Near-Fault Seismic Site Response

Near-Fault Seismic Site Response

by

Adrian Rodriguez-Marek

B.S. (Washington State University) 1994 M.S. (Washington State University) 1996

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Engineering-Civil Engineering

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge:

Professor Jonathan D. Bray, Chair Professor Juan M. Pestana

Professor Alexandre J. Chorin

Fall 2000

The dissertation of Adrian Rodriguez-Marek is approved:

________________________________________________ Chair Date

________________________________________________ Date

________________________________________________ Date

University of California, Berkeley

Fall 2000

1

Abstract

NEAR-FAULT SEISMIC SITE RESPONSE

by

Adrian Rodriguez Marek

Doctor of Philosophy in Engineering-Civil Engineering

University of California, Berkeley

Professor Jonathan D. Bray, Chair

The understanding of ground motions in the near-fault region is important for

seismic risk assessment in a number of populated areas that overlie fault zones.

Understanding the effect of local site conditions is important for a complete

characterization of near-fault ground motions. A geotechnically-based site classification

scheme that includes soil depth is proposed for the evaluation of site effects. This

classification scheme is used to evaluate site response for the 1989 Loma Prieta and 1994

Northridge earthquakes. Regression analyses resulted in estimates of ground motion with

lower uncertainties than those made with the simpler classification schemes typically

used in attenuation relationships. These results emphasize the need to better define the

baseline site category used in attenuation relationships. Site amplification factors for

each site category are proposed. These factors, however, are based on data recorded at

distances generally greater than 10 km from the fault and are not directly applicable to

pulse-like near-fault forward-directivity ground motions. Thus, a separate database of

near-fault forward-directivity ground motions is studied. Each record in the database is

characterized by a pulse period and its peak ground velocity (PGV). Regression analyses

2

on these parameters indicate that site response alters the characteristics of forward-

directivity pulses.

The large scatter of the near-fault ground motion data precludes definite

conclusions regarding site response based exclusively on empirical analyses. Numerical

site response studies are performed to develop a better understanding of near-fault site

response. A nonlinear site response methodology using a bounding surface plasticity

model developed by Borja and Amies (1994) is implemented in the finite element

program GeoFEAP. Site response analyses using pulse-like simplified velocity time-

histories as input motions are performed on generalized site profiles representing average

site conditions for different site categories. Site effects influence the characteristics of

input velocity-pulses by amplifying PGV and elongating the period of the input pulses.

The extent of PGV amplification and pulse period elongation is a function of site

condition, profile depth, and input pulse characteristics. The numerical results presented

agree with the observations made in the empirical analysis of near-fault ground motions.

____________________________ Jonathan D. Bray, Thesis Advisor ____________________________ Date

i

For Tina

ii

TABLE OF CONTENTS

ABSTRACT................................................................................................................... 1

TABLE OF CONTENTS ............................................................................................... ii

LIST OF TABLES ........................................................................................................ vi

LIST OF FIGURES....................................................................................................... ix

ACKNOWLEDGMENTS........................................................................................... xix

CHAPTER 1 - INTRODUCTION .............................................................................. 1

1.1 PROBLEM ..........................................................................................................1

1.2 SIGNIFICANCE .................................................................................................3

1.3 OBJECTIVE........................................................................................................4

1.4 OUTLINE............................................................................................................5

CHAPTER 2 - LITERATURE REVIEW .................................................................. 7

2.1 INTRODUCTION...............................................................................................7

2.2 SITE RESPONSE ...............................................................................................8

2.2.1 Site Response Studies and Code Development............................................8

2.2.2 Cyclic Soil Models and Site Response Methodologies..............................12

2.3 SITE EFFECTS IN ATTENUATION RELATIONSHIPS...............................18

2.4 SUMMARY ......................................................................................................24

CHAPTER 3 - CHARACTERIZATION OF SITE RESPONSE GENERAL SITE CATEGORIES ........................................................................................................... 30

3.1 INTRODUCTION.............................................................................................30

3.2 METHODOLOGY............................................................................................32

3.3 SITE CLASSIFICATION .................................................................................34

3.3.1 Classification Scheme ...............................................................................34

3.3.2 Site Classification......................................................................................37

3.4 GROUND MOTION DATA.............................................................................38

3.5 STATISTICAL ANALYSIS .............................................................................39

3.5.1 General......................................................................................................39

iii

3.5.2 Northridge Earthquake .............................................................................41

3.5.3 Loma Prieta Earthquake ...........................................................................44

3.5.4 Results .......................................................................................................45

3.6 EVALUATION OF RESULTS.........................................................................46

3.6.1 General......................................................................................................46

3.6.2 Comparison With a "Soil vs. Rock" Classification System .......................46

3.6.3 Comparison With a Code-Based Site Classification System.....................48

3.6.4 Subdivision of Site C .................................................................................50

3.6.5 Subdivision of Site D .................................................................................51

3.6.6 Effect of Depth to Basement Rock .............................................................51

3.6.7 Amplification Factors................................................................................52

3.6.8 Recommended Factors ..............................................................................54

3.7 SUMMARY ......................................................................................................56

3.7.1 Findings.....................................................................................................56

3.7.2 Recommendations......................................................................................58

CHAPTER 4 - EMPIRICAL CHARACTERIZATION OF NEAR-FAULT GROUND MOTIONS.............................................................................................. 103

4.1 INTRODUCTION...........................................................................................103

4.2 FREQUENCY DOMAIN REPRESENTATION OF NEAR-FAULT GROUND MOTIONS.....................................................................................106

4.3 TIME DOMAIN REPRESENTATION OF NEAR-FAULT GROUND MOTIONS.......................................................................................................108

4.3.1 General....................................................................................................108

4.3.2 Parameterization of velocity pulses ........................................................110

4.3.3 Statistical Evaluation of Equivalent Pulse parameters in the Fault Normal Direction ................................................................................................117

4.3.4 Characteristics of the fault parallel component of motion .....................128

4.3.5 Development of bi-directional simplified pulses for use as baseline input motions.......................................................................................................130

4.4 SUMMARY AND FINDINGS .......................................................................134

CHAPTER 5 - SITE RESPONSE ANALYSIS METHODOLOGY ................... 199

5.1 INTRODUCTION...........................................................................................199

iv

5.2 CONSTITUTIVE MODEL.............................................................................203

5.2.1 Selection of a constitutive model.............................................................203

5.2.2 Mathematical development .....................................................................206

5.3 FINITE ELEMENT IMPLEMENTATION ....................................................216

5.3.1 General....................................................................................................216

5.3.2 Local integration of the constitutive equations.......................................219

5.3.3 Radiation Boundary Conditions..............................................................224

5.3.4 Model Parameters ...................................................................................228

5.3.5 General comments...................................................................................230

5.4 VALIDATION ................................................................................................231

5.4.1 General....................................................................................................232

5.4.2 Lotung Array ...........................................................................................233

5.4.3 Chiba Downhole Array ...........................................................................239

5.4.4 Analysis of Shaking Table Tests ..............................................................241

5.4.5 General Comments..................................................................................246

CHAPTER 6 ....... - SITE RESPONSE ANALYSIS OF NEAR-FAULT GROUND MOTIONS ................................................................................................................ 284

6.1 INTRODUCTION...........................................................................................284

6.2 GENERALIZED SITE PROFILES.................................................................285

6.2.1 General....................................................................................................285

6.3 COMPARISON OF RESULTS FOR RECORDED AND SIMPLIFIED NEAR-FAULT MOTIONS.............................................................................292

6.3.1 General....................................................................................................292

6.4 SITE RESPONSE TO SIMPLIFIED PULSES ...............................................297

6.4.1 General....................................................................................................297

6.4.2 Stiff Soil Profiles (Sites C and D)............................................................300

6.4.3 Soft clay profile .......................................................................................307

6.4.4 General Observations .............................................................................310

6.5 FINDINGS ......................................................................................................314

CHAPTER 7 - CONCLUSION ............................................................................... 370

7.1 SUMMARY ....................................................................................................370

v

7.2 FINDINGS ......................................................................................................372

7.2.1 General....................................................................................................372

7.2.2 Characterization of site response............................................................372

7.2.3 Empirical study of near-fault, forward-directivity ground motions........373

7.2.4 Site response analyses to near-fault ground motions..............................375

7.3 RECOMENDATIONS FOR FURTHER RESEARCH..................................378

REFERENCES ......................................................................................................... 381

APPENDIX A - LIST OF GROUND MOTION SITES WITH CORRESPONDING SITE CLASSIFICATION .......................................................................................... 398

APPENDIX B - SITE VISITS TO SELECTED GROUND MOTION SITES......... 419

APPENDIX C - EQUATIONS TO OBTAIN COMBINED SPECTRAL ACCELERATION RATIOS FOR THE NORTHRIDGE AND LOMA PRIETA EARTHQUAKES ...................................................................................................... 441

vi

LIST OF TABLES Table 3.1 Geotechnical Site Categories (after Bray and Rodríguez-Marek 1997). .....60

Table 3.2 Sites located on the footwall (FW) and hanging-wall (HW) in the Northridge Earthquake (adapted from Abrahamson and Somerville 1996)......61

Table 3.3 Regression coefficients and Standard Error for spectral acceleration values at 5% damping for (a) Northridge Earthquake (b) Loma Prieta Earthquake.........................................................................................................62

Table 3.4 Standard deviations for the Northridge Earthquake compared with standard deviations from Somerville and Abrahamson (Somerville, personal comm.). Values of the standard deviation of the sample standard deviation are given in parenthesis. ....................................................................64

Table 3.5 Subdivision of sites classified according to the presented classification system by means of the 1997 UBC shear wave velocity-based classification system..........................................................................................65

Table 3.6 Comparison of standard errors at selected periods for an analysis based on the classification system presented herein and an analysis based on the 1997 UBC average shear wave velocity-based classification system. ..............66

Table 3.7 Spectral acceleration amplification factors with respect to Site B and standard deviations for corresponding soil type. (a) Geometric mean of the Loma Prieta and Northridge earthquakes. (b) Variance weighted mean of the Loma Prieta and Northridge earthquakes. ...................................................67

Table 3.8 Spectral acceleration amplification factors with respect to Site C and standard deviations for corresponding soil type. (a) Geometric mean of the Loma Prieta and Northridge earthquakes. (b) Variance weighted mean of the Loma Prieta and Northridge earthquakes. ...................................................69

Table 3.9 (a) Short-period (Fa) and mid-period (Fv) spectral amplification factors from the 1997 Uniform Building Code. (b) Average spectral amplification periods over the short-period range (0.1 s – 0.5 s) and the mid-period range (0.4 s – 2.0 s), denoted by Fa and Fv, respectively................71

Table 4.1. Modification to ground motion parameters to account for directivity effects. .............................................................................................................138

Table 4.2. Near-source factors from the 1997 Uniform Building Code (UBC).........139

Table 4.3. Earthquakes included in the study of near-fault ground motions. Fault parameters are obtained from Somerville et al. (1997). ..................................140

Table 4.4. Parameters used to define the simplified sine-pulse ground motions. ......141

Table 4.5. Stations included in the analysis of near-fault ground motions. ...............142

Table 4.6. Pulse parameter for stations included in this study. ..................................144

vii

Table 4.7. Ground motion recordings satisfying geometric requirements for forward-directivity conditions not included in this study. Listed are only stations for earthquakes with Mw ≥ 6.5 and R < 20 km...................................146

Table 4.8. Values of Tv-p/Tv for multiple-record events. ............................................147

Table 4.9. Number of half-cycle pulses (N) by event for the recordings considered in this study. Value in parenthesis is the number of half-cycle pulses that corresponds to a cut-off value of 33% of the PGV (as opposed to 50% used to define N)......................................................................................................148

Table 4.10. Parameters from the regression analyses for the period of the pulse of maximum amplitude, Tv, and the period corresponding to the maximum pseudo-velocity response spectral value, Tv-p (Equation 4.8)..........................149

Table 4.11. Inter-event error term from the random effects model for the attenuation relationship for pulse period.........................................................150

Table 4.12. Comparison of pulse periods between rock and soil stations for recordings in the Gilroy area in the 1989 Loma Prieta earthquake. ................151

Table 4.13. Parameters from the regression analyses for peak ground velocity (Equation 4.12)................................................................................................152

Table 4.14. Inter-event error term from the random effects model for the attenuation relationship for peak ground velocity. ..........................................153

Table 4.15. Ratios of pulse period in the fault parallel to pulse period in the fault normal direction. .............................................................................................154

Table 4.16. Ground motions included in the determination of typical pulse shapes. .............................................................................................................155

Table 5.1. Parameters needed for the model implementation. ...................................251

Table 5.2. Model and numerical parameters used for analysis of Lotung array. The shear-wave velocity profile is given in Figure 5.10. ................................252

Table 5.3. An example of the rate of convergence of the Residual Norm in the finite element implementation.........................................................................253

Table 5.4. Model and numerical parameters used in the analysis of the Chiba downhole array. The shear-wave velocity profile is given in Figure 5.16. ....254

Table 5.5. Model and numerical parameters used in the analysis of the Shaking Table tests. Shear-velocity profile is given in Figure 5.23.............................255

Table 5.6 Peak response values for the Shaking Table Runs. ....................................256

Table 6.1. Number of profiles for each Site Type (based on UBC classification scheme, Table A3) in the database of shear wave velocity profiles of Silva (personal comm.).............................................................................................319

Table 6.2. Model parameters used for the soils considered in the Stiff Soil profiles. The shear-wave velocity profile is given in Figure 6.1. ...................320

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Table 6.3. Model parameters used for the soils considered in the Soft Clay profiles. The shear-wave velocity and density profile is given in Figure 6.4....................................................................................................................321

Table 6.4. Input motions and results of site response analyses on recorded near-fault ground motions. ......................................................................................322

Table 6.5. Values of parameters used in the parametric study of site response to simplified pulse motions. ................................................................................323

ix

LIST OF FIGURES

Figure 2.1. PGA amplification in soft soils (from Idriss 1990). ..................................25

Figure 2.2. Spectral shapes for different site conditins (from Seed and Idriss 1983)..................................................................................................................26

Figure 2.3. Site Factors used in some current ground motion attenuation relationships for estimating spectral acceleration at 5% damping. ...................27

Figure 2.4. Site factors for varying levels of PGA from the Abrahamson and Silva (1997) spectral acceleration (5% damping) attenuation relationship. ...............28

