Development of a New Family of Normalized Modulus ... · normalized modulus reduction and material...

Copyright

by

MEHMET BARIS DARENDELI

2001

darendmb

DEVELOPMENT OF A NEW FAMILY OF NORMALIZED

MODULUS REDUCTION AND MATERIAL DAMPING

CURVES

by

MEHMET BARIS DARENDELI, B.S., M.S.

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

The University of Texas at Austin

August, 2001

Dedicated

To

My Parents,

My Wife and My Daughter

v

Acknowledgements

I would like to thank my supervising professor Dr. Kenneth H. Stokoe, II

for his guidance and support through the course of this study. His passion and

enthusiasm in his work has always inspired me. Our stimulating conversations

have made this study enjoyable.

Dr. Robert B. Gilbert’s assistance and guidance, which have made this

dissertation possible, is gratefully acknowledged. Besides his valuable input to

this work, he has influenced my perception of science and engineering with his

lectures on decision, risk and reliability.

I would also like to thank my dissertation committee members Dr. Jose M.

Roesset, Dr. Ellen M. Rathje, Dr. Alan F. Rauch and Dr. Mark F. Hamilton for

reviewing this dissertation in such a limited time frame and for their valuable

contributions to this work. Thanks are also extended to the rest of the former and

current geotechnical engineering faculty, Dr. Roy E. Olson, Dr. David E. Daniel,

and Dr. Stephen G. Wright for their lectures that broadened my knowledge.

The support from the California Department of Transportation, the

National Science Foundation, the Electric Power Research Institute, and Pacific

Gas and Electric Company is gratefully acknowledged for funding various stages

of the ROSRINE project. I would also like to acknowledge the contributions of

the National Institute of Standards and Technology, the United States Geological

Survey, the Department of Energy, the Westinghouse Savannah River

Corporation, Kajima Corporation, Geovision, Agbabian Associates, Fugro, Inc.,

Earth Mechanics, Inc., S&ME, Inc. in funding the research projects the results of

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which are utilized in this study. Encouragement and guidance from Dr. Clifford

Roblee, Dr. John Schneider, Dr. Walter Silva, Dr. Robert Pyke, Dr. Robert

Nigbor, Dr. David Boore, Prof. Mladen Vucetic and Dr. Richard Lee, who took

part in these research projects, are appreciated.

Thanks to my best friend Cem Akguner for always being there whenever I

needed him, to Dr. Brent L. Rosenblad for trying to teach me how to bat

whenever we overworked, to Dr. Ahmet Yakut for our stimulating card plays and

arguments regarding them that lasted for hours, and to Baris Binici for each and

every five minute coffee break at 100oF. You have kept me sane (although

everyone reading this paragraph will question it a little) for the past seven years.

I would also like to thank the former and current graduate students that I

have worked side by side. I enjoyed each and every day and night that I worked

together with Dr. James A. Bay, Dr. Seon-Keun Hwang, Farn-Yuh Menq, Brian

Moulin, Celestino Valle and Nicola Chiara. Thanks are also extended to other

graduate students of whom I had the pleasure of making acquaintance; Dr. Eric

Liedtke, Dr. Mike Kalinski, Jeffrey Lee, Paul Axtell, Jiun Chen, Cem Topkaya

and many others that I unfortunately omitted. I would also like to thank Teresa

Tice-Boggs and Alicia Zapata for their administrative support, and Frank Wise,

Gonzalo Zapata, Max Trevino and Paul Walters for their technical assistance over

the years.

vii

DEVELOPMENT OF A NEW FAMILY OF NORMALIZED

MODULUS REDUCTION AND MATERIAL DAMPING

CURVES

Publication No._____________

Mehmet Baris Darendeli, Ph.D.

The University of Texas at Austin, 2001

Supervisor: Kenneth H. Stokoe, II

As part of various research projects [including the SRS (Savannah River Site)

Project AA891070, EPRI (Electric Power Research Institute) Project 3302, and

ROSRINE (Resolution of Site Response Issues from the Northridge Earthquake)

Project], numerous geotechnical sites were drilled and sampled. Intact soil

samples over a depth range of several hundred meters were recovered from 20 of

these sites. These soil samples were tested in the laboratory at The University of

Texas at Austin (UTA) to characterize the materials dynamically. The presence of

a database accumulated from testing these intact specimens motivated a re-

evaluation of empirical curves employed in the state of practice. The weaknesses

of empirical curves reported in the literature were identified and the necessity of

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developing an improved set of empirical curves was recognized. This study

focused on developing the empirical framework that can be used to generate

normalized modulus reduction and material damping curves. This framework is

composed of simple equations, which incorporate the key parameters that control

nonlinear soil behavior. The data collected over the past decade at The University

of Texas at Austin are statistically analyzed using First-order, Second-moment

Bayesian Method (FSBM). The effects of various parameters (such as confining

pressure and soil plasticity) on dynamic soil properties are evaluated and

quantified within this framework. One of the most important aspects of this study

is estimating not only the mean values of the empirical curves but also estimating

the uncertainty associated with these values. This study provides the opportunity

to handle uncertainty in the empirical estimates of dynamic soil properties within

the probabilistic seismic hazard analysis framework. A refinement in site-specific

probabilistic seismic hazard assessment is expected to materialize in the near

future by incorporating the results of this study into the state of practice.

ix

TABLE OF CONTENTS

LIST OF TABLES ...............................................................................................xiii

LIST OF FIGURES............................................................................................xviii

CHAPTER 1 INTRODUCTION........................................................................ 1 1.1 Background ........................................................................................... 1 1.2 Dynamic Soil Properties........................................................................ 4 1.3 Ground Response Analysis ................................................................... 8 1.4 Objectives of Research........................................................................ 10 1.5 Organization of Dissertation ............................................................... 11

CHAPTER 2 LABORATORY TESTING EQUIPMENT............................... 13 2.1 Introduction ......................................................................................... 13 2.2 Combined Resonant Column and Torsional Shear Equipment........... 14 2.3 Torsional Resonant Column Test ........................................................ 16 2.4 Cyclic Torsional Shear Test ................................................................ 21 2.5 Summary ............................................................................................. 22

CHAPTER 3 PHYSICAL PROPERTIES OF TEST SPECIMENS ................ 23 3.1 Introduction ......................................................................................... 23 3.2 Undisturbed Soil Specimens from Northern California...................... 25 3.3 Undisturbed Soil Specimens from Southern California...................... 29 3.4 Undisturbed Soil Specimens from South Carolina ............................. 35 3.5 Undisturbed Soil Specimens from Lotung, Taiwan ............................ 38 3.6 Overview of The Database .................................................................. 39 3.7 Summary ............................................................................................. 53

CHAPTER 4 OBSERVED TRENDS IN DYNAMIC SOIL PROPERTIES .. 54 4.1 Introduction ......................................................................................... 54 4.2 Background ......................................................................................... 54 4.3 Nonlinear Dynamic Soil Properties..................................................... 56

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4.4 Effect of Duration of Confinement on Small-Strain Dynamic Soil Properties............................................................................................. 59

4.5 Effect of Effective Confining Pressure ............................................... 61 4.6 Effect of Overconsolidation Ratio....................................................... 70 4.7 Effect of Number of Cycles ................................................................ 74 4.8 Effect of Loading Frequency............................................................... 76 4.9 Effect of Soil Type .............................................................................. 81 4.10 Effect of Sample Disturbance ............................................................. 90 4.11 Summary ........................................................................................... 104

CHAPTER 5 EMPIRICAL RELATIONSHIPS ............................................ 107 5.1 Introduction ....................................................................................... 107 5.2 Hardin and Drnevich (1972) Design Equations ................................ 107 5.3 Empirical Relationships .................................................................... 113 5.4 Summary ........................................................................................... 129

CHAPTER 6 PROPOSED SOIL MODEL .................................................... 131 6.1 Introduction ....................................................................................... 131 6.2 Normalized Modulus Reduction Curve............................................. 132 6.3 Nonlinear Material Damping Curve.................................................. 134 6.4 Parametric Study of The Soil Model................................................. 147 6.5 Summary ........................................................................................... 152

CHAPTER 7 STATISTICAL ANALYSIS OF COLLECTED DATA USING FIRST-ORDER, SECOND-MOMENT BAYESIAN METHOD 154 7.1 Introduction ....................................................................................... 154 7.2 Bayesian Approach ........................................................................... 155 7.3 First-Order, Second-Moment Bayesian Method ............................... 164 7.4 Form of Proposed Equations ............................................................. 172 7.5 Summary ........................................................................................... 179

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CHAPTER 8 STATISTICAL ANALYSIS OF THE RCTS DATA.............. 180 8.1 Introduction ....................................................................................... 180 8.2 Analysis of Subsets of The Data ....................................................... 184 8.3 Analysis of All Credible Data ........................................................... 212 8.4 Summary ........................................................................................... 217

CHAPTER 9 PREDICTING NONLINEAR SOIL BEHAVIOR USING THE CALIBRATED MODEL................................................................... 220 9.1 Introduction ....................................................................................... 220 9.2 Calculation of Reference Strain, Curvature Coefficient, Small-

Strain Material Damping Ratio and the Scaling Coefficient............. 221 9.3 Estimation of Normalized Modulus Reduction and Material

Damping Curves................................................................................ 224 9.4 Effect of Overconsolidation Ratio, Loading Frequency and

Number of Loading Cycles on Nonlinear Soil Behavior .................. 228 9.5 Effect of Confining Pressure on Nonlinear Soil Behavior ................ 234 9.6 Effect of Soil Type on Nonlinear Soil Behavior ............................... 238 9.7 Effects of Confining Pressure and Soil Type on Stress-Strain

Curves................................................................................................ 242 9.8 Summary ........................................................................................... 248

CHAPTER 10 RECOMMENDED NORMALIZED MODULUS REDUCTION AND MATERIAL DAMPING CURVES ......................... 249 10.1 Introduction ....................................................................................... 249 10.2 Effect of PI at a Given Mean Effective Stress .................................. 250 10.3 Effect of Mean Effective Stress on a Soil with Given Plasticity ...... 250 10.4 Impact of Utilizing the Recommended Curves on Earthquake

Response Predictions of Deep Sites .................................................. 250 10.5 Summary ........................................................................................... 272

CHAPTER 11 UNCERTAINTY ASSOCIATED WITH THE MODEL PREDICTIONS.......................................................................................... 273 11.1 Introduction ....................................................................................... 273 11.2 Uncertainty in Nonlinear Soil Behavior............................................ 273

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11.3 Uncertainty in Predicted Ground Motions Due to the Uncertainty in Nonlinear Soil Behavior................................................................ 284

11.4 Summary ........................................................................................... 295

CHAPTER 12 SUMMARY AND CONCLUSIONS....................................... 296 12.1 Summary ........................................................................................... 296 12.2 Conclusions ....................................................................................... 301

APPENDIX A ..................................................................................................... 303

APPENDIX B ..................................................................................................... 306

APPENDIX C ..................................................................................................... 311

APPENDIX D ..................................................................................................... 338

REFERENCES.................................................................................................... 357

VITA ................................................................................................................... 363

xiii

LIST OF TABLES

Table 3.1 Physical properties of soils recovered from Oakland Outer Harbor and test pressures (Hwang, 1997) ..................................... 24

Table 3.2 Physical properties of soils recovered from Treasure Island and test pressures (Hwang and Stokoe, 1993b; and Hwang, 1997).............................................................................................. 25

Table 3.3 Physical properties of soils recovered from San Francisco Airport and test pressures (Hwang, 1997)..................................... 27

Table 3.4 Physical properties of soils recovered from Gilroy and test pressures (Hwang and Stokoe, 1993c; Hwang, 1997; and Stokoe et al., 2001)........................................................................ 27

Table 3.5 Physical properties of soils recovered from Garner Valley and test pressures (Stokoe and Darendeli, 1998) .......................... 28

Table 3.6 Physical properties of soils recovered from San Francisco-Oakland Bay Bridge Site and test pressures (Stokoe et al., 1998d)............................................................................................ 28

Table 3.7 Physical properties of soils recovered from Corralitos and test pressures (Stokoe et al., 2001) ................................................ 28

Table 3.8 Physical properties of soils recovered from Borrego and test pressures (Hwang, 1997)............................................................... 32

Table 3.9 Physical properties of soils recovered from Arleta and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997) ..... 32

Table 3.10 Physical properties of soils recovered from Kagel and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997) ..... 32

Table 3.11 Physical properties of soils recovered from La Cienega and test pressures (Darendeli and Stokoe, 1997; Darendeli, 1997; and Stokoe et al., 1998e) ............................................................... 33

Table 3.12 Physical properties of soils recovered from Newhall and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997) ..... 33

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Table 3.13 Physical properties of soils recovered from Sepulveda V.A. Hospital and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997)............................................................................ 34

Table 3.14 Physical properties of soils recovered from Potrero Canyon and test pressures (Stokoe et al., 1998e) ....................................... 34

Table 3.15 Physical properties of soils recovered from Rinaldi Receiving Station and test pressures (Stokoe et al., 1998e). .......................... 34

Table 3.16 Physical properties of soils recovered from North Palm Springs and test pressures (Stokoe et al., 2001) ............................ 35

Table 3.17 Physical properties of soils recovered from Imperial Valley College and test pressures (Stokoe et al., 2001)............................ 35

Table 3.18 Physical properties of soils recovered from Savannah River Site and test pressures (Hwang, 1997; and Stokoe et al., 1998a)............................................................................................ 37

Table 3.19 Physical properties of soils recovered from Daniel Island and test pressures (Stokoe et al., 1998b). ............................................. 37

Table 3.20 Physical properties of soils recovered from Lotung site and test pressures (Hwang and Stokoe, 1993a; and Hwang, 1997) ..... 39

Table 3.21 Distribution of soil samples according to the sample depth in each geographic region.................................................................. 41

Table 3.22 Distribution of collected according to the isotropic confining pressure in each geographic region ............................................... 42

Table 3.23 Distribution of soil samples according to the Unified Soil Classification System (USCS) designation and sample depth ...... 44

Table 4.1 Parameters that control nonlinear soil behavior and their relative importance in terms of affecting normalized modulus reduction and material damping curves based on general trends observed during the course of this study .......................... 105

Table 5.1 Parameters that control nonlinear soil behavior and their relative importance in terms of affecting shear modulus and material damping (Hardin and Drnevich, 1972b) ....................... 108

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Table 7.1 Prior information provided in the discrete example.................... 160

Table 7.2 Prior information regarding the model parameters in the FSBM example............................................................................ 165

Table 7.3 Prior covariance structure of the model parameters in the FSBM example............................................................................ 165

Table 7.4 Data used to calibrate the model parameters in the FSBM example ....................................................................................... 166

Table 7.5 Comparison of the prior and posterior information regarding the model parameters in the FSBM example .............................. 169

Table 7.6 Posterior covariance structure of the model parameters in the FSBM example............................................................................ 170

Table 7.7 Posterior covariance structure of the model parameters in the FSBM example............................................................................ 171

Table 8.1 Distribution of specimens with soil type and geographic location ........................................................................................ 181

Table 8.2 Distribution of specimens by soil group and geographic location ........................................................................................ 181

Table 8.3 Distribution of specimens with soil type and geographic location for the updated database ................................................ 182

Table 8.4 Distribution of specimens by soil group and geographic location for the updated database ................................................ 183

Table 8.5 Prior mean values and variances of the model parameters ......... 185

Table 8.6 Updated mean values and variances of the model parameters for the soils from Northern California......................................... 186

Table 8.7 Updated mean values and variances of the model parameters for the soils from Southern California......................................... 191

Table 8.8 Updated mean values and variances of the model parameters for the soils from South Carolina ................................................ 194

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Table 8.9 Updated mean values and variances of the model parameters for the South Carolina soil groups affected by change in the contents of the database............................................................... 198

Table 8.10 Updated mean values and variances of the model parameters for the soils from Lotung, Taiwan............................................... 200

Table 8.11 Updated mean values and variances of the model parameters for the four soil groups ................................................................ 207

Table 8.12 Comparison of the prior and updated mean values and variances of the model parameters for all the credible data ........ 214

Table 8.13 Covariance structure of the updated model parameters for all the credible data .......................................................................... 218

Table 10.1 Effect of PI on normalized modulus reduction curve: σo’ = 0.25 atm....................................................................................... 252

Table 10.2 Effect of PI on material damping curve: σo’ = 0.25 atm............. 252

Table 10.3 Effect of PI on normalized modulus reduction curve: σo’ = 1.0 atm......................................................................................... 254

Table 10.4 Effect of PI on material damping curve: σo’ = 1.0 atm............... 254

Table 10.5 Effect of PI on normalized modulus reduction curve: σo’ = 4.0 atm......................................................................................... 256

Table 10.6 Effect of PI on material damping curve: σo’ = 4.0 atm............... 256

Table 10.7 Effect of PI on normalized modulus reduction curve: σo’ = 16 atm............................................................................................... 258

Table 10.8 Effect of PI on material damping curve: σo’ = 16 atm................ 258

Table 10.9 Effect of σo’ on normalized modulus reduction curve: PI = 0 %.................................................................................................. 260

Table 10.10 Effect of σo’ on material damping curve: PI = 0 % ................... 260

xvii

Table 10.11 Effect of σo’ on normalized modulus reduction curve: PI = 15 %............................................................................................. 262

Table 10.12 Effect of σo’ on material damping curve: PI = 15 % ................. 262





Table 10.17 Effect of σo’ on normalized modulus reduction curve: PI = 100 %........................................................................................... 268

Table 10.18 Effect of σo’ on material damping curve: PI = 100 % ............... 268

Table 11.1 Predicted mean values and standard deviations accounting for uncertainty in the values of model parameters and variability due to modeled uncertainty ......................................................... 275

Table 11.2 Predicted covariance structure accounting for uncertainty in the values of model parameters and variability due to modeled uncertainty .................................................................... 276

Table 11.3 Predicted mean values and standard deviations accounting only for variability due to modeled uncertainty .......................... 277

Table 11.4 Predicted covariance structure accounting only for variability due to modeled uncertainty ......................................................... 278

Table 12.1 Parameters that control nonlinear soil behavior and their relative importance in terms of affecting normalized modulus reduction and material damping curves based on general trends observed during the course of this study .......................... 297

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LIST OF FIGURES

Figure 1.1 Evaluation of ground motion at a geotechnical site based on vertically propagating shear waves between the bedrock and ground surface ................................................................................. 2

Figure 1.2 Fourier amplitude of (a) the ground motion as a result of (b) the bedrock motion at the geotechnical site shown in Figure 1.1.................................................................................................... 3

Figure 1.3 Representation of a soil deposit in terms of dynamic soil properties in geotechnical earthquake engineering ......................... 4

Figure 1.4 Nonlinear stress-strain curve of soils and variation of secant shear modulus with shearing strain amplitude ................................ 5

Figure 1.5 Estimation of shear modulus and material damping ratio during cyclic loading....................................................................... 6

Figure 1.6 (a) Nonlinear shear modulus and (b) normalized modulus reduction curves .............................................................................. 7

Figure 1.7 Nonlinear material damping ratio curve.......................................... 7

Figure 1.8 Field curves representing nonlinear soil behavior........................... 9

Figure 2.1 Simplified diagram of the RCTS device (from Stokoe et al., 1999).............................................................................................. 14

Figure 2.2 Simplified cross-sectional view of the confining system (from Hwang, 1997) ...................................................................... 15

Figure 2.3 General Configuration of RCTS Equipment (after Hwang, 1997).............................................................................................. 17

Figure 2.4 Frequency response curve measured in the RC test (from Stokoe et al., 1999)........................................................................ 18

Figure 2.5 Material damping measurement in the RC test using the half-power bandwidth (from Stokoe et al., 1999)................................. 18

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Figure 2.6 Material damping measurement in the RC test using the free-vibration decay curve (from Stokoe et al., 1999).......................... 19

Figure 2.7 Calculation of shear modulus and material damping ratio in the TS test...................................................................................... 21

Figure 3.1 Map of Northern California showing the locations of the geotechnical sites in this area ........................................................ 26

Figure 3.2 Map of Southern California showing the locations of the three geotechnical sites outside the Los Angeles area .................. 30

Figure 3.3 Map of Los Angeles showing the locations of the seven geotechnical sites in this area ........................................................ 31

Figure 3.4 Map of South Carolina showing the locations of the geotechnical sites in this area ........................................................ 36

Figure 3.5 Map of Taiwan showing the location of Lotung site .................... 38

Figure 3.6 Distribution of soil samples with geographic region .................... 40

Figure 3.7 Distribution of the number of geotechnical sites with geographic region.......................................................................... 40

Figure 3.8 Distribution of soil samples according to the sample depth.......... 41

Figure 3.9 Distribution of confining pressures at which nonlinear measurements were performed...................................................... 42

Figure 3.10 Distribution of soil samples according to soil type as classified by the Unified Soil Classification System (USCS)....... 43

Figure 3.11 Distribution of soil samples according to soil plasticity in terms of the plasticity index, PI..................................................... 44

Figure 3.12 Distribution of soil samples according to total unit weight .......... 46

Figure 3.13 Distribution of soil samples according to dry unit weight ............ 46

Figure 3.14 Distribution of soil samples according to water content ............... 47

Figure 3.15 Distribution of soil samples according to void ratio ..................... 47

xx

Figure 3.16 Variation of dry unit weight with depth of (a) fine grained and (b) coarse grained soils included in this study........................ 48

Figure 3.17 Variation of water content with depth of (a) fine grained and (b) coarse grained soils included in this study .............................. 49

Figure 3.18 Variation of void ratio with depth of (a) fine grained and (b) coarse grained soils included in this study .................................... 50

Figure 3.19 Distribution of soil samples according to estimated overconsolidation ratio.................................................................. 51

Figure 3.20 Variation of estimated overconsolidation ratio with depth of (a) fine grained and (b) coarse grained soils included in this study .............................................................................................. 52

Figure 4.1 Linear elastic, nonlinear elastic and plastic strain ranges on (a) normalized modulus reduction and (b) material damping curves ............................................................................................ 57

Figure 4.2 Variation of (a) low-amplitude shear modulus, (b) low-amplitude material damping ratio, and (c) void ratio with magnitude and duration of isotropic confining pressure............... 60

Figure 4.3 Variation of (a) low-amplitude shear modulus, (b) low-amplitude material damping ratio, and (c) void ratio with effective isotropic confining pressure ........................................... 62

Figure 4.4 The effect of confining pressure on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as measured in the torsional resonant column ....................................................... 65

Figure 4.5 The effect of confining pressure on normalized modulus reduction curve (a) for soils with moderate plasticity, and (b) for non-plastic soils evaluated as part of the ROSRINE study (after Stokoe et al., 1999) .............................................................. 67

Figure 4.6 The effect of confining pressure on (a) normalized modulus reduction and (b) material damping curves of silty sands evaluated as part of the ROSRINE study (after Darendeli et al., 2001)........................................................................................ 68

xxi

Figure 4.7 Impact on nonlinear site response of accounting for the effect of confining pressure on dynamic soil properties (after Darendeli et al., 2001) ................................................................... 70

Figure 4.8 The effect of overconsolidation ratio on the variation of (a) shear modulus, (b) material damping ratio, and (c) void ratio with effective isotropic confining pressure as measured in the torsional resonant column ............................................................. 71

Figure 4.9 The effect of overconsolidation ratio on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as measured in the torsional resonant column ................................... 72

Figure 4.10 The effect of number of loading cycles on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as determined in the combined RCTS testing ................................... 75

Figure 4.11 The effect of loading frequency on (a) low-amplitude shear modulus, and (b) low-amplitude material damping ratio as determined in the combined RCTS testing ................................... 77

Figure 4.12 Comparison of the effect of loading frequency on low-amplitude shear modulus and low-amplitude material damping ratio (from Stokoe and Santamarina, 2000) ................... 78

Figure 4.13 The effect of loading frequency on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as determined in the combined RCTS testing ...................................................... 80

Figure 4.14 The effect of soil type on the variation of (a) low-amplitude shear modulus, and (b) low-amplitude material damping ratio with effective isotropic confining pressure as determined in the combined RCTS testing........................................................... 82

Figure 4.15 The effect of soil type on the variation of low-amplitude shear modulus with loading frequency as determined in the combined RCTS testing ................................................................ 84

xxii

Figure 4.16 The effect of soil type on the variation of low-amplitude material damping ratio with loading frequency as determined in the combined RCTS testing ...................................................... 85

Figure 4.17 The effect of soil type on the normalized modulus reduction curve as measured in the torsional resonant column..................... 86

Figure 4.18 The effect of soil type on the material damping curve determined at (a) N ~ 1000 cycles, (b) N = 1 cycle, and (c) N = 10 cycles from combined RCTS testing .................................... 87

Figure 4.19 The effect of soil type on normalized modulus reduction and material damping curves (after Stokoe et al., 1999) ..................... 88

Figure 4.20 Comparison of field and laboratory measurements of shear wave velocity at the La Cienega site in the ROSRINE project..... 91

Figure 4.21 Variation of sampling disturbance expressed in terms of Vs,

lab/Vs, field and Gmax, lab/Gmax, field with the in-situ shear wave velocity .......................................................................................... 93

Figure 4.22 Comparison of laboratory and field measurements of small strain material damping ratio (from Stokoe et al., 1999) .............. 95

Figure 4.23 Comparison of nonlinear soil properties back-calculated from the free-field downhole accelerations with the laboratory measurements (from Zeghal et al., 1995)...................................... 96

Figure 4.24 Comparison of the variation of (a) low-amplitude shear modulus, (b) low-amplitude material damping ratio, and (c) void ratio with effective isotropic confining pressure of intact (undisturbed) and reconstituted (remolded) specimens ................ 99

Figure 4.25 Comparison of the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain of intact (undisturbed) and reconstituted (remolded) specimens ................................................................. 100

Figure 4.26 Comparison of the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain measured using various equipment on companion soil samples (from Stokoe et al., 1999) .................... 102

xxiii

Figure 5.1 Hyperbolic soil model proposed by Hardin and Drnevich (1972b) ........................................................................................ 110

Figure 5.2 The normalized modulus reduction and material damping curves estimated based on the hyperbolic model ........................ 112

Figure 5.3 The effect of confining pressure on normalized modulus reduction curve for Toyoura Sand (Iwasaki et al., 1978)............ 114

Figure 5.4 The effect of confining pressure on (a) normalized modulus reduction, and (b) material damping curves for Toyoura Sand (Kokusho, 1980).......................................................................... 115

Figure 5.5 The effect of confining pressure on (a) normalized modulus reduction, and (b) material damping curves for non-plastic soils (Ni, 1987) ............................................................................ 116

Figure 5.6 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Seed et al. (1986).......................... 118

Figure 5.7 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Sun et al. (1988) for soils with plasticity ...................................................................................... 119

Figure 5.8 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Idriss (1990) ................................. 121

Figure 5.9 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Vucetic and Dobry (1991)............ 122

Figure 5.10 The effect of confining pressure on (a) normalized modulus reduction, and (b) material damping curves for non-plastic soils (Ishibashi and Zhang, 1993) ............................................... 124

Figure 5.11 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Ishibashi and Zhang (1993).......... 125

Figure 5.12 Variation in empirical (a) normalized modulus reduction, and (b) material damping curves with depth (EPRI, 1993c).............. 127

Figure 5.13 Variation in empirical (a) normalized modulus reduction, and (b) material damping curves with soil type (EPRI, 1993c)......... 128

xxiv

Figure 6.1 Normalized modulus reduction curve (of a silty sand at 1 atm effective confining pressure) represented using a modified hyperbolic model......................................................................... 133

Figure 6.2 Stress-strain curve (of a silty sand at 1 atm effective confining pressure) estimated based on a modified reference strain model ................................................................................. 135

Figure 6.3 Hysteresis loop estimated by modeling stress-strain reversals for two-way cyclic loading according to Masing behavior......... 137

Figure 6.4 Calculation of damping ratio utilizing a hysteresis loop............. 138

Figure 6.5 Variations of c1, c2 and c3 with curvature coefficient, a.............. 141

Figure 6.6 Damping curve estimated based on Masing behavior................. 143

Figure 6.7 Effect of high-amplitude cycling on low-amplitude shear modulus and material damping ratio (from Stokoe and Lodde, 1978) ............................................................................... 144

Figure 6.8 Comparison of the variation in F with shearing strain for different values of p..................................................................... 145

Figure 6.9 (a) Damping curve estimated based on Masing behavior, (b) adjusted curve using the scaling coefficient, and (c) shifted curve using the small-strain material damping ratio ................... 146

Figure 6.10 Effect of reference strain on (a) normalized modulus reduction, (b) stress-strain, and (c) material damping curves ..... 148

Figure 6.11 Effect of the curvature coefficient on the normalized modulus reduction curve............................................................................ 149

Figure 6.12 Effect of the curvature coefficient on the stress-strain curve (a) at small and intermediate strains, and (b) at high strains....... 149

Figure 6.13 Effect of the curvature coefficient on the material damping curve ............................................................................................ 150

Figure 6.14 Effect of Dmin on the material damping curve............................. 151

Figure 6.15 The effect of scaling coefficient on material damping curve...... 152

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Figure 7.1 Prior probability mass function for the discrete example ........... 159

Figure 7.2 Posterior probability mass function for the discrete example ..... 161

Figure 7.3 Imaginary correlation between model parameters upon updating prior information based on limited number of observations................................................................................. 170

Figure 7.4 Variation of standard deviation with normalized shear modulus ....................................................................................... 176

Figure 7.5 Standard deviation modeled for normalized modulus reduction curve............................................................................ 177

Figure 7.6 Variation of standard deviation with material damping ratio ..... 178

Figure 7.7 Standard deviation modeled for material damping curve ........... 178

Figure 8.1 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands from Northern California...................................... 188

Figure 8.2 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from Northern California....................... 188

Figure 8.3 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from Northern California ............................................................ 189

Figure 8.4 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from Northern California ............................................................ 189

Figure 8.5 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands from Southern California...................................... 192

Figure 8.6 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from Southern California....................... 192

xxvi

Figure 8.7 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from Southern California ............................................................ 193

Figure 8.8 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from Southern California ............................................................ 193

Figure 8.9 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands from South Carolina ............................................. 195

Figure 8.10 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from South Carolina .............................. 195

Figure 8.11 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from South Carolina .................................................................... 196

Figure 8.12 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from South Carolina .................................................................... 196

Figure 8.13 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from South Carolina (After Discarding Specimens UT-39-G and UT-39-M) ........................ 199

Figure 8.14 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from South Carolina (After Discarding Specimens UT-39-O and UT-39-S)............................................................................... 199

Figure 8.15 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from Lotung, Taiwan............................. 201

Figure 8.16 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from Lotung, Taiwan................................................................... 201

xxvii

Figure 8.17 (a) Normalized modulus reduction and (b) material damping curves estimated for a nonplastic silty sand using updated mean values of model parameters calibrated at different geographic locations.................................................................... 203

Figure 8.18 (a) Normalized modulus reduction and (b) material damping curves estimated for a moderate plasticity silt using updated mean values of model parameters calibrated at different geographic locations.................................................................... 204

Figure 8.19 (a) Normalized modulus reduction and (b) material damping curves estimated for a moderate plasticity clay using updated mean values of model parameters calibrated at different geographic locations.................................................................... 205

Figure 8.20 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands ............................................................................... 208

Figure 8.21 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content ................................................................ 208

Figure 8.22 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts ..... 209

Figure 8.23 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays ... 209

Figure 8.24 (a) Normalized modulus reduction and (b) material damping curves estimated using updated mean values of model parameters calibrated for different soil groups ........................... 211

Figure 8.25 All credible (a) normalized modulus data from the resonant column tests, and (b) material damping data from the resonant column and torsional shear tests utilized to calibrate the model parameters. ................................................................. 213

Figure 8.26 Comparisons of the measured and predicted values of normalized modulus for all credible data .................................... 215

Figure 8.27 Comparisons of the measured and predicted values of material damping for all credible data......................................... 216

xxviii

Figure 9.1 Estimation of reference strain for given values of PI, OCR and in-situ mean effective stress ................................................. 223

Figure 9.2 Estimation of scaling coefficient for a given value of number of loading cycles.......................................................................... 223

Figure 9.3 Estimation of small-strain material damping ratio for given values of PI, OCR, in-situ mean effective stress and loading frequency..................................................................................... 225

Figure 9.4 Estimated (a) normalized modulus reduction and (b) material damping curves for the soil type and loading conditions discussed in Section 9.2 .............................................................. 227

Figure 9.5 Effect of overconsolidation ratio on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model .......................................................................... 229

Figure 9.6 Effect of loading frequency on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model .......................................................................... 231

Figure 9.7 Effect of number of loading cycles on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model ............................................................... 232

Figure 9.8 Comparison of (a) normalized modulus reduction and (b) material damping curves predicted for resonant column and torsional shear tests ..................................................................... 233

Figure 9.9 Effect of confining pressure on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model .......................................................................... 235

Figure 9.10 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed for sands by Seed et al. (1986) .......... 236

Figure 9.11 Comparison of the effect of confining pressure on nonlinear soil behavior of sand (PI = 0 %) predicted by the calibrated model and empirical curves proposed for sands by Seed et al. (1986) .......................................................................................... 237

xxix

Figure 9.12 Effect of soil plasticity on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model ........................................................................................... 239

Figure 9.13 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Vucetic and Dobry (1991)............ 240

Figure 9.14 Comparison of the effect of soil plasticity on nonlinear soil behavior predicted by the calibrated model and empirical curves proposed by Vucetic and Dobry (1991)........................... 241

Figure 9.15 Comparison of the measured in-situ shear wave velocities and values predicted using Equation 9.4..................................... 244

Figure 9.16 Effect of confining pressure on stress-strain curve predicted by the calibrated model for shearing strains ranging (a) from γ = 0 to 1 % and (b) from γ = 0 to 0.01 %................................... 245

Figure 9.17 Effect of soil plasticity on stress-strain curve predicted by the calibrated model for shearing strains ranging (a) from γ = 0 to 1 % and (b) from γ = 0 to 0.01 % ................................................ 246

Figure 9.18 Comparison of the stress-strain curves of a sand and a moderate plasticity clay based on the calibrated model for shearing strains ranging (a) from γ = 0 to 1 % and (b) from γ = 0 to 0.01 % ............................................................................... 247

Figure 10.1 Effect of PI on (a) normalized modulus reduction and (b) material damping curves at 0.25 atm confining pressure............ 251

Figure 10.2 Effect of PI on (a) normalized modulus reduction and (b) material damping curves at 1.0 atm confining pressure.............. 253

Figure 10.3 Effect of PI on (a) normalized modulus reduction and (b) material damping curves at 4.0 atm confining pressure.............. 255

Figure 10.4 Effect of PI on (a) normalized modulus reduction and (b) material damping curves at 16 atm confining pressure............... 257

Figure 10.5 Effect of mean effective stress on (a) normalized modulus reduction and (b) material damping curves of a nonplastic soil ............................................................................................... 259

xxx

Figure 10.6 Effect of mean effective stress on (a) normalized modulus reduction and (b) material damping curves of a soil with PI = 15 %............................................................................................. 261



Figure 10.9 Effect of mean effective stress on (a) normalized modulus reduction and (b) material damping curves of a soil with PI = 100 %........................................................................................... 267

Figure 10.10 Shear wave velocity profile assumed for the 100-m thick silty sand deposit ................................................................................. 269

Figure 10.11 An example of utilizing the recommended normalized modulus reduction and material damping curves and its impact on estimated nonlinear site response ............................... 271

Figure 11.1 Mean values and standard deviations associated with the point estimates of (a) normalized modulus reduction and (b) material damping curves ............................................................. 280

Figure 11.2 Comparison of the correlated random realization of (a) normalized modulus reduction and (b) material damping curves relative to the mean curves and one standard deviation ranges shown in Figure 11.1 ....................................................... 283

Figure 11.3 Comparison of spectral accelerations calculated using perfectly correlated soil layers with µ, µ+σ and µ−σ normalized modulus reduction and material damping curves..... 286

Figure 11.4 Comparison of spectral accelerations calculated using perfectly correlated soil layers with 1) µ curves, 2) +σ normalized modulus reduction and −σ material damping curves, and 3) −σ normalized modulus reduction and +σ material damping curves........................................................ 288

xxxi

Figure 11.5 Fifty realizations of spectral acceleration computed using completely uncorrelated soil layers with randomly generated normalized modulus reduction and material damping curves..... 290

Figure 11.6 Histograms of spectral accelerations from fifty realizations presented in Figure 11.5 (a) at 0.1 sec and (b) at 0.3 sec ............ 291

Figure 11.7 Histograms of spectral accelerations from fifty realizations presented in Figure 11.5 (a) at 1 sec and (b) at 3 sec .................. 292

Figure 11.8 Distribution of fifty realizations of spectral acceleration presented in Figure 11.5 .............................................................. 293

Figure 11.9 Comparison of the spectral accelerations from the fifty realizations with the results computed utilizing mean normalized modulus reduction and material damping curves..... 294

Figure 12.1 Comparison of the effect of confining pressure on nonlinear soil behavior of sand (PI = 0 %) predicted by the calibrated model and empirical curves proposed for sands by Seed et al. (1986) .......................................................................................... 299

Figure 12.2 Comparison of the effect of soil plasticity on nonlinear soil behavior predicted by the calibrated model and empirical curves proposed by Vucetic and Dobry (1991)........................... 300

Figure 12.3 Mean values and standard deviations associated with the point estimates of (a) normalized modulus reduction and (b) material damping curves ............................................................. 302

1

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

In earthquake engineering, the energy released during an earthquake is

represented by stress waves propagating through the bedrock and surfacing at the

site of interest. In terms of the geotechnical characteristics of the site, the site is

typically modeled as a series of horizontal layers with varying properties. In most

cases, the site is represented by softer soils close to the surface and stiffer soils at

depth. The increase in stiffness with depth is due to the older age of deeper

material and the confining effect of the overburden. Because of the progressive

increase in stiffness with depth, stress waves coming from depth often surface in a

propagation direction that is almost vertical.

Often times, an earthquake analysis includes predicting the dynamic

response of a structure at the geotechnical site. Since structures are always

designed with a factor of safety to support a static load (its self weight and the live

load) as a result of 1g vertical acceleration, the vertical component of the ground

motion does not generally have as much an impact on earthquake resistant design

as the horizontal component for which less precaution is often taken in the static

design.

With vertically propagating shear waves and a higher susceptibility of

structures to horizontal motions, the ground motion in many earthquake problems

is simply modeled as horizontal shaking due to vertically propagating shear

2

waves. In such a model, the soil deposit acts like a filter that amplifies energy at

some frequencies while attenuating it at others. Therefore, the estimated ground

motion is a function of the earthquake event and the local soil conditions as

shown in Figure 1.1. Two acceleration-time records are presented in this figure.

One of these is the bedrock motion and the second is the ground motion estimated

based on the bedrock motion and characteristics of the soil deposit.

BEDROCK

SOIL LAYER 1

SOIL LAYER 2

SOIL LAYER ..

SOIL LAYER n

übedrock

üground

Time, sec

Bedrock Acceleration,

g

Ground Acceleration,

g

Time, sec

-0.5

0.0

0.5

6050403020100

-0.5

0.0

0.5

6050403020100 BEDROCK

SOIL LAYER 1

SOIL LAYER 2

SOIL LAYER ..

SOIL LAYER n

übedrock

üground

BEDROCK

SOIL LAYER 1

SOIL LAYER 2

SOIL LAYER ..

SOIL LAYER n

übedrock

üground

Time, sec


g


g

Time, sec

-0.5

0.0

0.5

6050403020100

-0.5

0.0

0.5

6050403020100 Time, sec


g


g

Time, sec

-0.5

0.0

0.5

6050403020100

-0.5

0.0

0.5

6050403020100

Figure 1.1 Evaluation of ground motion at a geotechnical site based on vertically propagating shear waves between the bedrock and ground surface

The filtering effect of the soil deposit is demonstrated in Figure 1.2 by

looking at the Fourier amplitude spectra of the two acceleration records. In this

figure, the acceleration components at different frequencies are shown for the

motions at the bedrock and ground surface. In this case, the low-frequency

motions (below 3 Hz) are amplified significantly. On the other hand, the high-

3

frequency motions are slightly attenuated. This effect can also be observed from

the comparison of the time records presented in Figure 1.1. Different cycles can

more easily be identified in the ground motion time record than in the bedrock

record.

0.010

0.008

0.006

0.004

0.002

0.000

FourierAmplitude,

g * sec

(a)

0.010

0.008

0.006

0.004

0.002

0.000

Fourier

1086420

Frequency, Hz

Amplitude,g * sec

(b)

Figure 1.2 Fourier amplitude of (a) the ground motion as a result of (b) the bedrock motion at the geotechnical site shown in Figure 1.1

4

1.2 DYNAMIC SOIL PROPERTIES

As discussed above, to analyze the response of structures during an

earthquake, it is necessary to characterize the ground motion underneath the

structure caused by the earthquake. Some of the most important ground motion

parameters are amplitude of motion (e.g., peak acceleration, peak velocity and

peak displacement), frequency content (e.g., Fourier spectra, response spectra,

predominant period, bandwidth) and duration. These parameters are primarily

affected by three factors: 1. source effects or the characteristics of the earthquake

(such as amount of energy released and type of faulting), 2. path effects (the

distance from the point of energy release to the site), and 3. site effects (such as

characteristics of the soil deposit, topography and other near-surface features).

This study focuses on characterization of the soil deposit. The properties that

typically need to be characterized are shear modulus, G, and material damping

ratio, D, as presented in Figure 1.3.

Shear Modulus, G Material

Damping Ratio, D

≈SOIL DEPOSIT

BEDROCK

Shear Modulus, G Material

Damping Ratio, D

≈SOIL DEPOSIT

BEDROCK

SOIL DEPOSIT

BEDROCK

Figure 1.3 Representation of a soil deposit in terms of dynamic soil properties in geotechnical earthquake engineering

5

Shear modulus, G, represents the shear stiffness of the soil. It is essentially

the slope of the relationship between shear stress (τ) and shearing strain (γ).

Because of the nonlinear nature of the stress-strain curve of soils, shear modulus

of soils change with strain amplitude as shown in Figure 1.4. The secant shear

modulus can also be approximated for the case of dynamic loading over a cycle of

loading at a given strain amplitude as shown in Figure 1.5. The stress-strain path

illustrated in this figure is called a hysteresis loop. The slope of the line that

connects the end points of the hysteresis loop represents the “average” shear

stiffness of the soil, hence the secant shear modulus.

1G1

γ1 γ2

1G2

ShearStress, τ

ShearingStrain, γ

1G1

γ1 γ2

1G2

ShearStress, τ

ShearingStrain, γ

Figure 1.4 Nonlinear stress-strain curve of soils and variation of secant shear modulus with shearing strain amplitude

6

1GShear

Stress, τ

Shearing Strain, γ

G = τ / γD = AL / (4 π AT)

AL

AT

1GShear

Stress, τ

Shearing Strain, γ

G = τ / γD = AL / (4 π AT)

AL

AT

Figure 1.5 Estimation of shear modulus and material damping ratio during cyclic loading

Material damping ratio, D, is a measure of the proportion of dissipated

energy to the maximum retained strain energy during each cycle at a given strain

amplitude as shown in Figure 1.5. The energy dissipated over a loading cycle is

represented by the gray area within the hysteresis loop (AL), and the maximum

retained strain energy is represented by the triangular area (AT) that is calculated

using peak shear stress and peak shearing strain. Material damping ratio is a result

of friction between soil particles, strain rate effects and nonlinearity of the stress-

strain relationship in soils.

As presented in Figure 1.4, soils exhibit nonlinear behavior in shear. The

secant shear modulus decreases with increasing strain amplitude as shown in

Figure 1.6a. Shear modulus at small strains, at which soil behavior is linear, is

referred to as small-strain shear modulus, Gmax. The relationship between shear

7

modulus and strain amplitude is typically characterized by a normalized modulus

reduction curve as shown in Figure 1.6b.

Gmax

120

80

40

00.001 0.01 0.1 1

Shearing Strain, γ , %

G,MPa

1.0

0.5

00.001 0.01 0.1 1


GGmax

(a) (b)

Gmax

120

80

40

00.001 0.01 0.1 1


G,MPa

1.0

0.5

00.001 0.01 0.1 1


GGmaxGmax

120

80

40

00.001 0.01 0.1 1


G,MPa Gmax

120

80

40

00.001 0.01 0.1 1


G,MPa

1.0

0.5

00.001 0.01 0.1 1


GGmax

1.0

0.5

00.001 0.01 0.1 1


GGmax

(a) (b)

Figure 1.6 (a) Nonlinear shear modulus and (b) normalized modulus reduction curves

The nonlinearity in the stress-strain relationship results in an increase in

energy dissipation and, therefore, an increase in material damping ratio with

increasing strain amplitude as presented in Figure 1.7. Material damping ratio at

small strains (in the linear range) is referred to as small-strain material damping

ratio, Dmin, herein.

DminD,%

16

8

00.001 0.01 0.1 1


DminD,%

16

8

00.001 0.01 0.1 1


Figure 1.7 Nonlinear material damping ratio curve

8

1.3 GROUND RESPONSE ANALYSIS

In analyzing ground motions due to small vibrations, soil behavior is

assumed to be linear. Each soil layer is assigned a shear modulus and a material

damping ratio. Since a horizontally layered system is being modeled, the task of

ground response analysis is reduced to a simple 1-D wave propagation problem

that has a closed-form solution (Kramer, 1996).

On the other hand, dynamic soil properties can be extremely nonlinear

when ground motions are caused by large vibrations (such as design level

earthquakes). As a result, the change in shear modulus and material damping ratio

with shearing strain amplitude must be accounted for in ground response analysis.

