depth z (kPa).

The duration of the strong ground motion for alluvium deposits can be estimated using Equation (2-35):

    5  dur0  dur   5 1  dur0     dur7   where:

1

5

 2 sec

(2-35a)

dur0 = 8.94 – 3.86M + 0.57M 2 + 0.07r r

(2-35b)

= Epicentral distance (km).

dur7 = [email protected] The median peak horizontal component of the ground velocity for alluvium sites can be estimated using Equation (2-36).

log v max   log Ao  3.059M  0.201M 2  9.8135; v max in (m / sec)

(2-36a)

where: log Ao =

r    1.4   50  

for r  75km

r    2.525  200  

for 75km  r  350km

; r is in km

37

(2-36b)

Capacity: By performing a regression analysis of Demands computed using Equation (2-34) to obtain a reasonable separation of data corresponding to liquefaction and no liquefaction, the energy Capacity for Trifunac No.3 is:  87.2  Capacity     N 1  0.95 

3.4

(2-37)

Factor of Safety: The factor of safety against liquefaction is given by Equation (2-38): FS 

N1  0.953.4  ( 'vo )1.5

(2-38)

2

3.961  10 6  v max  dur

Trifunac No.4 In Trifunac No.4, the energy of the earthquake motions at the site is computed from the Fourier amplitude of the velocity motions at a period of 0.39sec, which is near the plateau of the Fourier spectrum amplitudes for a broad range of magnitudes and site conditions. This expression is given as Equation (2-39).  ' vo1.5  Demand   2   FV 

where:

FV =

1

(2-39)

Fourier amplitude of the horizontal velocity motions at a period of 0.39sec (m).

’vo = Initial effective vertical stress at depth z (kPa).

Trifunac (1995) gives the following empirical relation to compute the median value of FV for alluvium sites,

log FV   log Ao  2.053M  0.1206M 2  0.000709r  7.657

(2-40)

where log Ao was given previously in Equation (2-36b), FV is in m and r is in km. For the M2 term, if M > 8.51, M = 8.51 should be used.

38

Capacity: By performing a regression analysis of Demands computed using Equation (2-39) to obtain a reasonable separation of data corresponding to liquefaction and no liquefaction, the following expression for Capacity was obtained for Trifunac No.4:  538.0  Capacity     N 1  1.05 

3 . 6

(2-41)

Factor of Safety against Liquefaction: The factor of safety against liquefaction is given by Equation (2-42): FS 

N1  1.053.6  ( 'vo )1.5

(2-42)

6.774  10 9  FV 2

Trifunac No.5 Demand: In Trifunac No.5, the energy of the earthquake motions at the site is computed from the peak ground velocity at the site, the small strain shear modulus, and the duration of strong ground motion at the site. This expression is given as Equation (2-43).    ' vo1.5 Demand     v max  Gmax  dur 

where:

1

(2-43)

Gmax = Small strain shear modulus (kPa), Equation (2-12). vmax = Peak ground velocity at the site (m/sec), Equation (2-36a). dur

= Duration of strong ground motion at the site (sec), Equation (2-35a).

’vo

= Initial effective vertical stress at depth z (kPa).

Capacity: By performing a regression analysis of Demands computed using Equation (2-43) to obtain a reasonable separation of data corresponding to liquefaction and no liquefaction, the following expression for Capacity was obtained for Trifunac No.5:

39

 0.313  Capacity     N 1  0.05 

2.1

(2-44)

Factor of Safety: The factor of safety against liquefaction is given by Equation (2-45): FS 

N1  0.052.1  ( 'vo )1.5

(2-45)

0.08723  v max  Gmax  dur

2.2.3.1.2 Arias intensity approaches In the following procedures, Arias intensity (Ih) is used as the quantitative measure of earthquake shaking intensity at the site. As given by Kayen and Mitchell (1997), Ih is “the sum of the two component energy per unit weight stored in a population of undamped linear oscillators evenly distributed in frequency, at the end of earthquake shaking.” Arias intensity is calculated by integrating the acceleration time histories, as given by Equation (2-46) (Arias 1970). Ih 



dur dur    a x2 (t ) dt   a y2 (t ) dt   0 2 g  0

(2-46) where:

Ih

= Arias intensity of the earthquake motion at the top of the soil profile (m/sec).

ax(t) = Horizontal acceleration time history in the x-direction (m/sec2). ay(t) = Horizontal acceleration time history in the y-direction (m/sec2). g

= Acceleration due to gravity (m/sec2).

dur = Duration of earthquake shaking (sec). 2.2.3.1.2a Kayen and Mitchell (1997) Kayen and Mitchell (1997) further developed the preliminary work of Egan and Rosidi (1991) by establishing correlations for the occurrence and non-occurrence of liquefaction

40

as functions of the Arias intensity of the earthquake motion and penetration resistance of the soil. Demand: To estimate the Arias intensity corresponding to the seismological parameters of the design earthquake and the local site conditions, Kayen and Mitchell (1997) proposed the following empirical relation: Ih = where:

M  3.8  2 log r *  0.63P

for alluvium

M  3.4  2 log r *

for soft soil

P

=

r* =

(m/sec)

(2-47)

Probit (= 1 for  standard deviation; = 0 for mean). r 2  2



=

Focal depth (km).

r

=

Distance from site to closest surface distance to the surface fault rupture at the focal depth (km).

r*, r, and  are defined graphically in Figure 2-14.

Closest Surface Distance, r site Focal Depth, 

r* Elevation View

Fault Rupture

Fault Rupture site

r* Focus

Plan View

Figure 2-14. Graphical illustration of the site-to-source distance r*. (Adapted from Kayen 1993). 41

Analogous to rd used in the stress-based procedure, Kayen (1993) and Kayen and Mitchell (1998) developed a factor that relates the Arias intensity of surface motions to the Arias intensity at depth in a profile. This relation is expressed by Equation (2-48).

I hb, eq  I h  rb where:

(2-48)

Ihb, eq

= Arias intensity of the earthquake motions at a given depth in the profile (i.e., Demand).

rb

= Arias intensity depth-of-burial reduction parameter.

Ih

= Arias intensity at soil surface, Equation (2-47).

A statistical analysis of the variation of rb as a function of depth in a soil profile is shown in Figure 2-15.

rb 0

0

0.2

0.4

0.6

0.8

1

SHAKE profile statistics

Depth (ft)

10

20

30

40

-1 (P=-1)

+1 (P=+1)

Mean (P=0)

50 Figure 2-15. Results of a statistical analysis of the variation of rb as a function of depth in a soil profile. (Adapted from Kayen and Mitchell 1997).

42

Capacity: Similar to the other procedures discussed previously, Kayen and Mitchell (1997) grouped case histories according to the observance or non-observance of liquefaction. Using Equation (2-48), the Demand imparted to soil was estimated for each. The computed Demands were plotted as functions of the corresponding (N1,60cs)km, where (N1,60cs)km is N1,60 corrected for fines content (FC). The correction factor fines content is given as follows:

N



km

1, 60cs

N 

 N1,60  N

(2-49a)

0

for FC  5%

7 ( FC  5) 30 7

for 5%  FC  35% ; where FC is in %

(2-49b)

for FC  35%

Equation (2-49) is an earlier version of the procedure used to correct for fines in the stress-based approach. Either Equation (2-49) or Equation (2-6) should be acceptable for use with the Kayen and Mitchell approach. The plot of Ihb,eq versus (N1,60cs)km is shown in Figure 2-16, where the boundary separating the points representing liquefaction and noliquefaction defines the Capacity of the soil (Ihb,l).

43

5

LIQUEFACTION

Liquefaction

Ihb (m/sec)

1

No Liquefaction Liquefaction ??

Clean-Sand Boundary NO LIQUEFACTION Ih < 0.10 m/s

0.1

0

5

10

15

(N1,60cs)

20

km

25

30

Figure 2-16. Liquefaction curve proposed by Kayen and Mitchell (1997). Kayen and Mitchell (1997) did not provide a mathematical expression for Capacity. However, Equation (2-50) is proposed as a reasonable approximation of the boundary.

log( I hb, l ) 1.234  10 6 ( N 1km ) 5  6.956  10 5 ( N 1km ) 4  0.001421 ( N 1km )3 , 60 cs , 60 cs , 60 cs  0.01132 ( N 1km ) 2  0.04162 ( N 1km )  0.6227 , 60 cs , 60 cs for

(2-50)

3  (N1,60cs)km  25

Factor of Safety: The factor of safety against liquefaction is computed as follows:

FS 

I hb, l

(2-51)

I hb, eq

where Ihb,eq can be determined from Equation (2-48) and Ihb,l can be determined from Equation (2-50) or Figure 2-16.

44

2.2.3.1.2b Running (1996) Running (1996) proposed a hybrid liquefaction evaluation procedure combining facets of the Kayen (1993) approach with critical state theory. Demand: Running uses the same definition for Demand as that proposed by Kayen and Mitchell (1997) and given by Equation (2-48). Capacity: From critical state theory, Running derived an expression for the shear energy capacity of soils at failure. This expression is given as Equation (2-52).

3000    ' m o sin  '  c cos  '

2

U sf 

0.126 Gmax

where:

1000   ' vo  1  K o  6 Gmax 2

 3 cos  sin  sin  ' 

2



2

(J / m3 )

’mo

=

Initial mean effective confining stress (kPa), Equation (2-13).

c

=

Cohesive intercept of the soil strength envelope (kPa).

Gmax

=

Small strain shear modulus of the soil (kPa), Equation (2-12).

’

=

Effective angle of internal friction of the soil, Equation (2-15).



=

Lode angle, which is a function of the load path (for horizontal earthquake motions  = 0.5210 rad).

Using the same case history data set as used by Kayen and Mitchell (1997), Running plotted Ihb,eq as a function of U sf and drew a boundary giving a reasonable separation of points representing liquefaction and no-liquefaction, as shown in Figure 2-17.

45

(2-52)

Ihb (m/s)

3 Liq. No Liq. Possible

2 Liquefaction 1

No Liquefaction 0

0

20

40

60

80 100 120 140 160

U sf (J/m3) Figure 2-17. Liquefaction curve proposed by Running (1996).

Running (1996) did not provide a mathematical expression for his proposed boundary, but Equation (2-53) provides a reasonable approximation. Equation (2-53) allows the shear energy capacity of the soil to be expressed in terms of Arias intensity required to induce liquefaction (Ihb,l = Capacity). I hb, l  0.005U sf  0.28

(2-53)

where Ihb,l is in m/sec, and U sf is in J/m3.

Factor of Safety: The factor of safety against liquefaction is computed as follows:

FS 

I hb, l

(2-54)

I hb, eq

where Ihb,eq can be determined from Equation (2-48) and Ihb,l can be determined from Equation (2-53) or Figure 2-17.

46

2.2.3.2 Procedures developed from laboratory data In all the energy-based procedures presented above, the Capacity of the soil was determined from the analyzing earthquake case histories. Alternatively, the two procedures discussed in this section rely on laboratory test data for determining the Capacity of the soil. 2.2.3.2.1 Alkhatib (1994) Demand: Alkhatib (1994) quantified Demand by the dimensionless parameter Normalized Maximum Energy (NME), which is computed by integrating the stress-strain time histories at depth in a soil profile. From a series of site response analyses using scaled, western United States acceleration time histories, Alkhatib proposed the relationship shown in Figure 2-18 between the maximum acceleration (amax) of the acceleration time

Maximum Acceleration, g

history and NME. 1

0.1

0.01 0.000001 0.00001 0.0001 0.001 0.01 Normalized Maximum Energy

0.1

Figure 2-18. Correlation of the dimensionless parameter Normalized Maximum Energy (NME) and maximum acceleration (amax). (Adapted from Alkhatib 1994). Alkhatib (1994) did not provide a mathematical expression for the correlation shown in Figure 2-18, but Equation (2-55) provides a reasonable approximation. log( NME )  2.933  log( amax ) 1.207

(2-55)

where amax is in g, and NME is dimensionless.

47

Capacity: From a series of cyclic triaxial tests conducted on Monterey sand subjected to earthquake type load functions, Alkhatib proposed the correlation shown in Figure 2-19 between the relative density (Dr) of the sand and energy ratio (ER). ER is the ratio of the energy computed by integrating the stress-strain hysteresis loops to the initial effective confining

Energy Ratio

stress and was used to quantify the Capacity of the soil.

10

10

2

3

0

20

40 60 80 Relative Density, %

100

Figure 2-19. Correlation between relative density (Dr) and energy ratio (ER). (Adapted from Alkhatib 1994).

Alkhatib (1994) did not provide a mathematical expression for the correlation shown in Figure 2-19, but Equation (2-56) provides a reasonable approximation.

log( ER)  0.01747  log( Dr )  3.6291

(2-56)

where Dr is in %, and ER is dimensionless. The following expressions can be used to relate penetration resistance to Dr.

Dr 

N 1.710   'vo 

where:

(Gibbs and Holtz 1957)

Dr =

Relative density in decimal.

N

Measured SPT N-value.

=

’vo = Initial vertical effective confining stress (psi).

48

(2-57)

Dr 15  N1,60 

0.5

where:

Dr

;

30%  Dr  90%

(Skempton 1986)

(2-58)

= Relative density (%).

N1,60 = Corrected SPT N-value. Factor of Safety: Using the definitions given above for Capacity and Demand, the author derived the following expression for the factor of safety against liquefaction: FS = alog{0.01747 log(Dr) – 2.933 log(amax) – 2.4221}

(2-59)

2.2.3.2.2 Liang (1995) Extensive research on energy-based liquefaction evaluations procedures has been conducted at Case Western Reserve University under the direction of Professor J.L. Figueroa: Figueroa, 1990; Dahisaria, 1991; Figueroa and Dahisaria, 1991; Figueroa, 1993; Figueroa et al., 1994a; Figueroa et al., 1994b; Figueroa et al., 1995; Liang, 1995; Liang et al., 1995a; Liang et al., 1995b; Kern, 1996; Kusky, 1996; Figueroa et al., 1997a; Figueroa et al., 1997b; Rokoff, 1999; Figueroa et al., 1999. The most complete presentation of this research is given in Liang (1995). The proposed liquefaction evaluation procedure requires both laboratory testing of soil and site response analyses to be performed. For the purposes of the present study, it is desired to develop an energybased procedure for designing remedial ground densification programs using simplified procedures (i.e., that do not require site response analyses). Also, obtaining truly undisturbed samples from the field is difficult. Accordingly, the author desires a procedure that correlates energy Capacity to an in-situ field test (e.g., SPT), as opposed to having to perform laboratory tests on undisturbed samples. Based on these conditions, the work of Figueroa and his associates is only of general interest but not considered for possible use in the design of remedial densification programs.

49

2.2.3.3 Other Approaches 2.2.3.3.1 Mostaghel and Habibaghi (1978, 1979) Mostaghel and Habibaghi (1978, 1979) is the earliest energy-based liquefaction evaluation procedure known to the author. In this procedure, an energy Capacity relationship is assumed. As discussed in Chapter 4, Section 4.2.2 and 4.3, the assumed Capacity relation is contrary to trends observed in the laboratory. 2.2.3.3.2 Moroto and Tanoue (1989) The liquefaction evaluation procedure proposed by Moroto and Tanoue (1989) and Moroto (1995) is similar to Davis and Berrill (1982) presented in Section 2.2.3.1.1a. 2.2.3.3.3 Ostadan, Deng, and Arango (1996, 1998) Ostadan et al. (1996, 1998) analyzed 150 cyclic laboratory tests performed on high quality undisturbed and reconstituted samples. They found that the strain-energy is a function of the relative density for clean sands and a function of the fines content and confining pressure for silty sands. Capacity curves analogous to that developed by Alkhatib (1994) shown in Figure 2-19 are presented. However, as opposed to Alkhatib (1994), Ostadan et al. do not normalize the strain-energy by the initial effective confining stress. 2.3 Overview of the Parameter Study 2.3.1 Objective A parameter study was performed using twelve of the procedures presented above: Stress-based procedure, Strain-based procedure, Davis and Berrill (1982), Berrill and Davis (1985), Law, Cao, and He (1990), Trifunac No.1, Trifunac No.3, Trifunac No.4, Trifunac No.5, Kayen and Mitchell (1997), Running (1996), and Alkhatib (1994). The two remaining procedures (i.e., Trifunac No.2 and Liang 1995) were not included because Trifunac No.2 requires as an input the Fourier amplitude spectrum of the strong motion acceleration and Liang (1995) requires both laboratory testing and a site response analysis to be performed. For each of the procedures examined, the factors of safety

50

versus depth (FS-profile) were computed for simple soil profiles. The following profiles were examined: 

100ft thick profile of clean sand having N1,60 = 15blws/ft for all depths. The depth to the groundwater table (gwt) is 0ft, 11ft, and 25ft.



100ft thick profile of clean sand with the groundwater table at a depth of approximately 11ft. The profile is assumed to have constant N1,60 with depth equal to 5, 10, and 15blws/ft.

In the first case, the profile is assumed to have a constant N1,60 with depth and the elevation of the water table is varied. This scenario was selected because for a soil having a given fabric, N1,60 correlates well with relative density (Dr) and the variation of the water table give different effective confining stress profiles. Numerous laboratory studies have examined the influence of effective confining stress on similar samples (i.e., same fabric and Dr). Accordingly, trends in the predicted FS-profiles can be evaluated against the observed behavior from laboratory studies, which are incorporated in stressbased procedure. In the second case, the elevation of the water table is held constant, but the penetration resistance is varied. Changes in N1,60 values are often used to measure the effectiveness of various remediation techniques, wherein increases in N1,60 reflects an increases in the FS against liquefaction. Accordingly, the sensitivity of predicted FSprofiles to changes in N1,60 is of practical interest. The profiles were assumed to be subjected to a M7.5 earthquake at a distance of 60km, where this distance is defined according to the requirements of each procedure (e.g., epicentral distance, center of energy release, etc.). Using the acceleration attenuation relation proposed by Trifunac (1976a), given below as Equation (2-60), amax is 0.13g.

log a max   M  log Ao  log( a' );

a max in (cm / sec 2 )

(2-60a)

where: log(a’) =

7.163 1.975M  0.186 M 2

for M  7.5

5.768 1.789 M  0.186 M 2

for 4.8  M  7.5

1.466

for M  4.8

51

(2-60b)

This expression gives the median peak horizontal acceleration for alluvium sites. An expression for Ao was given previously as Equation (2-36b). All other seismological parameters required by the various procedures (e.g., duration of strong motion, Arias intensity) were determined using the empirical relations presented along with the respective procedures. 2.3.2 Discussion of expected trends The stress-based procedure was included in the parameter study to use as a benchmark for assessing the other procedures. The stress-based procedure has been thoroughly scrutinized, tested, and continually revised based on new laboratory and field observations. Accordingly, the trends in the predicted FS are assumed to be correct. This is not to say that the stress-based procedure is 100% accurate in predicting liquefaction, but rather the variation in predicted factor of safeties tend in the correct direction as parameters known to influence liquefaction are changed. The strain-based procedure was also included in this study. Although this procedure only provides the FS against gross slippage across particle contact surfaces, and not the FS against liquefaction, it provides information as to whether excess pore pressures will be generated. This information may be used as a lower bound for expected FS (i.e., if gross slippage across the contact areas is not predicted, all the procedures should predict FS > 1). Because the absolute values of the predicted FS will be unique to each procedure (e.g., a FS = 1.2 for one procedure does not mean the same thing as it does for another procedure), the focus of the parameter study is to examine the trends in the FS-profiles as influenced by the variation of parameters. One of the things examined is the variation in critical depth (i.e., the depth below the gwt corresponding to the lowest FS). The FS-profiles for the stress- and strain-based procedures are shown in Figures 2-20 and 2-21. From examination of these figures, the following observations are made: 

The critical depth is unaffected by the changes in N1,60.

52



The critical depth shifts downward as the elevation of the ground water table is lowered.



The FS increases as N1,60 increases and as the gwt is lowered. Both of these trends correspond to the observed laboratory behavior that the Capacity of the soil increases with increasing relative density and increasing effective confining stress.

The strain-based procedure predicts a high FS towards the surface of the profile. This is believed to be due to limitations in the empirical correlation for Gmax, (G/Gmax), and ’ vs. N1,60 at low effective confining stresses and is not the expected behavior of the soil. The interpretation given to the results from the strain-based procedure is that excess pore pressures will be generated for the entire depths of all the profiles.

53

NCEER (1997) 0

1

0

a)

1

b) 2

5

5

10

Depth (m)

3

Approximate Critical Depth

2 10 3

15

15 N1,60 = 15

20

20 Approximate Critical Depths

25

25 5

30

0

1 2 3

10 N1,60 = 15

1 2 Factor of Safety

3

30

0

1

2

3

Factor of Safety

Figure 2-20. FS-profiles for the stress-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

54

Dobry et al. (1982) 0

0

a) 5

Depth (m)

b)

10

2

5

1

5

N1,60 = 15

10

1

10

2

3

3

1

Approximate Critical Depth

N1,60 = 15

2

15

15

3 Approximate Critical Depths

20

20 Critical Depths difficult to determine

25

30

25

0

1

2

3

Factor of Safety

30

0

1

2

3

Factor of Safety

Figure 2-21. FS-profiles for the strain-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

55

2.3.3 Results of parameter study and summary of observed trends The results from Davis and Berrill (1982), Berrill and Davis (1985), Law, Cao, and He (1990), Trifunac No.1, Trifunac No.3, Trifunac No.4, Trifunac No.5, Kayen and Mitchell (1997), Running (1996), and Alkhatib (1994) are shown in Figures 2-22 to 2-31, respectively. From examination of these figures, the following observations are made: 

Similar to the stress- and strain-based procedures, none of the approaches predict a variation in the critical depth due to changes in N1,60, with several of the procedures not showing a clearly identifiable critical depth.



Unlike the stress- and strain-based procedures, none of the approaches predict a variation in the critical depth due to changes in the water table (i.e., the critical depth always corresponds to the depth of the water table).



Davis and Berrill (1982), Berrill and Davis (1985), Trifunac No.1, Trifunac No.3, Trifunac No.4, Trifunac No.5, and Running (1996) predict a continually increasing FS with depth.



Law, Cao, and He (1990) and Alkhatib (1994) show no variation of FS with depth. Based on the formulations of these procedures, the predicted FS probably correspond to an initial vertical effective confining stress of 100kPa. Depending on the stratigraphy of the profile and elevation of the gwt, basing the FS on 100kPa may be under or overly conservative.



The predicted FS-profiles by Davis and Berrill (1982), Trifunac No.1, Trifunac No.3, and Trifunac No.4 are overly sensitive to variations in N1,60. Given the variability of N1,60 values in actual “uniform” layers or profiles (e.g., Elton and Hadj-Hamou 1990), the predicted FS using these procedures are unreliable.



Running (1996) shows a decrease in FS in response to an increase in N1,60. This is contrary to observed behavior in the laboratory that Capacity increases with increasing Dr.



The FS-profiles predicted by Law, Cao, and He (1990), Kayen and Mitchell (1997), and Alkhatib (1994) are insensitive to changes in the elevation of the water table, assuming a constant N1,60. This is contrary to observed behavior in the laboratory that Capacity varies as a function of effective confining stress.

56

Davis and Berrill (1982) 0

a)

1

1

0

Approximate Critical Depth

b) 2

2

5

5 3

3

Depth (m)

10

10 N1,60 = 15

15

15

20

20 Approximate Critical Depths

25

25 5

30

10

1

N1,60 = 15

0 10 20 30 40 50 60 70 Factor of Safety

30

0

2

3

10 20 30 40 50 60 70 Factor of Safety

Figure 2-22. FS-profiles for Davis and Berrill (1982): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

57

Berrill and Davis (1985) 0

1

1

0

a)

b)

Approximate Critical Depth

2

2

5

5 3

Depth (m)

10

3

10 N1,60 = 15

15

15

20

20

25

25 5

30

0

1

10

Approximate Critical Depths

N1,60 = 15

2 3 4 5 6 Factor of Safety

7

8

30

0

1

2

12 3

3 4 5 6 Factor of Safety

7

8

Figure 2-23. FS-profiles for Berrill and Davis (1985): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

58

Law, Cao, and He (1990) 0

0

a)

1 b) 2

5

5 3

Depth (m)

10

10 N1,60 = 15

15

15 FS constant Critical Depths can’t be determined

20

20

FS constant Critical Depths can’t be determined

1 2 25

3

25 5 10

30 0.0

N1,60 = 15 1.0 1.5 0.5 Factor of Safety

2.0

30 0.0

0.5

1.0

1.5

2.0

Factor of Safety

Figure 2-24. FS-profiles for Law, Cao, and He (1990): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

59

Trifunac No.1 0

0 Approximate Critical Depth

5

1

1

2

2

5 3

3

Depth (m)

10

10 N1,60 = 15

15

15 Approximate Critical Depths

20

20

25

25 5 10

30

0

1 2

N1,60 = 15 150

300

30 0

Factor of Safety

3

150

300

Factor of Safety

Figure 2-25. FS-profiles for Trifunac No.1: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

60

Trifunac No.3 0

a)

0

Approximate Critical Depth

5

1

1

2

2

b)

5 3

3

Depth (m)

10

10 N1,60 = 15

15

15

20

20

25

25 5

30

0

1 2

N1,60 = 15

10 10

Approximate Critical Depths

20

30

40

50 30 0

Factor of Safety

10

20

30

3 40

50

Factor of Safety

Figure 2-26. FS-profiles for Trifunac No.3: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

61

Trifunac No.4 0

0

a) Approximate Critical Depth

5

1

1

2

2

b)

5 3

3

Depth (m)

10

10 N1,60 = 15

15

15 Approximate Critical Depths

20

20

25

25 5

30

0

10

1 2 3

N1,60 = 15

10 20

30

40

50 30 0

Factor of Safety

10

20

30

40

50

Factor of Safety

Figure 2-27. FS-profiles for Trifunac No.4: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

62

Trifunac No.5 0

0

a)

1

1

Approximate Critical Depth

b) 2

2

5

5 3

3

Depth (m)

10

10 N1,60 = 15

15

15

20

20

25

25 5

30

0

1

Approximate Critical Depths

N1,60 = 15

10 2

3

4

5 30 0

Factor of Safety

1

2

12 3 3

4

5

Factor of Safety

Figure 2-28. FS-profiles for Trifunac No.5: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

63

Kayen and Mitchell (1997) 0

1

1

0

a) Approximate Critical Depth

b) 2

2

5

5 3

3

Depth (m)

10

10 N1,60 = 15

15

15

20

20

N1,60 = 15

25

25 Approximate Critical Depths

5 10 30 0

1 2 3

30 1 2 Factor of Safety

3

0

1 2 Factor of Safety

3

Figure 2-29. FS-profiles for Kayen and Mitchell (1997): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

64

Running (1996) 0

a)

1

1

0

b)

Approximate Critical Depth

2

2

5

5 3

3

Depth (m)

10

10 N1,60 = 15

15

15

20

20

25

25 N1,60 = 15

30

Approximate Critical Depths

1 2

10 5

3

30 0

4

8 12 16 Factor of Safety

20

0

4

8 12 16 Factor of Safety

20

Figure 2-30. FS-profiles for Running (1996): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

65

Alkhatib (1994) 0

1

0

a)

b) 2

5

5 3

Depth (m)

10

10 N1,60 = 15

15

15 FS constant Critical Depths can’t be determined

20

25

20

1 2 3

25 5

30

FS constant Critical Depths can’t be determined

0

5

N1,60 = 15

10 10

15

20

25

30

Factor of Safety

30

0

5

10

15

20

25

30

Factor of Safety

Figure 2-31. FS-profiles for Alkhatib (1994): a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

66

2.3.4 Commentary on procedures A detailed commentary is not given for each procedure, rather the results of the parameter study generally speak for themselves. However, general commentary is given for several of the procedures, and the Arias intensity procedure proposed by Kayen and Mitchell (1997) is examined in depth. 2.3.4.1 Gutenberg-Richter approaches One of the assumptions central to Davis and Berrill (1982), Berrill and Davis (1985), Trifunac No.1, Trifunac No.2, Trifunac No.3, Trifunac No.4, and Trifunac No.5 is that energy dissipation in the soil due to material damping is proportional to 1/(’vo)0.5. The basis for this assumption is the laboratory study by Hardin (1965). However, Hardin’s study was conducted on dry sands (i.e., the effective confining pressure remains constant for the duration of the test). During the process of liquefaction, the effective confining stress is continually changing. Accordingly, it would seem more reasonable to interpret Hardin’s results as energy dissipation due to material damping is proportional to 1/(’v)0.5, where ’v is the effective overburden stress at a specific time, not the initial effective overburden stress. Davis and Berrill (1982), Trifunac No.1, Trifunac No.2, Trifunac No.3, Trifunac No.4, and Trifunac No.5 all assume a linear relationship between dissipated energy and excess pore pressure generation. As discussed in more detail in Chapter 4 and as revised in Berrill and Davis (1985), excess pore pressure generation varies as a function of the square root of dissipated energy. It is unclear as to whether the amax value used in Alkhatib (1994) should be for the rock outcrop or the soil surface. The correlation shown in Figure 2-18 between amax and NME appears to have been developed from site response analyses using scaled acceleration time histories. Based on this, it is assumed that amax corresponds to that for a rock outcrop. However, from examining the case histories presented in Alkhatib (1994), it appears that soil surface amax values were used. In computing the FS-profile shown in Figure 2-31, it was assumed amax corresponds to the soil surface. Furthermore, the

67

correlation shown in Figure 2-18 between amax and NME was developed from total stress site response analyses (i.e., SHAKE), and the Capacity correlation shown in Figure 2-19 was based on laboratory tests, which are inherently effective stress. It cannot be expected that the dissipated energies computed from total stress analyses will correspond to those from effective stress analyses. A more detailed discussion on this subject is presented in Chapter 5, Section 5.6. 2.3.4.2 Arias intensity approaches Because Kayen and Mitchell’s Arias intensity approach was considered the most promising for remedial ground densification design, it is reviewed more in depth than the others. From the parameter study, it may be observed that unlike the results from stressbased procedure shown in Figure 2-20, the FS profiles for Kayen and Mitchell’s Arias intensity approach do not vary in response to changes in the groundwater table (Figure 229). As stated previously, N1,60 correlates to the relative density (Dr) of the soil (e.g., Equation (2-58)). Therefore, the scenario of a varying water table with constant N1,60 is analogous to laboratory specimens prepared to the same Dr, but subjected to different effective confining pressures. In order for N1,60 to remain constant for all elevations of the gwt, it is inherently implied that N60 (i.e., the SPT penetration resistance normalized to 60% of the hammer energy) will change, which may be unrealistic. Although this parameter study is academic in nature, it highlights the Arias intensity approach’s failure to properly account for changes in effective confining pressure, as well as several of the other energy-based liquefaction evaluation procedures. In addition to the observation from the parameter study discussed above, the following simple examination provides further insight into Kayen and Mitchell’s Arias intensity procedure. Consider a liquefiable sand layer lying below a stiff desiccated crust, such as shown in Figure 2-32a. Due to the stiffness of the crust, it can be assumed that it will respond as a rigid body during an earthquake (i.e., the accelerations experienced at the surface of the crust will be the same as the accelerations experienced at its base). The earthquake induced stresses acting on a soil element in the liquefiable layer can be

68

modeled in the laboratory using a cyclic simple shear device, as represented in Figure 232b. a(t)

a)

b)

 vo 

W

Stiff Desiccated Crust

W A

t)

z

t) Liquefiable Layer Figure 2-32. a) Soil profile having a liquefiable layer lying below a stiff desiccated crust. b) Duplication of in-situ stresses using simple shear device on a soil sample from the liquefiable layer. From Newton’s second law, it can be shown that the acceleration time history of the rigid desiccated crust, a(t), and the shear stress time history, (t), acting on the top of the liquefiable layer are related according to Equation (2-61).

 (t ) 

a(t )   vo g

(2-61)

A simple rearrangement yields: a (t ) 

 (t )  g  vo

(2-62)

Substitution of Equation (2-62) into the expression for Arias intensity, given previously as Equation (2-46) and repeated below for a single component of motion, allows the Arias intensity to be expressed as a function of shear stress time history. Ih  Ih 

 2g



dur 0

a 2 (t ) dt

(2-63)

dur g    2 (t ) dt 2 2 vo 0

(2-64)

69

Using Equation (2-64), the Arias intensity required to cause liquefaction (Ih,l) can be computed for a laboratory specimen subjected to cyclic simple shear. Assume that the laboratory specimen is subjected to a sinusoidal loading having amplitude max and frequency f. If the sample liquefies in n cycles of loading, the Arias intensity required to cause liquefaction is given by Equation (2-65). I h ,l 

2 ng  max  f 4 vo2

(2-65)

This expression is revealing in two ways. First, Ih,l is independent of effective confining stress. This verifies the results of the parameter study, which showed the procedure is insensitive to changes in the elevation of the groundwater table. Second, Ih,l (i.e., Capacity) is a function of the frequency of loading. However, laboratory studies have shown that for a given amplitude load, the number of cycles required to induce liquefaction is relatively independent of the frequency of the applied load (e.g., Arango 1994). The obvious question becomes “why does the field data shown in Figure 2-16 show a reasonable separation of points representing liquefaction and no-liquefaction?” The answer to this can be understood by reformulating the expression for Arias intensity. By observation, Equation (2-63) is similar in form to the equation for root mean square acceleration (arms) given as Equation (2-66).

a rm s  where:

dur 1   a 2 (t ) dt dur 0

(2-66)

arms

= Root mean square acceleration.

a(t)

= Acceleration time history.

dur

= Duration of earthquake motion.

A simple rearrangement yields:



dur 0

2 a 2 (t ) dt  dur  a rm s

(2-67)

70

Substituting Equation (2-67) into Equation (2-63) and multiplying by rb allows the Arias intensity of the earthquake motions at depth in a soil profile (Ihb,eq) to be expressed as a function of the duration of the earthquake shaking (dur) and arms:

I hb, eq 

 2g

2  dur  arm s  rb

(2-68)

Seismological studies have shown (Hanks and McGuire 1981):

2

a max 3 a rm s

1 For this illustration, it is assumed a rm s   a max . Substitution of this relation into 3 Equation (2-68) yields: I hb,eq 

0.056 2  dur  a max  rb g

(2-69)

Dobry et al. (1978) proposed the following relationship between the duration of strong motion (dur) and magnitude (M). dur = alog(0.432M – 1.83)

(2-70)

By empirical observation, the duration of strong motion may also be approximated as:   1  dur  25    MSFAndrus Stokoe 

2

(2-71)

MSFAndrus-Stokoe is the magnitude scaling factor relation proposed by Andrus and Stokoe (Youd and Noble 1997) for use in the stress-based liquefaction evaluation procedure. This relation was presented previously as Equation (2-5b) and is repeated below.

M  MSFAndrusStokoe     7.5 

3.3

(2-5b)

A comparison of Equations (2-70) and (2-71) is shown in Figure 2-33.

71

50 Equation (2-70) Equation (2-71)

Duration (sec)

40 30 20 10 0

5

6

7

8

Magnitude Figure 2-33. Comparison of expressions for duration of strong ground motion as a function of magnitude.

Substituting Equation (2-71) into Equation (2-69) yields: 2

  a  1  I hb,eq  1.4g   max   rb    g   MSFAndrusStokoe 

2

(2-72)

In reviewing the liquefaction case histories used by Kayen and Mitchell (1997) in developing their liquefaction chart, a reasonable average of the of the total vertical stresses at the critical depths may be assumed: vo = 200kPa. Based on this assumption, 1.4g  0.00255(0.65vo)2, where g has units of (m/sec2). Furthermore, as shown in Figure 2-34, rb  rd2, where rb and rd are the "depth reduction factors" for the Arias intensity and stress-based liquefaction evaluation procedures, respectively. The curves shown in this figure represent the means, about which considerable scatter exists. Although the curves in Figure 2-34 do not lie on top of each other, there is considerable overlap in the scatter corresponding to each of the curves; scatter not shown in figure.

72

rb and (rd)2 0

0

0.2

0.4

0.6

0.8

1

Depth (ft)

10

20 rb (rd)2 30

40

50 Figure 2-34. Comparison of the “depth reduction factors” rb for the Arias Intensity approach and (rd)2 from the stress-based approach.