Figure 3.1. Relationship between structural damage intensity and soil depth in the Caracas earthquake of 1967 (From Seed and Alonso 1974). ............................73

Figure 3.2a. Shear wave velocity versus depth for a generic stiff clay deposit. Shear wave velocity of underlying bedrock is 1220 m/s.c ................................74

Figure 3.2b. Spectral accelerations for the stiff soil deposit shown in Figure 3.2a, with PGA = 0.3 g for a Mw = 8.0 earthquake. ..................................................75

Figure 3.2c. Spectral acceleration amplification ratio for the stiff soil deposit shown in Figure 3.2a with PGA = 0.3 g for a Mw = 8.0 earthquake. The predominant period of the site is indicated by a circle. .....................................75

Figure 3.3a. Shear wave velocity profiles for generic sites. Shear wave velocity of underlying bedrock is 1220 m/s. ...................................................................76

Figure 3.3b. Spectral acceleration amplification ratio for the soil profiles shown in Figure 3b, with PGA = 0.3 g for a Mw = 8.0 earthquake. ............................76

Figure 3.4a. Distribution of data by site type for the Northridge Earthquake. .............77

Figure 3.4b. Distribution of data by site type for the Loma Prieta Earthquake............77

Figure 3.5. Number of recordings as a function of period. ...........................................78

Figure 3.6. Regression coefficients for the Northridge Earthquake. ............................79

Figure 3.7. Regression coefficients for the Loma Prieta Earthquake. The coefficient "c" is equal to 1.0 for all periods. ....................................................80

Figure 3.8. Comparison of response spectra before smoothing and after smoothing regression coefficients. Corresponds to the Northridge Earthquake at R = 10 km...................................................................................81

Figure 3.9. Response spectra for the Northridge Earthquake. Thick lines represent median values, thin lines represent ± one standard deviation............82

Figure 3.10. Response spectra for the Loma Prieta Earthquake. Thick lines represent median values, thin lines represent ± one standard deviation............83

Figure 3.11. Median spectral values vs. distance for the Northridge Earthquake........84

Figure 3.12. Median spectral values vs. distance for the Loma Prieta Earthquake......85

x

Figure 3.13. Comparison of results with an earthquake specific attenuation relationship by Somerville and Abrahamson (1998). Response spectra at 5% damping for the Northridge Earthquake at R = 20 km. ..............................86

Figure 3.14. Residuals with respect to regression analysis for Site C. All sites shown are classified as C sites in the classification system proposed in this study, but are differentiated with respect to their corresponding UBC classification based in the average shear wave in the upper 30 m. ...................87

Figure 3.15a. Residuals for Site C, Northridge Earthquake. Table gives mean of residuals for each subgroup...............................................................................88

Figure 3.15b. Residuals for Site C, Loma Prieta Earthquake. Table gives mean of residuals for each subgroup...............................................................................89

Figure 3.16a. Residuals for Site D, Northridge Earthquake. Table gives mean of residuals for each subgroup...............................................................................90

Figure 3.16b. Residuals for Site D, Loma Prieta Earthquake. Table gives mean of residuals for each subgroup...............................................................................91

Figure 3.17. Residuals for D sites within the Los Angeles Basin plotted as a function of depth to basement rock. ..................................................................92

Figure 3.18a. Amplification factors with respect to Site B for the Northridge Earthquake.........................................................................................................93

Figure 3.18b. Amplification factors with respect to Site C for the Northridge Earthquake.........................................................................................................94

Figure 3.18c. Amplification factors with respect to Site B for the Loma Prieta Earthquake.........................................................................................................95

Figure 3.18d. Amplification factors with respect to Site C for the Loma Prieta Earthquake.........................................................................................................96

Figure 3.19a. Amplification factors with respect to Site B. Geometric mean of the Northridge and Loma Prieta Earthquakes. ..................................................97

Figure 3.19b. Amplification factors with respect to Site C. Geometric mean of the Northridge and Loma Prieta Earthquakes. ..................................................98

Figure 3.20a. Amplification factors with respect to Site B. Weighted mean of the Northridge and Loma Prieta Earthquakes. ........................................................99

Figure 3.20b. Amplification factors with respect to Site C. Weighted mean of the Northridge and Loma Prieta Earthquakes. ......................................................100

Figure 3.21. Earthquake weighting scheme used for calculating spectral amplification factors. Shown here is an average of the weights used for all periods. Weights are inversely proportional to the sample variance..............101

xi

Figure 3.22. Short-period (Fa) and intermediate-period (Fv) spectral amplification factors. Dotted lines are code values (UBC 1997), and continuous lines are values obtained from this study. ................................................................102

Figure 4.1 Schematic diagram of rupture directivity effects for a vertical strike-slip fault. .........................................................................................................156

Figure 4.2. Definition of parameters used in defining rupture directivity conditions (adapted from Somerville et al. 1997). ..........................................157

Figure 4.3. Predictions from the Somerville et al. (1997) relationship for varying directivity conditions. .....................................................................................158

Figure 4.4. Simplified pulses used by other researchers. ...........................................159

Figure 4.5. Recorded and simplification of the dominant pulses for selected fault-normal velocity time-histories. Thick lines correspond to motions representing a simplification of the dominant pulses......................................160

Figure 4.6. Fault normal (FN) and fault parallel (FP) velocity time-histories and horizontal velocity-trace plots for two near-fault recordings. Both recordings have significant fault normal velocities, but the Meloland Overpass recording from the Imperial Valley earthquake has much lower fault parallel velocities than the West Pico Canyon Road record from the Northridge earthquake.....................................................................................161

Figure 4.7. Parameters needed to define the fault parallel and fault normal components of simplified velocity pulses. Subscripts N and P indicate fault normal and fault parallel motions, respectively. .....................................162

Figure 4.8. Comparison of two different measures of pulse period: weighted average period and period of maximum pulse (Tv). .......................................163

Figure 4.9. Determination of pulse period (Tv) for cases in which the pulse is preceded by a small drift in the velocity time-history. ..................................164

Figure 4.10. Determination of period corresponding to the peak pseudo-velocity response spectral value. The pseudo-velocity response spectra shown is for the Erzincan record of the Erzincan, Turkey, earthquake..........................165

Figure 4.11. Comparison of the period of the maximum pseudo-velocity response spectral value (Tv-p) with the period of the maximum pulse, Tv...................166

Figure 4.12. Normalized power spectral densities of velocity time histories for selected ground motions. The two ground motions on the left have lower Tv-p than Tv, while those on the right have roughly coinciding values of Tv-p and Tv.....................................................................................................167

Figure 4.13. Velocity time-histories for the 1966 Parkfield earthquake. The Temblor record corresponds to the fault normal direction. The seismograph at the Cholame 02 station recorded only acceleration in one direction that corresponds to 15° degrees from the fault normal direction. Dashed lines correspond to 50% and 33% of the PGV...................................168

xii

Figure 4.14. Fault normal velocity time-histories for the Pacoima Dam record in the 1971 San Fernando earthquake. Dashed lines correspond to 50% and 33% of the PGV. .............................................................................................169

Figure 4.15a. Fault normal velocity time-histories for the 1979 Imperial Valley earthquake. Dashed lines correspond to 50% and 33% of the PGV. .............170

Figure 4.15b. Fault normal velocity time-histories for the 1979 Imperial Valley earthquake. Dashed lines correspond to 50% and 33% of the PGV. .............171

Figure 4.16. Fault normal velocity time-histories for the 1984 Morgan Hill earthquake. Dashed lines correspond to 50% and 33% of the PGV. .............172

Figure 4.17. Fault normal velocity time-histories for the 1987 Superstition Hills earthquake. Dashed lines correspond to 50% and 33% of the PGV. .............173

Figure 4.18. Fault normal velocity time-histories for the 1989 Loma Prieta earthquake. Dashed lines correspond to 50% and 33% of the PGV. .............174

Figure 4.19. Fault normal velocity time-histories for the Erzincan record in the 1992 Erzincan, Turkey, earthquake. Dashed lines correspond to 50% and 33% of the PGV. .............................................................................................175

Figure 4.20. Fault normal velocity time-histories for the Lucerne record in the 1992 Landers earthquake. Dashed lines correspond to 50% and 33% of the PGV. ..........................................................................................................176

Figure 4.21a. Fault normal velocity time-histories for rock records from the 1994 Northridge earthquake. Dashed lines correspond to 50% and 33% of the PGV.................................................................................................................177

Figure 4.21b. Fault normal velocity time-histories for soil records from the 1994 Northridge earthquake. Dashed lines correspond to 50% and 33% of the PGV.................................................................................................................178

Figure 4.22. Fault normal velocity time-histories for records from the 1995 Kobe, Japan, earthquake. Dashed lines correspond to 50% and 33% of the PGV. ..179

Figure 4.23. Fault normal velocity time-histories for records from the 1999 Kocaeli, Turkey, earthquake. Dashed lines correspond to 50% and 33% of the PGV. ..........................................................................................................180

Figure 4.24. Attenuation relationship for pulse period (Tv). .....................................181

Figure 4.25. Comparison of results form regression analysis with relationships proposed by other researchers. The definition of pulse period of Somerville (1998) is similar to that used in this study (Tv). On the other hand, the pulse period of Alavi and Krawinkler (2000) is the period of the maximum pseudo-velocity response spectral value (Tv-p).............................182

Figure 4.26. Attenuation relationship for pulse period (Tv) showing one standard deviation band. ................................................................................................183

xiii

Figure 4.27a. Velocity time-histories and velocity-trace plots for sites in the Gilroy area recorded in the 1989 Loma Prieta earthquake. .............................184

Figure 4.27b. Pseudo-velocity response spectra for the sites in Figure 2.27a. Observe the significant difference in frequency content at long periods between the soil and the rock sites. .................................................................185

Figure 4.28. Distribution of near-fault sites considered in this study. .......................186

Figure 4.29. Attenuation relationship for PGV in the near-fault region. ...................187

Figure 4.29b. Comparison of results from regression analysis for PGV with relationships proposed by other researchers....................................................188

Figure 4.30. Dependence of ratio of PGV from soil to rock on magnitude and distance............................................................................................................189

Figure 4.31. Relationship between the ratio of fault parallel to fault normal peak ground velocity (PGVP/N) to fault normal peak ground velocity (PGV). Regression line and equation are for Rock sites..............................................190

Figure 4.32. Selected motions with one dominant half-cycle pulse (N = 1). .............191

Figure 4.33. Selected motions with a dominant full cycle of motion (N = 2)............193

Figure 4.34. Simplification of the dominant pulses in the recordings in Figure 4.33..................................................................................................................195

Figure 4.35a. Simplified sine-pulse representation of near-fault ground motions. The fault parallel PGV is set to 50% of the fault normal PGV. ......................197

Figure 5.1. Schematic representation of bounding surface plasticity model..............257

Figure 5.2. Fraction of critical damping versus frequency for Rayleigh damping. is the target fraction of critical damping. ........................................................258

Figure 5.3. Stress-strain loops for two different loading paths ..................................259

Figure 5.4. Schematic representation of site response problem. ................................260

Figure 5.5. Influence of hardening parameter h on modulus reduction and damping curves. R/Gmax = .02, m = 1, Ho = 0 (adapted from Borja and Amies 1994). ...................................................................................................261

Figure 5.6. Influence of hardening parameter m on modulus reduction and damping curves. R/ Gmax = .02, h/ Gmax = 1, Ho = 0 (adapted from Borja and Amies 1994). ..................................................................................262

Figure 5.7. Representative stress-strain loops at different amplitudes. .....................263

Figure 5.8. Finite element response when subjected to cyclic seismic loading R/Gmax = .02, m = 1, h/Gmax = 1, Ho = 001. The loading was the KJMA station in the Kobe earthquake. .......................................................................264

Figure 5.9. Influence of the slope of the rebound modulus in the model by Pestana and Lok...............................................................................................265

xiv

Figure 5.10. Shear wave velocity (Vs) profile for Lotung Site (Borja et al. 1999, EPRI 1993). .....................................................................................................266

Figure 5.11. Modulus degradation and damping curves for the Lotung site..............267

Figure 5.12. Comparison of recorded and computed ground motions at the surface for the Lotung array. ...........................................................................268

Figure 5.13. Recorded and computed acceleration time histories in the Lotung array. Thick lines are recorded accelerations. The input motion is at Elev. 47. ...................................................................................................................269

Figure 5.14. Recorded and computed velocity time histories in the Lotung array. Thick lines are recorded velocities. The input motion is at Elev. 47. ............270

Figure 5.15. Comparison of uni-directional and bi-directional analyses for the Lotung array. ...................................................................................................271

Figure 5.16. Shear wave velocity (Vs) profile for the Chiba site (Katayama et al. 1990)................................................................................................................272

Figure 5.17. Modulus degradation and damping curves for soils in the Chiba site. Thin lines correspond the curves predicted by the model. Model parameters are matched to curves from the indicated references. ...................273

Figure 5.18. Recorded and computed acceleration time histories in the Chiba array. Thick lines are recorded accelerations. The input motion is at Elev. 40. The accelerogram at 5 m depth did not trigger for this event. .................274

Figure 5.19. Recorded and computed velocity time histories in the Chiba array. Input motion is at Elev. 40. .............................................................................275

Figure 5.20. Acceleration response spectra for the analyses of the North-South component of motion in the Chiba downhole array. .......................................276

Figure 5.21. Strains predicted by SHAKE91 (Idriss and Sun 1992) and the GeoFEAP analysis for the North-South component of the Chiba downhole array. ..............................................................................................................277

Figure 5.22. Shear-wave velocity profile of the model clay soil used in the Shaking Table Test 2.53, including location of accelerograms.......................278

Figure 5.23. Modulus degradation and damping curves for model soil used in shaking table test. ............................................................................................279

Figure 5.24. Velocity response spectral for recorded and calculated motions in Test 2.16. .........................................................................................................280

Figure 5.25. Velocity response spectral for recorded and calculated motions in Test 2.53.. ........................................................................................................281


xv


Figure 6.1. Selected shear wave velocity profile for the Stiff Soil profile. Included in the graph are the median and ± one standard deviation for a database of 343 profiles (Silva, personal comm.) classified as Site D by the Uniform Building Code (e.g. 180 m/s < Vs ≤ 360 m/s)..................................324

Figure 6.2. Shear modulus degradation and equivalent viscous damping curves for the soils used in the Stiff Soil profile. .......................................................325

Figure 6.3. Dynamic undrained shear strength for the Stiff Soil profile. ...................326