The linear solution, which is applicable for small vibration levels, can be modified

to overcome this problem.

One approach to handling nonlinear soil behavior due to shaking during a

design level event is to perform linear analyses with dynamic soil properties that

are iterated in a manner consistent with an “effective” shearing strain induced in

the soil layer (Schnabel et al., 1972; and EduPro, 1998). This iterative approach is

called equivalent linear analysis.

The effective shearing strain is defined as a certain portion of the

maximum strain amplitude throughout the time history. The ratio of effective

shearing strain to maximum strain amplitude is typically related to the magnitude

of the earthquake event or the characteristics of the acceleration-time record

employed in the analysis. When a design level earthquake is analyzed, the ratio of

effective to maximum shearing strain is typically about 0.6.

9

The state of practice in equivalent linear analysis often employs empirical

normalized modulus reduction and a material damping curves. These empirical

curves are developed based on laboratory studies performed over the past three

decades.

The empirical normalized modulus reduction curve is scaled using an

estimate of the small-strain shear modulus, Gmax. The small-strain shear modulus

can be calculated using shear wave velocity, Vs, from in-situ seismic

measurements and mass density, ρ.

Gmax = ρ * Vs2 (1.1)

The curve calculated by scaling the empirical normalized modulus

reduction curve is called the field shear modulus curve (Figure 1.8). Since

material damping ratio can not be estimated accurately in-situ, the field material

damping curve is assumed to be identical to the empirical material damping curve

as shown in Figure 1.8.

Dfield = Dempirical

D,%

16

8

00.001 0.01 0.1 1


150

100

50

00.001 0.01 0.1 1


G,MPa

Gfield = Gmax, field *empirical

( )GGmax

Gmax, field

Dfield = Dempirical

D,%

16

8

00.001 0.01 0.1 1


Dfield = Dempirical

D,%

16

8

00.001 0.01 0.1 1


150

100

50

00.001 0.01 0.1 1


G,MPa


( )GGmax

Gmax, field150

100

50

00.001 0.01 0.1 1


G,MPa


( )GGmax

Gmax, field

Figure 1.8 Field curves representing nonlinear soil behavior

10

1.4 OBJECTIVES OF RESEARCH

As part of various research projects [including the SRS (Savannah River

Site) Project AA891070, EPRI (Electric Power Research Institute) Project 3302,

and ROSRINE (Resolution of Site Response Issues from the Northridge

Earthquake) Project] numerous sites were drilled and sampled. Intact soil samples

over a depth range of several hundred meters were recovered from 20 of these

sites. These soil samples were tested in the soil dynamics laboratory at The

University of Texas at Austin (UTA) to characterize the materials.

The presence of a database accumulated from testing these intact

specimens motivated a re-evaluation of empirical curves often employed in

seismic site response analyses. The weaknesses of empirical curves reported in

the literature were recognized and the necessity of developing an improved set of

empirical curves was acknowledged.

This study focuses on generating an improved set of empirical curves that

can be represented in the form of a set of simple equations. The data collected

over the past decade at The University of Texas at Austin are statistically

analyzed using the First-order, Second-moment Bayesian Method (FSBM). The

effects of various parameters (such as confining pressure and soil plasticity) on

dynamic soil properties are evaluated and quantified within this framework.

One of the most important aspects of this study is estimating not only the

mean values of the empirical curves but also the uncertainty associated with these

values. The handling of uncertainty in the empirical estimates of dynamic soil

11

properties is expected to result in a refinement of probabilistic seismic hazard

analysis.

1.5 ORGANIZATION OF DISSERTATION

A general overview of the dynamic laboratory test equipment used to

evaluate the nonlinear soil properties is presented in Chapter Two along with a

brief review of the theory upon which the laboratory testing is founded.

Information regarding the soil samples analyzed in this work is

summarized in Chapter Three. All testing was conducted at The University of

Texas at Austin over the past decade.

The sensitivity of dynamic soil properties to soil type and loading

conditions are described in Chapter Four. The general trends (in terms of how

these parameters affect nonlinear soil behavior) observed during the course of this

work and those reported in the literature are discussed.

The empirical relationships reported in the literature are summarized in

Chapter Five. The empirical normalized modulus reduction and material damping

curves proposed in the literature are evaluated in terms of capturing the general

trends discussed in Chapter Four.

A four-parameter soil model that describes the change in normalized shear

modulus and material damping ratio with shearing strain is presented in Chapter

Six along with a parametric study of the model. Two of these parameters,

reference strain and curvature coefficient, are utilized in describing the

normalized modulus reduction curve. Masing behavior is used as a criterion in

evaluating material damping. A scaling coefficient and small-strain material

12

damping ratio are utilized in describing the material damping curve relative to the

damping curve estimated from the normalized modulus reduction curve and

assuming Masing Behavior. The impact of soil type and loading conditions on the

model parameters are also described herein.

The First-order, Second-moment Bayesian method is briefly discussed in

Chapter Seven. The form of the equations that are used in relating model

parameters to soil type and loading conditions are discussed in this chapter.

Results of the statistical analysis are presented in Chapter Eight. Measured

and predicted curves are compared in order to evaluate the success of the model in

representing nonlinear soil behavior.

In Chapter Nine, the impact of soil type and loading conditions on model

parameters are quantified. Equations and graphical solutions that are utilized to

construct normalized shear modulus reduction and material damping curves for

different soil types and loading conditions are presented. These curves are

compared with other empirical curves reported in the literature.

In Chapter Ten, recommended normalized modulus reduction and material

damping curves are presented for soils with a broad range plasticity confined at

different mean effective stresses.

Uncertainty associated with the predicted normalized modulus reduction

and material damping curves is discussed in Chapter Eleven. Recommendations

for future work related with handling uncertainty in nonlinear soil behavior are

presented for probabilistic seismic hazard analysis.

A summary of the study and conclusions are presented in Chapter Twelve.

13

CHAPTER 2

LABORATORY TESTING EQUIPMENT

2.1 INTRODUCTION

Combined resonant column and torsional shear (RCTS) equipment was

employed in this work to evaluate the dynamic soil properties of undisturbed soil

specimens. This equipment was developed by Professor Stokoe and his graduate

students (Isenhower, 1979; Lodde, 1982; Ni, 1987; and Hwang, 1997) following

earlier designs by Hall and Richart (1963), Hardin and Music (1965), and

Drnevich (1967). Detailed information regarding the equipment, testing method,

theory and calibration is presented in Darendeli (1997).

The RCTS equipment uses a fixed-free configuration. The soil specimen

rests on a fixed bottom pedestal (fixed at the bottom) and is free at the top. At the

free end, four magnets are attached to the top cap and fixed coils surrounding the

magnets are used to excite the top of the specimen with torsional vibrations

without constraining the top of the specimen (hence the top of the specimen is

“free”). A simplified diagram of the combined RCTS equipment is presented in

Figure 2.1.

14

Proximitor ProbesAccelerometer

SupportPlate

Fluid Bath

SecuringPlate

Magnet

InnerCylinder

Specimen

PorousStone O-ring

RubberMembrane

Top Cap

Resonant or Slow CyclicTorsional Excitation

Counter Weight

DriveCoil

Base Plate

Proximitor TargetProximitor ProbesAccelerometer

SupportPlate

Fluid Bath

SecuringPlate

Magnet

InnerCylinder

Specimen

PorousStone O-ring

RubberMembrane

Top Cap

Resonant or Slow CyclicTorsional Excitation

Counter Weight

DriveCoil

Base Plate

Proximitor Target

Figure 2.1 Simplified diagram of the RCTS device (from Stokoe et al., 1999)

2.2 COMBINED RESONANT COLUMN AND TORSIONAL SHEAR EQUIPMENT

Combined RCTS equipment is capable of testing a soil specimen in two

different modes. These modes are: 1. low frequency cyclic testing, and 2. higher

frequency dynamic testing during resonance. Thus, the same specimen can be

tested using both modes and variability due to testing different specimens or

testing the same specimen after it has been subjected to a different stress history is

eliminated. The data collected from the two independent modes of testing can

effectively be compared in order to gain more insight regarding material behavior.

One of the testing modes is called the torsional resonant column (RC) test,

which is based on the theory of torsional wave propagation in a fixed-free

cylinder with a mass attached at the free end. In this mode, well-defined boundary

15

conditions and specimen geometry are utilized in evaluating the shear modulus

and material damping ratio in shear from measurements at first-mode resonance.

The second testing mode is called the cyclic torsional shear (TS) test,

which involves monitoring the applied torque and displacement at the top of the

specimen. The torque is converted into shear stress and the displacement is

converted into shearing strain. Thus, hysteresis loops, which are utilized in

evaluation of shear modulus and material damping ratio, are generated.

These tests are typically carried out while the specimen is confined

isotropically. The confining chamber is designed to handle pressures up to 40

atmospheres (4.1 MPa). A cross-sectional view of the confining system is

presented in Figure 2.2.

AirPressure

σ

Membrane

FixingRod

Top Plate

HollowCylinder

SiliconFluid Bath

σ

σ

O-Ring

Soil

ThinMetal Tube

Drainage

Figure 2.2 Simplified cross-sectional view of the confining system (from Hwang, 1997)

16

The soil specimen is tested using both the cyclic torsional shear and

resonance modes simply by changing: 1) the amplitude and frequency of the

current in the drive coils, and 2) the motion monitoring devices (shown in Figure

2.3) used to record the specimen response. These changes are performed outside

the confining chamber; hence, they can be done without changing the state of

stress on the specimen.

2.3 TORSIONAL RESONANT COLUMN TEST

In torsional RC testing, a forcing function with fixed amplitude and

varying frequency is applied at the top of a cylindrical soil specimen. The output

from the accelerometer on the drive plate (shown in Figure 2.3) is recorded versus

the vibration frequency during a frequency sweep. The graph of accelerometer

output versus vibration frequency is called the frequency response curve. A

typical response curve is shown in Figure 2.4. The frequency at which the

accelerometer output reaches a maximum during first-mode torsional resonance is

denoted as the resonant frequency, fr, and it is used in calculating the shear wave

velocity of the specimen. The value of accelerometer output, Ar, at this frequency

is then used in calculating the peak shearing strain amplitude during the test.

The frequency response curve is also utilized in evaluating the material

damping ratio at small shearing strains, γ, (γ < 0.005 %). The half-power points

are identified as the two points on the frequency response curve with an amplitude

of 1/√2 times the peak value. The frequencies associated with the half-power

points, f1 and f2, are used in evaluating the material damping ratio as presented in

Figure 2.5.

17

SupportPlate

CounterWeight

Drive Plate

Accelerometer

Magnet

Drive CoilHolder

A

A Proximitor Probe

(a) Top View

Leveling andS

LVDT

ProximitorTarget

Accelerometer

ecuring Screw

SupportPlate

Fluid Bath

SecuringPlate

Magnet

DriveCoil

InnerCylinder

Base Pedestal

ProximitorProbe

SupportPostProximitor

Holder

PorousStone

(b) Section AA

Drainage Line

Top Cap

Specimen

SupportPlate

CounterWeight

Drive Plate

Accelerometer

Magnet

Drive CoilHolder

A

A Proximitor Probe

(a) Top View

SupportPlate

CounterWeight

Drive Plate

Accelerometer

Magnet

Drive CoilHolder

A

A Proximitor Probe

(a) Top View

Leveling andS

LVDT

ProximitorTarget

Accelerometer

ecuring Screw

SupportPlate

Fluid Bath

SecuringPlate

Magnet

DriveCoil

InnerCylinder

Base Pedestal

ProximitorProbe


Holder

PorousStone

(b) Section AA

Drainage Line

Top Cap

Specimen

Leveling andS

LVDT

ProximitorTarget

Accelerometer

ecuring Screw

SupportPlate

Fluid Bath

SecuringPlate

Magnet

DriveCoil

InnerCylinder

Base Pedestal

ProximitorProbe


Holder

PorousStone

(b) Section AA

Drainage Line

Top Cap

Specimen

Figure 2.3 General Configuration of RCTS Equipment (after Hwang, 1997)

18

I/Io=(ωrL/Vs) tan(ωrL/Vs)

G = ρVs2

→γAr

f = / 2πrr ω

605550454035

120

80

40

0

Ar

Acc

eler

omet

er O

utpu

t, m

V Resonance

Frequency, f, Hz

Figure 2.4 Frequency response curve measured in the RC test (from Stokoe et al., 1999)

Figure 2.5 Material damping measurement in the RC test using the half-power bandwidth (from Stokoe et al., 1999)

19

Once the resonant frequency is identified, a second measurement of

material damping ratio can be performed using the free-vibration decay curve.

This method involves vibrating the specimen in steady-state, first-mode torsional

resonance and recording the decay of free vibrations after shutting off the driving

force. Figure 2.6 shows an example decay curve. Sh

earin

g St

rain

Am

plitu

de, γ

x 1

0-3, %

0.40.30.20.10.0Time, seconds

0

1

2

3

-1

-2

-3

(a)1

510

15

Steady State Free Vibration Decay

CycleNumber

Nor

mal

ized

Pea

k-to

-Pea

k A

mpl

itude

s

20151050Number of Cycles

Remolded Sandeo = 0.71δ = 0.0734D = 1.17 %

(b)

1.0

0.7

0.5

0.3

Figure 2.6 Material damping measurement in the RC test using the free-vibration decay curve (from Stokoe et al., 1999)

20

The logarithmic decrement, δ, is defined from the free-vibration decay

curve as:

=

+1

1ln1

nzz

nδ (2.1)

where n equals number of cycles between two peak points in the time record, and

z1 and zn+1 are the amplitudes of cycle 1 and cycle n+1, respectively (Richart, Hall

and Woods, 1970). Material damping ratio can then be calculated using Equation

2.2.

22

2

4 δδ

+Π=D (2.2)

The half-power bandwidth method is based on the theory of elasticity and

it is accurate during testing at small strains as noted above (γ < 0.005 %). Material

damping estimates based on this method are quite reproducible at small strains

since points around the peak output on the frequency response curve are utilized

in the calculations. On the other hand, background noise can have a more adverse

effect on the free-vibration decay curve.

At large strains, nonlinear behavior of soil results in the linear assumption,

on which the half-power bandwidth method is based, to become invalid. In this

case, the free-vibration decay curve is applied along with an adjustment of the

strain amplitude, which is constantly changing as the vibrations decay. In the free-

vibration decay method, material damping ratio is calculated using the first three

cycles of vibration in this study. As a result, the average amplitude of the first

three cycles of vibration is assumed to represent the shearing strain at which the

measurement is made (rather than the steady-state amplitude).

21

2.4 CYCLIC TORSIONAL SHEAR TEST

In cyclic TS testing, a slow torsional loading is applied at the top of the

specimen. The loading frequency used in TS testing is much lower than resonance

testing (at least 10 times less than the resonant frequency). The current in the

calibrated drive coils is monitored, and the torque applied to the specimen is

calculated. The displacement at the top of the specimen is also monitored using

proximitors. Based on the torque and displacement at the top of the specimen,

hysteresis loops are generated.

The secant shear modulus for each cycle of loading is evaluated by

calculating the slope of the line that connects the end points of the hysteresis loop

as illustrated in Figure 2.7.

1GShear

Stress, τ

Shearing Strain, γ

G = τ / γD = AL / (4 π AT)

AL

AT

1GShear

Stress, τ

Shearing Strain, γ

G = τ / γD = AL / (4 π AT)

G = τ / γD = AL / (4 π AT)

AL

AT

Figure 2.7 Calculation of shear modulus and material damping ratio in the TS test

22

Material damping ratio is evaluated by calculating the ratio of the area

within the hysteresis loop (AL) and the maximum potential energy stored in each

cycle of motion as represented by the triangular area (AT). The area AT is

calculated using the end point of the hysteresis loop as shown in Figure 2.7.

T

L

AADΠ

=4

(2.3)

2.5 SUMMARY

Information regarding the RCTS equipment is summarized in this chapter.

This equipment has been employed in evaluating nonlinear dynamic soil

properties of undisturbed soil specimens for more than two decades at The

University of Texas at Austin. Detailed information regarding the equipment,

testing method, theory and calibration is presented in Ni (1987), Hwang (1997)

and Darendeli (1997).

23

CHAPTER 3

PHYSICAL PROPERTIES OF TEST SPECIMENS

3.1 INTRODUCTION

Over the past decade, a total of 110 undisturbed soil samples, which were

taken from 20 geotechnical sites, were tested in the soil dynamics laboratory

using the combined RCTS equipment. In this study, dynamic properties of these

soils in the nonlinear range are analyzed. Information regarding the soil samples

and the confining pressures at which these samples were tested are tabulated in

this chapter. Also, the geotechnical reports that contain the original data are cited

herein.

The chapter has been divided into the following sections. Sections 3.2

through 3.5 describe samples taken from Northern California, Southern

California, South Carolina and Lotung, Taiwan, respectively. In each section,

information regarding specimens from each geotechnical site in a given

geographic region is presented in a separate table. As an example, Table 3.1

shows physical properties of soils recovered from Oakland Outer Harbor. The

publication that contains the original data is cited in the title of the table. The table

provides the following information on each specimen: 1) Specimen identification

(ID), 2) Depth of the soil sample, 3) Soil type determined according to the Unified

Soil Classification System (USCS) based on gradation and plasticity tests

performed at The University of Texas at Austin, 4) Percentage of fine material

(passing #200 sieve) by weight (listed as Fines Content), 5) Liquid limit (LL), 6)

24

Plasticity index (PI) which is equal to the difference between the liquid limit and

the plastic limit of the soil sample, 7) Water content of the specimen, 8) Total unit

weight of the specimen, 9) Void ratio of the specimen, 10) Estimated

overconsolidation ratio (Est. OCR) of the specimen based on characteristics of the

measured relationship between small-strain shear modulus and mean effective

confining pressure, and 11) Mean effective confining pressure (listed as Test

Pressure) at which data regarding nonlinear soil behavior were collected. These

11 items represent the 11 column headings in the tables.

Table 3.1 Physical properties of soils recovered from Oakland Outer Harbor and test pressures (Hwang, 1997)

Fines Water Total Void Est. TestSpecimen Depth Soil Type Content LL PI Content Unit Wt. Ratio, OCR Pressure

ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-33-A 34 CL 73 49 29 41 1.78 1.15 1 2.3UT-33-B 21 SM-SC 37 21 4 15 1.78 0.74 1 1.5UT-33-C 5 SP 4 NP NP 20 1.75 0.83 2 0.3UT-33-D 81 CH 100 62 37 15 1.67 0.87 1 5.4UT-33-E 144 SM-SC 40 25 7 18 2.05 0.52 1 8.2

* Research was funded by EPRI.

Section 3.6 contains a discussion on the distribution of samples in terms of

their geographic location, depth, soil type, plasticity index, void ratio and unit

weight. This discussion is an attempt to familiarize the reader with the

characteristics of the database that is utilized in generating a new set of empirical

curves and equations regarding nonlinear soil behavior. Knowing the contents of

the database that this study has utilized, the reader will be aware when an

application requires extrapolation of these empirical curves and equations so that

the results will be used with more caution under such circumstances.

25

3.2 UNDISTURBED SOIL SPECIMENS FROM NORTHERN CALIFORNIA

A total of 37 undisturbed soil samples from 7 sites in Northern California

tested as part of a number of research projects are included in this study. These

projects were funded by the Kajima Corporation, Geovision, Agbabian

Associates, EPRI (Electric Power Research Institute), Fugro, Inc., and Earth

Mechanics, Inc.

The geotechnical sites in Northern California are Corralitos, Garner

Valley, Gilroy, Oakland Outer Harbor, San Francisco Airport, San Francisco-

Oakland Bay Bridge and Treasure Island. A map of Northern California is

presented in Figure 3.1 showing the locations of these sites. Information regarding

the soil samples and the confining pressures at which nonlinear properties were

measured are tabulated in Tables 3.1 through 3.7. The references that contain the

original data are also cited in the titles of the tables.

Table 3.2 Physical properties of soils recovered from Treasure Island and test pressures (Hwang and Stokoe, 1993b; and Hwang, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-28-A 18 CH 50 51 26 50 1.73 1.34 1 1.2UT-28-B 52 CL 79 34 19 21 2.05 0.58 1 3.8UT-28-C 71 CL 67 48 30 33 1.84 0.95 1 5.1UT-28-D 40 CL 63 37 23 37 1.83 1.02 1 2.9UT-28-E 9.1 SP-SM 79 NP NP 21 1.92 0.67 1 0.7UT-28-F 27 CL 58 42 19 42 1.81 1.10 1 1.9UT-28-G 5.3 SM 80 NP NP 20 1.92 0.69 1 0.4UT-28-H 34 SP-SM 78 NP NP 22 1.83 0.76 1 2.2


26

Figure 3.1 Map of Northern California showing the locations of the geotechnical sites in this area

27

Table 3.3 Physical properties of soils recovered from San Francisco Airport and test pressures (Hwang, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-36-A 4.9 CL-ML 53 21 5 19 2.08 0.54 2 0.3UT-36-B 7.9 CL 54 30 13 18 2.10 0.50 2 0.5


Table 3.4 Physical properties of soils recovered from Gilroy and test pressures (Hwang and Stokoe, 1993c; Hwang, 1997; and Stokoe et al., 2001)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-24-A 6.1 CL 86 43 23 30 1.91 0.84 1 0.8UT-24-B 3.0 CL-ML 65 29 7 26 1.88 0.81 1 0.4UT-24-C 26 MH 100 47 17 31 1.94 0.82 1 2.4UT-24-D 37 ML 62 NP NP 20 2.12 0.48 1 3.3

UT-24-E++ 128 * * * * 14 2.18 0.41 1 8.7UT-24-F 64 SW-SM 8 NP NP 15 2.08 0.46 1 4.9UT-24-G 15 SP 2 NP NP 16 1.97 0.55 1 2.0UT-24-H 106 CL 65 35 13 24 2.04 0.63 1 8.6UT-24-I 6.1 CL 86 43 23 30 1.91 0.84 1 0.8UT-24-J 26 MH 100 47 17 31 1.94 0.82 1 2.4UT-24-K 37 ML 62 NP NP 20 2.12 0.48 1 3.3

UT-24-L++ 52 SM 13 NP NP 8 2.13 0.34 1 4.8UTA-18-I 3.4 SC 17 36 20 20 2.15 0.47 1 0.4UTA-18-J 16 SC 28 49 24 15 1.95 0.56 1 1.1

* Information is not available. ** Research was funded by EPRI and Kajima Corporation, Japan through Geovision. ++ UT-24-E and UT-24-L were stopped due to membrane leakage and are not included in this study.

28

Table 3.5 Physical properties of soils recovered from Garner Valley and test pressures (Stokoe and Darendeli, 1998)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-52-A 3.5 SM 26 NP NP 19 1.90 0.69 1 0.5UT-52-B 6.5 SM 15 NP NP 17 1.79 0.76 1 0.7UT-52-C 41 SM 36 NP NP 18 1.91 0.67 1 2.7UT-52-D 27 SM 19 NP NP 14 2.09 0.47 1 1.7

* Research was funded by Agbabian Associates.

Table 3.6 Physical properties of soils recovered from San Francisco-Oakland Bay Bridge Site and test pressures (Stokoe et al., 1998d)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-10-A 8.4 CH 93 63 36 50 1.71 1.38 1 0.5, 2.2UTA-10-B 11 CH 97 75 43 58 1.63 1.61 1 0.5, 2.2UTA-10-C 24 CH 96 89 53 57 1.70 1.50 8 1.1, 4.4UTA-10-D 71 CL 91 46 19 30 1.92 0.83 1 4.1, 16.3

* Research was funded by Fugro, Inc. and Earth Mechanics, Inc.

Table 3.7 Physical properties of soils recovered from Corralitos and test pressures (Stokoe et al., 2001)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-18-F 10 SW-SC 7 31 10 14 2.16 0.44 1 1.4

UTA-18-G** 3.3 SC 31 44 19 19 2.01 0.62 1 0.5UTA-18-K 46 ML 52 23 4 8 2.39 0.24 2 4.1

* Research was funded by Kajima Corporation, Japan through Geovision. ** UTA-18-G was reconstituted and is not included in this study.

29

Specimen UTA-18-G sampled from Corralitos was reconstituted.

Specimens UT-24-E and UT-24-L from Gilroy had to be stopped due to

membrane leakage before the TS data could be collected. As a result, these

samples are not included in this study, but are included in the tables for

completeness.

3.3 UNDISTURBED SOIL SPECIMENS FROM SOUTHERN CALIFORNIA

A total of 47 undisturbed soil samples from 10 sites in Southern California

tested as part of a number of research projects are included in this study. These

projects were funded by the ROSRINE (Resolution of Site Response Issues from

the Northridge Earthquake) Project, Kajima Corporation, Geovision and

Agbabian Associates.

The geotechnical sites in Southern California are Arleta, Borrego, Imperial

Valley College, Kagel, La Cienega, Newhall, North Palm Springs, Potrero

Canyon, Rinaldi Receiving Station and Sepulveda V.A. Hospital. A map of

Southern California is presented in Figure 3.2 showing the locations of the three

sites outside the Los Angeles area (Borrego, Imperial Valley College and North

Palm Springs). The remaining seven sites are presented on a map of Los Angeles

area in Figure 3.3.

Information regarding the soil samples and the confining pressures (at

which these samples were tested) are tabulated in Tables 3.8 through 3.17. The

references that contain the original data are also cited in the titles of the tables.

30

Figure 3.2 Map of Southern California showing the locations of the three geotechnical sites outside the Los Angeles area

Specimen UT-40-G from Borrego was stopped due to membrane leakage

before the TS data could be collected. Specimens UTA-9-I from Rinaldi

Receiving Station, and UTA-18-B and UTA-18-E from North Palm Springs were

reconstituted. As a result, these samples are not included in this study, but are

included in the tables for completeness.

31

Figure 3.3 Map of Los Angeles showing the locations of the seven geotechnical sites in this area

32

Table 3.8 Physical properties of soils recovered from Borrego and test pressures (Hwang, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-40-B 3.4 SM 84 NP NP 12 1.76 0.68 1 0.4UT-40-C 20 SP-SM 89 NP NP 11 1.83 0.62 1 2.5UT-40-E 49 SP-SM 91 NP NP 14 1.76 0.72 1 6.1UT-40-F 110 SP-SM 91 NP NP 17 1.78 0.75 1 13.6

UT-40-G** 146 SP-SM 95 NP NP 11 2.04 0.44 1 18.2 * Research was funded by Agbabian Associates. ** UT-40-G was stopped due to membrane leakage and is not included in this study.

Table 3.9 Physical properties of soils recovered from Arleta and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-1-C 31 SM 40 21 1 13 2.16 0.42 2 2.7, 10.9UTA-1-L 15 SM 28 NP NP 14 2.10 0.46 1 1.6

* Research was funded by ROSRINE Project.

Table 3.10 Physical properties of soils recovered from Kagel and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-1-E 8.5 SW-SM 10 NP NP 3 1.87 0.48 1 1.1, 4.4UTA-1-F 31 SW-SM 12 NP NP 13 2.11 0.44 1 3.3UTA-1-G 65 SW-SM 9 NP NP 10 2.16 0.38 1 5.4UTA-1-H 92 SP-SM 10 NP NP 13 2.07 0.46 1 6.8


33

Table 3.11 Physical properties of soils recovered from La Cienega and test pressures (Darendeli and Stokoe, 1997; Darendeli, 1997; and Stokoe et al., 1998e)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-1-J 4.9 CL 51 33 10 21 2.03 0.61 1 0.6, 2.4UTA-1-K 3.4 SC 35 42 20 15 1.98 0.57 2 0.5, 2.2UTA-1-M 7.9 SM 43 NP NP 22 2.07 0.58 1 1.1, 4.4UTA-1-N 6.1 CL 60 29 8 19 1.99 0.62 2 0.8, 3.3UTA-1-O 6.4 CL 57 32 10 20 2.05 0.58 1 0.8, 3.3UTA-9-J 28 CH 99 50 25 30 1.87 0.87 4 3.3UTA-9-K 34 CL 83 26 10 15 2.01 0.55 1 4.6UTA-9-L 36 CL 64 30 10 20 2.08 0.56 1 4.6UTA-9-M 95 SM 17 NP NP 16 2.07 0.51 1 13.6UTA-9-N 125 SM 30 NP NP 16 2.10 0.49 1 17.0UTA-9-O 186 ML 71 28 5 29 2.05 0.69 1 24.5UTA-9-P 241 SM 32 NP NP 18 2.05 0.55 1 27.2UTA-9-Q 52 CL 92 34 11 29 1.90 0.83 1 6.8UTA-9-R 107 CH 99 64 36 32 1.96 0.82 1 13.6UTA-9-S 150 CH 99 52 25 26 2.02 0.69 1 20.4UTA-9-T 218 CL 94 42 18 21 2.04 0.60 1 27.2


Table 3.12 Physical properties of soils recovered from Newhall and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-1-D 62 SC-SM 48 25 5 13 2.21 0.37 4 5.4UTA-1-I 21 SM 26 NP NP 18 1.92 0.65 1 1.6


34

Table 3.13 Physical properties of soils recovered from Sepulveda V.A. Hospital and test pressures (Darendeli and Stokoe, 1997; and Darendeli, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-1-A 3.4 CL 75 39 15 19 2.04 0.58 8 0.7UTA-1-B 3.1 CL 62 37 15 20 1.91 0.70 8 0.5UTA-1-P 14 ML 85 34 9 26 1.93 0.77 1 1.6UTA-1-Q 17 SM 41 NP NP 15 2.00 0.55 1 2.2UTA-1-R 37 CL 64 29 9 18 2.07 0.54 1 3.4UTA-1-S 59 CL 66 42 16 22 2.14 0.54 2 5.4UTA-1-T 2.4 CH 70 54 29 25 2.02 0.67 8 0.3, 1.4UTA-1-U 86 CL 77 35 12 17 2.11 0.50 1 6.8


Table 3.14 Physical properties of soils recovered from Potrero Canyon and test pressures (Stokoe et al., 1998e)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-9-C 8.5 SC-SM 20 19 5 11 2.15 0.39 1 1.1UTA-9-D 16 CL 73 26 9 16 2.13 0.47 1 1.9UTA-9-E 31 CL 83 32 12 10 2.32 0.28 1 4.4UTA-9-G 2.4 SM 41 20 2 21 1.88 0.74 1 0.4


Table 3.15 Physical properties of soils recovered from Rinaldi Receiving Station and test pressures (Stokoe et al., 1998e).


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-9-A 11 SM 22 23 1 15 2.04 0.51 1 1.4UTA-9-B 21 CL-ML 53 23 5 24 1.99 0.69 1 2.4UTA-9-F 2.4 CL-ML 51 22 4 22 2.03 0.62 1 0.4UTA-9-H 15 SM 41 NP NP 16 2.03 0.55 1 1.6

UTA-9-I** 7.6 SW-SM 9 NP NP 12 2.10 0.44 1 1.1 * Research was funded by ROSRINE Project. ** UTA-9-I was reconstituted and is not included in this study.

35

Table 3.16 Physical properties of soils recovered from North Palm Springs and test pressures (Stokoe et al., 2001)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-18-B** 72 SP 0 NP NP 16 1.93 0.59 1 4.8UTA-18-C 46 SM 48 29 3 23 1.99 0.64 1 3.1

UTA-18-E** 17 SW 2 NP NP 18 2.17 0.44 1 1.4 * Research was funded by Kajima Corporation, Japan through Geovision. ** UTA-18-B and UTA-18-E were reconstituted and are not included in this study.

Table 3.17 Physical properties of soils recovered from Imperial Valley College and test pressures (Stokoe et al., 2001)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-18-A 102 CL 100 46 29 21 1.93 0.69 1 6.5UTA-18-H 16 CL 95 49 28 26 2.08 0.64 8 1.1

* Research was funded by Kajima Corporation, Japan through Geovision.

3.4 UNDISTURBED SOIL SPECIMENS FROM SOUTH CAROLINA

A total of 18 undisturbed soil samples from 2 sites in South Carolina were

tested as part of two research projects funded by the Westinghouse Savannah

River Company and S&ME, Inc.

The geotechnical sites in South Carolina are Savannah River Site and

Daniel Island. A map of South Carolina is presented in Figure 3.4 showing the

locations of these sites. Information regarding the soil samples and the confining

pressures at which nonlinear properties were measured are tabulated in Tables

3.18 and 3.19. The references that contain the original data are also cited in the

titles of the tables.

36

Figure 3.4 Map of South Carolina showing the locations of the geotechnical sites in this area

37

Table 3.18 Physical properties of soils recovered from Savannah River Site and test pressures (Hwang, 1997; and Stokoe et al., 1998a).


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-39-A 3.2 SC 30 52 31 15 2.07 0.48 4 0.8UT-39-B 17 SM 20 NP NP 33 1.83 0.92 1 2.1UT-39-C 27 CH 63 80 53 53 1.62 1.55 1 2.8UT-39-D 7.0 SC 28 46 19 21 1.84 0.74 2 0.9UT-39-E 47 SP-SM 9 NP NP 26 1.94 0.73 1 4.0UT-39-F 57 SP-SM 11 NP NP 24 1.94 0.69 1 4.6UT-39-G 86 SM 24 NP NP 20 1.65 0.93 1 6.3UT-39-H 80 SM 20 NP NP 24 1.78 0.86 1 5.9UT-39-I 24 SC 23 61 34 31 1.83 0.9 1 2.4UT-39-K 13 SM 14 NP NP 27 1.81 0.85 1 1.6UT-39-L 32 SM 18 NP NP 28 1.83 0.86 1 3.1UT-39-M 263 SC 29 34 16 12 2.08 0.43 1 16.8UT-39-N 107 CH 87 51 27 21 2.02 0.61 1 7.6UT-39-O 226 CL 76 30 12 7 2.07 0.37 1 14.6UT-39-S 199 CL 70 39 14 16 2.12 0.45 1 13.0

* Research was funded by Westinghouse Savannah River Company. ** As discussed in Chapter Eight, specimens UT-39-G, UT-39-M, UT-39-O, and UT-39-S were removed from the database during the analysis because the resonant column results did not follow the general trends reported in the literature and observed during the course of this study while the torsional shear results did follow the general trends but were not of sufficient strain range to be included.

Table 3.19 Physical properties of soils recovered from Daniel Island and test pressures (Stokoe et al., 1998b).


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UTA-7-A 11 CH 93 122 79 84 1.51 2.31 1 0.4UTA-7-B 10 SP-SM 8 26 2 38 1.84 1.03 1 0.6UTA-7-C 20 MH 68 210 132 83 1.48 2.33 4 0.7, 2.7

* Research was funded by S&ME, Inc.

38

3.5 UNDISTURBED SOIL SPECIMENS FROM LOTUNG, TAIWAN

Eight samples from Lotung site in Taiwan were tested as part of a research

project funded by EPRI. Detailed information about this work can be found in

Hwang and Stokoe (1993a), and Hwang (1997).

A map of Taiwan is presented in Figure 3.5 showing the location of

Lotung. Information regarding the soil samples and the confining pressures at

which these samples were tested are tabulated in Table 3.20.

Lotung8 SamplesLotung8 Samples

Figure 3.5 Map of Taiwan showing the location of Lotung site

39

Table 3.20 Physical properties of soils recovered from Lotung site and test pressures (Hwang and Stokoe, 1993a; and Hwang, 1997)


ID (m) (USCS) (%) (%) (%) (%) (gr/cm3) e (atm)UT-37-A 34 ML 99 32 7 35 1.89 0.92 1 2.0UT-37-B 18 SM 40 NP NP 33 1.75 1.08 1 1.1UT-37-C 5.5 ML 52 NP NP 31 1.79 0.97 1 0.3UT-37-D 11 ML 85 NP NP 33 1.89 0.91 1 0.7UT-37-E 29 SM 30 NP NP 31 1.91 0.88 1 1.7UT-37-F 41 ML 98 33 8 31 1.88 0.93 1 2.4UT-37-G 45 ML 78 NP NP 24 2.05 0.64 1 2.7UT-37-H 25.0 ML 98 38 12 37 1.88 1.00 1 1.5


3.6 OVERVIEW OF THE DATABASE

In this section, the distribution of the specimens included in this study is

briefly discussed in terms of their various characteristics. Figure 3.6 shows the

number of samples taken from each geographic region (Northern California,

Southern California, South Carolina and Taiwan). In Figure 3.7, the number of

geotechnical sites in each of the four geographic regions is presented. It is

important to note that most of the samples in this database have come from

California (84 out of 110 samples or 76 %).

Figure 3.8 shows the distribution of soil samples with depth. The samples

in this database have been recovered from a depth range of 3 to 263 m. This depth

range has been divided into eight categories as noted in the legend. The number of

samples from each geographic region in each depth category is presented in Table

3.21.

40

Figure 3.6 Distribution of soil samples with geographic region

Figure 3.7 Distribution of the number of geotechnical sites with geographic region

41

Figure 3.8 Distribution of soil samples according to the sample depth

Table 3.21 Distribution of soil samples according to the sample depth in each geographic region

Geographic Region 0-5 5-10 10-20 20-30 30-50 50-100 100-200 200-263 TOTALNorthern California 5 8 4 6 7 5 2 - 37Southern California 8 5 7 4 8 7 6 2 47

South Carolina 1 1 4 3 2 3 2 2 18Taiwan - 1 2 2 3 - - - 8TOTAL 14 15 17 15 20 15 10 4 110

Depth Range (meters)

In Figure 3.9, information regarding the isotropic confining pressures at

which nonlinear measurements have been performed is presented. The test

pressures ranged from 0.3 to 27 atmospheres. The isotropic confining test

pressure is in most cases equal to the estimated in-situ mean effective stress

calculated based on the sample depth, location of water table and assuming 0.5 as

the coefficient of horizontal earth pressure at rest. However, some specimens

42

were tested at more than one state of stress. As a result, the total number of test

pressures at which nonlinear measurements are performed is slightly more than

the total number of specimens. The number of samples from each geographic

region in each confining pressure category is presented in Table 3.22.

Figure 3.9 Distribution of confining pressures at which nonlinear measurements were performed

Table 3.22 Distribution of collected according to the isotropic confining pressure in each geographic region

Geographic Region 0.3-0.5 0.5-1.0 1.0-2.0 2.0-4.0 4.0-8.0 8.0-16.0 16.0-27.2 TOTALNorthern California 7 6 7 12 6 2 1 41Southern California 5 5 11 12 13 4 5 55

South Carolina 1 4 1 5 5 2 1 19Taiwan 1 1 3 3 - - - 8TOTAL 14 16 22 32 24 8 7 123

Test Pressure Range (atmospheres)

43

Figures 3.10 and 3.11 show the distribution of soil samples according to

the Unified Soil Classification System (USCS) designation and in terms of their

plasticity, respectively. It is important to note that soils with a wide range of

plasticity are represented in this database. About half of the soils classify as fine-

grained soils. Coarse-grained soils included in this study are limited to sands. Due

to limitations on specimen size, gravelly soils were not tested as part of this work.

The number of samples in each sample depth category divided according to their

Unified Soil Classification System (USCS) designation is presented in Table 3.23.

This table shows whether a given soil type from a given depth range is

represented in the database or not. It is important to note that a variety of soil

types from a wide range of sampling depths are represented in this database.

Figure 3.10 Distribution of soil samples according to soil type as classified by the Unified Soil Classification System (USCS)

44

Figure 3.11 Distribution of soil samples according to soil plasticity in terms of the plasticity index, PI

Table 3.23 Distribution of soil samples according to the Unified Soil Classification System (USCS) designation and sample depth

Soil Type 0-5 5-10 10-20 20-30 30-50 50-100 100-200 200-263 TOTALCH 1 1 3 3 - 1 3 - 12CL 3 5 2 1 6 6 3 2 28

CL-ML 3 - - 1 - - - - 4MH - - - 3 - - - - 3ML - 1 2 1 6 - 1 - 11SC 3 1 1 1 - - - 1 7

SC-SM - 1 - 1 - 1 1 - 4SM 3 3 7 3 4 3 1 1 25SP 1 - 1 - - - - - 2

SP-SM - 1 1 1 3 2 1 9SW-SC - 1 - - - - - - 1SW-SM - 1 - - 1 2 - - 4TOTAL 14 15 17 15 20 15 10 4 110

Depth Range (meters)

45

Figures 3.12 through 3.15 present distributions of soil samples according

to total unit weight, dry unit weight, water content and void ratio, respectively. In

this study, specific gravity of sandy soils is assumed 2.65 and that of clayey soils

is assumed 2.70. It is important to note that most of the soils included in this study

are competent soils with low void ratios sampled from geotechnical sites that have

not liquefied during seismic activity. Consequently, none of these samples have

exhibited major changes in their stiffness (or normalized modulus reduction

curve) due to cycling at a given strain amplitude within the range of shearing

strains testing was performed. Normally consolidated soils with higher plasticity

are the exceptions that contribute to the void-ratio diversity of this database.

Figure 3.16 shows the variation of the dry unit weights of the soil samples

with depth. In this figure, dry unit weights of the coarse grained soils (sands) are

observed to form a relatively narrow band compared to that of fine grained soils

included in this study. Also, dry unit weights of both the fine-grained and coarse-

grained soils are observed to increase with depth due to higher confining

pressures at deeper soil layers. Figures 3.17 and 3.18 present variations of water

content and void ratio with depth, respectively. These figures indicate trends

consistent with those observed in Figure 3.16. Increase in dry unit weight

necessitates a decrease in the void space (and, therefore, a decrease in water

volume filling the voids) within the soil structure. As a result, void ratio and water

content of both the fine-grained and coarse-grained soils are observed to decrease

with depth. The coarse grained soils (sands) are again observed to form a

relatively narrower band than that of the fine grained soils included in this study.

46

Figure 3.12 Distribution of soil samples according to total unit weight

Figure 3.13 Distribution of soil samples according to dry unit weight

47

Figure 3.14 Distribution of soil samples according to water content

Figure 3.15 Distribution of soil samples according to void ratio

48

0

50

100

150

200

250

300

0.00 0.50 1.00 1.50 2.00 2.50

Dry Unit Weight, gr/cm3

Depth, m

(a)

Clayey SoilsSilty Soils

0.0

50.0

100.0

150.0

200.0

250.0

300.0

0.00 0.50 1.00 1.50 2.00 2.50

Dry Unit Weight, gr/cm3

Depth, m

(b)Sandy Soils

Figure 3.16 Variation of dry unit weight with depth of (a) fine grained and (b) coarse grained soils included in this study

49

0

50

100

150

200

250

300

0 20 40 60 80 100

Water Content, %

Depth, m

(a)


0.0

50.0

100.0

150.0

200.0

250.0

300.0

0 20 40 60 80 100

Water Content, %

Depth, m

(b)

Sandy Soils

Figure 3.17 Variation of water content with depth of (a) fine grained and (b) coarse grained soils included in this study

50

0

50

100

150

200

250

300

0.00 0.50 1.00 1.50 2.00 2.50

Void Ratio, e

Depth, m

(a)


0.0

50.0

100.0

150.0

200.0

250.0

300.0

0.00 0.50 1.00 1.50 2.00 2.50

Void Ratio, e

Depth, m

(b)

Sandy Soils

Figure 3.18 Variation of void ratio with depth of (a) fine grained and (b) coarse grained soils included in this study

51

In Figures 3.19 and 3.20, the distribution of samples according to

estimated overconsolidation ratio and the variation of estimated overconsolidation

ratio with depth are presented, respectively. The overconsolidated soils included

in this study are observed to be sampled from depths less than about 50 m.

Unfortunately, it was not feasible to perform consolidation tests on these samples

during the course of this work. Overconsolidation ratio of the samples are

estimated based on the characteristics of log Gmax – log σo’ relationships. The

effective isotropic confining pressure, at which a break in the log Gmax – log σo’

relationship is observed, is assumed to be the maximum mean effective stress that

the sample has experienced. Overconsolidation ratio is calculated by dividing this

pressure to the estimated mean effective stress. In most cases, the soils are

classified as normally consolidated when a clean break in the log Gmax – log σo’

relationship is not observed.

Figure 3.19 Distribution of soil samples according to estimated overconsolidation ratio

52

0

50

100

150

200

250

300

0 2 4 6 8 10

Estimated Overconsolidation Ratio, OCR

Depth, m

(a)


0.0

50.0

100.0

150.0

200.0

250.0

300.0

0 2 4 6 8 10

Estimated Overconsolidation Ratio, OCR

Depth, m

(b)

Sandy Soils

Figure 3.20 Variation of estimated overconsolidation ratio with depth of (a) fine grained and (b) coarse grained soils included in this study

53

3.7 SUMMARY

The data that is analyzed in order to evaluate nonlinear soil behavior is

presented herein. This database has been compiled over the past decade from

research work supported by various organizations and carried out by a number of

graduate students working in the soil dynamics laboratory at The University of

Texas at Austin. Information regarding specimens from each geotechnical site in a

given geographic region is presented in separate tables showing physical

properties of soils and citing the publication that contains the original data.

A discussion regarding the distribution of samples in terms of their

geographic location, depth, soil type, plasticity index, void ratio and unit weight is

also presented at the end of this chapter in an attempt to familiarize the reader

with the characteristics of the database that is utilized in this study.

54

CHAPTER 4

OBSERVED TRENDS IN DYNAMIC SOIL PROPERTIES

4.1 INTRODUCTION

Dynamic soil properties (in terms of G and D) and the parameters that

affect these properties are discussed in this chapter. The relative importance of

each parameter on G and D and the trends reported in the literature and/or

observed during the course of this work are presented. This discussion is

presented so that the shortcomings of the existing empirical curves presented in

Chapter Five can be assessed and the strengths and limitations of the improved

empirical relationships developed in this study can be easily recognized.