Using the above approximations, the Arias intensity of the earthquake motions at depth in a soil profile (Ihb,eq) can be written:   a 1  I hb,eq  0.00255   0.65  max   vo  rd  g MSFAndrusStokoe  

2

(2-73)

It may be observed that the expression inside the parentheses is the amplitude of the earthquake-induced cyclic stress adjusted to a M7.5 event (refer to Equations (2-1) and (2-4)).

I hb,eq  0.00255   M 7.5 

2

(2-74)

For the case of liquefaction where FS =1.0 (i.e., M7.5 = CRR’vo), Equation (2-74) becomes:

I hb,l  0.00255  CRR   ' vo 

2

(2-75)

73

Again, in reviewing the liquefaction case histories used by Kayen and Mitchell (1997) in developing their liquefaction chart, a reasonable average of the of the initial effective vertical effective stresses at the critical depths may be assumed: ’vo = 100kPa. Based on this assumption, CRR and Ihb,l are related as: CRR 

I hb,l

(2-76)

5

A comparison of the CRR curve, given previously as Equation (2-7), and 0.2(Ihb,l)0.5, where Ihb,l is quantified by Equation (2-50), is shown in Figure 2-35. 0.6

No Liquefaction Liquefaction CRR

0.5

I hb,l 5

CRR

0.4 0.3 0.2 0.1 0.0

0

5

10

15

20

25

30

N1,60cs Figure 2-35. Comparison of Arias intensity liquefaction curve and CRR.

Attention is now returned to the question “why does the field data shown in Figure 2-16 show a reasonable separation of points representing liquefaction and no-liquefaction?” The Arias intensity approach proposed by Kayen and Mitchell (1997) is an alternate formulation of the stress-based procedure. However, this alternate form introduces two errors: a relative insensitivity of the soil Capacity to effective confining stress and an

74

erroneous sensitivity to frequency. The magnitudes of these errors are probably small for the case histories used to develop the liquefaction chart. This is because for the cases analyzed there is not a large variation in the initial vertical effective stresses at the critical depth for liquefaction, and the frequency content of western US earthquakes probably does not vary that much. However, it should be expected that the induced errors may be significant if the procedure is used to perform liquefaction evaluations where conditions differ significantly from those of the data base used in deriving the procedure (e.g., the upstream toe of a dam located in the central or eastern US). Finally, it needs to be emphasized that Equation (2-73) and all subsequently derived expressions are only valid around the critical depth of liquefaction (i.e., vo  200kPa) and do not apply to other depths. This limitation can be understood by comparing the Ihb,eq values computed using Equations (2-72) and (2-73) at the surface of a profile. The Ihb,eq value computed using Equation (2-72) will be non-zero, while the value computed using Equation (2-73) will equal zero. This dichotomy results from the surface of the soil profile being a zero shear stress boundary (i.e., M7.5 = 0) but the Arias intensity of the surface motion being non-zero and typically greater than the Arias intensities of the motions at depth. However, because the data points used by Kayen and Mitchell to derive their Capacity curve correspond to the critical depths of liquefaction, the depth limitation of Equation (2-73) does not detract from the hypothesis: the reasonable separation of points representing liquefaction and no-liquefaction in Figure 2-16 is because the Arias intensity procedure is an alternate form of the stress-based procedure. 2.4 Conclusions A significant amount of work has been done in energy-based evaluations of liquefiable soils, and several pioneering evaluation procedures have been developed (e.g., Davis and Berrill 1982 and Kayen and Mitchell 1997). Although a lot can be learned from the procedures reviewed, none are considered comprehensive enough in their present state of development for the use in remedial ground densification design. This does not preclude the use of the procedures for performing earthquake liquefaction evaluations, or upon further development, possible use in remedial densification design.

75

Appendix 2a: Normalization of measured SPT N-values This appendix outlines the NCEER(1997) and Youd et al. (2001) recommended normalization factors for measured SPT N-values for overburden pressure, hammer energy, borehole diameter, rod length, and sampling method. Measured SPT N-values should be normalized as follows:

N1,60  N  C N  C E  C B  C R  Cs where:

(2a-1)

N

= Measured SPT N-value.

CN

= Normalization factor for overburden pressure.

CE

= Normalization factor for hammer energy.

CB

= Normalization factor for borehole diameter.

CR

= Normalization factor for rod length.

CS

= Normalization factor for sampler method.

Values for the different normalization factors in Equation (2a-1) are listed in the following table.

76

Table 2a-1. SPT Normalization and Correction Factors. Factor Test Variable Term

Correction 0 .5

1

Overburden Pressure

Energy Ratio

Borehole Diameter

Rod Length2

Sampling Method

CN

Donut Hammer

CE

 Pa     '  vo  CN  1.7

0.5 to 1.0

Safety Hammer

0.7 to 1.2

Automatic-Trip Donut-Type Hammer

0.8 to 1.3

65mm to 115mm

CB

1.0

150mm

1.05

200mm

1.15

< 3m

CR

0.75

3m to 4m

0.8

4m to 6m

0.85

6m to 10m

0.95

10m to 30m

1.0

Standard Sampler

CS

Sampler without Liners

1.0 0.1 to 1.3

(Modified from Skempton 1986 and Robertson and Wride 1998) 1

The effective overburden pressure should be the value corresponding to that at the time of drilling and testing. A higher groundwater level might be assumed for conservatism in the liquefaction resistance calculations.

2

Rod corrections were not applied for lengths greater than 3m in the formulation of the simplified procedure; therefore, corrections are not required in applying the procedure for lengths greater than 3m.

77

Chapter 3. Mechanism and Mathematical Representation of Energy Dissipation 3.1 Introduction In the previous chapter, a brief review of existing energy-based liquefaction evaluation procedures was presented. Expressions were given quantifying the energy Demand imparted to the soil by the earthquake, but physical interpretations of these expressions were limited. Also, various terms were used without full description (e.g., material damping and dissipated energy). Although the diverse formulations of the liquefaction evaluation procedures necessitated the presentation style of Chapter 2, the purpose of this chapter is to provide better physical insight into the energy imparted to the soil and the dissipation mechanisms. A brief treatise is also given to the equivalent linear technique for mathematical modeling of energy dissipation. Finally, a consistent set of expressions is derived for computing the energy dissipated in cyclic triaxial, simple shear, and hollow cylinder triaxial-torsional shear tests. 3.2 Mechanisms of Energy Dissipation in Sands As seismic waves propagate through soil, a portion of their energy dissipates, resulting in a reduction in the amplitude of the waves. With the exceptions of the Arias intensity approaches, all of the energy-based Demand expressions presented in the previous chapter attempt to quantify the portion of energy that dissipates in the soil as some fraction of the seismic wave energy arriving at the site. For cohesionless soils, the dominant mechanism of energy dissipation is the frictional sliding at grain-to-grain contact surfaces (Whitman and Dobry 1993). Additionally, if the soil is saturated, energy also dissipates from the viscous drag of the pore fluid moving relative to the soil skeleton. The contributions from other mechanisms, such as particle breakage, are relatively insignificant for most soils and are not discussed further. The conceptual relationship between the energy imparted to the soil and that dissipated by friction and viscous drag is shown in Figure 3-1, where the fraction attributed to viscous drag is exaggerated for illustrative purposes. The stored energy represents that portion of the input energy that continues to propagate through the soil column, and accordingly, at the end of the

78

shaking, is equal to zero. Each of the energy components shown in Figure 3-1 is examined in more detail. 12 10

Energy (10)

8 Stored Energy

6 4

Energy Dissipated by Friction

2

Energy Dissipated by Viscous Drag

0

0

10

20

30

40

50

Total Dissipated Energy

60

Time (sec) Figure 3-1. Conceptualization of the cumulative energy imparted to the soil by an earthquake and the portions dissipated by frictional and viscous mechanisms. (Loosely adapted from Hall and McCabe 1989).

3.2.1 Frictional Dissipation Mechanism In relation to liquefaction, the portion of the energy dissipated by friction is of considerable importance. This is because liquefaction requires the complete breakdown of the soil structure, which inherently involves slippage of contact surfaces as the particles rearrange. The physics of energy dissipation by friction can be understood by the interaction of two elastic spheres under the action of normal and shear forces, which has been study at depth by several investigators: Mindlin (1949); Mindlin et al. (1951); Mindlin (1954); Duffy and Mindlin (1956); Johnson (1961); Goodman and Brown (1962); Deresiewicz (1974); Dobry et al. (1982). The following discussion on the interaction of two elastic spheres is based largely on this work, unless otherwise noted.

79

The contact forces and corresponding stresses between two spheres are shown in Figure 3-2. The radius of the contact area (a) is a function of both the applied normal force and the elastic properties of the spheres. This is shown in Figure 3-3, along with the variation of the normal stress (c) across the contact area. As the tangential force T increases, there is a proportional increase in the lateral displacement () between the centers of the spheres. However, gross (or complete) sliding across the entire contact area does not occur until T = fN, where N is the normal force and f is the coefficient of friction of the contact surfaces. The progression of slippage across the contact area as T increases from 0 to fN is shown in Figure 3-4. As illustrated in this figure, sliding starts at the outer radius of the contact area and progresses inward, thus forming an annulus of slippage surrounding a zone of no slippage for 0 < T < fN. A series of laboratory experiments verified this theoretical behavior (e.g., Deresiewicz 1974). A photograph from one of the experiments is shown in Figure 3-5 in which two spheres were in contact and subjected to an oscillating tangential force 0 < T < fN. In this figure, the wear marks formed by the annulus of slippage can be clearly identified. From the above, it can be seen that energy is dissipated through friction, even before gross sliding across the entire contact surface occurs. Accordingly, scenarios can be hypothesized where the applied tangential force is 0 < T < fN, and an infinite amount of energy could be dissipated without the occurrence of liquefaction. The strain-based liquefaction evaluation procedure presented in Chapter 2 may be used to screen for these scenarios, which was the purpose for including it in the parameter study. This scenario can only exist if the induced strain () is less than the threshold strain (i.e.,  < th), where the threshold strain is that corresponding to T = fN. It can be further hypothesized that, for a transient earthquake type loading, energy is dissipated prior to the arrival of a pulse of sufficient amplitude to induce a shear strain that exceeds the threshold strain (i.e.,  >

th). From the examination of typical earthquake acceleration time histories, such as shown in Figure 3-6, it can be seen that the large amplitude shear waves arrive early in the record. From this, it is assumed that if the threshold strain is exceeded, it will occur early in the shaking, and little energy will be dissipated prior to its exceedance.

80

/2

c (no slip)

/2

a

R

c

N fc

c (slip)

N

T

T

T

T

c N

N

a

/2

/2

Figures 3-3 and 3-4 provide a further breakdown of the normal and shear stresses across the contact surfaces.

R

R = radius of spheres c = normal stress between the spheres N = normal force c = shear stress between the spheres T = tangential force a = radius of the contact area  = lateral displacement of spheres c = radius of the non-slip contact area f = coefficient of friction between the spheres Figure 3-2. Contact forces and stresses between two equal sized spheres of radius R.

81

c  3 (1   2 ) R N  a  4E  

1

3

N contact surfaces

N

c 

a



3N a2   3 2a



0.5



R = radius of spheres N = normal force a = radius of the contact area  = Poisson’s ratio

c = normal stress between the spheres  = distance from the center of the contact area E = Young’s modulus

Figure 3-3. Radius of the contact area (a) and the normal stress (c) across the contact area of the two spheres.

82

c (no slip)

/2

/2 fc

N T

T

N

c (slip)

N

T

T

/2

(slip) T N c=0 a

c a

a Gross slippage

No slippage Zone of slippage

T=0 Plan view of the contact area 3

c

/2

a, c

T c   1  f N a

fc N T

c N a

a, c

/2

/2

0 < T < fN

Zone of no slippage

T  fN

23 3(2   ) (1   ) f N   T       1 1 4Ea   f N  

c = normal stress between the spheres c = shear stress between the spheres

R = radius of spheres N = normal force T = tangential force  = lateral displacement of spheres f = coefficient of friction of the spheres E = Young’s modulus

a = radius of the contact area c = radius of the non-slip contact area  = Poisson’s ratio

Figure 3-4. Relative slippage of the spheres: no slippage at T = 0; gross slippage at T = fN. At intermediate values 0 < T < fN, there is an annulus of slippage surrounding a zone of no slippage.

83

Outer boundary of contact area

Zone of slippage c a

Zone of no slippage

Figure 3-5. Photograph of an actual sphere, which in contact with another sphere was subjected to an oscillating tangential force 0 < T < fN. Wear marks formed by the annulus of slippage between the spheres can be clearly identified. (Adapted from Deresiewicz 1974).

Approximate arrival time of shear waves

Acceleration (g)

0.1 Time (Seconds) 0

5

10

15

20

25

-0.1

Figure 3-6. Typical acceleration time history with the arrival of the large amplitude shear waves occurring early in the record.

84

30

3.2.2 Viscous Dissipation Mechanism Viscosity is the measure of a fluid’s resistance to flow. Viscous drag is the force resisting the relative movement of a fluid and a solid and is analogous to the frictional force between two solids. The theory by Biot (1956) may be used for the theoretical evaluation of energy dissipation by viscous mechanisms in soils. Hall (1962) and Hall and Richart (1963) outline the results of a laboratory study examining the influence of various parameters on the total energy dissipated in granular materials, including the viscosity of the pore fluid. By comparing the energy dissipated in dry samples to similar saturated samples, the relative contributions from friction and viscous drag can be discerned. In their study, Hall (1962) and Hall and Richart (1963) performed a series of resonant column tests where the specimens were excited at their first mode of vibration and then set in free vibration, while the amplitude of the rotations being recorded. The decay in the rotational amplitudes results from energy dissipation. Accordingly, by comparing the rotational amplitude decay in similar saturated and dry specimens, the relative contributions of the viscous and frictional energy dissipation mechanisms can be examined. Such comparisons are shown in Figures 3-7 and 3-8 for Ottawa sand and glass beads, respectively. In these figures, the vertical axes are the logarithmic decrement: ln(ui/ui+1), where ui is the peak rotational displacement of ith cycle, and the horizontal axes are the double amplitudes of the rotations. As may be observed from these figures, the logarithmic decrement for the saturated specimens shows less variation with rotational amplitude than the dry specimens (i.e., the curves for the saturated samples have a flatter slope than the curves for the dry samples). Therefore, the portion of energy dissipated by viscous mechanisms increases as the amplitude of the rotations decreases (Hall 1962). The specimens tested were 1.59in diameter and 10.8in long and were subjected to torsional oscillations. Accordingly, the amplitude of the induced shear strains varied across the diameter of the samples. Assuming the deformations in the samples were linearly distributed, the outer surfaces of the specimens were subjected to shear strains of

85

approximately 0.015%, for the largest amplitude oscillations. This is slightly larger than the threshold strain determined by Dobry et al. (1982), which was conservatively estimated as 0.01%. However, pre- and post-test measurements showed little-to-no differences in the void ratios, implying little-to-no change in density and effective confining stress in the dry and saturated samples, respectively. This confirms that the results shown in Figures 3-7 and 3-8 for dry and saturated samples are comparable. Unfortunately, similar comparisons between dry and saturated samples subjected to large shear strains do not exist due to the tendency of the samples to densify, resulting in increased density of dry samples and elevated pore pressures in saturated samples. However, from extrapolation of the trends shown in Figures 3-7 and 3-8, Whitman and Dobry (1993) pose a corollary interpretation to that of Hall (1962) stated above: the portion of energy dissipated by frictional mechanisms increases with increasing rotational amplitude (or increasing strain) and becomes the dominant mechanism in both saturated and dry specimens subjected to large strains, such as those of interest in earthquake

Logarithmic Decrement

engineering (Whitman and Dobry 1993).

0.2

691psf 1410psf Dr = 48%, saturated

0.1 677psf 1152psf 2320psf 5010psf Dr = 37%, dry

0.05

0.02 10-4

10-3 Double Amplitude, Radians

10-2

Figure 3-7. Comparison of the variation of logarithmic decrement with amplitude for dry and saturated Ottawa sand in torsional oscillation. (Adapted from Hall and Richart 1963).

86

Logarithmic Decrement

0.2 0.1 dry

saturated 734psf 1469psf 3600psf 7340psf Dr = 89%

0.05

0.02 10-5

10-3 10-4 Double Amplitude, Radians

Figure 3-8. Comparison of the variation of logarithmic decrement with amplitude for dry and saturated glass beads in torsional oscillation. (Adapted from Hall and Richart 1963).

3.3 Modeling of Energy Dissipation 3.3.1 Hysteresis loops As discussed above, if two spheres are in contact acting under a normal force (N) and tangential force (T), partial or total slippage of the contact area will occur. As a result, if the tangential force is applied, removed, and then re-applied, a plot of the resulting forcedisplacement relation scribes a hysteretic loop, such as shown in Figure 3-9a. The mathematical expressions defining the shape of the loop are given by the expressions in Figures 3-3 and 3-4. Similar to the interaction of two spheres, the force-displacement response of an assemblage of particles also scribes a hysteresis loop, which is often represented by bi-linear, hyperbolic, or Ramberg-Osgood models (e.g., bi-linear: Idriss and Seed 1968; hyperbolic: Lee and Finn 1978; Ramberg-Osgood: Streeter et al. 1973).

87

a)

T

b) Dissipated energy

 Dissipated energy per unit volume





Figure 3-9. Hysteresis loop resulting from: a) the application and removal of a tangential force T. b) the application and removal of a shear stress . The area bound by the hysteresis loop quantifies the energy dissipated in the system of particles. Similar to Figure 3-9a, a corresponding plot can be made in terms of stress () and strain (), as shown in Figure 3-9b. For this case, the area bound by the hysteresis loop quantifies the dissipated energy per unit volume of material (W). Laboratory studies have shown that the shape of the hysteresis loop is independent of the load rate for dry sands. This implies that for a given amplitude load, the quantity of energy dissipated by the frictional mechanism is independent of the frequency of the applied loading (Hardin 1965). On the contrary, energy dissipated by viscous mechanisms is directly proportional to the frequency of the applied loading. The frequency dependency of energy dissipated by viscous mechanisms can be understood from the viscous dampers often used on screen doors: if you close a screen door with a viscous damper quickly, it takes much more effort than if you close the same door slowly. For saturated, undrained samples, the area bound by the hysteresis loops represents the energy dissipated by all mechanisms, and the contributions from friction and viscous drag cannot be discerned. Hysteresis loops from a typical stress-controlled cyclic triaxial test conducted on a saturated, undrained sample are shown in Figure 3-10.

88

Deviator Stress (kPa)

80 60 40 20 0 -20 -40 -60

-6 -4 -2 0 2 4 6 Axial Strain (%)

8

Figure 3-10. Hysteresis loops from a stress-controlled cyclic triaxial test. Referring back to Figure 3-1, “stored energy” and the “total dissipated energy” are related to the hysteresis loop as shown in Figure 3-11.

 Total dissipated energy (W)

 Stored energy

Figure 3-11. Graphical definitions of stored energy and total dissipated energy.

3.3.2 Equivalent Lineariztion and Damping Ratios As mentioned above, the hysteretic response of an assemblage of particles is often approximated by bi-linear, hyperbolic, or Ramberg-Osgood models. However, even with these relations, a second order, non-linear partial differential equation is needed to describe the phenomena associated with wave propagation. Depending on the response

89

quantity of interest, a further simplification referred to as equivalent linearization may be employed. This technique is based on the idea of replacing a non-linear system by a related linear system in such a way that the difference between the two is minimized in some statistical sense (Jacobsen 1930, Iwan and Yang 1971, Dobry 1970, and Dobry et al. 1971). The equivalent linear model is used in the site response computer program SHAKE (Schnabel et al. 1972). The rheological models for the non-linear hysteretic and linearized hysteretic systems, and the corresponding hysteresis loops, are shown in Figure 3-12. Although the linearized model is based on a visco-elastic material, the viscous damping coefficient () can be set inversely proportional to the circular frequency of the applied loading ( ) to remove the frequency dependence of the hysteresis loop (Hardin 1965). In selecting a linearized hysteretic system, it is typical to equate the secant modulus (G) and the area bound by the hysteresis loop (W) to those of the non-linear hysteretic system. The secant modulus is commonly defined as the slope of a line drawn through the origin and the point of load reversal. However, the hysteresis loop scribed by the linearized hysteresis model is elliptical with no clear point of load reversal. For this case, the secant modulus is drawn through the origin and the point of maximum shear strain.

90

time





Gtan = f ()

time =

G

Non-linear Hysteretic Model

2GD 

Linearized Hysteretic Model





G

max

G



max



Point of maximum shear strain

Gtan Point of tangency Figure 3-12. Rheological models and corresponding hysteresis loops for hysteretic and equivalent linear materials.

The full expression for the damping coefficient used in the linearized hysteretic model is:



2GD

(3-1)



where:



=

Damping coefficient (units of stress  sec).

G

=

Secant shear modulus (units of stress).

D

=

Damping ratio (dimensionless).



=

Circular frequency (rad/sec).

More than any other quantity, reference is made to the damping ratio (D) when describing the soil’s ability to dissipate energy (i.e., material damping). Using the

91

definitions for W and W1 given in Figure 3-13 (Jacobsen 1960), the damping ratio (D) is commonly given as Equation (3-2).



G

max

max

W1



1 W   max  max 2

Figure 3-13. Quantities used in defining damping ratio (D). D

1 W1  4 W

where: D

(3-2) =

Damping ratio.

W1 =

Dissipated energy per unit volume in one hysteretic loop.

W

Energy stored in an elastic material having the same G as the visco-elastic material.

=

From Equation (3-1), it can be observed that the damping coefficient  is a function of both the shear modulus (G) and the damping ratio (D). In turn, both G and D are functions of the induced shear strain (). Extensive laboratory studies have been conducted to develop shear modulus and damping degradation curves, which empirically relate G and D to . The shear modulus degradation curves were introduced in Chapter 2 (Section 2.2.2) and are developed from strain controlled cyclic tests on saturated-drained, or dry specimens. By subjecting soil samples to cyclic shear strains of different amplitudes, the corresponding secant shear moduli are determined. The shear modulus degradation curve is a plot of the secant shear modulus as a function of shear strain amplitude. Typically, the shear moduli are normalized by the secant shear modulus for  = 10-4% (i.e., Gmax). This process is illustrated below. 92



G2



G1

2

1

1 < 2 < 3

G3



3

1.0

G/Gmax

0.8 0.6 0.4

G1/Gmax G2/Gmax G3/Gmax

0.2 0.0 0.0001

1 0.001

2

0.01 Shear Strain (%)

3 0.1

Figure 3-14. Development of shear modulus degradation curves.

93

1

Damping degradation curves are developed in a similar fashion. However, instead of computing the area bound by the hysteresis loops directly, which is required for determining the damping ratio per Equation (3-2), the amplitude decay of a sample in free vibration is often used. In this procedure, the soil sample is oscillated at its resonant frequency and then the exciting force is removed, while the free vibration motion of the sample is recorded. The damping ratio (D) is related to the peak displacements of sequential cycles of free vibrating soil as (Chopra 1995): D

 u  1  ln  i  2  u i 1 

for D  20%

(3-3)

where: D

=

Damping ratio.

ui

=

Peak displacement of cyclic i of the soil sample in free vibration.

ui+1 =

Peak displacement of cyclic i+1 of the soil sample in free vibration.

The term ln(ui/ui+1) in Equation (3-3) is referred to as the “logarithmic decrement” and was used as the vertical axis of the data plotted in Figures 3-7 and 3-8. As stated in Chapter 2, the Ishibashi and Zhang (1993) shear modulus degradation curves were used throughout this research. These curves were selected because they are presented in equation form and are expressed as functions of both effective confining stress and plasticity index (Ip). Expressions for the shear modulus degradation curves were given previously in Chapter 2 (i.e., Equation (2-16)). The following empirical expression is for the damping degradation curves:



0.333  1 e D( , I p )  2

0.0145I 1p.3

 0.586    

2   G  G     1.547   1  Gmax ( , I p )  Gmax ( , I p )  

(3-4)

Plots of the shear modulus and damping degradation curves for various mean initial effective confining pressures are shown in Figure 3-15 for a non-plastic soil.

94

0.35

Damping

0.30

’mo 

0.25 0.20 0.15 0.10 0.05 0.00 0.0001

0.001

0.01 Shear Strain (%)

0.1

1

0.1

1

1.0

G/Gmax

0.8 0.6 0.4

’mo 

0.2 0.0 0.0001

’mo = 13 psf ’mo = 67 psf ’mo = 173 psf ’mo = 333 psf

0.001

0.01 Shear Strain (%)

’mo = 547 psf ’mo = 813 psf ’mo = 1133 psf ’mo = 1507 psf

’mo = 1933 psf ’mo = 2413 psf ’mo = 2947 psf ’mo = 3533 psf

Figure 3-15. Plots of the Ishibashi and Zhang (1993) shear modulus degradation curves for various initial mean effective confining stresses. 3.3.3 Final Comments on D The damping ratio D used above differs from that of the same name used in conjunction with viscously damped oscillators. D is defined for the linearized hysteretic material without introducing the mass (m), and therefore is a material property. The definition for

95

the damping ratio () for viscously damped oscillators includes m, and is therefore a system property (Dobry et al. 1971). In relation to the energy dissipation,



1 W1  n   4 W 

where: 

=

(3-5) Damping ratio of viscously damped oscillators.

W1 =

Dissipated energy per unit volume in one hysteretic loop.

W

=

Energy stored in an elastic material having the same G as the visco-elastic material.



=

Frequency of the applied loading (rad/sec).

n =

Natural frequency of the SDOF system (rad/sec).

As may be observed from Equations (3-2) and (3-5), at resonance (i.e., n = ), D = . 3.4 Use of Dissipated Energy to Quantify Capacity As defined in this thesis, the energy Capacity of soil is the cumulative energy dissipated up to the point of liquefaction. In light of the above discussions on energy dissipation mechanisms, energy Capacity is re-examined. Given that the energy dissipated by the frictional mechanism is directly related to slippage of particle contact surfaces and is independent of the frequency of the applied loading (Hardin 1965), it seems an appropriate measure of soil Capacity. However, because the energy dissipated by the various mechanisms (e.g., viscous and frictional) cannot be easily discerned from the laboratory and field data, the energy dissipated by the frictional mechanism is approximated by the total dissipated energy. This is justified because at the large strain amplitudes of interest in earthquake engineering, the frictional mechanism is expected to dominate (Section 3.2.2). Accordingly, the total amount of energy dissipated up to the point of liquefaction should be relatively independent of frequency for large amplitude loads. Contrary to this, for small amplitude strains, the viscous mechanism of energy dissipation may be significant, and the amount of the energy dissipated by this mechanism is directly proportional to the frequency of the applied loading. For a load of arbitrary amplitude, it is expected that the total dissipated energy up to the point of

96

liquefaction is either independent or increases proportionally with the frequency of the applied loading. Now compare these trends with those expected using Arias intensity to quantify soil Capacity. As may be recalled from Chapter 2 (Section 2.3.4.2), the Arias intensity of the load required to induce liquefaction in a sample subjected to stress-controlled sinusoidal input motion is inversely proportional to the frequency of the applied load. This was shown by Equation (2-65), which is repeated below. I h ,l 

2 ng  max  f 4 vo2

where:

Ih,l

(2-65)

= Arias intensity required to induce liquefaction.

max = Amplitude of the applied loading. f

= Frequency of the applied loading.

g

= Acceleration due to gravity.

n

= Number of cycles to failure.

vo

= Total vertical stress at depth z.

Both total dissipated energy and Arias intensity are functions of the applied loading. However, for loads inducing large amplitude strains (e.g., earthquake loading), the total dissipated energy should be relatively insensitive to the frequency of the loading, while Arias intensity retains its frequency dependency. 3.5 Computing Dissipated Energy from Laboratory Tests In the above sections, the relationship between dissipated energy and the hysteresis loop was presented. However, the presentation was intentionally vague in specifying which stress-strain hysteresis loops should be used in computing the dissipated energy per unit volume of material. The hysteretic loop that should be used is a function of load path. Furthermore, from a review of the literature, there appears to be no clear consensus on which loops should be used, even for a given load path. For example, in computing W from cyclic triaxial test data, it appears that Simcock et al. (1983) used the axial stress versus axial strain hysteresis loops, Alkhatib (1994) used the deviator stress versus axial

97

strain hysteresis loops, and Ostadan et al. (1996) used 3/4 times the deviator stress versus axial strain hysteresis loops. The purpose of this section is to outline a fundamentally sound approach for computing dissipated energy for arbitrary load paths. Starting with general expressions for incremental work, a consistent set of equations is derived for computing the dissipated energy per unit volume for cyclic triaxial, cyclic simple shear, and cyclic torsional shear tests. The stress components acting on a cubical element are shown in Figure 3-16, using both tensor-suffix notation and engineering notation. x1

11 a)

b)

12

13

xy 31

21

32

x2

22

xz zx

33 yx

23 x3

x

x

zy

z

y

yz z

y

Figure 3-16. Stresses acting on a differential element a) tensor-suffix notation b) engineering notation. (Adapted from Schofield and Wroth 1968). The increment in the energy dissipated per unit volume of material in the cubical element using indicial notation is:

dW   ij d ij

(3-6)

In this expression ij and dij are the stress and incremental strain tensors, respectively, and are given as:

98

 11  12  13   ij   21  22  23   31  32  33 

 d 11 d 12 d ij  d 21 d 22 d 31 d 32

d 13  d 23  d 33 

(3-7)

Expansion of Equation (3-6) yields:

dW   11d 11   12 d 12   13d 13   21d 21   22 d 22   23d 23

(3-8)

  31d 31   32 d 32   33d 33 Assuming symmetric tensors (i.e., ij = ji and dij = dji), this expression reduces to: dW   11d 11   22 d 22   33d 33  2 12 d 12  2 13d 13  2 23 d 23

(3-9)

Using the following relations, Equation (3-9) can be written in engineering notation, as given by Equation (3-11).

 x   11

d x  d 11

 y   22

d y  d 22

 z   33

d z  d 33

 xy   12

d xy  2d 12

 xz   13

d xz  2d 13

 yz   23

d yz  2d 23

(3-10)

dW   x d x   y d y   z d z

(3-11)

  xy d xy   xz d xz   yz d yz

For an arbitrary load path, the cumulative energy dissipated per unit volume of material (W) can be computed by integrating either Equation (3-9) or (3-11): W   dW

(3-12)

99

3.5.1 Cyclic Triaxial Test The following expressions relate the common notation used in reference to cyclic triaxial tests with those of the tensor-suffix and engineering notations presented above.

 '1   11   x d a  d 11  d x

 '3   22   33   y   z

(3-13)

d h  d 22  d 33  d y  d z where:

’1

=

The major principal effective stress.

’3

=

The minor principal effective stress.

da

=

The increment in axial strain.

dh

=

The increment in lateral strain.

For the cyclic triaxial test, the following boundary conditions apply:

 21   12   23   32   13   31  0 Substitution of the boundary conditions into Equation (3-9) yields: dW   '1 d a  2 ' 3 d h

(3-14)

This expression can be further simplified by using the following relations for deviatoric stress (d) and Poisson’s ratio ().

 d   '1  ' 3

(3-15a)

h a

(3-15b)

 

Substituting these expressions into Equation (3-14) yields: dW   d d a   '3 (1  2 )d a

(3-16)

Finally, for a saturated undrained tests, = 0.5, and Equation (3-16) reduces to: dW   d d a

(3-17)

Using the trapezoidal rule to integrate Equation (3-17), the dissipated energy per unit volume of material (W) can be determined by:

100

W 

1 n 1  ( d ,i 1   d ,i )( a,i 1   a,i ) 2 i 1

where:

(3-18)

W =

Dissipated energy per unit volume of material up to the nth load increment.

d,i

=

The ith increment in deviatoric stress.

a,i

=

The ith increment in axial strain.

n

=

Total number of increments.

Equation (3-18) is illustrated in Figure 3-17 for an actual hysteresis loop from a stresscontrolled cyclic triaxial test.

Deviator Stress (kPa)

80 60 40 20

a,i+1 - a,i)

0

d,i d,i+1

-20 -40

d,id,i)

2

-60 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 Axial Strain (%) Figure 3-17. The dissipated energy per unit volume for a soil sample in cyclic triaxial loading is defined as the area bound by the deviator stress - axial strain hysteresis loops, Equation (3-18). 3.5.2 Cyclic Simple Shear Test Assuming the cyclic shear is applied in the 13 direction, the following boundary conditions apply for a sample subjected to cyclic simple shear.

 12   23   xy   yz  0 d11  d 22  d 33  d x  d y  d z  0

(3-19)

Using these boundary conditions, the general expressions for the increment in dissipated energy given in Equations (3-9) and (3-11) reduce to: 101

dW  2 13d 13   xz d xz

(3-20)

Appling the trapezoidal rule to integrate Equation (3-20), and dropping the xz subscripts, the cumulative energy dissipated per unit volume of material (W) can be determined as: W 

1 n 1  ( i 1   i )( i 1   i ) 2 i 1

where:

(3-21)

i

=

The ith increment in shear stress.

i

=

The ith increment in shear strain.

n

=

Total number of increments.

3.5.3 Hollow Cylinder Triaxial - Torsional Shear Test With the unique geometry of a hollow cylinder sample, the imposed stresses are shown in Figure 3-18. From examination of an element of soil in the sample, it can be observed that the stresses imposed on it are very similar to the cubical element shown in Figure 316.

’v

vh

’h

’h

’v ’h vh

’h ’h

’h Figure 3-18. Stress conditions for a hollow cylinder triaxial-torsional shear test. (Adapted from Towhata and Ishihara 1985). 102

The following expressions relate the notation used in Figure 3-18 to the tensor-suffix and engineering notations.

 ' v   11   x

 ' h   22   33   y   z  vh   xz   13 d a  d 11  d x

(3-22)

d h  d 22  d 33  d y  d z d vh  d xz  2d 13

Additionally, the following stress conditions exist on an element for soil in the sample:

 12   23   xy   yz  0

(3-23)

 y   z   22   33 Substituting these conditions into the general expressions of the increment in dissipated energy per unit volume of material, given as Equations (3-9) and (3-11), yields: dW   x d x  2 z d z   xz d xz   11d 11  2 33d 33  2 13d 13

(3-24)

Using the notation of Figure 3-18, Equation (3-24) can be equivalently written: dW   'v d a  2 'h d h   vhd vh

(3-25)

For isotropically consolidated samples subjected to torsional shear, the axial and lateral deformations are approximately zero: d a  d h  0

(3-26)

d x  d y  d z  0 The expression for the increment in dissipated energy per unit volume of material reduces to: dW   xz d xz  2 13d 13

(3-27)

or equivalently, dW   vh d vh

(3-28)

103

Appling the trapezoidal rule to integrate Equation (3-27) or (3-28), and dropping the subscripts, the cumulative energy dissipated per unit volume of material (W) can be determined by: W 

1 n 1  ( i 1   i )( i 1   i ) 2 i 1

where:

(3-29)

i

= The ith increment in shear stress.

i

= The ith increment in shear strain.

n

= Total number of increments.

3.5.4 Use of the Derived Equations Equations (3-18), (3-21), and (3-29) are a set of consistently derived expressions for computing the cumulative energy dissipated in soil subjected to cyclic triaxial, cyclic simple shear, and cyclic torsional shear loading, respectively. These expressions will be used in Chapters 4 and 5 for computing the energy dissipated in both laboratory specimens and in soil profiles via site response analyses.