Figure 6.4. Generic Soft Clay shear wave velocity and density profiles. The profile represents typical Bay Mud sites from the San Francisco Bay region...............................................................................................................327

Figure 6.5. Dynamic shear strength profile for the generic Soft Clay site. ................328

Figure 6.6. Shear modulus degradation and equivalent viscous damping curves for the Soft Clay soil. The Large-Strain model curve is given for the average strength value of the clay. The curves shift to the right for an increase in strength, and to the left for a decrease in strength.........................329

Figure 6.7. Shear modulus degradation and equivalent viscous damping curves for the Pleistocene clay used in the generic Soft Clay profile. The upper- and lower-bound model curve corresponds to the upper bound and lower bounds, respectively, for the strength parameter R. The upper bound curve is used when soil yielding is not of concern....................................................330

Figure 6.8. Selected shear wave velocity profile for the Very Stiff Soil profile (Site C). Included in the graph are the median and ± one standard deviation for a database including 227 profiles (Silva, personal comm.) classified as Site C by the Uniform Building Code (e.g. 360m/s < Vs ≤ 760 m/s)..................................................................................................................331

Figure 6.9. Power spectral densities, PSD (normalized by the maximum value of the PSD) for the ground motions used in the site response analyses. PSD are given for the full recorded ground motion, for the time interval of the ground motion containing the velocity pulse, and for the simplified representation of the velocity pulse.................................................................332

Figure 6.10. Site response analyses results. Pacoima Dam record for the 1971 San Fernando Earthquake................................................................................333

Figure 6.11. Site response analyses results. Gilroy Gavilan College record for the 1989 Loma Prieta Earthquake. ........................................................................334

Figure 6.12. Site response analyses results. Pacoima Dam downstream record for the 1994 Northridge Earthquake. ....................................................................335

Figure 6.13. Site response analyses results. KJMA record for the 1995 Kobe Earthquake.......................................................................................................336

xvi

Figure 6.14. Site response analyses results. Input (rock) and output (soil) particle velocity-trace plots. .........................................................................................337

Figure 6.15. Fault-normal acceleration and velocity time-histories for the Pacoima Dam record of the 1971 San Fernando earthquake. The dominant velocity pulse is highlighted to show that the maximum accelerations do not coincide with the fault-normal velocity pulse. ..........................................338

Figure 6.16. Calculated strains using the recorded velocity pulse and the simplified velocity pulse as input motions. .....................................................339

Figure 6.17. Response of a linear-elastic soil column subject to the simplified velocity pulses. The natural period of the soil deposit is 1.0 s.......................340

Figure 6.18. Amplification of input motions by a linear-elastic profile (ω is the frequency of the input motion, Vs is the shear wave velocity of the elastic soil, and H is the soil depth (from Kramer 1996)............................................341

Figure 6.19. Results from the site response analyses for the 60 m deep Stiff Soil profile. Particle velocity-trace plots for the input (rock) and output (soil) motions. Input motion set is Set 6. PGVp/n of input motion is 0.5. .............342

Figure 6.20. Results from the site response analyses for the 60 m deep Stiff Soil profile. Particle velocity-trace plots for the input (rock) and output (soil) motions. Input motion set is Set 7. PGVp/n of input motion is 0.5. .............343

Figure 6.21. Results from the site response analyses for the 60 m deep Stiff Soil profile. Particle velocity-trace plots for the input (rock) and output (soil) motions. Input motion set is Set 8. PGVp/n of input motion is 0.5. ............344.

Figure 6.22. Results of site response analyses for the 60 m deep Stiff Soil site for three different input motion sets. Results are plotted with fault normal input pulse period in the abscissa. ...................................................................345

Figure 6.23. Site response analyses results for the 30 m deep Stiff Soil profile. The input motion is Set 6, with a PGVp/n ratio of 0.5. ..................................346

Figure 6.24. Comparison of site response studies for two profiles with equal depth to bedrock (30 m) and varying profile stiffness. Input motion is Set 6 with a PGVp/n ratio of 0.5. ..........................................................................347

Figure 6.25. Comparison of site response analyses results for the Stiff Soil profile. The depth to bedrock is varied from 30 m to 200 m. The input motion is Set 6 with a PGVp/n ratio of 0.5. ....................................................348

Figure 6.26. Ratio of pulse period in soil to pulse period in rock for site response analyses on Stiff Soil profiles with varying depth to bedrock. The input motion is Set 6 with a PGVp/n ratio of 0.5. ....................................................349

Figure 6.27. Comparison of site response analyses for soils with different shear modulus reduction and damping curves. Results for input motion Set 6 with a PGVp/n ratio of 0.5. .............................................................................350

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Figure 6.28. Comparison of calculated ratio of soil pulse period to rock pulse period. Results for site response analyses for soils with different shear modulus reduction and damping curves. Input motion is Set 6, with PGVp/n ratios of 0.5........................................................................................351

Figure 6.29. Comparison of site response analyses results to input motions with varying levels of fault parallel velocity. Results for a 45 m deep Stiff Soil for PGVp/n ratios of 0, 0.25 and 0.75. Input fault-normal PGV is 75 cm/s...352

Figure 6.30. Comparison of site response analyses to input motions with different fault-parallel velocities. Results for the 45 m deep Stiff Soil profile, Input motion Set 6. ...................................................................................................353

Figure 6.31. Comparison of site response analyses results for the different input motion sets. The analyses are for the Stiff Soil profiles with varying depth. The input PGVp/n ratio is 0.25 for Sets 7 and 8, and 0.5 for Set 6.....354

Figure 6.32. Comparison of strain levels for runs with (a) equal PGV, and (b) equal PGA. Stiff Soil profile with a depth of 45 m. Input motion is Set 6 with PGVp/n = 0.5. .........................................................................................355

Figure 6.33. Calculated strains for various Stiff Soil profiles. Input motion is Set 6 with PGVp/n ratios of 0.5, and input PGV of 160 cm/s...............................356

Figure 6.34. Results of site response analyses. Particle velocity-trace plots of input and output motions. Soft Clay site, input motion is Set 6 with PGVp/n ratios of 0.5........................................................................................357

Figure 6.35. Results of site response analyses. Particle velocity-trace plots of input and output motions. Soft Clay site, input motion is Set 7 with PGVp/n ratios of 0.25......................................................................................358

Figure 6.36. Results of site response analyses. Particle velocity-trace plots of input and output motions. Soft Clay site, input motion is Set 8 with PGVp/n ratios of 0.25......................................................................................359

Figure 6.37. Calculated stresses and strains for the Soft Clay profile. Input motion is Set 6 with PGVp/n = 0.5 .................................................................360

Figure 6.38. Calculated profiles of displacement, velocity, and acceleration for the Soft Clay site. Input motion is Set 6 with PGVp/n of 0.5 and input PGV of 160 cm/s. ............................................................................................361

Figure 6.39. Comparison of results of site response analyses for two profiles with equal depth to bedrock but different soil profiles. Ratios of PGV in soil to PGV in rock. Input motion is Set 6 with PGVp/n of 0.5................................362

Figure 6.40. Comparison of site response analyses for the Soft Clay profile and the Stiff Soil profile with equal depth (60 m). Results shown are for all input motion sets with PGVp/n values ranging from 0.25 to 0.5....................363

xviii

Figure 6.41. Comparison of results of site response analyses for two profiles with equal depth to bedrock but different soil profiles. Ratios of pulse period in soil to pulse period in rock. Input motion is Set 6 with PGVp/n of 0.5. ........364

Figure 6.42. Effect of site response on the PGVp/n ratio. Results for three different profiles and for input motion sets 7 and 8. PGV of input motion is 75 cm/s.........................................................................................................365

Figure 6.43. Calculated amplification of peak ground velocity. Results for site response analyses for all the Stiff Clay profiles and all the simplified velocity poulse input motion sets. Large symbols are the predicted PGV amplification for the Stiff Soil site with a depth of 45 m using the indicated records as input motions. .................................................................366

Figure 6.44. Relationship between PGV in soil and PGV in rock. Results for the 45 m deep Stiff Soil deposit. Input motion is Set 6 with PGVp/n ratio of 0.5. Observe the dependence of PGV soil to rock ratio on pulse period (Tvp) and intensity of motion (PGVrock). ......................................................367

Figure 6.45. Calculated ratios of pulse period in the soil over pulse period in rock. Results from site response analyses on Stiff Clay profiles using all simplified velocity pulse input motions sets and PGVp/n ratios ranging from 0.25 to 0.75. ..........................................................................................368

Figure 6.46. Calculated ratio of pulse period in soil to pulse period in rock for the 45 m deep Stiff Soil profile. Results are shown for all the input motion sets with input PGV lower than 300 m/s. Results for the indicated ground motions are included for comparison with the results of the simplified velocity pulses. ................................................................................................369

xix

ACKNOWLEDGMENTS

This work would not have been possible without the help of a large number of

individuals to whom I feel deeply indebted. First, I would like to thank my advisor,

Professor Jonathan D. Bray, for his advice, support, and friendship. He has contributed to

this work by providing innumerable ideas and exciting intellectual challenges. Most of

all, I greatly appreciate his mentorship and his support throughout my doctoral studies.

I would also like to thank Professor Juan Manuel Pestana for reviewing this

dissertation, and especially for his teachings and insights on soil behavior; and Professor

Alexandre J. Chorin for reviewing this dissertation and allowing me to feed from his

great knowledge of numerical methods. My years at U.C. Berkeley have been the greatest

intellectual experience of my life. My deepest thanks go to the faculty in the Civil

Engineering Department for fostering the environment that made this intellectual

experience possible. In particular, I would like to mention Professor Nicholas Sitar for

his support during my first year in the doctoral program.

The discussions during meetings with Dr. Norman A. Abrahamson of Pacific Gas

and Electric Company planted the seeds for most of the research presented in this work.

In addition, I would also like to extend my thanks to Professor Robert L. Taylor for his

invaluable comments regarding time integration schemes and the finite element method;

Professor Govindjee for his guidance on constitutive model implementation; Dr. Walter

Silva for his assistance in providing the ground motion database used in this study; Dr.

François E. Heuze of the Lawrence Livermore National Laboratory for his assistance in

sharing their geotechnical database; Dr. Paul Somerville for providing event-specific

xx

attenuation relationships for the Northridge and Loma Prieta earthquake and for providing

useful comments on the near-fault studies in this work; Prof. Raymond B. Seed for his

insights into site response; Dr. Mladen Vucetic, Dr. Sands Figuers, and Dr. David Rogers

for providing essential geotechnical data for ground motion sites; Dr. Tomoya Iwashita

for his help in obtaining data from the Kobe region; Dr. Jonathan Stewart for the

information on the Oakland BART Station site and numerous sites in Southern

California, for the ground motion data of forward-directivity parameters, and for the

many instructive discussions; Dr. Susan Chang for her willingness to share knowledge

and information on ground motion site classification; Dr. Ellen Rathje for providing the

Kocaeli records; Dr. Philip Meymand, Dr. Thomas Lok and Dr. Michael Riemer for their

help in using the U.C. Berkeley Shaking Table test data; Mr. Ignacio Romero for

providing a linear-elastic brick element and helping me in the finite element

implementation; Mrs. Laurie Gaskins and Professor Steve Glaser for the data from

downhole arrays; Mrs. Catherine O'Sullivan for reviewing this manuscript; Dr.

Christopher Hunt for his jokes and his help using Matlab; and Dr. Laurent Luccioni for

his willingness to come to my help anytime it was needed.

I would also like to extend my thanks to my classmates and friends for their

friendship, support, and for all the great soil talks we had over the years. Each of them

has contributed to making the years in Berkeley not only instructive, but also fun. Even if

I were to try to list them all, I would invariably fail to mention all of them; their names

might be unlisted here, but their friendship will always remain alive.

Financial support was provided by the Pacific Gas and Electric Company, the

David and Lucile Packard Foundation, and the Pacific Earthquake Engineering Center.

xxi

My thanks go to my fellow researchers in PEER for their instructive comments at

research meetings.

Finally, I would like to acknowledge the love and support of my family. I would

not be here if it were not for my parents' support. I would not be the same if it were not

for my brothers' friendship. I would not have finished if it were not for my wife's love.

My greatest thanks go to her.

1

CHAPTER 1

INTRODUCTION

1.1 PROBLEM

A number of major metropolitan centers lie in close proximity to major faults.

For example, in the United States, the San Andreas fault system crosses densely

populated areas of the metropolitan Los Angeles and Bay Area regions, while other active

faults lie beneath cities in the Pacific Northwest. The proximity of these fault systems to

large population centers motivates research seeking a better understanding of ground

motions close to faults.

The estimation of ground motions in zones that are close to the causative faults of

medium to large magnitude earthquakes should account for the special characteristics of

near-fault ground motions. Of particular importance in the near-fault region are the

effects of forward-directivity. Forward-directivity conditions produce ground motions

characterized by a strong pulse or series of pulses of long period motions. In recent years,

many efforts have been devoted to the characterization of forward-directivity motions

(e.g., Somerville et al. 1997, Krawinkler and Alavi 1998, Sasani and Bertero 2000).

These studies have highlighted the important contribution of near-fault ground motions to

the seismic risk of structures. Building on this understanding, more research is needed to

account for the potential effects of local soil conditions on these motions.

2

The seismic site response problem has been studied extensively in recent years,

both by empirical analysis of the existing database of ground motions (e.g., Borcherdt

1994, Crouse and McGuire 1996) and by means of analytical site response tools (e.g.,

Seed et al. 1991, Dobry et al. 1994). The empirical characterization of site response is

constrained by the availability of data. Until the 1994 Northridge, 1995 Kobe, 1999

Kocaeli, and especially the 1999 Chi-Chi earthquakes, very few recordings at close

distances to the causative fault were available for study. Consequently, most of the

empirical study of site response has been concentrated on ground motions recorded at

more than 15 km from the zone of energy release. The same unavailability of near-fault

data also caused the direction of seismic site response studies to be driven towards

analyzing site response to "normal" ground motions, that is, ground motions recorded at

intermediate to long distances from the zones of energy release.

Near-fault ground motions, and in particular forward-directivity motions, have

specific characteristics that differentiate these motions from those recorded at

intermediate to long distances from the zone of energy release. These particular

characteristics raise the question of whether the current understanding of site response

applies directly to near-fault ground motions. This dissertation aims at helping to resolve

this question and as a consequence, provide a better understanding of the effect of local

site conditions on near-fault ground motions. In addition, a new site classification

scheme is introduced for the evaluation of site response at intermediate distances from the

zone of energy release. This classification scheme is used to evaluate data from the 1989

Loma Prieta and 1994 Northridge earthquakes.