4.2 BACKGROUND

Nonlinear dynamic soil properties are affected by a number of parameters

which have varying levels of importance. These parameters can be divided into

two groups: 1) parameters that relate to the static and dynamic loading conditions,

and 2) parameters that relate to the material type.

Important parameters related to the loading conditions which affect

nonlinear soil behavior are:

a) strain amplitude,

b) magnitude of the effective confinement state, often expressed by the

“equivalent” effective isotropic confining pressure,

c) duration of the effective confinement state, sometimes termed the

“long term time effect”,

55

d) number of loading cycles,

e) loading frequency (or strain rate), and

f) overconsolidation ratio (or loading history).

Soils are natural materials that can, and typically do, vary widely. The

behavior and performance of these materials tend to change significantly from

one soil to another. One of the challenges that a geotechnical engineer has to deal

with is the necessity to design an engineered structure with the material available

at a given site. As a result, for decades, geotechnical engineers have been

classifying different soils and associating their performance in various

applications with soil classes. This perspective has been utilized herein to analyze

the impact of material type on dynamic soil behavior. This study is an effort to

characterize nonlinear behavior of “competent” soils (soils that do not undergo

large volume changes during dynamic loading) at shearing strain amplitudes less

than 1 %. The results of this research are intended to be utilized in the analysis of

free-field ground motions during design level earthquakes.

Finally, the effect of sampling disturbance on dynamic soil properties is

briefly discussed in this chapter in order to show the importance of small-strain,

in-situ seismic measurements and to justify the emphasis in this work on

normalized modulus reduction curves rather than absolute values of stiffness.

Unfortunately, seismic methods have not been successfully used to date to

measure in-situ material damping ratio at any strain level on a routine basis.

Therefore, it is not possible to apply some type of laboratory-to-field

transformation to obtain field material damping curves. As a result, damping

56

curves measured in the laboratory are directly utilized as design curves in ground

motion analysis.

4.3 NONLINEAR DYNAMIC SOIL PROPERTIES

As discussed in Chapter One, soils exhibit nonlinear behavior. In other

words, secant shear modulus, G, decreases with increasing strain amplitude. Shear

modulus at small strains is referred to as small-strain shear modulus, Gmax or Go.

The relationship between shear modulus and shearing strain amplitude is typically

characterized by a normalized modulus reduction curve as shown in Figure 4.1a.

The nonlinearity in the stress-strain relationship results in an increase in

energy dissipation and therefore an increase in material damping ratio, D, with

increasing strain amplitude. Material damping ratio at small strains is referred to

as small-strain material damping ratio, Dmin. The relationship between material

damping ratio and strain amplitude is typically characterized by a material

damping curve as shown in Figure 4.1b. As noted above and as illustrated in

Figure 4.1b, D-log γ curve is expressed in absolute terms, not in normalized terms

(for instance D/Dmin-log γ or D/Dmax-log γ) because the nonlinear characteristics

of the D-log γ curve are related to the normalized modulus reduction curve of a

given soil rather than the value of material damping ratio at small or large strains.

These two curves can be broken into three strain ranges over which soils

behave differently. At small strains, γ < 0.001 %, soils exhibit linear elastic

behavior. The main source of energy dissipation is friction between particles

and/or viscosity. In other words, shear modulus is constant at a maximum value,

Gmax, and material damping ratio is constant at a minimum value, Dmin.

57

γte γt

c

N=10

N=11.0

0.5

00.001 0.01 0.1 1Shearing Strain, γ, %

~0.8G

Gmax

(a)

γte γt

c

D, %

16

8


∆D ~3 %

Dmin

(b)

γte = elastic treshold strain

γtc = cyclic treshold strain

Linear ElasticNonlinear ElasticPlastic

γte γt

c

N=10

N=11.0

0.5


~0.8G

Gmax

(a)

γte γt

c

N=10

N=11.0

0.5


~0.8G

Gmax

(a)

γte γt

c

D, %

16

8


∆D ~3 %

Dmin

(b)

γte γt

c

D, %

16

8


∆D ~3 %

Dmin

(b)








Figure 4.1 Linear elastic, nonlinear elastic and plastic strain ranges on (a) normalized modulus reduction and (b) material damping curves

58

The strain amplitude at which shear modulus decreases to 98 % of its

original value is commonly called the elastic threshold strain and is denoted by etγ . It is also called the nonlinearity threshold by Vucetic and Dobry (1991) and

Ishihara (1996). Above the elastic threshold strain, soils behave nonlinear but still

elastic. In other words, the stress-strain relationship is curved, but the

deformations are recoverable upon unloading. Due to the nonlinear stress-strain

relationship, an increase in material damping ratio is observed. The strain

amplitude at which deformations become irrecoverable is called the cyclic (or

plastic) threshold strain and is denoted by ctγ . It is also called the degradation

threshold by Vucetic and Dobry (1991) and Ishihara (1996). At this strain, shear

modulus has decreased to about 80 % of Gmax, and material damping ratio is about

3 % higher than Dmin (Stokoe et al., 1999).

Above the cyclic threshold strain, soils may change volume as they

deform. Soils exhibit different behavior when sheared depending on how dense

they are packed. Loose saturated soils tend to contract and/or develop positive

pore pressures while dense soils tend to dilate and/or develop negative pore

pressures. A change in pore pressure results in a change in effective stress and

normalized modulus reduction and material damping curves shift with each cycle

of loading and unloading, as presented in Figure 4.1. Normalized modulus

reduction curves of the soils analyzed in this study were observed to shift very

little (or not at all in most cases) with number of cycles while a considerable shift

in material damping curve was recognized. The effect of number of loading

cycles on dynamic soil behavior is discussed in more detail in Section 4.7.

59

4.4 EFFECT OF DURATION OF CONFINEMENT ON SMALL-STRAIN DYNAMIC SOIL PROPERTIES

Figure 4.2 shows the effects of magnitude and duration of isotropic

confining pressure on the small-strain shear modulus and material damping ratio

for a typical soil specimen (UTA-1-J in Table 3.11). The variation of void ratio

with magnitude and duration of isotropic confining pressure is also presented in

this figure.

As shown in Figure 4.2a, Gmax increases as the specimen consolidates at a

given confining pressure and it also increases with increasing confining pressure.

As shown in Figure 4.2b, Dmin decreases as the specimen consolidates at a given

confining pressure and it also decreases with increasing confining pressure.

The impact of magnitude and duration of confining pressure is smaller if

the soil specimen is in an overconsolidated state (in other words, if the specimen

has been subjected to a higher confining pressure in the past) compared with the

normally consolidated state for all soils. Furthermore, these effects decrease as

overconsolidation ratio increases. If the soil specimen is normally consolidated,

magnitude and duration of confining pressure is observed to have a larger effect

on clayey soils than on sandy soils.

60

200

150

100

50

0

Gmax,

(a)

MPa

Isotropic Confining Pressure0.14 atm 0.27 atm0.61 atm 1.22 atm2.45 atm

10

8

6

4

2

0

Dmin,

(b)

%

0.64

0.62

0.60

0.58

0.56

e

100 101 102 103 104

Duration of Confinement, t, min

Sandy Lean Clay (CL)

(c)

Figure 4.2 Variation of (a) low-amplitude shear modulus, (b) low-amplitude material damping ratio, and (c) void ratio with magnitude and duration of isotropic confining pressure

61

The sandy lean clay (CL) specimen in Figure 4.2 is expected to reach 99%

consolidation in a fraction of the time period over which a change in small-strain

dynamic properties is observed. Part of the increase in Gmax and the decrease in

Dmin results from the consolidation of the specimen under the applied pressure.

However, small-strain dynamic properties continue to change after consolidation

of the specimen (in other words after the applied pressure has become the

effective confining pressure). The impact of duration of confining pressure after

primary consolidation is called the long-term time effect (or creep) and is

discussed in detail by Anderson and Stokoe (1978).

4.5 EFFECT OF EFFECTIVE CONFINING PRESSURE

4.5.1 Small-Strain (Linear) Dynamic Soil Properties

The effect of effective confining pressure, σo’, on small-strain dynamic

soil properties has been documented by various investigators (e.g., Hardin and

Drnevich, 1972a and b; Hardin, 1978; Stokoe et al., 1994; and Stokoe et al.,

1999). This effect is studied by measuring values of Gmax and Dmin (and void ratio

for that matter) after the specimen (UTA-1-J in Table 3.11) has fully consolidated

at each confining pressure. Typical results illustrating the effect of σo’ are


62

10

100

1000

Gmax,

(a)

MPa

σpm'

0.1

1

10

Dmin,

(b)

% σpm'

0.64

0.62

0.60

0.58

0.56

e

0.1 1 10Effective Isotropic Confining Pressure, σo

′, atm

Sandy Lean Clay (CL)Time = 1 day

(c)σpm'

Figure 4.3 Variation of (a) low-amplitude shear modulus, (b) low-amplitude material damping ratio, and (c) void ratio with effective isotropic confining pressure

63

Small-strain shear modulus, Gmax, increases with increasing effective

confining pressure as shown in Figure 4.3a. Overconsolidated soils tend to exhibit

some “memory” of stress history and they can be recognized from their bilinear

log Gmax – log σo’ relationships. The effective confining pressure at which a

change in the slope of log Gmax – log σo’ relationship is observed is the maximum

mean effective stress that the soil sample has ever experienced in the past and it is

indicated with σpm’ in Figure 4.3.

In the normally consolidated range, the slope of log Gmax – log σo’

relationship for most competent soils falls in a range of about 0.5 to 0.6. In the

overconsolidated range, Gmax is less sensitive to σo’, resulting in log Gmax – log

σo’ relationships with slopes smaller than 0.5.

Small-strain material damping ratio, Dmin, decreases with increasing

confining pressure as shown in Figure 4.3b. As in the case of Gmax,

overconsolidated soils tend to exhibit a bilinear log Dmin – log σo’ relationship. A

change in slope is observed at σpm’. The slope of log Dmin – log σo’ relationship in

the normally consolidated range is slightly higher than the slope in the

overconsolidated range. The variation of void ratio, e, with confining pressure is

also presented in Figure 4.3.

4.5.2 Nonlinear Dynamic Soil Properties

Over the past three decades, numerous studies have been conducted

regarding dynamic soil properties and the parameters affecting them. Various

investigators have synthesized this work and proposed nonlinear generic curves

for use in earthquake analyses (e.g., Seed et al., 1986, for sands, and Vucetic and

64

Dobry, 1991, for soils with plasticity). Most of these generic curves proposed in

previous studies were derived from dynamic measurements at effective confining

pressures around one atmosphere.

The importance of effective confining pressure on the variation of shear

modulus, normalized shear modulus and material damping ratio with shearing

strain is illustrated in Figure 4.4. These measurements were performed on a

normally consolidated silty sand (SM) specimen (UTA-1-M in Table 3.11). The

specimen was tested at the estimated in-situ mean effective confining pressure of

0.5 atm. Then, the confining pressure was increased to four times the estimated

in-situ mean effective confining pressure and the specimen was tested at 2.0 atm

again in a normally consolidated state. All results shown in Figure 4.4 were

determined using the resonant column method; hence, each measurement

involved about 1000 cycles of loading in the frequency range of 43 to 94 Hz. The

effect of number of cycles on dynamic soil behavior is discussed in Section 4.7.

Figure 4.4 illustrates that normalized modulus reduction and material

damping curves become increasingly linear as confining pressure increases. Only

a few investigations (e.g., Iwasaki et al., 1978; Kokusho, 1980; Ni, 1987; and

Ishibashi and Zhang, 1993) have considered the effect of confining pressure on

dynamic soil properties. However, most of these studies were restricted to

pressures much less than 10 atmospheres.

65

150

100

50

0

G, MPa

Silty Sand (SM)

(a)

1.2

0.8

0.4

0.0

G/Gmax

(b)

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1

Shearing Strain, γ, %

(c)σo' ~ 0.5 atmσo' ~ 2.0 atm

Figure 4.4 The effect of confining pressure on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as measured in the torsional resonant column

66

As part of the ROSRINE project, numerous intact soil samples were

recovered over a depth range of 3 to 300 m. Some of these samples were tested

using combined resonant column and torsional shear (RCTS) equipment at

isotropic confining pressures ranging from 0.25 to 30 atmospheres. The results of

these tests show that depth as manifested through confining pressure, has a

significant impact on the shear modulus, normalized modulus reduction and

material damping curves for all soils (Stokoe et al., 1999). Typical representative

results from the ROSRINE project are illustrated in Figures 4.5 and 4.6. In

Figures 4.5a and b, average normalized modulus reduction curves for soils with

moderate plasticity and for nonplastic soils are presented, respectively. In Figure

4.6, the effect of confining pressure on normalized modulus reduction and

material damping curves is illustrated based on the results of the tests performed

on silty sands. However, only general trends were noted in the ROSRINE study.

Unfortunately, a quantitative model explaining these trends was not developed.

Hence, ROSRINE project formed the foundation for this study.

67

1.2

0.8

0.4

0.0

G/Gmax PI = 2 to 36 (%)Depth < 7.5 mDepth = 7.5 to 100 mDepth = 100 to 250 m

(a)

1.2

0.8

0.4

0.0

G/Gmax

0.0001 0.001 0.01 0.1 1Shearing Strain, γ, %

Non-Plastic SoilsDepth = 7.5 to 100 mDepth = 100 to 250 m

(b)

Figure 4.5 The effect of confining pressure on normalized modulus reduction curve (a) for soils with moderate plasticity, and (b) for non-plastic soils evaluated as part of the ROSRINE study (after Stokoe et al., 1999)

68

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


(b)

ROSRINE Study (Silty Sands) σo' = 0.25 atm σo' = 1.0 atm σo' = 4.0 atm σo' = 16 atm

Figure 4.6 The effect of confining pressure on (a) normalized modulus reduction and (b) material damping curves of silty sands evaluated as part of the ROSRINE study (after Darendeli et al., 2001)

69

Site response analyses were carried out to evaluate the impact of modeling

confining-pressure-dependent nonlinear soil properties on predicted ground

motions (Stokoe and Santamarina, 2000 and Darendeli et al., 2001). These

analyses indicate that utilizing a family of confining-pressure-dependent curves

results in larger intensity ground motions than those predicted with average

generic curves, particularly at periods less than about 1.0 sec. In Figure 4.7, a

comparison of the predicted ground motions in terms of spectral acceleration, Sa,

for a 120-m thick silty sand deposit shaken by the Topanga motion (Maximum

Horizontal Acceleration, MHA = 0.33g) is presented. The ratio of Sa predicted

using a family of confining-pressure-dependent curves (presented in Figure 4.6)

to those predicted with average generic curves is 81 % at a period of 0.3 sec and it

is 50 % at a period of 1.0 sec. This result is more pronounced for deeper sites

subjected to higher intensity input motions due to lower damping introduced by

the pressure-dependent curves. At longer spectral periods, the response is

dominated by the overall stiffness of the site. As a result, the confining-pressure-

dependent analyses tend to predict a smaller response at longer periods due to the

more linear response modeled by these curves.

70

2.5

2.0

1.5

1.0

0.5

0.0

S a, g

0.01 0.1 1 10

Period, T, sec

Pressure-DependentSoil PropertiesPressure-IndependentSoil PropertiesInput Motion

5 % Structural Damping

Figure 4.7 Impact on nonlinear site response of accounting for the effect of confining pressure on dynamic soil properties (after Darendeli et al., 2001)

4.6 EFFECT OF OVERCONSOLIDATION RATIO

Overconsolidation has an effect on the dynamic properties of soils,

particularly of those with plasticity. As an example, consider the resonant column

measurements on a kaolinite specimen presented in Figures 4.8 and 4.9. This

specimen has first been consolidated at 0.34 atm, tested at confining pressures

ranging from 0.09 to 1.36 atm in a loading sequence, and then unloaded to 0.34

atm and re-tested. These tests were performed in part to investigate the impact of

overconsolidation ratio on nonlinear behavior in both the initial loading and

unloading regions.

71

10

100

1000

Gmax,

Kaolinite Specimen

*MPa

(a)

0.1

1

10

Dmin, *

* Unloading

%

(b)

0.80

0.78

0.76

0.74

0.72

0.70

e

0.01 0.1 1 10Effective Isotropic Confining Pressure, σo', atm

Note: Specimen was consolidated at 0.34 atmbefore testing.

*

(c)

Figure 4.8 The effect of overconsolidation ratio on the variation of (a) shear modulus, (b) material damping ratio, and (c) void ratio with effective isotropic confining pressure as measured in the torsional resonant column

72

100

80

60

40

20

0

G, MPa

Kaolinite Specimen (a)

1.2

0.8

0.4

0.0

G/Gmaxσo' = 0.34 atm

Loading, OCR = 1.0Unloading, OCR = 4.0

(b)

15

10

5

0

D, %

10-5 10-4 10-3 10-2 10-1


Shearing strains in RC test werecorrected to the average of thefirst 3 free-vibration cycles.

(c)

Figure 4.9 The effect of overconsolidation ratio on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as measured in the torsional resonant column

73

As illustrated in Figure 4.8 and discussed in Section 4.5, overconsolidated

soils tend to exhibit some “memory” of stress history. As a result, Gmax is larger

and Dmin is smaller in the overconsolidated state. Therefore, small-strain dynamic

properties of overconsolidated soils are less sensitive to σo’. Once a specimen is

consolidated at a higher confining pressure (back to a normally consolidated

state), a substantial decrease in void ratio results in a change in overall soil

structure and therefore a change in dynamic behavior.

The difference in Gmax for the normally consolidated and overconsolidated

states results in different nonlinear shear modulus curves as shown in Figure 4.9a.

However, in the strain range that the measurements are performed, the normalized

modulus reduction curve exhibits only a slight difference for this material as

presented in Figure 4.9b. Material damping curves for the normally consolidated

and overconsolidated states are also observed to follow a similar trend, with the

D-log γ relationship shifting slightly to higher strain amplitudes along with a

slight decrease in Dmin as shown in Figure 4.9c. Nevertheless, overconsolidation

ratio should be expected to have some impact on nonlinear soil behavior, and it

should be accounted for in developing the next generation of normalized modulus

reduction and material damping curves.

74

4.7 EFFECT OF NUMBER OF CYCLES

The effect of number of cycles on G and D can be investigated using the

combined RCTS equipment. Figure 4.10 shows a comparison of the first and tenth

cycles (N = 1 and 10) of torsional shear tests, and resonant column test results

(assuming N ~ 1000 cycles) for a typical competent soil specimen that does not

undergo large volume changes during dynamic loading (UTA-1-M in Table 3.11).

As illustrated in Figure 4.10a, shear modulus stays constant and equal to

Gmax below an elastic threshold strain, which is nominally in the range of 0.001 %

to 0.01 %. The value of Gmax measured during RC testing is slightly higher than

Gmax measured during TS testing due to the effect of frequency in this strain range

as discussed in Section 4.8. As shearing strain increases above the elastic

threshold, G decreases nonlinearly with increasing γ. Shear modulus decreases in

a similar manner in both the RC and TS tests (Stokoe et al., 1999).

Number of loading cycles, N, has no effect on G until the cyclic threshold

strain (nominally in the range of 0.01 % and 0.1 %) is exceeded. Above the cyclic

threshold strain, G varies with γ and N. The value of G somewhat decreases with

increasing N at a constant γ. The effect of N on G can be influenced by soil type,

void ratio, confining pressure, degree of saturation and soil plasticity. However,

for the “competent” soils tested in this study, N has a minor impact on G as

shown in Figure 4.10a. The variation in normalized shear modulus, G/Gmax, with

the logarithm of shearing strain is shown in Figure 4.10b. The trends, which are

related to G, can also be easily observed in the G/Gmax – log γ curves.

75

100

80

60

40

20

0

G, MPa

Silty Sand (SM) (a)

1.2

0.8

0.4

0.0

G/Gmax

(b)

Note:σm' ~ 0.5 atm

20

15

10

5

0

D, %


(c)RC (~ 1000 Cycles)

TS 1st Cycle

TS 10th Cycle

Figure 4.10 The effect of number of loading cycles on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as determined in the combined RCTS testing

76

It is seen in Figure 4.10c that material damping is constant and equal to

Dmin at strains less than or equal to the elastic threshold strain, which is nominally

equal to or slightly less than that found for G. As with Gmax, there is a difference

between Dmin values determined in the RC and TS tests because of different

loading frequencies in the two tests. (This point is discussed in the following

section.) As γ increases above the elastic threshold, D increases significantly. A

cyclic threshold strain also exists for D. The cyclic threshold for D is observed to

be somewhat smaller than that found for G. (However, this result is assumed to

show that D is more sensitive to changes in γ around ctγ than G) Above the cyclic

threshold, D decreases as N increases with the importance of N increasing with γ.

Much of the decrease in D with increasing N occurs in the first 10 cycles as

shown in Figure 4.10c. When results of resonant column tests are compared with

the data collected during the tenth cycle of torsional shear testing, the effect of N

on G and D is observed to be overwhelmed by the effect of loading frequency

(discussed in Section 4.8). It is also interesting to note that N has a greater

influence on D than G (Stokoe et al., 1994; and Stokoe et al., 1999).

4.8 EFFECT OF LOADING FREQUENCY

The effect of excitation frequency, f, on Gmax and Dmin is shown in Figure

4.11 for an intact sandy lean clay (CL) specimen (UTA-1-J in Table 3.11). In this

case, the effect of excitation frequency on Gmax is small, averaging only about 10

% as frequency increases by an order of magnitude (for frequencies ranging from

1 Hz to 100 Hz) at a given confining pressure. On the other hand, the effect of

excitation frequency on Dmin is very significant above 1Hz, with Dmin increasing

77

by about 100 % over a log-cycle increase in excitation frequency. This effect is

clearly shown in Figure 4.11b, where all values of Dmin measured in the RC test

plot above values measured in the TS test at 1 Hz. It is also important to note that

the effect of f on Dmin is more pronounced at higher frequencies.

200

150

100

50

0

Gmax,RCTS

Isotropic Confining Pressure0.14 atm 0.27 atm0.61 atm 1.22 atm2.45 atm

(a)

MPa

10

8

6

4

2

0

Dmin, %

0.001 0.01 0.1 1 10 100 1000Loading Frequency, f, Hz


RCTS

(b)

Figure 4.11 The effect of loading frequency on (a) low-amplitude shear modulus, and (b) low-amplitude material damping ratio as determined in the combined RCTS testing

78

The effect of loading frequency on Gmax and Dmin can be easily compared

when the data collected over a frequency range are normalized with the value

measured at 1 Hz. Figure 4.12 shows such a generalized summary comparison

derived from testing numerous specimens. The relative widths of the bands

indicate how much more sensitive small-strain material damping ratio is to

frequency than small-strain shear modulus.

3

2

1

0

Gmax

Gmax 1Hz

or

Dmin

D min 1Hz

0.01 0.1 1 10 100Excitation Frequency, f, Hz

D min /D min 1Hz

Gmax /Gmax 1Hz

Inc.PI

Increasing PlasticityIndex, PI

Note:Intact Specimens of Soils with PIs = 0 to 35 %

3

2

1

0

Gmax

Gmax 1Hz

or

Dmin

D min 1Hz

0.01 0.1 1 10 100Excitation Frequency, f, Hz

D min /D min 1Hz

Gmax /Gmax 1Hz

Inc.PI

Increasing PlasticityIndex, PI

Note:Intact Specimens of Soils with PIs = 0 to 35 %

Figure 4.12 Comparison of the effect of loading frequency on low-amplitude shear modulus and low-amplitude material damping ratio (from Stokoe and Santamarina, 2000)

79

In Figure 4.13, the effect of excitation frequency on the variation of shear

modulus, normalized shear modulus and material damping ratio with shearing

strain are presented for the same clayey specimen. The G-log γ and G/Gmax-log γ

relationships are observed not to be very sensitive to frequency. On the other

hand, the effect of f on Dmin is observed to shift the D-log γ relationship over the

whole strain range that the sandy lean clay (CL) specimen (UTA-1-J in Table

3.11) is tested.

The effect of frequency is observed to be sensitive to soil type and

plasticity as discussed in the following section. Significant variability with soil

type has been observed in the effect of excitation frequency on Dmin as illustrated

by the wide band in Figure 4.12.

As a result, it may be important to capture the frequency dependence of

small-strain material damping in developing an empirical model to represent

dynamic soil behavior (particularly, for analyses that involve dynamic loading

with frequencies above 10 Hz). Unfortunately, this issue has not been addressed

in any of the generic relationships proposed in the literature.

80

100

80

60

40

20

0

G, MPa


(a)

1.2

0.8

0.4

0.0

G/Gmax

(b)

Note:σm' ~ 0.5 atm

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


(c)RC (~ 1000 Cycles)

TS 1st Cycle

TS 10th Cycle

Figure 4.13 The effect of loading frequency on the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain amplitude as determined in the combined RCTS testing

81

4.9 EFFECT OF SOIL TYPE

The effect of soil type is very important when considering linear and

nonlinear dynamic soil properties as discussed in Sections 4.9.1 and 4.9.2,

respectively. This effect also manifests itself in influencing the relative effect of

other parameters such as loading frequency, number of cycles, etc. as discussed

below.

4.9.1 Small-Strain Dynamic Soil Properties

Figure 4.14 shows a comparison of the variation in Gmax and Dmin with

effective isotropic confining pressure from RCTS testing of two specimens at

similar confining pressures. One of the specimens is a silty sand (SM) specimen

(UTA-1-M in Table 3.11) and the other is a sandy lean clay (CL) specimen

(UTA-1-J in Table 3.11).

As shown in Figure 4.14a, the sandy lean clay (CL) exhibits a memory of

loading history characterized by the bilinear log Gmax – log σo’ relationship while

the silty sand (SM) follows almost a straight line. RC and TS test results denoted

with solid and open symbols in this figure are observed to be very close. This

figure also confirms that Gmax is not very sensitive to excitation frequency.

On the other hand, the values of Dmin in the log Dmin – log σo’

relationships presented in Figure 4.14b are quiet different for the two material

types. The sandy lean clay (CL) has much higher damping than the silty sand

(SM). This finding is consistent with the general trends reported in the literature

such that small-strain material damping increases with increasing soil plasticity

(Stokoe et al., 1994; and Stokoe et al., 1999).

82

10

100

1000

G,

RC TS Soil TypeSilty Sand (SM)Sandy Lean Clay (CL)

(a)

MPa

0.1

1

10

Dmin ,%

0.1 1 10Effective Isotropic Confining Pressure, σo', atm

(b)

Figure 4.14 The effect of soil type on the variation of (a) low-amplitude shear modulus, and (b) low-amplitude material damping ratio with effective isotropic confining pressure as determined in the combined RCTS testing

Unfortunately, most of the generic curves used in state of practice (e.g.,

Vucetic and Dobry, 1991) are not accurate in terms of representing this trend in

Dmin. These generic curves were synthesized from studies generally performed

with relatively older cyclic testing equipment. Because of accuracy problems at

small strains, damping measurements were not performed and data were

extrapolated to represent the small-strain behavior.

83

It is also important to note that the Dmin values measured in the RC and TS

tests are observed to be quite different for both of these materials. This is due to

the effect of excitation frequency as discussed in Section 4.8. The effect of

frequency is more pronounced on the sandy lean clay (CL). Strain-rate effects

being more pronounced in plastic soils is a well-known phenomenon reported in

the geotechnical engineering literature.

Figures 4.15 and 4.16 show the impact of excitation frequency for these

two soils in more detail. Results of torsional shear tests performed at several

frequencies well below resonance and resonant column tests performed at a

relatively higher frequency are presented for the sandy lean clay (CL) and for the

silty sand (SM) in these figures.

Figures 4.15a and 4.16a show the Gmax and Dmin measurements,

respectively, while Figures 4.15b and 4.16b present the same data using a

different perspective. The data in Figures 4.15b and 4.16b have been normalized

with the TS measurements at 1Hz in order to indicate the sensitivity of the small-

strain dynamic properties to excitation frequency.

It is important to note the scales used in Figures 4.15b and 4.16b. An

increase in Gmax on the order of 5 % to 10 % is observed per order of magnitude

increase in frequency (between 10 Hz and 100 Hz) for the silty sand (SM) and the

sandy lean clay (CL), respectively. On the other hand, an increase in Dmin on the

order of 80 % to 120 % is presented in Figure 4.16b over the same frequency

range.

84

100

80

60

40

20

0

Gmax,

Silty Sand (SM)Sandy Lean Clay (CL)

RCTS

(a)

MPa

1.2

1.1

1.0

0.9

0.8

Gmax

0.01 0.1 1 10 100

Loading Frequency, f, Hz

RCTS

(b)

Gmax1Hz

Figure 4.15 The effect of soil type on the variation of low-amplitude shear modulus with loading frequency as determined in the combined RCTS testing

In general, frequency effects on modulus may be considered small to

negligible keeping in mind: 1) variability in soil conditions and 2) how

representative a small specimen may be in estimating field stiffness. On the other

hand, a considerable change in small-strain damping, which can potentially shift

the whole D – log γ relationship, should be captured in empirical curves for

certain frequency ranges.

85

5

4

3

2

1

0

Dmin ,%


RCTS

(a)

3

2

1

0

Dmin

0.01 0.1 1 10 100

Loading Frequency, f, Hz

RCTS

(b)

Dmin1Hz

Figure 4.16 The effect of soil type on the variation of low-amplitude material damping ratio with loading frequency as determined in the combined RCTS testing

4.9.2 Nonlinear Dynamic Soil Properties

In terms of nonlinear soil behavior, the G/Gmax – log γ and D – log γ

curves for the SM and CL specimens discussed in the previous section are shown

in Figures 4.17 and 4.18. The comparison of the nonlinear behavior of the two

soils does not represent a wide range of soil types. However, Figure 4.17 indicates

a shift in normalized modulus reduction curve with changing soil type.

86

1.2

0.8

0.4

0.0

G/Gmax



Note:σm' ~ 0.5 atm

Figure 4.17 The effect of soil type on the normalized modulus reduction curve as measured in the torsional resonant column

Material damping curves determined using both the resonant column and

the torsional shear methods are shown in Figure 4.18. The data collected during

the first and the tenth cycles of torsional shear test indicates that the silty sand

(SM) is more sensitive to number of cycles than the sandy lean clay (CL). Also,

comparison of the resonant column and torsional shear data illustrates higher

sensitivity of sandy lean clay (CL) to loading frequency than the silty sand (SM).

In Figure 4.19, normalized modulus reduction and material damping

curves of five different soils with a wide range of plasticity are presented. All of

these soils were tested at similar confining pressures and both sets of curves are

observed to shift to higher strains as plasticity index, PI, increases. This trend

agrees with all empirical curves presented in the literature, which show the effect

of PI on normalized modulus reduction and material damping curves, (e.g., Sun et

al., 1988; Idriss, 1990; Vucetic and Dobry, 1991; and Ishibashi and Zhang, 1993).

87

20

15

10

5

0

D, %


(a)

RC Test

20

15

10

5

0

D, %

(b)

Note:σm' ~ 0.5 atm

TS Test 1st Cyclef = 1 Hz

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


(c)TS Test 10th Cyclef = 1 Hz

Figure 4.18 The effect of soil type on the material damping curve determined at (a) N ~ 1000 cycles, (b) N = 1 cycle, and (c) N = 10 cycles from combined RCTS testing

88

1.2

0.8

0.4

0.0

G/Gmax

RC TestPI = 10 %PI = 15 %PI = 36 %PI = 79 %Peat

(a)

20

15

10

5

0

D, %

(b)σm' ~ 0.5 atmRC Test

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


(c)TS Test 10th Cyclef = 1 Hz

Figure 4.19 The effect of soil type on normalized modulus reduction and material damping curves (after Stokoe et al., 1999)

89

The values of Dmin increase with increasing PI as presented in Figure 4.19.

On the other hand, the values of D at high strains (γ ~ 0.1 %) decrease as PI

increases also as shown in this figure. This rather complex relationship between

the D - log γ curves for different soils is not shown in any empirical curves. It has

been presented in a general sense in Electric Power Research Institute, EPRI

(1993b and c), based on RCTS tests of intact soil specimens tested at UT. This

behavior has also been observed by Vucetic et al. (1998) following the EPRI

study.

The general switching in the relative positions of the D- log γ curves for

the different soil types is best shown in the TS tests at an excitation frequency of 1

Hz. The effect of excitation frequency impacts the RC measurements and the

effect of number of loading cycles impacts the tenth-cycle TS measurements.

The effect of σο’ on the normalized modulus reduction and material

damping curves is discussed in Section 4.5. The general trend shows both of the

curves shifting to higher strains. In case of D - log γ relationships, the curves

simultaneously shift downward. As a result, D decreases slightly at a given γ as

σο’ increases. In general terms, the largest shift is shown by nonplastic soils, with

the effect decreasing with increasing plasticity (Stokoe et al., 1994; and Stokoe et

al., 1999). Similar behavior is observed for normalized shear modulus, except no

shift occurs at small strains with increasing σο’ because the curve is normalized

with Gmax.

90

4.10 EFFECT OF SAMPLE DISTURBANCE

Although laboratory testing methods have been standardized and

laboratory testing equipment has been significantly improved over the past several

decades, estimating accurate engineering properties of soils has always been and

will always be a challenge for geotechnical engineers. The data collected in the

laboratory should always be evaluated in terms of its success in representing the

in-situ conditions. Beside scaling effects (due to characterizing a soil deposit with

specimens that are only a few 100 cubic centimeters in volume), effects of the

sampling operation on the laboratory measurements of an engineering property

have to be taken into account prior to utilizing a laboratory test result in design.

4.10.1 Effect of Disturbance on Gmax = ρVs2

Figure 4.20 shows the shear-wave velocity measurements at one of the

sites (La Cienega located in West Hollywood) characterized as part of the

ROSRINE project. Two sets of data are presented in this figure: 1) in-situ seismic

(OYO logger and crosshole) measurements, and 2) laboratory measurements.

Comparison of the two sets of data clearly shows a discrepancy between field and

laboratory values of shear wave velocity, Vs, with laboratory Vs values generally

lower than the field measurements.

91

300

250

200

150

100

50

0D

epth

, m

10008006004002000

Shear Wave Velocity, m/sec

OYO Logger MeasurementsCrosshole MeasurementsLaboratory Measurements

Figure 4.20 Comparison of field and laboratory measurements of shear wave velocity at the La Cienega site in the ROSRINE project

10

8

6

4

2

0

300200100

Vs, m/sec

92

It is important to note that these samples were recovered and tested

following procedures that should be considered high-quality, state-of-the-art

practice. The ratios of laboratory measurements to field measurements at this site

range from 0.63 to 1.07. Part of the difference between the field and laboratory

values should be attributed to variability of soil conditions at the site and how

representative a small test specimen can possibly be relative to the soil deposit.

However, an important reason for the discrepancy between laboratory and

field values of Vs is due to the fact that the sampling process itself causes a

reduction in the soil stiffness by “damaging” the existing structure and

cementation of the soil material that has occurred due to aging under some state of

stress for thousands to millions of years. This phenomenon is also discussed in

Anderson and Woods (1975) and shear wave velocities measured in the

laboratory are generally characterized as being slightly less to considerably less

than the in-situ values.

A summary of 40 comparisons from the ROSRINE study is presented in

Figure 4.21. The data indicate that sampling disturbance is more pronounced in

stiffer soils. It is important to note that the shear modulus is proportional to the

square of shear wave velocity. As a result, a reduction of 40 % in shear wave

velocity due to sampling disturbance means a reduction of 64 % in small-strain

shear modulus. This comparison indicates the need for in-situ measurement of

Gmax at critical sites that are being characterized for geotechnical earthquake

engineering purposes.

93

900

600

300

0

1.50.50.0

0.25 1.00 2.000.10 0.50 0.80 1.50

Modulus Ratio, Gmax, lab / Gmax, fieldV

S, fi

eld,

m/s

ec

Velocity Ratio, VS, lab / VS, field

Range fromROSRINE

Study

General Trend

1.0900

600

300

0

1.50.50.0

0.25 1.00 2.000.10 0.50 0.80 1.50

Modulus Ratio, Gmax, lab / Gmax, fieldV

S, fi

eld,

m/s

ec

Velocity Ratio, VS, lab / VS, field

Range fromROSRINE

Study

General Trend

1.0

Figure 4.21 Variation of sampling disturbance expressed in terms of Vs, lab/Vs, field and Gmax, lab/Gmax, field with the in-situ shear wave velocity

4.10.2 Effect of Disturbance on Dmin

Theoretically, it is possible to estimate the small-strain material damping

ratio from in-situ seismic measurements. Crosshole and downhole test results may

be used to evaluate the in-situ material damping ratio. The SASW method may be

extended to permit in-situ measurements of material damping ratio in addition to

shear wave velocity (Lai and Rix, 1998; and Rix et al., 2000). Response of

instrumented soil deposits to earthquakes and aftershocks can be analyzed in

order to estimate in-situ shear wave velocity and material damping ratio.

However, the accuracy of all of these methods is still questionable and no robust

field method exists today.

94

Most of these methods typically assume a horizontally layered system and

utilize attenuation of wave amplitude with distance from the source.

Backscattering of waves due to the contrast between soil layers and lateral

variability in the soil deposit causes significant uncertainty regarding the

estimates of in-situ material damping ratio. The quality of the estimate is further

reduced by geometric attenuation, which generally has a more significant impact

on attenuation with distance than material damping ratio.

As an example, Figure 4.22 shows independent measurements of material

damping ratio from field and laboratory tests. These field crosshole measurements

and the laboratory measurements were conducted by UT personnel (Fuhriman,

1993; and Hwang, 1997) and are presented in EPRI (1993a and b) along with a

discussion of the data collection and analysis procedures. The material damping

ratios from surface wave measurements were performed by Lai and Rix (1998)

and are generally less than those from crosshole testing possibly due to: 1)

different attenuation mechanisms which control at higher frequencies and produce

frequency-dependent damping ratios, 2) different volumes of soil sampled by the

methods, and 3) uncoupled analyses of Vs and Dmin. On the other hand, values of

damping ratio from the surface wave tests agree more closely with values from

resonant column and torsional shear laboratory tests than those estimated using

the crosshole method. These results seem to indicate that laboratory estimates of

Dmin can be used with some “judgment” for evaluation of soil deposits for

geotechnical earthquake engineering purposes. However, many more studies are

warranted in this area.

95

Damping Ratio, %

0

3

6

9

12

0 2 4 6 8 10

Surface WaveCrosshole (UT)Resonant Column (UT)Torsional Shear (UT)

Depth, m

Damping Ratio, %

0

3

6

9

12

0 2 4 6 8 10

Surface WaveCrosshole (UT)Resonant Column (UT)Torsional Shear (UT)

Depth, m

Figure 4.22 Comparison of laboratory and field measurements of small strain material damping ratio (from Stokoe et al., 1999)

4.10.3 Effect of Disturbance on Nonlinear Behavior

4.10.3.1 Comparison of Back-calculated Curves with Laboratory Test Results

Zeghal et al. (1995) back-calculated the variations of shear modulus and

material damping ratio with shearing strain amplitude using stress-strain histories

calculated from the free-field downhole accelerations at the Lotung site in Taiwan

from which specimens discussed in Section 3.5 were taken. The comparison of

the back-calculated nonlinear soil properties with the data collected at the

University of Texas at Austin is presented in Figure 4.23.

96

(a)(a)

(b)(b)

Figure 4.23 Comparison of nonlinear soil properties back-calculated from the free-field downhole accelerations with the laboratory measurements (from Zeghal et al., 1995)

Estimates before peak shearing strain

Statistical Fit

97

Figure 4.23a indicates a good correlation between the normalized modulus

reduction curves estimated based on in-situ seismic response and the data

collected in the laboratory. There is a significant difference between the “field”

and laboratory material damping curves in Figure 4.23b. It is felt that this

difference can be attributed to different attenuation mechanisms and different

volumes of soil sampled by the two methods. However, if these curves were

utilized in ground motion analysis for earthquake resistant design, it is important

to note that the material damping curve measured in the laboratory would result in

a more conservative design than the field estimate.

4.10.3.2 Comparison of Test Results on Intact and Reconstituted Specimens

Disturbance was defined above as “damaging” the existing structure and

cementation of the soil material that has been aging under some state of stress for

thousands to millions of years. A similar process can be simulated in the

laboratory by breaking an “undisturbed” soil sample into small pieces, destroying

the existing soil structure completely and reconstituting it. This extreme situation

associated with disturbance is called remolding. Comparison of the measurements

on undisturbed and remolded soils can be utilized as an indicator of the sensitivity

to disturbance of a given engineering property for a given soil type.

98

As an example, consider the comparisons of dynamic soil properties

shown in Figures 4.24 and 4.25. In these figures, data collected from RCTS

testing of an undisturbed specimen and a remolded specimen are presented. The

first specimen was trimmed from a poorly graded sand (SP-SM) sample from

Idaho Falls (Stokoe et al., 1998c). Then, this specimen was remolded and a

second specimen was reconstituted from this material at a similar unit weight.

In Figure 4.24, the variations of small-strain shear modulus, small-strain

material damping ratio and void ratio are presented. The values of Gmax and Dmin

are observed to exhibit very similar relationships although this comparison

represents an extreme case of disturbance. Gmax values at lower confining

pressures are smaller for the remolded specimen since the undisturbed specimen

has some “memory” of state of stress in the field which is characterized by the

bilinear log Gmax – log σo’ relationship. Dmin values are quite similar (considering

the limited accuracy of damping measurements, they are essentially identical).

Figure 4.25 shows the comparison of the nonlinear soil behavior of the

undisturbed and remolded specimens at a mean effective confining pressure of

0.82 atm. About 20 % difference between the shear modulus of the undisturbed

and remolded specimens is easily recognized in Figure 4.25a at small-strain

amplitudes. On the other hand, the normalized modulus reduction curves

(presented in Figure 4.25b) for these two specimens are almost identical. The D –

log γ relationship (shown in Figure 4.25c) is nearly the same up to γ of about 0.01

%. At higher strains, the reconstituted specimen exhibits increasingly higher

values of D.

99

10

100

1000

Gmax,

MPa

(a)

0.1

1

10

Dmin, Poorly Graded Sandwith Silt (SP-SM)

Intact (γ t = 1.64 gr/cm3)

Reconstituted (γ t = 1.65 gr/cm3)

%

(b)

0.8

0.7

0.6

0.5

0.4

e

0.1 1 10Effective Isotropic Confining Pressure, σo

′, atm

(c)

Figure 4.24 Comparison of the variation of (a) low-amplitude shear modulus, (b) low-amplitude material damping ratio, and (c) void ratio with effective isotropic confining pressure of intact (undisturbed) and reconstituted (remolded) specimens

100

200

150

100

50

0

G, MPa

(a)

σo' = 0.82 atm

1.2

0.8

0.4

0.0

G/Gmax

(b)

20

15

10

5

0

D, %

10-5 10-4 10-3 10-2 10-1 100


Poorly Graded Sandwith Silt (SP-SM)

Intact (γ t = 1.64 gr/cm3)

Reconstituted (γ t = 1.65 gr/cm3)

(c)

Figure 4.25 Comparison of the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain of intact (undisturbed) and reconstituted (remolded) specimens

101

4.10.3.3 Comparison of Test Results on Companion Specimens

As part of the ROSRINE project, companion samples from a site in

Southern California (La Cienega) were tested at the University of California at

Los Angeles (UCLA) and the University of Texas at Austin (UT).

Test results in terms of the shear modulus, normalized modulus reduction

and material damping curves are presented in Figure 4.26. The double-specimen

direct simple shear, DSDSS, (Doroudian and Vucetic, 1995) tests were performed

by Prof. Vucetic and students at UCLA. These results are shown by the solid

symbols in this figure. The resonant column and torsional shear test results

performed at UT are shown by open symbols. The value of shear modulus

estimated based on the in-situ seismic crosshole measurements by UT personnel

is also presented in Figure 4.26a for comparison purposes.

Each UCLA and UT companion specimen was recovered from the same

undisturbed sample, and each companion specimen was tested at equivalent

effective stresses based on an effective coefficient of earth pressure at rest of

about 0.5. It is important to note that the RCTS and DSDSS confinement states

are isotropic and anisotropic, respectively.

102

120

80

40

0

G, MPa

RC (UT)TS (UT)DSDSS (UCLA)CrossholeSeismic (UT)

(a)

1.2

0.8

0.4

0.0

G/Gmax

(b)

Equivalent σo' ~ 1.1 atm

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1 10


(c)Silty Sand (SM)

Figure 4.26 Comparison of the variation of (a) shear modulus, (b) normalized shear modulus, and (c) material damping ratio with shearing strain measured using various equipment on companion soil samples (from Stokoe et al., 1999)

103

The main difference between the results from the tests performed on

companion specimens exists in shear modulus. The G/Gmax – log γ relationships

are nearly identical as are the D - log γ relationships from RCTS and DSDSS

testing. The in-situ seismic value of Gmax is above values determined in the

laboratory due to sampling disturbance. In this case, the field value is about 50%

greater than the average value determined in the laboratory. Figure 4.26 shows

that normalized modulus reduction and material damping curves fall on top of

each other for the two tests with different stress states. These data seem to

indicate that as long as a soil specimen is tested at the in-situ mean effective

confining pressure, the nonlinear soil behavior (characterized by normalized

modulus reduction and material damping curves) should be determined with

reasonable precision even though the anisotropic state of stress in the field is not

duplicated. Thus, RCTS testing under isotropic confinement is a robust means of

characterizing nonlinear soil behavior.