104

Chapter 4. Energy-Based Excess Pore Pressure Generation Models 4.1 Introduction Corollary to liquefaction evaluation procedures are excess pore pressure generation models. This can be understood because liquefaction, as defined in this thesis, is the condition where the excess pore pressure (uxs) is equal to the initial effective overburden (’vo). Therefore, if parameters such as stress, strain, and energy can be related to one value of uxs (i.e., uxs = ’vo), it should be expected that the same parameters can also be related to other values of uxs, and in fact, they have. Both stress- and strain-based pore pressure models have been developed and are commonly used in numerical modeling of soils subjected to cyclic loading (e.g., Booker et al. 1976 and Martin et al. 1975). In addition to stress- and strain-based models, numerous energy-based excess pore pressure generation models have been developed. Nemat-Nasser and Shokooh (1979) established governing differential equations relating energy dissipation to the densification of dry samples and to the generation of excess pore pressures in saturated samples. Complementing Nemat-Nasser and Shokooh’s theoretical work are numerous empirical relations (e.g., Simcock et al. 1983, Law et al. 1990, Green et al. 2000). Although these pore pressure models provide credence to quantifying soil Capacity in terms of dissipated energy, the dependence of the models’ calibration parameters on factors other than the soil’s density, fabric, and stress state (i.e., state variables) needs to be examined. Because earthquake motions have very different amplitudes, frequency content, and durations in comparison to the motions imparted to the soil by various remediation techniques, the dependence of calibration parameters on the nature of the loading, as well as state variables, may inherently limit the usefulness of Capacity curves developed from earthquake case histories for remedial ground densification design. Several energy-based pore pressure generation models from literature are reviewed in this chapter. Additionally, from the analysis of numerous cyclic triaxial test data, a new expression is presented relating dissipated energy and pore pressure generation. The proposed model is referred to as the GMP Model to denote the authors of the publication

105

in which it was first proposed: Green, Mitchell, and Politio (Green et al. 2000). The GMP Model is similar in form to several of earlier proposed models, but only has one calibration parameter (PEC: pseudo energy capacity). Established correlations for PEC are presented and discussed, with particular emphasis on the variation of PEC with respect to the amplitude of the applied loading. 4.2 Energy-Based Pore Pressure Models from Published Literature A thorough literature review was performed and selected energy-based pore pressure models are presented below. The large number of the models necessitates a “cookbook” presentation style similar to that used in Chapter 2 in presenting the energy-based liquefaction evaluation procedures. With the exceptions of the models proposed by Nemat-Nasser and Shokooh (1979) and Mostaghel and Habibaghi (1979), all of the expressions relating excess pore pressure generation and dissipated energy are empirical curve fit equations of laboratory data. The models relate the dissipated energy per unit volume of material (W) to the residual excess pore pressures (uxs), where uxs is defined as the pore pressure in excess of hydrostatic conditions when the applied cyclic stress is zero. Where possible, the models are presented in terms of pore pressure ratio (ru), which is defined as uxs divided by the initial effective overburden stress (’vo). For each of the proposed models, an expression relating ru and W is given, followed by the values of the calibration parameters and information on the laboratory tests on which the models are based. For many of the models, procedures for determining the calibration parameters from laboratory data were not given in the referenced literature. Also, many of the details about the laboratory test data are incomplete. The validity of the models may be limited to the specific test conditions (e.g., load path and amplitude, effective confining stress) and soil types on which they are based. To provide a consistent presentation of the models, several of the expressions are presented in alternate forms from those in the referenced literature. However, the underlying assumptions on which the models are based are unchanged. Comparison of the models is not possible because few of the have correlations relating calibration parameters to soil properties

106

(e.g., Dr or N1,60). However, Section 4.3 provides limited commentaries on several of the proposed models. 4.2.1 N-NS Model (Nemat-Nasser and Shokooh 1977, 1979) Model: 1

 W e  e b  1a o min ru  1  1  ˆ u e   o where:

(4-1)

W = Dissipated energy per unit volume of material (psi). eo

= Initial void ratio of the soil.

emin = Minimum void ratio of the soil. a,b =

Calibration parameters. v'

û

=

w

= Bulk modulus of water (psi).

’

= Calibration parameter.

Calibration Parameters:

( a  1)   w

a = 2.5 b = 3.5 h uˆ

= 7.4110 5

From laboratory tests, the following relation may be used to determine h: W =

h  N  CSR

1



= 2.5

h

= Calibration parameter, specific value not given.

CSR = Cyclic stress Ratio. N Soil and Effective Confining stress: Monterey No. 0

= Number of cycles of loading. Ref. Lab data: DeAlba et al. (1976)

(emin = 0.564, emax = 0.852)

’vo = 30 to 55kPa

107

Dr = 54 and 68% Test Apparatus and Loading: Large scale simple shear tests Stress-controlled CSR = 0.104 to 0.185 4.2.2 MH Model (Mostaghel and Habibaghi 1978, 1979) Model: ru 

1 W e o  ' vo

where:

(4-2) W =

Dissipated energy per unit volume of material.

eo =

Initial void ratio of the soil.

’vo =

Initial vertical effective stress.

W and ’vo have the same units. Mostaghel and Habibaghi (1978, 1979) did not propose Equation (4-2) as a formal pore pressure generation model, but they assumed this relation in deriving their liquefaction evaluation procedure. 4.2.3 DB1 Model (Davis and Berrill 1982) Model: ru  

W

(4-3)

 ' vo

where:

W =

Dissipated energy per unit volume of material.

’vo =

Initial vertical effective stress.



Calibration parameter.

=

W and ’vo have the same units, and  is dimensionless. Calibration Parameter: 50    80 (Davis and Berrill 2001) Soil and Effective Confining stress:

Ref. lab data: Simcock et al. (1983)

New Brighton Sand Moist tamped

108

Isotropically consolidated

’o =50, 100, 150kPa Dr = 67 to 95% Test Apparatus and Loading: Cyclic triaxial Stress-controlled CSR = 0.127 to 0.293 4.2.4 BD Model (Berrill and Davis 1985) Model:

 W  ru        ' vo  where:



(4-4)

W =

Dissipated energy per unit volume of material.

’vo =

Initial vertical effective stress.

’, 

= Calibration parameters.

W and ’vo have the same units, and ’ and  are dimensionless. Calibration Parameter: (’,  ): (0.8, 0.6), (0.65, 0.5), and (0.5, 0.5) “...  = 0.5 appears to be optimal.” Soil and Effective Confining stress, Test Apparatus and Loading: Same as for DB1 Model 4.2.5 DB2 Model (Davis and Berrill 2001) Model: ru  1  exp( 

where:

W

 ' vo

(4-5)

)

W =

Dissipated energy per unit volume of material.

’vo =

Initial vertical effective stress.



Calibration parameter.

=

109

W and ’vo have the same units, and  is dimensionless. Calibration Parameter: 50    80 Soil and Effective Confining stress, Test Apparatus and Loading: Same as for DB1 Model 4.2.6 YTI Model (Yamazaki, Towhata, and Ishihara 1985) Model: 1 a

ru =

 b ' vo  1    W  a   W    ln    ln( b)   0.5 4    ' vo  

where:

for ru  0.5

(4-6) for 0.5  ru  1.0

W =

Dissipated energy per unit volume of material.

’vo =

Initial vertical effective stress.

a,b =

Calibration parameters.

W and ’vo have the same units, and a and b are dimensionless. Calibration Parameters:

Toyoura Sand: ’o = 294kPa

a = 0.848 b = 2.04 x 10-3

Dr = 40 to 50%

The calibration parameters can be determined from laboratory test data as shown in Figure 4-1.

110

1.0 Straight Line ru 

ru 0.5

   ln  W 4    ' vo

    ln( b)   0.5  

a

Logistic Curve 1

ru 

 b ' vo    W 

a

1 

b

0.0

W

(log scale)

 ' vo

Figure 4-1. Procedure for determining calibration parameters for YTI Model from laboratory test data. (Adapted from Yamazaki et al. 1985).

Soil and Effective Confining stress: Ref: Lab data Towhata and Ishihara (1985) Toyoura Sand Air pluviated samples Isotropically consolidated

’o = 294kPa Dr = 42 to 51% (torsional shear) Dr = 42 to 49% (triaxial) Test Apparatus and Loading: Hollow cylinder triaxial-torsional shear Stress-controlled loading Various load paths including torsional shear and triaxial CSR = 0.188 to 0.264 (torsional shear) CSR = 0.184 to 0.194 (triaxial) Load rate=0.14% strain/min

111

4.2.7 LCH Model (Law, Cao, and He 1990) Model: u xs

 ' ho



(4-7)

  WN

where:

WN = = W =

Dissipated energy parameter. F1 ( K c ) F2 ( D r )

W

 ' ho

Dissipated energy per unit volume of material.

F1(Kc) = Normalizing function to account for Kc. =

1   log( K c )

F2(Dr) = Normalizing function to account for Dr. =

10

 ( Dr 0.70 )

Kc =

’vo/’ho

Dr =

Relative Density.

’ho =

Initial horizontal effective stress.

’vo =

Initial vertical effective stress.

,,, = Calibration parameters. uxs =

Excess pore pressure.

W, uxs, ’vo, and ’ho have the same units, and , , , and  are dimensionless. Calibration Parameters:

 = 3.0  = -2.0 No values are given for  and 

Soil and Effective Confining stress: Fujian Standard sand Moist tamped Isotropically consolidated

’o =50, 100, 150kPa 112

Dr = 70% Anisotropic consolidated

’ho =100kPa Kc = 1.5, 1.75, and 2.0 Dr = 70% Test Apparatus and Loading: Cyclic triaxial and Hollow cylinder torsional shear Stress-controlled Load rate: 1.0hz 4.2.8 YS Model (Yanagisawa and Sugano 1994) Model: ru 

S'

s a  bS '

where:

(4-8) s

S’s = =

State parameter. dW

'

m

dW =

Increment in dissipated energy per unit volume of material.

’m =

Mean effective stress.

a,b =

Calibration parameters.

dW and ’m have the same units, and a and b are dimensionless. It is unclear whether the

’m is the initial mean confining stress or the mean confining stress at the time of the increment dW. Calibration Parameters:

No values for a and b are given.

Soil and Effective Confining stress and Test Apparatus and Loading: Strain-controlled cyclic triaxial Toyoura sand Air pluviated Isotropically consolidated 113

’o = 196, 294, 343kPa Dr = 40, 70%

cyc = 0.24, 0.38, 0.47, 0.50, 0.56, 0.60%, and random Load rate: 0.5hz Strain-controlled torsional shear Samples prepared by MSP Method (Miura and Toki 1982) Isotropically consolidated

’o = 196kPa Dr = 49, 81%

cyc = 0.2, 0.5, 1.0% Two-Directional Cubic Shear Ko consolidated

’vo = 49kPa Dr = 61% 4.2.9 Hsu Model (Hsu 1995) Model:

 W     ' vo 

b

ru  a 

where:

(4-9)

W =

Dissipated energy per unit volume of material.

’vo =

Initial vertical effective stress.

a,b =

Calibration parameters.

W and ’vo have the same units, and a and b are dimensionless. Calibration Parameters:

a  400  420 Dr  CSR b  1.9  1.25 Dr  CSR

2

1/ 2

CSR = Cyclic stress ratio. Dr

= Relative Density.

114

Soil and Effective Confining stress: Chu-An sand Air pluviated Isotropically consolidated

’o = 50, 100, 200kPa Dr = 35, 50, 80% Test Apparatus and Loading: Hollow cylinder apparatus Stress-controlled torsional shear CSR = 0.16, 0.19, and 0.22 Load rate: 1.0hz 4.2.10 Liang Model (Liang 1995) Model: ru  a 

where:

b W

(4-10)

c  W

W =

Dissipated energy per unit volume of material.

’vo =

Initial vertical effective stress.

a,b,c

= Calibration parameters.

W and ’vo have the same units, and a, b, and c are dimensionless. Calibration Parameters:

Dr (%)

a

b

c

Reid

54.9

0.03663 1.30619 0.36743

Bedford

58.3

0.00233 1.36265 0.47798

sand

67.5

0.00663 1.14189 0.35988

LSFD

60.0

0.05891

1.2216

0.228

silty-

71.2

0.00336

1.2847

0.208

sand

87.4

0.03

1.66

0.159

Soil and Effective Confining stress: Reid Bedford sand and Lower San Fernando Dam silty-sand (FC < 28%) 115

Dry tapped Isotropically consolidated

’o = 82.7kPa Dr = 54.9, 58.3, 67.5% (Reid Bedford) Dr = 59.9, 71.2, 87.4% (LSFD) Test Apparatus and Loading: Hollow cylinder torsional shear Stress-controlled Random earthquake type loading 4.2.11 OAY Model (Ogawa, Abe, and Yoshitsugu 1995) Model: u xs  W

0.5

where:

(4-11)

W =

Dissipated energy per unit volume of material.

uxs =

Pore pressure ratio.

W and uxs have the same units. Calibration Parameters: None Soil and Effective Confining stress: Toyoura sand Isotropically consolidated

’o = 98kPa Dr = 60% Test Apparatus and Loading: Hollow cylinder torsional shear Stress-controlled CSR = 0.2 Load rate: 0.1hz

116

4.2.12 FSKL Model (Figueroa, Saada, Kern, and Liang 1997) Model: ru   A W  A W  2 

where:

(4-12)

W =

Dissipated energy per unit volume of material (kPa).

A

Calibration parameter.

=

The above model was derived from an empirical expression presented in Figueroa et al. (1997) relating shear modulus degradation to W. In deriving this expression, it was assumed G = Gmax(1-ru)0.5. Calibration Parameters:

A = 151 .52  1.10 Dr  27.89   0.016  ' o (Kern 1996) A = 165 .6 1.44 Dr  9.09  (Figueroa et al. 1997) Dr =

Relative Density (%).

 = Shear strain (%). ’o = Initial effective confining stress (kPa). Soil and Effective Confining stress: Reid Bedford sand Dry tapped Isotropically consolidated

’o = 41.4, 82.7, 124.1kPa Dr = 50, 60, 70% Test Apparatus and Loading: Hollow cylinder torsional shear Strain-controlled

 = 0.15, 0.47, and 1.02% Load rate: 0.1hz

117

4.2.13 WTK Model (Wang, Takemura, and Kuwano 1997) Model: a ru 

W

 ' m o n

1 b

where:

(4-13)

W

 ' m o n W

= Dissipated energy per unit volume of material.

’mo = Initial mean effective confining stress (kPa). a,b,n = Calibration parameters. W and ’mo have the same units, and a, b, and n are dimensionless. Calibration Parameters:

Ip

2

5

10

30

FC(%)

8

12

16

37

a

70.4

65.9

12.5

3.39

b

65.9

62.9

13.7

4.87

Ip

= Plasticity index.

FC = Fines content. n

= Calibration parameter (0.6 to 0.8).

Soil and Effective Confining stress: Toyoura sand and Kawasaki clay mixtures Ip = 2, 5, 10, and 30 Consolidated from slurry Istropically consolidated

’o = 98, 196, 392kPa Test Apparatus and Loading: Cyclic triaxial Stress-controlled CSR: 0.006 to 0.154 (Ip = 2) 0.0009 to 0.171 (Ip = 5)

118

0.023 to 0.231 (Ip = 10) 0.029 to 0.251 (Ip = 30) Load rate: 0.01, 0.1, 0.2, and 1.0hz

Kuwano (2000)

4.3 Commentaries on the Models Although a detailed parameter study comparing the various pore pressure models cannot be performed due to the lack of correlations between the models’ calibration parameters and state variables, limited commentaries are given for several of the models. MH Model: Rearrangement of the MH Model yields: W  ru eo ' vo . From this expression, it can be seen that the MH Model predicts that it will require more energy to liquefy loose soil (i.e., large eo) than dense soil (i.e., small eo). This is contrary to observed behavior of soil. DB1 Model: The DB1 Model tends to over predict increases in pore pressures as the cumulative dissipated energy increases. BD Model: Berrill and Davis (1985) give three sets of values for the calibration parameters (’,  ): (0.8, 0.6), (0.65, 0.5), and (0.5, 0.5). Figure 2 in Berrill and Davis (1985) shows that these values cover the range of their laboratory data. However, the author was unable to duplicate Berrill and Davis’ Figure 2 using the published values for the calibration parameters in conjunction with the BD Model. It is uncertain whether this is due to an error in the published values of the calibration parameters or in the author’s implementation of the model. YTI Model: Towhata and Ishihara (1985) present the results of an extensive laboratory study showing a unique relationship between dissipated energy and pore pressure generation. In every

119

figure in this paper, a smooth curve is fit to the data without mention of how the curve was generated. Due to the commonality of the authors, subject matter, and dates of publishing, it is assumed that this curve is the YTI Model, the mathematical treatise of which is presented in Yamazaki, Towhata, and Ishihara (1985). LCH Model: As opposed to using the standard definition for pore pressure ratio (i.e., ru = uxs/’vo), Law et al. (1990) used uxs/’ho, without explanation. The most probable reason for this is that for anisotropically consolidated cyclic triaxial tests when Kc > 1, failure occurs in the sample when uxs/’ho = 1, while uxs/’vo < 1. This is due to the lack of lateral restraint on the sample and is the reason why anisotropically consolidated cyclic triaxial tests are not used to model in-situ conditions for level ground. Rather these tests represent conditions of initial static shear stress and are used to develop the K used in the stress-based liquefaction evaluation procedure (Chapter 2, Section 2.2.1). As shown in Seed and Harder (1990), the influence of initial static shear stress on soil Capacity is a function of both the amplitude of the static shear stress and the soil’s relative density (Dr). However, the F1(Kc) normalizing function proposed by Law et al. (1990) is independent of Dr. 4.4 GMP Model (Green, Mitchell, and Polito 2000) From analyzing numerous cyclic triaxial test data, the following empirical expression, denoted as the GMP Model, provides a simple and accurate relationship between residual excess pore pressure generation and the energy dissipated per unit volume of soil:

ru 

W PEC

(4-14)

where ru and W were defined previously and PEC (i.e., pseudo energy capacity) is a calibration parameter. By setting the calibration parameters ’and in the BD Model (Section 4.2.4) equal to 1/PEC0.5 and 0.5, respectively, it can be seen that the GMP Model is actually a special case of the more general BD Model. Other models having similar forms include the LCH, Hsu, and OAY Models.

120

PEC can be determined from cyclic test data by plotting ru versus the square root of W. The square root of PEC is the value on the horizontal axis corresponding to the intersection of a straight line drawn through the origin and the point of ru = 0.65 and a horizontal line drawn at ru = 1.0. This process of determining PEC is illustrated

1.0

Measured ru ru = 0.65

0.5

0.0 0.0

PEC

Residual Excess Pore Pressure, ru

graphically in Figure 4-2.

0.5

1.0

1.5 W

2.0

2.5

Figure 4-2. Graphic illustration of how PEC is determined from cyclic test data. The data shown in this figure is from a cyclic triaxial test conducted on Yatesville clean sand. PEC can also be determined numerically using Equation (4-15).

PEC 

W ru 0.65

(4-15)

0.4225

where Wru=0.65 is the value of W corresponding to ru = 0.65. The term “pseudo energy capacity” or PEC is used to label the calibration parameter because it is approximately equal to W at the point of initial liquefaction of the sample. Comparisons of measured and predicted pore pressures for various samples subjected to cyclic triaxial and cyclic torsional loading, both stress and strain controlled, are shown in Figures 4-3a and 4-3b. As can be seen in these figures, the computed values of ru closely match the measured values for a variety of soils and test conditions. These comparisons

121

are representative of those for several hundred cyclic tests, mainly stress-controlled cyclic triaxial tests.

Residual Excess Pore Pressure Ratio, ru

1.0 0.8 0.6 0.4 Yatesville Sand, FC = 0%, Dr=68.3% Yatesville Silt, FC = 100%, Dr=85.6% Yatesville Sand, FC = 7%, Dr=34.6% Yatesville Sand, FC = 17%, Dr=18.0% Monterey Sand, FC = 0%, Dr=81.1%

0.2 0.0

0

10

20

40 30 Time (seconds)

50

60

70

Figure 4-3a. Comparison of measured and computed residual excess pore pressures in various silt-sand mixtures having varying densities. Symbols are values computed using the GMP Model and the lines are the measured values. All samples were run in stress controlled cyclic triaxial tests. (Cyclic triaxial data is from Polito 1999).

122

Residual Excess Pore Pressure Ratio, ru

1.0 0.8 0.6 0.4 Monterey Sand, strain controlled Plastic Coal Mine Tailings, stress controlled cyclic Turkish Silty-Sand, stress controlled triaxial New Castle Sand, stress controlled Fine Sand, stress controlled cyclic torsional

0.2 0.0

0.0

10.0

20.0 30.0 Time (seconds)

40.0

50.0

Figure 4-3b. Comparison of measured and computed residual excess pore pressures in samples tested in various configurations. Symbols are values computed using the GMP Model and the lines are the measured values. (Cyclic data is from Bonita 2000, Koester 1992, and Polito 1999).

From stress-controlled cyclic triaxial test data, correlations were developed relating CSR, PEC, and Dr for Monterey and Yatesville sands (i.e., medium and fine grained sands, respectively). All the samples were moist tamped, isotropically consolidated, and subjected to either 0.5hz or 1hz sinusoidal loading. The computed PEC values are listed in Table 5-2 in Chapter 5. Additional details about the laboratory tests can be found in Polito (1999). Equation (4-16) and (4-17) define the best-fit surfaces for the Monterey silt-sand and Yatesville silt-sand data, respectively. Based upon test results for specimens with non-plastic silt contents up to approximately 30%, the presence of the silt appears to have little or no influence on the computed PEC. This is in line with the findings of Polito (1999) who hypothesizes that the apparent increase in the Capacity of high fines content sand is actually attributed to the decrease in the penetration resistance of the silty sands. Given that relative density, and not penetration resistance, was used in the PEC correlations, silt content does not influence the correlation.

123

Monterey sand:

PEC =

-3.5DrCSR + 310.7CSR2 – 129.7CSR + 14.3 + 0.7144Dr + 0.01132Dr2 for CSR  5.63210-3 Dr + 0.2087

(4-16)

1.46310-3Dr2 – 0.01613Dr + 0.7644 for CSR > 5.63210-3 Dr + 0.2087

Yatesville sand:

PEC =

-1.64DrCSR + 310.7CSR2 – 110.2CSR + 10.3 + 0.305Dr + 0.0023Dr2 for CSR  2.63910-3Dr + 0.1773

(4-17)

1.35910 Dr + 0.01416Dr + 0.5285 for CSR > 2.63910-3Dr + 0.1773 -4

2

where PEC is in kPa, CSR is the cyclic stress ratio of the applied loading, and Dr is in percent. Plots of Equations (4-16) and (4-17) are shown as contours of constant Dr in Figures 4-4 and 4-5, respectively, superimposed on laboratory data. Curve fitting in three dimensions (i.e., two independent variables and one independent variable) is as much an art as it is a mathematical process. In selecting the surfaces to fit the data, represented by Equations (4-16) and (4-17), an attempt was made to keep the geometry of the surfaces as simple as possible. In this vein, the same general shape was used for both the Yatesville and Monterey data. By doing this, the trends from one soil type were used to extrapolate the surface to regions of sparse data for the other soil type. As may be observed from these figures, there is a great deal of scatter in the data, implying that the two independent variables used in the correlation (i.e., Dr and CSR) do not completely account for the observed variation of PEC. Additional studies are required to refine the correlations shown in these figures.

124

Pseudo Energy Capacity (PEC) kPa

40 80% < Dr < 90% 60% < Dr < 70% 50% < Dr < 60% 40% < Dr < 50% 10% < Dr < 20% -5% < Dr < 5%

35 30 25 20 15

Dr = 15% Dr = 85%

10

Dr = 65% Dr = 55% Dr = 45%

5 Dr = 0% 0 0.0

0.2

0.6 1.0 0.4 0.8 Cyclic Stress Ratio (CSR) Figure 4-4. Correlations among CSR, PEC, and Dr for Monterey silt-sand mixtures (FC  30%).

Pseudo Energy Capacity (PEC) kPa

4.0 65% < Dr < 75% 45% < Dr < 55% 25% < Dr < 35% 15% < Dr < 25% -5% < Dr < 5%

3.5 3.0 2.5

Dr = 70%

2.0

Dr = 50%

1.5 1.0

Dr = 30% Dr = 20% Dr = 0%

0.5 0.0 0.0

0.1

0.2 0.3 Cyclic Stress Ratio (CSR)

0.4

0.5

Figure 4-5. Correlations among CSR, PEC, and Dr for Yatesville silt-sand mixtures (FC  30%).

125

4.5 Load Dependency of Calibration Parameters As stated in the introduction to this chapter, the dependence of the calibration parameters on the amplitude and frequency of the loading may limit the use of energy-based Capacity curves in the design of ground densification techniques. Conflicting conclusions are drawn from review of the published literature as to whether a unique relation exists between dissipated energy and pore pressure generation (i.e., if the calibration parameters are independent of factors other than the state variables of the soil, then a unique relation exists between dissipated energy and pore pressure generation). Only two of the energy-based models presented in Section 4.2 provide correlations for their calibration parameters (i.e., Hsu and FSKL Models), and both expressions are functions of the amplitude of loading. Additionally, in presenting the results of a laboratory study conducted to validate the DB1 Model, Simcock et al. (1983) state: “...it appears that pore pressure generation and dissipated energy are functionally related, but this relationship appears to depend strongly on the level of cyclic deviator stress.” Because the soils, test apparatus, and control parameter for the loading (i.e., stress controlled versus strain controlled loading) differed among these studies, the load dependency of the calibration constants cannot be attributed to specific test conditions. On the contrary, Towhata and Ishihara (1985), Yanagisawa and Sugano (1994), and Wang et al. (1997) determined that the relationship between W and ru is independent of load path and amplitude of loading. The only commonality in these studies is the soil tested (i.e., Toyoura fine sand). From examination of the correlations developed for the GMP Model’s calibration parameter, given by Equations (4-16) and (4-17) and plotted in Figures 4-4 and 4-5, it can be seen that PEC increases rapidly below a certain CSR, which varies as a function of Dr. Alternate plots of the correlation for the Yatesville and Monterey silt-sand mixtures are shown in Figure 4-6a and 4-6b, respectively. Even though the same shape surface was used to fit the Monterey and Yatesville data, the dependency of PEC on CSR is more prominent in the medium grained Monterey sand than for the fined grained Yatesville

126

sand. There is no apparent reason for this dichotomy in the behavior of fine grained sands (e.g., Toyoura and Yatesville) and medium grained sands (e.g., Monterey). The load dependence of PEC may be attributed to viscous drag. For a given frequency of applied loading (e.g., 1.0hz), the contribution of energy dissipated due to viscous drag increases as the amplitude of the applied loading decreases (Hall 1962). Of the three studies that showed load independency, Towhata and Ishihara (1985) used an extremely slow loading rate 0.14% stain/min, which would minimize the energy lost due to viscous drag regardless of the amplitude of the applied loading. Yanagisawa and Sugano (1994) used 1.0hz, and Wang et al. (1997) used a loading rate ranging from 0.01 to 1.0hz (Kuwano 2000). Additional laboratory data is required to resolve the load dependency issue.

Pseudo Energy Capacity (PEC) kPa

3.0 2.5

CSR = 0.15

0.2

0.25

0.3 0.35 0.4

2.0 1.5 1.0 0.5 0.0 -40

-20

0

20

40

60

80

Relative Density (%) Figure 4-6a. Alternate plot of the correlation relating PEC, Dr, and CSR for Yatesville fine grained sand-silt mixtures.

127

Pseudo Energy Capacity (PEC) kPa

0.40 0.35

CSR = 0.2

0.3

0.4

0.5

0.30 0.25 0.6

0.20 0.15

0.7

0.10 0.05 0.00 -10 0

20 40 60 80 Relative Density (%)

100

Figure 4-6b. Alternate plot of the correlation relating PEC, Dr, and CSR for Monterey medium grained sand-silt mixtures.

4.6 Summary A review of published excess pore pressure generation models was presented. Particular emphasis was placed on examining the dependence (or independence) of the models’ calibration parameters on factors other than the soil’s state variables. The reason for this is that the dependence of the calibration parameters on such things as the amplitude of the loading, as well as state variables, may inherently limit the use of liquefaction curves developed from earthquake case histories for remedial ground densification design. Conflicting conclusions are drawn from the review of the literature and from the analyses of numerous cyclic tests. The only tentative conclusion is that the calibration parameters for fines sands appear to be less dependent on the amplitude and frequency of loading than for medium grained sands. Additional studies are required to determine why. In short, no conclusion can be drawn from the author’s review excess pore pressure generation models as to the universality of dissipated energy for quantify the Capacity of cohesionless soil.

128

Chapter 5. Proposed Energy-Based Liquefaction Evaluation Procedure 5.1 Introduction A critique of existing energy-based liquefaction evaluation procedures for possible use in remedial ground densification design was presented in Chapter 2. Unfortunately, the conclusion of the review was that none of the procedures were adequate for this purpose. In this chapter, a new energy-based liquefaction evaluation procedure is proposed. As will be shown, this new approach is a conceptual and mathematical unification of the stress- and strain-based procedures, both of which were outlined in Chapter 2. The Capacity curve for the proposed procedure was developed from the analyses of earthquake case histories and compared with normalized data from numerous cyclic triaxial tests. Because the Capacity curve provides the critical link between earthquake liquefaction evaluation and remedial ground densification design, the same parameter study used to critique the existing energy-based liquefaction evaluation procedures in Chapter 2 (Section 2.3) is used to evaluate the proposed procedure. As part of the proposed energy-based liquefaction evaluation procedure, a new correlation was developed relating the number of equivalent cycles, earthquake magnitude, and site-to-source distance. Aside from the focus of this thesis, in Appendix 5c, this new correlation is used as the basis for a new set of magnitude scaling factors (MSF) for use in the stress-based liquefaction evaluation procedure. As opposed to the current NCEER (1997) recommended MSF, presented in Chapter 2 (Figure 2-2, Section 2.2.1), the new set of MSF are functions of both earthquake magnitude and site-to-source distance. 5.2 Mathematical Expression for Computing Dissipated Energy An expression was presented in Chapter 3 defining the damping ratio (D) in terms of the energy dissipated per unit volume of material in a single cycle of loading (Equation (3-2), Section 3.3.2). This expression and related figure are repeated below as Equation (5-1) and Figure 5-1.

129



G





W1



1 W  2

Figure 5-1. Quantities used in defining damping ratio (D).

D

1 W1  4 W

where: D

(5-1) = Damping ratio.

W1 =

Dissipated energy per unit volume of material in one cycle of loading.

W

Maximum energy stored in an elastic material having the same G as the visco-elastic material.

=

Simple rearrangement of Equation (5-1) and substitution of the definition of W given in Figure 5-1 yields:

W1  2D

(5-2)

Using this expression in conjunction with the equivalent-number-of-cycles concept (e.g., Annaki and Lee 1977), the energy dissipated in the soil for the entire duration of the earthquake motion can be estimated as:

W  W1  N eqv  (2D )  N eqv where: W =

(5-3)

Dissipated energy per unit volume for entire earthquake motion.

W1 =

Dissipated energy per unit volume for one equivalent cycle of earthquake motion.

Neqv =

Number of equivalent cycles in the earthquake motion.

D

=

Damping ratio.



=

Shear stress.

130



=

Shear strain.

By appropriately selecting  and  in Equation (5-3), the stress- and strain-based liquefaction evaluation procedures can be unified. Expressing the shear strain () in terms of the shear stress (), i.e.,  =  /G, Equation (5-3) can be re-written as:  2 D  2    N eqv W    G 

(5-4)

The determination of , G, D, and Neqv will be discussed in order. 5.2.1 Determination of  As presented in Section 2.2.1, alternative to site response analyses, the average or effective amplitude of the earthquake-induced shear stress (ave) at depth z in a soil profile can be determined by the following expression:

 ave  0.65  where:

a max   vo  rd g

(5-5)

ave

= Effective amplitude of the earthquake-induced shear stress.

amax

=

g

= Acceleration due to gravity.

vo

= Total vertical stress at depth z.

rd

= Dimensionless parameter that accounts for the stress reduction due to soil column deformability.

Maximum soil surface acceleration.

5.2.2 Determination of G and D Because the damping ratio (D) and shear modulus (G) are functions of the induced shearstrain, the iterative procedure used in the strain-based approach (Section 2.2.2) is employed to determine the shear strain () corresponding to ave. Starting with the relation between stress and strain  =  /G, Dobry et al. (1982) derived the following expression for determining the earthquake-induced shear strain at depth in a soil deposit:

131

a max   vo  rd g   G   Gmax    Gmax   0.65 

where:

Gmax

(5-6)

= shear modulus corresponding to  = 10-4%.

(G /Gmax) = ratio of shear moduli corresponding to  and  = 10-4%. This expression was previously given as Equation (2-11) in Chapter 2, and is solved iteratively as illustrated in Figure 5-2. For the first iteration, a value of G/Gmax is assumed and  computed. In the second iteration, the ratio of G/Gmax corresponding to  computed in the first iteration is used. The process is repeated until the assumed and computed ratios are within a tolerable error.

1.0 Assumed values of G/Gmax Computed values of G/Gmax

G/Gmax

iteration 1

(G/Gmax)

10-4%

iteration 2 final iteration

tolerable error

 (log Scale) 

1%

Figure 5-2. Iterative solution of Equation (5-6) to determine the effective shear-strain () at a given depth in a soil profile.

Once  is determined, the damping and shear modulus ratios are easily determined from the respective degradation curves, as illustrated in Figure 5-3.

132

G/Gmax (G/Gmax)

D D



10-4%

 (log scale)

1%

Figure 5-3. The determination of shear modulus and damping ratios from the respective degradation curves.

Using the above expressions for , G, and D, the energy dissipated at depth in a soil deposit for one equivalent cycle of loading is given as:

2 D

a   W1   0.65  max   vo  rd  g  G     Gmax    Gmax  

2

(5-7)

Shear Stress

The various components of this expression are illustrated in Figure 5-4.

 G   G  G max   G  max  

ave

W1 

2 2D  ave



G

Shear Strain

Figure 5-4. Graphical representation of the dissipated energy per unit volume for an equivalent cycle of loading.

133

5.2.3 Determination of Neqv The final variable needed to determine the energy dissipated in the soil for the entire duration of the earthquake is the number of equivalent cycles (Neqv). The equivalentnumber-of-cycles concept resulted from the inherent difficulties of imposing random earthquake type motions to laboratory specimens. The concept is based on equating transient earthquake motions, involving peaks of varying amplitudes, to a uniform cyclic load having Neqv cycles. Lee and Chan (1972) determined the equivalent number of cycles (Neqv) from earthquake motions by computing the weighted sum of the number of spikes in an acceleration time history, where the weighting factors were based on a normalized laboratory Capacity curve. Using this same procedure, Seed et al. (1975) developed the correlation shown in Figure 5-5. As may be seen in this figure, there is a large amount of scatter in the data, which has brought criticism of the equivalent-number-of-cycles concept. However, an alternative energy-based correlation relating Neqv, M, and site-to-source distance developed as part of this thesis shows much less scatter. This new correlation is based on equating the energy dissipated in the layers of a soil profile, as determined from site response analyses, to the energy dissipated in an equivalent cycle of loading multiplied by Neqv. This procedure is depicted graphically in Figure 5-6.

134

A-1

40

Mean +1 Standard S-I Deviation

30

Neqv at ave

Neqv  26 Mean

20 Neqv  15 Mean -1 Standard Deviation

Neqv  10

10

5

6

M8.5

M7.5

M6-3/4

0

M6

Neqv  6

8

7

9

Earthquake Magnitude Figure 5-5. A correlation relating earthquake magnitude and equivalent number of cycles. The data points labeled S-I and A-1 are assumed to be from a different study than the rest of the data. Also shown in this figure are several M-Neqv pairs that are commonly presented in tabular form in published literature. (Adapted from Seed et al. 1975).

135

Dissipated energy from site response analysis

Soil Profile

Dissipated energy for an equivalent cycle times Neqv







Layer 1

=



=



Acceleration



ave

 





Layer 3







Layer 2

G

ave

=

ave





x

Neqv

x

Neqv

x

Neqv

G

G

Time (sec)

Figure 5-6. Illustration of the procedure used to develop the Neqv correlation. In this procedure, the dissipated in a layer of soil, as computed from a site response analysis, is equated to the energy dissipated in an equivalent cycle of loading multiplied by Neqv.

An option was added to the site response computer program SHAKE91 (Idriss and Sun 1992) to compute the energy dissipated in each layer of a soil deposit subjected to an earthquake loading. This modified version of SHAKE91 is herein referred to as SHAKEVT, with Figure 5-7 being an example of hysteresis loops output by the program. A parametric study was performed by propagating 24 pairs of horizontal motions from various magnitude earthquakes through 12 different soil profiles (i.e., 288 site response analyses). The normalized energy demand (NED = W/’mo) was computed for each layer in the profiles. The acceleration time histories used in the parametric were recorded on stiff soil and rock sites at varying site-to-source distances. Appendix 5a and Appendix 5b list the earthquake records and profiles, respectively, used in the parametric study.