3

1.2 SIGNIFICANCE

The relevance and importance of correctly characterizing seismic site response

and in particular local soil condition effects cannot be over-stated. Local soil conditions

have played a major role in the damage and loss of life in earthquakes such as the 1985

Mexico City, 1989 Loma Prieta, 1994 Northridge, and 1995 Kobe earthquakes.

Moreover, large population centers across the globe can potentially be struck by an

earthquake originating on nearby faults. A better understanding of the mechanisms that

control near-fault ground motions and the interaction of near-fault ground motions with

soil deposits is imperative to evaluate better seismic risk in these regions. This

understanding, in turn, will permit the development of adequate code-based regulations

that ensure the safety of engineered structures.

The assessment of seismic demand on a structure within the framework of

performance-based design requires not only an estimation of the median expected levels

of ground motion intensity, but also the standard error associated with this estimated

median. A complete assessment of site conditions, if sufficient data are available, should

reduce the uncertainty associated with the predictions of ground motions. However, a

fine balance is necessary when dealing with sparse data sets, such as the ground motion

database. Site classification schemes that are too elaborate would result in a data set that

is too small to yield statistically meaningful results. On the other hand, a site

classification scheme that is too simplistic might not capture trends in ground motions

that could significantly reduce the uncertainty associated with ground motion prediction.

The use of adequate classification schemes should result in better estimates of ground

4

motion for engineering design. A reduction in uncertainty associated with better site

classification schemes could also lead to better risk assessment methodologies useful for

the private sector (e.g., insurance companies), as well as the public sector (e.g., advanced

emergency planning).

A better understanding of site amplification and, in particular, site amplification in

the near-fault region, is useful beyond its application in the estimation of seismic demand

on structures. The recent earthquakes in Kocaeli, Turkey, and Chi-Chi, Taiwan, have

increased the available database of strong motions recordings by an order of magnitude,

particularly in the near-fault region. Undoubtedly, the larger database will result in a

push for new attenuation relationships that can directly account for near-fault effects. A

proper understanding of the influence of local site conditions on ground motions will be

of paramount importance in the evaluation of the current database of strong motions

recordings. This dissertation will contribute toward this goal by providing both a

qualitative and a quantitative evaluation of seismic site response to near-fault ground

motions.

1.3 OBJECTIVE

The objectives of this dissertation are to address the site response problem in

general, and the site response to near-fault ground motions in particular. These objectives

can be summarized as follows:

5

• Development and evaluation of a site classification scheme that accounts for both

profile depth and material stiffness. This classification scheme will be used to

evaluate the effect of profile depth on site response.

• Evaluation of the empirical database of near-fault ground motions, with emphasis on

the development simplified pulses that can be used for seismic site response analyses,

and on the evaluation of differences in ground motions recorded on rock from those

recorded on soil.

• Evaluation of the implementation of a dynamic constitutive model into a finite

element computer code for the analysis of site response to near-fault ground motions.

• Quantification of trends in site response to near-fault ground motions by means of

finite element analysis of the dynamic response of different site profiles.

1.4 OUTLINE

This dissertation is organized into seven chapters. The first chapter outlines the

problem of near-fault seismic site response and presents the motivating factors that led to

this dissertation. Chapter Two presents a brief review of the existing literature on site

response with particular emphasis on current methodologies used to solve the site

response problem.

Chapter Three consists of an exhaustive analysis of site response for ground

motions recorded during two important recent earthquakes, the 1989 Loma Prieta and

1999 earthquakes. This chapter presents and evaluates a site classification scheme that is

used to evaluate the effect of profile depth on reducing the uncertainties associated with

6

ground motion prediction. Site amplification factors based on the classification scheme

are developed from ground motions recorded during the 1989 Loma Prieta and 1994

Northridge earthquakes. It is found that the database does not include many ground

motion recordings in the near-fault region and is applicable only to sites at intermediate

distances from the zone of energy release.

Chapter Four presents an empirical analysis of the database of motions that are

located in the near-fault and that experienced forward-directivity effects. The study

concentrates on developing a parameterization of this type of motions. Differences

between motions recorded in rock and in soil are analyzed and the observed trends are

discussed.

Chapter Five contains the description of a finite element implementation of a

constitutive model used to solve the site response problem. The constitutive model used

is first presented in detail. The finite element implementation is then validated through a

series of case studies, with an emphasis on its application to near-fault ground motions.

Chapter Six presents a series of site response analyses using the soil model and its

finite element implementation presented in Chapter Five. The analyses are performed for

recorded near-fault motions, as well as for simplified motions developed from the

empirical analysis in Chapter Four. The trends in seismic site response noted in Chapter

Four are used to validate the results obtained in this chapter.

Chapter Seven presents a summary of the results and the findings resulting from

this work. It includes recommendations for future research.

7

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

The present study involves contributions from a variety of disciplines. The

empirical study of site response in Chapters Three and Four behooves an up-to-date

assessment of many of the issues relating to site response and an understanding of the

state of the art in attenuation relationships. A significant amount of research on near-fault

ground motions has been published in the seismological literature, and many of the issues

raised in the analysis of near-fault site response (Chapters Four and Six) are intrinsically

tied to this research. Additionally, the presentation of the finite element implementation

of the soil model utilized to analyze the site response problem feeds from a great amount

of work in finite element technology and the development of constitutive models of soil

response to dynamic loading. A comprehensive review of all these areas is beyond the

scope of this research project. Attempts to cover such a variety of topics in a

comprehensive manner would undoubtedly fail to cover adequately all of the important

research performed in each of these individual areas of study. In this chapter, only a brief

synopsis of recent work in seismic site response analysis and the treatment of site

amplification by current ground motion attenuation relationships are presented. The

objective is to provide the reader with a general knowledge of the foundations upon

which this work is grounded.

8

2.2 SITE RESPONSE

2.2.1 Site Response Studies and Code Development

The influence of local site conditions on ground motions has been observed since

the early days of earthquake engineering. Observations from as early as the 1800s exist in

the literature indicating the effects of local geology on ground motions (EPRI 1993).

Traditionally, site amplification had been studied by seismologists as part of the larger

problem of seismic wave propagation (e.g., Sezawa and Kanai 1932, Kanai 1950,

Thompson 1950, Haskell 1960). Seismologists have traditionally treated soil as a linear

material and rarely considered soil nonlinearity in their assessment of site conditions

(Finn 1991). The pioneering work of Seed and Idriss (1969) brought attention to the

nonlinear behavior of soils during seismic shaking. This work stemmed form

observations of the earthquakes in Niigata and Alaska in 1964, and the 1967 Caracas

earthquake. Since then, site response has become an integral part of geotechnical

earthquake engineering.

Following the pioneering work of the late Professor H. B. Seed and his

colleagues, site response has been studied in a large number of earthquakes since the

1960s. The 1985 Michoacan Earthquake is particularly significant due to the large levels

of spectral amplification recorded in the soft lakebed sediments in Mexico City. Prior to

the 1985 Michoacan earthquake, soft soils were thought to de-amplify motions at peak

ground accelerations (PGAs) larger than 0.1 to 0.2 g (e.g., Seed and Idriss 1983), while

motions at stiff soils were thought to be largely unaffected by the ground motion

intensity. However, the response of the high plasticity Mexico City clays was more linear

9

than expected for the observed levels of ground motion. This resulted in a reassessment

of the amplification of PGAs in soft ground, as reflected in Figure 2.1 (Idriss 1990). The

Mexico City earthquake also brought attention to the need for a better understanding of

the dynamic properties of soft clays (Finn 1991).

The development of design codes has followed the advancements in

understanding of site response. Seed et al. (1976) and Mohraz (1976) developed average

spectral shapes for various soil conditions based on a statistical study of more than 100

records from 21 earthquakes (Figure 2.2). While the spectral shapes in Figure 2.2 are

relatively independent of site condition at short periods, they are largely site dependent at

periods longer than 0.5 s. This led to the adoption of simplified response spectra shapes

by the Applied Technology Council (1978) and later, with some variations, by the

Uniform Building Code (1988). The use of spectral shapes without amplification factors

for peak acceleration reflected the observations by Seed et al. (1976) that accelerations in

soils and rocks were approximately equal.

The 1989 Loma Prieta earthquake provided a large database of strong ground

motion recordings in a variety of site conditions. This database was analyzed by a

number of authors (e.g., Silva and Stark 1992, Borcherdt 1994, Dickenson 1994, Chang

and Bray 1995, Boatwright and Seekings 1997, among others). Borcherdt (1994)

introduced a classification system based uniquely on the average shear wave velocity of

the upper 30 m of the profile at the site, sV . This classification system was used to

develop spectral amplification factors with respect to a baseline condition. The

amplification factors were obtained by performing a statistical regression analysis on the

10

Loma Prieta data using a continuous function of sV . These factors are applicable mainly

to low intensity motions as most of the recordings of the 1989 earthquake were located at

significant distances (i.e. > 20 km.) from the causative fault.

The Loma Prieta earthquake, following in the wake of the devastating 1985

Michoacan earthquake, served as a catalyst for the update of building codes. This effort

was initiated in a workshop convened by the National Center for Earthquake Engineering

Research (now MCEER: Multidisciplinary Center for Earthquake Engineering Research)

in 1991 (Whitman 1992). At a later workshop in Los Angeles in 1992 (Martin 1994), a

consensus was reached that resulted in the adoption of the 1994 National Earthquake

Hazard Reduction Program Provisions (NEHRP 1994), which later were adopted by the

1997 UBC, and are unaltered in the 2000 International Building Code (Dobry et al. 2000).

The 1994 NEHRP provisions were based on the shear wave velocity based site

classification system by Borcherdt (1994). The low intensity amplification factors for the

Loma Prieta earthquake developed by Borcherdt (1994) were extrapolated to larger

intensities using linear and nonlinear site response analyses (Seed et al. 1991, Dobry et al.

1994). The design spectra in the 1994 NEHRP Provisions differ from the previous codes

in that two separate site-dependent coefficients are used to represent the spectral response

over the short period and long period spectral region. This step is taken as recognition of

the potential differences in amplification between the short and the long period regions of

the spectra. An extensive review of the development of code provisions up to the 1997

NEHRP code is presented in Dobry et al. (2000). Further discussion of these is also

included in Chapter Three.

11

Crouse and McGuire (1996) used an extended database to validate the site factors

adopted in the 1994 NEHRP provisions. They performed a regression analysis on sites C

and D in the UBC classification scheme (See Table A3 in Chapter Three) using a

weighting scheme proposed by Campbell (1990), which gave the same total weight to

recordings from each earthquake within each magnitude and distance interval. The

regression lines for sites C and D where then scaled to fit data of sites B and E using a

least squares fit. The scaling factor was assumed to be independent of ground motion

intensity. The authors concluded that the extended database supports the factors

implemented in the NEHRP provisions. However, the large degree of nonlinearity of

short period amplification factors in the NEHRP Provisions was not observed in the

larger database.

The use of the average shear wave velocity over the upper 30 m of a profile was

reviewed by a number of authors (Day 1996, Anderson et al. 1996, Darragh and Idriss

1998). Darragh and Idriss (1998) performed equivalent-linear analysis on the Gilroy

Array in Northern California with records from the Loma Prieta earthquake. The authors

concluded that the use of sV is justified for input levels near 0.45 g. Anderson et al.

(1996) performed wave propagation analyses on horizontal elastic layers with depths

extending to 5 km and concluded that the upper 30 m have a large influence on

amplifications of peak amplitudes and root mean square acceleration. However, the

ground motions at the surface also depend on the attenuation structure of deeper deposits.

The site classification scheme adopted in the 1994 NEHRP Provisions does not

directly include soil depth as a classification parameter, as did earlier codes with the

12

direct use of site period for site classification (UBC 1976). Studies using generalized soil

profiles with varying depth to bedrock have indicated the important effect of soil depth in

amplification factors (EPRI 1993, Silva 1998). Chang and Bray (1995) and Chang et al.

(1997) observed that the ground motions at deep stiff soil sites were larger than those

predicted by the corresponding sV -based site class.

2.2.2 Cyclic Soil Models and Site Response Methodologies

Ever since the nonlinear behavior of soil was recognized as an important factor in

site response, the development of cyclic soil models used to represent the dynamic

response of soil has attracted a great deal of attention from engineers. Fundamental to the

development of cyclic soil models was the improvement in the understanding of cyclic

response of soil through laboratory experimentation (e.g., Seed and Idriss 1970, Hardin

and Drnevich 1972, Seed et al. 1984, Sun et al. 1988). In later years, the improvement of

local strain measurement techniques has increased the understanding of the small-strain

behavior of soils. A review of recent work in dynamic testing is presented in Bray et al.

(1999). For a compilation of testing data on small strain soil properties see Lanzo and

Vucetic (1999). More recently, data from downhole arrays has also been used to

characterize the nonlinear stress-strain behavior of soils (Zeghal et al. 1995). The use of

downhole array data has the advantage of avoiding sample disturbance problems, but

introduces additional problems such as definition of boundary conditions.

The site response problem is generally solved either by equivalent-linear analysis

(Seed and Idriss 1969), or by a fully nonlinear approach. The site response code used

most frequently in engineering practice is the equivalent-linear program SHAKE

13

(Schnabel et al. 1972, Idriss and Sun 1992). Details on the equivalent-linear method are

given in Chapter 5. In the following paragraph, some of the site response codes and

cyclic soil models in current use are summarized.

Lok (1999) classifies nonlinear models into three separate classes: Mechanical, in

which soil behavior is represented by single mechanical elements, such as springs,

dashpots, and sliders, placed in series or in parallel; empirical, which use empirically-

derived functions to fit the nonlinear stress-strain behavior of soils; and plasticity models,

based on the framework of plasticity theory.

Mechanical models are commonly of the type described by Iwan (1967). Iwan-

type models consist of elastoplastic elements placed either in series or in parallel. Each

element consists of a linear spring and a Coulomb slider. Any stress-strain curve can be

described within a specified degree of accuracy by increasing the number of elastoplastic

elements. The series model, more apt to describe strain in terms of stress, was used by

Joyner and Chen (1975) and the parallel model, apt for strain-driven problems, was used

by Taylor and Larkin (1978).

Empirical models that are commonly used include the Ramberg-Osgood

(Ramberg and Osgood 1943), the Davidenkov, and the hyperbolic model (Kondner 1963).