4.10.3.4 Final Remarks on Effects of Disturbance

At this point in time, it is not possible to conduct field measurements to

evaluate shear modulus and material damping at working strains during design

level ground shaking. The data presented in this section strongly support the state-

of-practice of scaling normalized modulus reduction curve from laboratory with

in-situ Gmax. The state-of-practice of utilizing the laboratory material damping

curve without any modification obviously needs more investigation and should be

used prudently at this time.

104

However, a geotechnical engineer always has to consider the

consequences of attempting to estimate engineering properties from small

samples. Problems associated with in-situ seismic testing (backscattering of

waves due to the contrast between soil layers and lateral variability in the soil

deposit) are not errors in testing procedures. These phenomena are a part of the

dynamic response of a soil deposit too complex (and therefore too expensive) to

model in most geotechnical investigations. A discrepancy will always exist

between a model based on engineering properties measured in the laboratory and

actual field performance. Not being able to model some phenomenon does not

endorse ignoring it. Instead, the engineer has to overcome such challenges by

using judgment based on facts and experience.

4.11 SUMMARY

In this chapter, dynamic soil properties (G and D) and parameters that

affect them are discussed. Table 4.1 shows a list of these parameters and their

relative importance in terms of affecting normalized modulus reduction and

material damping curves. Trends reported in the literature and observations made

during the course of this work are summarized. The importance of accounting for

the impact of soil type and loading conditions in developing a new generation of

design curves is addressed.

105

Table 4.1 Parameters that control nonlinear soil behavior and their relative importance in terms of affecting normalized modulus reduction and material damping curves based on general trends observed during the course of this study

Parameter Impact on Normalized Modulus Reduction Curve

Impact on Material Damping Curve

Strain Amplitude *** ***Mean Effective Confining Pressure *** ***Soil Type and Plasticity *** ***Number of Loading Cycles *+ ***++

Frequency of Loading (above 1 Hz) * **Overconsolidation Ratio * *Void Ratio * *Degree of Saturation * *Grain Characteristics, Size, Shape, Gradation, Mineralogy * *

*** Very Important + On competent soils included in this study ** Important ++ Soil Type Dependent * Less Important

The effect of sampling disturbance on measured dynamic soil properties is

also discussed. Regarding the dynamic response of soil deposits, the data indicate

that: 1) in-situ measurement of Gmax is needed, 2) the normalized shear modulus

curves measured in the laboratory are not very sensitive to disturbance, and 3) the

material damping curves measured in the laboratory are the only estimates of

nonlinear material damping in the field at this point in time and should be used

cautiously.

106

The state-of-practice in geotechnical earthquake engineering involves

scaling a normalized modulus reduction curve from the laboratory by the in-situ

Gmax and utilizing the laboratory material damping curve as is. The findings

presented in this chapter support the adequacy of the state-of-practice provided

that the engineer accounts for discrepancies that might arise as a result of scaling

effects and variability of soil conditions at the site.

107

CHAPTER 5

EMPIRICAL RELATIONSHIPS

5.1 INTRODUCTION

Empirical curves which represent G/Gmax – log γ and D – log γ are widely

used in geotechnical earthquake engineering practice. The most common of these

curves are reviewed in this chapter. The strengths and weaknesses of the

empirical curves in estimating nonlinear soil behavior are discussed. This

discussion is presented in order to justify the need for an improved set of

empirical curves and equations that can be utilized in earthquake ground response

analyses, soil dynamics applications regarding base isolation problems, design of

machine (dynamically loaded) foundations, and in many other cases that require

prediction of strains under working loads.

5.2 HARDIN AND DRNEVICH (1972) DESIGN EQUATIONS

The first comprehensive study in which the parameters that control

nonlinear soil behavior were identified was the study by Hardin and Drnevich

(1972a and b). This study was published in the University of Kentucky reports

UKY 26-70-CE2 (Hardin and Drnevich, 1970a) and UKY 27-70-CE3 (Hardin

and Drnevich, 1970b). Table 5.1 shows the list of these parameters and their

relative importance in terms of their effect on shear modulus and material

damping based on their research.

108

Table 5.1 Parameters that control nonlinear soil behavior and their relative importance in terms of affecting shear modulus and material damping (Hardin and Drnevich, 1972b)

Clean Sands

Cohesive Soils

Clean Sands

Cohesive Soils

Strain Amplitude *** *** *** ***Mean Effective Confining Pressure *** *** *** ***

Void Ratio *** *** *** ***

Number of Loading Cycles + * *** ***

Degree of Saturation * *** ** -

Overconsolidation Ratio * ** * **

Effective Strength Envelope ** ** ** **

Octahedral Shear Stress ** ** ** **Frequency of Loading (above 0.1 Hz) * * * **Other Time Effects (Thixotropy) * ** * **Grain Characteristics, Size, Shape, Gradation, Mineralogy * * * *

Soil Structure * * * *Volume Change Due to Shearing Strain below 0.5 % - * - *

Impact on Modulus Impact on DampingParameter

*** Very Important ** Less Important * Relatively Unimportant + Relatively Unimportant Except for Saturated Sand - Unknown

109

Hardin and Drnevich (1972b) also proposed that a hyperbolic relationship

can be used to relate shear stress and shearing strain in modeling dynamic soil

behavior. The Hyperbolic model, illustrated in Figure 5.1a, can be expressed as:

maxmax

1τ

γγτ+

=

G

(5.1)

where: τ = shear stress,

γ = shearing strain,

Gmax = small-strain shear modulus, and

τmax = shear strength of the soil.

In this model, reference strain is defined as:

max

max

Grτ

γ = (5.2)

By dividing both sides of Equation 5.1 by γ, the secant shear modulus, G, is

obtained:

maxmax

11

τγ

+=

G

G (5.3)

The normalized modulus reduction curve can be evaluated from Equation

5.3 by rearranging the equation as follows:

r

GG

γγ

+=

1

1

max

(5.4)

110

1

G

1Gmax

(γr,τmax)τ

γ

τ =γ

1 γGmax τmax

+1

G

1Gmax

(γr,τmax)τ

γ

τ =γ

1 γGmax τmax

+

SAND

CLAY

HYPERBOLIC

τ

γ

SAND

CLAY

HYPERBOLIC

τ

γ

a. Hyperbolic stress-strain relationship

b. Effect of soil type on stress-strain relationship

1

G

1Gmax

(γr,τmax)τ

γ

τ =γ

1 γGmax τmax

+1

G

1Gmax

(γr,τmax)τ

γ

τ =γ

1 γGmax τmax

+

SAND

CLAY

HYPERBOLIC

τ

γ

SAND

CLAY

HYPERBOLIC

τ

γ

a. Hyperbolic stress-strain relationship

b. Effect of soil type on stress-strain relationship

Figure 5.1 Hyperbolic soil model proposed by Hardin and Drnevich (1972b)

111

Hardin and Drnevich (1972b) also proposed an approximate shape for the

material damping curve as:

r

r

DD

γγ

γγ

+=

1max (5.5)

where Dmax is the maximum damping ratio of the soil that depends on soil type,

confining pressure, number of cycles and loading frequency.

Also as shown in Figure 5.1b, Hardin and Drnevich (1972b) observed that

soil type has an impact on the stress-strain relationship. Measured stress-strain

curves deviate from the simple mathematical model depending on the soil type.

As a result, they proposed to approximate observed soil behavior by distorting the

strain scale to make the measured stress-strain curve have a hyperbolic shape. For

this purpose, they defined a hyperbolic strain, hγ , which replaces the rγγ / term

in Equations 5.4 and 5.5. Hyperbolic strain is defined as:

−+=

rrh ba

γγ

γγγ *exp*1 (5.6)

where “a” and “b” are coefficients that adjust the shape of the stress-strain curve

for soil type, number of cycles and loading frequency. Figure 5.2 shows the

normalized modulus reduction and material damping curves estimated based on

the hyperbolic model.

112

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

0.01 0.1 1 10 100Hyperbolic Strain, γh

1.0

0.8

0.6

0.4

0.2

0.0

D/Dmax

G/Gmax = 1 / (1 + γh)D/Dmax = 1 - G/Gmax

Figure 5.2 The normalized modulus reduction and material damping curves estimated based on the hyperbolic model

The empirical equations proposed by Hardin and Drnevich (1972b)

account for the effects of plasticity index, overconsolidation ratio and confining

pressure mainly through adjusting reference strain. Effects of soil type, number of

loading cycles, loading frequency and saturation are taken into consideration by

adjusting Dmax in Equation 5.5 and the “a” and “b” coefficients in Equation 5.6.

Hardin and Drnevich (1972b) proposed graphs and equations based on their

research and experience. The complexity of the procedure in calculating the

normalized modulus reduction and material damping curves limited utilization of

this work in practice. However, their work represented an enormous step forward

in characterizing dynamic soil behavior.

113

5.3 EMPIRICAL RELATIONSHIPS

Numerous other researchers have been influenced by the Hardin and

Drnevich (1972a and b) work and have attempted to refine, improve and

generalize their results. In these other studies, “average” normalized modulus

reduction and material damping curves have been presented. Many of these

curves are widely accepted and utilized in practice. In this section, the strengths

and weaknesses of these empirical curves in estimating nonlinear soil behavior

are discussed.

The effect of mean effective confining pressure on normalized modulus

reduction curves is presented in Figure 5.3 based on resonant column and

torsional shear tests using hollow specimens (Iwasaki et al., 1978). These tests

were performed on saturated clean sand specimens under drained conditions. The

confining pressure used in RCTS testing ranged between 0.25 atm and 2.0 atm.

The results reported in Iwasaki et al. (1978) are consistent with the general trends

outlined in Chapter Four. However, the normalized modulus reduction curves are

found to be somewhat more linear compared to those observed during the course

of this study. The discrepancy is believed to result from the uniform grain size

distribution of the clean sand relative to the natural soils tested as part of this

work.

114

Iwasaki et al. (1978)0.25 atm0.5 atm1.0 atm2.0 atm

0.0001 0.001 0.01 0.1 1Shearing Strain,γ, %

1.2

0.8

0.4

0.0

G/Gmax




1.2

0.8

0.4

0.0

G/Gmax


1.2

0.8

0.4

0.0

G/Gmax

Figure 5.3 The effect of confining pressure on normalized modulus reduction curve for Toyoura Sand (Iwasaki et al., 1978)

Results of cyclic triaxial tests on specimens made of the same saturated

clean sand are presented in Figure 5.4 (Kokusho, 1980). The effect of mean

effective confining pressure (cell pressure) was quantified for material damping

curves as well as for normalized modulus reduction curves by Kokusho (1980).

As in the case of Iwasaki et al. (1978), the range of cell pressures is restricted to

low confining stresses.

Ni (1987) also reported results of tests on clean sands (shown in Figure

5.5) using RCTS equipment at University of Texas at Austin.

The results reported in Iwasaki et al. (1978), Kokusho (1980) and Ni

(1987) are consistent with each other showing that RCTS and cyclic triaxial test

methods yield similar measurements for clean sand specimens. The normalized

modulus reduction and material damping curves are observed to shift to higher

strains and Dmin is observed to decrease with increasing confining pressure

consistent with the general trends outlined in Chapter Four.

115

20

15

10

5

0

D , %


Kokusho (1980)0.2 atm0.5 atm1.0 atm

(b)

2.0 atm3.0 atm

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D , %



(b)

2.0 atm3.0 atm

20

15

10

5

0

D , %



(b)

2.0 atm3.0 atm

1.2

0.8

0.4

0.0

G/Gmax

(a)1.2

0.8

0.4

0.0

G/Gmax

(a)

Figure 5.4 The effect of confining pressure on (a) normalized modulus reduction, and (b) material damping curves for Toyoura Sand (Kokusho, 1980)

116

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D , %

0.0001 0.001 0.01 0.1 1Shearing Strain, γ , %

Ni (1987) 0.41 atm 0.82 atm 1.63 atm 3.27 atm

(b)

Figure 5.5 The effect of confining pressure on (a) normalized modulus reduction, and (b) material damping curves for non-plastic soils (Ni, 1987)

117

The sand curves, first proposed by Seed and Idriss (1970), and then re-

analyzed and re-proposed by Seed et al. (1986), are shown in Figure 5.6. These

sand curves are found to be more consistent with results measured in this study.

The upper and lower ranges in Seed et al. (1986) can be attributed to: 1)

variability in the characteristics of the granular particles (shape, size, gradation

and mineralogy), 2) variability in nonlinear soil behavior, 3) accuracy in

measurements, and 4) effect of confining pressure. The upper and lower ranges

are observed to correspond to silty sand behavior at confining pressures ranging

from about 0.25 atm to about 4 atm. The data, which Seed et al. (1986)

synthesized, are the results of tests performed on natural sands in this pressure

range.

The curves for soils with plasticity which were proposed by Sun et al.

(1988) are presented in Figure 5.7. The normalized modulus reduction curves

proposed by Sun et al. (1988) account for the effect of plasticity on nonlinear soil

behavior while the material damping curves are presented in terms of one mean

curve and a generalized range in the data over the range of soil plasticities. The

data, which Sun et al. (1988) synthesized, are also the results of tests performed at

confining pressures ranging from about 0.25 atm to about 4 atm. The lack of

correlation they found in material damping is due, at least in part, to the

difficulties that exist in performing material damping measurements, especially at

moderate to small strains.

118

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D, %


Seed et al., (1986)Average for SandsRange

(b)

Figure 5.6 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Seed et al. (1986)

119

1.2

0.8

0.4

0.0

G/Gmax

Sun et al.,(1988)PI = 5 -10 %PI = 10 -20 %PI = 20 -40 %PI = 40 -80 %PI = >80 %

(a)

20

15

10

5

0

D, %


Sun et al.,(1988)Average for ClaysRange

(b)

Figure 5.7 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Sun et al. (1988) for soils with plasticity

120

The curves proposed by Idriss (1990) are presented in Figure 5.8. Two

“average” normalized modulus reduction curves are proposed by Idriss (1990): 1)

for sands and 2) for clays. On the other hand, a single material damping curve is

proposed for all soil types. The normalized modulus reduction curve proposed for

sands is consistent with Seed et al. (1986) while the modulus reduction curve

proposed for clays represents a high (about 50 %) plasticity clay based on the

curves proposed by Vucetic and Dobry (1991) which are discussed below. The

proposed material damping curve is similar to the lower bound curve proposed by

Seed et al. (1986). A unified material damping curve for all soil types can also be

attributed to uncertainty in damping measurements.

The curves proposed by Vucetic and Dobry (1991) are presented in Figure

5.9. The normalized modulus reduction and material damping curves proposed in

their study account for the effect of plasticity on nonlinear soil behavior.

However, the values of small-strain damping, Dmin, have been left somewhat

undefined due to the lack of small-strain data. As shown in Figure 5.9, the value

of Dmin is predicted to decrease with increasing soil plasticity, while the opposite

trend is observed during the course of this study as discussed in Chapter Four.

The data, which Vucetic and Dobry (1991) synthesized, are also the results of

tests performed at confining pressures ranging from about 0.25 atm to about 4

atm.

121

1.2

0.8

0.4

0.0

G/GmaxIdriss (1990)

For SandsFor Clays

(a)

20

15

10

5

0

D, %


Idriss (1990)For Sands and Clays

(b)

Figure 5.8 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Idriss (1990)

122

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D, %


Vucetic and Dobry (1991)Non-PlasticPI = 15 %PI = 30 %PI = 50 %PI = 100 %PI = 200 %

(b)

Figure 5.9 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Vucetic and Dobry (1991)

123

The curves proposed by Ishibashi and Zhang (1993) are presented in

Figures 5.10 and 5.11. In their study, a set of equations, which generate

normalized modulus reduction and material damping curves changing with

confining pressure, oσ , and soil plasticity, PI, are proposed. The equations

associated with the normalized modulus reduction curve are: omPIm

oPIKG

G −= ),(

max),( γσγ (5.7)

where:

3.10145.04.0

000556.0lntanh1272.0),( PIo emPIm −

−=−

γγ (5.8)

++=

492.0)(000102.0lntanh15.0),(

γγ PInPIK (5.9)

707015150

0

10*7.210*0.710*37.3

0.0

)(

115.15

976.17

404.16

>≤<≤<

=

=

−

−

−

PIPIPI

PI

PIPIPI

PIn (5.10)

Ishibashi and Zhang (1993) proposed to associate the material damping curve

with the normalized modulus reduction curve as follows:

( )

+

−

+=

−

1547.1586.02

1333.0

max

2

max

0145.0 3.1

GG

GGeD

PI (5.11)

124

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D, %


Ishibashi and Zhang (1993)

0.25 atm1.00 atm4.00 atm16.0 atm

(b)

Figure 5.10 The effect of confining pressure on (a) normalized modulus reduction, and (b) material damping curves for non-plastic soils (Ishibashi and Zhang, 1993)

125

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D, %


Ishibashi and Zhang (1993)

Non-PlasticPI = 50 %

(b)

Figure 5.11 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Ishibashi and Zhang (1993)

126

The data, which Ishibashi and Zhang (1993) synthesized, are the results of

tests performed at confining pressures less than 10 atm. Unfortunately, the

proposed set of equations are observed to give unrealistic relationships at higher

pressures, as illustrated in Figure 5.10 by the curves at oσ = 16 atm. At high

confining pressures, these equations predict normalized shear modulus values

exceeding 1.0 at intermediate strains. At the same time, negative values of

material damping ratio may be predicted using Equation 5.11 under the same

circumstances.

The equations proposed by Ishibashi and Zhang (1993) are also observed

not to be accurate in representing the general trends related with Dmin. These

equations ignore the effect of confining pressure on Dmin, and the value of Dmin is

predicted to decrease with increasing soil plasticity, while the opposite trend is

presented in Chapter Four.

As part of a research project funded by Electric Power Research Institute,

EPRI, a total of 35 undisturbed soil samples from 5 geotechnical sites (Treasure

Island, Gilroy, Oakland Outer Harbor, San Francisco Airport and Lotung) were

tested in the soil dynamics laboratory using the combined RCTS equipment at

The University of Texas at Austin. These samples were taken from a depth range

of 3 m to 150 m and were tested over a wide range of confining pressures. The

results of these tests are also utilized in this study as discussed in Chapter Three.

The normalized modulus reduction and material damping curves based on the

EPRI (1993c) study are presented in Figures 5.12 and 5.13.

127

Figure 5.12 Variation in empirical (a) normalized modulus reduction, and (b) material damping curves with depth (EPRI, 1993c)

(a)

(b)

128

Figure 5.13 Variation in empirical (a) normalized modulus reduction, and (b) material damping curves with soil type (EPRI, 1993c)

(a)

(b)

129

In Figure 5.12, the shift in both the normalized modulus reduction and

material damping curves to higher strain levels and the decrease in Dmin with

increasing depth is consistent with the general trends regarding the effect of

confining pressure on nonlinear soil behavior outlined in Chapter Four.

In Figure 5.13, the effect of soil type and plasticity on the normalized

modulus reduction and material damping curves at moderate confining pressures

(at which bulk of the EPRI data is collected) is presented. As discussed in Section

4.9, an increase in Dmin and a simultaneous shift of the material damping curve to

higher strain levels with increasing PI is not shown in any empirical curves except

for the EPRI (1993c) study.

5.4 SUMMARY

The comprehensive study performed by Hardin and Drnevich (1972a and

b) is introduced at the beginning of this chapter in order to familiarize the reader

with the basis of generic curves widely used in the state-of-practice. The generic

curves presently utilized in practice are then discussed. It is important to note that

the generic curves proposed by Seed et al. (1986), Sun et al. (1988), Idriss (1990),

and Vucetic and Dobry (1991) are based on data collected at around 1-atm

confining pressure and these curves do not capture the effect of confining pressure

on nonlinear soil behavior.

Although Iwasaki et al. (1978) and Kokusho (1980) studied the impact of

confining pressure, these studies were limited to observations on clean sands

tested at low pressures.

130

The set of equations proposed by Ishibashi and Zhang (1993) account for

both soil plasticity and confining pressure on nonlinear behavior. However, these

equations are based on data collected at confining pressures less than 10 atm and

are observed to give unrealistic values at higher pressures. Also, the effect of soil

plasticity on Dmin is not represented accurately in any of the generic curves widely

used in the state-of-practice.

The empirical curves from the EPRI (1993c) study are based on data

collected over a relatively wider range of confining pressures and are consistent

with the general trends outlined in Chapter Four. Although the EPRI (1993c)

study is one of the most comprehensive studies of nonlinear soil behavior, the

effects of some of the factors such as loading frequency and number of cycles are

not accounted for as part of this work.

The new empirical curves based on the four-parameter soil model

discussed in Chapter Six are formulated to accurately represent (consistent with

the general trends outlined in Chapter Four) the effects of soil type, confining

pressure, loading frequency and number of cycles on the normalized modulus

reduction and material damping curves. These factors are shown to be the key

factors affecting nonlinear dynamic soil behavior for “competent” soils as

summarized in Table 4.1.

131

CHAPTER 6

PROPOSED SOIL MODEL

6.1 INTRODUCTION

In this chapter, a four-parameter model that can be used to characterize

normalized modulus reduction and material damping curves is proposed. This

model is used to develop empirical curves that are based on the hyperbolic soil

model originally developed by Hardin and Drnevich (1972b). The basic

hyperbolic relationship between stress and strain is slightly modified in order to

accommodate a better fit to the modulus reduction curves measured in the

laboratory.

The equation for the material damping curve is related to the shape of the

modulus reduction curve assuming the validity of Masing behavior (Masing,

1926) combined with two modifying parameters. To start, Masing behavior is

used to calculate material damping by evaluating the hysteresis loops that should

form for a given modulus reduction curve and two-way stress reversals. This

material damping curve is then modified using two parameters to fit the

laboratory data as discussed below.

A parametric study is also presented in this chapter to assist the reader in

becoming familiar with the modified hyperbolic model.

132

6.2 NORMALIZED MODULUS REDUCTION CURVE

The hyperbolic model proposed by Hardin and Drnevich (1972b) and the

modification of the model is discussed in this section. A modified hyperbolic

model is utilized to evaluate and model dynamic soil properties in this study.

As discussed in Chapter Five, a normalized modulus reduction curve

based on the hyperbolic model can be expressed as:

r

GG

γγ

+=

1

1

max

(6.1)

It is easy to see that reference strain, γr, corresponds to the strain amplitude when

shear modulus reduces to one half of Gmax. If one uses this approach in defining

γr, reference strain for any given normalized modulus reduction curve can be

easily evaluated from laboratory measurements as long as G/Gmax values around

0.5 are measured during testing. In fact, γr = γG/Gmax=0.5 is a key characteristic of

the hyperbolic model as employed in this research.

In this study, a relatively simple approach is utilized to fit measured stress-

strain curves. A curvature coefficient, a, is integrated into the normalized modulus

reduction curve (Darendeli, 1997) as follows:

a

r

GG

+

=

γγ1

1

max

(6.2)

The curvature coefficient, as the name implies, has an impact on the curvature of

the normalized modulus reduction curve. The reference strain still corresponds to

the strain amplitude when shear modulus reduces to one half of Gmax. The

advantage of this modification is its simplicity. However, depending on the value

133

of the curvature coefficient, the calculated stress-strain curve may not be

asymptotic to the horizontal line defined by τmax. Since this study is an effort to

model strain amplitudes far below failure, the success of this equation in

modeling soil behavior when full shear strength is mobilized is of lesser concern.

Figure 6.1 shows a normalized modulus reduction curve calculated using

this modification to the Hardin and Drnevich (1972b) hyperbolic model; that is

using Equation 6.2 with γr = γG/Gmax=0.5 = 0.03 % and a = 0.90.

The modified hyperbolic relationship can be used to approximate the

normalized modulus reduction curve of all competent soil types at strains even in

excess of 0.3 % as shown in Chapter Eight.

1.0

0.5

0.0

G/Gmax


γ r = γG/Gmax=0.5 = 0.03 %

a = 0.90

γ r

Figure 6.1 Normalized modulus reduction curve (of a silty sand at 1 atm effective confining pressure) represented using a modified hyperbolic model

134

6.3 NONLINEAR MATERIAL DAMPING CURVE

Material damping can be assumed to result from at least two separate

phenomena. Part of the energy applied to a soil body is attenuated due to the

friction and/or viscous losses at the contact surfaces between particles. In other

words, regardless of the strain amplitude, some energy loss and, therefore,

equivalent damping should be anticipated. On the other hand, soils are extremely

nonlinear in the strain range of interest (0.0001 % < γ < 1 %) during a design-

level earthquake. The nonlinearity in the stress-strain relationship results in

energy loss in a system. Thus, a second component has to be considered. In this

study, these two sources of damping are handled separately and added to each

other in order to evaluate equivalent damping of soils.

Since material damping resulting from nonlinearity of the stress-strain

relationship is the major component, it is discussed first in this section. As

discussed in Section 6.2, the stress-strain and modulus reduction curves are

directly related to each other for two-way cyclic loading (complete stress

reversals). Once a soil is characterized in terms of its normalized modulus

reduction curve and small-strain shear modulus, it is possible to predict the stress-

strain path that the soil is expected to follow under monotonic loading. Figure

6.2a shows the normalized modulus reduction curve (Figure 6.2a is the same

curve as the one shown in Figure 6.1). This curve is scaled to Gmax = 45 MPa in

Figure 6.2b in order to evaluate the G – log γ relationship. Shear stress can be

expressed as:

γτ *G= (6.3)

135

1.2

0.8

0.4

0.0

G/Gmax

(a)

γ r = 0.03 %a = 0.90

50

40

30

20

10

0

G, MPa


(b)

0.015

0.010

0.005

0.000

τ,

1.00.80.60.40.20.0Shearing Strain, γ , %

(c)

MPa

Figure 6.2 Stress-strain curve (of a silty sand at 1 atm effective confining pressure) estimated based on a modified reference strain model

136

and the stress-strain curve is easily calculated from the G – log γ relationship. The

resulting stress-strain curve is presented in Figure 6.2c. This curve is theoretically

the stress-strain path under monotonic loading for the material characterized by

the normalized modulus reduction curve in Figure 6.2a and Gmax = 45 MPa.

Masing (1926) assumed that the stress-strain path during cyclic loading

could be related to the monotonic loading stress-strain path, which is also called

the backbone curve. His first attempt to relate these two stress-strain paths is

typically called “Masing behavior”. As presented in Figure 6.3, Masing behavior

assumes that hysteresis loops for two-way cyclic loading can be constructed by

scaling the backbone curve by a factor of two. After initial loading, the scaled

curve is flipped on the horizontal and vertical axes, respectively, and placed at the

end of the backbone curve to represent the unloading path. In order to represent

reloading, the scaled curve is placed at the end of the unloading path. As

unloading and reloading is continued, the same stress-strain path is followed. This

simple approach is powerful in explaining the mechanism that causes energy loss

and the formation of hysteresis loops. However, its direct application in modeling

nonlinear soil behavior is limited and has been shown to perform poorly in

various strain ranges as discussed below.

137

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

τ,MPa

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3


-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

τ,MPa

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3


Figure 6.3 Hysteresis loop estimated by modeling stress-strain reversals for two-way cyclic loading according to Masing behavior

One of the major shortcomings of the Masing behavior assumption is due

to the fact that the forcing function during an earthquake is not sinusoidal and a

few more rules have to be adapted in order to utilize this approach in modeling

loadings that do not cycle between two equal and opposite values of stress (two-

way cyclic loading). Since this study concentrates on relating the material

damping curve to modulus reduction curve, the assumption of cyclic loading is

appropriate and an irregular forcing function is not relevant to this work. An

algebraic expression can be written assuming sinusoidal loading conditions to

evaluate equivalent viscous damping as a function of strain amplitude.

138

As discussed in Section 2.4 and presented in Figure 6.4, damping ratio can

be calculated using the ratio of the dissipated energy to stored strain energy in one

complete cycle of motion. Assuming Masing behavior, the area inside the

hysteresis loop (AL) can be calculated by integrating the stress-strain curve over

one loading cycle.

1GShear

Stress, τ

Shearing Strain, γ

G = τ / γD = AL / (4 π AT)

AL

AT

1GShear

Stress, τ

Shearing Strain, γ

G = τ / γD = AL / (4 π AT)

G = τ / γD = AL / (4 π AT)

AL

AT

Figure 6.4 Calculation of damping ratio utilizing a hysteresis loop

With Equations 6.2 and 6.3, the stress-strain curve can be expressed as:

max*

1

Ga

r

+

=

γγ

γτ (6.4)

The area inside the hysteresis loop can be related to the integral of the stress-strain

curve as follows:

−= ∫ τγγτ

21*8 dAL (6.5)

139

Equivalent viscous damping is expressed as:

T

Leq A

ADΠ

=4

(6.6)

where: Π = pi (= 3.1416),

AT = stored strain energy (τγ/2),

AL = dissipated energy, and

Deq = equivalent viscous damping.

By combining Equations 6.5 and 6.6, Masing-behavior damping can be written as

a function of strain amplitude as follows:

τγ

τγγτ

214

21*8

sinΠ

−

=∫ d

D gMa (6.7)

By substituting Equation 6.4 into Equation in 6.7, Masing-behavior

damping can be rewritten as follows:

a

r

a

r

a

rgMa

d

D

+

+

−

+

Π=

∫

γγ

γγγ

γγ

γγ

γ

1

121

1*4

2

2

sin (6.8)

For a curvature coefficient, “a”, equal to 1.0, Equation 6.8 reduces to:

r

r

r

rrr

agMaD

γγ

γγγ

γγ

γγγγγ

+

+−

+−

Π==

1

121ln

*42

2

0.1,sin (6.9)

140

and further rearrangement of this equation results in:

−

+

+−

Π== 2

ln4100(%) 20.1,sin

r

r

rr

agMaD

γγγ

γγγ

γγ (6.10)

Unfortunately, the integration in Equation 6.8 can not be evaluated

algebraically for most curvature coefficient values other than 1.0. As a result, a

numerical approach is utilized to calculate functions that represent Masing

damping for different values of the curvature coefficient.

First, damping based on Masing behavior for any given strain amplitude is

assumed to be a function of: 1) Masing damping at that strain for a curvature

coefficient equal to 1.0, and 2) the value of curvature coefficient. Thus, the

expression below is assumed to be valid: 3

0.1,sin32

0.1,sin20.1,sin1sin === ++= agMaagMaagMagMa DcDcDcD (6.11)

In other words, Masing damping is assumed to be a polynomial function of

Masing damping for curvature coefficient equal to 1.0, and c1, c2 and c3 are

assumed to be functions of curvature coefficient.

Masing damping for curvature coefficients ranging from 0.7 to 1.3 are

calculated through numerical integration using the trapezoid rule. These damping

curves are fitted using the expression in Equation 6.11. Thus, c1, c2 and c3 values

are calculated for sixty damping curves that are calculated based on different

values of the curvature coefficient. In Figure 6.5, variations of c1, c2 and c3 with

the curvature coefficient are presented. The solid curves passing through the data

points are the best-fit polynomial relationships.

141

1.1

1.0

0.9

0.8

0.7

c1

(a)

c1 = -1.1143a2 + 1.8618a + 0.2523

R2 = 0.9997

0.04

0.02

0.00

-0.02

c2

(b)

c2 = 0.0805a2 - 0.0710a - 0.0095

R2 = 1.0000

-0.0003

-0.0002

-0.0001

0.0000

0.0001

0.0002

c3

1.31.21.11.00.90.80.7

Curvature Coefficient, a

(c)

c3 = -0.0005a2 + 0.0002a + 0.0003

R2 = 0.9996

Figure 6.5 Variations of c1, c2 and c3 with curvature coefficient, a

142

In this way, a simple algebraic expression for a damping curve based on a

modified hyperbolic stress-strain curve and Masing behavior was derived. This

result is expressed as:

3

0.1,sin32

0.1,sin20.1,sin1sin === ++= agMaagMaagMagMa DcDcDcD

where:

−

+

+−

Π== 2

ln4100(%) 20.1,sin

r

r

rr

agMaD

γγγ

γγγ

γγ (6.12)

0.2523 + 1.8618a + -1.1143a21 =c

0.0095 - 0.0710a - 0.0805a 22 =c

0.0003 + 0.0002a + 0.0005a- 23 =c

Figure 6.6 shows a damping curve estimated using Equation 6.12. It is

important to note that this curve indicates larger damping ratios at high strains

than experimental results observed in the course of this study and values reported

in the literature (e.g., Seed et al., 1986; Vucetic and Dobry, 1991; etc.). Also,

damping calculated based on Masing Behavior lacks small-strain damping, Dmin.

As a result, further work is required to fit the mathematical expressions to the

experimental observations. The corrections on the damping curve calculated

based on Masing behavior can be summarized as follows:

minsin* DDFD gMa += (6.13)

where F is a function that adjusts damping at high strains.

143

60

40

20

0

D, %


γ r = 0.03 %a = 0.90

Damping values calculated through numerical intagrationDamping values calculated using polynomial approximation

Figure 6.6 Damping curve estimated based on Masing behavior

Stokoe and Lodde (1978) have shown that small-strain properties are

affected by prior high-amplitude cycling. Gmax tends to decrease slightly and Dmin

tends to increase slightly after high-amplitude cycling as shown in Figure 6.7. The

decrease in Gmax should be expected to result in a considerable decrease in the

nonlinear component of material damping. Also, on various geotechnical

earthquake engineering applications (e.g., Stokoe et al., 1998a), a cap on high-

amplitude damping is required in seismic evaluations of critical structures (in

order to be conservative). As a result, function F in Equation 6.13 is replaced with

a function of G/Gmax in the form of:

p

GGbF

=

max* (6.14)

where b and p are parameters that control the characteristics of this function.

144

Figure 6.7 Effect of high-amplitude cycling on low-amplitude shear modulus and material damping ratio (from Stokoe and Lodde, 1978)

Figure 6.8 shows the variation of this function with shearing strain amplitude for

different values of “p”. Using the value of 0.1 for the “p” parameter was observed

to fit the experimental data. In order to simplify the model, the “p” parameter was

replaced with this constant value. Thus, damping adjusted using function F is

expressed as:

gMaAdjusted DG

GbD sin

1.0

max**

= (6.15)

where: b = scaling coefficient,

G/Gmax = normalized modulus, and

DAdjusted = scaled and capped material damping.

145

1.0

0.8

0.6

0.4

0.2

0.0

F


F = b * (G/Gmax)p

b = 0.56

p = 0p = 0.1p = 0.25

Figure 6.8 Comparison of the variation in F with shearing strain for different values of p

This function acts as a damping cap at very high strains. At the same time,

within this function, a parameter, which is called the scaling coefficient, is

introduced to the model. The scaling coefficient is, in a sense, the ratio of the

measured damping to the damping value which is estimated from Masing

behavior at intermediate strain amplitudes. Hence, the adjusted damping curve in

Figure 6.9b is estimated.

As discussed earlier in this chapter, small-strain damping is also accounted

for in this model. Dmin is added to the adjusted damping curve and the whole

curve is shifted by this amount as shown in Figure 6.9c. The effect of high-

amplitude cycling on Dmin is ignored in this model due to two reasons: 1) for

design purposes, it is always conservative to underestimate material damping

ratio, and 2) even a 50 % increase in Dmin (as shown in Figure 6.7 at γ = 0.1 %)

has a negligible impact on the material damping values at strain levels produced

during design level shaking.

146

60

40

20

0

D, %

(a)

γ r = 0.03 %a = 0.90

25

20

15

10

5

0

D, %

(b)

F = b * (G/Gmax)0.1

b = 0.56

25

20

15

10

5

0

D, %


(c)

Masing BehaviorAdjusted CurveShifted Curve

Dmin = 0.90 %

Figure 6.9 (a) Damping curve estimated based on Masing behavior, (b) adjusted curve using the scaling coefficient, and (c) shifted curve using the small-strain material damping ratio

147

6.4 PARAMETRIC STUDY OF THE SOIL MODEL

In this section, a parametric study of the soil model is presented so that the

effect of each parameter on the normalized modulus reduction curve and the

material damping curve can be easily visualized. The ability of this four-

parameter model in representing the trends discussed in Chapter Four is also

discussed herein.

6.4.1 Reference Strain, γr

The effect of reference strain on the normalized modulus reduction, stress-

strain and material damping curves is presented in Figure 6.10. As discussed in

Sections 4.5 and 4.9, increasing confining pressure and increasing soil plasticity

both cause shifts to higher strain levels in the normalized modulus reduction and

material damping curves. This trend is accounted for by adjusting this parameter

as shown in Figure 6.10.

6.4.2 Curvature Coefficient, a

The effect of the curvature coefficient on the normalized modulus

reduction, stress-strain and material damping curves is presented in Figures 6.11

through 6.13. Hardin and Drnevich (1972b) have pointed out a relationship

between the shape of the stress-strain curve and soil type, which can be accounted

for with this parameter as shown in these figures.

148

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax


Reference Strain, γ r1.00E-2 %3.00E-2 %1.00E-1 %

(a)

0.03

0.02

0.01

0.00

τ,


MPa

(b)

30

20

10

0

D, %


(c)

Figure 6.10 Effect of reference strain on (a) normalized modulus reduction, (b) stress-strain, and (c) material damping curves

149

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax


Curvature Coefficient, a0.81.01.2

γ r

Figure 6.11 Effect of the curvature coefficient on the normalized modulus reduction curve

0.008

0.006

0.004

0.002

0.000

τ,

0.040.030.020.010.00Shearing Strain, γ , %

MPa

(a)

0.03

0.02

0.01

0.00

τ,


MPa

(b)

Figure 6.12 Effect of the curvature coefficient on the stress-strain curve (a) at small and intermediate strains, and (b) at high strains

150

30

20

10

0

D, %


Figure 6.13 Effect of the curvature coefficient on the material damping curve

6.4.3 Small-Strain Material Damping Ratio, Dmin

The effect of Dmin on the material damping curve is presented in Figure

6.14. As discussed in Sections 4.5, 4.8 and 4.9, confining pressure, loading

frequency and soil plasticity affect the small-strain material damping ratio. The

impact of all of these parameters on the material damping curve can be accounted

for with Dmin, which results in a general shifting of the material damping curve.

The largest relative impact of this parameter on the value of D is, of course, at

small strains.

151

30

20

10

0

D, %


Small Strain Damping, Dmin0.0 %0.9 %3.0 %

Figure 6.14 Effect of Dmin on the material damping curve

6.4.4 Scaling Coefficient, b

The effect of the scaling coefficient on the material damping curve is

presented in Figure 6.15. As discussed in Section 4.7, the number of loading

cycles affects the damping curve, which can be accounted for with this parameter

by varying b with number of cycles.

152

30

20

10

0

D, %


Scaling Coefficient, b0.400.560.70

Figure 6.15 The effect of scaling coefficient on material damping curve

6.5 SUMMARY

In this chapter, a set of equations based on the Hardin and Drnevich

(1972b) hyperbolic model is proposed to represent the general trends of dynamic

soil properties outlined in Chapter Four.

This empirical model utilizes a modified hyperbolic soil model in order to

represent the stress-strain relationships and, therefore, the normalized modulus

reduction curves. The modified hyperbolic soil model utilizes two parameters

(reference strain and curvature coefficient) and can be expressed as:

a

r

GG

+

=

γγ1

1

max

(6.16)

153

In case of the material damping curve, the normalized modulus reduction

curve and Masing behavior are employed as a starting point. A material damping

curve is calculated by evaluating the hysteresis loops that form for a given

modulus reduction curve assuming Masing behavior. A function, F, is defined as: p

GGbF

=

max* (6.17)

which utilizes the normalized modulus reduction curve and two parameters.

Parameter “p” is replaced by a constant value of 0.1 to simplify the model. Thus,

the damping curve estimated from Masing behavior is scaled to fit the

experimental observations using the normalized modulus reduction curve and a

parameter called the scaling coefficient. The adjusted (scaled) damping curve is

then shifted by a second parameter, Dmin. This way, using two additional

parameters, the material damping curve is expressed as:

minsin

1.0

max** DD

GGbD gMa +

= (6.18)

Hence, a four-parameter model (γr, a, b and Dmin) is used to characterize

the normalized modulus reduction and material damping curves.

A parametric study is also presented to assist the reader in becoming

familiar with the modified hyperbolic model and the impact of each of the

parameters. The general ability and usefulness of this four-parameter model in

representing the trends discussed in Chapter Four is also discussed.

154

CHAPTER 7

STATISTICAL ANALYSIS OF COLLECTED DATA USING

FIRST-ORDER, SECOND-MOMENT BAYESIAN METHOD

7.1 INTRODUCTION

In this chapter, the statistical analysis carried out as part of this study is

presented. The statistical tool utilized in this study is briefly explained and the

advantages of using this approach in geotechnical engineering are discussed.

Bayes’ theorem is introduced early in the chapter in order to familiarize

the reader with the concept of systematically utilizing both experience and

observations in statistical analysis. Then, application of the theorem to discrete

and continuous problems is briefly discussed. A discrete example is presented in

order to clarify the methodology. The computational drawbacks in direct

application of Bayes’ theorem are pointed out and an iterative method, which is

used in analyzing the collected data, is presented.

Each parameter of the soil model (presented in Chapter Six) is expressed

in the form of an equation. The equations are formulated so that the impact of

strain amplitude, effective isotropic confining pressure, loading history and

loading frequency on dynamic soil behavior can be properly modeled following

the general trends discussed in Chapter Four. These equations and the form of the

anticipated covariance structure, which describes the deviations between the data

and the model in terms of their magnitude and interrelationships (correlations),

are also presented herein.

155

7.2 BAYESIAN APPROACH

Whenever an engineer faces a problem, he or she attempts to analytically

structure the problem and represent the physical phenomenon by a mathematical

relationship. Once the parameters of the mathematical relationship are

determined, calculations are performed. The solution of the problem is achieved

when the numerical result is expressed as a physical quantity. This method is the

way a structural engineer decides the required reinforcement for a concrete beam

and it is the same method a geotechnical engineer follows to determine the

required pile length for a deep foundation. If the parameters of the mathematical

relationship are estimated accurately, the outcome is a success. The engineer can

develop a safe and economic solution.

In statistical analysis, accurate estimates of parameters require large

amounts of data. In most engineering applications, observed data are limited due

to the fact that data are an expensive commodity. As a result, using judgment and

intuition has always been an essential part of decision making in engineering.

Complex engineering problems require combining information based on

experience with observational data. The Bayesian method is a systematic way of

achieving this goal.

By modeling the unknown parameters as random variables, uncertainty

associated with the estimation of the parameters can be combined with the

inherent variability of the basic random variable through Bayes’ theorem. Thus,

engineering judgment and experience can be incorporated with observed data to

obtain a balanced estimation (Ang and Tang, 1975 and 1990).

156

7.2.1 Bayes’ Theorem

The numerical measure of the likelihood of occurrence of an event, Ei,

relative to all possible outcomes is called the probability of this event, and it is

denoted as P(Ei).

If the probability of an event depends on the occurrence of another event,

the associated probability is called conditional probability. The conditional

probability of E1 assuming E2 has occurred is denoted as P(E1/E2) and it is

expressed as:

)()()/(

2

2121 EP

EEPEEP = (7.1)

where P(E1E2) denotes probability of occurrence of both events.

P(E1E2) can also be expressed in terms of the conditional probability of E2

assuming E1 has occurred using the same form presented in Equation 7.1 as

follows:

)()()/(

1

2112 EP

EEPEEP = (7.2)

By substituting Equation 7.2 into Equation 7.1, the following equation can

be written:

)()(*)/()/(

2

11221 EP

EPEEPEEP = (7.3)

Considering n mutually exclusive and collectively exhaustive events, E1,

E2, …. , En; and another event A in the same sample space, we can write:

∑=

=n

iiEPAPAP

1)(*)()( (7.4)

157

Using the definition of conditional probability, Equation 7.4 can be

expressed as follows.

∑=

=n

iii EPEAPAP

1)(*)/()( (7.5)

Therefore, if the event A has occurred, the probability of occurrence of a

particular event Ei can be expressed by Equation 7.6, which is also known as

Bayes’ theorem, and is expressed as:

∑=

= n

jjj

iii

EPEAP

EPEAPAEP

1)(*)/(

)(*)/()/( (7.6)

7.2.2 Discrete Case

Application of Bayes’ theorem for a discrete problem in order to update an

initial estimate in light of new observations is described in this section.

If a variable X can only take on certain discrete values of x, it is called a

discrete random variable. The outcome of rolling a dice represents such a

variable. It is possible to roll only integer values between 1 and 6. In this case,

the function expressing P(X = x) for all x is utilized to describe the probability

distribution of X. This function is called a probability mass function (PMF).

If a parameter X is often used by an engineer, he or she would have some

prior knowledge of possible values and values more likely to occur than others

based on experience or intuition. This initial information can be represented in

terms of a PMF if X is a discrete random variable. Once additional information

becomes available, the prior assumptions on the parameter are modified or

158

updated for the specific problem at hand. This task can be systematically achieved

by employing Bayes’ theorem using:

∑=

==

==== n

iii

iii

xXPxXP

xXPxXPxXP

1)(*)/(

)(*)/()/(

ε

εε (7.7)

where )/( ixXP =ε = the conditional probability of obtaining a

particular experimental outcome, ε, assuming

that parameter X is equal to xi,

)( ixXP = = the probability of parameter X being equal to xi

before the availability of experimental

information, ε, and

)/( εixXP = = the probability of parameter X being equal to xi

after the initial assumptions are revised in the

light of the experimental outcome, ε.

If the prior and posterior probabilities are denoted as P’(X = xi) and P’’(X

= xi), respectively, Equation 7.7 is expressed as follows:

∑=

==

==== n

jjj

iii

xXPxXP

xXPxXPxXP

1)('*)/(

)('*)/()(''

ε

ε (7.8)

This equation represents the posterior PMF in light of the experimental outcome.