136

The shear modulus and damping degradation curves proposed by Ishibashi and Zhang

300 200 100 -0.05

Shear Stress (psf)

(1993) were used in the site response analyses.

Shear Strain (%) 0.025 0.05

-0.025 -100

-200 -300 Figure 5-7. Shear-stress shear-strain hysteresis loops, output by SHAKEVT, for a given depth in a soil profile subjected to an earthquake acceleration time history.

To compute the number of equivalent cycles corresponding to total motions experienced at a site, the following procedure was used. 

Using Equation (5-7), the normalized energy dissipated in one equivalent cycle of loading (i.e., NED1 = W1/’mo) was computed for each layer in a profile. In solving Equation (5-7), amax and rd values from the respective site response analysis were used in conjunction with the Ishibashi and Zhang (1993) shear modulus and damping degradation curves.



The number of equivalent cycles for the motion in each layer is determined by: N eqvi 

where:

NEDi NED1i

(5-8a)

Neqv i = Number of equivalent of cycles of the motion in layer i. NEDi = Normalized energy dissipated in layer i, as computed by SHAKEVT. NED1i = The normalized energy dissipated in one equivalent cycle of loading, as computed by Equation (5-7). 137



Neqv for the entire profile subjected to a single component of horizontal motion was computed as a weighted average of the Neqv i for layers lying between the groundwater table and a depth of about 60ft, where the Neqv i values were weighted according to the thickness of the corresponding layer: N eqv   N eqv i  i

where:

Neqv

hi 60 ft  d gwt

(5-8b)

= Number of equivalent of cycles for entire profile.

Neqv i = Number of equivalent of cycles of the motion in layer i. hi

= Thickness of layer i.

d gwt

= Depth to the groundwater table.

A comparison of the NED values computed by SHAKEVT and Equation (5-7) multiplied by Neqv is shown in Figure 5-8. 

Finally, the Neqv values for each horizontal component in a pair were summed to give the total Neqv of the motions experienced at the site. 0

0

a)

10

10

20

20 amax = 0.31g Neqv = 6.1

40

30

Depth (ft)

Depth (ft)

30

50 60 70

40

amax = 0.22g Neqv = 8.2

50 60 70

80

80

W1  Neqv ’mo SHAKEVT

90 100

b)

0

90 100

5

W1  Neqv ’mo SHAKEVT

10 15 20 0 2 4 6 NED  104 NED  104 Figure 5-8. A comparison of the NED values computed by SHAKEVT and Equation (5-7) multiplied by Neqv. Figures a) and b) correspond to the two horizontal components of motion at the same site. The sum of the Neqv for each component represents the Neqv for total motions experienced at the site.

138

For each of the 12 profiles shown in Appendix 5b, the computed Neqv values for the 24 pairs of time histories were plotted as functions of epicentral distance (ED) and Richter magnitude (M). Although epicentral distance and Richter magnitude are not the best measures of site-to-source distance and earthquake magnitude, respectively, they were used because these were the only measures available for the majority of the liquefaction case histories. Figures 2-8a and 2-8b show the various definitions of site-to-source distance and a comparison of various magnitude scales, respectively. Regression analyses were performed to fit surfaces to the computed Neqv values for each profile. For three dimensional regression analyses, an unlimited array of surfaces may be employed.

Neqv

The simplest surface that provided a good fit of the data is shown in Figure 5-9.

Richter Magnitude

Epicentral Distance (km)

Figure 5-9. A fit surface to the computed Neqv for one of the profiles shown in Appendix 5b. The black dots shown in this plot are the computed Neqv values. All the surfaces fit to the data sets for the 12 profiles have the same general shapes but vary slightly in the “steepness” of their inclines. An average of the 12 surfaces was computed, a contour plot of which is shown in Figure 5-10. An approximation of the average surface is given by Equation (5-9).

f ( ED)

for

f ( ED)  f ( M )

f (M )

for

f ( ED)  f ( M )

Neqv =

139

(5-9a)

where: 9.633M 2  110.453M  325.172

for

M  5.7

8.5

for

M  5.7

0.011ED 2  0.012 ED  8.487

for

ED  0.55km

f(M) =

(5-9b)

(5-9c)

f(ED) = 8.5

for

ED  0.55km

As a further examination of the proposed Neqv correlation, the near field - far field boundary proposed by Krinitzsky et al. (1993) is superimposed on the contour surface shown in Figure 5-10. The close coincidence of the “elbows” in the contours and the near field - far field boundary suggests that the two are related. It is hypothesized that the elbows result from setting the amplitude of ave proportional to a fixed percentage of amax (i.e., 0.65 amax). In the far field, the acceleration time histories are more regular due to the attenuation of the high frequencies motions, and therefore, a fixed fraction of amax adequately represents the entire time history. However, in the near field the time histories are much more chaotic, and often amax is associated with a high frequency spike, which is not representative of the entire time history. The random nature of the relationship between amax (or a fixed fraction of amax) and the rest of the acceleration time history lead to the effective-peak-acceleration concept used by Newmark and Hall (1982) in scaling design spectra.

140

Epicentral Distance (km)

100 80 20

60

25

30

50 45 55 35 40

15 Neqv = 10

40

FAR FIELD

20 NEAR FIELD

0

5

6 7 Richter Magnitude (M)

8

Figure 5-10. Contour plot of the average fit surface for each of the 12 profiles used in the parameter study. The contours are of constant Neqv as a function of epicentral distance and Richter magnitude. The near field - far field boundary superimposed on the contour plot is that proposed by Krinitzsky et al. (1993); note this boundary is only given up to M7.5.

In comparing the Neqv correlation in Figures 5-10 to that developed by Seed et al. (1975) shown in Figure 5-5, it needs to be realized that the correlation shown in Figure 5-5 is for a single component of horizontal motion, while Figure 5-10 is for both components. The vertical portion of the contours shown in Figure 5-10 indicates that Neqv is independent of epicentral distance in the far field, similar to the correlation proposed by Seed et al. (1975). However, the horizontal portion of the contours indicates that Neqv is independent of earthquake magnitude in the near field. Again, this near field independence is hypothesized to be a consequence of setting the amplitude of ave proportional to a fixed percentage of the rather chaotic amax (i.e., 0.65 amax). Aside from the focus of this thesis, in Appendix 5c, the correlation shown in Figure 5-10 is used in the derivation of a new set of magnitude scaling factors (MSF) for use in the stress-based liquefaction evaluation procedure. As opposed to the current NCEER

141

(1997) recommended MSF, presented in Chapter 2 (Figure 2-2, Section 2.2.1), the new set of MSF is a function of both earthquake magnitude and site-to-source distance. 5.3 Capacity Curve Similar to the liquefaction evaluation procedures presented in Chapter 2, a Capacity curve was developed by analyzing liquefaction case histories. The Demand imposed on the soil by the earthquake can be estimated as: NED 

W1  N eqv  'm o 2

2  D

a     0.65 max  vo rd   N eqv g  G      ' m o  Gmax    Gmax  

(5-10)

This expression was derived by substituting the values of , G, D, and Neqv, determined as outlined above, into Equation (5-4) and normalizing the result by the initial mean effective confining stress (’mo). Using this expression, the Demand (i.e., NED) imposed on the soil was estimated for the 126 earthquake case histories listed in Table 5-1. The data in this table came from liquefaction databases assembled by Liao and Whitman (1986) and Fear and McRoberts (1995), where the Fear and McRoberts’ database is a compendium and re-interpretation of the case histories listed in the databases of Seed et al. (1984) and Ambraseys (1988). Furthermore, the Fear and McRoberts’ database provides the available boring logs for each case history. Only the Liao and Whitman database includes epicentral distances. The variables listed in Table 5-1 are those given by Seed et al. (1984), with the epicentral distances coming from Liao and Whitman (1986), and the N1,60cs values were computed using the latest NCEER (1997) recommended procedure presented in Chapter 2. Because not all the case histories listed in Seed et al. (1984) could be matched to corresponding case histories in Liao and Whitman (1986), the data point numbers listed in Table 5-1, as assigned by Seed et al. (1984), are not completely sequential. To facilitate the cross reference between the two

142

databases, the case numbers assigned by Liao and Whitman (1986) are listed in the last column of the table. Per Ambraseys (1988), the case histories were categorized as: Liq 0 – No Liquefaction; Liq 1 – Marginal Widespread Liquefaction; Liq 2 – Sporadic Liquefaction; and Liq 3 – Complete Liquefaction. A plot of the data is shown in Figure 5-11. For the purposes of drawing a boundary separating the points corresponding to liquefaction and no liquefaction, liquefaction was assumed to occur for all the case histories categorized as Liq1, 2, and 3. Analogous to the procedures presented in Chapter 2, this boundary quantifies the Capacity of the soil as a function of the penetration resistance and may be approximated by Equation (5-11). NEC 1.195  10 4 exp 0.185  N1,60 cs 

where:

NEC

for 3  N1,60cs  27

=

Normalized energy capacity (i.e., Capacity).

N1,60cs =

SPT N-values corrected per NCEER (1997) recommendations.

(5-11)

The boundary shown in Figure 5-11 and defined by Equation (5-11) was drawn slightly “less conservative” (i.e., to left and up) than may be desired for liquefaction evaluations. However, the position of the boundary was selected because its eventual use is for remedial ground densification design, and its placement is such as to ensure enough energy is imparted in the ground to induce liquefaction. Accordingly, a curve that is “unconservative” for liquefaction evaluations is “conservative” for remedial ground densification design.

143

0.020

53

84 108 85

107

109

114

No Liquefaction (Liq:0) Marginal Widespread Liquefaction (Liq:1)

0.018

Sporadic Liquefaction (Liq:2) Complete Liquefaction (Liq:3) Capacity Curve

0.016

83

Normalized Energy Demand (NED)

0.014 1

91

0.012

18 13 55

0.010

A9

37

2 36 54

86 7

87

0.008

89

94

32

27 42 105

0.004

82

61

A7

3

8

59

A8

0.006

92 35

12 43 98

97 45

33

4 96

102

103

90

100

99 88

111 101

46 104 64 See Figure 5-11b 30 106 22 for enlargement of 15 25 A1 26 63 this area A2 117 62 23 29 A5 6 14 0.002 A3 48 51 49 119 120 112 44 52 57 20 113 73 31 122

0.000

0

5

10

66

15

20

25

28

30

35

40

45

50

N1,60cs Figure 5-11a. Energy-based Capacity curve developed from 126 liquefaction field case histories.

144

0.0040

No Liquefaction (Liq:0) Marginal Widespread Liquefaction (Liq:1) 93

66

Sporadic Liquefaction (Liq:2)

0.0035

Complete Liquefaction (Liq:3) Capacity Curve

110

5 24

21

30

9

64

95 15

0.0030 63

22

Normalized Energy Demand (NED)

A6 62

0.0025

A2 14 118

29

6

0.0020 48

51 10

0.0015 65

58

50 49

0.0010

119

44

57 52

112 11 40,68

0.0005

20

72

81

70 78

76 47 121 115 41 116 75 69

31

0

2

4

6

8

10

80

39

38

67 60

123

0.0000

74

71

79

77

12

14

N1,60cs Figure 5-11b. Energy-based Capacity curve developed from 126 liquefaction field case histories.

145

15

Seed et al. Critical Depth of Data Pt # Depth (ft) gwt (ft) 1 33 3 2 23 7 3 20 6 4 17 8 5 23 13 6 14 13 7 27 3 8 17 3 9 17 10 10 27 6 11 20 6 12 17 7 13 12 2 14 10 8 15 23 7 18 13 4 20 10 8 21 23 3 22 23 3 23 23 6 24 33 3 25 33 3 26 33 6 27 14 0 28 20 4 29 20 8 30 15 2 (psf) 3960 2760 2400 2040 2630 1550 3240 2040 2040 3095 2270 2040 1440 1120 2760 1560 1075 2760 2760 2760 3960 3960 3960 1680 2400 2400 1800

vo

Table 5-1. Liquefaction case histories.

’vo (psf) 2090 1760 1530 1480 2000 1490 1740 1170 1600 1845 1460 1420 820 1000 1760 1000 950 1510 1510 1700 2090 2090 2270 810 1400 1650 990 FC (%) 0 5 3 4 10 22 1 5 20 25 2 10 27 30 5 0 3 2 2 2 2 2 2 10 0 0 0

146

N1,60cs (blws/ft) 25.4 11.8 24.0 15.0 11.4 6.3 19.7 14.5 10.3 13.1 7.8 10.5 6.1 7.8 11.8 10.6 4.3 9.5 14.3 22.4 10.7 17.1 22.7 6.8 30 13.7 7.9

amax (g) 0.32 0.32 0.32 0.32 0.2 0.2 0.2 0.2 0.2 0.2 0.16 0.2 0.2 0.2 0.16 0.4 0.19 0.16 0.16 0.18 0.16 0.16 0.18 0.16 0.18 0.18 0.16

Richter M 7.9 7.9 7.9 7.9 7.9 7.9 7.9 7.9 7.9 6.3 6.3 8.0 8.0 8.0 8.0 7.3 5.3 7.3 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5

ED (km) 30 30 30 30 72 72 72 72 72 24 24 165 165 165 200 6 6 51 51 51 51 51 51 51 51 51 51 NED 0.01328 0.01012 0.00772 0.00583 0.00345 0.00207 0.00841 0.00727 0.00323 0.00151 0.00069 0.00494 0.01142 0.00227 0.00305 0.01208 0.0004 0.00336 0.0028 0.0025 0.00337 0.00283 0.00297 0.00536 0.00267 0.00208 0.00333

Liq 3 3 3 3 2 2 3 3 1 0 0 3 3 3 3 3 3 3 1 0 3 1 0 3 0 0 3

L&W Case # 0301 0302 0303 0304 0702 0703 0704 0705 0707 1005 1003 1201 1202 1203 1204 1304 1502 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823

Seed et al. Critical Depth of Data Pt # Depth (ft) gwt (ft) 31 3 3 32 13 3 33 20 7 35 13 3 36 20 10 37 20 15 38 19.7 6.6 39 42.7 6.6 40 20.3 5 41 9.8 6.6 42 32.8 6.6 43 33.8 6.6 44 30 5 45 27 5 46 27 5 47 27 5 48 34 5 49 15 8 50 38 8 51 35 12.5 52 7.5 4.5 53 23 3.3 54 17.5 10 55 17.4 3 57 6.7 5 58 20 4 59 20 3 60 30 13 61 23 5

vo (psf) 330 1560 2400 1560 2250 2330 2325 5290 2505 1180 4055 4180 3675 3300 3300 3300 2910 1150 3210 2860 875 2740 2040 2130 760 2440 2455 3555 2800

’vo (psf) 330 940 1590 940 1800 2000 1505 3038 1550 980 2420 2480 2115 1930 1930 1930 1800 705 1860 1490 685 1510 1570 1230 655 1440 1395 2495 1675 FC (%) 0 20 5 5 20 >50 ? ? ? ? ? ? 67 ? 48 ? 3 3 3 0 ? ? 10 20 12 12 ? 1 3

147

N1,60cs (blws/ft) 2.8 11.0 30 22.6 10.9 6.7 10.4 11.8 6.5 9.6 7.8 8.1 12.4 10.6 20.1 8.7 4.7 8.6 11.7 13.9 14 4.4 30 23.5 14.7 13.1 10.2 11.1 11.4

amax (g) 0.13 0.2 0.23 0.23 0.45 0.45 0.1 0.1 0.1 0.13 0.2 0.2 0.13 0.2 0.2 0.1 0.135 0.135 0.135 0.135 0.2 0.35 0.5 0.35 0.2 0.13 0.2 0.07 0.2

Richter M 6.3 7.9 7.9 7.9 6.6 6.6 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.5 7.5 7.5 7.5 7.8 7.8 7.6 7.6 7.6 7.6 7.6 7.6 7.6

ED (km) 56 290 190 190 14 14 90 104 95 60 55 55 55 53 50 90 170 170 170 170 43 25 1 20 80 90 90 110 110 NED 0.00007 0.00655 0.00614 0.00804 0.00971 0.00992 0.00035 0.0004 0.00052 0.00034 0.00493 0.00479 0.00118 0.00488 0.00387 0.00055 0.00167 0.00109 0.00122 0.00162 0.00110 0.04219 0.00915 0.01061 0.00113 0.00133 0.00705 0.00013 0.00574

Liq 3 3 0 0 3 3 0 0 3 3 3 3 3 3 3 0 3 1 1 0 3 3 0 3 3 3 3 0 3

L&W Case # 2001 2201 2207 2208 2603 2601 2803 2805 2806? 2810 2811 2812 2813 2809 2814 2815 2901 2902 2902 2903/4 3001 3002 3006 3007 3009 3010 3011 3012 3013

Seed et al. Critical Depth of Data Pt # Depth (ft) gwt (ft) 62 27 15 63 39 22 64 12 4 65 10 7 66 17 6 67 21 3 68 11 2 69 11 4 70 18 6 71 14 3 72 14 6 73 11 4 74 14 1 75 21 14 76 21 8 77 11 10 78 13 8 79 17 8 80 20 8 81 13 5 82 21 3 83 11 3 84 11 2 85 11 4 86 18 6 87 14 3 88 18 7 89 14 6 90 11 4

vo (psf) 2970 3190 1320 1100 1870 2520 1320 1320 2060 1680 1680 1320 1680 2220 2390 1300 1560 2040 2400 1560 2520 1320 1320 1320 2060 1680 2160 1680 1320

’vo (psf) 2220 2130 820 910 1180 1400 760 880 1310 990 1180 880 870 1780 1580 1240 1250 1480 1650 1060 1400 820 760 880 1310 990 1470 1180 880 FC (%) 3 ? 4 3 50 0 5 4 60 0 10 7 12 5 4 5 10 20 3 10 0 4 5 4 60 0 0 2 7

148

N1,60cs (blws/ft) 10.2 8.7 13.1 12.5 11.0 12.4 6.5 8.4 7.5 13.3 5.4 17.8 12.2 9.1 8.6 11.2 7.6 14.1 13.7 5.8 12.4 26.6 6.5 8.4 7.5 13.3 22.2 5.4 17.8

amax (g) 0.2 0.2 0.2 0.2 0.2 0.1 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.14 0.14 0.14 0.14 0.14 0.14 0.12 0.2 0.32 0.32 0.32 0.24 0.24 0.24 0.24 0.24

Richter M 7.4 7.4 7.4 7.4 7.4 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 6.7 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4

ED (km) 70 70 70 70 70 140 115 115 115 115 115 115 115 77 77 77 75 75 75 75 115 112 112 112 115 115 115 115 115 NED 0.00251 0.00285 0.00337 0.00135 0.00387 0.00028 0.00052 0.00029 0.00041 0.00037 0.00035 0.00021 0.00056 0.00032 0.0006 0.00015 0.00033 0.00036 0.00044 0.00036 0.0058 0.01439 0.03956 0.01916 0.00883 0.00825 0.00456 0.00753 0.0046

Liq 3 3 0 0 3 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 2 3 2 0 2 3

L&W Case # 3201 3203 3204 3205 3206 3401 3403 3404 3405 3406 3408 3409 3410 3412 3413 3414 3417 3419 3421 3424 3501 3502 3503 3504 3505 3506 3507 3508 3509

Seed et al. Critical Depth of Data Pt # Depth (ft) gwt (ft) 91 14 1 92 24 4 93 21 14 94 21 8 95 11 10 96 20 10 97 20 8 98 13 8 99 23 8 100 17 8 101 15 8 102 20 8 103 20 8 104 23 12 105 13 5 106 20 5 107 12 6 108 12 6 109 14 6 110 6 1 111 14 1 112 11 7 113 7.5 7 114 7 5 115 20 3 116 47 3 117 16 3 118 14 9 119 6 1

vo (psf) 1680 2880 2220 2390 1300 2400 2400 1560 2760 2040 1800 2400 2320 2660 1560 2400 1410 1410 1660 735 1735 1270 830 800 2260 5310 1960 1615 735

’vo (psf) 870 1630 1780 1580 1240 1780 1680 1250 1820 1480 1360 1650 1570 1970 1060 1460 1035 1035 1160 425 930 1020 800 675 1200 2560 1150 1305 425 FC (%) 12 17 5 4 5 0 10 10 5 20 26 3 11 12 10 10 25 29 37 80 18 75 30 31 13 27 40 92 80

149

N1,60cs (blws/ft) 12.2 24.3 9.1 8.6 11.2 28 12.4 7.6 25.3 14.1 17 13.7 21.2 21.9 5.8 21.5 30 6 23.3 9.8 18.6 7.8 20.5 9.4 8.8 8.8 17.7 11.6 9.8

amax (g) 0.24 0.24 0.24 0.24 0.28 0.28 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.2 0.2 0.78 0.78 0.78 0.24 0.24 0.2 0.2 0.51 0.13 0.13 0.26 0.32 0.21

Richter M 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 7.4 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.6 6.1 6.1 5.6 5.6 5.6

ED (km) 115 115 92 92 95 95 85 85 85 85 85 85 85 85 80 80 10 10 10 42 42 45 45 26 15 15 10 3 15 NED 0.01329 0.0082 0.0038 0.00733 0.00316 0.00564 0.00522 0.00402 0.00499 0.00435 0.00347 0.0053 0.00475 0.00362 0.00416 0.00356 0.09502 0.21985 0.15075 0.00348 0.00442 0.00089 0.00028 0.01939 0.0004 0.00037 0.00249 0.0021 0.00111

Liq 2 0 2 2 2 0 1 2 0 2 1 3 0 0 3 0 0 3 0 3 3 3 0 3 0 0 3 3 0

L&W Case # 3510 3511 3512 3513 3514 3515 3516 3517 3518 3519 3520 3521 3522 3523 3524 3525 3901 3902 3903 3904 3905 3918 3919 3920/1? 4101 4103 4206-11? 4214-17? 4204

vo (psf) 1735 1270 830 800

’vo (psf) 930 1020 800 675 FC (%) 18 75 30 31

N1,60cs (blws/ft) 18.6 7.8 20.5 9.4

amax (g) 0.21 0.2 0.2 0.09

Richter M 5.6 5.6 5.6 5.6

ED (km) 15 14 14 28

FC (%) ? ? 50 35 1 3 10 ?

N1,60cs (blws/ft) 18.6 11.2 16.9 17.2 9.2 7 9.7 8.7

amax (g) 0.16 0.16 0.45 0.24 0.2 0.2 0.35 0.35

Richter M 7.5 7.5 6.6 6.5 6.9 6.9 6.7 6.7

ED (km) 51 51 14 25 50 50 23 23

150

Seed et al. Data Pt #: Data point number assigned by Seed et al. (1984) Critical Depth (ft): Depth in the profile most susceptible to liquefaction Depth of gwt (ft): Depth of the ground water table vo (psf): Total vertical stress at the critical depth ’vo (psf): Initial vertical effective stress at the critical depth FC (%): Fines content of the soil at the critical depth N1,60cs (blws/ft): Corrected penetration resistance at the critical depth amax (g): Maximum horizontal acceleration at the soil surface Richter M: Richter magnitude ED (km): Epicentral distance from the site to the source NED: The normalized energy demand (i.e., Demand) imposed on the soil by the earthquake L&W Case #: Designation number for the case history assigned by Liao and Whitman (1986)

Critical Depth of vo ’vo Depth (m) gwt (m) (kg/cm2) (kg/cm2) 9.7 0.5 1.8 0.92 9.5 2.0 1.8 1.07 16.5 16.5 3.22 3.22 5.8 1.5 1.1 0.67 4.3 1.2 0.8 0.49 8 1 1.49 0.79 4 2.1 0.72 0.53 4.9 2.1 0.9 0.6

Column headings:

Data Pt # A1 A2 A3 A5 A6 A7 A8 A9

Additional case histories listed in Fear and McRoberts (1995), but not listed in Seed et al. (1984).

Seed et al. Critical Depth of Data Pt # Depth (ft) gwt (ft) 120 14 1 121 11 7 122 7.5 7 123 7 5

NED 0.00307 0.00238 0.00159 0.00257 0.00278 0.00549 0.00644 0.01005

NED 0.00148 0.00048 0.00015 0.00002

Liq 0 3 3 0 3 3 3 3

Liq 0 3 0 0

L&W Case # 3602 3801? 3801? 4001? 4001?

L&W Case # 4205 4218 4219 4221

5.4 Liquefaction Evaluation To perform a liquefaction evaluation, the Demand (i.e., NED) imposed on the soil by the earthquake is determined from the iterative solution of Equation (5-10). The normalized energy capacity (NEC) is then determined graphically using Figure 5-11 or numerically using Equation (5-11). If NED < NEC, liquefaction is not predicted to occur, and if NED  NEC liquefaction is predicted. The factor of safety (FS) against liquefaction is given by Equation (5-12).

FS 



Capacity Demand NEC Equation (5  11) NED Equation (5  10)

(5-12)

5.5 Parameter Study The proposed liquefaction procedure was evaluated in the same way as the procedures presented in Chapter 2, where the factors of safety versus depth (FS-profile) are computed for simple soil profiles. The profiles were assumed to be subjected to a M7.5 earthquake at an epicentral distance of 60km, resulting in amax = 0.13g. The following profiles were examined: 

100ft thick profile of clean sand having N1,60 = 15blws/ft at all depths. The depth of the ground water table (gwt) is equal to 0ft, 11ft, and 25ft.



100ft thick profile of clean sand with the gwt at a depth of approximately 11ft. The profile is assumed to have constant N1,60 with depth equal to 5, 10, and 15blws/ft.

The resulting FS-profiles for the proposed procedure are shown in Figure 5-12. For comparison purposes, the FS-profiles computed using both the stress- and strain-based procedures (Figures 2-20 and 2-21 in Chapter 2) are presented below as Figures 5-13 and 5-14, respectively. Because the stress-based procedure has continually evolved with new insights into soil behavior, the trends in the stress-based FS-profiles are assumed to be

151

correct, although the absolute values of the FS may not be. From examination of the stress-based FS-profiles, the following observations are made: 

The critical depth is unaffected by the changes in N1,60.



The critical depth shifts downward as the elevation of the ground water table is lowered.



The FS increases as N1,60 increases and as the gwt is lowered. Both of these trends correspond to the observed laboratory behavior that the Capacity of the soil increases with increasing relative density and increasing effective confining stress.

The critical depth is defined as the depth below the gwt corresponding to the lowest FS. As may be recalled from Chapter 2, from FS-profiles for the strain-based procedure excess pore pressures are expected for the entire depths of all the profiles. The high FS towards the surface of the profiles shown in Figure 5-14 are attributed to the limitations in the empirical correlation for Gmax, (G/Gmax), and ’ vs. N1,60 at low effective confining stresses and are not the expected behavior of the soil. Because the proposed energy-based liquefaction evaluation procedure is a unification of the stress- and strain-based approaches, the FS-profiles shown in Figure 5-12 have the characteristics of those shown in Figures 5-13 and 5-14. Most notably: 

Except for the profile with the gwt at the surface, the critical depths are essentially the same as those of the stress-based procedure.



The critical depth is unaffected by the changes in N1,60.



The critical depth shifts downward as the elevation of the ground water table is lowered.



The FS are over predicted at low effective confining stresses (i.e., near the surface of the profiles).

Even with the last observation, the proposed liquefaction evaluation procedure is deemed acceptable for remedial ground densification design. As a final comment, it may be observed from comparisons of the FS-profiles that the absolute values of the predicted FS

152

are unique to each procedure (e.g., a FS = 1.2 for one procedure does not mean the same thing as it does for another procedure). a)

0

b)

0

1 2

5

5

10

Depth (m)

1

Approximate Critical Depth

10

3 2 3 N1,60 = 15

15

15

20

20

25

25 5

30

0

Approximate Critical Depths

1

N1,60 = 15

10 5

10

15

30

Factor of Safety

0

5

2

3 10

15

Factor of Safety

Figure 5-12. FS-profiles computed using the proposed energy-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

153

NCEER (1997) 0

0

a)

b)

1

1

2 5

5

10

Depth (m)

3

Approximate Critical Depth

2 10 3

15

15 N1,60 = 15

20

20 Approximate Critical Depths

25

25 5

30

0

1 2 3

10 N1,60 = 15

1 2 Factor of Safety

3

30

0

1

2

3

Factor of Safety

Figure 5-13. FS-profiles for the stress-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

154

Dobry et al. (1982) 0

a) 5

b)

10

2

5

1

5

N1,60 = 15

10

Depth (m)

1

0

10

3

N1,60 = 15

2

15

15

3

20 Critical Depths difficult to determine

25

30

3

1

Approximate Critical Depth

20

2

Approximate Critical Depths

25

0

1

2

3

Factor of Safety

30

0

1

2

3

Factor of Safety

Figure 5-14. FS-profiles for the strain-based procedure: a) Profiles have constant N1,60 = 5, 10, 15blws/ft and gwt at approximately 11ft. b) Profiles have constant N1,60 = 15blws/ft and gwt depths 0, 11, and 25ft.

155

5.6 Comparison of Energy-Based Capacity Curve Derived from Field Data with Laboratory Test Data The Capacity curve shown in Figure 5-11, derived from the analysis of earthquake case histories, cannot be directly compared to the cumulative energy dissipated in undrained laboratory specimens subjected to cyclic loading. This is because the Capacity curve was developed using numerical analyses that did not account for softening of the soil due to increases in pore pressures (herein referred to as total stress analyses), while such softening inherently occurs in the undrained cyclic loading of saturated sands. The influence of pore pressure increases on energy dissipation can be understood by

Deviator Stress (kPa)

comparing the respective hysteresis loops shown in Figure 5-15.

60 40

40

a)

20 0 -20 -40 -60 -0.3 -0.15 0 0.15 Axial Strain (%) 140

b)

Deviator Stress (kPa)

Deviator Stress (kPa)

80

60

20 0 -20 -40 -60

-6 -4 -2 0 2 4 6 Axial Strain (%)

0.3

c)

100 60 20 0 -20 -60 -0.2

8

-0.1 0.0 0.1 Axial Strain (%)

Figure 5-15. Stress-strain hysteresis loops for a) total stress numerical analysis, b) stress controlled cyclic triaxial test and c) strain controlled cyclic triaxial test.

156

0.2

The hysteresis loops shown in Figure 5-15a are those expected if an element of soil was subjected to sinusoidal loading, and the boundary conditions were such that density of the sample and the effective confining stress were not allowed to change (e.g., total stress numerical analysis). Both the shear modulus and damping ratio remain constant for each cycle of loading, and the resulting loops are the same for stress- and strain-controlled loadings. On the contrary, for saturated samples subjected to undrained cyclic loading, the shear modulus for each subsequent loop is less than the previous loop, and the damping ratio for each subsequent loop is greater than the previous loop. As may be seen in Figures 5-15b and 5-15c, the area bound by hysteresis loops increases as the loading progresses for stress-controlled loading, while it decreases for strain-controlled loading. Accordingly, the cumulative dissipated energy determined by a total stress numerical analysis significantly differs from that which actually dissipates in undrained cyclic laboratory specimens. As may be recalled from Chapter 2, several of the existing energybased procedures were criticized for the improperly comparing the energies from total stress numerical analyses to undrained cyclic laboratory data. A normalization technique is proposed for removing (or compensating for) the dissipated energy resulting from the softening of the soil due to increases in pore pressures. Such a procedure allows the dissipated energy computed from undrained cyclic laboratory tests to be compared to the Capacity curve derived from total stress numerical analyses (i.e., Figure 5-11). Two idealized hysteresis loops are shown in Figure 5-16 for a stress controlled cyclic laboratory test, where loop 1 is measured earlier in the test than loop 2 (i.e., ’m1 > ’m2).

157

 W1



1 W2

2



’m1 > ’m2

Figure 5-16. Two idealized hysteresis loops for a stress-controlled, undrained cyclic test, where loop 1 occurs earlier in the test than loop 2.

Starting with Equation (5-2), the ratio of the areas (or dissipated energies) bound by loop 2 and loop 1 (ER) may be written: W 2 W1

ER 



2D 2 2 2D1 1



(5-13)

D 2 G1 D1G 2

where: ER =

Energy ratio.

W1 =

Area bound by loop 1 (i.e., dissipated energy per unit volume associated with loop 1).

W2 =

Area bound by loop 2 (i.e., dissipated energy per unit volume associated with loop 2).



=

Amplitude of the shear stress for stress controlled loading.

1

=

Amplitude of shear strain for loop 1.

2

=

Amplitude of shear strain for loop 2.

D1 =

Damping ratio for loop 1.

D2 =

Damping ratio for loop 2.

158

G1 =

Shear modulus for loop 1.

G2 =

Shear modulus for loop 2.

Using the Ishibashi and Zhang (1993) shear modulus and damping degradation curves, ER can be computed for different values of ’m1 and ’m2, different amplitudes of the applied loading (CSR), and different relative densities of the soil (Dr). From examining the ER values for several combinations of CSR and Dr, it was observed that these parameters have relatively little influence on ER, so the following discussion is limited to the influence of ’m1 and ’m2. Additionally, inherent to using the degradation curves in computing ER is the assumption that changes in pore pressure only affects the effective confining stress on the soil (i.e., undrained behavior can be approximated as drain, with appropriate changes in effective confining pressure). This assumption is reasonable and is one of the underlying assumptions of the commonly used strain-based pore pressure generation model proposed by Martin et al. (1975). Setting ’m1 = ’mo and ’m2 = ’m, ER is plotted as a function of the ratio ’m /’mo in Figure 5-17 for CSR = 0.1, ’mo = 100kPa, and two different relative densities (Dr). For isotropically consolidated specimens, the ratio ’m/’mo is related to the excess pore pressure ratio (ru) by: ru = 1-’m/’mo. Accordingly, when ’m /’mo = 1.0, ru =0, and when ’m/’mo = 0, ru = 1.0 (i.e., liquefaction). From Figure 5-17, it can be observed that ER increases as ’m/’mo decreases. This implies that for a stress controlled cyclic test, the area bound by a hysteresis loop increases as the pore pressures increase, which is consistent with observed behavior in laboratory test data such as shown in Figure 5-15b. In addition to ER curves, the function: (’m /’mo)-1.65 is also plotted in Figure 5-17. As may be observed, the curve corresponding to this function reasonably approximates the ER curves. Based on this approximation (i.e., ER  (’m2/’m1)-1.65), the dissipated energies associated with the hysteresis loops shown in Figure 5-16 can be related as:

159

W1 

W2 ER

 '  W2   m 2   ' m1

where:

ER

=

  

1.65

(5-14) Energy Ratio.

W1 =

Dissipated energy per unit volume for loop 1.

W2 =

Dissipated energy per unit volume for loop 2.

’m1 =

Mean effective confining stress corresponding to loop 1.

’m2 =

Mean effective confining stress corresponding to loop 2.

25 CSR = 0.1, ’mo = 100kPa Dr = 40% Dr = 70%

Energy Ratio (ER)

20

  'm      'm o 

1.65

15

10

5 ER = 1 0

0.2

ru = 0.8

0.3

0.4

0.5

0.6

’m /’mo

0.7

0.8

0.9

1.0 ru = 0.0

Figure 5-17. The ratio of the energies dissipated during two cycles of loading in an undrained stress controlled cyclic test.

160

Equation (5-14) is quite significant and forms the basis of the proposed normalization technique for removing (or compensating for) the dissipated energy resulting from the softening of the soil due to increases in pore pressures. In short, if the energy dissipated in a sample confined at ’m2 is known, the energy that would have been dissipated in the same sample had it been confined at another effective confining stress (’m1) can be determined. An expression for the cumulative energy dissipated in a cyclic triaxial specimen was derived previously (i.e., Equation (3-18)) and is presented below as being divided by initial mean effective confining stress. W 1    ( d , i 1   d , i )( a ,i 1   a ,i )  ' m o 2 ' m o i

where:

(5-15a)

W =

The cumulative energy dissipated in a cyclic triaxial sample.