These models are typically used to describe first loading and the backbone curve. The

hysteresis loops observed in cyclic tests of soil can be easily constructed from the

backbone curve by the use of the Masing rules. These rules are stated as (Pyke 1979): (1)

The shear modulus on each loading reversal assumes a value equal to the initial tangent

modulus for the initial loading curve; and (2) the shape of the unloading or reloading

14

curves is the same as that of the initial loading curve. Pyke (1979) discusses the

extension of monotonic models into cyclic models and suggests adding two more

Masing-type rules to adequately represent cyclic soil behavior: (3) The unloading and

reloading curves should follow the initial loading curve if the previous maximum shear

strain is exceeded; and (4) if the current loading or unloading curve intersects the curve

described by a previous loading or unloading curve, the stress-strain relationship follows

that previous curve (Newmark and Rosenblueth 1971, Rosenblueth and Herrera 1964).

Alternatively, stress-strain curves compatible with the Masing rules are obtained using

plasticity based models or appropriate interpolation rules (Pyke 1979).

Empirical models are used in a number of implementations of site response

analyses. CHARSOIL (Streeter et al. 1974) uses the Ramberg-Osgood model and solves

the site response problem using the method of characteristics. Martin and Seed (1982)

used both the Ramberg-Osgood and the Davidenkov model to perform one-dimensional

ground response analyses. A hyperbolic model is used by Lee and Finn (1978, 1991) in

the program DESRA. A variation of the constitutive model by Lee and Finn (Matasovic

and Vucetic 1993) is used in the program D-MOD. Pyke (1992) uses a hyperbolic model

and the Cundall-Pyke hypothesis (Pyke 1979) in the finite difference code TESS.

Plasticity-based models provide the most flexibility in representing details of soil

behavior, including yielding, pore pressure generation, and soil response to multi-

directional loading paths. However, plasticity-based models are not always amenable to a

robust numerical implementation. Since the inception of critical state soil mechanics

(Schofield and Wroth 1968), critical state models have been used successfully to model

15

monotonic soil behavior. Plasticity based models have also been incorporated into site

response procedures (e.g., Scott 1985, Finn 1988, Li et al. 1998, Borja et al. 1999).

Typically, plasticity models are implemented in finite element or finite difference codes.

The basic elements of a plasticity formulation are (Prévost 1977): (1) the yield

condition, specifying states of stress for which plastic flow occurs; (2) a flow rule,

indicating the direction of the stress increment for a given plastic strain increment; and

(3) a hardening rule, specifying the evolution of the loading surface. In addition, the

conditions for loading and unloading from the yield surface must be specified (Simo and

Hughes 1998), as well as definition of the soil response before the state of stress reaches

the yield condition (typically elastic). In cyclic models, condition (3) must ensure that

plastic strains accumulate immediately after stress reversal, as observed in laboratory

cyclic tests. Prévost (1977) proposed the use of a combination of isotropic and kinematic

plastic hardening rules to represent soil response. Additionally, Prévost (1977) uses a

field of shear moduli (originally proposed by Mróz 1967) defined in stress space by a

collection of nested yield surfaces that define regions of constant shear moduli. The

multiple yield surface model was initially proposed in connection to metal plasticity by

Mróz (1967). The innermost surface in the collection of nested surfaces can be

represented as a single point in order to model the plastic strains observed in soils at the

onset of loading. On the other hand, the outermost surface plays the role of a failure

surface, outside which states of stress are not possible. The model by Prévost (1977)

permits the modeling of essential elements of cyclic soil behavior, such as small strain

nonlinearity and hysteretic dissipation of energy. It also automatically satisfies the

extended Masing rules (Pyke 1979). The model by Prévost is applied in the site response

16

program DYNA1D (Prévost 1989). Other multiple yield surface models were proposed

by Mróz, Norris, and Zienkiewicz (1978) and Hirai (1987). Mróz et al. (1981) presented

an improvement of the multi-surface model by introducing an infinite number of nested

loading surfaces. In this way, the plastic moduli were made continuous in the whole

domain enclosed by the outer surface.

Small strain nonlinearity can also be described through what is known as

Bounding Surface Plasticity. The bounding surface plasticity concept was independently

proposed by Dafalias and Popov (1975, 1976) and Krieg (1975). Bounding surface

plasticity rests on the idea that any stress point inside a bounding surface has a unique

'image' point on the surface, defined by means of a specific rule. The instantaneous value

of the plastic modulus depends on the distance between the stress point and its 'image'

(Dafalias and Herrmann 1982). Bounding surface plasticity models have been used by a

number of authors, including Dafalias and Herrmann (1982), Dafalias (1986), Bardet

(1989), Mróz et al. (1979), Borja and Amies (1994), Li et al. (1998), and others. These

models vary in the definition of the elastic region or in the type of bounding surface used.

Bounding surface models incorporated into site response analysis include SPECTRA

(Borja et al. 1999) and SUMDES (Wang et al. 1990).

Other models used to describe cyclic soil behavior are based on the concept of

hypoplasticity and endochronic theory. Hypoplasticity provides general expressions for

the constitutive tensors satisfying all necessary requirements (Zienkiewicz et al. 1999).

Darve and Labanieh (1982) suggested that the constitutive tensor can be interpolated from

constitutive tensors defined along specified loading paths. Other hypoplasticity models

17

include the model by Dafalias (1986). This model, mentioned earlier along with the

bounding surface plasticity models, presents extensions to the bounding surface plasticity

model within the framework of hypoplasticity. Endochronic models describe nonlinearity

by a sequence of events leading to successive states of the material (Finn 1982).

Endochronic theory was developed by Valanis, and applied to liquefaction problems by

Bazant and Krizek (1976) and Zienkiewicz et al. (1978).

Several of the plasticity models listed in the above paragraphs are able to account

for pore pressure generation leading to liquefaction in loose soil deposits. The

liquefaction problem, however, requires the use of models able to reproduce contractive

behavior of soils, and in some cases, performed coupled mechanic-pore pressure

dissipation analysis. For a review of such models, see Zienkiewicz et al. (1999).

The ability of constitutive models in representing cyclic soil behavior can be

evaluated by comparing model predictions to laboratory data. Typically, model

predictions of shear modulus reduction and damping curves can be compared to

laboratory curves. In many cases, model parameters that are used to match shear moduli

curves fail to adequately match damping curves. Lok (1999) presents an enhancement of

a hysteretic model introduced by Whittle and Kavvadas (1994, Pestana and Whittle 1999)

within a generalized plasticity model. This model permits a simultaneous match of shear

modulus reduction and material damping curves. Some details of the model are given in

Chapter Five.

18

2.3 SITE EFFECTS IN ATTENUATION RELATIONSHIPS

Attenuation relationships are used widely to predict ground motion levels for risk

assessment and seismic design. The use of attenuation relationships permits a more

flexible assessment of seismic hazard in design as opposed to the fixed levels of 2% and

10% probability of occurrence in fifty years traditionally used in the codes. Moreover,

within the context of probabilistic seismic hazard assessment, the estimate of the level of

uncertainty of a ground motion is as important as the prediction of its median values.

Attenuation relationships vary mainly from the database used in their development, the

differing definitions of distance, and the assumptions made in the statistical analysis. A

complete review of current attenuation relationships is presented by Abrahamson and

Shedlock (1997). Most attenuation relationships include a 'site factor' that accounts for

site amplification effects. The following discussion focuses specifically on the way in

which site effects are incorporated into some of the most widely used attenuation models.

A comparison of all the predicted amplification factors is given in Figure 2.3.

Abrahamson and Silva (1997)

The relationship by Abrahamson and Silva (1997) is developed for shallow crustal

earthquakes in active tectonic regions. The site factor (f5) is included as an additive term

to the natural logarithm of the corresponding spectral value. The amplification term is a

function of the expected PGA for rock (PGArock). The form of the equation was also used

by Youngs (1993), but the parameter c5 is included by Abrahamson and Silva (1997):

f5 (PGArock) = a10 + a11 ln (PGArock + c5), (2.1)

19

where a10, a11, and c5 are parameters of the regression analysis. Amplification factors as

a function of period and PGA in the rock are given in Figure 2.4. The site factor f5

decreases with increasing PGA in the rock for spectral periods lower than 1.0 s, however,

it increases with intensity for spectral periods larger than 1.0 s. This is consistent with the

expected shift in site period from nonlinear soil effects (Chapter Three). However, the

rock PGA at which attenuation of accelerations are observed is lower than that suggested

by other researchers (Seed et al. 1991, Dickenson 1994, Chang et al. 1997). Note that

although rock and shallow (< 20m) soils are grouped in a single category, it is likely that

a site with 20 m of soil will have a motion different to that of a rock site (Abrahamson

and Silva 1997). Moreover, Abrahamson and Silva (1997) indicate that the site factor

could also have magnitude dependence.

Boore, Joyner and Fumal

Joyner and Boore (1981, 1982) proposed an attenuation relationship for shallow

crustal earthquakes in active tectonic regions. Further reviews were published by Boore,

Joyner and Fumal (1993, 1994a and b) and are summarized in Boore et al. (1997). These

reviews will be denoted by BJF 1994a and BJF 1994b.

In BJF 1994a the site factor was changed from a constant for each site condition

to a continuous function of average shear wave velocity over the upper 30 m ( sV ). The

site factor is determined as an additive term to the natural log of the ground motion

parameter, and is given by:

20

SITE FACTOR = bv ln ( sV /Va), (2.2)

where both bv and Va are regression parameters. The same functional form for

amplification factors was used by Borcherdt (1994). The amplification factors from 'soil'

to 'rock' are given in Figure 2.3. 'Soil' is defined by sV = 310 m/s and 'Rock' is defined by

sV = 620 m/s (Boore et al. 1997). The site effect was first obtained as an average site

amplification for USGS Site B (soil with 360 m/s < sV ≤ 750 m/s) and USGS Site C (180

m/s < sV < 360 m/s) relative to Site A ( sV > 750) while accounting for distance and

magnitude dependence. (BJF 1993). Residuals with respect to USGS Site A for

recordings made at sites with shear wave velocity measurements were used in BJF 1994a

to develop parameters for the sV relationship (Equation 2.2). Boore, Joyner and Fumal

(1997) indicate that a better parameter than sV is the average shear-wave velocity over a

quarter of the wavelength of the motion for each spectral period. Similarly, they suggest

that a 'depth of the attenuating layer' should be included in the relationships to explain

systematic differences in motion depending on site condition. Site factors are obtained

from stations located at long distances from the fault and are assumed to be independent

of intensity (i.e., the BJF relationship does not account for soil nonlinearity). For short

distances, the data are controlled by soil sites in the Imperial Valley, thus motions for

rock at short distances are obtained from the same factors obtained for long distances.

Therefore, results for rock at short distances from the fault are not considered reliable.

21

Toro, Abrahamson and Schneider (1997)

The attenuation relationship by Toro, Abrahamson, and Schneider (1997) was

developed for shallow crustal earthquakes in stable continental regions. The soil factors

used were developed by Silva et al. (in EPRI 1993) and are shown in Figure 2.3. The

authors recommend that when calculating uncertainty in soil motions, the uncertainty for

rock motions should be used as a conservative estimate of soil-site uncertainty. This is

counter-intuitive if one thinks of the uncertainty of site amplification being applied on top

of the uncertainty of rock motion predictions. However, Abrahamson and Sykora (1993)

show from dense arrays that uncertainty in soil is lower than rock. Toro et al. (1997)

propose that either soil is a homogenizing factor, or uncertainty in bedrock motions is

lower than in outcrop motions. Moreover, the apparent paradox might occur because

uncertainties in soil and rock are mutually dependent. The decrease in uncertainty for

soils is also evident for high intensity motions. Toro et al. (1997) observed from

equivalent-linear analyses a decrease in uncertainty for high intensity motions. This

decrease apparently offsets the increased uncertainty associated with high-strain dynamic

properties of soils.

Youngs et al. (1997)

The relationship of Youngs et al. (1997) was originally developed for subduction

zone earthquakes. The authors initially divided the database into Deep Stiff Soil, Shallow

Stiff Soil, and Rock. The results, however, are only presented for Rock and Deep Stiff

Soils. In an initial analysis, the ratio of soil to rock PGA increased as PGA increased,

which contradicts intuitive soil behavior. To correct for this apparent error, the

22

relationship was constrained so that soil and rock PGA are equal for small distances to

compensate for sparse soil data at small distances.

The attenuation relationship is calculated separately for soil and rock. The site

effect can be reduced to a series of additive terms to the constant term, the magnitude

scaling term (both linear and cubic), and the distance scaling term in the rock attenuation

relationship. The stiff soil to intermediate rock factors shown in Figure 2.3 are in general

greater than factors derived from other attenuation relationships (note that the Atkinson

and Boore (1995) and the EPRI (1993) site factors are for stiff soil relative to hard rock).

Campbell (1997)

The Campbell (1997) ground motion attenuation relationship was developed for

shallow earthquakes in active tectonic regions. The database is divided into Hard Rock,

Soft Rock and Firm Soil categories. The baseline attenuation relationship is developed

for Firm Soil, with factors for Hard Rock and Soft Rock. The relationship includes a

depth-to-basement term that defines a depth to crystalline basement. The attenuation

relationship is developed for PGA and normalized spectral periods. The depth-to-

basement term affects only the normalized spectra.

Site factors are given as functions of distance in the attenuation relationship for

PGA. For normalized spectral periods, the site factor is added as a parameter that varies

with period but not with distance. When the depth-to-basement term is ignored, the site

amplification with respect to hard rock is independent of period, that is, the long period

site amplification is a direct function of depth-to-basement. The ratio of ground motions

23

in the Firm Soil to the Soft Rock categories is included in Figure 2.3 for a depth-to-

basement of 1 km.

Sadigh et al. (1997)

The attenuation model by Sadigh et al. (1997) is developed for shallow crustal

earthquakes in active tectonic regions. The database is divided into rock (bedrock is at

least 1 m from surface) and deep soil (a minimum of 20 m of firm soil). Soft soils are

excluded from the database. Sadigh et al. (1997) indicate that a significant number of

'rock' sites have sV < 750 m/s. The relationship is regressed independently for PGA and

for normalized spectra. Two different sets of coefficients are given, resulting in both a

PGA and a magnitude dependence in the site amplification factors.