In order to clarify Bayesian approach, a simple example is presented

below:

To determine the insurance rate for a five-story reinforced concrete residential building in Istanbul, Turkey, an insurance company is in need of an estimate of seismic risk of the structure during a design level ground motion. Unfortunately, available information is very limited. There are two groups of reinforced concrete

159

buildings in Istanbul. Group 1 represents 30 percent of the buildings and they do not meet the local building design code. Forty percent of these buildings are expected to collapse during the design level shaking. The remaining buildings meet the code and only 1% collapse is expected for this second group. Survival of a building during any given earthquake does not necessitate good performance during the following one. Given the fact that the building of interest has survived two earthquakes in 1999 (resulting in design level shaking at the site), what is the probability of collapse for this structure during the next earthquake?

Once the problem is structured analytically, the building of interest is

expected to fall into one of the two categories. The category of the building is a

discrete variable and the probability distribution for the two categories is the prior

PMF as shown in Figure 7.1. If the building is a member of Category 1, it is more

likely to collapse during design level shaking as tabulated in Table 7.1.

30%

70%

0%

25%

50%

75%

100%

Category 1 Category 2

Building Category

Prob

abili

ty

Figure 7.1 Prior probability mass function for the discrete example

160

Table 7.1 Prior information provided in the discrete example

P’(Category = 1) 30 %P’(Category = 2) 70 %P(collapse / Category = 1) 40 %P(collapse / Category = 2) 1 %

Based on this information, the probability of collapse for a particular

building (not knowing whether or not it was built following the code) can be

calculated as follows:

p’(collapse) = 0.30 * 0.40 + 0.70 * 0.01 (7.9a)

p’(collapse) = 12.7 % (7.9b)

The statement regarding survival of the building during two earthquakes in 1999

is observational data. Assuming survival of the building at two instances are

statistically independent:

P(success) = 1 – P(collapse) (7.10)

P(success 2 consecutive earthquakes) = P(success)2 (7.11)

Based on Equation 7.11, the prior PMF can be updated using Equation 7.12 based

on Bayes’ theorem as:

∑=

==

==== 2

1)i Category ('*)i Category /(

)1 Category ('*)1 Category /()1 Category (''

iPP

PPPε

ε (7.12a)

∑=

==

==== 2

1)i Category ('*)i Category /(

)2 Category ('*)2 Category /()2 Category (''

iPP

PPPε

ε (7.12b)

161

Figure 7.2 illustrates the posterior (updated) PMF based on independent

observations regarding performance of the structure during two earthquakes in

1999. Using the posterior PMF, the probability of collapse for a particular

building can be recalculated as:

p’’(collapse) = 0.136 * 0.40 + 0.864 * 0.01 (7.13a)

p’’(collapse) = 6.3 % (7.13b)

13.60%

86.40%

0%

25%

50%

75%

100%

Category 1 Category 2

Building Category

Prob

abili

ty

Figure 7.2 Posterior probability mass function for the discrete example

In the light of the observational data, the likelihood of the building of

interest being a member of Category 2 has increased and, as a result, the

probability of collapse updated based on this data (6.3 %) is smaller than the

estimate based on prior information (12.7 %).

162

7.2.3 Continuous Case

In most cases, the variable of interest does not take on certain discrete

values. Instead, it can take on any value within a continuum. These kinds of

parameters are called continuous random variables. The unit weight of a soil

represents such a variable. In this case, probabilities are associated with intervals.

In other words, the probability of X taking a value between x1 and x2 is relevant.

Thus, a probability density function (PDF) is defined for continuous random

variables. If fx(x) is the PDF of X, the probability of X taking a value between x1

and x2 is:

∫=≤<2

1

)()( 21

x

xx dxxfxXxP (7.14)

In other words,

dxxfdxxXxP x )()( =+≤< (7.15)

Bayes’ theorem can be employed in the same fashion for the continuous

case as in the discrete case as follows:

∫∞

∞−

=

θθθε

θθεθdfP

fPf)(')/(

)(')/()('' (7.16)

where )/( θεP is the conditional probability of obtaining a particular

experimental outcome, ε, assuming that the parameter of interest is equal to θ.

Since )/( θεP is a function of θ it is also called the likelihood function of θ and is

denoted as L(θ). Noting that the denominator in Equation 7.16 is independent of

θ, )('' θf can also be expressed as:

)(')()('' θθθ fkLf = (7.17)

163

where k is a normalizing constant scaling )('' θf to become a proper PDF and

L(θ) is the likelihood of observing the experimental outcome ε assuming a given

value, θ.

In most engineering problems, more than one parameter needs to be

characterized. Even in the simplest case of a linear fit, two parameters are needed,

the intercept and slope. As a result, Equation 7.16 always becomes a bit more

complicated because the denominator has to be integrated for each parameter as

follows:

∫∫∞

∞−

∞

∞−

=

nddfP

fPfφφφφε

φφεφ.....)(')/(......

)(')/()(''

1

rrr

rrrr (7.18)

where a vector of model parameters, φr

, needs to be updated based on a number

of observations.

The integration in Equation 7.18 may become extremely troublesome,

especially when a nonlinear model is being calibrated. The moments of the

updated distribution may also become impossible to compute analytically.

Although the Bayesian approach is a powerful tool, the lack of an analytical

solution makes direct application of the Bayesian approach impractical for most

engineering applications, specifically in those cases dealing with nonlinear

problems.

164

7.3 FIRST-ORDER, SECOND-MOMENT BAYESIAN METHOD

An analytical approach called the First-order, Second-moment Bayesian

Method (FSBM) is proposed by Gilbert (1999) to deal with nonlinear problems.

This method utilizes analytical approximations in updating model parameters

based on experimental observations.

Instead of presenting the mathematical description of the FSBM (which

can be found in Gilbert, 1999), an example application is presented in this section

to assist the reader in obtaining a feel for this method. This example application

introduces the methodology of applying the FSBM to calibrate a simple model

that represents the shear wave velocity profile at a geotechnical site.

The assumed shear wave velocity profile model is presented in Equation

7.19. The model relates the mean estimate of shear wave velocity, µ, to depth, z.

The covariance structure, COV(Vsi,Vsj), is formulated to account for magnitude of

the scatter (or deviations from the mean estimate) and correlation between shear

wave velocities at similar depths anticipating a horizontally layered soil deposit. 25.0

21 zφφµ += (7.19a)

)2exp(),(4

23 φ

φ jisjsi

zzVVCOV

−−= (7.19b)

where:

1φ is the mean shear wave velocity at the surface,

2φ is the empirical constant relating mean shear wave velocity to depth,

3φ is the standard deviation of shear wave velocity, and

4φ is the scale of fluctuation.

165

Based on past experience, the following prior information is generated.

The expected values, µφ, of model parameters are tabulated in Table 7.2. A rather

large coefficient of variation, δφ, is used to represent a low confidence in the

initial guess (prior information). The model parameters are assumed to be

independent from each other and the resulting covariance structure, Cφ, for the

model parameters is shown in Table 7.3.

Table 7.2 Prior information regarding the model parameters in the FSBM example

Parameter µφ δφ

φ1 50 0.50φ2 100 0.50φ3 20 0.50φ4 5 0.50

Table 7.3 Prior covariance structure of the model parameters in the FSBM example

Cφ φ1 φ2 φ3 φ4

φ1 625 0 0 0φ2 0 2500 0 0φ3 0 0 100 0φ4 0 0 0 6.25

166

The covariance structure within the model should not be confused with the

covariance structure of the model parameters. The covariance structure within the

model expresses the discrepancy between the shear wave velocity profile and the

mathematical representation (model), and it also acknowledges the

interrelationship between shear wave velocities at similar depths and therefore the

interrelationship between closely spaced measurements (in terms of depth). Most

soil deposits are horizontally layered systems and soils sampled from similar

depths are likely to be taken from the same layer resulting in the engineering

properties of these samples to be similar. On the other hand, the covariance

structure of the model parameters describes the uncertainty associated with the

model parameters and how they are related to each other.

The shear wave velocity profile model discussed above is calibrated using

the data presented in Table 7.4 collected at the geotechnical site of interest using

an in-situ seismic method.

Table 7.4 Data used to calibrate the model parameters in the FSBM example

1 1.0 125

2 5.0 1503 5.5 1604 10.0 200

Measurement Depth (m)

Shear Wave Velocity (m/s)

167

In this application, a multivariate normal likelihood function is used with

FSBM to calibrate the model which can be formulated as:

−−−= − )()(

21exp1)\( 1

21 εεε

ε

µεµεφε rrrrrr CC

L T (7.20)

where:

=

nε

ε

ε::1

r ,

=

nµ

µ

µε ::1

r ,

=

2,

21,

2,1

21,1

....::::::::::::

....

εε

εε

ε

σρσρ

σρσρ

nnn

n

C ,

εC = the matrix determinant of Cε, and T = superscript denoting the matrix

transpose.

The FSBM is based on using a second-order Taylor series to approximate

the likelihood function. The Taylor series is expanded about a set of values of

model parameters, *φr

, at which the natural logarithm of the likelihood function,

)(φr

g , is maximized as follows:

[ ])\(ln)( φεφrrr

Lg = (7.21)

{ }

{ } { }

−

∂∂∂

−+

∂∂

−+

≡*

2*

**

*

*

21

)(

)(

φφφφ

φφ

φφφφ

φ

φ

φ

rrrr

rrr

r

r

r

ji

T

i

T

g

gg

g (7.22)

The values of model parameters maximizing the natural logarithm of

Equation 7.20 are estimated by evaluating the following matrices:

168

=

200160150125

εr (7.23a)

+−+−+−+−

=−

)10*(200)5.5*(160

)5*(150)1*(125

25.0*2

*1

25.0*2

*1

25.0*2

*1

25.0*2

*1

φφφφφφφφ

µε εrr

(7.23b)

−−−

−−−

−−−

−−−

=

2*3*

4

2*3*

4

2*3*

4

2*3

*4

2*3

2*3*

4

2*3*

4

2*3

*4

2*3*

4

2*3

2*3*

4

2*3

*4

2*3*

4

2*3*

4

2*3

2*3

)5.42exp()52exp()92exp(

)5.42exp()5.02exp()5.42exp(

)52exp()5.02exp()42exp(

)92exp()5.42exp()42exp(

φφ

φφ

φφ

φ

φφφ

φφ

φφ

φφ

φφφ

φφ

φφ

φφ

φφφ

εC (7.23c)

After taking the natural logarithms of both sides of Equation 7.17 and

substituting Taylor’s approximation for )(φr

g (Equation 7.22), the updated

(posterior) model parameters can be estimated using: [ ] [ ] { }[ ]''' *1

// GGCC +−≡ − φµµ φφεφεφ

rrr (7.24a)

[ ][ ] 11/ ''

−− −≡ GCC φεφ (7.24b)

where:i

gGφφ

∂∂

=)('r

= vector of first partial derivatives of )(φr

g ,

ji

gGφφφ

∂∂∂

=)(''

2 r

= matrix of second partial derivatives of )(φr

g ,

φC = prior covariance matrix of model parameters,

εφ /C = posterior covariance matrix of model parameters,

φµr = prior mean vector of model parameters,

εφµ /r = posterior mean vector of model parameters, and

169

*φr

= expansion point of Taylor series approximation.

The calibrated model parameters are tabulated in Table 7.5. The

uncertainty in the estimates of all model parameters is observed to decrease. More

improvement is observed in the case of φ2 and φ3 parameters suggesting that the

data have provided more information regarding these parameters. If a

measurement had been performed at the ground surface, a lot of improvement in

the φ1 parameter would have been observed. The scale of fluctuation, φ4, requires

more data points to be evaluated with certainty and as a result little improvement

is observed in this parameter.

Table 7.5 Comparison of the prior and posterior information regarding the model parameters in the FSBM example

Parameter µφ' δφ' µφ'' δφ''φ1 50 0.50 38.05 0.34φ2 100 0.50 85.15 0.12φ3 20 0.50 10.84 0.16φ4 5 0.50 3.89 0.37

Prior Posterior

Table 7.6 presents the covariance structure of the model parameters in

light of the experimental observations. The negative correlation between φ1 and

φ2, and the positive correlation between φ3 and φ4 are results of working with the

same limited data set in evaluating the model parameters. As the intercept of the

mean of the profile increases, the slope decreases. Similarly, as the variability

around the mean increases, the scale of fluctuation increases with the variability.

The results do not necessarily indicate a physical correlation between the model

parameters in such an application.

170

Table 7.6 Posterior covariance structure of the model parameters in the FSBM example

Cφ φ1 φ2 φ3 φ4

φ1 289.81 -191.19 0.00 0.00φ2 -191.19 139.06 0.00 0.00φ3 0.00 0.00 10.12 0.98φ4 0.00 0.00 0.98 3.38

Figure 7.3 illustrates this point with a simple example using a linear fit.

The best linear fit to this data set is: y = 34.875x + 38.506. Another reasonable

linear fit to this data set is expected to have a smaller slope for a larger intercept

(or a larger slope for a smaller intercept). This relationship between the intercept

and slope is a result of working with a limited set of data with certain

characteristics. Thus, an imaginary correlation between model parameters is

observed upon updating prior information based on limited number of

observations.

y = 0.1348x + 60

y = 34.875x + 38.506

0

50

100

150

0 0.2 0.4 0.6 0.8 1

Independent Variable, x

Dependent Variable, y

Figure 7.3 Imaginary correlation between model parameters upon updating prior information based on limited number of observations

171

Table 7.7 tabulates the estimates of shear wave velocity at a depth of 10 m

using prior and posterior model parameters. The uncertainties associated with

these estimates are also presented in this table. Upon calibration of the model with

field observations, a tremendous reduction in the uncertainty regarding model

parameters takes place. As a result, the standard deviation associated with the

estimate of shear wave velocity at a depth of 10 m decreases significantly from 95

m/s to 13 m/s. An experienced engineer familiar with the geotechnical site of

interest might have a better idea about the model parameters prior to in-situ

testing and, under such circumstances, the reduction in uncertainty would not be

as significant.

Table 7.7 Posterior covariance structure of the model parameters in the FSBM example

Prior PosteriorExpected Value of Vs at 10 m 227.83 189.47

Variance of Vs at 10 m 9030.69 177.09Standard Deviation of Vs at 10 m 95.03 13.31Variance from Model Uncertainty 8630.69 59.68Variance from Random Variability 400.00 117.41Contribution of Model Uncertainty 0.96 0.34Contribution of Random Variability 0.04 0.66

172

7.4 FORM OF PROPOSED EQUATIONS

A four-parameter (reference strain, curvature coefficient, scaling

coefficient and small-strain material damping ratio) soil model was presented in

Chapter Six. In Chapter Four, the impact of soil type and loading conditions on

nonlinear soil behavior were discussed.

In this section, the problem of accounting for the impact of soil type and

loading conditions on nonlinear soil behavior is structured analytically. The

parameters of the soil model are related to soil plasticity, void ratio, confining

pressure, overconsolidation ratio, loading frequency and number of loading

cycles. The parameters (denoted as φi in these equations) need to be calibrated

based on the experimental observations (resonant column and torsional shear test

results). The resulting equations can then be utilized to estimate the mean

normalized modulus reduction and material damping curves for a given soil type

and loading conditions.

In an effort to represent the trends regarding the scatter of the data, a

second set of equations for standard deviation are proposed in this section. The

modeled correlation structure of the data is also presented herein.

7.4.1 Equations for Mean

Equations used in representing the impact of soil type and loading

conditions on nonlinear soil behavior are described in this section. Proposed

forms of these equations are based on the trends reported in the literature and

experimental observations in the course of this study.

173

As discussed in Section 5.2, the hyperbolic soil model originally proposed

by Hardin and Drnevich (1972b) is slightly modified and this modified

relationship is utilized to represent a normalized modulus reduction curve as:

a

r

GG

+

=

γγ1

1

max

(7.25)

where: G/Gmax = normalized modulus,


γr = reference strain, and

a = curvature coefficient.

The two parameters in this model (γr and a) can be related to soil type and

loading conditions (σo’ and OCR) as follows: 43 '*)**( 21

φφ σφφγ or OCRPI+= (7.26a)

5φ=a (7.26b)

where: σo’ = mean effective confining pressure (atm),

PI = soil plasticity (%),

OCR = overconsolidation ratio, and

φ1 through φ5 = parameters that relate the normalized modulus reduction

curve to soil type and loading conditions.

These equations can model the shifting of the normalized modulus

reduction curve with increasing soil plasticity, overconsolidation and confining

pressure. When subsets of the data are evaluated, the sensitivity of the curvature

coefficient to soil type and loading conditions can also be studied.

174

In Section 5.3, the material damping curve is calculated assuming Masing

behavior. The calculated material damping curve is then scaled and shifted using:

minsin* DDFD gMa += (7.27)

where: 3

0.1,sin32

0.1,sin20.1,sin1sin === ++= agMaagMaagMagMa DcDcDcD ,

−

+

+−

Π== 2

ln41

20.1,sin

r

r

rr

agMaD

γγγ

γγγ

γγ,

0.2523 + 1.8618a + -1.1143a21 =c ,

0.0095 - 0.0710a - 0.0805a 22 =c ,

0.0003 + 0.0002a + 0.0005a- 23 =c ,

1.0

max*

=

GGbF ,

b = scaling coefficient,

G/Gmax = normalized modulus,

DMasing = damping estimated based on Masing Behavior, and

Dmin = small-strain material damping ratio.

The two parameters in this equation (Dmin and b) can be related to soil type

and loading conditions as follows:

[ ])ln(*1*'*)**( 1076min98 frqOCRPID o φσφφ φφ ++= (7.28a)

)ln(*1211 Nb φφ += (7.28b)



OCR = overconsolidation ratio,

175

frq = loading frequency,

N = number of loading cycles, and

φ6 through φ12 = parameters that relate material damping curve to soil type


Since the material damping curve is directly related to the normalized

modulus reduction curve, any shift in the normalized modulus reduction curve

(due to increasing soil plasticity, overconsolidation and confining pressure) is also

captured in the material damping curve. Small-strain trends in material damping

(discussed in Chapter Four) are accounted for by modeling Dmin separately as a

function of soil plasticity, overconsolidation, confining pressure and loading

frequency. Finally, the impact of number of loading cycles on the material

damping curve is captured in the equation for scaling coefficient.

7.4.2 Equations for Standard Deviation and Covariance Structure

Scatter of the data around the mean estimate is modeled considering the

characteristics of the normalized modulus reduction and material damping curves.

In the case of the normalized modulus reduction curve, less scatter around

the mean is expected at small strains (at which G/Gmax is about 1.0) and at rather

large strains (at which G/Gmax is less than 20%). Uncertainty maximizes around

the reference strain (at which G/Gmax is equal to 0.5). To capture these

characteristics, the standard deviation for a given normalized modulus reduction

curve is modeled as:

)exp()5.0/(

)exp(25.0)exp(

14

2max

1413 φφ

φσ−

−+=GG

NG (7.29)

where: σNG = standard deviation for normalized modulus reduction curve,

176

G/Gmax = estimated normalized shear modulus, and

φ13 and φ14 = parameters that relate standard deviation to mean estimate of

normalized shear modulus.

Figure 7.4 shows the variation of standard deviation with normalized shear

modulus predicted with Equation 7.29. This equation predicts smaller values of

standard deviation for G/Gmax values close to 1.0 and to 0.0. The scatter relative to

an estimated normalized modulus reduction curve is presented in Figure 7.5. This

scatter pattern looks a lot like the lower bound and upper bound curves proposed

by Seed et al. (1986) for sands shown in Figure 4.22.

0.5

0.4

0.3

0.2

0.1

0.0

σNG

1.00.80.60.40.20.0G/Gmax

Figure 7.4 Variation of standard deviation with normalized shear modulus

177

1.2

0.8

0.4

0.0

G/Gmax


Figure 7.5 Standard deviation modeled for normalized modulus reduction curve

Similarly, in case of the material damping curve, less scatter around the

mean is expected at small strains (at which D is close to Dmin) and uncertainty

increases with increasing shearing strain. As a result, standard deviation for

material damping ratio is modeled as follows:

DD *)exp()exp( 1615 φφσ += (7.30)

where: σD = standard deviation for material damping curve,

D = estimated material damping ratio, and

φ15 and φ16 = parameters that relate standard deviation to the mean

estimate of material damping ratio.

Figure 7.6 shows the variation of standard deviation with material

damping ratio predicted with Equation 7.30. This equation predicts larger values

of standard deviation for higher values of D. The scatter relative to an estimated

material damping curve is presented in Figure 7.7. This scatter pattern also looks

like the lower bound and upper bound curves proposed by Seed et al. (1986) for

sands shown in Figure 4.22.

178

4

3

2

1

0

σD

2520151050D, %

Figure 7.6 Variation of standard deviation with material damping ratio

25

20

15

10

5

0

D , %


Figure 7.7 Standard deviation modeled for material damping curve

Random variability and correlation between measurements are also

modeled in this study.

Due to equipment and operator errors, limited resolution of test equipment

and electronic noise that reduces measurement quality and precision, the result of

a test may not exactly be reproduced even though another test is performed under

identical circumstances. Measurements performed on the same specimen at the

179

same confining pressure using the same testing method are assumed to be

correlated. This correlation is modeled through using scale of fluctuation about

the mean estimate. Measurements performed at similar strain amplitudes are

modeled to be highly correlated with each other. The covariance structure is

formulated using:

))exp(

lnlnexp(*)

)exp(1exp(

1817, φ

γγ

φρ ji

ji

−−−= (7.31)

where: ρi,j = correlation coefficient,

γi = shearing strain at which measurement i is performed,

φ17 = random variability of data, and

φ18 = scale of fluctuation of data about the mean estimate.

7.5 SUMMARY

In this chapter, the First-order, Second-moment Bayesian Method is

introduced. This method is an approximation of Bayes’ theorem, which is a

systematic way of utilizing experience and observations in statistical analysis.

Equations that relate each parameter of the soil model to soil type and

loading conditions are also presented herein. The equations are formulated so that

the impact of strain amplitude, effective isotropic confining pressure, loading

history and loading frequency on dynamic soil behavior can be properly modeled

following the trends discussed in Chapter Four.

Finally, the equations formulated to represent the scatter and covariance

structure of the data are presented near the end of the chapter.

180

CHAPTER 8

STATISTICAL ANALYSIS OF THE RCTS DATA

8.1 INTRODUCTION

The statistical analysis of the RCTS data is performed using the First-

order, Second-moment Bayesian Method (FSBM) discussed in Chapter Seven. A

computer program that utilizes FSBM originally written by Dr. Robert B. Gilbert

is used in the analysis to calibrate the set of equations presented in Section 7.4.

The program was customized for this application through the C++ header and

source files presented in Appendices A through D.

Since the effect of number of cycles, N, and loading frequency, f, on

normalized modulus reduction curve is negligible in the case of the competent

soils that were investigated in the course of this study, the proposed equations in

Section 7.4 ignore the effect of N and f on G/Gmax. As a result, only resonant

column data (which is typically collected over a wider range of shearing strain

than the torsional shear test results) are utilized in the analysis of modulus

reduction. In case of material damping ratio, the proposed equations are calibrated

using the first and tenth cycles of torsional shear data along with the resonant

column data in an effort to model the effect of N and f on material damping ratio.

Since soil type is expected to be one of the most important parameters that

impact nonlinear soil behavior, the data was analyzed in several subsets according

to soil type and geographic location. Table 8.1 presents the distribution of the

specimens within the database according to soil type and geographic location. The

181

soil types can be categorized into four groups: 1) “Clean” Sands (sands with fines

content less than 12%), 2) Sands with High Fines Content (sands with fines

content greater than 12%), 3) Silts, and 4) Clays. The distribution of the

specimens according to these soil groups is tabulated in Table 8.2.

Table 8.1 Distribution of specimens with soil type and geographic location

Soil Type Northern California Southern California South Carolina Lotung, Taiwan TOTALSW-SM 1 3 - - 4SW-SC 1 - - - 1SP-SM 2 4 3 - 9

SP 2 - - - 2SM 5 13 5 2 25

SC-SM 2 2 - - 4SC 2 1 4 - 7ML 3 2 - 6 11MH 2 - 1 - 3

CL-ML 2 2 - - 4CL 10 16 2 - 28CH 5 4 3 - 12

TOTAL 37 47 18 8 110

Geographic Location

Table 8.2 Distribution of specimens by soil group and geographic location

Soil Group Soil Type Northern California Southern California South Carolina Lotung, Taiwan TOTALSW-SMSW-SCSP-SM

SPSM

SC-SMSCMLMH

CL-MLCLCH

16

36

18

40

61

5 -

Silts

Clays

6 7

9

7

15

16

4

20

Geographic Location

"Clean" Sands

Sands with High Fines

Content

3 -

9 2

182

The test results of all specimens from each soil group within a geographic

location were analyzed separately in order to study the effect of geology on model

parameters. Four specimens from South Carolina (specimens UT-39-G, UT-39-

M, UT-39-O, and UT-39-S) were removed from the database following the

analysis because the resonant column results did not follow the general trends

reported in the literature and observed during the course of this study. The

torsional shear results for these soils did follow the general trends but were not of

sufficient strain range to be included. As a result, a second set of analyses was

performed on two soil groups from which data had been discarded. Tables 8.3 and

8.4 present the distribution of the specimens within the database after the four

specimens have been discarded.

Table 8.3 Distribution of specimens with soil type and geographic location for the updated database

Soil Type Northern California Southern California South Carolina Lotung, Taiwan TOTALSW-SM 1 3 - - 4SW-SC 1 - - - 1SP-SM 2 4 3 - 9

SP 2 - - - 2SM 5 13 4 2 24

SC-SM 2 2 - - 4SC 2 1 3 - 6ML 3 2 - 6 11MH 2 - 1 - 3

CL-ML 2 2 - - 4CL 10 16 - - 26CH 5 4 3 - 12

TOTAL 37 47 14 8 106

Geographic Location

183

Table 8.4 Distribution of specimens by soil group and geographic location for the updated database

Soil Group Soil Type Northern California Southern California South Carolina Lotung, Taiwan TOTALSW-SMSW-SCSP-SM

SPSM

SC-SMSCMLMH

CL-MLCLCH

6 18

Clays 15 20 3 - 38

Silts 7 4 1

16

Sands with High Fines

Content9 16 7 2 34

Geographic Location

"Clean" Sands 6 7 3 -

The test results of all specimens from each soil group (regardless of its

geographic location) were also analyzed in order to study the effect of soil type on

model parameters. These analyses were performed on the updated database (after

discarding test results of the four specimens from South Carolina).

After concluding that the data were being stretched too thin to calibrate the

model using the subsets, the complete database was utilized in the analysis.

Section 8.3 presents the analysis of all credible data (within the updated database),

which forms the basis of the following chapters regarding the predictions based

on the calibrated model.

184

8.2 ANALYSIS OF SUBSETS OF THE DATA

As discussed in Section 8.1, the data were analyzed in several subsets

according to soil type and geographic location. This section presents the results of

these analyses in both graphical and tabular forms. Table 8.5 presents the prior

mean values and variances of the model parameters that were utilized in the

analysis. Prior mean values of the model parameters are initial guesses based on

general trends reported in the literature and observed during the course of this

study. Variances of the model parameters reflect the confidence in these initial

guesses.

8.2.1 Sorted by Location and Soil Group

The analyses of fourteen subsets of the data are presented in this section.

In most cases, a very limited number of points (for a meaningful analysis from a

statistical standpoint) are utilized in the analysis presented herein as a result of

dividing the data into many subsets. Consequently, the results of these analyses

are presented only for qualitative purposes to study the effect of geology on

model parameters.

8.2.1.1 Samples from Northern California

The results of the analysis of the four soil groups from Northern California

are presented in this section. The updated mean values and variances of the model

parameters for the soil groups are tabulated in Table 8.6.

185

Table 8.5 Prior mean values and variances of the model parameters

Mean Variance Mean Variance Mean Variance Mean Varianceφ1 3.50E-02 1.00E-04 3.50E-02 1.00E-04 3.50E-02 1.00E-04 3.50E-02 1.00E-04φ2 1.00E-03 6.25E-06 1.00E-03 6.25E-06 1.00E-03 6.25E-06 1.00E-03 6.25E-06φ3 2.50E-01 1.00E-02 2.50E-01 1.00E-02 2.50E-01 1.00E-02 2.50E-01 1.00E-02φ4 5.00E-01 1.00E-02 5.00E-01 1.00E-02 5.00E-01 1.00E-02 5.00E-01 1.00E-02φ5 8.50E-01 2.25E-02 9.00E-01 2.25E-02 1.00E+00 2.25E-02 1.05E+00 2.25E-02φ6 8.00E-01 2.50E-01 8.50E-01 2.50E-01 1.00E+00 2.50E-01 1.10E+00 2.50E-01φ7 1.00E-02 2.50E-05 1.00E-02 2.50E-05 1.00E-02 2.50E-05 1.00E-02 2.50E-05φ8 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03φ9 -1.50E-01 1.00E-02 -1.50E-01 1.00E-02 -1.50E-01 1.00E-02 -1.50E-01 1.00E-02φ10 2.00E-01 1.00E-02 2.00E-01 1.00E-02 2.00E-01 1.00E-02 2.00E-01 1.00E-02φ11 7.50E-01 1.00E-02 7.00E-01 1.00E-02 6.50E-01 1.00E-02 6.00E-01 1.00E-02φ12 -3.00E-02 1.00E-04 -2.00E-02 1.00E-04 -1.00E-02 1.00E-04 0.00E+00 1.00E-04φ13 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00φ14 4.50E+00 4.00E+00 4.50E+00 4.00E+00 4.50E+00 4.00E+00 4.50E+00 4.00E+00φ15 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00φ16 -1.00E+00 1.00E+00 -1.00E+00 1.00E+00 -1.00E+00 1.00E+00 -1.00E+00 1.00E+00φ17 4.00E+00 9.00E+00 4.00E+00 9.00E+00 4.00E+00 9.00E+00 4.00E+00 9.00E+00φ18 2.00E+00 1.00E+00 2.00E+00 1.00E+00 2.00E+00 1.00E+00 2.00E+00 1.00E+00

Model Parameters*

"Clean" Sands Sands with High Fines Content Silts Clays

* Model parameters were defined in Equations 7.25 through 7.31 in Section 7.4.

a

r

GG

+

=

γγ1

1

max

43 '*)**( 21φφ σφφγ or OCRPI+=

5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=

[ ])ln(*1*'*)**( 1076min98 frqOCRPID o φσφφ φφ ++=

)ln(*1211 Nb φφ +=

186

Table 8.6 Updated mean values and variances of the model parameters for the soils from Northern California

Mean Variance Mean Variance Mean Variance Mean Varianceφ1 4.26E-02 9.50E-06 2.76E-02 2.05E-06 3.97E-02 8.49E-06 3.39E-02 3.65E-05φ2 -2.28E-03 1.20E-07 -3.42E-05 8.90E-09 -3.27E-06 4.25E-08 1.75E-03 5.30E-08φ3 2.50E-01 1.00E-02 2.50E-01 1.00E-02 2.50E-01 1.00E-02 2.97E-01 4.62E-03φ4 2.53E-01 1.90E-03 2.49E-01 1.26E-03 1.50E-01 1.41E-03 2.78E-01 1.07E-03φ5 9.41E-01 6.28E-04 8.83E-01 2.33E-04 1.05E+00 8.37E-04 9.93E-01 4.66E-04φ6 9.95E-01 1.25E-02 1.26E+00 2.19E-02 9.54E-01 6.99E-03 1.05E+00 1.30E-02φ7 1.24E-02 2.37E-05 1.61E-02 2.21E-05 8.04E-03 1.95E-05 8.20E-04 1.05E-05φ8 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03 -9.95E-02 2.50E-03 -1.01E-01 2.50E-03φ9 1.25E-01 5.15E-03 -1.73E-01 3.56E-03 -2.30E-01 2.40E-03 -1.72E-01 1.67E-03φ10 2.85E-01 3.37E-03 2.19E-01 2.09E-03 2.90E-01 3.59E-03 3.23E-01 2.98E-03φ11 7.50E-01 1.96E-03 6.71E-01 1.09E-03 4.11E-01 6.93E-04 4.40E-01 5.99E-04φ12 -2.83E-02 3.40E-05 -1.50E-02 3.28E-05 1.58E-02 1.81E-05 2.33E-02 1.61E-05φ13 -3.70E+00 7.49E-02 -3.35E+00 1.76E-02 -5.00E+00 9.00E+00 -5.28E+00 1.27E-01φ14 4.71E+00 3.07E-01 5.11E+00 3.50E+00 4.31E+00 6.40E-02 3.89E+00 4.45E-02φ15 -5.03E+00 8.97E+00 -5.06E+00 8.90E+00 -5.01E+00 8.99E+00 -5.22E+00 8.75E+00φ16 -5.02E-01 1.39E-02 -4.29E-01 1.14E-02 -9.62E-01 1.01E-02 -6.70E-01 6.16E-03φ17 4.85E+00 2.14E-01 3.79E+00 9.24E-02 4.03E+00 8.98E+00 7.04E+00 6.66E-01φ18 2.95E+00 6.66E-02 2.67E+00 5.89E-02 1.94E+00 4.60E-02 2.18E+00 2.52E-02

Silts ClaysModel Parameters*

"Clean" Sands Sands with High Fines Content


a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

187

Since most of the specimens from Northern California are normally

consolidated, the updated mean values and variances of φ3 and φ8 (which

represent the effect of overconsolidation ratio on reference strain and small-strain

material damping ratio, respectively) are almost identical to prior values. Hence,

the data have not provided much information regarding these parameters. The

values of φ1 (which represents the reference strain of a nonplastic soil at 1 atm

confining pressure), φ4 (which represents the effect of confining pressure on

reference strain), φ6 (which represents the small-strain material damping ratio of a

nonplastic soil at 1 atm confining pressure deformed at 1 Hz loading frequency)

and φ10 (which represents the effect of loading frequency on small-strain material

damping ratio) are observed to be consistent between soil groups.

The comparisons of the measurements with the predicted values based on

the calibrated models for the four soil groups are presented in Figures 8.1 through

8.4. Significantly less error is observed in the prediction of normalized shear

modulus than in the prediction of material damping predictions ratio for all soil

groups. This difference can be attributed to material damping ratio being sensitive

to the characteristics of the complete stress-strain loop while normalized shear

modulus is only related to the end points of the stress-strain loop. Consequently,

measurement and prediction of material damping ratio is more complicated than

measurement and prediction of normalized shear modulus.

188

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0

Measured Normalized Modulus

(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050

Measured Material Damping

(b)

Figure 8.1 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands from Northern California

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.2 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from Northern California

189

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.3 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from Northern California

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.4 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from Northern California

190

8.2.1.2 Samples from Southern California

The updated mean values and variances of the model parameters for the

four soil groups from Southern California are tabulated in Table 8.7. As in the

case of Northern California, the data have not provided much information

regarding the φ3 and φ8 parameters. The φ1, φ4, φ6 and φ10 parameters are again

observed to be consistent between soil groups. φ5 (which is the curvature

coefficient) is observed to slightly increase with decreasing particle size. The

comparisons of the measured and predicted values based on the calibrated models

are presented in Figures 8.5 through 8.8.

8.2.1.3 Samples from South Carolina

8.2.1.3.1 Analysis of Test Results from All (Eighteen) Specimens


four soil groups from South Carolina are tabulated in Table 8.8. The model

parameters are observed to be extremely inconsistent. Part of the problem is

believed to be the result of analyzing a very small dataset in the case of “clean”

sands and silts.

The comparisons of the measured and predicted values based on the

calibrated models are presented in Figures 8.9 through 8.12. Test results from four

specimens do not agree with the observed trends and reduce the quality of the

predictions for two soil groups, sands with high fines content and clays. As a

result, these specimens were identified and discarded from the database.

191

Table 8.7 Updated mean values and variances of the model parameters for the soils from Southern California

Mean Variance Mean Variance Mean Variance Mean Varianceφ1 2.53E-02 6.44E-06 3.51E-02 5.30E-06 5.18E-02 4.53E-05 3.52E-02 5.76E-06φ2 1.00E-03 6.25E-06 1.34E-03 8.41E-08 5.96E-05 9.19E-07 7.07E-04 9.12E-09φ3 2.50E-01 1.00E-02 2.62E-01 9.44E-03 2.50E-01 1.00E-02 3.69E-01 7.30E-03φ4 4.62E-01 2.97E-03 5.04E-01 1.35E-03 4.26E-01 9.34E-04 2.97E-01 5.31E-04φ5 8.34E-01 6.37E-04 8.58E-01 3.62E-04 9.40E-01 1.41E-03 9.50E-01 2.25E-04φ6 8.42E-01 3.56E-02 8.26E-01 1.19E-02 7.75E-01 7.47E-03 1.01E+00 4.39E-03φ7 1.00E-02 2.50E-05 1.29E-02 2.36E-05 8.55E-03 2.41E-05 3.92E-04 1.06E-05φ8 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03 -1.01E-01 2.49E-03φ9 -2.90E-01 6.32E-03 -4.18E-01 2.99E-03 -1.30E-01 1.50E-03 -1.97E-01 9.35E-04φ10 2.53E-01 5.59E-03 2.37E-01 2.70E-03 1.97E-01 2.90E-03 3.74E-01 1.76E-03φ11 7.62E-01 2.08E-03 7.70E-01 1.61E-03 6.67E-01 2.24E-03 5.18E-01 4.84E-04φ12 -2.67E-02 4.19E-05 -2.41E-02 3.12E-05 -1.95E-02 3.56E-05 1.68E-02 1.18E-05φ13 -5.05E+00 8.93E+00 -5.00E+00 9.00E+00 -5.02E+00 8.98E+00 -5.68E+00 3.93E-02φ14 4.51E+00 6.15E-02 3.27E+00 3.49E-02 4.92E+00 8.97E-02 4.39E+00 1.98E-02φ15 -5.00E+00 9.00E+00 -5.04E+00 8.96E+00 -5.00E+00 9.00E+00 -5.03E+00 8.97E+00φ16 -5.97E-01 1.38E-02 -1.81E-01 7.62E-03 -1.23E+00 1.54E-02 -8.40E-01 2.98E-03φ17 3.99E+00 9.00E+00 4.00E+00 9.00E+00 4.06E+00 8.93E+00 4.23E+00 8.80E+00φ18 2.19E+00 5.19E-02 2.80E+00 2.65E-02 1.53E+00 8.08E-02 2.08E+00 1.31E-02

"Clean" Sands Sands with High Fines Content Silts ClaysModel

Parameters*


a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

192

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.5 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands from Southern California

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.6 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from Southern California

193

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.7 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from Southern California

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.8 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from Southern California

194

Table 8.8 Updated mean values and variances of the model parameters for the soils from South Carolina

Mean Variance Mean Variance Mean Variance Mean Varianceφ1 6.83E-02 3.33E-05 3.22E-02 2.32E-05 3.47E-02 9.99E-05 3.20E-02 9.90E-05φ2 5.82E-03 5.21E-06 -5.03E-04 7.03E-08 9.34E-04 9.82E-09 2.79E-03 2.90E-07φ3 2.50E-01 1.00E-02 2.50E-01 1.00E-02 2.83E-01 9.88E-03 2.50E-01 1.00E-02φ4 -3.88E-01 2.85E-03 -3.80E-02 3.52E-03 2.30E-01 1.98E-03 7.04E-02 1.92E-03φ5 8.38E-01 6.73E-03 8.96E-01 1.87E-03 1.17E+00 2.18E-03 1.09E+00 5.85E-03φ6 1.57E-01 5.13E-05 2.82E-01 1.47E-03 5.13E-01 1.40E-01 -2.18E-01 8.59E-02φ7 1.01E-02 2.49E-05 1.68E-02 1.00E-05 4.90E-03 9.65E-06 5.70E-03 1.24E-05φ8 -1.00E-01 2.50E-03 -9.87E-02 2.50E-03 -1.12E-01 2.10E-03 -1.00E-01 2.50E-03φ9 -1.46E-01 6.68E-03 4.71E-02 6.16E-03 -1.25E-01 4.22E-03 -2.93E-01 7.83E-03φ10 2.27E-01 1.26E-03 3.05E-01 8.95E-03 9.62E-02 8.47E-04 1.42E-01 5.69E-03φ11 8.87E-01 2.36E-03 6.68E-01 3.05E-03 4.35E-01 1.12E-03 3.53E-01 4.25E-03φ12 -1.26E-02 3.76E-05 -2.05E-02 4.69E-05 2.32E-02 3.22E-05 -2.08E-02 7.07E-05φ13 -6.57E+00 6.41E-01 -5.04E+00 8.97E+00 -5.84E+00 3.42E-01 -4.65E+00 1.01E-01φ14 5.38E+00 2.88E-01 2.44E+00 7.23E-02 5.98E+00 3.14E-01 3.94E+00 2.88E-01φ15 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00 -5.02E+00 8.98E+00 -5.00E+00 9.00E+00φ16 -1.80E+00 1.43E-02 -1.20E-01 1.75E-02 -1.59E+00 1.11E-02 8.52E-01 3.96E-02φ17 3.36E+00 5.19E-01 4.00E+00 9.00E+00 4.01E+00 8.99E+00 4.03E+00 8.98E+00φ18 5.66E-01 8.76E-02 2.39E+00 5.09E-02 -3.42E-03 8.17E-02 2.82E+00 1.00E-01

"Clean" Sands Sands with High Fines Content Silts ClaysModel

Parameters*


a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

195

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.9 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands from South Carolina

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.10 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from South Carolina

196

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.11 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from South Carolina

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.12 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from South Carolina

197

8.2.1.3.2 Analysis of Test Results from Fourteen Specimens

After discarding the four specimens that were identified not to be

consistent with the observed trends, the two soil groups (sands with high fines

content and clays) that were affected by the process of changing the contents of

the database were reevaluated. Considerable improvement in predicted values is

achieved. The updated mean values and variances of the model parameters for

these two soil groups from South Carolina are tabulated in Table 8.9 and the

comparisons of the measured and predicted values based on the calibrated models

are presented in Figures 8.13 through 8.14.

8.2.1.4 Samples from Lotung, Taiwan


two soil groups from Lotung, Taiwan are tabulated in Table 8.10. Inconsistencies

between the model parameters are attributed to analyzing a very small dataset in

the case of sands with high fines content. The comparisons of the measured and

predicted values based on the calibrated models are presented in Figures 8.15 and

8.16.

198

Table 8.9 Updated mean values and variances of the model parameters for the South Carolina soil groups affected by change in the contents of the database

Mean Variance Mean Varianceφ1 2.88E-02 7.22E-06 3.25E-02 9.87E-05φ2 -3.62E-05 7.87E-09 2.64E-03 6.85E-08φ3 2.48E-01 1.00E-02 2.50E-01 1.00E-02φ4 6.36E-01 4.38E-03 1.89E-02 1.36E-03φ5 8.78E-01 7.74E-04 1.29E+00 3.12E-03φ6 4.37E-01 1.85E-03 9.66E-01 3.93E-02φ7 1.23E-02 1.56E-06 1.08E-02 1.06E-05φ8 -8.35E-02 2.43E-03 -1.00E-01 2.50E-03φ9 -7.45E-02 7.99E-03 -3.84E-02 2.43E-03φ10 3.25E-01 5.92E-03 2.89E-01 1.59E-03φ11 7.92E-01 2.54E-03 5.09E-01 2.98E-03φ12 -3.66E-02 3.96E-05 6.75E-03 5.86E-05φ13 -5.02E+00 8.98E+00 -5.07E+00 1.59E-01φ14 4.31E+00 5.55E-02 4.93E+00 2.33E-01φ15 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00φ16 -6.92E-01 1.11E-02 -1.62E+00 1.79E-02φ17 4.00E+00 8.98E+00 3.55E+00 1.09E+00φ18 1.93E+00 4.71E-02 1.40E+00 1.91E-01

Sands with High Fines Content ClaysModel

Parameters*


a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

199

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.13 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from South Carolina (After Discarding Specimens UT-39-G and UT-39-M)

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.14 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays from South Carolina (After Discarding Specimens UT-39-O and UT-39-S)

200

Table 8.10 Updated mean values and variances of the model parameters for the soils from Lotung, Taiwan

Mean Variance Mean Varianceφ1 2.90E-02 9.78E-06 4.52E-02 1.08E-05φ2 1.00E-03 6.25E-06 1.69E-04 1.78E-07φ3 2.50E-01 1.00E-02 2.50E-01 1.00E-02φ4 5.02E-01 8.88E-03 1.53E-01 1.97E-03φ5 8.02E-01 6.48E-04 1.02E+00 1.08E-03φ6 5.46E-01 1.48E-02 5.73E-01 4.88E-03φ7 1.00E-02 2.50E-05 1.05E-02 2.17E-05φ8 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03φ9 -1.64E-01 9.79E-03 -1.39E-01 5.10E-03φ10 2.37E-01 7.16E-03 1.02E-01 2.83E-03φ11 8.07E-01 3.16E-03 7.22E-01 2.07E-03φ12 -3.84E-02 5.39E-05 -1.70E-02 2.89E-05φ13 -7.12E+00 4.55E+00 -5.01E+00 8.99E+00φ14 4.79E+00 3.14E-01 4.56E+00 5.02E-02φ15 -5.00E+00 9.00E+00 -5.01E+00 8.99E+00φ16 -8.14E-01 5.27E-02 -9.93E-01 8.88E-03φ17 3.99E+00 8.97E+00 4.00E+00 9.00E+00φ18 2.74E+00 2.01E-01 9.02E-01 4.99E-02

Sands with High Fines Content SiltsModel

Parameters*


a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

201

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.15 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content from Lotung, Taiwan

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.16 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts from Lotung, Taiwan

202

8.2.1.5 Comparison of the Nonlinear Behavior of Soils from Different Locations

In order to evaluate the impact of geology on nonlinear soil behavior, the

normalized modulus reduction and material damping curves for a given soil type

under given loading conditions are predicted utilizing the updated model

parameters for different geographic locations. Figures 8.17 through 8.19 present

the comparison of these predictions.