’mo =

Initial mean effective confining stress.

d,i

=

The ith increment in deviatoric stress.

a,i

=

The ith increment in axial strain.

By applying the approximation ER  (’m /’mo)-1.65 (i.e., ’m1 = ’mo and ’m2 = ’m) to remove the effects of strain softening due to increased pore pressures, the above expression becomes: W 'mo

 'm o where:

  ' m ,i 1  ' m ,i 1    ( d , i 1   d , i )( a ,i 1   a ,i ) 2 ' m o i 2 ' m o 

  

1.65

(5-15b)

W’mo = Energy dissipated in sample if effective confining stress remains ’mo.

’m,i

= The mean effective confining stress at the ith increment of loading.

Finally, if the above expression is used to integrate the stress-strain hysteresis loops up to initial liquefaction, the normalized energy capacity (NEC) of the specimen, for which the effects of strain softening due to increased pore pressures are removed, can be determined from the following expression.

161

1.65

n 1  'm , i   ' 1  NEC    ( d , i 1   d , i )( a , i 1   a ,i ) m ,i 1 2 'm o i 1 2  ' m o  

where:

(5-15c)

NEC =

Normalized energy capacity (dimensionless).

n

Number of load increments up to failure (e.g., liquefaction).

=

The NEC determined from this expression can be compared to the Capacity curve shown in Figure 5-11, which was developed from total stress numerical analyses of earthquake case histories. The validity of Equation (5-15c) can be assessed from analyzing cyclic triaxial test data. For a soil sample subjected to a stress controlled, sinusoidal loading wherein strain softening occurs due to pore pressure increases, the rate of energy dissipation will increase with time. This can be understood from examining Figure 515b. The hysteresis loops shown in this figure were scribed at a constant rate and increase in size as the loading progresses. Accordingly, if the cumulative energy dissipated in the sample is plotted as a function of time, the curve would have a slope that progressively increases with time (the slope represents the rate of energy dissipation). Using Equation (5-15a), such a plot was computed and is shown in Figure 5-18. Correspondingly, for a total stress numerical analysis of the same soil sample (e.g., using SHAKE), the energy will dissipate at a constant rate. This can be understood from examining Figure 5-15a, wherein the hysteresis loops remain unchanged from one cycle to the next and are scribed at a constant rate. The corresponding plot of the cumulative energy dissipated as a function of time will be a straight line with the slope representing the rate of energy dissipation. Accordingly, the cumulative energy dissipated as a function of time for cyclic laboratory data computed using Equation (5-15b) should also plot as a straight line, if the proposed normalization technique is valid. The same cyclic triaxial test data analyzed using Equation (5-15a) were reanalyzed using Equation (515b), the results of which are shown in Figure 5-18. As may be seen in this figure, the results of the reanalysis plot approximately as a straight line and is typical of the data analyzed similarly for several hundred cyclic triaxial tests.

162

ru = 1.0

Normalized Dissipated Energy

Normalized per Equation (5-15a) Normalized per Equation (5-15b)

0.04

W@ liq

 'm o

0.02 NEC 0.00 0

5

10

20 15 Time (sec)

25

30

Figure 5-18. Normalized dissipated energies computed from cyclic triaxial test data using Equations (5-15a) and (5-15b). The data shown in this figure is that listed as m0e76c28 in Table 5-2a.

As stated previously, several researchers inappropriately compared the energies from total stress numerical analyses to undrained cyclic laboratory data. The significance of the error in this can be seen in Figure 5-18 by comparing the energies at initial liquefaction as determined by Equations (5-15a) and (5-15b). As may be seen in this figure, the Capacity of the soil determined from Equation (5-15a) is significantly higher than that computed using Equation (5-15c). The appropriate procedure for computing Capacity depends on how the Demand is determined, and errors are only introduced when the two are determined inconsistently. For example, if total stress numerical analyses are used to compute the Demand (e.g., using SHAKE), then the Equation (515c) should be used to compute a comparable Capacity from undrained cyclic laboratory test data. However, if effective stress numerical analyses (i.e., analyses that include strain softening due to pore pressure increases) are used to compute the Demand, then Equation (5-15a) should be used to compute the Capacity of the soil from undrained cyclic laboratory data. This latter approach was used in the energy-based liquefaction evaluation procedure proposed by Professor J.L. Figueroa and his associates at Case Western Reserve University (e.g., Liang (1995)).

163

Using Equation (5-15c), NEC was computed from several hundred isotropically consolidated, stress controlled cyclic triaxial data. The specimens were reconstituted, moist tamped, non-plastic silt-sand mixtures having varying fines contents (FC) and relative densities (Dr). The specimens were reconstituted using Monterey and Yatesville sands and non-plastic silt, where the Monterey sand is medium grained and the Yatesville sand is fine grained. The full details of the testing program and how Dr was determined for silty samples are given in Polito (1999) and Polito and Martin (2001). As shown in Polito (1999) and Polito and Martin (2001), the non-plastic silt has no influence on the strength of the soil up to the limiting silt content of the base sand. The limiting silt content is defined as that required to fill the voids of the sand skeleton. Above the limiting silt content, the sand particles are suspended in the silt matrix and have no influence on the soil behavior, while the silt particles control the behavior of the specimen. For relative densities less than about 80%, the limiting silt content is between 25% and 35% for Monterey sand and between 37% and 50% for the Yatesville sand. Polito (1999) and Polito and Martin (2001) do not specify the limiting silt contents as functions of relative density. However, from reanalysis of their data, the author hypothesizes that for samples less than about Dr = 80%, the soil particles can rearrange so that the sand grains become in contact and control the behavior of the soil, even if in the initial undisturbed state the silt lies between the sand grain contact surfaces. However, at higher relative densities, the sand particles cannot easily rearrange, and the silt lying between the sand grains prevents grain-to-grain contact of the sand, thus the silt controls the soil behavior. More refined estimates of limiting silt contents as functions of relative density and fines content cannot be determined from the available laboratory data. The specimens were divided into two groups according to whether their silt contents were above or below the limiting values of the sand. The computed NEC values are plotted as functions of Dr in Figures 5-19 and 5-20 for samples having FC below and above the limiting silt contents, respectively. Failure was defined as either initial liquefaction (ru = 1.0) or a double amplitude strain of 5% (DA = 5%), whichever occurred first. Tables 52a and 5-2b list the failure modes and the NEC values for the Monterey and Yatesville silt-sand mixtures, respectively.

164

For comparison, the energy-based Capacity curve from Figure 5-11 is also shown in Figures 5-19 and 5-20. To convert N1,60 to Dr, the following relation was used: Dr 15  N1,60 (%)

(Skempton 1986)

(5-16)

As can be seen in Figure 5-19, the field based curve gives an approximate lower bound of the normalized energy capacities determined from laboratory data for samples below the limiting silt content. This is consistent with the selected position of the boundary separating the field data corresponding to liquefaction and no liquefaction. From examining the NEC values listed in Table 5-2a, it may be observed that samples subjected to higher amplitude CSR loadings have lower computed NEC values, similar to the PEC discussed in Chapter 4. This trend could result from energy being dissipated in the test equipment and the membrane surrounding the sample. At higher amplitude loadings the energy lost due to these mechanisms may be small as compared to energy dissipated due to friction between the sand grains. However, as the amplitude of the loading decreases, these mechanisms may become important. Also, a viscous mechanism due the relative movement of the pore water and the sand grains (i.e., viscous drag) may become more prominent at lower amplitude loadings (Hall 1962). A detailed laboratory study is required to examine the load dependence of NEC.

165

Normalized Energy Capacity (NEC)

0.12 0.10

Yatesville Lab Data (FC37%) Monterey Lab Data (FC25%) Field Based Liquefaction Curve

0.08

0.06

0.04

0.02

0.00 -60 -40 -20

0 20 40 60 80 100 120 Relative Density (%)

Figure 5-19. NEC values computed from stress controlled cyclic triaxial tests for samples having silt contents below the limiting value. Also shown is the capacity curve derived from field case histories. The negative relative densities were generated by moist tamping the laboratory samples, thus creating specimens having larger void ratios than the maximum predicted by the index tests using dry soil.

From Figure 5-20, it can be observed that the field based curve is considerably greater than the Capacities of samples having a silt content above the limiting values. A similar trend was identified by Polito (1999) and Polito and Martin (2001) from the evaluation of silt-sand mixtures using cyclic stress ratio (CSR) as the measure of Capacity and Demand. As hypothesized by Polito (1999) and Polito and Martin (2001), above the limiting silt content, the sand particles are suspended in the silt matrix and have no influence on the soil behavior, while the silt particles control the behavior of the specimen. The data presented in Figure 5-20 show that silt is less resistant to liquefaction than sand having the same relative density (i.e., the zone representing liquefaction in Figure 5-20 is larger than the corresponding zone for sands shown in Figure 5-19).

166

Normalized Energy Capacity (NEC)

0.12 0.10

Yatesville Lab Data (FC>37%) Monterey Lab Data (FC25%) Monterey Lab Data (FC>25%) Field Based Liquefaction Curve

0.08

FC=20% 20% 25% 25%

0.06

20%

0.04

25%

0.02

0.00 -60 -40 -20

0 20 40 60 80 100 120 Relative Density (%)

Figure 5-20. NEC values computed from stress controlled cyclic triaxial tests for samples having silt contents above the limiting value. Also shown is the capacity curve derived from field case histories.

In addition to the NEC values, Tables 5-2a and 5-2b also list the dissipated energy per unit volume of soil (W), W/’3o, and PEC. Liang (1995) used W to define the soil capacity, while Alkhatib (1994) used W/’3o.

167

This Page Was Intentionally Left Blank to Ensure Proper Pagination

168

File Name m0e68c30 m0e68c35 m0e68c40 m0e72c27 m0e72c30 m0e72c35 m0e73c25 m0e73c31 m0e73c37 m0e74c22 m0e74c25 m0e74c30 m0e75c20 m0e75c25 m0e75c30 m0e76c22 m0e76c25 m0e76c28 m0e84c20 m0e84c21 m5e61c32 m5e61c35 m5e61c40 m5e65c27 m5e65c30 m5e65c33

Dr (%) 81.1 81.1 81.1 65.3 64.2 64.2 57.4 58.4 58.4 46.3 46.8 46.8 47.4 47.4 47.9 42.6 42.6 42.9 -4.7 -3.7 82.1 79.9 81.6 64.7 64.4 63.3

CSR 0.511 0.598 0.691 0.439 0.505 0.590 0.399 0.525 0.597 0.362 0.389 0.522 0.314 0.408 0.484 0.343 0.395 0.465 0.179 0.18 0.525 0.591 0.681 0.426 0.488 0.552

T (sec) 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 (kPa) 99.74 99.84 100.2 100.4 99.11 99.53 100 99.11 99.74 98.16 100.8 97.74 100 99.63 99.21 100.4 99.95 98.79 99.53 99.95 101.1 100.8 100.9 101.4 100.5 99.53

’3o Nliq 168.1 30.1 12.1 73.1 58.1 12.6 351.0 15.1 5.1 108.1 50.6 6.1 457.0 34.6 7.1 87.0 36.0 12.1 3.9 5.0 46.1 28.1 15.1 83.1 22.6 10.1

169

W (kPa) 15.22 11.33 10.34 7.68 8.26 7.52 16.42 5.02 4.18 4.18 3.88 3.52 8.35 4.02 3.59 6.26 4.19 3.37 1.04 0.48 12.32 12.16 11.53 10.17 7.11 7.45 W /’3o 0.153 0.113 0.103 0.076 0.083 0.076 0.164 0.051 0.042 0.043 0.039 0.036 0.084 0.04 0.036 0.062 0.042 0.034 0.0104 0.0048 0.1218 0.1209 0.1143 0.1005 0.0708 0.0748

Table 5-2a. Cyclic Triaxial Laboratory Test Data: Monterey Sand – Silt Mixtures.

NEC 0.0469 0.0289 0.027 0.021 0.0273 0.0184 0.071 0.0149 0.01 0.0136 0.0102 0.0086 0.0318 0.0117 0.0079 0.0187 0.0112 0.0087 0.0011 0.0008 0.0301 0.0297 0.0283 0.0314 0.016 0.0149

PEC (kPa) 22.46 12.66 10.6 0.07 11.29 8.51 31.97 6.3 6 5.99 4.6 4.03 14.5 5.13 4.55 8.46 4.86 4.52 0.54 0.38 12.71 11.46 10.42 13.61 6.88 6.44

FC (%) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5

Failure Mode ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

File Name m5e67c22 m5e67c23 m5e67c25 m5e67c38 m5e68c20 m5e68c25 m5e68c30 m10e53c30 m10e53c35 m10e53c40 m10e57c25 m10e57c35 m10e60c20 m10e60c25 m10e60c30 m10e68c20 m10e68c22 m10e68c25 m15e48c30 m15e48c35 m15e48c38 m15e53c23 m15e53c28 m15e68c13 m15e68c15 m15e68c20 m20e36c25 m20e36c35 m20e36c40

Dr (%) 53.4 54 54.5 55.6 49.7 49.7 53.4 81.0 79.7 80.6 62.9 62.9 52.2 51.7 51.7 53.3 54.1 55.2 81.7 82.1 81.7 62.5 62.9 10.8 11.2 10.8 96.8 97.5 96.8

CSR 0.349 0.366 0.408 0.478 0.308 0.401 0.516 0.514 0.601 0.695 0.403 0.595 0.306 0.398 0.495 0.3 0.340 0.412 0.501 0.579 0.673 0.358 0.469 0.178 0.215 0.306 0.417 0.614 0.701

T (sec) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (kPa) 99.63 100.7 99.21 97.42 98.9 99.84 93.74 98.48 99.84 99.95 100.3 98.69 99.53 99.21 99.53 99.11 100.2 99.95 100.9 99.84 99.53 98.58 98.58 98.58 99.32 99.21 99 97.84 99.42

’3o Nliq 444.3 163.1 25.0 9.0 416.3 38.1 10.1 109.1 44.0 18.0 112.1 9.1 173.1 35.0 8.0 44.1 21.1 6.1 49.0 23.1 20.1 45.1 10.1 99.0 38.0 8.0 338.2 56.0 42.7

170

W (kPa) 18.94 9.38 5.03 4.89 20.97 5.81 6.17 22.2 17.74 14.35 11.08 8.37 7.61 5.71 5.17 3.69 3.40 2.81 12.38 11.51 15.34 6.13 4.49 2.69 2.61 2.47 27.59 21.07 34.97 W /’3o 0.1899 0.0933 0.0507 0.0502 0.212 0.0582 0.0659 0.2258 0.1778 0.1434 0.1108 0.0847 0.0766 0.0576 0.052 0.0373 0.0338 0.0281 0.1226 0.1151 0.1542 0.0622 0.0456 0.0273 0.0263 0.0249 0.2788 0.2149 0.3525 NEC 0.0677 0.025 0.0107 0.0104 0.0667 0.0131 0.0127 0.0628 0.0455 0.0368 0.029 0.0187 0.0201 0.0141 0.0103 0.0089 0.0073 0.0063 0.0307 0.0273 0.0427 0.0161 0.0106 0.0061 0.0059 0.0046 0.0705 0.0533 0.0782

PEC (kPa) 34.55 13.33 4.6 4.49 36.64 5.98 4.64 25.9 18.23 11.69 13.66 6.74 9.52 5.88 3.73 4.23 3.2 3.05 11.69 9.08 13.59 7.28 4 2.87 2.79 2 33.77 18.89 20.11

FC (%) 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 20 20 20

Failure Mode ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 DA = 5%

File Name m20e46c18 m20e46c20 m20e46c25 m20e46c28 m20e68c13 m20e68c17 m25e35c30 m25e35c40 m25e35c50 m25e42c20 m25e42c25 m25e42c28 m25e47c17 m25e47c20 m25e47c30 m25e68c09 m25e68c12 m25e68c15 m35e61c11 m35e61c13 m35e61c15 m35e68c10 m35e68c13 m35e68c15 m50e68c14 m50e68c16 m50e68c18 m50e68c20

Dr (%) 60.8 61.1 60.8 60.4 -2.1 -1.1 97.9 97.9 98.3 75.9 76.2 75.5 60.7 60 61.4 -2.1 -0.3 -1.7 55.8 69.2 62.5 64.4 67.6 72.8 73.9 76.9 77.3 79.6

CSR 0.261 0.317 0.407 0.452 0.167 0.247 0.516 0.709 0.903 0.305 0.407 0.47 0.251 0.311 0.450 0.114 0.149 0.226 0.081 0.110 0.119 0.075 0.110 0.119 0.112 0.125 0.160 0.148

T (sec) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 (kPa) 99.42 99.32 98.9 99.32 97.74 98.37 99.53 99.21 99.11 99.95 99.21 99.63 98.6 99.21 99.42 99.11 95.95 97.42 100.2 99.42 98.79 100.5 99.42 98.79 97 98.9 98.37 98.68

’3o Nliq 310.2 55.0 14.1 12.1 55.3 8.0 52.0 25.9 11.8 124.1 21.0 14.1 60.0 29.1 6.1 155.3 55.0 8.0 42.4 11.5 11.3 1198.0 20.0 11.3 466.0 9.4 20.0 20.0

171

W (kPa) 10.40 5.98 4.24 4.54 2.10 2.10 19.74 27.65 21.42 8.80 5.32 5.16 3.65 4.23 3.62 2.20 2.29 2.26 0.56 1.00 0.92 0.03 0.57 0.92 1.25 1.89 1.37 1.85 W /’3o 0.1048 0.0603 0.043 0.0458 0.0215 0.0214 0.1979 0.2788 0.2162 0.0881 0.0537 0.0518 0.0372 0.0427 0.0365 0.0222 0.0239 0.0232 0.0056 0.01 0.0093 0.00027 0.00571 0.0093 0.0129 0.01914 0.01396 0.01868 NEC 0.0289 0.0138 0.0094 0.0107 0.0049 0.004 0.0388 0.0654 0.0621 0.0266 0.0117 0.0115 0.0096 0.01 0.0073 0.0052 0.0043 0.0034 0.0011 0.001 0.0013 0.00017 0.00095 0.0013 0.0014 0.00161 0.00283 0.00197

PEC (kPa) 13.85 6.11 3.72 4.21 2.21 1.66 12.8 15.06 20.84 12.43 4.65 4.28 4.45 4.34 2.8 2.45 1.99 1.47 0.45 0.31 0.47 0.171 0.311 0.473 0.388 0.442 0.616 0.51

FC (%) 20 20 20 20 20 20 25 25 25 25 25 25 25 25 25 25 25 25 35 35 35 35 35 35 50 50 50 50

Failure Mode ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 DA = 5% DA = 5% ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 DA = 5% DA = 5% ru = 1 DA = 5% ru = 1 ru = 1 DA = 5% DA = 5% ru = 1 ru = 1 ru = 1

T CSR (sec) 0.136 1 0.159 1 (kPa) 98.58 100.5

’3o Nliq 13.7 6.4

W (kPa) 1.66 1.59 W /’3o 0.0169 0.0158

File Name y0e76c30 y0e76c33 y0e76c35 y0e90c19 y0e90c22 y0e90c25 y0e11c10 y0e11c12 y0e12c09 y0e12c10 y0e12c11 y0e12c13 y0e12c14 y4e86c17 y4e86c18 y4e86c20 y4e86c23 y4e76c27 y4e76c30

Dr (%) 68 70 68.3 27.3 24.8 25.6 -10.6 -10.4 -32.7 -44.5 -37.2 -32.3 -32 27.5 22.9 28.2 19.6 52.1 52.4

CSR 0.286 0.323 0.342 0.137 0.168 0.224 0.076 0.09 0.066 0.063 0.0668 0.087 0.110 0.136 0.148 0.163 0.211 0.249 0.283

T (sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

’3o (kPa) 99.1 99.63 98.58 99.42 101.3 98.05 98.58 99.42 100.6 99.74 98.47 99.84 98.26 99.53 98.16 98.16 100.3 99.84 100.8 Nliq 23.1 11.1 11.1 36.1 14 5 31.4 14.5 43.3 78.5 60.4 11.3 2.3 8.6 6.3 3.2 15 33.1 11.1

172

W (kPa) 2.46 2.59 2.54 0.49 0.62 1.42 0.17 0.32 0.3 0.37 0.37 0.53 0.49 0.38 0.67 0.74 0.88 2.56 1.74 W /’3o 0.0248 0.026 0.0257 0.0049 0.0061 0.0144 0.00172 0.00322 0.00302 0.00372 0.00371 0.00531 0.00497 0.00381 0.00683 0.00756 0.00881 0.0256 0.01724

Table 5-2b. Cyclic Triaxial Laboratory Test Data: Yatesville Sand – Silt Mixtures.

File Dr Name (%) m75e68c16 92 m75e68c20 90.5

NEC 0.0053 0.00556 0.00554 0.00102 0.00094 0.0015 0.000504 0.00057 0.000606 0.000799 0.000716 0.000866 0.0000998 0.000784 0.000939 0.00102 0.001576 0.005406 0.003722

NEC 0.0015 0.0018

PEC (kPa) 2.195 2.06 2.179 0.452 0.429 0.62 0.222 0.226 0.258 0.362 0.306 0.313 0.604 0.334 0.301 0.408 0.706 2.449 1.537

PEC (kPa) 0.511 0.686

FC (%) 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 4 4

FC (%) 75 75

Failure Mode ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

Failure Mode DA = 5% DA = 5%

File Name y4e76c35 y4e83c22 y4e83c25 y4e83c28 y4e97c12 y4e97c14 y4e97c15 y4e11c10 y4e11c12 y4e11c14 y7e76c26 y7e76c27 y7e76c30 y7e76c32 y7e78c25 y7e78c31 y7e78c34 y7e92c12 y7e92c15 y7e92c18 y7e10c11 y7e10c12 y7e10c14 y7e10c16 y12e70c35 y12e70c38 y12e70c41 y12e67c28 y12e67c31

Dr (%) 52.1 29.5 30.8 29.5 -10.9 -9 -9.6 -36.5 -35.5 -30.2 34 34.6 33.7 33.7 27.3 29.4 30.7 -9 -8.7 -5.5 -20.4 -31.7 -25.3 -25.3 45.2 42.4 43.9 54.4 55.1

CSR 0.335 0.188 0.229 0.243 0.094 0.113 0.127 0.077 0.081 0.111 0.241 0.251 0.283 0.307 0.225 0.297 0.323 0.0859 0.124 0.161 0.084 0.0884 0.113 0.132 0.339 0.37 0.386 0.255 0.284

T (sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (kPa) 99.21 100.8 99.31 100.6 96.89 99.53 96.79 97.74 99.84 98.89 97.21 98.47 100.6 98.05 100.5 98.05 100.9 99.84 99.74 96.47 100.9 102.5 97.63 98.05 99.1 100.4 102.4 99.63 98.89

’3o Nliq 7.1 54 19.1 11.1 28.3 12 7.2 30.3 20.3 6.3 20.1 25.1 9.1 6 58.1 15.1 7.1 65 16.8 4.2 29.4 29.6 8.3 4.3 13.1 15.1 12.1 11.7 11.1

173

W (kPa) 2.01 1.36 1 1.21 0.43 0.42 0.41 0.31 0.58 0.61 1.95 2.04 1.33 1.77 2.44 1.93 1.81 0.49 0.55 0.59 0.51 0.33 0.51 0.61 2.78 3.48 3.89 2.81 1.59 W /’3o 0.02025 0.01349 0.01003 0.01196 0.00447 0.00418 0.00419 0.00322 0.00576 0.00616 0.02014 0.02074 0.01318 0.01803 0.0243 0.01963 0.01789 0.00494 0.00548 0.00616 0.00502 0.00324 0.00519 0.00625 0.02801 0.03463 0.03805 0.0283 0.01607 NEC 0.004107 0.002902 0.002185 0.002771 0.000746 0.000691 0.000713 0.000632 0.000753 0.000846 0.00398 0.004408 0.002986 0.003546 0.005938 0.004692 0.004277 0.000986 0.000848 0.000953 0.000785 0.000745 0.000797 0.000934 0.006327 0.008331 0.009533 0.00423 0.003844

PEC (kPa) 1.384 1.468 0.99 1.181 0.296 0.299 0.291 0.263 0.261 0.327 1.77 1.911 1.247 1.212 2.896 1.987 1.611 0.46 0.349 0.406 0.324 0.328 0.321 0.352 2.454 3.131 4.071 1.578 1.637

FC (%) 4 4 4 4 4 4 4 4 4 4 7 7 7 7 7 7 7 7 7 7 7 7 7 7 12 12 12 12 12 ru = 1

DA = 5%

Failure Mode ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

File Name y12e67c33 y12e76c23 y12e76c26 y12e76c28 y12e84c18 y12e84c22 y12e84c25 y12e92c12 y12e92c13 y12e92c15 y12e92c17 y17e59c36 y17e59c39 y17e59c42 y17e70c25 y17e70c30 y17e70c32 y17e73c22 y17e73c25 y17e73c27 y17e76c22 y17e76c25 y17e76c28 y17e84c12 y17e84c15 y17e84c18 y26e63c30 y26e63c33 y26e63c36

Dr (%) 56.9 25.7 24.7 27.5 3.4 2.8 4.6 -11.1 -11.8 -9.6 -9.9 76.1 77.8 76.3 37.7 41.0 38.3 25.3 26.0 27.7 18.0 18.0 17.0 -10.7 -7.7 -6.7 40.3 42.8 43.2

CSR 0.312 0.201 0.231 0.267 0.153 0.188 0.224 0.09 0.097 0.116 0.144 0.349 0.376 0.406 0.102 0.156 0.245 0.201 0.212 0.248 0.198 0.222 0.263 0.091 0.128 0.159 0.283 0.301 0.33

T (sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (kPa) 99 100.3 99.53 98.89 101.1 100.6 101 98.68 100.9 99.31 98.89 98.89 100.6 100.3 99.0 98.37 102.8 98.68 100.3 100.2 98.26 100.1 99.31 97.21 97.32 97.74 99.95 99.74 99.42

’3o Nliq 7.1 47.1 12.6 7 29 10.9 4 28.3 31.4 9.3 3.4 15.7 16.7 15.1 118.8 16.7 2.5 46.6 25 7 38 14 6 62.4 20.9 5.2 11.1 14.1 14.1

174

W (kPa) 1.56 2.12 1.24 1.79 0.78 1.65 2.14 0.41 0.43 0.61 1.05 6.63 7.17 7.08 0.61 0.98 2.09 1.5 1.04 1.2 1.44 1.31 1.67 0.64 0.8 0.88 1.34 2.1 2.68 W /’3o 0.01579 0.0211 0.01245 0.01816 0.00772 0.0164 0.02113 0.00416 0.00424 0.00617 0.01065 0.067 0.0713 0.07068 0.00614 0.00991 0.02029 0.01518 0.01036 0.01202 0.01469 0.01309 0.01682 0.00662 0.00818 0.00899 0.0134 0.021 0.02696 NEC 0.004209 0.005565 0.003142 0.002898 0.002089 0.00152 0.001704 0.000825 0.00095 0.000834 0.00128 0.00939 0.01215 0.015383 0.000694 0.00219 0.002856 0.004733 0.002369 0.002149 0.003407 0.002409 0.002706 0.001185 0.001063 0.001212 0.002904 0.004474 0.005865

PEC (kPa) 1.542 2.688 1.425 0.998 1 0.642 0.706 0.339 0.403 0.332 0.66 1.968 3.358 3.645 0.233 0.75 1.071 1.972 0.938 0.836 1.587 1.027 0.985 0.511 0.443 0.458 1.147 1.703 2.132

FC (%) 12 12 12 12 12 12 12 12 12 12 12 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 26 26 26 ru = 1 ru = 1 ru = 1

DA = 5%

ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

DA = 5% DA = 5%

ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

DA = 5%

Failure Mode ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

File Name y26e67c25 y26e67c27 y26e67c29 y26e71c19 y26e71c22 y26e71c24 y26e76c15 y26e76c18 y26e76c20 y37e62c25 y37e62c28 y37e62c31 y37e67c14 y37e67c15 y37e67c17 y37e69c19 y37e69c20 y37e69c22 y37e76c10 y37e76c12 y37e76c14 y50e55c18 y50e55c20 y50e55c23 y50e67c14 y50e67c16 y50e67c18 y50e76c09

Dr (%) 26.9 28.1 28.5 24.9 31.0 27.7 -10.2 -9 -9 25.1 25.8 24.4 4.5 8.4 7.7 1 0 3.8 -24.7 -11.1 -16.4 81.1 80.0 91.2 72.2 69.8 71.6 63.4

CSR 0.19 0.249 0.273 0.161 0.192 0.219 0.126 0.151 0.169 0.224 0.234 0.285 0.112 0.119 0.141 0.116 0.178 0.194 0.0779 0.0918 0.118 0.151 0.168 0.205 0.111 0.132 0.154 0.068

T (sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (kPa) 100.2 98.89 100.9 100.5 99.0 97.95 99.21 100.9 99.74 102.6 99.63 100.6 99.84 99.42 99.95 98.79 98.47 99.63 95.32 98.47 96.68 100.4 100.1 100.2 99.31 99.1 98.79 99.84

’3o Nliq 69 13 5.1 21 14.1 9 30.1 9.4 7.1 12.1 10 4.1 35.7 17.9 9.1 29 7 3.4 46.3 19.3 7.7 18.1 9.7 3.8 12.5 7.5 4.2 48.2

175

W (kPa) 1.12 1.35 1.06 1.1 1.1 1.58 0.58 1.24 0.91 0.75 1.08 2.48 0.49 0.74 0.6 0.4 0.89 1.83 0.32 0.58 0.88 1.07 2.18 1.87 0.74 1.37 1.14 0.65 W /’3o 0.01119 0.0136 0.01049 0.01095 0.0111 0.01608 0.00585 0.01229 0.00911 0.00736 0.01082 0.02468 0.00492 0.00742 0.00598 0.00404 0.00906 0.0184 0.00338 0.00592 0.00912 0.01068 0.0218 0.0187 0.00747 0.01384 0.01152 0.0065 NEC 0.002688 0.002937 0.002224 0.001876 0.002609 0.002422 0.001262 0.001449 0.001447 0.001423 0.002196 0.004575 0.001104 0.001011 0.00092 0.001062 0.00126 0.001872 0.000658 0.000755 0.001158 0.000789 0.00193 0.00395 0.000583 0.001546 0.001417 0.00085

PEC (kPa) 1.005 1.24 0.883 0.846 1.149 0.93 0.556 0.526 0.585 0.592 0.876 1.004 0.493 0.434 0.385 0.39 0.463 0.436 0.278 0.299 0.35 0.265 0.59 2.386 0.227 0.409 0.573 0.34

FC (%) 26 26 26 26 26 26 26 26 26 37 37 37 37 37 37 37 37 37 37 37 37 50 50 50 50 50 50 50 DA = 5%

ru = 1 ru = 1 ru = 1

DA = 5% DA = 5%

ru = 1

DA = 5%

ru = 1 ru = 1

DA = 5%

Failure Mode ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

File Name y50e76c13 y50e76c15 y50e78c10 y50e78c12 y50e78c14 y75e76c13 y75e76c15 y75e76c17 y75e85c14 y75e85c16 y75e85c18 y75e90c10 y75e90c13 y75e90c15 se68c13 se68c40 se68c43 se76c25 se76c35 se76c38 se91c18 se91c20 se91c22 se95c10 se95c13 se95c15 se99c10 se99c13

Dr (%) 62.2 58.6 62 54.4 69.3 85.5 86.6 85.8 79.7 78 78.2 75.1 76.9 74.2 105.1 104.3 103.3 97.5 95.9 98.5 85.6 86.2 87.2 87.6 88.4 87.2 84.3 84.6

CSR 0.107 0.125 0.076 0.095 0.114 0.105 0.121 0.121 0.12 0.132 0.152 0.078 0.106 0.121 0.103 0.364 0.423 0.231 0.340 0.365 0.151 0.171 0.194 0.079 0.098 0.119 0.078 0.104

T (sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (kPa) 98.68 96.37 99.95 99.53 100.5 102.5 97.95 103.1 96.47 97.53 96.89 99.53 99.84 99.42 99.95 101.1 100.6 98.79 99.53 99.31 98.47 100.9 98.47 100.3 99.95 100.1 99.63 99.31

’3o Nliq 11.2 5.7 46 15.2 6.3 26.2 11.3 13.4 14.3 9.5 5.4 36.2 9.3 5.3 602.5 15.2 21.3 180.2 17.2 3.2 18.2 9.8 3.7 52.2 20.8 8.8 46.1 11.2

176

W (kPa) 0.73 1.13 1.1 0.74 0.9 0.53 1.01 1.18 1.11 1.28 1.57 0.85 1.04 1.01 3.71 3.07 5.02 8.32 2.8 2.33 0.84 1.84 1.8 0.98 1.07 1.38 0.88 1 W /’3o 0.0074 0.0117 0.01106 0.0074 0.009 0.00519 0.0103 0.0115 0.0115 0.01317 0.01619 0.0086 0.0104 0.0101 0.03712 0.03032 0.04991 0.08432 0.0281 0.02346 0.00853 0.0182 0.0182 0.0097 0.0107 0.0138 0.00882 0.0101 NEC 0.00078 0.00187 0.000952 0.00126 0.00156 0.001037 0.00167 0.00092 0.00146 0.001702 0.002326 0.00103 0.00178 0.00218 0.00887 0.009069 0.018926 0.025773 0.008895 0.006764 0.00203 0.00244 0.00289 0.00122 0.00148 0.00176 0.001121 0.00181

PEC (kPa) 0.21 0.555 0.394 0.42 0.499 0.499 0.573 0.363 0.593 0.618 0.865 0.375 0.603 1.16 4.43 10.218 19.308 14.232 5.608 4.076 1.018 0.986 1.325 0.516 0.574 0.757 0.473 0.8

FC (%) 50 50 50 50 50 75 75 75 75 75 75 75 75 75 100 100 100 100 100 100 100 100 100 100 100 100 100 100 = = = = =

5% 5% 5% 5% 5%

DA = 5%

ru = 1

DA DA DA DA DA

ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1 ru = 1

DA = 5% DA = 5% DA = 5%

ru = 1 ru = 1

DA = 5% DA = 5% DA = 5%

ru = 1

DA = 5% DA = 5%

ru = 1

DA = 5% DA = 5%

Failure Mode

File Name se99c15

Dr (%) 84.7

T CSR (sec) 0.114 1 (kPa) 100.7

’3o Nliq 7.8

177

W (kPa) 2.3 W /’3o 0.0229 NEC 0.0025

PEC (kPa) 0.973

FC (%) 100 DA = 5%

Failure Mode

Appendix 5a: Earthquake time histories used in the parameter study to develop the correlation relating Richter magnitude (M), epicentral distance (ED) from the site to the source, and number of equivalent cycles (Neqv). All the time histories were recorded on either Site Class a or b profiles, where Site Class a has an average shear wave velocity (vs) for the upper 30m of the profile greater than 1500m/s, and Site Class b: 760m/s  vs  1500m/s. San Francisco Earthquake (also called Daly City Earthquake), March 22,1957, M5.3 1) Golden Gate Park (ct 077): Lat.N. 37.770, Long W. 122.478, ED =11km Site Class a Azimuth 1: 10 Azimuth 2: 100 Sitka Earthquake (Alaska), July 30, 1972, M7.5 2) Sitka magnetic observatory: Lat. N. 57.060, Long W. 135.320, ED = 86km Site Class a Azimuth 1: 180 Azimuth 2: 90 Coyote Lake Earthquake, August 6, 1979, M5.7 3) Gilroy Array 1 (cdmg 47379): Lat. N. 36.973, Long W. 121.572, ED = 16.7km Site Class b Azimuth 1: 320 Azimuth 2: 230 Morgan Hill Earthquake, April 24, 1984, M6.1 4) Gilroy Array 1 (cdmg 47379): Lat. N. 36.973, Long W. 121.572, ED = 38.6km Site Class b Azimuth 1: 67 Azimuth 2: 337 Loma Prieta Earthquake, October 18, 1989, M7.0 5) Cherry Flat Reservoir (usgs 1696): Lat. N. 37.396, Long W. 121.756, ED = 41.1km Site Class a Azimuth 1: 360 Azimuth 2: 270 6) Hollister Sago Vault (usgs 1032): Lat. N. 36.765, Long W. 121.446, ED = 49.1km Site Class b Azimuth 1: 360 Azimuth 2: 270 7) Gilroy Array 1 (cdmg 47379): Lat. N. 36.973, Long W. 121.572, ED = 28.4km Site Class b Azimuth 1: 90 Azimuth 2: 0 8) Monterey City Hall (cdmg 47377): Lat. N. 36.597, Long W. 121.897, ED = 49.0km Site Class b Azimuth 1: 90 Azimuth 2: 0 9) San Fran Sierra Point (cdmg 58539): Lat. N. 37.674, Long W. 122.388, ED = 83.7km Site Class b Azimuth 1: 205 Azimuth 2: 115

178

Petrolia Earthquake (also called Cape Mendocino Earthquake), April 25, 1992, M7.0 10) Bunker Hill FAA (usgs 1584): Lat. N. 40.498, Long W. 124.294, ED = 15.0km Site Class b Azimuth 1: 360 Azimuth 2: 270 Landers Earthquake, June 28, 1992, M7.3 11) Twenty Nine Palms (cdmg 22161): Lat. N. 34.021, Long W. 116.009, ED = 44.2km Site Class b Azimuth 1: 90 Azimuth 2: 0 12) Silent Valley (cdmg 12206): Lat. N. 33.851, Long W. 116.852, ED = 54.7km Site Class b Azimuth 1: 90 Azimuth 2: 0 13) Amboy (cdmg 21081): Lat. N. 34.560, Long W. 115.743, ED = 75.2km Site Class b Azimuth 1: 90 Azimuth 2: 0 14) Whitewater Canyon (usgs 5072): Lat. N. 33.989, Long W. 116.655, ED = 31.0km Site Class b Azimuth 1: 270 Azimuth 2: 180 Northridge Earthquake, January 17, 1994, M6.7 15) Lake Hughes 9 (cdmg 24272): Lat. N. 34.608, Long W. 118.558, ED = 44.7km Site Class b Azimuth 1: 90 Azimuth 2: 360 16) Littlerock – Brainard Canyon (cdmg 23595): Lat. N. 34.486, Long W. 117.980, ED = 60.1km Site Class b Azimuth 1: 90 Azimuth 2: 180 17) Mt. Baldy – Elem. Sch (cdmg 23572): Lat. N. 34.233, Long W. 117.661, ED = 81.0km Site Class a Azimuth 1: 90 Azimuth 2: 180 18) Mt. Wilson (cdmg 24399): Lat. N. 34.224, Long W. 118.057, ED = 44.6km Site Class a Azimuth 1: 90 Azimuth 2: 360 19) Rancho Cucamnga – Deer Canyon (cdmg 23598): Lat. N. 34.169, Long W. 117.579, ED = 88.7km Site Class b Azimuth 1: 90 Azimuth 2: 180 20) Rancho Palos Verdes (cdmg 14404): Lat. N. 33.746, Long W. 118.396, ED = 53.2km Site Class b Azimuth 1: 90 Azimuth 2: 0 21) Wrightwood – Jackson Flat (cdmg 23590): Lat. N. 34.381, Long W. 117.737, ED = 76.4km Site Class b Azimuth 1: 90 Azimuth 2: 180

179

22) 8510 Wonderland Ave, LA (usc 17): Lat. N. 34.114, Long W. 118.380, ED = 18.2km Site Class b Azimuth 1: 185 Azimuth 2: 95 23) 1250 Howard Rd., Burbank (usc 59): Lat. N. 34.204, Long W. 118.302, ED = 22.0km Site Class b Azimuth 1: 330 Azimuth 2: 60 24) Griffith Observatory (usgs 141): Lat. N. 34.118, Long W. 118.299, ED = 24.3km Site Class b Azimuth 1: 360 Azimuth 2: 270

The station designations are as follows: cdmg: California Division of Mines and Geology Strong Motion Instrumentation Program ct: California Institute of Technology Civil Engineering Department usc: University of Southern California Civil Engineering Department usgs: United States Geological Survey National Strong Motion Program

180

Appendix 5b: Profiles used in the parametric study to develop the correlation relating Richter magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv). The profiles used in the parametric study were selected to be representative of actual profiles. The profiles used had constant, decreasing, and increasing N1,60 with depth. The water table was placed at the soil surface and at a depth of approximately 11ft. Both soft and medium stiff profiles were examined. The following expressions were used to relate N1,60 to Gmax.