There is a large degree of nonlinearity for PGA, in fact, PGA in soil is lower than

PGA in rock for PGA values in rock greater than about 0.2 g. The same degree of

nonlinearity was inferred by Abrahamson and Silva (1997). In general, amplification

factors are close to those obtained by Abrahamson and Silva (1997).

Spudich et al. (1997): Extensional regimes

The relationship of Spudich et al. (1997) for extensional regimes is a modification

of the attenuation relationship of Boore et. al. (1997). Site factors are constant for each

period. Relatively low amplification levels at long periods are predicted compared to the

other relationships (Figure 2.3).

24

Summary

The treatment of site response in the ground motion attenuation relationships

discussed herein is summarized in Figure 2.3. The site amplification factors predicted by

the different models have relatively large variability; however, all relationships predict

the same trends with varying period, namely, larger site factors for longer periods. The

relationships that include a dependence on intensity of motion (e.g. Abrahamson and

Silva 1997, Sadigh et al. 1997, Campbell 1997, EPRI 1993, Youngs et al. 1997) predict

lower amplifications for more intense ground motions. The variability in the estimates of

site response illustrated in Figure 2.3 highlights the need for further study of seismic site

response.

2.4 SUMMARY

A brief review of the relevant work in the areas of site response, constitutive

modeling of cyclic soil behavior, and treatment of soil response in attenuation

relationships was presented. Additionally, this dissertation feeds from advancements in

finite element technology (e.g., Zienkiewicz and Taylor 1989, Taylor 1998),

computational mechanics (e.g., Simo and Hughes 1998), and seismology (e.g. Somerville

et al. 1997). Research of particular relevance to this work in these areas is presented

when appropriate in the later Chapters of this dissertation.

25

Figure 2.1. PGA amplification in soft soils (from Idriss 1990).

26

Figure 2.2. Spectral shapes for different site conditins (from Seed and Idriss 1983).

27

Figure 2.3. Site Factors used in some current ground motion attenuation relationships for estimating spectral acceleration at 5% damping. Factors are from "Rock" to "Soil". In Campbell (1997), the factor is from "Firm Soil" to "Soft Rock" and the Depth to Basement term equal to 1 km. In Boore et al. (1997), "Soil" is defined as sV = 310 m/s

and "Rock" as sV = 620 m/s. The EPRI (1993) curves are given for a profile with an average depth of 76 m. When necessary, the magnitude used to get the PGA at the baseline site condition was set to 7.0. Amplification factors for Atkinson and Boore (1995) and EPRI (1993) are with respect to harder Eastern North America rock.

b) Baseline PGA = 0.4 g

0

0.5

1

1.5

2

2.5

0.01 0.1 1 10

Period (s)

Site

Fac

tor


Atkinson and Boore (1995)

Boore et al. (1997)

Campbell (1997)


Spudich et al. (1997)

Youngs et al. (1997)

EPRI (1993)

a) Baseline PGA = 0.1 g

0

0.5

1

1.5

2

2.5

0.01 0.1 1 10

Period (s)

Site

Fac

tor


Atkinson and Boore (1995)

Boore et al. (1997)

Campbell (1997)


Spudich et al. (1997)

EPRI (1993)

b) Baseline PGA = 0.4 g

28

Figure 2.4. Site factors for varying levels of PGA from the Abrahamson and Silva (1997) spectral acceleration (5% damping) attenuation relationship.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.01 0.1 1 10

Period (s)

Fac

tor,

f5

PGA = 0.1 gPGA = 0.2 g

PGA = 0.3 g

PGA = 0.4 g

29

BLANK

30

CHAPTER THREE

CHARACTERIZATION OF SITE RESPONSE

GENERAL SITE CATEGORIES

3.1 INTRODUCTION

The effect of local site conditions on the amplification of ground motions has long

been recognized (e.g., Seed and Idriss 1982). Recent earthquakes, such as the 1985

Mexico City, 1989 Loma Prieta, 1994 Northridge, and 1995 Kobe earthquakes have

resulted in significant damage associated with amplification effects due to local geologic

conditions (e.g., Seed et al 1987, Chang et al. 1996). While potentially other factors lead

to damage (such as topographic and basin effects, liquefaction, ground failure, or

structural deficiencies), these events emphasize the need to characterize the potential

effect of local soil deposits on the amplification of ground motions.

Extensive studies of seismic site response have been performed over the last thirty

years. Recently, Borcherdt (1994) developed intensity-dependent, short and long period

amplification factors based on the average shear wave velocity measured over the upper

30 m of a site. Concurrently, Seed et al. (1991) developed a geotechnical site

classification system based on shear wave velocity, depth to bedrock, and general

geotechnical descriptions of the soil deposits at a site. Seed et al. (1991) then developed

intensity-dependent site amplification factors to modify the baseline "rock" peak ground

acceleration (PGA) to account for site effects. With this site PGA value and a site-

31

dependent normalized acceleration response spectra, a site-dependent design spectra can

be developed. Work by these researchers along with work by Dobry (Dobry et al. 1994)

has been incorporated into the 1997 Uniform Building Code (UBC) based primarily on the

site classification system and amplification factors developed by Borcherdt (1994). A

shear wave velocity based classification system, however, has two important limitations:

(a) it requires a relatively extensive field investigation, and (b) it overlooks the potential

importance of depth to bedrock as a factor in site response. Recent work completed at

the University of California at Berkeley based on results from the Northridge and Loma

Prieta earthquakes reflects the importance of introducing a measure of depth in a site

classification system (Chang and Bray 1995, Chang et al. 1997). Moreover, the Borcherdt

(1994) site amplification factors are based primarily on observations from the 1989 Loma

Prieta Earthquake, which shows significant nonlinear site response effects; whereas,

observations from the 1994 Northridge Earthquake indicate that site amplification factors

should not decrease as rapidly with increasing ground motion intensity. Hence, the

current code site factors may be unconservative, and this should be re-evaluated using the

extensive Northridge ground motion database.

A probabilistic seismic hazard assessment requires not only an estimation of the

median expected levels of ground motion intensity, but also the standard error associated

with such a median estimation. Current ground motion attenuation relationships provide

this information (e.g., Abrahamson and Silva 1997, Campbell 1997, Sadigh et al. 1997,

Boore et al. 1997). However, most current attenuation relationships have a simplified

classification scheme for site conditions in which all sites are divided into two or three

broad classifications, e.g., rock/shallow soils, deep stiff soils, and soft soils. A notable

32

exception is the attenuation relationship by Boore et al. (1997). In this relationship, the

factor that accounts for site response (site factor) is a continuous function of the average

shear wave velocity measured over the upper 30 m of a site. However, Boore et al.

(1997) ignore the effects of ground motion intensity on the site factor, which contradicts

measured observations of nonlinear site response (e.g., Trifunac and Todorovska 1996).

Conversely, studies involving a more elaborate site classification scheme encompassing

stiffness, depth, and intensity of motion, currently lack an appropriate estimate of the

statistical uncertainty involved (e.g., Seed et al. 1991).

The significant quantity of ground motion data recorded in the 1994 Northridge

and 1989 Loma Prieta earthquakes provides an opportunity to assess and to improve

empirically based predictions of seismic site response. The objective of this work is to

develop site amplification factors that are both intensity-dependent and frequency-

dependent. The site amplification factors will be estimated based on a new proposed site

classification system that includes soil stiffness and soil depth as key parameters. The

uncertainty levels resulting from the proposed classification system will be compared with

those resulting from a simplified "rock vs. soil" classification system and the more

elaborate code-based system which uses average shear wave velocity measured over the

upper 30 m of a site.

3.2 METHODOLOGY

The following three steps constitute the methodology used in the development of

the proposed empirically based site-dependent amplification factors:

33

(1) A site classification scheme was developed with the objective of encompassing the

factors that have the greatest influence on seismic site response. The proposed

scheme utilizes only general geological and geotechnical information, including

depth to bedrock or to a significant impedance contrast. More elaborate

measurements, such as average shear wave velocity ( Vs ), are utilized only as a

guideline and are not essential to the classification system. The classification

scheme will be described in detail in the next section.

(2) Two major recent earthquakes, the Loma Prieta Earthquake of October 17, 1989,

and the Northridge Earthquake of January 17, 1994, were considered in this study.

The strong motion sites that recorded these earthquakes were classified according

to the site classification scheme developed in this study. Distance-dependent

attenuation relationships for 5% damped elastic acceleration response spectra were

developed for each earthquake and for each site condition. For simplicity,

hereinafter, any reference to response spectral values will imply linear elastic

acceleration response spectra at 5% damping.

(3) These attenuation relationships were utilized to develop site-dependent

amplification factors with respect to the baseline site condition, Site Class B,

"California Rock." The site-dependent amplification factors are a function of both

spectral period and intensity of motion. Amplification factors estimated for the

Northridge and Loma Prieta earthquakes were combined to develop

recommendations that can be generalized to other events.

34

3.3 SITE CLASSIFICATION

3.3.1 Classification Scheme

The amplification of ground motions at a nearly level site is significantly affected

by the natural period of the site (Tn = 4H/Vs; where Tn = natural period, H = soil depth,

and Vs = shear wave velocity; i.e., both dynamic stiffness and depth are important). Other

important seismic site response factors are the impedance ratio between surficial and

underlying deposits, the material damping of the surficial deposits, and how these seismic

site response characteristics vary as a function of the intensity of the ground motion, as

well as other factors. To account partially for these factors, a site classification system

should include a measure of the dynamic stiffness of the site and a measure of the depth of

the deposit. Although earlier codes made use of natural period as a means to classify site

conditions (e.g., 1976 UBC), recent codes such as the 1997 UBC disregard the depth of

the soil deposit and use mean shear wave velocity over the upper 30 m as the primary

parameter for site classification.

Both analytical studies and observation of previous earthquakes indicate that depth

is indeed an important parameter affecting seismic site response. Figure 3.1 shows a

measure of building damage as a function of site depth in the Caracas Earthquake of 1967.

Damage is concentrated in buildings whose natural period matches the natural period of

the soil deposit (Seed and Alonso 1974). To illustrate the effect of soil profile depth on

surface ground motions, a one-dimensional wave propagation analysis was performed

using the equivalent-linear program SHAKE91 (Idriss and Sun 1992). A synthetic motion

for an earthquake of moment magnitude 8.0 (Mw = 8.0) on the San Andreas Fault in the

35

San Francisco Bay was used as an input outcropping rock motion for a soil profile with

varying thickness. The input rock motion was modified to match the Abrahamson and

Silva (1997) attenuation relationship for an earthquake of moment magnitude 7.5 (Mw =

7.5) at a distance of 30 km. The soil profile represents a generic stiff clay site. The upper

30 m of the profile was kept constant, while the depth of the profile was varied between

30 m and 150 m (Figure 3.2a). Figure 3.2b shows the resulting surface linear elastic

acceleration response spectra, and Figure 3.2c shows the corresponding spectral

amplification factors. Observe that an increase in depth shifts the fundamental period,

where amplification is most significant, toward higher values. This results in significantly

different surface motions as a function of the depth to bedrock. An increase in depth also

results in a longer travel path for the waves through the soil deposit. This accentuates the

effect of soil material damping, resulting in greater attenuation of high frequency motion.

However, the significantly higher response at longer periods for deep soil deposits is an

important expected result that should be accommodated in a seismic site response

evaluation.

The same input motion was applied to the four profiles illustrated in Figure 3.3a.

The depth to bedrock for the four profiles is kept constant at 30 m. The four different

profiles correspond to a dense sand, a stiff clay, a loose sand, and a soft clay profile. The

shear modulus reduction curves proposed by Iwasaki et al. (1976) were used for the dense

and loose sand, along with the damping curves for sand proposed by Seed and Idriss

(1970). The Vucetic and Dobry (1991) shear modulus reduction and damping curves for

clays with PI = 30 were used for the stiff clays, whereas for the soft clays the shear

modulus reduction and damping curves for Holocene Bay Mud proposed by Sun et al.

36

(1988) were used. Figure 3.3b illustrates the resulting spectral amplification factors.

Observe that the effect of different average dynamic shear wave velocities over the upper

30 m is similar to the effect of changing the depth to bedrock, as observed in Figure 3.2;

that is, the peak spectral amplification factor shifts toward higher periods. Hence, case

records and analytical studies support a site classification scheme that captures both the

important influences of soil stiffness and soil depth on seismic site response and resulting

damage.

Seismic site response is also a function of the intensity of motion due to the

nonlinear stress-strain response of soils. The effect of nonlinearity is largely a function of

soil type (e.g., Vucetic and Dobry 1991). Factors such as cementation and geologic age

may also affect the nonlinear behavior of soils. The effect of soil nonlinearity is two-fold:

(a) the site period shifts toward longer values, as illustrated in the previous example, and

(b) material damping levels in the soils at a site increase. The increased damping levels

result in lower spectral amplifications for all periods. The effect of damping, however, is

more pronounced for high frequency motion. Hence, PGA is more significantly affected

by soil damping. The consequences of the shift toward longer site periods depend on the

soil type and the input motion. For some sites, the site period may be shifted toward

periods containing high-energy input motion, resulting in large spectral amplification

factors with an associated increase in PGA. Conversely, the site period may be shifted to

periods where the energy of the input motion is low, resulting in large spectral

amplification at long periods associated with a decrease of amplification for short periods.

This may result in lower levels of PGA, and possibly even in attenuation of PGA.

37

The site classification system proposed herein is an attempt to encompass the

factors affecting seismic site response while minimizing the amount of data required for

site characterization. The site classification system is based on two primary parameters

and two secondary ones. The primary parameters are:

(1) Type of deposit, i.e., hard rock, competent rock, weathered rock, stiff soil, soft

soil, and potentially liquefiable sand. These general divisions introduce a

measure of stiffness (i.e., average shear wave velocity) to the classification

system. However, a generic description of a site is sufficient for classification,

without the need for measuring shear wave velocity over the upper 30 m.

(2) Depth to bedrock (defined by Vs ≥ 760 m/s) or to a significant impedance

contrast between surficial soil deposits and material with Vs ≈ 760 m/s. .

The secondary parameters are depositional age and soil type. The former divides

soil sites into Holocene or Pleistocene groups, the latter into primarily cohesive or

cohesionless soils. These subdivisions are introduced to capture the anticipated different

nonlinear responses of these soils. Table 3.1 summarizes the site classification scheme.