In Figure 8.17, predicted normalized modulus reduction and material

damping curves for a silty sand are shown. The soil is selected to be nonplastic

and normally consolidated in order to analyze a representative material within the

database. The confining pressure is selected to be 1 atm in the same fashion. Ten

cycles of loading at 1 Hz is chosen so that the loading conditions represent the

characteristics of an earthquake. In this figure, the effect of geographic location

(and geology for that matter) on dynamic soil behavior is observed to be

negligible.

Figure 8.18 shows the comparison of predicted nonlinear soil behavior for

a moderate plasticity silt. The characteristics of the soil are again selected so that

a representative material within the database is evaluated. Loading conditions are

selected to be the same as those in the case of the silty sand. In Figure 8.18a,

normalized modulus reduction curves for the two predictions are observed to be

quite similar. On the other hand, the predicted material damping ratios shown in

Figure 8.18b are somewhat different at shearing strains above 0.1 %. This

divergence can be investigated in future studies using a larger database.

203

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

(a)

Silty Sand from Northern CaliforniaSilty Sand from Southern CaliforniaSilty Sand from South Carolina

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1

Shearing Strain, γ ,%

(b)

PI = 0 %OCR = 1f = 1 HzN =10σo' = 1 atm

Figure 8.17 (a) Normalized modulus reduction and (b) material damping curves estimated for a nonplastic silty sand using updated mean values of model parameters calibrated at different geographic locations

204

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

(a)

Silt from Northern CaliforniaSilt from Lotung, Taiwan

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


(b)

PI = 15 %OCR = 1f = 1 HzN =10σo' = 1 atm

Figure 8.18 (a) Normalized modulus reduction and (b) material damping curves estimated for a moderate plasticity silt using updated mean values of model parameters calibrated at different geographic locations

205

Figure 8.19 shows the comparison of predicted nonlinear soil behavior for

a moderate plasticity clay evaluated for identical loading conditions. As in the

case of the silty sand, the effect of geographic location and geology on dynamic

soil behavior is observed to be negligible.

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

(a)

Clay from Northern CaliforniaClay from Southern California

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


(b)

PI = 15 %OCR = 1f = 1 HzN =10σo' = 1 atm

Figure 8.19 (a) Normalized modulus reduction and (b) material damping curves estimated for a moderate plasticity clay using updated mean values of model parameters calibrated at different geographic locations

206

Comparison of the results for the subsets of the data sorted according to

geographic location does not indicate a strong correlation between geology and

nonlinear soil behavior. As a result, soils within a soil group from different

geographic locations shall be analyzed together in the following sections.

8.2.2 Sorted by Soil Group

Since the effect of geology on nonlinear soil behavior is observed not to

be very significant, soils from different geographic locations are combined into

larger datasets. This section presents the results of the analysis of all soils within

each soil group. Since four specimens from South Carolina were previously

identified to be inconsistent with the general trends, the data associated with these

specimens are not included in these analyses.

Table 8.11 presents the updated mean values and variances of the model

parameters calibrated for the four soil groups. Most of the model parameters are

observed to be consistent between soil groups. The value of φ5 (which is the

curvature coefficient) is observed to be slightly different for the coarse grained

and fine grained soils. This trend is consistent with the observations documented

by Hardin and Drnevich (1972b). The values of φ2 and φ7 (which represent the

effect of plasticity on reference strain and small-strain material damping ratio,

respectively) are observed to be quite different for the coarse grained and fine

grained soils. This difference is believed to result from the smaller range in

plasticity and the fewer number of plastic soils sampled in the case of coarse

grained soils. The comparisons of the measured and predicted values based on the

calibrated models are presented in Figures 8.20 through 8.23.

207

Table 8.11 Updated mean values and variances of the model parameters for the four soil groups

Mean Variance Mean Variance Mean Variance Mean Varianceφ1 4.74E-02 9.62E-06 3.34E-02 2.06E-06 4.16E-02 5.18E-06 2.58E-02 5.68E-06φ2 -2.34E-03 1.63E-07 -5.79E-05 8.09E-09 6.89E-04 7.74E-09 1.95E-03 1.84E-08φ3 2.50E-01 1.00E-02 2.49E-01 9.94E-03 3.21E-01 7.56E-03 9.92E-02 1.64E-03φ4 2.34E-01 1.08E-03 4.82E-01 7.46E-04 2.80E-01 8.63E-04 2.26E-01 3.48E-04φ5 8.95E-01 4.30E-04 8.45E-01 1.49E-04 1.00E+00 4.10E-04 9.75E-01 1.60E-04φ6 6.88E-01 7.82E-03 8.89E-01 5.86E-03 7.12E-01 3.55E-03 9.58E-01 2.93E-03φ7 1.22E-02 2.43E-05 2.02E-02 1.91E-05 3.03E-03 2.65E-06 5.65E-03 2.79E-06φ8 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03 -1.00E-01 2.50E-03φ9 -1.27E-01 4.00E-03 -3.72E-01 1.83E-03 -1.89E-01 1.95E-03 -1.96E-01 5.21E-04φ10 2.88E-01 3.14E-03 2.33E-01 1.35E-03 2.34E-01 2.60E-03 3.68E-01 1.19E-03φ11 7.67E-01 1.59E-03 7.76E-01 7.71E-04 5.92E-01 8.09E-04 4.66E-01 2.69E-04φ12 -2.83E-02 2.79E-05 -2.94E-02 1.70E-05 -7.67E-04 1.61E-05 2.23E-02 7.13E-06φ13 -4.14E+00 4.17E-02 -3.98E+00 1.82E-02 -5.02E+00 8.98E+00 -5.65E+00 3.37E-02φ14 3.61E+00 5.97E-02 4.32E+00 3.30E-02 3.93E+00 2.47E-02 4.00E+00 1.21E-02φ15 -5.15E+00 8.80E+00 -5.34E+00 8.55E+00 -5.20E+00 8.76E+00 -5.00E+00 9.00E+00φ16 -2.32E-01 7.56E-03 -2.66E-01 3.40E-03 -6.42E-01 4.78E-03 -7.25E-01 1.92E-03φ17 5.15E+00 6.91E-02 4.92E+00 3.74E-02 4.06E+00 8.96E+00 7.67E+00 3.51E-01φ18 3.12E+00 2.88E-02 2.68E+00 1.38E-02 1.94E+00 1.98E-02 2.16E+00 8.08E-03

ClaysModel Parameters*

"Clean" Sands Sands with High Fines Content Silts


a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

208

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.20 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for “clean” sands

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.21 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for sands with high fines content

209

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.22 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for silts

1.2

1.0

0.8

0.6

0.4

0.2

0.0Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


(a)25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


(b)

Figure 8.23 Comparisons of the measured and predicted values of (a) normalized modulus and (b) material damping ratio for clays

210

In order to evaluate the impact of soil type on nonlinear soil behavior, the

normalized modulus reduction and material damping curves for a given soil under

given loading conditions are predicted utilizing the updated model parameters for

different soil groups. Figure 8.24 presents the comparison of these predictions.

The coarse grained soils are selected to be nonplastic and the fine grained

soils are selected to be of moderate plasticity in order to analyze a representative

material within each soil group. The confining pressure is selected to be 1 atm for

the same reason. Ten cycles of loading at 1 Hz is again chosen so that the loading

conditions represent the characteristics of an earthquake.

In Figure 8.24, the difference in the nonlinear behavior of the soils from

the four soil groups is observed to be small. From a qualitative standpoint, “clean”

sands are observed to be relatively linear (normalized modulus reduction and

material damping curves located at higher strain amplitudes) compared to sands

with high fines content. This trend is consistent with the discrepancy between

normalized modulus reduction and material damping curves reported for

uniformly graded sand specimens (Iwasaki et al., 1978; Kokusho, 1980; and Ni,

1987) and for natural materials (Seed et al., 1986; Sun et al., 1988; Vucetic and

Dobry, 1991; Hwang, 1997; and Darendeli et al., 2001).

The comparison of the predictions in Figure 8.24 indicates that fines

content (the soil group) does not have a very significant impact on nonlinear soil

behavior. Thus, a model calibrated for the complete data set can be successfully

utilized in developing a new family of normalized modulus reduction and material

damping curves.

211

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

(a)

"Clean" Sand (PI = 0 %)Sand with High Fines Content (PI = 0 %)Silt (PI = 15 %)Clay (PI = 15 %)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


(b)

OCR = 1f = 1 HzN =10σo' = 1 atm

Figure 8.24 (a) Normalized modulus reduction and (b) material damping curves estimated using updated mean values of model parameters calibrated for different soil groups

212

8.3 ANALYSIS OF ALL CREDIBLE DATA

The predictions based on the calibrated models from the subsets of the

database indicate that the effects of geology (analyzed through geographic

location) and fines content (analyzed through soil groups) on the model

parameters are not very pronounced. As a result, all credible data (after removal

of the four specimens from South Carolina from the database) are analyzed as one

complete data set herein. The recommended values of the model parameters and

recommended nonlinear curves discussed in the following chapters are based on

the analysis presented in this section. As discussed in Section 8.1, only resonant

column data are utilized in the analysis of normalized shear modulus, and, in an

effort to model the effect of N and f on material damping, first and tenth cycles of

torsional shear data along with the resonant column data are utilized in the

analysis of material damping ratio. All credible data used in the analysis are


Table 8.12 presents the prior and the updated mean values and variances

of the model parameters calibrated for all the credible data presented in Figure

8.25. The table indicates considerable reduction in uncertainty (in the form of

variance) in the model parameters. The only exceptions are parameters φ8 (which

represents the impact of overconsolidation ratio on small-strain material damping

ratio) and φ15 (which represents the scatter of material damping ratio at small-

strains). In the case of these two parameters, very little information is gathered

indicating that the quality of the predictions (illustrated in Figures 8.26 and 8.27)

associated with the calibrated model is not very sensitive to these parameters.

213

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

Measured Normalized Shear Modulus

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Measured Material Damping Ratio (b)

Figure 8.25 All credible (a) normalized modulus data from the resonant column tests, and (b) material damping data from the resonant column and torsional shear tests utilized to calibrate the model parameters.

214

Table 8.12 Comparison of the prior and updated mean values and variances of the model parameters for all the credible data

Mean Variance Mean Varianceφ1 3.50E-02 1.00E-04 3.52E-02 9.99E-07φ2 1.00E-03 6.25E-06 1.01E-03 4.16E-09φ3 2.50E-01 1.00E-02 3.25E-01 2.85E-03φ4 5.00E-01 1.00E-02 3.48E-01 2.20E-04φ5 9.00E-01 2.25E-02 9.19E-01 6.78E-05φ6 8.50E-01 2.50E-01 8.01E-01 1.73E-03φ7 1.00E-02 2.50E-05 1.29E-02 3.82E-06φ8 -1.00E-01 2.50E-03 -1.07E-01 2.49E-03φ9 -1.50E-01 1.00E-02 -2.89E-01 4.96E-04φ10 2.00E-01 1.00E-02 2.92E-01 7.66E-04φ11 7.00E-01 1.00E-02 6.33E-01 2.23E-04φ12 -2.00E-02 1.00E-04 -5.66E-03 5.02E-06φ13 -5.00E+00 9.00E+00 -4.23E+00 5.38E-03φ14 4.50E+00 4.00E+00 3.62E+00 7.05E-03φ15 -5.00E+00 9.00E+00 -5.00E+00 9.00E+00φ16 -1.00E+00 1.00E+00 -2.50E-01 1.06E-03φ17 4.00E+00 9.00E+00 5.62E+00 1.53E-02φ18 2.00E+00 1.00E+00 2.78E+00 3.86E-03

Prior UpdatedModel Parameters*


a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

215

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Pred

icte

d N

orm

aliz

ed M

odul

us

1.21.00.80.60.40.20.0


Figure 8.26 Comparisons of the measured and predicted values of normalized modulus for all credible data

216

25

20

15

10

5

0

Pred

icte

d M

ater

ial D

ampi

ng

2520151050


Figure 8.27 Comparisons of the measured and predicted values of material damping for all credible data

217

Table 8.13 shows the covariance structure of the model parameters. This

information can be utilized in calculating model uncertainty associated with the

normalized modulus reduction and material damping curves estimated based on

this model as discussed in Chapter Eleven. Since the model is calibrated to a

rather large database, most of the uncertainty however results from the variability

within the database modeled using parameters φ13 through φ18 (Section 7.4.2).

8.4 SUMMARY

In order to study the effect of soil type and geology, the data was first

analyzed in several subsets according to soil group (“clean” sands, sands with

high fines content, silts and clays) and geographic location (Northern California,

Southern California, South Carolina and Lotung, Taiwan).

Four specimens from South Carolina (specimens UT-39-G, UT-39-M,

UT-39-O, and UT-39-S) were removed from the database following the analysis

because the resonant column results did not follow the general trends reported in

the literature and observed during the course of this study. The torsional shear

results for these specimens did follow the general trends but were not of sufficient

strain range to be included. At this point, the analyses affected by the change in

the content of the database were repeated and the rest of the analyses were carried

out without utilizing the data associated with these four specimens.

The test results of all specimens from each soil group (regardless of

geographic location) were also analyzed in order to study the effect of soil type on

model parameters.

218

Table 8.13 Covariance structure of the updated model parameters for all the credible data

φi* φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 φ10 φ11 φ12 φ13 φ14 φ15 φ16 φ17 φ18

φ1 1.00 -0.10 0.03 -0.39 -0.25 -0.05 0.05 0.00 0.05 0.03 0.51 -0.10 -0.06 0.03 0.00 -0.07 -0.06 -0.12

φ2 -0.10 1.00 -0.55 -0.27 -0.10 0.03 -0.18 0.00 0.06 0.05 0.15 0.06 0.03 0.00 0.00 -0.04 0.01 -0.04

φ3 0.03 -0.55 1.00 0.24 -0.01 -0.06 0.12 0.00 -0.02 -0.01 0.00 -0.01 -0.03 -0.03 0.00 0.03 -0.01 0.02

φ4 -0.39 -0.27 0.24 1.00 -0.01 0.01 0.05 0.00 -0.23 -0.03 -0.09 0.08 0.00 -0.13 0.00 0.14 0.03 0.14

φ5 -0.25 -0.10 -0.01 -0.01 1.00 -0.06 -0.02 0.00 -0.05 0.02 -0.51 0.00 0.11 -0.17 0.00 0.12 0.02 0.09

φ6 -0.05 0.03 -0.06 0.01 -0.06 1.00 -0.22 -0.01 -0.20 -0.56 0.24 -0.12 -0.02 0.08 0.00 -0.30 0.04 -0.10

φ7 0.05 -0.18 0.12 0.05 -0.02 -0.22 1.00 0.00 -0.18 -0.11 0.06 -0.08 -0.03 -0.01 0.00 -0.03 -0.06 0.01

φ8 0.00 0.00 0.00 0.00 0.00 -0.01 0.00 1.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

φ9 0.05 0.06 -0.02 -0.23 -0.05 -0.20 -0.18 0.01 1.00 -0.01 0.02 0.01 -0.05 0.12 0.00 -0.18 -0.04 -0.16

φ10 0.03 0.05 -0.01 -0.03 0.02 -0.56 -0.11 0.00 -0.01 1.00 -0.15 0.28 -0.09 0.09 0.00 -0.10 -0.15 -0.17

φ11 0.51 0.15 0.00 -0.09 -0.51 0.24 0.06 0.00 0.02 -0.15 1.00 -0.54 -0.07 0.08 0.00 -0.20 -0.01 -0.12

φ12 -0.10 0.06 -0.01 0.08 0.00 -0.12 -0.08 0.00 0.01 0.28 -0.54 1.00 -0.03 0.03 0.00 -0.04 -0.05 -0.06

φ13 -0.06 0.03 -0.03 0.00 0.11 -0.02 -0.03 0.00 -0.05 -0.09 -0.07 -0.03 1.00 0.12 0.00 0.37 0.42 0.37

φ14 0.03 0.00 -0.03 -0.13 -0.17 0.08 -0.01 0.00 0.12 0.09 0.08 0.03 0.12 1.00 0.00 -0.56 -0.17 -0.65

φ15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00

φ16 -0.07 -0.04 0.03 0.14 0.12 -0.30 -0.03 0.00 -0.18 -0.10 -0.20 -0.04 0.37 -0.56 0.00 1.00 0.42 0.86

φ17 -0.06 0.01 -0.01 0.03 0.02 0.04 -0.06 0.00 -0.04 -0.15 -0.01 -0.05 0.42 -0.17 0.00 0.42 1.00 0.34

φ18 -0.12 -0.04 0.02 0.14 0.09 -0.10 0.01 0.00 -0.16 -0.17 -0.12 -0.06 0.37 -0.65 0.00 0.86 0.34 1.00 * Model parameters, φi, were defined in Equations 7.25 through 7.31 in Section 7.4.

a

r

GG

+

=

γγ1

1

max


5φ=a

minsin

1.0

max** DD

GGbD gMaAdjusted +

=


)ln(*1211 Nb φφ +=

219

After concluding that the effects of geology (analyzed through geographic

location) and fines content (analyzed through soil groups) on the model

parameters were not very pronounced based on the analysis of subsets of the

database, all credible data (within the updated database) were analyzed as one

complete data set to calibrate the model. Calculation of mean normalized modulus

reduction and material damping curves and handling uncertainty associated with

these curves are discussed in the following chapters.

220

CHAPTER 9

PREDICTING NONLINEAR SOIL BEHAVIOR USING THE

CALIBRATED MODEL

9.1 INTRODUCTION

Proposed equations discussed in Section 7.4 have been calibrated using all

credible data in Section 8.3. The updated mean values of the model parameters

presented in Table 8.12 can be used to estimate normalized modulus reduction

and material damping curves for a broad range of soil types and loading

conditions.

Since the predictions are based on the model calibrated using all credible

data, the effects of a number of parameters regarding soil type (geology, fines

content, particle size, particle stiffness, etc.) are ignored in this model. The only

indicator of soil characteristics utilized in the estimation of nonlinear behavior is

plasticity index, PI.

In this chapter, estimation of nonlinear curves for a given soil plasticity

and loading condition is presented. Additionally, general trends based on these

estimated curves and their consistency with previous studies are discussed.

221

9.2 CALCULATION OF REFERENCE STRAIN, CURVATURE COEFFICIENT, SMALL-STRAIN MATERIAL DAMPING RATIO AND THE SCALING COEFFICIENT

The equations presented in Section 7.4.1 can be utilized to calculate

reference strain, curvature coefficient, small-strain material damping ratio and the

scaling coefficient by replacing parameters (φi) with their updated mean values

presented in Table 8.12 as follows: 43 '*)**( 21


5φ=a (9.1b)

[ ])ln(*1*'*)**( 1076min98 frqOCRPID o φσφφ φφ ++= (9.1c)

)ln(*1211 Nb φφ += (9.1d)



OCR = overconsolidation ratio,

frq = loading frequency,

N = number of loading cycles,

φ1 = 0.0352,

φ2 = 0.0010,

φ3 = 0.3246,

φ4 = 0.3483,

φ5 = 0.9190,

φ6 = 0.8005,

φ7 = 0.0129,

φ8 = -0.1069,

222

φ9 = -0.2889,

φ10 = 0.2919,

φ11 = 0.6329, and

φ12 = -0.0057.

In this way, the relationship between the four model parameters (reference

strain, curvature coefficient, small-strain material damping ratio and the scaling

coefficient) discussed in Chapter Six, and soil plasticity and loading conditions

can be established based on statistical analysis of the database. These

relationships are also presented in a graphical form to assist the reader in

understanding the model characteristics and in utilizing the model.

Figure 9.1 shows a graphical tool that can be used to estimate reference

strain for given values of PI, OCR and in-situ mean effective stress. An example

solution is presented utilizing a clayey soil with PI = 60 % and moderate

overconsolidation (OCR = 4) subjected to 4 atm in-situ mean effective stress.

Starting with the PI and OCR of the soil, reference strain of the material is

estimated as if it were subjected to 1 atm confining pressure. This value is then

adjusted for the effect of confining pressure in the graphical solution.

Figure 9.2 shows the graphical solution for the scaling coefficient, b.

Calculation of the value for ten cycles of loading is presented as an example in

this figure.

223

100 80 60 40 20 0

Plasticity Index, %

OCR = 1

OCR = 4

OCR = 16

0.30

0.25

0.20

0.15

0.10

0.05

0.00

Ref

eren

ce S

train

at 1

atm

, %

0.01 0.1 1

Reference Strain, %

σ o' =

1 a

tm

σ o' =

4 a

tm

σ o' =

16

atm

σ o' =

0.2

5 at

m

Figure 9.1 Estimation of reference strain for given values of PI, OCR and in-situ mean effective stress

0.64

0.63

0.62

0.61

0.60

0.59

0.58

Scal

ing

Coe

ffici

ent,

b

1 10 100 1000

Number of Loading Cycles

Figure 9.2 Estimation of scaling coefficient for a given value of number of loading cycles

224

Calculation of the small-strain material damping ratio is presented in

graphical form in Figure 9.3. As in the case of reference strain, small-strain

material damping ratio can be estimated graphically for given values of PI, OCR

in-situ mean effective stress and loading frequency. An example solution is

presented utilizing a clayey soil with PI = 60 % and moderate overconsolidation

(OCR = 4) subjected to 4 atm in-situ mean effective stress loaded at 10 Hz.

Starting with the PI and OCR of the soil, the small-strain material damping ratio

is estimated as if it were subjected to 1 atm confining pressure loaded at 1 Hz.

This value is adjusted for the effect of confining pressure and then for loading

frequency in the graphical solution.

9.3 ESTIMATION OF NORMALIZED MODULUS REDUCTION AND MATERIAL DAMPING CURVES

Once the four model parameters (reference strain, curvature coefficient,

small-strain material damping ratio, and scaling coefficient) are calculated for the

soil plasticity and loading conditions, the equations presented in Chapter Six can

be utilized to estimate the normalized modulus reduction and material damping

curves as follows:

a

r

GG

+

=

γγ1

1

max

(9.2a)

minsin

1.0

max** DD

GGbD gMaAdjusted +

= (9.2b)

225

100 80 60 40 20 0

Plasticity Index, %

OCR = 1

OCR = 4

OCR = 16

Dm

in a

t 1 a

tm a

nd a

t 1 H

z, %

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.80.1 1 10

Dmin at 1 Hz, %

σ o' =

1 a

tmσ o

' = 4

atm

σ o' =

16

atm

σ o' =

0.2

5 at

m

0.1

1

10

Dm

in, %

f = 10 Hz

f = 1 Hz

f = 100 Hz

Figure 9.3 Estimation of small-strain material damping ratio for given values of PI, OCR, in-situ mean effective stress and loading frequency

226

where:maxGG = normalized shear modulus,

γ = shearing strain (%),

γr = reference strain (%),

a = curvature coefficient,

Dmin = small-strain material damping ratio (%),

b = scaling coefficient,

DAdjusted = scaled and capped material damping (%), 3

0.1,sin32

0.1,sin20.1,sin1sin === ++= agMaagMaagMagMa DcDcDcD (%),

−

+

+−

Π== 2

ln4100

20.1,sin

r

r

rr

agMaD

γγγ

γγγ

γγ (%),

0.2523 + 1.8618a + -1.1143a21 =c ,

0.0095 - 0.0710a - 0.0805a 22 =c , and

0.0003 + 0.0002a + 0.0005a- 23 =c .

Figure 9.4 shows the normalized modulus reduction and material damping

curves for the soil type and loading conditions presented in Section 9.2, a clayey

soil with PI = 60 % and moderate overconsolidation (OCR = 4) subjected to 4 atm

in-situ mean effective stress and ten cycles of loading at 10 Hz.

227

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax G/Gmax Predictionγ r = 0.212 %a = 0.92

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Predictionγ r = 0.212 %a = 0.92Dmin = 1.65b = 0.62

(b)

Figure 9.4 Estimated (a) normalized modulus reduction and (b) material damping curves for the soil type and loading conditions discussed in Section 9.2

228

It is important to note that the nonlinear behavior predicted by the model

is based on data collected over shearing strain amplitudes ranging from 1x10-5 %

to less than 1 %. As a result, extrapolation of the curves to higher strain

amplitudes is not recommended. Also, predicted material damping ratios at strain

amplitudes over 10 % will decrease to smaller values because of the damping

adjustment that introduces a cap on material damping. Consequently, the model

should never be utilized in modeling soil behavior at such high strain levels unless

the results are verified by additional tests performed at high strain amplitudes.

9.4 EFFECT OF OVERCONSOLIDATION RATIO, LOADING FREQUENCY AND NUMBER OF LOADING CYCLES ON NONLINEAR SOIL BEHAVIOR

The effects of overconsolidation ratio, loading frequency and number of

loading cycles have been included in the model calibrated in Chapter Eight. The

results indicate that the effects of these variables on dynamic soil behavior are not

pronounced for the competent soils (that do not exhibit large volume change when

sheared at strains less than 1 %) investigated in this study.

Figure 9.5 presents the effect of overconsolidation ratio on nonlinear soil

behavior predicted by the calibrated model. Increasing overconsolidation ratio is

observed to result in a slight shift of the normalized modulus reduction and

material damping curves to higher strain amplitudes, along with a slight decrease

in small-strain material damping ratio. This effect is more pronounced for high

plasticity materials. These trends are consistent with those proposed by Hardin

and Drnevich (1972b).

229

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( σo' = 1 atm, PI = 15 %, N = 10 cycles, f = 1 Hz )

OCR = 1OCR = 4OCR = 16

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Prediction( σo' = 1 atm, PI = 15 %, N = 10 cycles, f = 1 Hz )

OCR = 1OCR = 4OCR = 16

(b)

Figure 9.5 Effect of overconsolidation ratio on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model

230

In Figures 9.6 and 9.7, the effects of loading frequency and number of

loading cycles are shown. As discussed in Section 8.3, the model has been

formulated ignoring the effect of these two variables on the normalized modulus

reduction curve based on general trends observed during the course of this study.

Figure 9.6a and 9.7a are presented to clarify this issue.

Figure 9.6b presents the effect of loading frequency on the material

damping curve predicted by the calibrated model. An increase in small-strain

material damping ratio with increasing loading frequency is observed in this

figure. This effect is consistent with the trends reported in Stokoe et al. (1999).

In Figure 9.7b, the effect of number of loading cycles on the material

damping curve is presented. Increasing number of cycles results in a slight

decrease in the scaling coefficient causing a slight decrease of material damping

ratio at high strains. This general trend is also consistent with the trends reported

in Hardin and Drnevich (1972b) and Stokoe et al. (1999).

The database utilized to calibrate the four-parameter model consists of the

results of the first and tenth cycles of torsional shear tests performed at 1 Hz and

resonant column tests performed at the resonant frequency of the specimen that is

typically on the order of around 100 Hz. During resonant column testing, the

specimen is cycled about 1000 times. Thus, the combined effect of loading

frequency and number of loading cycles is presented in Figure 9.8 showing the

predicted difference between hypothetical test results.

231

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( σo' = 1 atm, PI = 15 %, N = 10 cycles, OCR = 1 )

f = 1 Hzf = 10 Hzf = 100 Hz

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Prediction( σo' = 1 atm, PI = 15 %, N = 10 cycles, OCR = 1 )

f = 1 Hzf = 10 Hzf = 100 Hz

(b)

Figure 9.6 Effect of loading frequency on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model

232

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( σo' = 1 atm, PI = 15 %, f = 1 Hz, OCR =1 )

N = 1 cyclesN = 10 cyclesN = 1000 cycles

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Prediction( σo' = 1 atm, PI = 15 %, f = 1 Hz, OCR =1 )

N = 1 cyclesN = 10 cyclesN = 1000 cycles

(b)

Figure 9.7 Effect of number of loading cycles on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model

233

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( σo' = 1 atm, PI = 15 %, OCR =1 )

f = 1 Hz, N = 1 cyclesf = 1 Hz, N = 10 cyclesf = 100 Hz, N = 1000 cycles

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Prediction( σo' = 1 atm, PI = 15 %, OCR =1 )

f = 1 Hz, N = 1 cyclesf = 1 Hz, N = 10 cyclesf = 100 Hz, N = 1000 cycles

(b)

Figure 9.8 Comparison of (a) normalized modulus reduction and (b) material damping curves predicted for resonant column and torsional shear tests

234

9.5 EFFECT OF CONFINING PRESSURE ON NONLINEAR SOIL BEHAVIOR

The effect of confining pressure on normalized modulus reduction and

material damping curves predicted by the calibrated four-parameter model is

presented in Figure 9.9. The model shows the shift of normalized modulus

reduction and material damping curves to higher strain amplitudes with increasing

confining pressure along with a simultaneous decrease in small-strain material

damping ratio.

In Figure 9.10, the empirical curves proposed by Seed et al. (1986) are

presented. The comparison of the normalized modulus reduction curves predicted

by the calibrated model (Figure 9.9a) and the empirical curves proposed by Seed

et al. (1986) (Figure 9.10a) are presented in Figure 9.11a. The fact that the

nonlinear curves analyzed in Seed et al. (1986) were collected at low confining

pressures is supported by the close agreement between the mean Seed et al.

(1986) curve and the calibrated model curve for 1 atm. However, the comparison

of the material damping curves in Figure 9.11b shows that the material damping

values proposed by Seed et al. (1986) are higher than those encountered in the

course of this study. The discrepancy is believed to result from accuracy problems

in material damping measurements arising from the use of older generation cyclic

triaxial equipment employed in previous studies.

235

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax G/Gmax Prediction( PI = 0 %, N = 10 cycles, f = 1 Hz, OCR = 1 )

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Predictionσo' = 0.25 atmσo' = 1 atmσo' = 4 atmσo' = 16 atm

(b)

Figure 9.9 Effect of confining pressure on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model

236

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D, %


Seed et al., (1986)Average for SandsRange

(b)

Figure 9.10 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed for sands by Seed et al. (1986)

237

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( PI = 0 %, N = 10 cycles, f = 1 Hz, OCR = 1 )

(a)

Seed et al. (1986)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 9.11 Comparison of the effect of confining pressure on nonlinear soil behavior of sand (PI = 0 %) predicted by the calibrated model and empirical curves proposed for sands by Seed et al. (1986)

238

9.6 EFFECT OF SOIL TYPE ON NONLINEAR SOIL BEHAVIOR

The effect of soil plasticity on normalized modulus reduction and material

damping curves predicted by the calibrated four-parameter model is presented in

Figure 9.12. The model shows shifts in the normalized modulus reduction and

material damping curves to higher strain amplitudes with increasing soil plasticity

along with a simultaneous increase in the small-strain material damping ratio.

In Figure 9.13, the empirical curves proposed by Vucetic and Dobry

(1991) are presented. Comparison of the normalized modulus reduction curves

predicted by the calibrated model and the empirical curves proposed by Vucetic

and Dobry (1991) is presented in Figure 9.14a. As seen in the figure, the general

trend presented by Vucetic and Dobry (1991) agrees with this work. However, the

effect of soil plasticity presented by Vucetic and Dobry (1991) is more

pronounced than observed in this study.

Comparison of the material damping curves predicted by the calibrated

model and the empirical curves proposed by Vucetic and Dobry (1991) is

presented in Figure 9.14b. As seen in the figure, the material damping curves

proposed by Vucetic and Dobry (1991) also indicate a more pronounced effect of

soil plasticity. Also, as discussed in Section 5.3, the Vucetic and Dobry (1991)

damping curves do not accurately model the observed increase in small-strain

material damping ratio with increasing soil plasticity. As in the case of Seed et al.

(1986), the discrepancy is believed to be a result of accuracy problems in damping

measurements arising from the use of older generation equipment in previous

studies.

239

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax G/Gmax Prediction( σo' = 1 atm, N = 10 cycles, f = 1 Hz, OCR = 1 )

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping PredictionPI = 0 %PI = 15 %PI = 30 %PI = 50 %PI = 100 %

(b)

Figure 9.12 Effect of soil plasticity on (a) normalized modulus reduction and (b) material damping curves predicted by the calibrated model

240

1.2

0.8

0.4

0.0

G/Gmax

(a)

20

15

10

5

0

D, %


Vucetic and Dobry (1991)Non-PlasticPI = 15 %PI = 30 %PI = 50 %PI = 100 %PI = 200 %

(b)

Figure 9.13 Empirical (a) normalized modulus reduction, and (b) material damping curves proposed by Vucetic and Dobry (1991)

241

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( σo' = 1 atm, N = 10 cycles, f = 1 Hz, OCR = 1 )

Vucetic and Dobry (1991)

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 9.14 Comparison of the effect of soil plasticity on nonlinear soil behavior predicted by the calibrated model and empirical curves proposed by Vucetic and Dobry (1991)

242

9.7 EFFECTS OF CONFINING PRESSURE AND SOIL TYPE ON STRESS-STRAIN CURVES

The effects of confining pressure and soil type on normalized modulus

reduction curves predicted by the calibrated model have been evaluated in

Sections 9.5 and 9.6. Since the normalized modulus reduction curves analyzed as

part of this study are actually secant shear moduli scaled down using small-strain

values, stress-strain curves can be evaluated using the calibrated model.

The relationship between shear stress and secant shear modulus is:

γτ *G= (9.3)

where; τ = shear stress (MPa),


G = Gmax*

maxGG = shear modulus (MPa),

a

r

GG

+

=

γγ1

1

max

,

Gmax = ρ*Vs2 = small-strain shear modulus (MPa),

ρ = mass density (kg/m3), and

Vs = shear wave velocity (m/sec).

243

Equation 9.3 illustrates that, in order to estimate stress-strain curves based on the

four-parameter model, the impact of confining pressure and soil type on shear

wave velocity has to be evaluated. The relationship between these parameters and

in-situ shear wave velocity measurements can be assessed utilizing the same

database that was used to calibrate the four-parameter model. Eighty seven of the

specimens within the database were sampled from sites where in-situ shear wave

velocity measurements had been performed (Chiara, 2001). A least-squares fit to

part of the data suggests the following relationship:

PIV os *44.2'*300 27.0 −= σ (9.4)

where; σo’ = mean effective confining pressure (atm), and

PI = plasticity index (%).

The comparison of the predicted values of shear wave velocity and

measured values is presented in Figure 9.15 so that the reader can evaluate the

quality of the fit. Equation 9.4 is not part of the calibrated model recommended

for evaluating dynamic soil behavior and is utilized only as a starting point to

generate stress-strain curves presented in this section.

The effects of confining pressure and soil type on stress-strain curves

predicted using the calibrated four-parameter model are presented in Figures 9.16

and 9.17, respectively. Figures 9.16a and 9.17a show the predictions for shearing

strains up to 1 %. Figures 9.16b and 9.17b show the predictions for shearing

strains up to 0.01 % so that the characteristics of the curves at smaller strains can

be presented to the reader.

244

1000

800

600

400

200

0

Pred

icte

d Sh

ear W

ave

Vel

ocity

, m/se

c

10008006004002000

Measured Shear Wave Velocity, m/sec

Figure 9.15 Comparison of the measured in-situ shear wave velocities and values predicted using Equation 9.4

The model successfully predicts the increase in shear strength with

increasing confining pressure as shown in Figure 9.16. Figure 9.17 shows a

comparison of predicted stress-strain curves for soils with different plasticities.

Figure 9.18 shows part of the data in Figure 9.17 in an effort to compare the

behavior of a sand (PI = 0 %) and a moderate plasticity clay (PI = 30 %). It is

important to note the similarity between this figure and Figure 5.1b from Hardin

and Drnevich (1972b).

245

300

250

200

150

100

50

0

τ, kPa

1.00.80.60.40.20.0


σo' = 0.25 atmσo' = 1 atmσo' = 4 atm

(a)

10

8

6

4

2

0

τ, kPa

0.0100.0080.0060.0040.0020.000


σo' = 0.25 atmσo' = 1 atmσo' = 4 atm

(b)

Figure 9.16 Effect of confining pressure on stress-strain curve predicted by the calibrated model for shearing strains ranging (a) from γ = 0 to 1 % and (b) from γ = 0 to 0.01 %

246

100

80

60

40

20

0

τ, kPa

1.00.80.60.40.20.0


PI = 0 %PI = 15 %PI = 30 %PI = 50 %

(a)

10

8

6

4

2

0

τ, kPa

0.0100.0080.0060.0040.0020.000


PI = 0 %PI = 15 %PI = 30 %PI = 50 %

(b)

Figure 9.17 Effect of soil plasticity on stress-strain curve predicted by the calibrated model for shearing strains ranging (a) from γ = 0 to 1 % and (b) from γ = 0 to 0.01 %

247

100

80

60

40

20

0

τ, kPa

1.00.80.60.40.20.0


PI = 0 %PI = 30 %

(a)

10

8

6

4

2

0

τ, kPa

0.0100.0080.0060.0040.0020.000


PI = 0 %PI = 30 %

(b)

Figure 9.18 Comparison of the stress-strain curves of a sand and a moderate plasticity clay based on the calibrated model for shearing strains ranging (a) from γ = 0 to 1 % and (b) from γ = 0 to 0.01 %

248

9.8 SUMMARY

In this chapter, equations for the calibrated model are presented along with

graphical solutions that can be utilized in predicting normalized modulus

reduction and material damping curves for a given soil type and a given loading

condition.

The general trends of the predicted curves are briefly discussed. The

results indicate that soil plasticity and mean effective confining pressure are the

two most important parameters that control nonlinear behavior of “competent”

soils strained up to γ = 1 %. The comparison of the predicted curves with those

presented in the literature indicates a general agreement with the trends proposed

by other researchers. However, this comparison also highlights discrepancies in

material damping measurements resulting from the limitations of older generation

testing equipment utilized in previous studies.

Since stress-strain curves are related to normalized shear modulus curves,

a brief discussion regarding the predicted stress-strain curves based on the

calibrated model is also presented herein in an effort to bridge the gap between

traditional geotechnical engineering and soil dynamics. The findings indicate that

the results of dynamic tests can also be utilized in the traditional geotechnical

engineering applications that require modeling soil behavior at working strains.

249

CHAPTER 10

RECOMMENDED NORMALIZED MODULUS REDUCTION

AND MATERIAL DAMPING CURVES

10.1 INTRODUCTION

Mean values of the normalized shear modulus and the material damping

ratio (predicted by the calibrated model) at strain amplitudes ranging from 1x10-5

% to 1 % are presented in this chapter. As discussed in Chapter Nine, the mean

values of model parameters can be utilized to construct normalized modulus

reduction and material damping curves for different soil types and loading

conditions. However, the reader must use caution when a soil type or loading

condition not represented in the database is to be evaluated with these equations.

Since the impact of overconsolidation ratio is relatively small and ten

cycles at 1 Hz loading frequency closely represents the characteristics of

earthquake shaking, these parameters are fixed for the recommended curves. In

this chapter, recommended normalized modulus reduction and material damping

curves are presented for soils with a broad range of plasticities confined at a broad

range of mean effective stresses.

These curves are presented from two different perspectives so that the

reader can interpolate the data for different values of soil plasticity and confining

pressure. If the reader has to extrapolate for soil plasticities and confining

pressures not represented in the database, use of caution is suggested.

250

10.2 EFFECT OF PI AT A GIVEN MEAN EFFECTIVE STRESS

Figures 10.1 through 10.4 show the effect of PI on nonlinear soil behavior

at 0.25, 1.0, 4.0 and 16 atm, respectively. These normalized modulus and material

damping curves are presented so that the reader can interpolate these relationships

for soils with different plasticities. Also, these curves are tabulated in Tables 10.1

through 10.8. The figures and tables are organized so that the G/Gmax – log γ and

D – log γ curves are followed on the next page by the associated tables.

10.3 EFFECT OF MEAN EFFECTIVE STRESS ON A SOIL WITH GIVEN PLASTICITY

Figures 10.5 through 10.9 show the effect of mean effective stress on

nonlinear behavior of soils with 0, 15, 30, 50 and 100 % plasticity, respectively.

These normalized modulus and material damping curves are presented so that the

reader can interpolate these relationships for soil layers at different depths

confined under different mean effective stresses. Also, these curves are tabulated

in Tables 10.9 through 10.18. The figures and tables are organized so that the

G/Gmax – log γ and D – log γ curves are followed on the next page by the

associated tables.

10.4 IMPACT OF UTILIZING THE RECOMMENDED CURVES ON EARTHQUAKE RESPONSE PREDICTIONS OF DEEP SOIL SITES

The impact of utilizing the recommended curves when assigning nonlinear

soil properties in site response analyses of deep (>50 m) soil sites is discussed in

this section. This point is addressed because site response analyses are often

performed using average, pressure-independent generic curves.

251

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax G/Gmax Prediction( σo' = 0.25 atm, N = 10 cycles, f = 1 Hz, OCR = 1 )

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 10.1 Effect of PI on (a) normalized modulus reduction and (b) material damping curves at 0.25 atm confining pressure

252

Table 10.1 Effect of PI on normalized modulus reduction curve: σo’ = 0.25 atm

Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 0.999 0.999 1.000 1.000 1.0002.20E-05 0.998 0.999 0.999 0.999 1.0004.84E-05 0.996 0.997 0.998 0.998 0.9991.00E-04 0.993 0.995 0.996 0.997 0.9982.20E-04 0.986 0.990 0.992 0.994 0.9964.84E-04 0.971 0.979 0.983 0.987 0.9911.00E-03 0.944 0.959 0.968 0.975 0.9832.20E-03 0.891 0.919 0.936 0.949 0.9664.84E-03 0.799 0.847 0.876 0.900 0.9321.00E-02 0.671 0.739 0.783 0.822 0.8762.20E-02 0.497 0.579 0.637 0.692 0.7744.84E-02 0.324 0.400 0.459 0.521 0.6251.00E-01 0.197 0.255 0.303 0.358 0.4612.20E-01 0.107 0.142 0.174 0.213 0.2934.84E-01 0.055 0.074 0.093 0.116 0.1671.00E+00 0.029 0.040 0.050 0.063 0.093

Table 10.2 Effect of PI on material damping curve: σo’ = 0.25 atm


253

1.2

1.0

0.8

0.6

0.4

0.2

0.0


(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)


254





255

1.2

1.0

0.8

0.6

0.4

0.2

0.0


(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)


256





257

1.2

1.0

0.8

0.6

0.4

0.2

0.0


(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 10.4 Effect of PI on (a) normalized modulus reduction and (b) material damping curves at 16 atm confining pressure

258

Table 10.7 Effect of PI on normalized modulus reduction curve: σo’ = 16 atm


Table 10.8 Effect of PI on material damping curve: σo’ = 16 atm


259

1.2

1.0

0.8

0.6

0.4

0.2

0.0


(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 10.5 Effect of mean effective stress on (a) normalized modulus reduction and (b) material damping curves of a nonplastic soil

260

Table 10.9 Effect of σo’ on normalized modulus reduction curve: PI = 0 %

Shearing Strain (%) σo' = 0.25 atm σo' = 1.0 atm σo' = 4.0 atm σo' = 16 atm1.00E-05 0.999 0.999 1.000 1.0002.20E-05 0.998 0.999 0.999 1.0004.84E-05 0.996 0.998 0.998 0.9991.00E-04 0.993 0.995 0.997 0.9982.20E-04 0.986 0.991 0.994 0.9964.84E-04 0.971 0.981 0.988 0.9921.00E-03 0.944 0.964 0.976 0.9852.20E-03 0.891 0.928 0.952 0.9694.84E-03 0.799 0.861 0.906 0.9381.00E-02 0.671 0.761 0.832 0.8852.20E-02 0.497 0.607 0.706 0.7894.84E-02 0.324 0.428 0.538 0.6451.00E-01 0.197 0.277 0.374 0.4822.20E-01 0.107 0.157 0.225 0.3114.84E-01 0.055 0.083 0.123 0.1791.00E+00 0.029 0.044 0.067 0.101

Table 10.10 Effect of σo’ on material damping curve: PI = 0 %


261

1.2

1.0

0.8

0.6

0.4

0.2

0.0


(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 10.6 Effect of mean effective stress on (a) normalized modulus reduction and (b) material damping curves of a soil with PI = 15 %

262





263

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/GmaxG/Gmax Prediction( PI = 30 %, N = 10 cycles, f = 1 Hz, OCR = 1 )

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)


264





265

1.2

1.0

0.8

0.6

0.4

0.2

0.0


(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)


266





267

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1.0

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0.6

0.4

0.2

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(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)


268





269

To illustrate the impact of utilizing the recommended curves on site

response analyses, a 100-m thick silty sand (SM) deposit was modeled in twenty

six layers and analyzed using the shareware version of ProShake (EduPro, 1998).

A confining-pressure-dependent shear wave velocity, Vs, profile was used (as

shown in Figure 10.10) along with 1500-m/sec Vs at the half space. The Topanga

motion (Maximum Horizontal Acceleration, MHA, = 0.33 g) recorded during the

1994 Northridge earthquake was used as the input “rock” motion.

100

80

60

40

20

0

Depth, m

10008006004002000

Vs, m/sec

Figure 10.10 Shear wave velocity profile assumed for the 100-m thick silty sand deposit

270

In Figure 10.11, the acceleration response spectra from two analyses are

presented: 1) using the average generic curves (Seed et al., 1986) to model all

layers, and 2) using the recommended nonlinear curves interpolated for each soil

layer. The response spectrum of the input motion is also shown in this figure. The

response spectra indicate that the recommended nonlinear curves produce an

MHA much higher than that predicted by the average generic curves (0.54 g vs.