 '  20  N1,60 0.5  20

(Hatanaka and Uchida 1996)

K o  1  sin( ' )

 1  2K o     ' vo  3 

 'mo  

G max  440 N 1, 60 

1/ 3

 '  Pa1  m o   Pa 2 

0.5

(Seed et al. 1986)

where Pa1 and Pa2 are atmospheric pressure having the same units as Gmax and ’mo, respectively. The shear modulus and damping degradation curves proposed by Ishibashi and Zhang (1993) were used in the site response analyses. Figures 5b-1 through 5b-4 show the profiles used in the parametric study.

181

0

a)

b)

10

10

20

20

30

30

40

40

Depth (ft)

Depth (ft)

0

50

50

60

60

70

70

80

80

90 100

Profile 1 Profile 5 Profile 9

t = 120pcf

Profile 1 Profile 5 Profile 9

90 100 0

0 200 400 600 800 1000 Shear Wave Velocity (fps)

6

12 18 N1,60

24

rock = 140pcf, Gmax =59521700psf (Site Class b) Gmax =108695700psf (Site Class a) Figure 5b-1. Profiles 1, 5, and 9 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

182

30

0

a)

b)

10

10

20

20

30

30

40

40

Depth (ft)

Depth (ft)

0

50

50

60

60

70

70

80

80

90 100

Profile 2 Profile 6 Profile 10

t = 125pcf

Profile 2 Profile 6 Profile 10

90 100 0

0 200 400 600 800 1000 Shear Wave Velocity (fps)

6

12 18 N1,60

24

rock = 140pcf, Gmax =59521700psf (Site Class b) Gmax =108695700psf (Site Class a) Figure 5b-2. Profiles 2, 6, and 10 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

183

30

0

0

b)

10

10

20

20

30

30

40

40

Depth (ft)

Depth (ft)

a)

50

50

60

60

70

70

80

80

90 100

Profile 3 Profile 7 Profile 11

t = 120pcf

Profile 3 Profile 7 Profile 11

90 100 0

0 200 400 600 800 1000 Shear Wave Velocity (fps)

6

12 18 N1,60

24

rock = 140pcf, Gmax =59521700psf (Site Class b) Gmax =108695700psf (Site Class a) Figure 5b-3. Profiles 3, 7, and 11 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

184

30

0

0

b)

10

10

20

20

30

30

40

40

Depth (ft)

Depth (ft)

a)

50

50

60

60

70

70

80

80

90 100

Profile 4 Profile 8 Profile 12

t = 125pcf

Profile 4 Profile 8 Profile 12

90 100 0

0 200 400 600 800 1000 Shear Wave Velocity (fps)

6

12 18 N1,60

24

rock = 140pcf, Gmax =59521700psf (Site Class b) Gmax =108695700psf (Site Class a) Figure 5b-4. Profiles 4, 8, and 12 used in the parameter study to develop the correlation relating Richter Magnitude (M), epicentral distance from the site to the source (ED), and number of equivalent cycles (Neqv): a) Profiles in terms of shear wave velocity, b) Profiles in terms of N1,60.

185

30

Appendix 5c: Energy-based magnitude scaling factors (MSF) As noted in Chapter 2 (Section 2.2.1), magnitude scaling factors (MSF) are used in the stress-based procedure to account for the duration of earthquake motions and are defined as: MSF 

CSR CSR M 7.5

(5c-1)

where CSRM7.5 is the cyclic stress ratio associated with a M7.5 earthquake and CSR is the cyclic stress ratio associated with an earthquake of magnitude M. The earliest MSF were developed from laboratory liquefaction curves similar to the one shown in Figure 5c-1. In this figure, the amplitude of the applied loading, as quantified by CSR, is plotted as a function of the number of cycles required cause liquefaction (Nliq). Cyclic triaxial and cyclic simple shear test apparatus were primarily used for developing these curves.

Cyclic Stress Ratio (d/2’o)

0.5

Dr = 60% ’o =100 kPa Monterey No. 0 Sand Moist Tamped Samples

0.4 0.3 0.2 0.1 1

3 10 30 Number of Cycles to Initial Liquefaction, Nliq

100

Figure 5c-1. Capacity curve developed from cyclic triaxial laboratory tests. Each point on the plot represents a separate test. Each test is conducted on similar samples subjected to varying amplitude CSR. For isotropically consolidated cyclic triaxial tests, the CSR = d/2’o, where d is the applied deviator stress, and ’o is the initial effective confining stress.

As illustrated in Figure 5c-2, the MSF are the factors required to flatten the laboratory liquefaction curve. Fifteen cycles was used as the base value (i.e., the Nliq value at which

186

MSF = 1.0) because early correlations equated 15 equivalent cycles to M7.5 (e.g., Figure 5-5). This was the earthquake magnitude for which the largest amount of liquefaction case histories existed when the MSF were first proposed. Using laboratory liquefaction curves in conjunction with correlations relating the number of equivalent cycles (Neqv) to earthquake magnitude, the Demand imposed on the soil can be normalized to a M7.5 earthquake. As the liquefaction case history database increased in size, MSF were developed directly from field data (e.g., Ambraseys 1988). However, laboratory liquefaction curves can still play an important role in developing MSF by complementing the field case histories.

0.4

Representative Laboratory Curve

0.3

CSR MSF

0.2 0.1 1

M7.5

Cyclic Stress Ratio (d/2’o)

0.5

3 10 15 30 Number of Cycles to Initial Liquefaction, Nliq

100

Figure 5c-2. Illustration of how MSF were developed from laboratory liquefaction curves. As shown in Annaki and Lee (1977) and Idriss (1997), laboratory liquefaction curves often plot as straight lines on log-log scales having a slope of -m. This is illustrated in Figure 5c-3 and allows the MSF to be expressed in terms of Neqv and –m.

187

Cyclic Stress Ratio (d/2’o)

1.0 Representative Laboratory Curve (log-log scale) 0.3

CSR MSF

0.1 1

M7.5

-m

3 10 15 30 Number of Cycles to Initial Liquefaction, Nliq

100

Figure 5c-3. Illustration of liquefaction curve plotted on log-log scales.

The following is a derivation of an expression for the MSF in terms of the number of equivalent cycles (Neqv) and the slope of the liquefaction curve (-m):

 m

m

log( CSR1 )  log( CSR2 ) log( N liq 1 ) log( N liq 2 )

log( CSR 2 )  log( CSR1 ) log( N liq 1 )  log( N liq 2 )

 CSR 2   log  CSR1     N liq 1   log  N  liq 2    N liq 1      log  CSR 2  m  log    N   CSR1   liq 2 

188

CSR2  N liq 1   CSR1  N liq 2 

m

(5c-2)

By setting CSR1 = CSRM7.5, Nliq 1 = Neqv M7.5, CSR2 = CSR, and Nliq 2 = Neqv, Equation (5c-2) reduces to:

 CSR   MSF   CSR M 7 .5  

m

 N eqv M 7.5   MSF    N  eqv  

m

where:

MSF

(5c-3) = Magnitude scaling factor.

CSR M7.5 = Cyclic stress ratio of an earthquake of magnitude M7.5. CSR

= Cyclic stress ratio of an earthquake of magnitude M.

Neqv M7.5 = Number of equivalent cycles for magnitude M7.5 earthquake motions. Neqv

= Number of equivalent cycles for magnitude M earthquake motions.

m

= Slope of the laboratory curve plotted on log-log scales.

As opposed to early correlations which expressed Neqv M7.5 as a function of magnitude only, the correlation developed as part of this research shows Neqv M7.5 is dependent on the site-to-source distance, as well as magnitude. This is illustrated in Figure 5c-4, which shows the correlation presented previously in Figure 5-10.

189

Epicentral Distance (km)

100 80 20

60

25

30

50 45 55 35 40

15 Neqv = 10

40 20 0

5

6 7 Richter Magnitude (M)

8

Figure 5c-4. Contour plot of Neqv as a function of epicentral distance and Richter magnitude. Because most of the liquefaction case histories are for far field, Equation (5c-4) is proposed as a new expression for MSF.  N eqv M 7.5 far field MSF    N eqv 

where:

   

m

(5c-4)

Neqv M7.5 far field = Number of equivalent cycles for magnitude M7.5 earthquake motions in the far field. Neqv

= Number of equivalent cycles for magnitude M earthquake motions in the near or far field.

m

= Slope of the laboratory curve plotted on log-log scales.

In applying Equation (5c-4), the near field-far field boundary is defined by the elbows of the contours shown in Figure 5c-4. Attention is now focused on the slope of the laboratory liquefaction curve, m. A literature review resulted in the following range of m values:

190

m = 0.23

Idriss (1997), data from DeAlba et al. (1976)

m = 0.27

Seed et al. (1975) (approximately)

m = 0.34

Idriss (1997), data from Yoshimi et al. (1984)

m = 0.5

Arango (1996) (from theory, not lab data)

As may be observed, there is a large variation is published values of m. However, as opposed to using any of these values directly, for this study, m was selected such that the resulting MSF curve, using Equation (5c-4) and the Neqv correlation shown in Figure 5c-4, matched closest the MSF curve proposed by Ambraseys (1988) in the far field. This avoids any reliance on laboratory data because Ambraseys’ MSF were developed directly from field case histories, predominately far field case histories. Additionally, the Ambraseys’ MSF are close to the average of the NCEER (1997) recommended range for MSF, as shown in Figure 2-2 in Chapter 2. From the calibration of Equation (5c-4) to Ambraseys’ MSF in the far field, m = 0.48, which is with in the limits of the values given in the literature. A comparison between the proposed MSF and Ambraseys’ MSF is shown in Figure 5c-5. As may be seen in this figure, there are two significant differences between MSF curves. The first is related to the proposed Neqv correlation for near and far field conditions. As indicated by the vertical and horizontal portions of the contours shown in Figure 5c-4, respectively, Neqv are only dependent on magnitude in the far field and only dependent on epicentral distance in the near field. The proposed MSF are similarly dependent. The near field dependency results in the series of horizontal lines corresponding to different epicentral distances shown Figure 5c-5. The intersections of the horizontal lines with the curved line defines the near field-far field boundary for the corresponding magnitudes. For example, the near field-far field boundary for M6.75 earthquake is approximately 30km. For far field conditions, the proposed MSF and Ambraseys’ MSF are approximately equal because m in Equation (5c-4) was selected accordingly. The proposed MSF could be matched to any of the MSF shown in Figure 2-2 for far field conditions by appropriately selecting m. The reasonableness of the resulting m can be

191

judged by comparing it to the range of values taken from published literature presented above (i.e., 0.23  m  0.5). The second significant difference between Ambraseys’ and the proposed MSF curves is for magnitudes less than M6.0, the proposed MSF reach a low magnitude plateau. This results from the interaction of setting the stress amplitude of the equivalent cycle proportional to 0.65amax and the duration of the earthquake motions. It is interesting to note that Idriss (1997) proposed MSF also having a low magnitude plateau. The basis for his plateau is related to the minimum fraction of a loading cycle required to induce dynamic liquefaction in a laboratory sample as opposed to failing the sample monotonically. 4.0 Ambraseys’ MSF curve Proposed MSF curve

Magnitude Scaling Factor

3.5 3.0 2.5

Epicentral Distance = 10km

2.0

20km near field

1.5

0.5 5.0

M6.75

1.0 5.5

6.0

far field

6.5 7.0 Magnitude

7.5

30km 40km 50km 60km

8.0

Figure 5c-5. A comparison of Ambraseys’ MSF curve and the set of curves developed in this study. The implications of the differences between the currently used MSF and the proposed MSF are that the former may under predict the Demand imposed on the soil at low magnitudes (i.e., M 0) ranges approximately from 0.09 to 0.15kg0.33/m. However, for optimal densification efficiency, Kok (1981) specifies usd/’  0.8. Using usd/’ = 0.8 as the criterion for densification, the corresponding range of HN is 0.16 to 0.39kg0.33/m, with an average of 0.27kg0.33/m. These values were computed using Equation (7-13) and are shown in Figure 7-41. For comparison purposes, Ivanov (1967) specifies a range of HN from 0.2 to 0.4 kg0.33/m for densification, also shown in Figure 741. The lower value of Ivanov’s range corresponds to fine-grained sand having 0  Dr  0.2, and the upper value is for medium-grained sand having 0.3  Dr  0.4. By knowing the HN required for densification, guidelines can be set for acceptable R-W combinations, such as shown as the dark shaded zone in Figure 7-42. Little to no densification is expected for R-W combinations falling to the right of the acceptable zone, and excessive vibrations and increased likelihood of cratering the ground surface occurs for R-W combinations falling to the left of the acceptable zone.

292

W (equivalent kg TNT)

R-W combinations which may cause excessive vibrations, cratering, etc. 100

acceptable R-W combinations for densification 0.39 0.27

80 60 40

3

W  0.16 R Little to No Densification  u sd   0 .8    ' 

20 0

0

10

20 30 R (m)

40

50

Figure 7-42. Optimal R-W combination required for densification is shown as the dark shaded zone.

Again from field case histories, Kok (1981) gives the correlation shown in Figure 7-43 relating vertical strain (H/H) and HN. 3

2

H (%) H

1.7

0 0.0

0.27

1

0.2

0.4

0.6 3

Hopkinson’s Number,

0.8

1.0

W  kg  , R  m  3

Figure 7-43. Correlation relating HN and surface settlement expressed in terms of vertical strain. (Adapted from Kok 1981).

293

An average fit of the data shown in Figure 7-43 is given by Equation (7-14).

0

H (%) = H

where:

3 W 2.73  0.9  ln   R

for

u sd < 0.8 '

  for  

u sd  '  0.8

(average)

H

= Surface settlement as a result of blasting.

H

= Initial thickness of loose deposit (same units as H).

(7-14)

usd = Semi-dynamic pore pressure.

’

= Original effective stress (same units as usd).

W

= Weight of charge in equivalent kg of TNT (kg).

R

= Distance from center of charge (m).

Using Equation (7-14) and Figure 7-42, the required R-W combination for a targeted densification may be selected. In practice, spacing of the charges typically ranges from 1 to 2 times R (Narin van Court and Mitchell 1994b). Before moving on to the next approach for determining the R-W combination, attention is focused on the pore pressure ratio usd/’. Studer and Kok (1980) make the following statements about semi-dynamic pore pressure (usd): “When a transient loading is severe the grain structure of the sand may, under certain conditions, collapse and consolidation takes place. Porewater is squeezed out and this process is mainly governed by the permeability of the sand. The porewater pressure dissipates and is measured as a slowly proceeding phenomenon, taking place in a time range from several minutes to several days.” If it is assumed that ’ (referred to as “the original effective stress” in Kok and Trense 1979) is the initial effective overburden (’vo), then by all accounts usd/’ is the same as the residual excess pore pressure ratio (ru) used in earthquake liquefaction studies. However, if usd/’ = ru, it is uncertain why usd/’ reaches values well above 1.0, the upper limit of ru. In this regard, Kok (1981) states: “For a specific site it is valid that 294

usd/’ can reach a value of approximately 1.7. This is due to the fact that the pore pressure measuring device is not able to discriminate actual excess porewater pressure and a ‘heavy’ liquid.” The author is uncertain what this statement means and hypothesizes that the data showing usd/’  1.0 (i.e., five points in Figure 7-41) resulted from pore pressure transducers becoming uncalibrated from the high intensity shock wave or result from some sort of residual increase in the total pressure from the blast generated gas bubbles. To gain a better understanding of the relationship between usd/’ and ru, the following simple comparison is made. Using Equation (7-13a), HN = 0.27 was computed for usd/’ = 0.8. Using Figure 7-43 or Equation (7-14), H/H = 1.7% for HN = 0.27. From the correlation presented in Ishihara and Yoshimine (1992) relating observed settlements during earthquakes to the maximum induced shear strain, H/H = v = 1.7% falls in the range denoted as initial liquefaction for Dr = 40%, as shown in Figure 7-44. Although this comparison does not conclusively prove that usd/’ = ru, it supports the

Volumetric strain due to consolidation following liquefaction, v, (%)

hypothesis that the two parameters are essentially the same.

5

Clean Sands

Dr = 40%

Initial Relative Density of Soil

4 50%

Initial Liquefaction

3 2

60% 70% 80% 90%

1.7

1 0 0

2 4 6 8 10 12 14 16 Maximum Amplitude of Shear Strain, max, (%)

Figure 7-44. Comparison of the vertical strain corresponding to usd/’ = 0.8 for blast data to the vertical strain from earthquake data. (Adapted from Ishihara and Yoshimine 1992).

295

7.4.2.2 Gohl et al. Approach Gohl et al. (2000) outline a procedure relating the fraction of the maximum achievable vertical strain to charge weight. Starting with Equation 7-15a for vertical strain in a saturated soil and using the expression for relative density (Equation 7-15b), Gohl et al. (2000) derived an expression for the maximum achievable vertical strain, given as Equation 7-15c. In deriving this expression, Gohl et al. assumed that the maximum achievable relative density from blasting is 80%.

v 

e 1 e

Dr 

(7-15a)

emax  e emax  emin

 v , max 

(7-15b)

0.8  Dr ,i (1  emax )  Dr , i (emax  emin )

(7-15c)

v, max = Maximum volumetric strain.

where:

Dr,i

= Initial relative density.

emax

= Maximum void ratio.

emin

= Minimum void ratio.

e

= Void ratio.

e

= Change in void ratio.

Dr

= Relative density.

For typical sand properties (e.g., emax = 1 and emin = 0.5), Equation (7-15c) can be approximated as:

 v , max 

0.8  Dr 4  Dr

(7-15d)

The following empirical relation expresses the fraction of the maximum achievable volumetric strain, given by Equation (7-15d), to blast parameters (Gohl et al. 2000).

296

W  Eff  k     where:

0.5

1

(7-16)

0.5

h R

Eff

= The fraction of maximum achievable vertical strain.

k

= Site-specific attenuation factor.

W

= Weight of charge.



= Mass density of the explosive.

R

= The radius of a circle having an area equal to the tributary area of the charge.

h

= Depth of burial.

Unfortunately, Gohl et al. (2000) do not specify the units of all the variables in Equation (7-16), a requirement for using empirical expressions. Several attempts were made to ascertain the units from the data presented in the paper but without success. Regardless, once Eff is determined from Equation (7-16), the volumetric strain resulting from blasting can be approximated as:

 v  Eff   v, max

(7-17)

The vertical strains observed during several field densification programs are plotted as functions of the initial relative densities of the soil in Figure 7-45. Also shown in this figure are the maximum predicted strains determined from Equation (7-15d) and from the plateau of the curves shown in Figure 7-44. From this figure, it can be seen that settlements observed in the field range between those predicted using Ishihara and Yoshimine (1992) correlation and those computed using Equation (7-15d).

297

10 Molikpaq 1 Molikpaq 2 Coldwater Trans - X Quebec Sato Kogyo Kitimat Kalowna Elliot Lake

Estimated maximum achievable compaction

9

Vertical Strain (%)

8 7 6 5 4 3 Maximum settlement following liquefaction (data after Ishihara & Yoshimine 1992)

2 1 0

0

10

20

30

40 50 60 70 Initial Relative Density (%)

80

90

100

Figure 7-45. Comparison of observed strains to maximum values predicted by Equation (7-15d) and from Figure 7-44. (Adapted from Gohl et al. 2000).

7.4.2.3 Narin van Court and Mitchell Approach The final procedure reviewed is that proposed by Narin van Court and Mitchell (1998), which expresses post-blast penetration resistance as a function of the initial penetration resistance and blast parameters. The empirical expression developed by Narin van Court and Mitchell (1998) is given as Equation (7-18).

q1, f

 W   0.450    2i   Ri 

where:

0.321

 q10,.o5

(7-18)

q1,f

= Final normalized tip resistance in the given depth interval (MPa).

Wi

= Weight of the individual charges surrounding the given interval (grams).

298

Ri

= Vector distance between the closest point of the individual charges and the middle of the given depth interval (m).

q1,o

= Initial normalized tip resistance in the given depth interval (MPa).

In this expression, the normalized tip resistance is the average of a given depth interval, normalized to 1ton/ft2 effective overburden pressure. It is assumed that q1,f corresponds to the normalized penetration resistance determined at a sufficient enough time after blasting that increases in the soil strength and stiffness have stabilized. Comparisons of measured and predicted final normalized tip resistances for several field case histories are shown in Figure 7-46. The large amount of scatter in the data shown in this figure is testament to the complexity of explosive compaction and the influence of

Measured Final Normalized Tip Resistance (MPa)

parameters such as site conditions.

40 +2

35 30

+1

mean

25 20

-1

15

-2

10 5 0

0

5 10 15 20 25 Predicted Final Normalized Tip Resistance (MPa)

Gdansk Bordeaux Molikpaq Zeebrugge Site A St. Petersburg – No Delay St. Petersburg – Delay Chicopee Norway Lausitz South Platte

30

Figure 7-46. Comparison of measured and predicted final normalized tip resistances. (Adapted from Narin van Court and Mitchell 1998).

299

7.5 Comparison of the Mechanical Energy Input by the Densification Techniques For comparison purposes, the following simple calculations are presented for estimating the mechanical energy required to densify a unit volume of soil using vibro-compaction, deep dynamic compaction, and explosive compaction. The term “Mechanical energy” refers to the energy that is available to do mechanical work, as opposed to energy expended in other forms (e.g., heat). The distinctions between the energies can be understood by considering deep dynamic compaction. The total energy expended during deep dynamic compaction could be quantified in terms of the fuel consumed by the crane that lifts the tamper. However, to avoid consideration of such things as the efficiency of the crane’s combustion engine, the potential energy of the tamper at its drop height is used to approximate the (mechanical) energy per drop imparted to the soil. In the following analyses, the mechanical energies (per unit volume of soil) required to densify loose clean sand (Zone 1 soil, Figure 7-29) by vibrocompaction, deep dynamic compaction, and explosive compaction are computed and compared. 7.5.1 Vibro-Compaction As described in Brown (1977) and D’Appolonia (1953), for electrically driven motors the current draw of the vibrator is used as an indicator of the compaction process: the current draw increases as the soil densifies. When the current draw “peaks,” the vibroflot is raised to the next location, at which point, the current draw drops and compaction begins again. This process is illustrated in the current log shown in Figure 7-47 (Degen and Hussin 2001). As may be observed from this figure, the vibroflot rapidly penetrates the soil profile to the desired treatment depth of 8m, with one up-down flushing of the machine after reaching 4m. The penetration time was just over one minute. After reaching 8m, the compaction process begins and is designated in this figure as t = 0min. The probe is raised in 0.5m intervals and held at each position for about 45sec. The average rate of work (i.e., power) performed on a soil by a vibroflot can be estimated as: P  I  E  pf  eff 

3 1000

(e.g., Puchstein et al. 1954)

300

(7-19)

where:

P

= Average rate of work performed by vibroflot (i.e., power) (kW, kJ/sec).

I

= Average line current (Amps).

E

= Phase-to-phase voltage requirement of vibrator (volts).

pf

= Average power factor ( 0.8).

eff

= Efficiency of electric motor (i.e., portion of the electrical power consumed by the motor that is available to do mechanical work,  0.9).

Current (Amps) 100

130

160

0

5

10

Extraction of probe

10

15

70

Depth (m)

5

Flushing of machine

8m

2.5m

5.5m

Insertion of probe

140Amps

115Amps

0

Time (minutes)

1min

Figure 7-47. Current log recorded during vibrocompaction. (Adapted from Degen and Hussin 2001).

301

Based on the average current draw and the amplitude of the peaks, the profile may be considered as consisting of two layers: 2.5 – 5.5m and 5.5 – 8m. The average current draws for the top and bottom layers are estimated to be about 140Amps and 115Amps, respectively. For the Vibro V23 vibrator (i.e., 440volts) and using Equation (7-19), the rates of work (P) performed by vibroflot on the top and bottom layers are estimated to be about 77 and 63kW, respectively. Knowing the compaction rate, the rate of work performed, and tributary area per compaction point, the mechanical energy required to treat a unit volume of soil can be determined. From Figure 7-47, the compaction rate is estimated to be about 0.37m/min (i.e., (8m – 2.5m)/15min; the probe was withdrawn from the ground at 2.5m). This is in reasonable agreement with the typical rate of 0.3m/min given in Mitchell (1981). From Figure 7-10 the tributary area per compaction point is estimated to be about 80ft2 (7.5m2). Finally, the range in the mechanical energy expended to treat a unit volume of soil in the profile corresponding to the current log shown in Figure 7-47 is:

63 to 77kW   

min 60 sec  1   1362 to 1665kJ / m 3  2  0.37 m min  7.5m

7.5.2 Deep Dynamic Compaction From Table 7-4, the mechanical energy required to densify Zone 1 soils ranges from 200 to 250kJ/m3. 7.5.3 Explosive Compaction The quantity of explosive required to treat a unit volume of soil by deep explosive compaction is given in Van Impe and Madhav (1995) as ranging from 15 to 35g/m3. Similarly, from the case histories listed in Ivanov (1967), a range of 8 to 28g/m3 can be reasonably assumed. From calorimeter measurements, the energy density of TNT is approximately 4560J/g. However, upon detonation, only about 67% of this energy is transformed into mechanical energy (Kennedy 1996). From the values stated above, the mechanical energy required to treat a unit volume of soil by explosive compaction is estimated to range from 22 to 100kJ/m3.

302

7.5.4 Discussion In summary, the following ranges of mechanical energy per unit volume of treated soil are: Vibro-Compaction:

1362 to 1665kJ/m3

Deep Dynamic Compaction:

200 to 250kJ/m3

Explosive Compaction:

22 to 100kJ/m3

From comparison of these ranges, explosive compaction appears to be the most efficient and vibrocompaction the least efficient (i.e., explosive compaction requires less mechanical energy to treat a unit volume of soil than the other techniques, etc.). The probable reason for the resulting efficiency rankings is the mode in which the energy is transferred to the soil by each of the remediation techniques. In vibrocompaction, the energy is imparted over a relatively long time span, during which the properties of the soil are continually changing. When liquefaction is induced in the soil immediately surrounding the probe, little energy is transferred from the probe to the outer, nonliquefied soil, during which time the majority of the imparted energy is expended inducing vibrations in the already liquefied soil. Furthermore, as may be recalled from Section 7.1, vibrocompaction improves the ground by both densifying the soil and increasing the lateral confining pressure. The latter improvement largely results from the lateral compaction of backfill. Accordingly, the energy range listed above reflects both the energy required to induce liquefaction in the virgin profile and the energy expended to laterally compact the backfill material. For deep dynamic compaction, the energy is rapidly imparted to the soil. However, a large portion of the energy is likely carried by surface waves (Figure 7-35), which have little effect in breaking down the soil structure at depth in the profile. This is in contrast to deep explosive compaction, wherein the initial mechanical energy imparted to the soil is carried by body waves (primarily compression-extension waves), which are more effective in breaking down the soil structure than surface waves. Analogous to deep dynamic compaction is explosive compaction where the charge is placed at the surface of

303

the soil profile, as opposed to buried deep within the profile. From the case histories listed in Ivanov (1967), the quantity of explosives required to treat a unit volume of soil by surface blasting is approximately five to ten times greater than required for deep blasting, thus densification by surface blasting has an efficiency comparable to deep dynamic compaction. As may be recalled from Chapter 5, the ability of the soil to resist liquefaction (i.e., Capacity) was quantified in terms of dissipated energy per unit volume, which is computed by integrating the stress-strain hysteresis loops up to initial liquefaction. From the Capacity curve derived from earthquake case histories (i.e., Equation (5-11) or Figure 5-11), the dissipated energy per unit volume required to induce liquefaction can be determined. For soils confined at an effective pressure of 100kPa and having N1,60 from 5 to 15blws/ft, the dissipated energy required to induce liquefaction ranges from 0.03 to 0.192kJ/m3. This range is several orders of magnitude less than the ranges listed above for the mechanical energy required to densify a unit volume of soil by vibrocompaction, deep dynamic compaction, or explosive compaction. Ultimately, all the mechanical energy imparted to the soil by the densification techniques “dissipates,” but much of it by radiating away from the immediate zone being treated (i.e., radiation damping). Accordingly, proper use of the earthquake Capacity curve for remedial ground densification design requires knowledge of the spatial distribution of the dissipation of the mechanical energy imparted to the soil during remedial ground densification. In the next chapter, first order numerical models are proposed for computing such spatial distributions. 7.6 Summary In this chapter, vibro-compaction, deep dynamic compaction, and explosive compaction techniques were reviewed. Discussions were presented on both the mechanisms associated with these techniques for breaking down the soil structure and the empirical design procedures for implementing the techniques. Finally, a comparison was made of the mechanical energy imparted to the soil by the densification techniques to treat a unit volume of soil. Although this comparison allowed the ranking of the relative energy

304

efficiencies of the remediation techniques, the calculations used to compute the imparted mechanical energies did not provide any information on the how the energy is spatially dissipated in the soil. Such information is needed if comparisons are to be made between the dissipated energy required to induce liquefaction during remedial ground densification and during earthquakes. In the next chapter, first order numerical models are proposed for computing the spatial distribution of the energy dissipated in the soil during treatment. This information is used in conjunction with the energy-based Capacity curve to predict the spatial extent of induced liquefaction. Based on the hypothesis that liquefaction is a requisite for the densification of saturated sands, the predicted extent of liquefaction predicted using proposed numerical models are compared with the spatial extent of improvement predicted using the empirical expressions and guidelines presented in this chapter.

305

Chapter 8. First Order Numerical Models of Remedial Ground Densification Techniques 8.1 Introduction In the previous chapter, simple calculations were presented comparing the total mechanical energy required to treat a unit volume of soil by vibrocompaction, deep dynamic densification, and explosive compaction. However, these calculations did not provide any information on the how the imparted energy is spatially dissipated in the soil. Such information is needed to make comparisons between the dissipated energy required to induce liquefaction during remedial ground densification and during earthquakes, which leads to the focus of this chapter: the development of first order numerical models for computing the spatial distribution of the energy dissipated in the soil during treatment. The proposed models are relatively simple and require a spreadsheet, or other general math program, for implementation; Mathcad 8 (MathSoft, Inc. 1998) was used by the author. In conjunction with the energy-based earthquake Capacity curve, the proposed models are used to predict the spatial extent of liquefaction induced in simple profiles during treatment. Based on the hypothesis that liquefaction is a requisite for the densification of saturated sands, the predicted extent of liquefaction using the proposed numerical models are compared with the spatial extent of improvement predicted using the empirical expressions and guidelines presented in Chapter 7. The purpose of the comparisons, and the ultimate goal of this thesis, is to assess the applicability of using the energy-based Capacity curve derived from earthquake case histories in remedial ground densification design. Before the proposed models are outlined for each of the remedial densification techniques, a brief discussion is presented on the verification and validation of numerical models.

306

8.2 Numerical Model Verification and Validation The term “numerical model” refers to the mathematical representation of a physical system or process. Numerical modeling techniques range in sophistication and complexity. In general, simpler models (or lower order models) require a greater understanding of the system or process being represented, while more sophisticated models (or higher order models) require a greater understanding of the specific modeling technique being employed (e.g., finite element method). A good philosophy is to start simple and progressively use more sophisticated models until a comfort level is reached that the actual system or process is being adequately represented. In doing so, the results from lower order models are used to check the appropriateness of the specifics of the higher order models (e.g., type of elements used, boundary conditions, constitutive equations), and the higher order models are used to check the appropriateness simplifying assumptions and the inherent limitations of the lower order models. Only after the results from the lower and higher order models are reconciled, and most importantly, only after the computed results are favorably compared with observations from well-documented field case histories, can any confidence be placed in the appropriateness of the models. Alternatively to the above approach, simple models are often evaluated by direct comparisons with field observations, without the benefit of comparisons with higher order model predictions. For processes for which an abundant amount of welldocumented field case histories are available, thorough evaluations of the simple models can be performed by this approach. However, for processes for which limited field case histories are available, this evaluation approach should be used with caution. First, if field observations are only available for complex profiles, unfavorable comparisons between predicted and observed results may lead to the model being summarily labeled inaccurate, when in actuality, the model may be suitable for analyses of less complex profiles. Second, if field observations are available for simple profiles, favorable comparisons between predicted and observed results may lead to over confidence in the predictive capabilities of the model, without a complete understanding of the model’s limitations.