3.3.2 Site Classification

The list of sites with the corresponding site classification based on the proposed

classification system is given in Appendix A (Tables A-1 and A-2). The sites are also

classified according to the 1997 UBC and the Seed et al. (1991) systems (Tables A-3 and

A-4 in Appendix A). The references used for the classification of each site are also

included in Appendix A. Due to the lack of consistent information for all the sites, the

38

subdivision of Site D into a very deep site sub-category (D3) was omitted. Additionally,

sufficient information was not available to categorize sites by a precise depth to bedrock

parameter, so that a regression analysis could not be performed using this parameter as a

continuous variable.

The references listed in Appendix A were complemented with site visits for some

of the sites where information was incomplete. A list of the visited sites is given in

Appendix B. Note that an important source of information, particularly for sites belonging

to the University of Southern California, was the database of Vucetic and Doroudian

(1995). The shear wave velocity values presented in this database have recently been

challenged (e.g., Wills 1998, Boore and Brown 1998). In light of these observations, the

shear wave velocities for these sites were used, whenever possible, only as a secondary

reference. For those sites where the only data available was those in the Vucetic-

Doroudian database, these shear wave velocity data were used after incorporating the

comments made by Boore and Brown (1998).

3.4 GROUND MOTION DATA

Ground motion data from two recent earthquakes, the 1989 Loma Prieta

Earthquake and the 1994 Northridge Earthquake, were used in this study. The ground

motion recordings were obtained from a database provided by Dr. Walter Silva from

Pacific Engineering and Analysis (personal comm. 1998). The database consists of

computed elastic spectral acceleration values at 5% damping.

39

The ground motion database provided by Dr. Walter Silva was complemented with

four additional motions for the Northridge Earthquake and eleven motions for the Loma

Prieta Earthquake. The baseline corrected motions were obtained from the Internet from

sites supported by the institutions in charge of the instruments (see Appendix A). A total

of 149 and 70 recorded "free-field" ground motions were used from the 1994 Northridge

and 1989 Loma Prieta earthquakes, respectively.

The ground motion recordings used in the study are listed in Appendix A. The

number of recordings is a function of spectral period, because of the acceptable filtering

parameters used in the processing of the data. The response spectral values are only used

if the frequency is greater than 1.25 times the high-pass-corner frequency and less than

1/1.25 times the low pass-corner frequency (Abrahamson and Silva 1997). The

distribution of recorded motions with distance as a function of site type for spectral

periods between 0.055 seconds and 1.0 seconds is given in Figure 3.4 for each earthquake.

The number of recordings as a function of period is given in Figure 3.5 for each

earthquake.

3.5 STATISTICAL ANALYSIS

3.5.1 General

The ground motion sites were divided into the major categories indicated in the

site classification scheme (Table 3.1). A regression analysis was performed to develop

event and site specific attenuation relationships for acceleration response spectral values

40

(5% damping) at selected periods. A basic form of an attenuation relationship was

selected for this study, that is,

ln[Sa] = a + b ln(R + c) + ε (3.1)

where ln[Sa] is the natural logarithm of the spectral acceleration at a specified

period, T, R is the closest distance to the rupture zone (Sadigh et al. 1993), ε is an error

term, and a, b, and c are regression coefficients. Equation (3.1) is used for various

spectral periods, thus the regression coefficients a, b, and c, and the error term ε are

functions of period. This functional form was previously used by Idriss (1994) and

Somerville (personal comm.). Note that this functional form does not capture the effects

of directivity (Somerville et al. 1997) and the hanging-wall effect (Abrahamson and

Somerville 1996). Forward directivity effects result in larger response spectral values at

periods larger than one second for near-fault sites and for the component of motion

perpendicular to the fault orientation. Similarly, the hanging wall effect increases the

motion of sites located directly above the rupture zone. Directivity and hanging-wall

effects introduce a systematic bias for the affected near-fault sites. This bias, however,

generally affects all sites regardless of their type, thus the effect on amplification factors

should not be significant. A list of sites located over the hanging-wall and the footwall for

the Northridge Earthquake is provided in Table 3.2.

The regression coefficients were estimated by means of a maximum likelihood

estimate (Benjamin and Cornell 1970). The error term ε in Equation (3.1) is assumed to

41

be normally distributed with mean zero and standard deviation σ. With this assumption,

maximum likelihood estimates are equivalent to ordinary least squares estimates (Benjamin

and Cornell 1970). The natural logarithm of the likelihood function is thus given by:

∑=

−N

i

measai

estai

S

S

1

2

,

,ln

5.0exp1

lnσσ

(3.2)

where σ is the standard deviation, ln(Sai)meas is the natural logarithm of the measured

spectral acceleration and ln(Sai)est is the mean value given by Equation (3.1). Similar to

Equation (3.1), Equation (3.2) is applied at each spectral period. The values of the

coefficients a, b, and c from Equation (3.1) and the standard deviation σ result from the

minimization of the negative of the natural logarithm of the likelihood function (Equation

3.2). The regression analysis was performed using the software JMP (SAS Institute, Inc.

1995). This analysis was performed separately for each earthquake and for selected

periods. In the following sections, the details of the analyses for each event are described.

3.1.2 Northridge Earthquake

The data distribution by site type is shown in Figure 3.4a. Initially, a separation

between sites C1, C2 and C3 (weathered/soft rock, shallow stiff soil, and intermediate

depth stiff soil, respectively) was assumed, but no significant differences were observed in

the resulting attenuation relationships. Consequently, the subdivision of Site C was

ignored in the preliminary analysis. Similarly, differences for deep soil sites based on age

42

and soil type (i.e., Holocene or Pleistocene and primarily cohesive or cohesionless) were

also not considered in the preliminary analysis.

The response of potentially liquefiable sand deposits (Site F) is mainly a function of

whether or not liquefaction is triggered or partially triggered (i.e., significant pore pressure

generation develops) at the site. Triggering of liquefaction is a function of the intensity

and duration of ground motion, the relative density of the soil, the permeability of the soil,

the fines content of the soil, as well as other factors. If liquefaction is triggered or nearly

triggered, ground motion is a function of a number of parameters, including rate of excess

pore pressure generation, dissipation of pore pressure, reduction of effective stress, shear

modulus degradation, duration of motion, as well as other factors. The analysis of these

sites is beyond the scope of this project; thus, ground motion sites that are classified as

Site F will be excluded from the analysis.

Most of the ground motion sites are concentrated between 20 and 70 km of the

zone of energy release (Figure 3.4a). Accordingly, the resulting attenuation relationships

are judged to be appropriate for sites located within this distance range from an active

fault. Of all the sites located closer than 20 km from the rupture plane, most sites are C

and D sites, and only one Site B (California rock) is located within 20 km.

Equation (3.1) is defined for all distance values only if the coefficient c is non-

negative. Accordingly, this coefficient was assumed to be non-negative for all periods.

Moreover, initial analyses yielded a large standard deviation for the coefficient c, implying

that changes in this coefficient did not result in an increase of the overall standard

deviation in Equation (3.1). The coefficient c was held constant across site conditions to

43

avoid the coupling of uncertainty in the coefficient c with uncertainty in the amplification

factors. This is consistent with a number of previous studies (e.g., Somerville, personal

comm.; Abrahamson and Silva 1997).

Preliminary results yielded spectral accelerations at long periods larger at rock sites

(Site B) than at soil sites (Sites C and D) for distances greater than 70 km, a result that

contradicts both previous analyses (Abrahamson and Silva 1997, Borcherdt 1994) and

theoretical considerations (Dobry et al. 1997). This result is thought to be primarily a

consequence of the poor sampling for Site B across all distances. Data for Site B are

concentrated within a distance range of 20 to 40 km (See Figure 3.4a), thus there is

limited data to constrain adequately all of the coefficients in Equation (3.1). The approach

taken was to assume that the coefficient a is equal for both B and C sites.

In summary, the regression analysis for the Northridge Earthquake proceeded in three

steps:

(1) The value of coefficient c was determined using the whole data set (Sites B, C

and D). The values of the coefficient c obtained in this manner for different

spectral periods were fitted to a piece-wise linear function.

(2) Using the values obtained in step 1 for the coefficient c, the coefficient a for

site types B and C was obtained using the data for these two site types.

(3) Finally, the remaining coefficients were determined using the data set for each

site type separately.

44

3.1.3 Loma Prieta Earthquake

The distribution of data by site type was given in Figure 3.4b. Due to the limited

quantity of data, the distinction between C1, C2, and C3 sites (i.e., weathered/soft rock,

shallow stiff soil, and intermediate depth soil, respectively) and the difference in age or soil

type (i.e., Holocene or Pleistocene and primarily cohesive or cohesionless sites) were not

considered in the preliminary stage of the analysis. Similar to the Northridge Earthquake,

potentially liquefiable sand and peat deposits (Site F) were excluded from the analysis.

Separate regression analyses were performed for sites B, C, D, and E. Most of the data

are concentrated within a distance range of 10 km to 90 km from the zone of energy

release. Moreover, there are only two sites located within 10 km of the fault rupture

plane, implying that the resulting attenuation relationships are poorly constrained for close

distances to the zone of energy release. Consequently, the results presented in this report

will not reflect the localized effects of near-fault ground motions on seismic site response.

Numerical simulations will be required to provide insight into the near-fault seismic

response of soil sites, and this is the objective of another ongoing research project by the

authors.

Only three rock sites (Site B) are located within 20 km of the zone of energy

release. This poor sampling implies that the attenuation relationship is not well

constrained for short distances. This is especially important since Site B is taken as the

baseline site for developing amplification factors. The low number of soft clay sites (7

sites) and the poor distribution with distance (see Figure 3.4) results in a poorly

constrained attenuation relationship for this site class. Previous studies (e.g., Seed et al.

45

1991) complemented this lack of empirical data with numerical simulations. Since the

objective of this work was the development of empirically based amplification factors, soft

clay sites were excluded from the analysis.

As previously indicated for the analysis of the Northridge data, the coefficient c in

Equation (3.1) was constrained to be non-negative. Unconstrained regression analysis

yielded a negative value of c for all periods. Consequently, the coefficient c was set to 1

for all periods. As a result of the better sampling in the Loma Prieta data set, there was no

need to constrain the parameter a in the analysis.

3.1.4 Results

The coefficients a, b, and c found for each earthquake were smoothed by a

convolution with a triangular function with a window-width of three. The convolution

was repeated until no further improvement was obtained. The smoothed coefficients are

illustrated in Figures 3.6 and 3.7 and listed in Table 3.3. A comparison between the

resulting smoothed and non-smoothed spectra is shown in Figure 3.8 for the Northridge

earthquake. The resulting attenuation relationships are illustrated in Figures 3.9 and 3.10

for selected distances. Spectral acceleration values as a function distance for selected

periods are shown in Figures 3.11 and 3.12.

46

3.6 EVALUATION OF RESULTS

3.6.1 General

The attenuation relationships obtained using the classification system introduced in

this work are compared with results from a simplified "Rock vs. Soil" classification

system, as well as with a more elaborate code-based classification system (1997 UBC).

Residuals for sites C and D were evaluated to judge whether a further subdivision is

justified.

3.6.2 Comparison With a "Soil vs. Rock" Classification System

Most current attenuation relationships use a broad and general site classification,

dividing sites in either rock/shallow soil or deep stiff soil, in addition to deep soft clay sites

(e.g., Abrahamson and Silva 1997). This classification is also often applied in design

practice (Abrahamson, personal comm.). Results from this study, however, show that this

classification is an oversimplification, and further division into additional categories is

warranted.

As a basis for comparison, the earthquake specific attenuation model developed by

Somerville and Abrahamson (Somerville, personal comm.) will be compared with the

model developed in this study. The Somerville and Abrahamson model will be denoted as

S&A. This model divides sites into rock/shallow soil (rock) and deep stiff soils (soil).

Deep soft clay sites are excluded. Figure 3.13 shows a comparison of the results at a

distance of 20 km. Note that the spectra for soil sites in S&A generally match the spectra

for Site D (deep stiff soils). However, the spectra for rock sites in S&A generally match

47

the spectra for Site C (shallow and intermediate depth soils and weathered/soft rock).

This result reflects the fact that for the joint database of rock and shallow soil sites, 83%

of the sites are shallow soil or weathered rock sites, and only 17% of these sites actually

belong to the Site B classification (competent rock sites). Note that the spectrum for Site

B falls significantly below that for Site C (20% to 40% lower, 30% on average). Similar

results were obtained independently by Idriss and Silva (personal comm. 1999). These

authors performed an analysis of strong motions at competent rock and weathered rock

sites and concluded that response spectra at weathered rock sites is 20% lower on average

than response spectra at competent rock sites.

Attenuation relationships are commonly used in engineering practice to predict ground

motions at baseline "outcropping rock" sites. These baseline motions are often later

modified for local soil conditions. The results presented herein point to the need of

redefining the baseline "outcropping rock" condition into a more restrictive category.

Many "rock" attenuation relationships used in practice actually correspond to a mix site

condition that is dominated by shallow stiff soil and soft/weathered rock records due to

the preponderance of these sites in the ground motion database. Hence, most "rock"

attenuation relationships reflect the amplified response of soft/weathered rock and shallow

stiff soil sites. The acceleration response spectrum at 5% damping for the true baseline

rock condition (California rock) is consistently overestimated by that developed using

currently available "rock attenuation relationships. Thus, updated "rock" attenuation

relationships that use a more restrictive baseline rock category are required.

48

A significant difference in response spectra was observed between the proposed

site categories (see Figures 3.9 and 3.10). Again, Site B (California rock) data plot

significantly below that for Site C (weathered rock/shallow stiff soil), which illustrates that

a further subdivision from the 'rock' vs. 'soil' classification is warranted. More significant,

however, is the reduction of uncertainty that results from the proposed classification

system. Table 3.4 compares the standard deviations from the S&A relationships with

those from the relationships proposed in this report. The decrease in the standard

deviation for Site B compared with S&A rock sites is between 30% and 40%. A similar

reduction is observed for soil sites (S&A Soil vs. Site D). Standard deviations for Site C,

however, remain high and are only marginally lower than standard deviations for rock in

the S&A model. A reduction in the uncertainty bands for sites B and D reflects the more

selective grouping criteria applied in this study.

3.6.3 Comparison With a Code-Based Site Classification System

The data set for both earthquakes was also divided according to the 1997 UBC

(i.e., using the average shear wave velocity measured over the upper 30 m of the site).