0.37 g). Additionally, larger spectral accelerations (typically 30 % to 50 % higher)

are calculated at all periods less than 1 sec for the analysis utilizing the

recommended nonlinear curves.

As discussed in Darendeli et al. (2001) the impact of utilizing a family of

confining-pressure-dependent curves is expected to be more pronounced for

deeper sites subjected to higher intensity input motions due to lower damping

introduced by the confining-pressure-dependent curves. At longer spectral periods

(T > 1 sec), the response is dominated by the overall stiffness of the site. As a

result, the confining-pressure-dependent analyses may tend to predict a smaller

response at longer periods due to the more linear response modeled by these

curves.

271

2.5

2.0

1.5

1.0

0.5

0.0

Spec

tral A

ccel

erat

ion,

S a ,

g

0.01 0.1 1 10Period, T, sec

This Study (a family of mean curves for PI = 0 %)Seed et al., 1986 (mean curve for sands)Input Motion

5 % Damping

Figure 10.11 An example of utilizing the recommended normalized modulus reduction and material damping curves and its impact on estimated nonlinear site response

272

10.5 SUMMARY

In this chapter, recommended normalized modulus reduction and material

damping curves are presented for soils with a broad range of plasticities confined

over a broad range of mean effective stresses.

The impact of utilizing the recommended curves when assigning nonlinear

soil properties in site response analyses is illustrated by analyzing a 100-m thick

silty sand (SM) deposit using average generic curves (Seed et al., 1986) to model

all twenty six layers, and the recommended nonlinear curves interpolated for each

soil layer. Larger spectral accelerations (typically 30 % to 50 % higher) are

calculated at all periods less than 1 sec for the analysis utilizing the recommended

nonlinear curves than those calculated for the analysis utilizing average generic

curves.

273

CHAPTER 11

UNCERTAINTY ASSOCIATED WITH THE MODEL

PREDICTIONS

11.1 INTRODUCTION

In this chapter, uncertainty associated with the normalized modulus

reduction and material damping curves predicted by the calibrated model is

briefly discussed.

Calculation of standard deviation associated with a point estimate of

normalized shear modulus or material damping ratio, and the covariance structure

of the predicted curves are presented.

Utilization of the calibrated model in probabilistic seismic hazard

assessment is also discussed herein. Integration of random shear-wave velocity

profiles and normalized modulus reduction and material damping curves into

ground motion analysis is recommended. An example regarding incorporation of

the modeled uncertainty (in nonlinear soil behavior) into site response analysis is

also presented.

11.2 UNCERTAINTY IN NONLINEAR SOIL BEHAVIOR

As presented in Chapter Nine, the calibrated model can be utilized to

construct normalized modulus reduction and material damping curves for various

soil types and loading conditions by using the updated mean values of model

parameters (φ1 through φ12) presented in Table 8.12. However, these predicted

274

curves represent average nonlinear curves and the actual data fall into a band of

scatter around these estimates.

At this point, it is important to note that there are two sources of

uncertainty associated with the predicted curves. First, there is uncertainty in the

values of the model parameters. As shown in Table 8.12, where a comparison of

the prior and updated variances of the model parameters are presented, this

component reduces significantly upon calibration of the model.

The second source of uncertainty is the modeled variability (discussed in

Section 7.4.2) of the physical phenomenon. In order to analyze this component of

uncertainty, some of the model parameters (φ13 through φ18) were utilized in

defining the standard deviation and covariance structure of the data (Section

7.4.2) and were simultaneously calibrated utilizing the First-Order, Second-

Moment Bayesian Method.

Table 11.1 presents the predicted mean values and standard deviations

while Table 11.2 presents the covariance structure for a nonplastic soil confined at

1 atm mean effective stress and loaded with ten cycles at 1 Hz accounting for both

components of uncertainty. Tables 11.3 and 11.4 show the same information

predicted by only accounting for modeled variability.

275

Table 11.1 Predicted mean values and standard deviations accounting for uncertainty in the values of model parameters and variability due to modeled uncertainty

Mean Standard Deviation Mean Standard Deviation1 1.00E-05 0.99945 0.01836 0.80434 0.707662 2.20E-05 0.99896 0.01979 0.80816 0.709303 4.84E-05 0.99759 0.02254 0.81958 0.714224 1.00E-04 0.99546 0.02553 0.83859 0.722325 2.20E-04 0.99067 0.03026 0.88404 0.741326 4.84E-04 0.98108 0.03683 0.98172 0.780587 1.00E-03 0.96353 0.04523 1.17383 0.852508 2.20E-03 0.92753 0.05699 1.60166 0.994029 4.84E-03 0.86113 0.07116 2.47409 1.23299

10 1.00E-02 0.76094 0.08438 3.95297 1.5560611 2.20E-02 0.60663 0.09454 6.57873 2.0048512 4.84E-02 0.42765 0.09556 10.18430 2.4924213 1.00E-01 0.27720 0.08784 13.78800 2.8987514 2.20E-01 0.15670 0.07407 17.19890 3.2365815 4.84E-01 0.08259 0.05961 19.56450 3.4514716 1.00E+00 0.04417 0.04818 20.71610 3.55137

Point Number

Normalized Shear Modulus Material Damping Ratio, %Shearing Strain, %

276

Table 11.2 Predicted covariance structure accounting for uncertainty in the values of model parameters and variability due to modeled uncertainty

Point Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1.00 0.95 0.90 0.86 0.82 0.78 0.75 0.71 0.68 0.65 0.62 0.59 0.56 0.54 0.51 0.492 0.95 1.00 0.95 0.91 0.86 0.82 0.79 0.75 0.71 0.68 0.65 0.62 0.59 0.56 0.54 0.513 0.90 0.95 1.00 0.95 0.91 0.86 0.83 0.79 0.75 0.72 0.68 0.65 0.62 0.59 0.56 0.544 0.86 0.91 0.95 1.00 0.95 0.90 0.86 0.82 0.78 0.75 0.71 0.68 0.65 0.62 0.59 0.565 0.82 0.86 0.91 0.95 1.00 0.95 0.91 0.86 0.82 0.79 0.75 0.71 0.68 0.65 0.62 0.596 0.78 0.82 0.86 0.90 0.95 1.00 0.95 0.91 0.86 0.83 0.79 0.75 0.72 0.68 0.65 0.627 0.75 0.79 0.83 0.86 0.91 0.95 1.00 0.95 0.90 0.86 0.82 0.78 0.75 0.71 0.68 0.658 0.71 0.75 0.79 0.82 0.86 0.91 0.95 1.00 0.95 0.91 0.86 0.82 0.79 0.75 0.71 0.689 0.68 0.71 0.75 0.78 0.82 0.86 0.90 0.95 1.00 0.95 0.91 0.86 0.83 0.79 0.75 0.72

10 0.65 0.68 0.72 0.75 0.79 0.83 0.86 0.91 0.95 1.00 0.95 0.90 0.86 0.82 0.78 0.7511 0.62 0.65 0.68 0.71 0.75 0.79 0.82 0.86 0.91 0.95 1.00 0.95 0.91 0.86 0.82 0.7912 0.59 0.62 0.65 0.68 0.71 0.75 0.78 0.82 0.86 0.90 0.95 1.00 0.95 0.91 0.86 0.8313 0.56 0.59 0.62 0.65 0.68 0.72 0.75 0.79 0.83 0.86 0.91 0.95 1.00 0.95 0.90 0.8614 0.54 0.56 0.59 0.62 0.65 0.68 0.71 0.75 0.79 0.82 0.86 0.91 0.95 1.00 0.95 0.9115 0.51 0.54 0.56 0.59 0.62 0.65 0.68 0.71 0.75 0.78 0.82 0.86 0.90 0.95 1.00 0.9516 0.49 0.51 0.54 0.56 0.59 0.62 0.65 0.68 0.72 0.75 0.79 0.83 0.86 0.91 0.95 1.00

277

Table 11.3 Predicted mean values and standard deviations accounting only for variability due to modeled uncertainty

Mean Standard Deviation Mean Standard Deviation1 1.00E-05 0.99945 0.01836 0.80434 0.705072 2.20E-05 0.99887 0.02003 0.80892 0.707053 4.84E-05 0.99766 0.02243 0.81897 0.711394 1.00E-04 0.99546 0.02553 0.83859 0.719785 2.20E-04 0.99067 0.03026 0.88404 0.738856 4.84E-04 0.98094 0.03692 0.98321 0.778827 1.00E-03 0.96353 0.04523 1.17383 0.850368 2.20E-03 0.92753 0.05699 1.60166 0.992189 4.84E-03 0.86113 0.07116 2.47409 1.2315110 1.00E-02 0.76093 0.08438 3.95297 1.5548811 2.20E-02 0.60663 0.09454 6.57873 2.0039412 4.84E-02 0.42765 0.09556 10.18426 2.4916813 1.00E-01 0.27720 0.08784 13.78804 2.8981114 2.20E-01 0.15670 0.07406 17.19895 3.2360115 4.84E-01 0.08259 0.05961 19.56452 3.4509316 1.00E+00 0.04417 0.04818 20.71612 3.55085

Point Number

Normalized Shear Modulus Material Damping Ratio, %Shearing Strain, %

278

Table 11.4 Predicted covariance structure accounting only for variability due to modeled uncertainty

Point Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1.00 0.95 0.90 0.86 0.82 0.78 0.75 0.71 0.68 0.65 0.62 0.59 0.56 0.54 0.51 0.492 0.95 1.00 0.95 0.91 0.86 0.82 0.79 0.75 0.71 0.68 0.65 0.62 0.59 0.56 0.54 0.513 0.90 0.95 1.00 0.95 0.91 0.86 0.83 0.79 0.75 0.72 0.68 0.65 0.62 0.59 0.56 0.544 0.86 0.91 0.95 1.00 0.95 0.90 0.86 0.82 0.78 0.75 0.71 0.68 0.65 0.62 0.59 0.565 0.82 0.86 0.91 0.95 1.00 0.95 0.91 0.86 0.82 0.79 0.75 0.71 0.68 0.65 0.62 0.596 0.78 0.82 0.86 0.90 0.95 1.00 0.95 0.91 0.86 0.83 0.79 0.75 0.72 0.68 0.65 0.627 0.75 0.79 0.83 0.86 0.91 0.95 1.00 0.95 0.90 0.86 0.82 0.78 0.75 0.71 0.68 0.658 0.71 0.75 0.79 0.82 0.86 0.91 0.95 1.00 0.95 0.91 0.86 0.82 0.79 0.75 0.71 0.689 0.68 0.71 0.75 0.78 0.82 0.86 0.90 0.95 1.00 0.95 0.91 0.86 0.83 0.79 0.75 0.72

10 0.65 0.68 0.72 0.75 0.79 0.83 0.86 0.91 0.95 1.00 0.95 0.90 0.86 0.82 0.78 0.7511 0.62 0.65 0.68 0.71 0.75 0.79 0.82 0.86 0.91 0.95 1.00 0.95 0.91 0.86 0.82 0.7912 0.59 0.62 0.65 0.68 0.71 0.75 0.78 0.82 0.86 0.90 0.95 1.00 0.95 0.91 0.86 0.8313 0.56 0.59 0.62 0.65 0.68 0.72 0.75 0.79 0.83 0.86 0.91 0.95 1.00 0.95 0.90 0.8614 0.54 0.56 0.59 0.62 0.65 0.68 0.71 0.75 0.79 0.82 0.86 0.91 0.95 1.00 0.95 0.9115 0.51 0.54 0.56 0.59 0.62 0.65 0.68 0.71 0.75 0.78 0.82 0.86 0.90 0.95 1.00 0.9516 0.49 0.51 0.54 0.56 0.59 0.62 0.65 0.68 0.72 0.75 0.79 0.83 0.86 0.91 0.95 1.00

279

The second set of tables (Tables 11.3 and 11.4) can be obtained by

replacing the model parameters in the equations presented in Section 7.4 with

updated mean values in Table 8.12. However, the first set of tables (Tables 11.1

and 11.2) requires a relatively complicated procedure that incorporates the

updated variance of the model parameters in Table 8.12, updated covariance

structure of the model parameters presented in Table 8.13, and derivatives of

equations utilized in modeling mean values and covariance structure presented in

Section 7.4 with respect to each model parameter. The details of this procedure

are beyond the scope of this study and can be found in Ang and Tang (1990).

The comparison of Tables 11.1 and 11.3 indicates that uncertainty in the

value of model parameters has a negligible impact on point estimates. The errors

introduced by calculating mean values and standard deviations without

accounting for uncertainty in the model parameters are less than about 0.1 % for

the mean values and less than about 1 % for the standard deviations. As a result,

the equations presented in Section 7.4 can be used in calculation of mean values

and standard deviations without introducing significant error due to ignoring

uncertainty regarding the model parameters.

Figure 11.1 shows the predicted mean normalized modulus reduction and

material damping curves and standard deviations of the point estimates tabulated

in Table 11.1. These mean curves are identical to the recommended curves

presented in Chapter Ten for deterministic design applications.

280

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Predictionσo' = 1 atm, PI = 0 %,N = 10 cycles, f = 1 Hz, OCR = 1

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Predictionσo' = 1 atm, PI = 0 %,N = 10 cycles, f = 1 Hz, OCR = 1

(b)

Figure 11.1 Mean values and standard deviations associated with the point estimates of (a) normalized modulus reduction and (b) material damping curves

281

Besides calculating mean normalized modulus reduction and material

damping curves, the additional information generated by analysis of the data

using the Bayesian approach (rather than an ordinary multivariate nonlinear

optimization procedure) can be utilized in predicting nonlinear curves (random

realizations) consistent with the database for probabilistic site response analysis.

As in the case of mean values and standard deviations, the covariance

structures presented in Tables 11.2 and 11.4 are also quite similar. The errors

introduced by calculating covariance structure without accounting for uncertainty

in the model parameters are less than about 0.1 %. As a result, Equation 7.31

presented in Section 7.4.2 can be used in calculation of correlation coefficients

without introducing significant error due to ignoring uncertainty regarding the

model parameters. At the same time, the covariance structure presented in Table

11.2 is unique to the calibrated model and only changes with the number and

relative amplitudes of shearing strains at which normalized modulus reduction

and material damping curves are generated. In other words, as long as nonlinear

curves are generated at the same sixteen shearing strain amplitudes presented in

Table 11.1, the same covariance structure can be utilized regardless of soil type


It is also important to note that the covariance structure in Table 11.2 is

not sensitive to the kind of modeled dynamic soil property. In other words, the

covariance structure does not change significantly whether normalized shear

modulus or material damping ratio is being investigated. The difference between

the covariance structure for the normalized modulus reduction curve and the

282

covariance structure for the material damping curve is less than about 0.1 % and

is a result of added uncertainty regarding the additional parameters utilized in

modeling material damping curve.

At the same time, the model calibration was performed assuming no

correlation between normalized modulus reduction and material damping curves,

although the mean G/Gmax and D curves are coupled to each other. In other words,

the calibrated model relates the material damping curve to the normalized

modulus reduction curve in terms of the average estimates, however, where the

point estimates of material damping ratio are relative to the mean material

damping curve is modeled to be independent from where the point estimates of

normalized shear modulus are relative to the mean normalized modulus reduction

curve.

Briefly, correlated random realizations of normalized modulus reduction

and material damping curves can be separately generated utilizing the covariance

structure calculated based on Equation 7.31 and the mean values of the φ17 and φ18

parameters in Table 8.12. The procedure to generate correlated random

realizations consistent with a given model is beyond the scope of this study and

can be found in Ang and Tang (1990).

A realization of normalized modulus reduction and material damping

curves is presented in Figure 11.2 for the same soil type and loading conditions in

Figure 11.1. The mean curves and one standard deviation ranges of normalized

modulus reduction and material damping curves are shown for comparison

purposes. It is important to note that the scatter of the point estimates is not

283

completely random. Any given point estimate is affected by the location of the

neighboring points relative to the mean curve. This is the result of utilizing a

covariance structure in the random realization process.

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/GmaxMean Prediction+/- One Standard DeviationRandom Realization

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Mean Prediction+/- One Standard DeviationRandom Realization

(b)

Figure 11.2 Comparison of the correlated random realization of (a) normalized modulus reduction and (b) material damping curves relative to the mean curves and one standard deviation ranges shown in Figure 11.1

284

11.3 UNCERTAINTY IN PREDICTED GROUND MOTIONS DUE TO THE UNCERTAINTY IN NONLINEAR SOIL BEHAVIOR

Seismic hazard analysis involves the quantitative estimation of ground

shaking at a particular site. If the analysis is carried out assuming a particular

scenario, it is called deterministic seismic hazard analysis. Traditionally, if the

uncertainties in earthquake size, location and time of occurrence are considered in

the analysis, it is called probabilistic seismic hazard analysis. It is important to

note that although probabilistic seismic hazard analysis is site specific,

uncertainties regarding the shear-wave velocity profile have been overlooked in

most cases and uncertainties regarding nonlinear soil behavior at different layers

have been ignored due to lack of data.

It has been established that although the soil profile constitutes a minute

fraction of the travel path from the point of rupture to the ground surface, the

characteristics of the soil layers have a major impact on the amplitude and

frequency content of the ground motion at a geotechnical site. Therefore,

incorporation of uncertainties in soil characteristics into probabilistic seismic

hazard assessment should be expected to result in significant improvement of the

estimated design ground motions.

This study provides the key data, in terms of mean design curves and

uncertainties associated with these curves, required for such an investigation.

However, existing computer programs have to be improved to automatically

incorporate uncertainties in nonlinear soil behavior and shear-wave velocity

profiles in site response analysis. Once such a program becomes available, a

285

number of realizations (varying shear-wave velocity profile and nonlinear curves

at each layer) at a given site utilizing a suite of input motions can be

accomplished with the readily available computational power.

In order to emphasize the impact of modeling uncertainties regarding

nonlinear soil behavior, the example presented in Section 10.4 is reevaluated

using the same input motion and identical shear-wave velocity profile for a

number of realizations.

Due to lack of data, the possible correlation of nonlinear soil behavior

between layers has not been resolved in this study. As a result, two extreme

scenarios regarding the correlation between the twenty five layers within the silty

sand deposit are evaluated: 1) perfectly correlated nonlinear curves, and 2)

completely uncorrelated nonlinear curves. The resulting spectral accelerations

computed using these two extreme cases do not necessarily bound the amplitude

of possible spectral accelerations for design purposes. However, these results

should assist the reader in visualizing the consequences of utilizing different

correlation structures between soil layers.

Figure 11.3 shows the comparison of three spectral accelerations

calculated using perfectly correlated soil layers. The first case is essentially the

same result as shown in Figure 10.11, which was calculated using the mean (µ)

normalized modulus reduction and material damping curves. The other two cases

presented in Figure 11.3 are analyses utilizing normalized modulus reduction and

material damping curves one standard deviation above the mean curves (µ+σ) and

one standard deviation below the mean curves (µ−σ).

286

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Spec

tral A

ccel

erat

ion,

S a ,

g

0.01 0.1 1 10Period, T, sec

Utilizing Mean CurvesUtilizing 1 Standard Deviation Above Average CurvesUtilizing 1 Standard Deviation Below Average Curves

5 % Damping

Figure 11.3 Comparison of spectral accelerations calculated using perfectly correlated soil layers with µ, µ+σ and µ−σ normalized modulus reduction and material damping curves

287

Increasing (or decreasing) the stiffness of all layers in the profile (as a

result of simultaneously shifting normalized modulus reduction curves) should be

expected to have a major impact on site period. In the µ+σ case, an increase in

damping accompanied with an increase in stiffness results in a decrease in

estimated ground motion. In the µ−σ case, the modeled normalized modulus

reduction curves are relatively nonlinear and higher strains are generated at deep

layers. Since material damping increases with strain amplitude, more energy is

dissipated at depth, and estimated ground motion turns out to be generally lower

than that estimated using the mean nonlinear curves.

Figure 11.4 also shows comparison of spectral accelerations calculated

using perfectly correlated soil layers with the result calculated using the mean (µ)

curves shown in Figure 10.11. One of the remaining two spectral accelerations in

this figure is computed utilizing µ+σ normalized modulus reduction curve with

µ−σ material damping curve. The third spectral acceleration is computed

utilizing µ−σ normalized modulus reduction curve with µ+σ material damping

curve. This way, if a layer is modeled to be linear relative to the mean curve in

terms of shear modulus, it is also modeled linear relative to the mean curve in

terms of material damping. In the case of the analysis performed on relatively

linear soil layers, changes in the site period can be identified and the spectral

acceleration is observed to be higher at some lower frequencies. Since the other

case involves relatively nonlinear modulus and damping curves, the resulting

ground motions are lower due to increase in energy dissipation.

288

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Spec

tral A

ccel

erat

ion,

S a ,

g

0.01 0.1 1 10Period, T, sec

Utilizing Mean CurvesUtilizing 1 Standard Deviation Linear CurvesUtilizing 1 Standard Deviation Non-Linear Curves

5 % Damping

Figure 11.4 Comparison of spectral accelerations calculated using perfectly correlated soil layers with 1) µ curves, 2) +σ normalized modulus reduction and −σ material damping curves, and 3) −σ normalized modulus reduction and +σ material damping curves

289

Fifty realizations utilizing completely uncorrelated nonlinear curves are

presented in Figure 11.5. Although completely uncorrelated soil layers are

unlikely to be a common scenario, the reader must keep in mind the possibility of

missing a thin soft layer during real-life site investigations.

Since the equivalent linear analysis program (EduPro, 1998) utilized in the

analysis is not designed for random realizations, an internal file that contains the

nonlinear curves had to be modified before each run. Although computation time

required for each run was merely about 10 seconds, modification of this file

significantly slowed the process. A number of improvements in site response

analysis programs will be required if they are to be utilized in probabilistic

seismic assessment as recommended.

In Figure 11.5, significantly different spectral accelerations are presented

for a given input motion and shear-wave velocity profile. A single relatively

nonlinear soil layer was observed to reduce surface motions drastically in some of

these realizations. As a result, graphing all spectral accelerations for a large

number of realizations or analyzing histograms of spectral accelerations at certain

periods (Figures 11.6 and 11.7) is recommended rather than analyzing µ, µ+σ,

or µ+2σ spectral accelerations as shown in Figure 11.8. However, a comparison

of the information presented in Figure 11.8 with the result computed utilizing the

mean curves (Figure 11.9) shows that depending on the consequences of failure,

the design acceleration response spectrum may have to be selected much higher

than the deterministic spectrum (estimated utilizing the mean curves) even though

the input motion and the shear-wave velocity profile were fixed in this example.

290

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Spec

tral A

ccel

erat

ion,

S a ,

g

0.01 0.1 1 10Period, T, sec

5 % DampingRandom Realization

Figure 11.5 Fifty realizations of spectral acceleration computed using completely uncorrelated soil layers with randomly generated normalized modulus reduction and material damping curves

291

05

101520253035

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Acceleration at 0.1 sec Period, g

Freq

uenc

y(a)

05

101520253035

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5Acceleration at 0.3 sec Period, g

Freq

uenc

y

(b)

Figure 11.6 Histograms of spectral accelerations from fifty realizations presented in Figure 11.5 (a) at 0.1 sec and (b) at 0.3 sec

292

05

101520253035

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Acceleration at 1 sec Period, g

Freq

uenc

y(a)

05

101520253035

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Acceleration at 3 sec Period, g

Freq

uenc

y

(b)

Figure 11.7 Histograms of spectral accelerations from fifty realizations presented in Figure 11.5 (a) at 1 sec and (b) at 3 sec

293

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Spec

tral A

ccel

erat

ion,

S a ,

g

0.01 0.1 1 10Period, T, sec

Mean of Random RealizationsMean +1 Standard Deviation of Random RealizationsMean +2 Standard Deviation of Random Realizations

5 % Damping

Figure 11.8 Distribution of fifty realizations of spectral acceleration presented in Figure 11.5

294

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Spec

tral A

ccel

erat

ion,

S a ,

g

0.01 0.1 1 10Period, T, sec

Mean of Random RealizationsMean +1 Standard Deviation of Random RealizationsMean +2 Standard Deviation of Random RealizationsUtilizing Mean Curves

5 % Damping

Figure 11.9 Comparison of the spectral accelerations from the fifty realizations with the results computed utilizing mean normalized modulus reduction and material damping curves

295

It is important to point out that the discussion and accompanying figures

presented above are related to observations on a single input motion and a

uniform silty sand deposit with increasing shear wave velocity with depth. The

consequences of ignoring the uncertainty in nonlinear soil behavior are expected

to be more pronounced in the case of higher intensity input motions and more

complicated soil deposits.

11.4 SUMMARY

The uncertainty associated with the recommended normalized modulus

reduction and material damping curves is discussed in this chapter. The impact of

such uncertainty on estimated ground motions for a given input motion and shear-

wave velocity profile is presented.

For probabilistic seismic hazard analysis applications, the recommended

procedure for handling the uncertainty in the shear-wave velocity profile and the

nonlinear soil behavior are also discussed.

Since this study provides the only available data regarding uncertainty

associated with the recommended normalized modulus reduction and material

damping curves, the results presented in this chapter are believed to be a very

important contribution to state of the art in geotechnical earthquake engineering.

296

CHAPTER 12

SUMMARY AND CONCLUSIONS

12.1 SUMMARY

In this study, the effects of soil type and loading conditions on dynamic

soil properties (presented in terms of normalized shear modulus and material

damping curves) have been quantified based on the data that has been collected at

the University of Texas at Austin over the past decade. Information regarding the

laboratory testing equipment used to collect the data and a general description of

the properties of the specimens included in the database are presented in Chapters

Two and Three, respectively.

The general trends regarding nonlinear soil behavior observed during the

course of this study and reported in the literature are presented in Chapter Four.

Parameters that control nonlinear soil behavior and their relative importance in

terms of affecting normalized modulus reduction and material damping curves

based on general trends observed during the course of this study are presented in

Table 12.1.

Based on the general trends, the successes and shortcomings of various

empirical relationships utilized in the state of practice are evaluated in Chapter

Five and a four-parameter (reference strain, curvature coefficient, small-strain

material damping ratio and the scaling coefficient) soil model that is capable of

capturing these general trends is proposed in Chapter Six (Equation 12.1).

297

Table 12.1 Parameters that control nonlinear soil behavior and their relative importance in terms of affecting normalized modulus reduction and material damping curves based on general trends observed during the course of this study

Parameter Impact on Normalized Modulus Reduction Curve

Impact on Material Damping Curve

Strain Amplitude *** ***Mean Effective Confining Pressure *** ***Soil Type and Plasticity *** ***Number of Loading Cycles *+ ***++

Frequency of Loading (above 1 Hz) * **Overconsolidation Ratio * *Void Ratio * *Degree of Saturation * *Grain Characteristics, Size, Shape, Gradation, Mineralogy * *

*** Very Important + On competent soils included in this study ** Important ++ Soil Type Dependent * Less Important

a

r

GG

+

=

γγ1

1

max

(12.1a)

minsin

1.0

max** DD

GGbD gMaAdjusted +

= (12.1b)

The First-Order, Second-Moment Bayesian Method utilized in the

statistical analysis of the data is discussed in Chapter Seven. An eighteen

parameter model that relates reference strain, curvature coefficient, small-strain

material damping ratio and scaling coefficient to soil type and loading conditions,

298

and that characterizes the covariance structure of the predicted normalized

modulus reduction and material damping curves is also presented in this chapter

(Equation 12.2). 43 '*)**( 21


5φ=a (12.2b)

[ ])ln(*1*'*)**( 1076min98 frqOCRPID o φσφφ φφ ++= (12.2c)

)ln(*1211 Nb φφ += (12.2d)

)exp()5.0/(

)exp(25.0)exp(

14

2max

1413 φφ

φσ−

−+=GG

NG (12.2e)

DD *)exp()exp( 1615 φφσ += (12.2f)

))exp(

lnlnexp(*)

)exp(1exp(

1817, φ

γγ

φρ ji

ji

−−−= (12.2g)

Statistical analysis of various subsets of the data and the model calibration

process are briefly described in Chapter Eight.

Chapters Nine and Ten present the equations, graphical solutions, plots

and tables of recommended normalized modulus reduction and material damping

curves for deterministic site-specific analysis. The proposed curves are also

compared with empirical curves widely accepted in state of practice in Chapter

Nine (as shown in Figures 12.1 and 12.2).

Finally, uncertainty associated with the recommended curves is discussed

in Chapter Eleven.

299

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( PI = 0 %, N = 10 cycles, f = 1 Hz, OCR = 1 )

(a)

Seed et al. (1986)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 12.1 Comparison of the effect of confining pressure on nonlinear soil behavior of sand (PI = 0 %) predicted by the calibrated model and empirical curves proposed for sands by Seed et al. (1986)

300

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Prediction( σo' = 1 atm, N = 10 cycles, f = 1 Hz, OCR = 1 )

Vucetic and Dobry (1991)

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1



(b)

Figure 12.2 Comparison of the effect of soil plasticity on nonlinear soil behavior predicted by the calibrated model and empirical curves proposed by Vucetic and Dobry (1991)

301

12.2 CONCLUSIONS

A new family of normalized modulus reduction and material damping

design curves is proposed utilizing a four-parameter model calibrated to a rather

large database of resonant column and torsional shear test results. One of the

unique features of this study is the consideration for uncertainty associated with

the recommended curves. Figure 12.3 shows mean values predicted using the

calibrated model and uncertainty associated with these point estimates.

This study is believed to be a valuable contribution to the state of the art

because it provides the means to incorporate the uncertainty in nonlinear soil

behavior into probabilistic seismic hazard analysis. However, utilization of this

work requires improvement of site response analysis programs available at this

point in time.

302

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G/Gmax

G/Gmax Predictionσo' = 1 atm, PI = 0 %,N = 10 cycles, f = 1 Hz, OCR = 1

(a)

25

20

15

10

5

0

D, %

0.0001 0.001 0.01 0.1 1


Material Damping Predictionσo' = 1 atm, PI = 0 %,N = 10 cycles, f = 1 Hz, OCR = 1

(b)

Figure 12.3 Mean values and standard deviations associated with the point estimates of (a) normalized modulus reduction and (b) material damping curves

303

APPENDIX A

HEADER FILE

FOR

FIRST ORDER SECOND MOMENT

BAYESIAN ANALYSIS

OF

RESONANT COLUMN

AND

TORSIONAL SHEAR

TEST RESULTS

304

// Modelh.h : Header file for RCTS data // class ModelStructure : public NormLike { public: int nCOV; // Data Indices int nlocation,nsoil,nspecimen,nswv,ndisturbance,npressure,ntest,nPI,nOCR,ne,nconpre,nfrq, nN,nstr,ncorrstr,nTYPE; // Model Indices int iphi1; int iphi2; int iphi3; int iphi4; int iphi5; int iphi6; int iphi7; int iphi8; int iphi9; int iphi10; int iphi11; int iphi12; int istdGa,istdGb,istdDa,istdDb,ithetanugget; int ntheta; iarray itheta; double scalar; void Initialize(DataStructure &Data, int nPhi, int nconst, double *xconst); void Feasible(double *x); void OutputModel(DataStructure &Data, double *x, CString outfilename); void PrintYMean(DataStructure &Data, double *x, CString output); void CalculateYMeanC(DataStructure &Data, double *x, darray &YMeanC, iarray &index); void CalculatedYMeanCiMM(int iv, DataStructure &Data, double *x, darray &dYMeani, iarray &index); void Calculated2YMeanCijMM(int iv, int jv, DataStructure &Data, double *x, darray &dYMeani, darray &dYMeanj, darray &d2YMeanij, iarray &index);

305

void CalculateYCOVC(DataStructure &Data, double *x, darray &YMean, CovMatrix &YCOV, iarray &index); void CalculatedYCOVCiMM(int iv, DataStructure &Data, double *x, darray &YMean, darray &dYMeani, CovMatrix &YCOV, smatrixsolve &dYCOVi, iarray &index); void Calculated2YCOVCijMM(int iv, int jv, DataStructure &Data, double *x, darray &YMean, darray &dYMeani, darray &dYMeanj, darray &d2YMeanij, CovMatrix &YCOV, smatrixsolve &dYCOVi, smatrixsolve &dYCOVj, smatrixsolve &d2YCOVij, iarray &index); double CalculateYrhoab(DataStructure &Data, double *x, int ka, int kb); double YCOVrho(double *tau, double *x); double dYCOVrhoi(int iv, double *tau, double *x); double d2YCOVrhoij(int iv, int jv, double *tau, double *x); };

306

APPENDIX B

MODEL FILE

FOR


BAYESIAN ANALYSIS

OF

RESONANT COLUMN

AND

TORSIONAL SHEAR

TEST RESULTS

307

// Model.cpp : Model for RCTS Data // #include "stdafx.h" #include <afxwin.h> #include <iostream.h> #include <fstream.h> #include <math.h> #include <direct.h> #include <time.h> #include "machh.h" #include "compareh.h" #include "dblash.h" #include "_arrayh.h" #include "_array2h.h" #include "_array3h.h" #include "matrixh.h" #include "smatrixh.h" #include "gmatrixh.h" #include "covmatrixh.h" #include "dblash.h" #include "goldenh.h" #include "rqph.h" #include "Datah.h" #include "NormalLikeh.h" #include "Modelh.h" void ModelStructure::Initialize(DataStructure &Data, int nPhi, int nconst, double *xconst) { /* Data Structure: Data.d[0] = location # (1=Northern CA, 2=Southern CA, 3=South Carolina, 4=Lotung, Taiwan) Data.d[1] = soil type ( 1=sands with fines < 12%, 2=sands with fines > 12% [< 50%], 3=silts, 4=clays) Data.d[2] = specimen # (3 or 4 digit specimen ID UTA-1-C=103, UT-24-F=2406)

308

Data.d[3] = in-situ shear wave velocity (not utilized in this study, a value of 500 is assigned to all specimens) Data.d[4] = shear wave velocity ratio (not utilized in this study, a value of 1.0 is assigned to all specimens) Data.d[5] = pressure # (some specimens are tested at multiple confining pressures) Data.d[6] = test type ( 1.05-1.2=LA_TS, 2=HA_TS1, 3=HA_TS10, 4=HA_RC) Data.d[7] = plasticity index, PI (%) Data.d[8] = overconsolidation ratio, OCR Data.d[9] = void ratio, e Data.d[10] = isotropic effective confining pressure, conpre (atm) Data.d[11] = loading frequency, frq (Hz) Data.d[12] = number of loading cycles, N Data.d[13] = peak strain for modulus, str (%) Data.d[14] = corrected strain for damping (diffrent only in high amplitude RC), corr_str (%) Data.d[15] = indicator of data type ( 0 for normalized modulus, NG 1 for material damping ratio, D) Data.d[16] = experimental observation ( normalized modulus, NG or material damping ratio, D, %)*/ nlocation = 0; /* index in Data.d of sample location */ nsoil = nlocation + 1; /* index in Data.d of soil type */ nspecimen = nsoil + 1; /* index in Data.d of specimen # */ nswv = nspecimen + 1; /* index in Data.d of in-situ shear wave velocity */ ndisturbance = nswv + 1; /* index in Data.d of shear wave velocity ratio (disturbance) */

309

npressure = ndisturbance + 1; /* index in Data.d of pressure # */ ntest = npressure + 1; /* index in Data.d of test type */ nPI = ntest + 1; /* index in Data.d of PI */ nOCR = nPI + 1; /* index in Data.d of OCR */ ne = nOCR + 1; /* index in Data.d of e */ nconpre = ne + 1; /* index in Data.d of conpre */ nfrq = nconpre + 1; /* index in Data.d of frq */ nN = nfrq + 1; /* index in Data.d of N */ nstr = nN + 1; /* index in Data.d of str */ ncorrstr = nstr + 1; /* index in Data.d of corr_str */ nTYPE = ncorrstr + 1; /* index in Data.d of Data Type (G versus D) */ //------------------------------------------- censorcheck = 0; /* 0 = all point measurments, 1 = some censored measurements */ ncflag = 99; /* index in Data.d of censor flag (indicates if measurement is censored, if censorcheck = 1) */ nydown = 99; /* index in Data.d of lower bound measurements, if censorcheck = 1 */ ny = nTYPE + 1; /* index in Data.d of point/upper bound measurements */ ny0 = 0; /* index of first useable data point */ //------------------------------------------- nx = nPhi; /* total number of parameters */ nCOV = nx; /* number of separate (not also used for mean) variance parameters */ nmu = nPhi - nCOV; /* number of separate (not also used for variance) mean parameters */ /* Indices in ModelStructure for the model parameters */ iphi1 = 0; iphi2 = iphi1 + 1; iphi3 = iphi2 + 1; iphi4 = iphi3 + 1; iphi5 = iphi4 + 1; iphi6 = iphi5 + 1; iphi7 = iphi6 + 1; iphi8 = iphi7 + 1; iphi9 = iphi8 + 1;

310

iphi10 = iphi9 + 1; iphi11 = iphi10 + 1; iphi12 = iphi11 + 1; istdGa = iphi12 + 1; /* ln */ istdGb = istdGa + 1; /* ln */ istdDa = istdGb + 1; /* ln */ istdDb = istdDa + 1; /* ln */ ithetanugget = istdDb + 1; /* ln */ ntheta = 5; /* ln */ itheta.construct(ntheta); itheta[0] = ithetanugget + 1; itheta[1] = itheta[0] + 1; itheta[2] = itheta[1] + 1; itheta[3] = itheta[2] + 1; itheta[4] = itheta[3] + 1; /* Scale measurements to avoid numerical precision problems in calculations */ scalar = 1.0; if (censorcheck == 1) dscal(Data.nMeas,scalar,Data.d[nydown],1); dscal(Data.nMeas,scalar,Data.d[ny],1); rhozero = 1.0e-2; /* effective zero value for correlation */ } void ModelStructure::Feasible(double *x) {} void ModelStructure::OutputModel(DataStructure &Data, double *x, CString outfilename) { PrintYMean(Data,x,outfilename); }

311

APPENDIX C

FILE USED IN ESTIMATING MEAN VALUES

FOR


BAYESIAN ANALYSIS

OF

RESONANT COLUMN

AND

TORSIONAL SHEAR

TEST RESULTS

312

// RCTSYMean.cpp : Proposed Equations for RCTS data // #include "stdafx.h" #include <afxwin.h> #include <iostream.h> #include <fstream.h> #include <math.h> #include <direct.h> #include "machh.h" #include "compareh.h" #include "dblash.h" #include "_arrayh.h" #include "_array2h.h" #include "_array3h.h" #include "matrixh.h" #include "smatrixh.h" #include "gmatrixh.h" #include "covmatrixh.h" #include "dblash.h" #include "goldenh.h" #include "rqph.h" #include "Datah.h" #include "NormalLikeh.h" #include "Modelh.h" void ModelStructure::PrintYMean(DataStructure &Data, double *x, CString output) { int n,k; int ytype; double yk; n = Data.nMeas; darray YMean(n); iarray index(n); for (k = 0; k < n; k++) index[k] = k; CalculateYMeanC(Data,x,YMean,index); CovMatrix YCOV(n); CalculateYCOVC(Data,x,YMean,YCOV,index); ofstream out(output);

313

out << "Point" << "\t" << "ytype" << "\t" << "yk" << "\t" << "YMean" << "\t" << "YCOV" "\n"; for (k = 0; k < n; k++) { yk = Data.d[ny][k]; ytype = int(Data.d[nTYPE][k]); out << k << "\t" << ytype << "\t" << yk/scalar << "\t" << YMean[k]/scalar << "\t" << YCOV.G.xptr[k] << "\n"; } out.close(); } void ModelStructure::CalculateYMeanC(DataStructure &Data, double *x, darray &YMeanC, iarray &index) { int kindex,nindex,k; double dbl_par_phi[25];// par stand for model parameters double dbl_atr_str, dbl_atr_corr_str, dbl_atr_PI, dbl_atr_OCR, dbl_atr_e, dbl_atr_Fe, dbl_atr_conpre, dbl_atr_frq, dbl_atr_N; // atr stands for attributes double dbl_refstr, dbl_a, dbl_NG, dbl_NG_corrstr, dbl_DMasing, dbl_c1, dbl_c2, dbl_c3, dbl_Dmin, dbl_b, dbl_D; // Dependent intermediate variables int datatype; double dbl_con_pi = 3.1415926535; //Constant PI nindex = index.n; // total number of nearby data points // assign values to model parameters dbl_par_phi[1] = x[iphi1]; dbl_par_phi[2] = x[iphi2]; dbl_par_phi[3] = x[iphi3]; dbl_par_phi[4] = x[iphi4]; dbl_par_phi[5] = x[iphi5]; dbl_par_phi[6] = x[iphi6]; dbl_par_phi[7] = x[iphi7]; dbl_par_phi[8] = x[iphi8]; dbl_par_phi[9] = x[iphi9]; dbl_par_phi[10] = x[iphi10]; dbl_par_phi[11] = x[iphi11];

314

dbl_par_phi[12] = x[iphi12]; // Step through nearby data set and get YMean for each point for (kindex = 0; kindex < nindex; kindex++) { k = index[kindex]; // assign values to attributes dbl_atr_str = Data.d[nstr][k]; dbl_atr_corr_str = Data.d[ncorrstr][k]; dbl_atr_PI = Data.d[nPI][k]; dbl_atr_OCR = Data.d[nOCR][k]; dbl_atr_e = Data.d[ne][k]; dbl_atr_conpre = Data.d[nconpre][k]; dbl_atr_frq = Data.d[nfrq][k]; dbl_atr_N = Data.d[nN][k]; datatype = int(Data.d[nTYPE][k]); // calculation of normalized modulus and damping values // for given atributes and model parameters // although the effect of void ratio is not accounted for in this study // the code is written so that an F(e) term such as the one below // can be included in the future // dbl_atr_Fe=0.3 + 0.7 * pow (dbl_atr_e,2); dbl_atr_Fe=1.0; dbl_refstr=(dbl_par_phi[1]+dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])) *dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4]); dbl_a=dbl_par_phi[5]; dbl_NG=1.0/(1+pow((dbl_atr_str/dbl_refstr),dbl_a)); dbl_NG_corrstr=1.0/(1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)); dbl_DMasing=(100.0/dbl_con_pi)*(4*(dbl_atr_corr_str-dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr))

315

/(pow(dbl_atr_corr_str,2)/(dbl_atr_corr_str+dbl_refstr))-2); dbl_c1= -1.1143*pow(dbl_a,2)+1.8618*dbl_a+0.2523; dbl_c2= 0.0805*pow(dbl_a,2)-0.0710*dbl_a-0.0095; dbl_c3= -0.0005*pow(dbl_a,2)+0.0002*dbl_a+0.0003; dbl_Dmin= (dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])) *pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq)); dbl_b=dbl_par_phi[11]+dbl_par_phi[12]*log(dbl_atr_N); dbl_D=dbl_Dmin+dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c1*dbl_DMasing+dbl_c2*pow(dbl_DMasing,2) +dbl_c3*pow(dbl_DMasing,3)); if (datatype == 0) { YMeanC[kindex] = scalar*dbl_NG; } else { YMeanC[kindex] = scalar*dbl_D; } } // for (kindex = 0; kindex < nindex; kindex++) } void ModelStructure::CalculatedYMeanCiMM(int iv, DataStructure &Data, double *x, darray &dYMeani, iarray &index) { int k,kindex,nindex;

316

double dbl_par_phi[25], dbl_par_dphi[25]; // par stand for model parameters double dbl_atr_str, dbl_atr_corr_str, dbl_atr_PI, dbl_atr_OCR, dbl_atr_e, dbl_atr_Fe, dbl_atr_conpre, dbl_atr_frq, dbl_atr_N; // atr stands for attributes double dbl_refstr, dbl_a, dbl_NG, dbl_NG_corrstr, dbl_DMasing, dbl_c1, dbl_c2, dbl_c3, dbl_Dmin, dbl_b, dbl_D; // Dependent intermediate variables double dbl_drefstr, dbl_da, dbl_dNG, dbl_dNG_corrstr, dbl_dc1, dbl_dc2, dbl_dc3, dbl_dDMasing, dbl_dDmin, dbl_db, dbl_dD; // First Order Partial Derivative of dependent intermediate variables int datatype; int int_loopcounter_i,int_loopcounter_j; double dbl_con_pi = 3.1415926535; //Constant PI nindex = index.n; // assign values to model parameters dbl_par_phi[1] = x[iphi1]; dbl_par_phi[2] = x[iphi2]; dbl_par_phi[3] = x[iphi3]; dbl_par_phi[4] = x[iphi4]; dbl_par_phi[5] = x[iphi5]; dbl_par_phi[6] = x[iphi6]; dbl_par_phi[7] = x[iphi7]; dbl_par_phi[8] = x[iphi8]; dbl_par_phi[9] = x[iphi9]; dbl_par_phi[10] = x[iphi10]; dbl_par_phi[11] = x[iphi11]; dbl_par_phi[12] = x[iphi12]; // set all dphi equal to zero for (int_loopcounter_j=0;int_loopcounter_j<nx;int_loopcounter_j++) { dbl_par_dphi[int_loopcounter_j]=0; } // calculation of first order derivatives for iv int_loopcounter_i=iv+1; // calculate derivatives for phi[int_loopcounter_i]