307

Noting the shortcomings of the alternative approach to model verification, the results of the comparisons of the predictions made using the proposed first order models and the empirical expressions and guidelines should be viewed as preliminary. While favorable comparisons will lend credence to using the energy-based Capacity curve derived from earthquake case histories in remedial ground densification design, such comparisons will not conclusively validate the concept of “energy-based remediation of liquefiable soils.” Finally, in line with the philosophy of evaluating simple models by comparisons with higher order models, as well as favorable comparisons with field case histories, a limited discussion is given at the end of the chapter on higher order numerical models. 8.3 Numerical Modeling of Vibro-Compaction 8.3.1 Proposed Model As discussed in Chapter 7, the vibroflot-soil interaction is very complex. The horizontal/torsional vibrations and the introduction of both water and backfill during vibrations do not lend themselves to being simply modeled. However, with a few simplifying assumptions, the vibratory probe method may be reduced, as a first order approximation, to a rather simple system. For this reason, a first order model is only proposed for the vibratory probe method and not for vibrocompaction. The mechanisms leading to the breakdown of the soil structure during vibratory probing were briefly discussed in Chapter 7 (Section 7.2.2) and illustrated in Figure 7-13. Although Rayleigh and P-waves contribute, it is assumed that the soil structure is primarily broken down by radially propagating, vertically polarized, shear waves (SVwaves). This assumption is similar to that inherent in the first order model proposed by Holeyman (1997). Based on this assumption, the simple model illustrated in Figure 8-1 is used to compute the amount of energy that is dissipated in the soil surrounding the probe. In this figure, it is shown that the amplitude of the SV-waves decreases as a function of distance from the probe. The decay in the of amplitude of the SV-waves with distance can be attributed to geometrical spreading (i.e., conservation of energy) and to material (or hysteretic) damping in the soil, where the latter is related to the breakdown of the soil structure (i.e., see Chapter 3).

308

The shear strains resulting from the radially propagating shear waves is given by the following expression (e.g., Newmark 1968, Dowding 1996).

 

v vs

where:

(8-1)



= Shear strain from SV-wave.

v

= Particle velocity of SV-wave.

vs

= Shear wave velocity of soil.

In this expression, v and vs have the same units, resulting in  being dimensionless. The peak particle velocity (v) as a function of distance from the probe (r) can be determined from empirical attenuation relations, which will be discussed subsequently in Section 8.3.2. However, first attention is given to vs. Because vs is a function of the induced shear strain (), Equation (8-1) has to be solved iteratively. A similar procedure to that used in the strain-based liquefaction procedure (Chapter 2, Section 2.2.2) may be employed. The shear wave velocity (vs) is related to shear modulus (G) by the following expression.

G

v s2   t g

where:

(e.g., Richart et al. 1970) G

=

Shear modulus.

vs

=

Shear wave velocity.

t

=

Total unit weight of the soil.

g

=

Acceleration due to gravity.

309

(8-2)

Vertical Exciting Force

v(r) r

 

v vs (G/Gmax)

D

D

 (log scale)

G/Gmax

Figure 8-1. First order model of the vibratory probe method.

From substituting Equation (8-2) into Equation (8-1), the resulting expression provides a direct relationship between shear strain () and the shear modulus ratio (G/Gmax).

 

v

(8-3)

G v s max  G max

where:



=

Shear strain.

v

=

Particle velocity of shear wave.

vs max

= Small strain shear wave velocity of the soil (i.e.,   10-6) .

G/Gmax = Shear modulus ratio. The iterative solution of Equation (8-3) is shown graphically in Figure 8-2, where in the first iteration a value of G/Gmax is assumed and  computed. In the second iteration, the ratio of G/Gmax corresponding to  computed in the first iteration is used. The process is

310

repeated until the assumed and computed ratios are with in a tolerable error. The Ishibashi and Zhang (1993) shear modulus degradation curves, given previously as Equation (2-16), were used to solve Equation (8-3).

1.0 Assumed values of G/Gmax Computed values of G/Gmax

G/Gmax

iteration 1

(G/Gmax)

10-4%

iteration 2 final iteration

tolerable error

 (log Scale) 

1%

Figure 8-2. Iterative solution of Equation (8-3) to determine the shear-strain () at a given distance from the vibratory probe.

Once  is determined, the damping and shear modulus ratios are readily determined from the respective degradation curves, as illustrated in Figure 8-3. An expression for the Ishibashi and Zhang (1993) damping degradation curves was given previously as Equation (3-4).

311

G/Gmax (G/Gmax)

D D



10-4%

 (log scale)

1%

Figure 8-3. The determination of shear modulus and damping ratios from the respective degradation curves. Knowing G/Gmax, D, and , two important sets of information can be determined. First, using the threshold strain concept presented in Chapter 2 (Section 2.2.2), the maximum possible extent of liquefaction around the probe can be defined by the locus of points where the induced strain equals the threshold strain ( = th; th  0.01 to 0.015%). Second, the cumulative energy dissipated per unit volume of soil, normalized by the initial mean effective confining stress, (i.e., normalized energy demand, NED: Section 5.2.3) imparted to the soil by the vibratory probe can be determined. The following series of algebraic manipulations and substitutions results in an expression, Equation (84), used to compute NED. For reference purposes, the section and equation numbers of expressions and terms presented earlier in this thesis are given.

W1  2  D    

(5-2)  G     G  max  

  G    G max  

 G     2 W1  2  D  G max    G max  

312

NED 

W1 N  'm o

(5-10)

N  f  dur  G  1    2  f  dur  NED  2  D  G max    'm o  G max  

 G  1    2    dur  NED  D  G max    'm o  G max  

where:

(8-4)

NED

= Normalized energy demand (Section 5.2.3).

W1

= Dissipated energy per unit volume is one cycle of loading (Section 3.3.2).

D

= Damping ratio (Section 3.3.2).

Gmax

= Small strain shear modulus (Section 2.2.2)

G/Gmax = Shear modulus ratio.



= Shear strain.

N

= Number of cycles of loading.

f

= Frequency of exciting force (hz).



= Frequency of exciting force (rad/sec).

dur

= Duration of exciting force (sec).

’mo

= Initial mean effective confining stress (Section 2.2.2).

Using the above expression for NED in conjunction with the expression for the normalized energy capacity of the soil (NEC: Equation (5-11), Figure 5-11), the extent of induced liquefaction can be predicted as a function of the duration of vibrations. NEC is defined as the cumulative energy dissipated per unit volume of soil, normalized by the initial mean effective confining stress, required to induce liquefaction. 8.3.2 Attenuation Relationships In order to solve Equation (8-4), attention is returned to attenuation relations for determining peak particle velocity (v) as a function of distance from the probe (r). Using empirical attenuation relationships is the most direct method for determining v. 313

Attenuation relations are of two general types depending of the required information needed to predict the amplitude of vibrations (Woods and Jedele 1985). The first type requires the source energy to be known. The second type requires the amplitude of vibrations to be known at one point to predict the amplitudes of vibrations at other points. Inherent to both types of relations are empirical constants, which are dependent on parameters such as the soil/site conditions. Clearly, using attenuation relations that predict peak particle velocities (v) as functions of the source energy would fit best with the theme of this thesis. Unfortunately, the empirical constants associated with this class of attenuation relations have primarily been established for impact type energy sources such as used in impact pile driving, deep dynamic compaction, and explosive compaction and have not been developed for vibratory type sources. The most likely reason for this is that the source energy for vibratory probes is not constant but rather increases as the soil densifies. This phenomenon was briefly discussed in Chapter 7 in relation to current draw of electric vibrators. As a result, it is difficult to readily quantify the source energy by a single value, and therefore, it is difficult to establish the empirical constants for the energysource type of attenuation relations. On the contrary, attenuation relations requiring the amplitude of vibrations to be known at one point in order to predict the amplitudes of vibrations at other points have been used for both impact type and vibratory sources. Considering only geometrical spreading, these attenuation relations take the form: r  w2  w1   1   r2 

where:

n

(e.g., Woods and Jedele 1985)

(8-5)

w1 = Known amplitude of vibration at distance r1 from the source. w2 = Unknown amplitude of vibration at distance r2 from the source. n

= Geometric damping coefficient; see Table 8-1.

314

Table 8-1. Geometric damping coefficients. (Adapted from Gutowski and Dym 1976 and Kim and Lee 1998). Source Location

Source Type

Point

Example Sources Friction pile driving, train loading (16 passenger cars)

Body Wave

Dynamic compaction

Surface Wave

Train loading (16 passenger cars)

Body Wave

Surface Infinite Line

Surface Wave

Point At-Depth

Induced Wave

At-depth Blasting

n

Monitoring Location

2.0

Surface

1.0

At-Depth

0.5

Surface

1.0

Surface

0.5

At-Depth

0

Surface

1.0

At-Depth

0.5

At-Depth

Body Wave

Infinite Line

To account for material damping, an additional term is added to Equation (8-5): n

r  w2  w1   1   exp    ( r2  r1 )  r2 

where:

(8-6)

 = Empirical coefficient of attenuation that varies as a function of soil/site conditions (1/distance); see Table 8-2.

315

Table 8-2. Proposed Classification of Earth Materials by Attenuation Coefficient. (Adapted from Woods and Jedele 1985). Attenuation Coefficient 

Class 5hz: I

50hz:

5hz: II 50hz:

5hz: III 50hz:

5hz: IV 50hz:

Description of Material

0.01 to 0.03 (1/m) 0.003 to 0.01 (1/ft) 0.10 to 0.30 (1/m) 0.03 to 0.10 (1/ft) 0.003 to 0.01 (1/m) 0.001 to 0.003 (1/ft) 0.03 to 0.10 (1/m) 0.01 to 0.03 (1/ft) 0.0003 to 0.003 (1/m) 0.0001 to 0.001 (1/ft) 0.003 to 0.03 (1/m) 0.001 to 0.01 (1/ft)

Weak or Soft Soils – lossy soils, dry or partially saturated peat and muck, mud, loose beach sand, and dune sand, recently plowed ground, soft spongy forest or jungle floor, organic soils, top soils. (shovel penetrates easily) Competent Soils – most sands, sandy clays, silty clays, gravel, silts, weathered rock. (can dig with shovel)

Hard Soils – dense compacted sand, dry consolidated clay, consolidated glacial till, some exposed rock. (cannot dig with shovel, must use pick to break up)

< 0.0003 (1/m) Hard, Competent Rock – bedrock, freshly exposed hard rock. (difficult to break with hammer)

< 0.0001 (1/ft) < 0.003 (1/m) < 0.001 (1/ft)

The  values presented in Table 8-2 were established from field data collected over a twenty year period (Woods and Jedele 1985). As discussed in Richart et al. (1970),  is frequency dependant. The following expression relates  and the frequency of the vibrations.

   where:

2D f f   Q  vs vs

(8-7)

 = Empirical coefficient of attenuation that varies as a function of 316

soil/site conditions (1/distance); see Table 8-2. f

= Frequency of vibrations (hz).

vs = Shear wave velocity of the soil. Q = Quality factor. =

1 2 D

D = Damping ratio of the soil (Chapter 3, Section 3.3.2). From Equation (8-7), the  values for two different frequencies may be related by the following expression:  f2    f1 

 2   1   where:

(8-8)

1 = Known  value corresponding to frequency f1 (1/distance). 2 = Unknown  value corresponding to frequency f2 (1/distance).

Table 8-2 gives the  values for two different frequencies: 5 and 50hz. Using Equation (8-8),  values for other frequencies can be determined. As a final observation, Equation (8-7) shows that  increases proportionally to the damping ratio of the soil (D). Accordingly, it is expected that loose soil deposits will have larger  values than denser deposits; this trend is confirmed by the empirical data presented in Table 8-2. The expressions representing attenuation due to geometrical spreading only and geometrical spreading plus material damping (i.e., Equations (8-5) and (8-6), respectively) are plotted in Figure 8-4. Also shown in this figure are peak particle velocities measured during deep dynamic compaction: W = 6tons, H = 27m. As may be observed from this figure, the influence of material damping can be significant.

317

Peak Vertical Velocity (in/sec)

10

1 Equation (8-5)

Equation (8-6) 0.1

0.01

10

100 1,000 Distance from Source (ft)

10,000

Figure 8-4. Comparison of attenuation relations given by Equation (8-5) (i.e., geometrical spreading only) and given by Equation (8-6) (i.e., combined geometrical spreading and material damping). The data points shown in the plot are peak particle velocities resulting from the dropping of a 6ton tamper from 27m. (Adapted from Woods and Jedele 1985).

A limited literature review was performed concerning attenuation relations specific to vibro-compaction. In regards to the vibratory probe method, Van Impe et al. (1994) state: “The attenuation coefficient  for vibratory compaction varies in saturated clean sands typically between 0.05 and 0.10m-1. In partially saturated to dry sands  can rise to 0.5m-1.” However, no mention is given to the frequencies corresponding to the stated  values. Figure 8-5 shows measured peak accelerations as a function of distance from Yand double-Y probes for two different excitation frequencies. Although the data points shown in this figure are not distinguished as a function of the excitation frequency, it appears to have little influence on the peak accelerations. However, as may be observed

318

from the following expression, the corresponding peak particle velocities (v) for the two excitation frequencies will be very different.

v

ag 2 f

where:

(8-9) v

= Peak particle velocity (consistent units with g).

a

= Peak acceleration (g).

g

= Acceleration due to gravity (same units as v).

f

= Excitation frequency (hz).

8

Acceleration (g)

6 probe

probe

4 2 0

0

1

2

3

5 0 1 2 4 Distance from the probe (m)

3

4

5

Figure 8-5. Attenuation of peak ground acceleration with distance from the vertically vibrating probe excited at 16 and 25hz. (Adapted from Van Impe et al. 1994).

The  values corresponding to the curves shown in Figure 8-5 were not given by Van Impe et al. (1994), but from simple trial and error, the author found that  = 0.45 and 0.4m-1 provide reasonable approximations for the Y- and double-Y curves, respectively. Using the form of Equation (8-6), the curves shown in Figure 8-5 may be approximated by the following expressions. Y-Probe: a max

 0.74   3.1     r 

0.5

 exp  0.45  (r  0.74)

319

(8-10a)

Double-Y Probe:  0.97  a max  2.5     r 

where:

0.5

 exp  0.40  (r  0.97)

(8-10b)

amax = Peak acceleration (g). r

= Distance from the probe (m).

Based on the statements of Van Impe et al. (1994) presented above regarding the range of

 for different soil conditions, it is likely that the data shown in the above plots were measured in partially saturated or dry sand. Holeyman (1997) proposed the following attenuation relation for peak accelerations as a function of distance from a Y-probe. “ Ar  (d .(1  b.d )) where:

1

2

.e  a.d

(8-11a)

Ar

= Amplitude of vibration at radial distance r

d

= (r2 + h2)1/2, with h = probe ribs equivalent focal depth

b

= geometric damping accentuation factor

a

= intrinsic soil damping”

Holeyman (1997) goes on to give the following values for the variables in this expression: h = 0.7m, b = 1/30m-1, and a = 0. 03m-1. Additionally, a plot of the above expression is shown in comparison to measured acceleration values for several excitation frequencies, which is reproduced as Figure 8-6. The author is uncertain how to interpret Equation (8-11a). If it is interpreted as: Ar 

1 d  (1  bd )

 e  ad

(8-11b)

where Ar is assumed to be in g and the horizontal axis in Figure 8-6 is assumed to be r, then the resulting plot does not match that given in Holeyman (1997), as shown in Figure 8-6. Several other interpretations of Equation (8-11a) were made but none resulted in a match of the curve presented in Holeyman (1997). Keeping the same form of Equation (8-11b), the following expression was found to give a good approximation of the curve presented in Holeyman (1997); see Figure 8-6.

320

1

Ar 

d  (1  bd )  9.81

where:

 e  ad

(8-11c)

Ar

= Amplitude of vibration at radial distance r (g).

d

= (r2 + h2)1/2

r

= Radial distance from probe (m).

h

= 0.7

b

= 1 /4.75 (as opposed to 1/30 as proposed in Holeyman 1997)

a

= 0.03

Alternatively, the data shown in Figure 8-6 may be fit by the following expression, as shown in Figure 8-6. a max

 2.66   0.14     r 

where:

0.5

 exp  0.05  (r  2.66)

(8-11d)

amax = Peak acceleration (g). r

= Radial distance from the probe (m).

The only clear trend that can be observed from the data shown in Figure 8-6 is that at greater radial distances the peak accelerations tend to be less for the 10.5hz data than for the two other excitation frequencies. Similar to the data shown in Figure 8-5, no clear trend can be observed in acceleration values for 15.6 and 21.0hz. However, the corresponding velocities will differ. For comparison purposes, the  values presented in Table 8-2 range from 0.01 to 0.03m-1 for Class I soils subjected to a 5hz excitation. Using Equation (8-8), the range becomes 0.04 to 0.12m-1 for a 20hz excitation, which is consistent with  = 0.05m-1 in Equation (8-11d).

321

1

Plot of Equation (8-11a) as interpreted by the author, i.e., Equation (8-11b)

Equation (8-11d)

Acceleration (g)

0.1 Plot of Equation (8-11a) as shown in Figure 2 of Holeyman (1997). Plot of Equation (8-11c) 0.01 Meas. acc @ 10.5hz Meas. acc @ 15.6hz Meas. acc @ 21.0hz 0.001

0.1

1

10

100

Radial Distance (m) Figure 8-6. Comparison of attenuation expressions to measured field data for three excitation frequencies. (Adapted from Holeyman 1997).

The final set of attenuation data reviewed is that presented in Baez (1995), which is shown in Figure 8-7. These data were recorded during the placement of stone columns using a Keller bottom-feed “S” vibrator with a maximum 20ton centrifugal force and a 165hp motor (stone column: vibrocompaction with gravel backfill).

322

4.0 Approx. Face of Vibrator 3.5

Max. Accel. Steady-State Accel.

Acceleration (g)

3.0 2.5

Equation 8-12b

2.0 1.5 1.0 Equation 8-12a

0.5 0.0

0

1

2 3 4 5 6 7 8 9 Horizontal Distance from Center Line of Stone Column (ft)

10

Figure 8-7. Measured vibration levels at a depth of 14ft during the placement of stone columns. (Adapted from Baez 1995).

The data presented in Baez (1995) is unique in that Baez distinguishes between peak and steady state vibration amplitudes, where the steady state amplitude is defined as that lasting 5 consecutive seconds or longer. Assuming the form of the attenuation relation of Equation (8-6), the following expressions provide a reasonable fit of the steady state and peak accelerations, respectively. a steadystate

a max

 3  1.7    r

 3  2.7    r

0.5

0.5

 exp  0.2  (r  3)

 exp  0.2  (r  3)

323

(8-12a)

(8-12b)

where:

asteady state = Steady state acceleration (g). amax

= Peak acceleration (g).

r

= Radial distance from the probe (ft).

As may be observed from Equations (8-12a) and (8-12b) and the data plotted in Figure 87, the ratio of asteady state to amax is essentially independent of distance from the probe and is approximately 0.633. Finally, assuming oscillatory motions, the accelerations computed by any of the above expressions are related to the corresponding particle velocity by:

v

ag

(8-13)



where:

v

= Particle velocity.

a

= Acceleration (g).

g

= Acceleration due to gravity.



= Excitation frequency (rad/sec).

In this expression, v and g have consistent units. 8.3.3 Comparison of Proposed First Order Model Predictions with Empirical Guidelines General trends in the predictions of the first order model proposed above are compared with the empirical guidelines presented in Chapter 7 (i.e., the typical spacing of compaction points for the vibratory probe method ranges from 1 to 2m (Broms 1991)). For a triangular grid pattern, the distance from a compaction point to the center of the compaction points is approximately 0.6S, as shown in Figure 8-8, where S is the spacing between the compaction points. Based on this, the lateral extent of improvement for a single compaction point is approximated as 0.6S, resulting in a range of about 2 to 4ft.

324

Compaction Points S

Center of Compaction Points

0.6S

Figure 8-8. For a triangular grid pattern, the distance from a compaction point to the center of the compaction points is approximately 60% of the distance between the compaction points. Using the proposed first order model, the lateral extent of liquefaction was estimated for a clean sand profile having a constant N1,60 = 5blws/ft with depth, t = 125pcf, and a depth to the groundwater table of 10ft. The vibratory probe was assumed to be excited at 20hz, and Equations (8-11d) and (8-13) were used to compute the peak particle velocity as a function of distance from the probe. Assuming a threshold strain of 0.01%, the maximum extent of liquefaction was determined by the iterative solution of Equation (83), the results of which are shown in Figure 8-9a. The boundary shown in this figure defines the distance from the probe to the locus of points in the profile where the induced shear strain equals 0.01%. The Demand and Capacity expressions (Equations (8-4) and (5-11), respectively) can be used to estimate the duration of the vibrations required to induce liquefaction. Such calculations were performed, and the results are shown in Figure 8-9b. In this figure, the extent of liquefaction (or liquefaction front) is shown for 0.5, 1, 5, 10, and 15sec of vibrations, where at 15sec the liquefaction front reaches its final location. In addition to the model predictions, the empirical range of the lateral extent of improvement is superimposed on Figures 8-9a and 8-9b.

325

0

0

a) 50hz

b) 0.5

20hz 10

10  > 0.01%

1

Excitation frequency 20

20  = 0.01%

Depth (ft)

Depth (ft)

5 30

30

 < 0.01% 40

40

Empirical range of improvement

Empirical range of improvement

50

50

10 60

0

5 10 Distance from probe (ft)

60

15

0

Duration of vibrations (sec) 15

5 10 Distance from probe (ft)

15

Figure 8-9. Simple model predictions using peak particle velocities: a) The maximum extent of liquefaction for 20 and 50hz excitation frequencies, b) Liquefaction fronts for 0.5, 1, 5, 10, and 15sec of vibrations at 20hz. Two immediate observations can be made concerning the plots shown in Figure 8-9. First, the model’s prediction for the maximum extent of liquefaction is larger than the empirical range of improvement. Second, the maximum extent of liquefaction is predicted to be reached in 15sec, which is rather quick. The most likely reason for both of these observations is that the empirical attenuation relation used is for peak particle velocity and not steady state particle velocity. To account for this, the ratio of peak to steady state accelerations determined from the data presented in Figure 8-7 (i.e., 0.633) was used to adjust the peak particle velocities computed using Equation (8-11d). Using

326

the adjusted velocities, analogous results to those shown in Figure 8-9 are shown in Figure 8-10.

0

50hz

0

a)

 > 0.01%

b) 1

20hz 10

10 Excitation frequency

20

 = 0.01%

Depth (ft)

Depth (ft)

20

5

30

Empirical range of improvement

40

Empirical range of improvement

30

40 25

 < 0.01%

50

60

0

2 4 6 8 Distance from probe (ft)

Duration of vibrations (sec)

50

60

10

0

2 4 6 8 Distance from probe (ft)

10

Figure 8-10. Simple model predictions using steady state particle velocities: a) The maximum extent of liquefaction for 20 and 50hz excitation frequencies, b) Liquefaction fronts for 1, 5, and 25sec of vibrations at 20hz. As may be observed from Figure 8-10, using the steady state particle velocities results in a reduction in predicted extent of liquefaction and an increase in the required vibration time for the liquefaction front to reach its final position. However, the predicted extent of liquefaction now under-predicts the extent of improvement in comparison with the empirical range. Although it may seem that the steady state particle velocities assumed are too low, this might not be the cause of the under-prediction. As may be recalled from

327

Chapter 7, improvement from vibro-compaction results from both the densification of the soil and from increased lateral confining pressure. The proposed simplified model does not account for the latter mechanism. Accordingly, the extent of improvement is expected to extend beyond the boundary of predicted liquefaction. Additionally, although the time required for the liquefaction front to reach its final position (i.e., 25sec) may still seem short, this may not be the case. In typical densification programs, the process of repeated insertion and withdrawal is employed, which extends the vibration time per compaction point over that required to induce liquefaction in a virgin profile. Several other observations can be made concerning the model predictions in Figure 8-10. As shown in Figure 8-10b, soils at depth are predicted to require greater vibration times for the liquefaction front to reach its final position. This is consistent with field observations. Furthermore, the conical shape of the predicted liquefaction zone is also consistent with field observations. For liquefied soil, settlements typically range from about 2 to 10% of the depth of the liquefied deposit (Tokimatsu and Seed 1987). Based on this, the magnitude of settlement would decrease with increasing distance from the probe, resulting in a conical surface depression. As may be seen in Figure 8-11, such conical depressions occur in the field. Finally, in Figures 8-9a and 8-10a, the boundaries for the extent of liquefaction are predicted for 50hz excitation frequency, in addition to the 20hz boundaries discussed above. The boundaries corresponding to 50hz are considerably less than those for 20hz. The significance of this is the following. To avoid getting the probe stuck in the profile, densification is undesirable during initial insertion of the probe to the desired depth of improvement; compaction occurs as the probe is withdrawn. From the results shown in Figures 8-9a and 8-10a, it can be seen that initial penetration at high a frequency and withdrawal at a lower frequency will achieve this goal.

328

Figure 8-11. Conical surface depression created by vibrocompacting without backfill. (Photo courtesy of W.S. Degen, Vibro Systems, Inc.).

8.4 Numerical Modeling of Deep Dynamic Compaction The first order model proposed for deep dynamic compaction (DDC) consists of two parts: the approximation of the input motion and the response of the soil at depth in the profile. Each will be presented in turn, starting with the approximation of the input motion. 8.4.1 Approximate Input Motion for Deep Dynamic Compaction The approach used to compute the motion resulting from a tamper impacting the surface of a soil profile is similar to that used in machine foundation design (i.e., dynamic soil structure interaction). In this approach, the soil profile is represented by a viscously damped, single degree of freedom oscillator, such as shown in Figure 8-12. The spring stiffness and the damping coefficient are determined by the expressions proposed by Lysmer and Richart (1966), Equations (8-14a) and (8-14b), respectively.

329

m

(t ), u (t ), u (t ) u

c

k

Figure 8-12. Viscously damped, single degree of freedom oscillator used to represent the soil profile.

k

4Gr 1 

(8-14a)

c

3.4 2 G t r  1  g

(8-14b)

where:

k

=

Spring stiffness.

G

=

Shear modulus of the soil.

r

=

Equivalent radius of the tamper.



=

Poisson’s ratio.

c

=

Viscous damping coefficient.

g

=

Acceleration due to gravity.

t

=

Total unit weight of the soil.

Assuming that upon impact the mass moves in unison with the soil profile and ignoring the influence of gravity acting on the tamper after impact, the governing differential equation for the motion of the mass is given by the following expression. mu(t )  cu (t )  ku (t )  0

where:

(e.g., Chopra 1995)

u(t )

=

Acceleration of mass.

u (t )

=

Velocity of mass.

u(t)

=

Displacement of mass.

m

=

Mass of tamper plus participating soil mass.

330

(8-15)

By dividing through by m, Equation (8-15) for the motion of the mass can be written in alternate form:

u(t )  2 n u (t )   n2 u(t )  0 where:

n

= Natural frequency of oscillator (rad/sec). =



(8-16)

k m

= Damping ratio (see Equation (3-5)). =

c 2 km

For a less than critically damped system, the general solution for Equation (8-16) is given by the following expression (e.g., Chopra 1995).   u   n u o u (t )  e   nt  u o cos( D t )  o sin( D t ) D  

where:

D

(8-17)

= Damped natural frequency of oscillator (rad/sec). =

n 1  2

uo

= Displacement of oscillator at t = 0.

u o

= Velocity of oscillator at t = 0.

Inherent to the solution given by Equation (8-17), m is assumed only equal to the mass of the tamper and the mass of the participating soil is not included. Given the first order approximation of the proposed model, this assumption is not considered to decrease the accuracy of the computed results beyond the desired limit. Roesset et al. (1994) give a more detailed discussion on the significance of this assumption. Finally, by designating t = 0 as the time of impact, the following initial boundary conditions apply. Impact velocity:

u o  2 gH

Initial displacement:

uo  0

331

Substituting the above initial boundary conditions into Equation (8-17) and differentiating results in the following expressions for displacement, velocity, and acceleration time histories.

u (t ) 

u (t ) 

u(t ) 

u o

D u o

D u o

D

 e  nt  sin( D t )

(8-18)



   n e  nt  sin( D t )   D e  nt  cos( D t )





 e  nt   2 n2  sin( D t )  2  n D  cos( D t )   D2  sin( D t )

(8-19)



(8-20)

The above expressions are for an elastic single degree of freedom oscillator. However, it is known that significant plastic deformations occur around the tamper impact zone (i.e., craters form). To account for the soil non-linearity, degraded values for the shear modulus (G) are used in computing the spring and damping constants (k and c, respectively). The degraded G is determined iteratively using shear modulus degradation curves, which requires a procedure for computing the deviatoric shear strain induced in the soil from the impacting tamper. The following procedure is proposed. Starting with the small strain soil properties, the maximum acceleration for the soil profile is determined as: a max  max u(t )

(8-21)

From amax and the dimensions of the tamper, the contact stress between the impacting tamper and soil surface can be determined as:

Fmax A ma max  A Wa max  gB 2

 max 

(8-22)

332

where:

max

= Maximum normal stress.

amax

= Peak acceleration.

Fmax

= Maximum force imposed by the impacting tamper.

A

= Contact area between tamper and soil profile.

W

= Weight of tamper.

g

= Acceleration due to gravity.

B

= Length of square contact area of tamper and soil.

For axi-symmetric conditions, axial stress and axial strain are related by Hooke’s law:

a 

a Ey

where:



2 h Ey

(e.g., Davis and Selvadurai 1996)

a

=

Axial strain.

a

=

Axial stress.

Ey

=

Young’s modulus.



=

Poisson’s ratio.

h

=

Lateral stress.

(8-23a)

However, the use of this expression requires the knowledge of the variation of lateral stress (h) in response to changes in the axial stress (a) induced by the impacting tamper. To avoid the complexities involved in solving Equation (8-23a) and associated equations, cone models are often used in dynamic soil structure interaction analyses (e.g., Wolf 1994). In such models, the elastic half space is replaced by a truncated cone, wherein the lateral stress is assumed equal to zero (h = 0); see Figure 8-18. This effectively reduces the problem to one dimension, and for the condition of maximum induced stress, Equation (8-23a) becomes:

 a max  where:

 max

(8-23b)

E'

a max = Maximum induced axial strain. a

= Maximum induced axial stress.

E’

= Soil modulus, the value of which is discussed subsequently.

333

The deviatoric shear strain resulting from the axial loading of the one-dimensional model is given by the following expression.

 dev   a   h where:

(8-24)

dev

=

Deviatoric shear strain.

a

=

Axial strain.

h

=

Lateral strain.

The axial and lateral strains are related by Poisson’s ratio ():



 h

a

  h   a

(8-25)

Finally, through a series of substitutions, the maximum induced deviatoric shear strain can be expressed as follows:

 dev max  (1   )

Wa max gB 2 E '

(8-26)

In this expression, Poisson’s ratio () is an intrinsic material property and the soil modulus (E’ ) is a both a material property and a function of the boundary/stress conditions. The influence of the boundary conditions on E’ can be observed from the two extremes: zero lateral stress and zero lateral strain. For the case of zero lateral stress, shown in Figure 8-13a, E’ is equal to Young’s modulus (Ey), which is given by Equation (8-27). a)

0.5h

b)

a

h = 0

h = 0

E’=Ey

a E’=Ec

Figure 8-13. The influence on boundary conditions on E’. a) Condition of zero lateral stress, b) Condition of zero lateral strain.

E y  2G(1   ) where:

(8-27)

Ey

=

Young’s modulus.

G

=

Shear modulus.

334



=

Poisson’s ratio.

Similarly, for the condition of zero lateral strain, shown in Figure 8-13b, E’ is equal to the constrained modulus (Ec), which is given by Equation (8-28).

Ec 

2G (1   ) 1  2

where:

(8-28)

Ec

=

Constrained modulus.

G

=

Shear modulus.



=

Poisson’s ratio.

As Poisson’s ratio approaches zero, Ec and Ey become equal, as shown in Figure 8-14. However, as Poisson’s ratio increases, Ec and Ey differ significantly. The question now becomes: which value of E’ is appropriate for the boundary conditions governing the response of a mass impacting the surface of a soil profile? No clear consensus is given in the literature on dynamic soil structure interaction, but the value determined as follows is consistent with the previous assumptions made in the development of this model.

10

Normalized Moduli

8 Ec G

6 4 2 0

ELa G 0.0

0.1

0.2 0.3 0.4 Poisson’s Ratio ()

Ey G 0.5

Figure 8-14. Comparison of Ey, Ec, and ELa as a function of Poisson’s ratio.

335

From wave propagation theory, it can be shown that an infinitely long rod can be terminated and replaced with a dashpot having a damping coefficient (c) given by Equation (8-29), and no reflection of a propagating wave will occur (e.g., Kramer 1996). This is shown conceptually in Figure 8-15. c  VA

where:

(8-29) c

=

Damping coefficient.



=

Mass density of rod.

V

=

Propagation velocity (e.g., shear wave velocity).

A

=

Cross-sectional area of rod.

c  VA Figure 8-15. An infinitely long rod may be modeled using an appropriately selected damper.

Substituting Equation (8-14b) into Equation (8-29) and solving for V results in the following expression, referred to as Lysmer’s velocity, VLa, (Holeyman 1985, Dobry and Gazetas 1986). V La 

3.4 2 G t 1 r   1  g A

(8-30)  Vs

3. 4  (1   )

The modulus corresponding to VLa is:

336

E La  VLa2   2

 3.4  V     ( 1   )   2 s

 3.4   G   (1   ) 

(8-31)

2

Comparisons of ELa with Ey and Ec are shown in Figure 8-14. Substitution of the above expression for ELa into Equation (8-26) results in an expression for maximum deviatoric shear strain:

 dev max  0.85  (1   ) 2 (1   ) where:

Wa max gB 2 G

(8-32)

dev max = Maximum deviatoric shear strain. 

= Poisson’s ratio.

W

= Weight of tamper.

amax

= Peak acceleration.

g

= Acceleration due to gravity.

B

= Length of square contact area of tamper and soil.

G

= Shear modulus.

The above expression is solved iteratively using the shear modulus degradation curve for the soil, similar to the procedure shown in Figure 8-2. As throughout this thesis, the Ishibashi and Zhang (1993) shear modulus degradation curves, given previously as Equation (2-16), are recommended for use. These curves are functions of the plasticity index (Ip) of the soil and the effective confining pressure. For cohessionless soil, Ip = 0. The confining pressure is assumed to be that of a depth of about 10ft. This depth is selected to be below the depth of the impact crater and is judged to represent the soil most directly influencing the response of the tamper. The final parameter that needs to be discussed is Poisson’s ratio (), which was defined previously by Equation (8-25). From the theory of elasticity it can be shown that 337

Poisson’s ratio can range from -1 to 0.5, where 0.5 corresponds to an incompressible material. Few materials have negative Poisson’s ratios, with typical values ranging from 0 to 0.5 (Davis and Selvadurai 1996). For saturated soils subjected to rapid loading, the range is typically smaller, with the lower bound being greater than zero, depending on the drainage conditions, and the upper bound remaining 0.5. Laboratory experience has shown that it is difficult to determine reliable values of Poisson’s ratio directly from experiments. Accordingly, Poisson’s ratio is often determined indirectly from other parameters, which can readily and reliably be determined from field and/or laboratory tests. Several relations relating Poisson’s ratio and other common parameters are:



M  2G 2( M  G )

(8-33a)



3K  M 3K  M

(8-33b)



3K  2G 2(3K  G)

(8-33c)

v 1  2 s v  p   v 2 1   s   v p 

where:

   

2

   

2

(8-33d)

   



=

Poisson’s ratio.

M

=

Constrained modulus.

G

=

Shear modulus.

K

=

Bulk modulus.

vs

=

Shear wave velocity.

vp

=

P-wave velocity.