317

dbl_par_dphi[int_loopcounter_i]=1; for (kindex = 0; kindex < nindex; kindex++) { k = index[kindex]; // assign values to attributes dbl_atr_str= Data.d[nstr][k]; dbl_atr_corr_str= Data.d[ncorrstr][k]; dbl_atr_PI= Data.d[nPI][k]; dbl_atr_OCR= Data.d[nOCR][k]; dbl_atr_e= Data.d[ne][k]; dbl_atr_conpre= Data.d[nconpre][k]; dbl_atr_frq= Data.d[nfrq][k]; dbl_atr_N= Data.d[nN][k]; datatype = int(Data.d[nTYPE][k]); // calculation of normalized modulus and damping values // for given atributes and model parameters // although the effect of void ratio is not accounted for in this study // the code is written so that an F(e) term such as the one below // can be included in the future // dbl_atr_Fe=0.3 + 0.7 * pow (dbl_atr_e,2); dbl_atr_Fe=1.0; dbl_refstr=(dbl_par_phi[1]+dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])) *dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4]); dbl_a=dbl_par_phi[5]; dbl_NG=1.0/(1+pow((dbl_atr_str/dbl_refstr),dbl_a)); dbl_NG_corrstr=1.0/(1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)); dbl_DMasing=(100.0/dbl_con_pi)*(4*(dbl_atr_corr_str-dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr))

318

/(pow(dbl_atr_corr_str,2)/(dbl_atr_corr_str+dbl_refstr))-2); dbl_c1= -1.1143*pow(dbl_a,2)+1.8618*dbl_a+0.2523; dbl_c2= 0.0805*pow(dbl_a,2)-0.0710*dbl_a-0.0095; dbl_c3= -0.0005*pow(dbl_a,2)+0.0002*dbl_a+0.0003; dbl_Dmin= (dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])) *pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq)); dbl_b=dbl_par_phi[11]+dbl_par_phi[12]*log(dbl_atr_N); dbl_D=dbl_Dmin+dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c1*dbl_DMasing+dbl_c2*pow(dbl_DMasing,2) +dbl_c3*pow(dbl_DMasing,3)); //"FIRST ORDER DERIVATIVES" CODE FOR phi[int_loopcounter_i] //------------------------------------------------------- dbl_drefstr=dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_dphi[1] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_dphi[2] +dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*log(dbl_atr_OCR)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_dphi[3] +(dbl_par_phi[1]+dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3]))*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_dphi[4]; dbl_da=dbl_par_dphi[5];

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dbl_dNG=(1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_drefstr +(-1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*log(dbl_atr_str/dbl_refstr)*dbl_da; dbl_dNG_corrstr=(1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_drefstr +(-1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*log(dbl_atr_corr_str/dbl_refstr)*dbl_da; dbl_dc1=(-2.2286*dbl_a+1.8618)*dbl_da; dbl_dc2=( 0.1610*dbl_a-0.0710)*dbl_da; dbl_dc3=(-0.0010*dbl_a+0.0002)*dbl_da; dbl_dDMasing=(-400)*((log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)*dbl_atr_corr_str +2*dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr) -2*dbl_atr_corr_str)/(pow(dbl_atr_corr_str,2)*dbl_con_pi)) *dbl_drefstr; dbl_dDmin=pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_dphi[6] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_dphi[7] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_dphi[8] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_dphi[9]

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+(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_dphi[10]; dbl_db=dbl_par_dphi[11]+log(dbl_atr_N)*dbl_par_dphi[12]; dbl_dD=dbl_dDmin +pow(dbl_NG_corrstr,0.1)*dbl_DMasing*(dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))*dbl_db +0.1*dbl_b*dbl_DMasing*((dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))/pow(dbl_NG_corrstr,0.9))*dbl_dNG_corrstr +dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_dDMasing +dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_dc1 +dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_dc2 +dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,3)*dbl_dc3; //------------------------------------------------------- if (datatype == 0) { dYMeani[kindex] = scalar*dbl_dNG; } else { dYMeani[kindex] = scalar*dbl_dD; } } // for (kindex = 0; kindex < nindex; kindex++) } void ModelStructure::Calculated2YMeanCijMM(int iv, int jv, DataStructure &Data, double *x,

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darray &dYMeani, darray &dYMeanj, darray &d2YMeanij, iarray &index) { int k,kindex,nindex; double dbl_par_phi[25], dbl_par_diphi[25], dbl_par_djphi[25]; // par stand for model parameters double dbl_atr_str, dbl_atr_corr_str, dbl_atr_PI, dbl_atr_OCR, dbl_atr_e, dbl_atr_Fe, dbl_atr_conpre, dbl_atr_frq, dbl_atr_N; // atr stands for attributes double dbl_refstr, dbl_a, dbl_NG, dbl_NG_corrstr, dbl_DMasing, dbl_c1, dbl_c2, dbl_c3, dbl_Dmin, dbl_b, dbl_D; // Dependent intermediate variables double dbl_direfstr, dbl_dia, dbl_diNG, dbl_diNG_corrstr, dbl_dic1, dbl_dic2, dbl_dic3, dbl_diDMasing, dbl_diDmin, dbl_dib, dbl_diD; // diNG stands for first order partial derivative of NG with respect to phi[i] double dbl_djrefstr, dbl_dja, dbl_djNG, dbl_djNG_corrstr, dbl_djc1, dbl_djc2, dbl_djc3, dbl_djDMasing, dbl_djDmin, dbl_djb, dbl_djD; // djNG stands for first order partial derivative of NG with respect to phi[j] double dbl_d2refstr, dbl_d2a, dbl_d2NG, dbl_d2NG_corrstr, dbl_d2c1, dbl_d2c2, dbl_d2c3, dbl_d2DMasing, dbl_d2Dmin, dbl_d2b, dbl_d2D; // d2NG stands for second order partial derivative of NG with respect to phi[i]=phi[j] int datatype; int int_loopcounter_k,int_loopcounter_l,int_loopcounter_m,int_loopcounter_n; double dbl_con_pi = 3.1415926535; //Constant PI nindex = index.n; // assign values to model parameters dbl_par_phi[1] = x[iphi1]; dbl_par_phi[2] = x[iphi2]; dbl_par_phi[3] = x[iphi3]; dbl_par_phi[4] = x[iphi4]; dbl_par_phi[5] = x[iphi5]; dbl_par_phi[6] = x[iphi6]; dbl_par_phi[7] = x[iphi7]; dbl_par_phi[8] = x[iphi8]; dbl_par_phi[9] = x[iphi9]; dbl_par_phi[10] = x[iphi10];

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dbl_par_phi[11] = x[iphi11]; dbl_par_phi[12] = x[iphi12]; // set all diphi equal to zero for (int_loopcounter_l=0;int_loopcounter_l<nx;int_loopcounter_l++) { dbl_par_diphi[int_loopcounter_l]=0; } // set all djphi equal to zero for (int_loopcounter_n=0;int_loopcounter_n<nx;int_loopcounter_n++) { dbl_par_djphi[int_loopcounter_n]=0; } // calculation of second order derivatives (d/diphi[int_loopcounter_k]*djphi[int_loopcounter_m]) // calculation of second order derivatives for iv, jv int_loopcounter_k=iv+1; int_loopcounter_m=jv+1; // calculate derivatives for iphi[int_loopcounter_k] dbl_par_diphi[int_loopcounter_k]=1; // calculate derivatives for jphi[int_loopcounter_m] dbl_par_djphi[int_loopcounter_m]=1; for (kindex = 0; kindex < nindex; kindex++) { k = index[kindex]; // assign values to attributes dbl_atr_str= Data.d[nstr][k]; dbl_atr_corr_str= Data.d[ncorrstr][k]; dbl_atr_PI= Data.d[nPI][k]; dbl_atr_OCR= Data.d[nOCR][k]; dbl_atr_e= Data.d[ne][k]; dbl_atr_conpre= Data.d[nconpre][k]; dbl_atr_frq= Data.d[nfrq][k]; dbl_atr_N= Data.d[nN][k]; datatype = int(Data.d[nTYPE][k]);

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// calculation of normalized modulus and damping values // for given atributes and model parameters // although the effect of void ratio is not accounted for in this study // the code is written so that an F(e) term such as the one below // can be included in the future // dbl_atr_Fe=0.3 + 0.7 * pow (dbl_atr_e,2); dbl_atr_Fe=1.0; dbl_refstr=(dbl_par_phi[1]+dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])) *dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4]); dbl_a=dbl_par_phi[5]; dbl_NG=1.0/(1+pow((dbl_atr_str/dbl_refstr),dbl_a)); dbl_NG_corrstr=1.0/(1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)); dbl_DMasing=(100.0/dbl_con_pi)*(4*(dbl_atr_corr_str-dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)) /(pow(dbl_atr_corr_str,2)/(dbl_atr_corr_str+dbl_refstr))-2); dbl_c1= -1.1143*pow(dbl_a,2)+1.8618*dbl_a+0.2523; dbl_c2= 0.0805*pow(dbl_a,2)-0.0710*dbl_a-0.0095; dbl_c3= -0.0005*pow(dbl_a,2)+0.0002*dbl_a+0.0003; dbl_Dmin= (dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])) *pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq)); dbl_b=dbl_par_phi[11]+dbl_par_phi[12]*log(dbl_atr_N);

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dbl_D=dbl_Dmin+dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c1*dbl_DMasing+dbl_c2*pow(dbl_DMasing,2) +dbl_c3*pow(dbl_DMasing,3)); //"FIRST ORDER DERIVATIVES" CODE FOR iphi[int_loopcounter_k] //------------------------------------------------------- dbl_direfstr=dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_diphi[1] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_diphi[2] +dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*log(dbl_atr_OCR)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_diphi[3] +(dbl_par_phi[1]+dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3]))*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_diphi[4]; dbl_dia=dbl_par_diphi[5]; dbl_diNG=(1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_direfstr +(-1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*log(dbl_atr_str/dbl_refstr)*dbl_dia; dbl_diNG_corrstr=(1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_direfstr +(-1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*log(dbl_atr_corr_str/dbl_refstr)*dbl_dia; dbl_dic1=(-2.2286*dbl_a+1.8618)*dbl_dia; dbl_dic2=( 0.1610*dbl_a-0.0710)*dbl_dia; dbl_dic3=(-0.0010*dbl_a+0.0002)*dbl_dia;

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dbl_diDMasing=(-400)*((log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)*dbl_atr_corr_str +2*dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr) -2*dbl_atr_corr_str)/(pow(dbl_atr_corr_str,2)*dbl_con_pi)) *dbl_direfstr; dbl_diDmin=pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[6] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[7] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[8] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[9] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_diphi[10]; dbl_dib=dbl_par_diphi[11]+log(dbl_atr_N)*dbl_par_diphi[12]; dbl_diD=dbl_diDmin +pow(dbl_NG_corrstr,0.1)*dbl_DMasing*(dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))*dbl_dib +0.1*dbl_b*dbl_DMasing*((dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))/pow(dbl_NG_corrstr,0.9))*dbl_diNG_corrstr +dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_diDMasing +dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_dic1

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+dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_dic2 +dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,3)*dbl_dic3; //------------------------------------------------------- //"FIRST ORDER DERIVATIVES" CODE FOR jphi[int_loopcounter_m] //------------------------------------------------------- dbl_djrefstr=dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_djphi[1] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_djphi[2] +dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*log(dbl_atr_OCR)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_djphi[3] +(dbl_par_phi[1]+dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3]))*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_djphi[4]; dbl_dja=dbl_par_djphi[5]; dbl_djNG=(1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_djrefstr +(-1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*log(dbl_atr_str/dbl_refstr)*dbl_dja; dbl_djNG_corrstr=(1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_djrefstr +(-1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*log(dbl_atr_corr_str/dbl_refstr)*dbl_dja; dbl_djc1=(-2.2286*dbl_a+1.8618)*dbl_dja; dbl_djc2=( 0.1610*dbl_a-0.0710)*dbl_dja; dbl_djc3=(-0.0010*dbl_a+0.0002)*dbl_dja;

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dbl_djDMasing=(-400)*((log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)*dbl_atr_corr_str +2*dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr) -2*dbl_atr_corr_str)/(pow(dbl_atr_corr_str,2)*dbl_con_pi)) *dbl_djrefstr; dbl_djDmin=pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_djphi[6] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_djphi[7] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_djphi[8] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_djphi[9] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_djphi[10]; dbl_djb=dbl_par_djphi[11]+log(dbl_atr_N)*dbl_par_djphi[12]; dbl_djD=dbl_djDmin +pow(dbl_NG_corrstr,0.1)*dbl_DMasing*(dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))*dbl_djb +0.1*dbl_b*dbl_DMasing*((dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))/pow(dbl_NG_corrstr,0.9))*dbl_djNG_corrstr +dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_djDMasing +dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_djc1

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+dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_djc2 +dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,3)*dbl_djc3; //------------------------------------------------------- //"SECOND ORDER DERIVATIVES" CODE FOR iphi[int_loopcounter_k],jphi[int_loopcounter_m] // Requires first order derivatives with respect to i and j //------------------------------------------------------- dbl_d2refstr=dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_diphi[1]*dbl_par_djphi[4] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*log(dbl_atr_OCR)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_diphi[2]*dbl_par_djphi[3] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_diphi[2]*dbl_par_djphi[4] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*log(dbl_atr_OCR)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_diphi[3]*dbl_par_djphi[2] +dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*pow(log(dbl_atr_OCR),2)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*dbl_par_diphi[3]*dbl_par_djphi[3] +dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*log(dbl_atr_OCR)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_diphi[3]*dbl_par_djphi[4] +dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_diphi[4]*dbl_par_djphi[1] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_diphi[4]*dbl_par_djphi[2] +dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])*log(dbl_atr_OCR)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*log(dbl_atr_conpre)*dbl_par_diphi[4]*dbl_par_djphi[3] +(dbl_par_phi[1]+dbl_par_phi[2]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[3])

329

)*dbl_atr_Fe*pow(dbl_atr_conpre,dbl_par_phi[4])*pow(log(dbl_atr_conpre),2)*dbl_par_diphi[4]*dbl_par_djphi[4]; dbl_d2a=0; dbl_d2NG=dbl_a*((pow((dbl_atr_str/dbl_refstr),(2*dbl_a))*dbl_a -pow((dbl_atr_str/dbl_refstr),dbl_a)*dbl_a -pow((dbl_atr_str/dbl_refstr),dbl_a) -pow((dbl_atr_str/dbl_refstr),(2*dbl_a))) /(pow(dbl_refstr,2)*pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),3)))*dbl_direfstr*dbl_djrefstr +((-2*pow(pow(dbl_atr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_atr_str) +2*pow(pow(dbl_atr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_atr_str) -pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_atr_str) -pow(dbl_atr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_refstr) +pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a) +pow(dbl_atr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))) /(dbl_refstr*pow((1+pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)),3)))*dbl_direfstr*dbl_dja +((-2*pow(pow(dbl_atr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_atr_str)

330

+2*pow(pow(dbl_atr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_atr_str) -pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_atr_str) -pow(dbl_atr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_refstr) +pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a) +pow(dbl_atr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))) /(dbl_refstr*pow((1+pow(dbl_atr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)),3)))*dbl_dia*dbl_djrefstr +(-pow(log(dbl_atr_str/dbl_refstr),2)*(-pow((dbl_atr_str/dbl_refstr),(2*dbl_a)) +pow((dbl_atr_str/dbl_refstr),dbl_a)) /pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),3))*dbl_dia*dbl_dja +((1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_d2refstr +(-1/pow((1+pow((dbl_atr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_str/dbl_refstr),dbl_a)*log(dbl_atr_str/dbl_refstr)*dbl_d2a); dbl_d2NG_corrstr=dbl_a*((pow((dbl_atr_corr_str/dbl_refstr),(2*dbl_a))*dbl_a -pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*dbl_a -pow((dbl_atr_corr_str/dbl_refstr),dbl_a) -pow((dbl_atr_corr_str/dbl_refstr),(2*dbl_a)))

331

/(pow(dbl_refstr,2)*pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),3)))*dbl_direfstr*dbl_djrefstr +((-2*pow(pow(dbl_atr_corr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_atr_corr_str) +2*pow(pow(dbl_atr_corr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_atr_corr_str) -pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_corr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_atr_corr_str) -pow(dbl_atr_corr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_refstr) +pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a) +pow(dbl_atr_corr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))) /(dbl_refstr*pow((1+pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)),3)))*dbl_direfstr*dbl_dja +((-2*pow(pow(dbl_atr_corr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_atr_corr_str) +2*pow(pow(dbl_atr_corr_str,2),dbl_a)*pow((1/pow(dbl_refstr,2)),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_atr_corr_str) -pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)*dbl_a*log(dbl_refstr) +pow(dbl_atr_corr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_atr_corr_str)

332

-pow(dbl_atr_corr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))*dbl_a*log(dbl_refstr) +pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a) +pow(dbl_atr_corr_str,(2*dbl_a))*pow((1/dbl_refstr),(2*dbl_a))) /(dbl_refstr*pow((1+pow(dbl_atr_corr_str,dbl_a)*pow((1/dbl_refstr),dbl_a)),3)))*dbl_dia*dbl_djrefstr +(-pow(log(dbl_atr_corr_str/dbl_refstr),2)*(-pow((dbl_atr_corr_str/dbl_refstr),(2*dbl_a)) +pow((dbl_atr_corr_str/dbl_refstr),dbl_a)) /pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),3))*dbl_dia*dbl_dja +((1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*(dbl_a/dbl_refstr)*dbl_d2refstr +(-1/pow((1+pow((dbl_atr_corr_str/dbl_refstr),dbl_a)),2))*pow((dbl_atr_corr_str/dbl_refstr),dbl_a)*log(dbl_atr_corr_str/dbl_refstr)*dbl_d2a); dbl_d2DMasing=400*((pow(dbl_atr_corr_str,2) -2*dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)*dbl_atr_corr_str -2*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)*pow(dbl_refstr,2) +2*dbl_atr_corr_str*dbl_refstr) /(dbl_con_pi*dbl_refstr*(dbl_atr_corr_str+dbl_refstr)*pow(dbl_atr_corr_str,2))) *dbl_direfstr*dbl_djrefstr +(-400)*((log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)*dbl_atr_corr_str +2*dbl_refstr*log((dbl_atr_corr_str+dbl_refstr)/dbl_refstr)

333

-2*dbl_atr_corr_str)/(pow(dbl_atr_corr_str,2)*dbl_con_pi)) *dbl_d2refstr; dbl_d2c1=-2.2286*dbl_dia*dbl_dja+(-2.2286*dbl_a+1.8618)*dbl_d2a; dbl_d2c2= 0.1610*dbl_dia*dbl_dja+( 0.1610*dbl_a-0.0710)*dbl_d2a; dbl_d2c3=-0.0010*dbl_dia*dbl_dja+(-0.0010*dbl_a+0.0002)*dbl_d2a; dbl_d2Dmin=pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[6]*dbl_par_djphi[9] +pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_diphi[6]*dbl_par_djphi[10] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[7]*dbl_par_djphi[8] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[7]*dbl_par_djphi[9] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_diphi[7]*dbl_par_djphi[10] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[8]*dbl_par_djphi[7] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(log(dbl_atr_OCR),2)*pow(dbl_atr_conpre,dbl_par_phi[9])*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[8]*dbl_par_djphi[8] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OC

334

R)*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[8]*dbl_par_djphi[9] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_diphi[8]*dbl_par_djphi[10] +pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[9]*dbl_par_djphi[6] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[9]*dbl_par_djphi[7] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[9]*dbl_par_djphi[8] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*pow(log(dbl_atr_conpre),2)*(1+dbl_par_phi[10]*log(dbl_atr_frq))*dbl_par_diphi[9]*dbl_par_djphi[9] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*log(dbl_atr_frq)*dbl_par_diphi[9]*dbl_par_djphi[10] +pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_diphi[10]*dbl_par_djphi[6] +dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_diphi[10]*dbl_par_djphi[7] +dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8])*log(dbl_atr_OCR)*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_frq)*dbl_par_diphi[10]*dbl_par_djphi[8] +(dbl_par_phi[6]+dbl_par_phi[7]*dbl_atr_PI*pow(dbl_atr_OCR,dbl_par_phi[8]))*pow(dbl_atr_conpre,dbl_par_phi[9])*log(dbl_atr_conpre)*log(dbl_atr_frq)*dbl_par_diphi[10]*dbl_par_djphi[9]; dbl_d2b=0;

335

dbl_d2D=0.1*dbl_DMasing*((dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))/pow(dbl_NG_corrstr,0.9))*dbl_dib*dbl_djNG_corrstr +pow(dbl_NG_corrstr,0.1)*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_dib*dbl_djDMasing +pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_dib*dbl_djc1 +pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_dib*dbl_djc2 +pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,3)*dbl_dib*dbl_djc3 +0.1*dbl_DMasing*((dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))/pow(dbl_NG_corrstr,0.9))*dbl_diNG_corrstr*dbl_djb +(-0.09)*dbl_b*dbl_DMasing*((dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))/pow(dbl_NG_corrstr,1.9))*dbl_diNG_corrstr*dbl_djNG_corrstr +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_diNG_corrstr*dbl_djDMasing +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*dbl_DMasing*dbl_diNG_corrstr*dbl_djc1 +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*pow(dbl_DMasing,2)*dbl_diNG_corrstr*dbl_djc2 +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*pow(dbl_DMasing,3)*dbl_diNG_corrstr*dbl_djc3 +pow(dbl_NG_corrstr,0.1)*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_diDMasing*dbl_djb +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_diDMasing*dbl_djNG_corrstr +2*dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c2+3*dbl_c3*dbl_DMasing)*dbl_diDMasing*dbl_djDMasing +dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_diDMasing*dbl_djc1

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+2*dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_diDMasing*dbl_djc2 +3*dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_diDMasing*dbl_djc3 +pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_dic1*dbl_djb +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*dbl_DMasing*dbl_dic1*dbl_djNG_corrstr +dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_dic1*dbl_djDMasing +pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_dic2*dbl_djb +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*pow(dbl_DMasing,2)*dbl_dic2*dbl_djNG_corrstr +2*dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_dic2*dbl_djDMasing +pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,3)*dbl_dic3*dbl_djb +0.1*(dbl_b/pow(dbl_NG_corrstr,0.9))*pow(dbl_DMasing,3)*dbl_dic3*dbl_djNG_corrstr +3*dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_dic3*dbl_djDMasing +dbl_d2Dmin +pow(dbl_NG_corrstr,0.1)*dbl_DMasing*(dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))*dbl_d2b +0.1*dbl_b*dbl_DMasing*((dbl_c1+dbl_c2*dbl_DMasing+dbl_c3*pow(dbl_DMasing,2))/pow(dbl_NG_corrstr,0.9))*dbl_d2NG_corrstr +dbl_b*pow(dbl_NG_corrstr,0.1)*(dbl_c1+2*dbl_c2*dbl_DMasing+3*dbl_c3*pow(dbl_DMasing,2))*dbl_d2DMasing +dbl_b*pow(dbl_NG_corrstr,0.1)*dbl_DMasing*dbl_d2c1

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+dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,2)*dbl_d2c2 +dbl_b*pow(dbl_NG_corrstr,0.1)*pow(dbl_DMasing,3)*dbl_d2c3; //------------------------------------------------------- if (datatype == 0) { dYMeani[kindex] = scalar*dbl_diNG; } else { dYMeani[kindex] = scalar*dbl_diD; } if (datatype == 0) { dYMeanj[kindex] = scalar*dbl_djNG; } else { dYMeanj[kindex] = scalar*dbl_djD; } if (datatype == 0) { d2YMeanij[kindex] = scalar*dbl_d2NG; } else { d2YMeanij[kindex] = scalar*dbl_d2D; } } // for (kindex = 0; kindex < nindex; kindex++) }

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APPENDIX D

FILE USED IN ESTIMATING COVARIANCE STRUCTURE

FOR


BAYESIAN ANALYSIS

OF

RESONANT COLUMN

AND

TORSIONAL SHEAR

TEST RESULTS

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// RCTSYCOV.cpp : Covariance Structure for RCTS Data // #include "stdafx.h" #include <afxwin.h> #include <iostream.h> #include <fstream.h> #include <math.h> #include <time.h> #include <direct.h> #include "machh.h" #include "compareh.h" #include "dblash.h" #include "_arrayh.h" #include "_array2h.h" #include "_array3h.h" #include "matrixh.h" #include "smatrixh.h" #include "covmatrixh.h" #include "gmatrixh.h" #include "dblash.h" #include "goldenh.h" #include "rqph.h" #include "Datah.h" #include "NormalLikeh.h" #include "Modelh.h" void ModelStructure::CalculateYCOVC(DataStructure &Data, double *x, darray &YMean, CovMatrix &YCOV, iarray &index) { // Fill and Invert Conditional COV Matrix int ka,kb,n,kaindex,kbindex; double Ga,Gb,Da,Db; int datatypea; n = index.n; Ga = exp(x[istdGa]); Gb = exp(x[istdGb]); Da = exp(x[istdDa]); Db = exp(x[istdDb]);

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for (kaindex = 0; kaindex < n; kaindex++) { ka = index[kaindex]; datatypea = int(Data.d[nTYPE][ka]); if (datatypea == 0) { YCOV.G.xptr[kaindex] = scalar*(Ga + pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)); } else { YCOV.G.xptr[kaindex] = scalar*(Da + Db*pow(YMean[kaindex],0.5)); } for (kbindex = 0; kbindex < kaindex; kbindex++) { kb = index[kbindex]; YCOV.R.a[kaindex][kbindex] = CalculateYrhoab(Data,x,ka,kb); } YCOV.R.a[kaindex][kaindex] = 1.0; } YCOV.Decompose(0); } void ModelStructure::CalculatedYCOVCiMM(int iv, DataStructure &Data, double *x, darray &YMean, darray &dYMeani, CovMatrix &YCOV, smatrixsolve &dYCOVi, iarray &index) { int ka,kb,n,kaindex,kbindex; double Ga,Gb,Da,Db; double dGa,dGai,dGb,dGbi; double dDa,dDai,dDb,dDbi; double dYCOVipartsGka,dYCOVipartsGkb; double dYCOVipartsRkakb; int datatypea,datatypeb; darray tau; tau.construct(ntheta);

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n = index.n; Ga = exp(x[istdGa]); dGa = Ga; dGai = double(iv == istdGa)*dGa; Gb = exp(x[istdGb]); dGb = Gb; dGbi = double(iv == istdGb)*dGb; Da = exp(x[istdDa]); dDa = Da; dDai = double(iv == istdDa)*dDa; Db = exp(x[istdDb]); dDb = Db; dDbi = double(iv == istdDb)*dDb; for (kaindex = 0; kaindex < n; kaindex++) { ka = index[kaindex]; datatypea = int(Data.d[nTYPE][ka]); if (datatypea == 0) { dYCOVipartsGka = scalar*( dGai + dGbi*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb))*(1/Gb) + dYMeani[kaindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)) *(YMean[kaindex]-0.5)/Gb); } else { dYCOVipartsGka = scalar*( dDai + dDbi*pow(YMean[kaindex],0.5) + dYMeani[kaindex]*0.5*Db/pow(YMean[kaindex],0.5)); }

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for (kbindex = 0; kbindex < kaindex; kbindex++) { kb = index[kbindex]; datatypeb = int(Data.d[nTYPE][kb]); if (datatypeb == 0) { dYCOVipartsGkb = scalar*( dGai + dGbi*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*(1/Gb) + dYMeani[kbindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb); } else { dYCOVipartsGkb = scalar*( dDai + dDbi*pow(YMean[kbindex],0.5) + dYMeani[kbindex]*0.5*Db/pow(YMean[kbindex],0.5)); } tau[0] = fabs(log(Data.d[nstr][ka])-log(Data.d[nstr][kb])); tau[1] = double (Data.d[nTYPE][ka] != Data.d[nTYPE][kb]); tau[2] = double (Data.d[nspecimen][ka] != Data.d[nspecimen][kb]); tau[3] = double (Data.d[ntest][ka] != Data.d[ntest][kb]); tau[4] = double (Data.d[npressure][ka] != Data.d[npressure][kb]); dYCOVipartsRkakb = dYCOVrhoi(iv,tau.xptr,x); dYCOVi.a[kaindex][kbindex] = dYCOVipartsGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex]

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+ YCOV.G.xptr[kaindex]*dYCOVipartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*dYCOVipartsGkb; } // kbindex = kaindex kb = index[kbindex]; datatypeb = int(Data.d[nTYPE][kb]); if (datatypeb == 0) { dYCOVipartsGkb = scalar*( dGai + dGbi*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*(1/Gb) + dYMeani[kbindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb); } else { dYCOVipartsGkb = scalar*( dDai + dDbi*pow(YMean[kbindex],0.5) + dYMeani[kbindex]*0.5*Db/pow(YMean[kbindex],0.5)); } dYCOVipartsRkakb = 0.0; dYCOVi.a[kaindex][kbindex] = dYCOVipartsGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*dYCOVipartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*dYCOVipartsGkb; } }

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void ModelStructure::Calculated2YCOVCijMM(int iv, int jv, DataStructure &Data, double *x, darray &YMean, darray &dYMeani, darray &dYMeanj, darray &d2YMeanij, CovMatrix &YCOV, smatrixsolve &dYCOVi, smatrixsolve &dYCOVj, smatrixsolve &d2YCOVij, iarray &index) { int ka,kb,n,kaindex,kbindex; double Ga,Gb,Da,Db; double dGa,dGai,dGaj,dGb,dGbi,dGbj; double dDa,dDai,dDaj,dDb,dDbi,dDbj; double d2Ga,d2Gaij,d2Gb,d2Gbij; double d2Da,d2Daij,d2Db,d2Dbij; double dYCOVipartsGka,dYCOVjpartsGka,dYCOVipartsGkb,dYCOVjpartsGkb; double d2YCOVijGka,d2YCOVijGkb; double dYCOVipartsRkakb,dYCOVjpartsRkakb; double d2YCOVijRab; int datatypea,datatypeb; darray tau; tau.construct(ntheta); n = index.n; darray d2YCOVijG(n); Ga = exp(x[istdGa]); dGa = Ga; d2Ga = Ga; dGai = double(iv == istdGa)*dGa; dGaj = double(jv == istdGa)*dGa; d2Gaij = double(iv == istdGa)*double(jv == istdGa)*d2Ga; Gb = exp(x[istdGb]); dGb = Gb; d2Gb = Gb; dGbi = double(iv == istdGb)*dGb; dGbj = double(jv == istdGb)*dGb; d2Gbij = double(iv == istdGb)*double(jv == istdGb)*d2Gb; Da = exp(x[istdDa]); dDa = Da; d2Da = Da; dDai = double(iv == istdDa)*dDa;

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dDaj = double(jv == istdDa)*dDa; d2Daij = double(iv == istdDa)*double(jv == istdDa)*d2Da; Db = exp(x[istdDb]); dDb = Db; d2Db = Db; dDbi = double(iv == istdDb)*dDb; dDbj = double(jv == istdDb)*dDb; d2Dbij = double(iv == istdDb)*double(jv == istdDb)*d2Db; for (kaindex = 0; kaindex < n; kaindex++) { ka = index[kaindex]; datatypea = int(Data.d[nTYPE][ka]); if (datatypea == 0) { dYCOVipartsGka = scalar*( dGai + dGbi*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb))*(1/Gb) + dYMeani[kaindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)) *(YMean[kaindex]-0.5)/Gb); dYCOVjpartsGka = scalar*( dGaj + dGbj*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb))*(1/Gb) + dYMeanj[kaindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)) *(YMean[kaindex]-0.5)/Gb); d2YCOVijGka = scalar*( d2Gaij + d2Gbij *(1/Gb)* ( (-1/(4*pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),1.5)))*pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),2) + (0.5 * pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5) ) )

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+ dGbi*dYMeanj[kaindex] *(1/Gb)* ( (-1/(2*pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),1.5)))*((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb))*((YMean[kaindex]-0.5)/Gb) + (1/(pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)))*((YMean[kaindex]-0.5)/Gb) ) + dYMeani[kaindex]*dGbj *(1/Gb)* ( (-1/(2*pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),1.5)))*((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb))*((YMean[kaindex]-0.5)/Gb) + (1/(pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)))*((YMean[kaindex]-0.5)/Gb) ) + dYMeani[kaindex]*dYMeanj[kaindex] * ( ((-1/(pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),1.5)))*( pow((YMean[kaindex]-0.5),2)/pow(Gb,2) )) - (1/( pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)*Gb)) ) + d2YMeanij[kaindex] * (-1/pow(((0.25/Gb)-(pow((YMean[kaindex]-0.5),2)/Gb)),0.5)) *(YMean[kaindex]-0.5)/Gb ); } else { dYCOVipartsGka = scalar*( dDai + dDbi*pow(YMean[kaindex],0.5) + dYMeani[kaindex]*0.5*Db/pow(YMean[kaindex],0.5)); dYCOVjpartsGka = scalar*( dDaj + dDbj*pow(YMean[kaindex],0.5) + dYMeanj[kaindex]*0.5*Db/pow(YMean[kaindex],0.5)); d2YCOVijGka = scalar*( d2Daij + d2Dbij*pow(YMean[kaindex],0.5) + dDbi*dYMeanj[kaindex] * (1/(2*pow(YMean[kaindex],0.5))) + dYMeani[kaindex]*dDbj * (1/(2*pow(YMean[kaindex],0.5))) + dYMeani[kaindex]*dYMeanj[kaindex]*(-0.25)*Db/pow(YMean[kaindex],1.5)

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+ d2YMeanij[kaindex] * 0.5*Db/pow(YMean[kaindex],0.5) ); } for (kbindex = 0; kbindex < kaindex; kbindex++) { kb = index[kbindex]; datatypeb = int(Data.d[nTYPE][kb]); if (datatypeb == 0) { dYCOVipartsGkb = scalar*( dGai + dGbi*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*(1/Gb) + dYMeani[kbindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb); dYCOVjpartsGkb = scalar*( dGaj + dGbj*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*(1/Gb) + dYMeanj[kbindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb); d2YCOVijGkb = scalar*( d2Gaij + d2Gbij *(1/Gb)* ( (-1/(4*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),2) + (0.5 * pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5) ) ) + dGbi*dYMeanj[kbindex] *(1/Gb)* ( (-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*((YMean[kbindex]-0.5)/Gb) + (1/(pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)))*((YMean[kbindex]-0.5)/Gb) )

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+ dYMeani[kbindex]*dGbj *(1/Gb)* ( (-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*((YMean[kbindex]-0.5)/Gb) + (1/(pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)))*((YMean[kbindex]-0.5)/Gb) ) + dYMeani[kbindex]*dYMeanj[kbindex] * ( ((-1/(pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*( pow((YMean[kbindex]-0.5),2)/pow(Gb,2) )) - (1/( pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)*Gb)) ) + d2YMeanij[kbindex] * (-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb ); } else { dYCOVipartsGkb = scalar*( dDai + dDbi*pow(YMean[kbindex],0.5) + dYMeani[kbindex]*0.5*Db/pow(YMean[kbindex],0.5)); dYCOVjpartsGkb = scalar*( dDaj + dDbj*pow(YMean[kbindex],0.5) + dYMeanj[kbindex]*0.5*Db/pow(YMean[kbindex],0.5)); d2YCOVijGkb = scalar*( d2Daij + d2Dbij*pow(YMean[kbindex],0.5) + dDbi*dYMeanj[kbindex] * (1/(2*pow(YMean[kbindex],0.5))) + dYMeani[kbindex]*dDbj * (1/(2*pow(YMean[kbindex],0.5))) + dYMeani[kbindex]*dYMeanj[kbindex]*(-0.25)*Db/pow(YMean[kbindex],1.5) + d2YMeanij[kbindex] * 0.5*Db/pow(YMean[kbindex],0.5) ); }

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tau[0] = fabs(log(Data.d[nstr][ka])-log(Data.d[nstr][kb])); tau[1] = double (Data.d[nTYPE][ka] != Data.d[nTYPE][kb]); tau[2] = double (Data.d[nspecimen][ka] != Data.d[nspecimen][kb]); tau[3] = double (Data.d[ntest][ka] != Data.d[ntest][kb]); tau[4] = double (Data.d[npressure][ka] != Data.d[npressure][kb]); dYCOVipartsRkakb = dYCOVrhoi(iv,tau.xptr,x); dYCOVjpartsRkakb = dYCOVrhoi(jv,tau.xptr,x); dYCOVi.a[kaindex][kbindex] = dYCOVipartsGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*dYCOVipartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*dYCOVipartsGkb; dYCOVj.a[kaindex][kbindex] = dYCOVjpartsGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*dYCOVjpartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*dYCOVjpartsGkb; d2YCOVijRab = d2YCOVrhoij(iv,jv,tau.xptr,x); d2YCOVij.a[kaindex][kbindex] = d2YCOVijGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex] + dYCOVipartsGka*dYCOVjpartsRkakb*YCOV.G.xptr[kbindex] + dYCOVipartsGka*YCOV.R.a[kaindex][kbindex]*dYCOVjpartsGkb + dYCOVjpartsGka*dYCOVipartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*d2YCOVijRab*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*dYCOVipartsRkakb*dYCOVjpartsGkb + dYCOVjpartsGka*YCOV.R.a[kaindex][kbindex]*dYCOVipartsGkb + YCOV.G.xptr[kaindex]*dYCOVjpartsRkakb*dYCOVipartsGkb + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*d2YCOVijGkb;

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} // kbindex = kaindex kb = index[kbindex]; datatypeb = int(Data.d[nTYPE][kb]); if (datatypeb == 0) { dYCOVipartsGkb = scalar*( dGai + dGbi*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*(1/Gb) + dYMeani[kbindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb); dYCOVjpartsGkb = scalar*( dGaj + dGbj*(-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5))) *((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*(1/Gb) + dYMeanj[kbindex]*(-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb); d2YCOVijGkb = scalar*( d2Gaij + d2Gbij *(1/Gb)* ( (-1/(4*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),2) + (0.5 * pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5) ) ) + dGbi*dYMeanj[kbindex] *(1/Gb)* ( (-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*((YMean[kbindex]-0.5)/Gb) + (1/(pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)))*((YMean[kbindex]-0.5)/Gb) ) + dYMeani[kbindex]*dGbj *(1/Gb)* ( (-1/(2*pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb))*((YMean[kbindex]-0.5)/Gb) + (1/(pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)))*((YMean[kbindex]-0.5)/Gb) )

351

+ dYMeani[kbindex]*dYMeanj[kbindex] * ( ((-1/(pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),1.5)))*( pow((YMean[kbindex]-0.5),2)/pow(Gb,2) )) - (1/( pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)*Gb)) ) + d2YMeanij[kbindex] * (-1/pow(((0.25/Gb)-(pow((YMean[kbindex]-0.5),2)/Gb)),0.5)) *(YMean[kbindex]-0.5)/Gb ); } else { dYCOVipartsGkb = scalar*( dDai + dDbi*pow(YMean[kbindex],0.5) + dYMeani[kbindex]*0.5*Db/pow(YMean[kbindex],0.5)); dYCOVjpartsGkb = scalar*( dDaj + dDbj*pow(YMean[kbindex],0.5) + dYMeanj[kbindex]*0.5*Db/pow(YMean[kbindex],0.5)); d2YCOVijGkb = scalar*( d2Daij + d2Dbij*pow(YMean[kbindex],0.5) + dDbi*dYMeanj[kbindex] * (1/(2*pow(YMean[kbindex],0.5))) + dYMeani[kbindex]*dDbj * (1/(2*pow(YMean[kbindex],0.5))) + dYMeani[kbindex]*dYMeanj[kbindex]*(-0.25)*Db/pow(YMean[kbindex],1.5) + d2YMeanij[kbindex] * 0.5*Db/pow(YMean[kbindex],0.5) ); } dYCOVipartsRkakb = 0.0; dYCOVjpartsRkakb = 0.0; dYCOVi.a[kaindex][kbindex] = dYCOVipartsGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex]

352

+ YCOV.G.xptr[kaindex]*dYCOVipartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*dYCOVipartsGkb; dYCOVj.a[kaindex][kbindex] = dYCOVjpartsGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*dYCOVjpartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*dYCOVjpartsGkb; d2YCOVijRab = 0.0; d2YCOVij.a[kaindex][kbindex] = d2YCOVijGka*YCOV.R.a[kaindex][kbindex]*YCOV.G.xptr[kbindex] + dYCOVipartsGka*dYCOVjpartsRkakb*YCOV.G.xptr[kbindex] + dYCOVipartsGka*YCOV.R.a[kaindex][kbindex]*dYCOVjpartsGkb + dYCOVjpartsGka*dYCOVipartsRkakb*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*d2YCOVijRab*YCOV.G.xptr[kbindex] + YCOV.G.xptr[kaindex]*dYCOVipartsRkakb*dYCOVjpartsGkb + dYCOVjpartsGka*YCOV.R.a[kaindex][kbindex]*dYCOVipartsGkb + YCOV.G.xptr[kaindex]*dYCOVjpartsRkakb*dYCOVipartsGkb + YCOV.G.xptr[kaindex]*YCOV.R.a[kaindex][kbindex]*d2YCOVijGkb; } } double ModelStructure::CalculateYrhoab(DataStructure &Data, double *x, int ka, int kb) { double rho; darray tau; tau.construct(ntheta); if (ka != kb) { tau[0] = fabs(log(Data.d[nstr][ka])-log(Data.d[nstr][kb]));

353

tau[1] = double (Data.d[nTYPE][ka] != Data.d[nTYPE][kb]); tau[2] = double (Data.d[nspecimen][ka] != Data.d[nspecimen][kb]); tau[3] = double (Data.d[ntest][ka] != Data.d[ntest][kb]); tau[4] = double (Data.d[npressure][ka] != Data.d[npressure][kb]); rho = YCOVrho(tau.xptr,x); } else { rho = 1.0; } return(rho); } double ModelStructure::YCOVrho(double *tau, double *x) { double rho,lnrho,thetanugget; darray theta; theta.construct(ntheta); int i; thetanugget = exp(x[ithetanugget]); for (i = 0; i < ntheta; i++) { theta[i] = exp(x[itheta[i]]); } lnrho = -1.0/thetanugget; for (i = 0; i < ntheta; i++) { lnrho = lnrho + -tau[i]/theta[i]; } rho = exp(lnrho); return(rho); } double ModelStructure::dYCOVrhoi(int iv, double *tau, double *x)

354

{ double lnrho; double thetanugget,dthetanugget,dthetanuggeti; darray theta; theta.construct(ntheta); darray dtheta; dtheta.construct(ntheta); darray dthetai; dthetai.construct(ntheta); double drhoi,dlnrhoi; int i; thetanugget = exp(x[ithetanugget]); dthetanugget = thetanugget; dthetanuggeti = double(iv == ithetanugget)*dthetanugget; for (i = 0; i < ntheta; i++) { theta[i] = exp(x[itheta[i]]); dtheta[i] = theta[i]; dthetai[i] = double(iv == itheta[i])*dtheta[i]; } lnrho = -1.0/thetanugget; dlnrhoi = 1.0/thetanugget/thetanugget*dthetanuggeti; for (i = 0; i < ntheta; i++) { lnrho = lnrho + -tau[i]/theta[i]; dlnrhoi = dlnrhoi + tau[i]/theta[i]/theta[i]*dthetai[i]; } drhoi = exp(lnrho)*dlnrhoi; return(drhoi); } double ModelStructure::d2YCOVrhoij(int iv, int jv, double *tau, double *x) { double lnrho; double thetanugget,dthetanugget,dthetanuggeti,dthetanuggetj;

355

double d2thetanugget,d2thetanuggetij; darray theta; theta.construct(ntheta); darray dtheta; dtheta.construct(ntheta); darray dthetai; dthetai.construct(ntheta); darray dthetaj; dthetaj.construct(ntheta); darray d2theta; d2theta.construct(ntheta); darray d2thetaij; d2thetaij.construct(ntheta); double dlnrhoi,dlnrhoj; double d2lnrhoij,d2rhoij; int i; thetanugget = exp(x[ithetanugget]); dthetanugget = thetanugget; dthetanuggeti = double(iv == ithetanugget)*dthetanugget; dthetanuggetj = double(jv == ithetanugget)*dthetanugget; d2thetanugget = thetanugget; d2thetanuggetij = double(iv == ithetanugget)*double(jv == ithetanugget)*d2thetanugget; for (i = 0; i < ntheta; i++) { theta[i] = exp(x[itheta[i]]); dtheta[i] = theta[i]; dthetai[i] = double(iv == itheta[i])*dtheta[i]; dthetaj[i] = double(jv == itheta[i])*dtheta[i]; d2theta[i] = theta[i]; d2thetaij[i] = double(iv == itheta[i])*double(jv == itheta[i])*d2theta[i]; } lnrho = -1.0/thetanugget;; dlnrhoi = 1.0/thetanugget/thetanugget*dthetanuggeti; dlnrhoj = 1.0/thetanugget/thetanugget*dthetanuggetj; d2lnrhoij = -2*1.0/thetanugget/thetanugget/thetanugget*dthetanuggetj*dthetanuggeti

356

+ 1.0/thetanugget/thetanugget*d2thetanuggetij; for (i = 0; i < ntheta; i++) { lnrho = lnrho + -tau[i]/theta[i]; dlnrhoi = dlnrhoi + tau[i]/theta[i]/theta[i]*dthetai[i]; dlnrhoj = dlnrhoj + tau[i]/theta[i]/theta[i]*dthetaj[i]; d2lnrhoij = d2lnrhoij + -2*tau[i]/theta[i]/theta[i]/theta[i]*dthetaj[i]*dthetai[i] + tau[i]/theta[i]/theta[i]*d2thetaij[i]; } d2rhoij = exp(lnrho)*dlnrhoj*dlnrhoi + exp(lnrho)*d2lnrhoij; return(d2rhoij); }

357

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