To assess Equation (8-20) and the proposed procedure for determining the degraded shear modulus, a comparison is made between predicted and recorded acceleration time 338

histories. The recorded time histories are those presented in Mayne and Jones (1983) for a 20.9tonnes tamper (B = 1.9m) dropped from 18.3m on to coal mine spoil, having  = 0.37. The recorded and predicted time histories are shown in Figure 8-16. As may be observed in this figure, the computed time history using the degraded soil properties gives a very reasonable approximation of the recorded time histories, both in peak amplitude and period, especially given the gross simplifying assumptions made. To highlight the influence of soil non-linearity, a time history was computed using the small strain shear modulus, which is also shown Figure 8-16. This time history is significantly greater in amplitude and shorter in period than the recorded time histories.

Deceleration of Tamper or Acceleration of soil surface (g)

400 Computed acceleration time history using small strain soil properties

300

Computed acceleration time history using degraded soil properties

200

Recorded acceleration time histories

100 0 -100

0

20

40 60 80 Time (milliseconds)

100

120

Figure 8-16. A comparison of computed and recorded acceleration time histories. The recorded time histories are from Mayne and Jones (1983). Mayne and Jones (1983) also present peak acceleration values as a function of the drop height of the tamper. A comparison of this data with predicted peak accelerations is shown in Figure 8-17. As may be observed from this figure, the predicted accelerations are in reasonable agreement with the field observations.

339

90

Procedure proposed by author

80

Peak Acceleration (g)

70 60

Equation (8-34)

50 40

vs = 880 ft/sec W = 20.9 tonnes ro = 1.07m

30 20

1tonnes = 2205lbs 1ft = 0.3048m

10 0

0

10

20

30 40 50 Drop Height (ft)

60

70

Figure 8-17. Predicted and measured peak acceleration values as a function of drop height of the tamper. (Measured data from Mayne and Jones 1983).

Alternative to the procedure developed by the author, Mayne and Jones (1983) proposed the following expression for estimating peak acceleration.

a max HB  v s max  g W where:

(8-34)

amax

=

Peak acceleration of the soil surface.

W

=

Weight of tamper (lbs).

H

=

Drop height (ft).

B

=

Dimension of tamper (ft).

vs max =

Small strain shear wave velocity of the soil (ft/sec).

g

Acceleration due to gravity.

=

Although all are not explicitly stated in Mayne and Jones (1983), the following assumptions are inherent in the above expression:

340

Triangular shaped acceleration time history. No participation of soil mass. G

=

0.1Gmax (based on field observations made by Hansbo 1977, 1978).



= 0.37

t

= 110lb/ft3

k

=

4Gr 1 

All of these assumptions are reasonable, and Equation (8-34) gives results very similar to those computed using the author’s model (as shown in Figure 8-17), with much less computational effort. Accordingly, unless the field conditions deviate considerably from the above stated assumptions or unless the entire acceleration time history is required, the expression proposed by Mayne and Jones (1983) is recommended for use. 8.4.2 Response of the Soil at Depth for Deep Dynamic Compaction The relationship between the maximum impact force on the soil surface and the maximum normal stress induced at depth in the profile is assumed to be:

 max ( z )  where:

Fmax

(8-35)

(B  z) 2

max(z) = Maximum induced normal stress at depth z in the profile. Fmax

= Maximum induced force on the surface of the profile.

B

= Dimension of tamper.

z

= Depth in the profile at which stress is being computed.

This expression inherently assumes a 2:1 stress distribution with depth, which is commonly assumed in static foundation design and was assumed in the deep dynamic compaction models proposed by Pearce and Scott (1975) and Mayne and Jones (1983). The relationship between axial stress and axial strain given previously as Equation (823b) applies equally to the soils at the surface as to the soils at depth for the assumed truncated cone model. However, the soil modulus (E’) will likely vary with depth. As discussed in the previous section, at the surface E’ assumed is equal to ELa. However, as

341

the stresses spread with depth in the profile, a condition is eventually reached where the lateral strain is approximately zero for the soils directly beneath the tamper. At this depth, E’ is equal to the constrained modulus (Ec). Such a variation of E’ is only required because a one-dimensional cone model is being used to represent a three-dimensional process. If a full three-dimensional model is developed, the imposed stress conditions account for the variation in the apparent soil stiffness. Based on a limited parameter study conducted by the author, E’ is assumed to vary parabolically from ELa at a depth of 10ft to Ec at a depth of 40ft. Above a depth of 10ft, E’ is assumed equal to ELa. Although a more detailed study, both numerical and experimental, is required to confirm this assumption, the assumed variation of E’ does not greatly influence the soil response for 0    0.4. It is noted that both Holeyman (1985) and Wolf (1994) use variants of the truncated cone model proposed by the author. However, both assume E’ is constant with depth: Holeyman (1985) assumed E’ = ELa and Wolf (1994) assumed E’ = Ec, which are the lower and upper bounds assumed by the author. The spreading of the stress front and the variation of the soil modulus (E’) are conceptually illustrated in Figure 8-18.

342

r m ELa, VLa

a

1

40ft (assumed)

truncated cone

2

h = 0 boundary

Ec, Vp

CL Figure 8-18. Truncated cone used to model induced stresses from impacting tamper on the surface of a profile. The soil modulus is assumed to vary parabolically from ELa at a depth of 10ft to Ec at a depth of 40ft, as shown in Figure 8-19b. Above a depth of 10ft, the soil modulus is assumed equal to ELa.

The following expression defines the parabolically varying E’ as a function of depth (z). E' ( z)  V ( z) 2 

where:

t

(8-36a)

g

E’(z)

= Soil Modulus at depth z in the profile.

V(z)

= Propagation velocity of compression-extension wave at depth z in the profile. VLa =

for z < z1

(V p  V La ) 

z  z1  V La z 2  z1

Vp

for z1  z  z2 for z > z2

343

(8-36b)

z1

= 10ft assumed.

z2

= 40ft assumed.

z

= Depth in the profile.

t

= Total unit weight of the soil.

g

= Acceleration due to gravity.

Plots of both V(z) and E’(z) (i.e., Equations (8-36b) and (8-36a), respectively) are shown in Figure 8-19 for  = 0.37 and vs = 880 ft/sec. In addition to E’ and V varying as a result of the changing boundary conditions with depth, the relationship between axial and lateral strain is also assumed to vary as a result of the changing stress conditions. This can be understood from examining Figure 8-13, where at depths when E’ = Ec, the ratio h/a = 0. This variation in the ratio h/a is accounted for by the parameter C. From substitution of Equation (8-25) into Equation (824), the deviatoric shear strain is given by the following expression.

dev = (1+)a = Ca where:

(8-37)

dev

=

Deviatoric shear strain.



=

Poisson’s ratio.

a

=

Axial strain.

C

=

Constant of proportionality relating deviatoric and axial strain.

344

0

0 b)

10

10

20

20

Depth (ft)

Depth (ft)

a)

30

30

40

40 VLa

Vp

ELa

50 1400 1600 1800 2000 Compression-Extension Wave Velocity (ft/sec)

50 0.6

Ec

0.8 1.0 1.2 1.4 Modulus 107 (lb/ft2)

Figure 8-19. Variation of: a) propagation velocity of a compression-extension wave, and b) soil modulus, as a function of depth in the profile. In Equation (8-37), the term (1+) is replaced by the constant of proportionality C, which is assumed to range from (1   )  C  1 to account for the variation of the ratio of h/a. Assuming a linear variation of C with E’ results in the following expression.

C( z)  1 

 E La  Ec

 E' ( z) 

Ec E La  Ec

(8-38)

A plot of Equation (8-38) is shown in Figure 8-20. As with the assumed relationships expressing the variation of E’ and V with depth, both numerical and experimental studies are required to verify Equation (8-38). However, from the limited parameter study

345

conducted by the author, the soil response is not greatly influenced by the assumed variation of the ratio h/a for 0    0.4. 0

Depth (ft)

10 20 30 40 50 1.0 1.1 1.2 1.3 1.4 1.5 C(z) Figure 8-20. Variation of the constant of proportionality relating deviatoric and axial strains as a function of depth.

Finally, through a series of substitutions and algebraic manipulations, the deviatoric shear strain is given by the following expression.

 dev ( z )  where:

C ( z ) v s max HBW  E' ( z) (B  z) 2

(8-39)

dev(z) = Deviatoric shear strain at depth z (dimensionless). E’(z)

= Soil modulus at depth z (Equation (8-36a)) (lbs/ft2).

C(z)

= Constant of proportionality relating deviatoric and axial strain (Equation (8-38)) (dimensionless).

W

= Weight of tamper (lbs).

H

= Drop height (ft).

B

= Dimension of tamper (ft).

z

= Depth in the profile at which stress is being computed (ft).

vs max

= Small strain shear wave velocity of the soil (ft/sec).

346

Inherent to Equation (8-39) are all the assumptions listed for Equation (8-34), for the expressions E’(z) and C(z), and the assumption of a uniform stress distribution across the contact area between the tamper and soil. From a limited parameter study conducted by the author, none of the imposed assumptions strongly influence the computed results, and all the assumptions are reasonable to a first order approximation. To account for soil non-linearity, the soil modulus, E’(z), is assumed to vary as a function of the induced deviatoric shear strain. Employing the equivalent linear approach, Equation (8-39) is solved iteratively using the shear modulus degradation curve corresponding to each depth of interest. The general axial stress-strain behavior of soil subjected to impact loading is shown in Figure 8-21 (Scott and Pearce 1975 and Sinitsin 1983). As shown in this figure, at high levels of axial strain the soil strain hardens. The author found, quite serendipitously, that strain hardening could be approximately modeled by slightly modifying the Ishibashi and Zhang (1993) shear modulus degradation curves. The modification entails simply assuming that the shear modulus ratio remains constant above a designated level of shear strain. This is shown in Figure 8-22a, wherein the shear modulus ratio is assumed constant for shear strains above 1%. The resulting axial stress-strain curve is shown in Figure 8-22b. For comparison purposes, the commonly used hyperbolic stress-strain relation is also shown in Figure 822b.

347

Axial Stress

Strain hardening

Strain softening

Axial Strain Figure 8-21. General axial stress-strain behavior of soil subjected to impact loading. (Adapted from Scott and Pearce 1975).

1.0 0.8

Strain softening

G 0.6 Gmax 0.4

Strain hardening

0.2 0 10-4

10-3

10-2 10-1 Shear Strain (%)

100

101

Figure 8-22a. Modified shear modulus degradation curve to incorporate strain hardening effects.

348

200 Strain softening; approx. hyperbolic (dev < 1.0%)

Axial Stress (kPa)

150

100

Strain hardening (dev > 1.0%)

No strain hardening

50

0

0

0.5 1.0 Axial Strain (%)

1.5

Figure 8-22b. Axial stress-strain relation corresponding to modified Ishibashi and Zhang shear modulus degradation curve shown in Figure 822a. For comparison, the commonly used hyperbolic stress-strain relation is shown, which does not account for strain hardening effects. Once the induced deviator shear strain is known as a function of depth in the soil profile, the normalized dissipated energy can be estimated using the following expression.  G  2 N eqv   dev NED  2D G max  G  'm o  max  

where:

NED

= Normalized energy demand (Section 5.2.3).

D

= Damping ratio (Section 3.3.2; Equation (3-4)).

Gmax

= Small strain shear modulus (Section 2.2.2; Equation (2-12)).

G/Gmax = Shear modulus ratio (Equation (2-16)).

dev

= Deviatoric shear strain (Equation (8-39)).

Neqv

= Number of equivalent cycles of loading.

’mo

= Initial mean effective confining stress (Section 2.2.2; Equation (2-13)).

349

(8-40)

Following the approach outlined in Chapter 5 (Section 5.2.3), acceleration time histories computed using Equation (8-20) were analyzed, and it was determined that Neqv is approximately 0.75 cycles. Using the above expression for NED in conjunction with the expression for the normalized energy capacity of the soil (NEC: Equation (5-11), Figure 5-11), the depth of induced liquefaction was estimated for the case where a 20.9tonnes tamper is dropped 18.3m onto the surface of a clean sand profile having a constant N1,60 = 5blws/ft and the groundwater table at 10ft below the soil surface. The results of these calculations are shown in Figure 8-23 for three different values of Poisson’s ratio: 0.2, 0.37, and 0.495, wherein liquefaction is predicted at depths where NED > NEC. As shown in this figure, the maximum depth of predicted liquefaction decreases as the assumed value of Poisson’s ratio increases. In line with the values assumed by Mayne and Jones (1983) and Pan and Selby (2000) in modeling deep dynamic compaction, a Poisson’s ratio of around 0.35 is likely appropriate. Finally, the kinks in the NED curves result from the modifications made to the Ishibashi and Zhang shear modulus degradation curves to account for strain hardening effects (i.e., at depths shallower than the kinks, dev  1.0%). Also shown in Figure 8-23 are the maximum depths of improvement computed using the empirical expression presented in Chapter 7 (i.e., Equation (7-6)) for three different n values: 0.3, 0.5, and 0.8. Assuming n = 0.5 and  = 0.37, the maximum depth of improvement predicted using the empirical expression is in reasonable agreement with the maximum depth of liquefaction predicted using the proposed first order model. A closer agreement between the two predicted depths is realized if it is considered that densification will occur slightly beyond the maximum depth of induce liquefaction. Although the favorable comparison between the predicted depths of improvement and induced liquefaction lend credence to the validity of the proposed numerical model, additional parameter studies are required to more fully evaluate the model and the inherent assumptions used in its development.

350

0 0.495

0.3  WH

20 0.37

 = 0.2

0.5  WH

Depth (ft)

40 Depths of Predicted Liquefaction

0.8  WH

60

NEC

80 NED 100 10-8

10-6

N1,60 = 5 blws/ft

10-4 100 10-2 Normalized Energy

102

104

Figure 8-23. Comparison of the predicted depth of induced liquefaction using the numerical model proposed by the author to the predicted depth of improvement using empirical expressions.

8.5 Numerical Modeling of Explosive Compaction The proposed model for explosive compaction is based on cavity expansion theory and is similar to the vibro-compaction model in that both use empirical attenuation relations in computing NED. Also, the proposed model for explosive compaction is similar to the deep dynamic compaction model in that the iterative solution for the degraded soil properties is based on the induced deviatoric shear strain. As discussed in Chapter 7 (Section 7.4.1), several mechanisms lead to the breakdown of the soil structure during blasting. However, the proposed first order model inherently assumes that the high

351

intensity shockwave is the dominant mechanism, which is similar to the inherent assumption in the higher order model proposed by Wu (1995, 1996). The governing differential equation for the dynamic response of a homogeneous, isotropic, elastic whole-space surrounding a spherical cavity is given by the following expression (Sharpe 1942).   2 2    2    2 (  2G )   2  R R  t  R

where:

(8-41)



= Lame’s parameter.

G

= Shear modulus of wholespace.



= Displacement potential, discussed below.



= Mass density of the whole space.

R

= Radial distance from the center of the charge.

t

= Time.

Sharpe (1942) derived a solution for the above differential equation for a pressure function of the form p  Po  e  't acting on the wall of a spherical cavity of radius a in a wholespace having a Poisson’s ratio () equal to 0.25. Wu (1995) generalized Sharpe’s solution for wholespaces having different ; this generalize solution is:

 (T ) 

 Po a  e  oT 2 R [  ( o   ' ) ] 

where:

2 o

(T) =

    '  cos( o T )  o sin( oT )  e  'T  o   

Displacement potential as a function of T.

v p  (1  2 )

o

=

o

=

T

=

t

vp

=

P-wave velocity.

a

=

Radius of spherical charge/cavity.

a  (1   ) v p  (1  2 ) 0.5 a  (1   )

( R  a) vp

352

(8-42)



=

Poisson’s ratio.

R

=

Radial distance from the center of the charge.

t

=

Time.

’

=

Decay parameter of load.



=

Mass density of the whole space.

Po

=

Amplitude of pressure at R = a and t = 0.

The radial displacement in the wholespace (uR) is related to the displacement potential by:

u R (T ) 

 (T ) R

(Sharpe 1942)

(8-43a)

Differentiating Equation (8-42) with respect to R, gives: for T < 0

0 Po a  R   [  ( o   ' ) 2 ] 2

2 o

     '  e  oT cos( o T )  o sin( o T ) o   

uR(T) =

e 

 'T

 R   oT   e   ' cos( o T )  vp  

(8-43b) for T  0

   o2   o2   o ' sin( o T )   ' e  'T  o  

Similarly, the radial velocity in the wholespace ( u R ) is related to the displacement potential by:

 2 Rt u  R t

u R (T ) 

(8-44a)

Differentiating Equation (8-43b) with respect to t, gives:

353

0

for T < 0

Po a  R   [  ( o   ' ) 2 ] 2

u R (T ) =

2 o

  oT    o2   o2   o ' sin( o T ) e  ' cos( o T )  o     R   oT  2 2   ' e  'T    e ( o   o  2 o ' ) cos( o T ) v   p  

(8-44b) for T  0

   o o2   '  o2   o3   ' o2 sin( o T )   ' 2 e  'T  o  

From cavity expansion theory, the deviatoric shear strain induced in the wholespace is given by the following expression.

 dev 

1   rr     2

where:

dev

= Deviatoric shear strain.

rr

= Radial strain. =



(e.g., Hryciw 1986)

(8-45a)

u  2 = R 2 vp r

= Tangential strain. =

u 1  = R r r r

In alternate form, the above expression may be written:

 dev 

1  u R u R     2  v p R 

(8-45b)

The relative contributions of the terms inside the brackets of the above expression (i.e.,

rr and ) can be examined using Equations (8-43b) and (8-44b). The time histories for rr and  are shown in Figure 8-24 at two distances from the center of the charge (R) for a wholespace having vs = 150m/sec and vp = 281m/sec (i.e.,  = 0.3). As may be 354

observed from this figure, as R increases, the relative contribution of  (= uR/R) rapidly decreases (i.e.,   0 at R = 1.0m). Accordingly, the deviatoric shear strain can be reasonably approximated as:

 dev 

1 u R 2 vp

(8-45c)

Components of Shear Strain

0.03

rr (=

0.02 0.01

u R ) at R = 0.2m vp

 (=

uR ) at R = 0.2m R

rr at R = 1.0m

0.00

  0 at R = 1.0m -0.01 0.000

0.001

0.002 0.003 Time (sec)

0.004

0.005

Figure 8-24. Relative contribution of the rr and  as for two distances from the cavity wall.

As with the proposed model for deep dynamic compaction, the deviatoric shear strain is used in conjunction with the shear modulus degradation curves to iteratively determine the degraded soil properties. However, the assumptions inherent to Equations (8-44b) for radial velocity do not apply to actual soil profiles (i.e., homogeneous, isotropic, elastic wholespace). As a result, empirical attenuation relations for radial velocity are used. Many blast attenuation relations have the form: u peak

 R   C1   m  W 

where:

n

(8-46)

u peak

= Peak particle velocity at distance R from the center of the charge (m/sec).

R

= Radial distance from the center of the charge (m).

355

W

= Weight of the charge (kg).

m

= Constant that is a function of the geometry of the charge.

C1

= Empirically determined constant.

n

= Empirically determined constant.

Values for m, C1, and n from various studies are listed in Table 8-3, and the relations listed for deep, concentrated charges are plotted in Figure 8-25. As may be observed from this figure, a great deal of scatter exists among the various relations. Table 8-3. Coefficients for blast attenuation expression. (Adapted from Narin van Court and Mitchell 1994b). Test Type

C1

m

n

0.6

1/2

1.35

Long et al. (1981)

11.6

1/2

1.67

Duvall et al. (1967)

7.2

1/2

1.15

Sanders (1982)

1.1 to 3.6

1/2

1.15

Charlie (1985)

12

1/3

1.50

Charlie et al. (1985a)

5.6

1/3

1.50

Drake and Little (1983)

8.75

1/3

2.06

Charlie et al. (1992)

8.0

1/3

2.30

Charlie et al. (1985b)

1.1 to 3.6

1/3

2.30

Charlie (1985)

Underwater blasting, loose sand

11.9

1/3

1.45

Schure (1990)

17.7

1/3

1.45

Schure (1990)

Underwater blasting, medium sand

16.6

1/3

1.45

Schure (1990)

13.9

1/3

1.45

Schure (1990)

12.8

1/3

1.5

Bretz (1989)

Deep blasting, Columnar charges

Deep blasting, Concentrated charges

Underwater blasting, dense sand

Reference

Besides the variability in site conditions and explosive types, some of the scatter shown in Figure 8-25 may be due to the specifics of the attenuation relations (e.g., location of the recorded motions, peak particle velocity being represented). Due to the complexities

356

of wave propagation in actual soil profiles, it is expected that the peak particle velocities recorded on the surface of the profile will differ from those recorded at depth. Also, the peak particle velocities represented by the relations in Table 8-3 are not necessarily the same. As discussed in Mayne (1985), attenuation relations commonly represent: the maximum peak velocity for any one of the three orthogonally-oriented components of motion; the square root of the sum of the squares of the peaks from each of the three orthogonally-oriented components of motion; or the true vector sum, which is the maximum of the square root of the sum of the squares the three orthogonally-oriented components at a given time t.

10 Charlie et al. (1985a)

Peak Particle Velocity (m/sec)

1

0.1

Drake and Little (1983)

0.01 Charlie et al. (1992)

Charlie (1985) 10-3

10-4

Charlie et al. (1985b)

1

10

100 1/3

1/3

Scaled Distance, (R/W ), (m/kg ) Figure 8-25. Empirical attenuation relations for deep, concentrated charges; refer to Table 8-3.

357

For computing the blast induced deviatoric shear strain using Equation (8-45c), the peak radial (or longitudinal) velocity measured at depth in the profile is needed. Unfortunately, the details of all the relations listed in Table 8-3 could not be ascertained by the author. As a first order approximation, the attenuation relation proposed by Charlie et al. (1992) is used. As may be observed from Figure 8-25, this relation lies in the middle of the range of relations listed in Table 8-3 for deep, concentrated charges. The relationship between the P-wave velocity (vp) for the degraded soil profile and the small strain shear wave velocity (vs max) is:  G  2(1   )   v p  v s max   1  2  Gmax  

where:

(8-47)

vp

= P-wave velocity for the degraded soil profile.

vs max

= Small strain shear wave velocity.



= Poisson’s ratio.

G/Gmax = Ratio of degraded to small strain shear moduli. Substitution of the above equation into Equation (8-45c) results in the following expression for deviatoric shear strain in terms of the shear modulus ratio G/Gmax. Similar to the procedure outlined for deep dynamic compaction, this expression is solved iteratively using the modified Ishibashi and Zhang shear modulus degradation curves.

 dev 

u R peak  G  2(1   )   2  v s max   1  2  Gmax  

(8-48)

Once the induced deviator shear strain is known, the normalized dissipated energy can be estimated using the following expression.  G  2 N eqv   dev NED  2D G max   'm o  G max  

where:

NED

= Normalized energy demand (Section 5.2.3).

D

= Damping ratio (Section 3.3.2; Equation (3-4)).

358

(8-49)

Gmax

= Small strain shear modulus (Section 2.2.2; Equation (2-12)).

G/Gmax = Shear modulus ratio (Equation (2-16)).

dev

= Deviatoric shear strain (Equation (8-48)).

Neqv

= Number of equivalent cycles of loading.

’mo

= Initial mean effective confining stress (Section 2.2.2; Equation (2-13)).

Following the approach outlined in Chapter 5 (Section 5.2.3), velocity time histories computed using Equation (8-44b) were analyzed, and it was determined that Neqv is approximately 0.75 cycles, similar to Neqv for deep dynamic compaction. Using the above expression for NED in conjunction with the expression for the normalized energy capacity of the soil (NEC: Equation (5-11), Figure 5-11), the radial extent of induced liquefaction was estimated for 5.5kg charge detonated at 12m in a clean sand profile having a constant N1,60 = 5blws/ft and the groundwater table at 3.05m (10ft) below the soil surface. The results of these calculations are shown in Figure 8-26 for three different values of Poisson’s ratio: 0.2, 0.37, and 0.495, wherein liquefaction is predicted when NED > NEC. Similar to the trend noted for deep dynamic compaction, the radial extent of liquefaction decreases as the assumed value of Poisson’s ratio increases. Also shown in Figure 8-26 are the empirically determined ranges for the radial extent of improvement given by Kok (1981) and Ivanov (1967), which were previously presented in Chapter 7 (Section 7.4.2). Assuming a Poisson’s ratio of 0.37, the radial extents of induced liquefaction and of ground improvement, predicted using the proposed numerical model and the empirical ranges given by Kok (1981) and Ivanov (1967), respectively, are in good agreement. However, for blast loading of saturated sands, Poisson’s ratio is likely to be closer to 0.495 than 0.37. For example, the author calculated the Poisson’s ratio used in the numerical studies of Awad (1990) and Wu (1996) to be approximately 0.46 and 0.496, respectively. Awad’s analyses under predicted the radial extent of induced liquefaction 359

by a factor of about two, similar to the author’s model if 0.495 is assumed for Poisson’s ratio. Although Wu’s predictions were close to those observed in the field, Wu’s model includes both a viscous damping term and a non-linear stress-strain relation. The inclusion of the viscous damping term provides an additional mechanism for energy dissipation beyond the hysteretic mechanism inherent to the non-linear soil behavior. Wu selected the damping coefficients such that the predicted results and field observations were in good agreement (i.e., Wu calibrated his model using existing data such that the model can be used for predicting future events). Although possibly not as fundamentally correct as Wu’s approach, the energy dissipation in the author’s proposed model can be achieved by using a lower Poisson’s ratio than the soil parameters dictate (i.e., using  = 0.37, as opposed to 0.495). 107 106

N1,60 = 5 blws/ft W = 5.5kg z = 12m

105 104

Normalized Energy

103 102

NED

101 Kok (1981)

100 10-1

Ivanov (1967)

-2

10

NEC

10-3 10-4  = 0.495

0.37 0.2

-5

10

10-6

Extent of Predicted Liquefaction

0 5 10 15 Radial Distance from Center of Charge, R, (m)

Figure 8-26. Comparison of the predicted radial extent of induced liquefaction using the numerical model proposed by the author to the predicted radial extent of improvement using empirical guidelines proposed by Ivanov (1967) and Kok (1981).

360

8.6 Higher Order Numerical Modeling for Remedial Ground Densification As discussed in Section 8-2, the reconciliation of results computed using lower and higher order models, as well as favorable comparisons of computed results with field observations, is probably the best approach for verifying and validating numerical models. Alternatively to this approach, in this chapter, the results from proposed first order models were compared directly to empirical expressions and guidelines, without the benefit of comparisons with higher order model predictions. Accordingly, many of the assumptions inherent in the proposed first order models have not been fully validated. In attempt to more fully evaluate the proposed first order models, the author developed both lumped mass and continuum higher order models to compute the spatial distribution of energy dissipation in the soil during treatment. However, the predictive capabilities of the models were limited, with the continuum model having the greatest potential for further development. Due to the assumptions inherent in the higher order models developed by the author (e.g., constitutive model, boundary conditions, source loading), the credibility given to the computed results was less than that given to the proposed first order models. Accordingly, the details of the models are not deemed necessary to report. However, Table 8-4 lists several higher order models for remedial ground densification techniques.

361

Table 8-4. Higher order models for remedial ground densification. Ground Improvement Technique

Model Description

Reference

Vibro-compaction

 Non-linear model, specific details not known

Fellin (2000)

Deep Dynamic Compaction

Explosive Compaction

 1D-Lumped mass model  Single phase  Hyperbolic non-linear constitutive model  2D-Finite element model  Single phase  Mohr-Coulomb elasto-plastic constitutive model

Holeyman (1985)

Pan and Selby (2000)

 3D-Lumped mass model  Single phase  von Mises elasto-plastic constitutive model

Ang (1966)*

 2D-Finite element model  Two phase, fully coupled pore pressure generation  Non-linear cap model

Awad (1990)

 1D-Finite element model  Single phase, uncoupled pore pressure generation  Hyperbolic non-linear constitutive model

Wu (1995, 1996)

*Originally developed for modeling blast wave propagation in soil.

8.7 Summary and Conclusions The interactions between soil and remedial densification systems are very complex. However, based on a series of simplifying assumptions, first order models for the vibratory probe method, deep dynamic compaction, and explosive compaction are

362

proposed. All of the proposed models assume a single phase medium and use an iterative, equivalent linear type approach to account for soil non-linearity. Both the vibro-compaction and explosive compaction models use empirical attenuation relations to estimate the strain induced in the soil during treatment, while the deep dynamic compaction model assumes a 2:1 stress distribution with depth. Once the amplitude of the induced strain is known, the shear modulus and damping degradation curves are used to compute the energy dissipated in the soil as a function of distance from the source. In comparing the model predictions with empirical guidelines for the vibratory probe method, the predicted lateral extent induced liquefaction is slightly less than empirical guidelines for the lateral extent of improvement. An inherent assumption was made in regards to the relation between peak and steady state amplitudes of vibrations, which in turn was used to compute the induced shear strain. However, the under-prediction of the extent of improvement is hypothesized to be related to improvement mechanisms and not due to the assumed attenuation relation. As discussed in Chapter 7, both increased density and increased lateral effective confining pressure contribute to the soil improvement. Accordingly, the lateral extent of improvement should extend beyond the zone of induced liquefaction. In comparing model predictions with empirical expressions for deep dynamic compaction, the predicted depth of liquefaction is in reasonable agreement predicted depth of improvement computed using an empirical expression. One of the most influential model parameters is Poisson’s ratio (), which inherently is a function of drainage conditions for the given load. The predicted depth of liquefaction decreases as  increases. Based on the work of Mayne and Jones (1983) and Pan and Selby (2000),   0.35 is assumed to be reasonable and yields good results. However, further study is required to refine this estimated value. In comparing model predictions with empirical guidelines for explosive compaction, the predicted radial extent of liquefaction for   0.37 is in reasonable agreement predicted radial extent of improvement per empirical guidelines. The assumed value of Poisson’s

363

ratio (i.e.,  = 0.37) is likely smaller than the soil properties, load rate, and drainage conditions dictate. However, use of a smaller than dictated Poisson’s ratio appears to be a simple way to account for energy dissipation mechanisms not explicitly accounted for in the proposed model. Based on the analyses and comparisons presented in this chapter and summarized above, the cumulative energy dissipated in a unit volume of soil, normalized by the initial mean effective confining stress, up to the point of initial liquefaction appears to be comparable for both earthquake type and remedial densification loadings. The implication being that the energy-based Capacity curve derived from earthquake case histories is applicable for use in remedial ground densification design. Additional research is required to verify and/or refine the proposed first order models and to develop design procedures for remedial ground densification programs that incorporate energy-based concepts. The ultimate goal being that such design procedures would lead to improved feasibility assessments of the various remediation techniques for given field conditions and improved initial designs over those that can be achieved by the currently used empirical expressions and guidelines.

364

Chapter 9. Summary and Conclusions 9.1 Restatement of Research Objective The states-of-practice for performing earthquake liquefaction analyses and remedial ground densification designs have evolved relatively independent of each other. This is in spite of the fact that liquefaction is typically induced in saturated sands as part of the remedial ground densification process. The goal of this research is to assess the feasibility of using the vast amount of data collected over the years on earthquake induced liquefaction for remedial ground densification design via energy-based concepts. 9.2 Overview of Research The energy dissipated by frictional mechanisms during the relative movement of sand grains is hypothesized to be directly related to the ability of a soil to resist liquefaction (i.e., Capacity). Assuming a linearized hysteretic model, a “simplified” expression was derived for computing the energy dissipated in the soil during an earthquake (i.e., Demand). Using this expression, the cumulative energy dissipated per unit volume of soil and normalized by the initial mean effective confining stress (i.e., normalized energy demand: NED) was calculated for 126 earthquake case histories for which the occurrence or non-occurrence of liquefaction is known. By plotting the computed NED values as a function of their corresponding SPT penetration resistance, a correlation between the normalized energy capacity of the soil (NEC) and SPT penetration resistance was established by the boundary giving a reasonable separation of the liquefaction / no liquefaction data points. NEC is the cumulative energy dissipated per unit volume of soil up to initial liquefaction, normalized by the initial mean effective confining stress, and the NEC correlation with SPT penetration resistance is referred to as the Capacity curve. The assessment of the feasibility of using the Capacity curve in remedial ground densification design was two fold. First, because earthquake motions and the motions induced in the soil by remedial ground densification techniques differ in amplitude, duration, and frequency content, the dependence of NEC on such things needed to be determined (i.e., if NEC varied as a function of the amplitude, duration, and frequency

365

content of the induced motions, then the NEC Capacity curve derived from earthquake case histories may not apply to remedial ground densification techniques). Towards this end, the calibration parameters for energy-based pore pressure generation models were examined for their dependence on the amplitude of the applied loading. The premise being that if the relationship between dissipated energy and pore pressure generation is independent of the amplitude of loading, then the energy required to generate excess pore pressures equal to the initial effective confining stress should also be independent of the load amplitude. However, no conclusive statement could be made from results of this review. Next, first order numerical models were developed for computing the spatial distribution of the energy dissipated in the soil during treatment using the vibratory probe method, deep dynamic compaction, and explosive compaction. In conjunction with the earthquake Capacity curves, the models were used to predict the spatial extent of induced liquefaction during soil treatment and compared with the predicted spatial extent of improvement using empirical expressions and guidelines. Although the proposed numerical models require further validation, the predicted extent of liquefaction and improvement are in very good agreement, thus giving credence to the feasibility of using the Capacity curve for remedial ground densification design. Although further work is required to develop energy-based remedial densification design procedures, the potential benefits of such procedures are as follows. By using the Capacity curve, the minimum dissipated energy required for successful treatment of the soil can be determined. Because there are physical limits on the magnitude of the energy that can be imparted by a given technique, such an approach may lead to improved feasibility assessments and initial designs of the densification programs. 9.3 Summary of Major Findings 

From the critical review of existing energy-based liquefaction evaluation procedures presented in Chapter 2, it was determined that none of the existing procedures are at the required stage of development for use in remedial densification design. This does not necessarily preclude the use of these

366

procedures for performing earthquake liquefaction evaluations, but such use is cautioned. 

The author proposed a new energy-based excess pore pressure generation model in Chapter 4. The model accurately predicted the excess pore pressures generated in a numerous cyclic laboratory tests on various silt-sand mixtures. The proposed model has a single calibration parameter for which preliminary correlations with relative density of medium- and fine-grained sands were presented.



A new energy-based liquefaction evaluation procedure is presented in Chapter 5. Similar to the stress-based procedure, the proposed energy-based procedure uses a Capacity curve, which was derived from analyzing earthquake case histories. The purpose for developing the energy-based procedure was for use in remedial densification design, which will be discussed subsequently. However, for earthquake analyses, this proposed energy-based liquefaction evaluation procedure offers several advantages over the commonly used stress-based procedure, especially when used in conjunction with total stress site response analyses (e.g., SHAKE). SHAKE is often used in conjunction with the stress-based procedure to compute refined estimates of the amplitude of the shear stresses induced in the soil column. Although such site response analyses give a refined estimate of the amplitude of the induced shear stress, magnitude scaling factors (MSF) are still relied on to account for the duration effects of the earthquake motion (i.e., Demand is a function of both the amplitude and duration of the applied loading). On the contrary, by quantifying Demand in terms of dissipated energy, as is done in the proposed energy-based liquefaction evaluation procedure, integration of the shear stress-strain time histories computed from SHAKE analyses inherently accounts for the amplitude, duration, and frequency content of the design motion.

367



As part of the proposed energy-based liquefaction evaluation procedure, a new procedure was developed for computing number of equivalent cycles (Neqv) for earthquake motions. Using this procedure, a correlation was developed relating Neqv to earthquake magnitude and site-to-source distance, as opposed to earlier correlations, which express Neqv only as a function earthquake magnitude. Aside from the main focus of this thesis, the proposed Neqv correlation was used to derive magnitude scaling factors (MSF) for the stress-based liquefaction evaluation procedure. The newly derived MSF are functions of both earthquake magnitude and site-to-source distance. In comparing the currently used MSF and the proposed MSF, the former may under predict the Demand imposed on the soil at low magnitudes (i.e., M