The UBC classification system is presented in Table 3.4 in Appendix A. Differences in the

classification of ground motion sites using both systems are shown in Table 3.5. For

simplicity, sites classified according to the system presented in this work (Table 3.1) will

simply be denoted by Site X, while sites classified according to the UBC system will be

denoted UBC X. Note that in the Northridge database, there is a significant number of C

sites that correspond to either UBC B or UBC D sites. The former are weathered rock

sites lying on top of harder, intact rock (such as Lake Hughes #9), and the latter are either

49

shallow soil or weathered rock sites with depth to bedrock ranging from 25 to 60 m.

There are also five D sites in Northridge and one in Loma Prieta that classify as UBC C

sites. These sites correspond to stiff clay or sand deposits with shear wave velocities only

slightly larger than the boundary values determined by the UBC classification system (such

as Sepulveda VA Hospital). This overlap results in different attenuation relationships

depending on the classification system. Because shear wave velocity measurements were

not taken at all ground motion stations used in this study, the classification of sites

according to the scheme presented in this work probably is more accurate than the

classification of sites according to their average shear wave velocity value. The same

finding carries over to the results of the regression analyses.

Table 3.6 compares the standard deviations at selected periods resulting from the

regression analysis using both classification systems. For the Loma Prieta Earthquake,

standard deviations for both classification systems are comparable. This is expected

because there is little overlap between classification systems for the Loma Prieta database

(Table 3.5). For the Northridge Earthquake, standard deviations vary slightly from one

classification system to the other. Standard deviations for Site B are slightly lower for the

proposed classification system. For Sites C and D, standard deviations are equal for a

period of 0.3 seconds, but vary slightly at a period of one second. With the exception of

Site D at a period of one second, the differences of the standard deviations resulting from

both classification systems are within the ranges of the estimates. Given that the spectral

amplification factors change significantly with depth at a period close to one second

(Figure 3.2), the exclusion of sites shallower than 60 m from Site D in the proposed

classification system result in a reduction of the scatter in the data.

50

3.6.4 Subdivision of Site C

For the Northridge Earthquake, the standard deviations of Site C at long periods

are larger than those of Sites B and D. This observation motivated a closer examination

of the results for Site C. Figure 3.14 shows the residuals for Site C for the Northridge

Earthquake, along with the mean value of the residuals. Each site is identified by its

corresponding UBC classification. Note that well-defined trends are observed for periods

larger than 0.1 seconds. UBC D sites plot significantly above the median while UBC C

sites plot below the median, illustrating that a further subdivision for Site C according to

shear wave velocity may be warranted. Similarly, these results demonstrate that whereas

for deep stiff soil sites and rock sites the additional expense of a shear wave velocity

characterization may not be justified, for intermediate depth soil sites characterization

using average shear wave velocity may reduce the uncertainty in the prediction of ground

motions.

A subdivision of Site C as indicated in Table 3.1 was also studied. Residuals for

Site C are plotted in Figure 3.15a for the Northridge Earthquake. Observe that no specific

trends for sites C1, C2, and C3 are observed, as opposed to the trend observed when C

sites were divided according to an average shear wave velocity-based classification

system. This observation implies that for shallow and intermediate depth soils, the

average shear wave velocity may be the discriminating additional factor.

51

3.6.5 Subdivision of Site D

A further subdivision for deep soil sites (Site D) according to age and soil type is

also studied. As shown in the classification system (Table 3.1), Site D is subdivided as

either Holocene or Pleistocene, or as primarily clayey or sandy. Figure 3.16a shows the

residuals for sites D for the Northridge Earthquake. Mean residuals consistently greater

than zero are observed for clay sites at all periods. These mean residuals are considered

important, however, not overly significant when compared with the standard deviation for

the entire distribution of around 0.4. This trend is magnified when only Pleistocene sites

(D2) are considered. However, since the number of such sites is low, further studies are

needed to confirm this trend. No apparent trend based solely on the age of the deposit is

observed. The same trends are observed in the Loma Prieta Earthquake (Figure 3.16b),

but the small number of sites precludes any definite finding in this regard.

In general, it appears that greater amplification can occur at clay sites, especially if

Pleistocene, and this is consistent with the concept that higher plasticity soils have higher

threshold strains and hence exhibit less shear modulus reduction and less material damping

at intermediate levels of ground motion. However, until additional ground motion and site

classification data are obtained, the limited number of sites and records, and the level of

scatter associated with Site D, precludes further subdivision at this time.

3.6.6 Effect of Depth to Basement Rock

In an effort to assess the ability of a depth to basement rock term to capture

seismic site effects, sites within the Los Angeles basin were investigated. The depth to

52

basement bedrock was obtained from a map by Conrey (1967), and is defined as the depth

to the top of Pliocene bedrock. All of the selected sites are classified as Site D, with the

exception of the USC 54 site (LA Centinela), which is a C3 site. Residuals for periods of

one and two seconds as a function of depth to bedrock are plotted in Figure 3.17 for sites

located in the Los Angeles basin. No trend is observed for sites shallower than 2000 m,

but residuals are higher than zero for sites deeper than 2000 m. Positive residuals may be

due to basin effects rather than to local site amplification.

3.6.7 Amplification Factors

The attenuation relationships developed in this work are event-specific relations

that cannot be generalized to other events. To extend the applicability of the results

presented in this work, amplification factors with respect to a baseline site condition were

obtained. The baseline site condition was taken to be rock (Site B). However, since

current "rock" attenuation relationships reflect mostly site condition C, factors with

respect to Site C will also be presented. The amplification factors are obtained by dividing

Equation (3.1) for the two site conditions. The resulting relationship can be written as

( ) )ln(ln /// cRbaF BCBCBC ++= (3.3)

where FC/B is the spectral acceleration amplification factor for Site C with respect

to Site B, R is the closest distance to the rupture plane, and aC/B and bC/B are coefficients

defined by:

53

aC/B = a(Site C) - a(Site B) (3.4a)

bC/B = b(Site C) - b(Site B) (3.4b)

The regression coefficient a was assumed to be independent of site conditions and

therefore no site-dependent ratio is necessary for this coefficient. These two coefficients

were smoothed for all periods. The smoothed coefficients are listed in Appendix C. The

resulting amplification factors are shown in Figure 3.18 for each earthquake. For the

Loma Prieta Earthquake, a reduction in spectral amplification factors for increasing levels

of base rock motion is observed for periods shorter than one second. This trend is

consistent with nonlinear soil behavior. At periods greater than one second, spectral

amplification values do not necessarily decrease with increasing levels of base rock

motion, as soil response nonlinearity would also tend to increase the response at larger

periods as the site softened. Other issues may have affected the data in this period range,

such as basin effects and surface waves. In addition, rather than a reflection of soil

response, these observations may be a result of the significant scatter of the data at long

periods. Moreover, for high values of PGA, the attenuation relationships are not well

constrained due to the lack of near-fault data for the Loma Prieta Earthquake.

Amplification factors from the Northridge Earthquake do not show the same

degree of nonlinearity, as do the results from Loma Prieta. Because the current UBC is

based mainly on observational data from the Loma Prieta Earthquake (e.g., Borcherdt

54

1994), amplification factors presented in the UBC may lead to underestimation of high

amplitude ground motions.

3.6.8 Recommended Factors

The spectral amplification factors from each earthquake were combined to develop

a set of recommended amplification factors. The factors were combined at equal PGA

values. Note that because event-specific attenuation relationships are used for each

earthquake, the relationship between PGA and distance is not unique for both earthquakes.

Two different weighting schemes were utilized. One weighting scheme gives equal weight

to each earthquake, while the other gives a weight inversely proportional to the variance

of the sample mean. The equations and coefficients used to determine the amplification

factors are given in Appendix C. The resulting amplification factors are shown in Figures

3.19 and 3.20, and are given in Tables 3.7 and 3.8. The standard deviations for each site

condition were averaged using the same weighting schemes, and are also presented in

Tables 3.7 and 3.8.

For long periods (T > 1.0 s) the difference in amplification factors between

earthquakes is significantly smaller than the difference in amplification factors between site

type. For shorter periods, however, differences between earthquakes are comparable to

differences due to site type.

Amplification factors with respect to Site B (Figures 3.19a and 3.20a) show a

significant degree of nonlinearity. On the other hand, spectral amplification factors from

Site D to Site C are nearly linear, mainly because of the linearity observed in the

Northridge data (Figure 3.18b). This effect is increased when weighting factors inversely

55

proportional to sample variance are applied (Figure 3.20b) as a result of the larger number

of Site C and Site D data points in the Northridge Earthquake (see Figure 3.21 for the

weights for each earthquake).

A comparison of Figures 3.20a and 3.20b illustrates the dramatic difference in

spectral amplification factors that results from taking either rock (Site B) or weathered-

soft rock/shallow stiff soil (Site C) as the baseline site condition. Current practice takes an

intermediate site condition as reference. The large differences in behavior between these

site conditions illustrated in this work serve to highlight the need to define a unique

baseline site condition.

For the sake of comparison with current code provisions, the spectral amplification

factors were averaged over a range of periods to obtain short-period and mid-period

amplification factors. The period range for the short-period amplification factor (Fa) is 0.1

to 0.5 seconds, and the period range for the mid-period amplification factor (Fv) is 0.4 to

2.0 seconds (Borcherdt 1994). The mean factors were averaged from a double

logarithmic plot of amplification factors versus period. The values of the code factors

(UBC 1997) and the factors obtained in this work are given in Table 3.9, and are

presented graphically in Figure 3.22.

The short-period amplification factors (Fa) obtained in this work are larger than

the code values. This is due in large part to the larger levels of motion observed in the

Northridge earthquake, which was not included in the studies that led to the adoption of

the 1997 UBC factors. Additionally, the site classification scheme adopted for the 1997

UBC differs from that proposed in this study, so that some sites are classified differently

56

(see Section 3.6.2 and Table 3.5). In addition, the intermediate-period amplification

factors from the proposed site classification show less nonlinearity than the code factors.

This is consistent with "soil" to "rock" amplification factors obtained by Abrahamson and

Silva (1997) using a much broader database. Overall, the differences in the amplification

factors are generally less than 25%, which is acceptable for design. However, as

additional data become available, these two site classification systems warrant further

comparisons.

3.7 SUMMARY

3.7.1 Findings

The strong ground motion data from the Loma Prieta and Northridge earthquakes

were analyzed and used to evaluate a proposed new site classification scheme developed

to account for site effects in probabilistic seismic hazard assessments. The proposed

classification system is based on a general geotechnical characterization of the site that

includes soil depth and stiffness.

The proposed classification scheme results in a significant reduction in standard

error when compared with a simpler "rock vs. soil" attenuation relationship approach.

Additionally, the generic "outcropping rock" category used in many current attenuation

relationships groups competent rock and weathered soft rock/shallow stiff soils. The

results shown herein indicate a significant difference in the seismic responses of these two

site classes, witch competent rock spectral ordinates being about 30% lower than the

corresponding spectral ordinates for weathered soft rock/shallow stiff soil sites for the

57

same magnitude and distance. Moreover, "rock" attenuation relationships used in practice

are dominated by the weathered soft rock/shallow stiff soil site category due to the

preponderance of these sites in the database. The results presented herein point to the

need of redefining the baseline "outcropping rock" condition into a more restrictive site

category.

The standard errors resulting from the proposed classification system are

comparable to the standard errors obtained using the 1997 UBC shear wave velocity-

based classification system. However, for profiles within the same UBC classification (i.e.

equal average shear wave velocity), the inclusion of soil depth leads to a significant

reduction of uncertainty. Conversely, for sites with a shallow soil depth (i.e. < 60 m),

refinement of the estimate of profile stiffness in the form of average shear wave velocity

reduces uncertainty levels. For deep stiff soil sites, measurement of the shear wave

velocity over the upper 30 m of the site does not significantly reduce uncertainty in the

prediction of seismic site response. These observations indicate that a classification

scheme including both soil stiffness and depth leads to better estimates of ground motion.

The relative merits of subdividing deep stiff clay sites (Site D) by soil type (e.g.,

either sand or clay) and depositional age (e.g., either Holocene or Pleistocene) was also

evaluated. Results indicate a trend of higher response for clay sites, especially if

Pleistocene. This is consistent with the concept that higher plasticity soils have higher

threshold strains and hence, exhibit less shear modulus reduction and less material

damping at intermediate levels of ground motion.

58

Spectral amplification factors with respect to competent bedrock (PEER 1998)

and Site C (i.e., "rock" classification for most attenuation relationships, e.g., Abrahamson

and Silva 1997) are presented. The intensity-dependent, period-dependent spectral

acceleration amplification factors can be obtained using the formulas given in Appendix C.

Overall, the spectral amplification factors do not vary significantly from the factors

presented in the 1997 UBC; however, the difference is sufficient to warrant further

reevaluation of the code factors. Additionally, the spectral amplification factors presented

herein can be used in probabilistic seismic hazard assessments, because, unlike the code

site factors, the proposed sit amplification factors include quantification of the underlying

uncertainty in the site dependent ground motion estimate. However, caution should be

exercised when using these factors because they are obtained from a data set containing

only two earthquake. Hence, intra-event scatter could be assessed for these two

earthquakes, but inter-event scatter could not be evaluated satisfactorily. Moreover, due

to the scarcity of the data, results are not well defined for near-fault conditions. This issue

will be addressed in detail in Chapter Four.

3.7.2 Recommendations

Based on the results of this analysis of the Loma Prieta and Northridge earthquake

ground motion databases, it is judged that the site-dependent, period-dependent

amplification factors given in Tables 3.7 and 3.8 and in equation form in Appendix C, can

be used in general probability seismic hazard assessments. However, caution should be

exercised when using these factors, because they are obtained from a data set containing

only two earthquakes. Hence, intra-event scatter could be assessed for these two

59

earthquakes, but inter-event scatter could not be evaluated satisfactorily based on only

two earthquake events. Moreover, due to the scarcity of the data, results are not well

defined at high acceleration levels. Amplification factors are given with respect to site

condition B (rock) and Site C (i.e., "rock" classification for most attenuation relationships,

e.g., Abrahamson and Silva 1997). Current attenuation relationships (i.e., Abrahamson

and Silva 1997) and probabilistic maps use an intermediate baseline site condition of B-C.

Due to the relative scarcity of data of B sites relative to C sites, their “rock” sites are more

closely analogous to Site C. This should be taken into consideration when applying the

recommended amplification factors to current attenuation relationship and probabilistic

map values.

The results of this study strongly support the development of an attenuation

relationship based on the proposed site classification scheme. With this new relationship,

spectral acceleration values for a site could be estimated directly without the use of

amplification factors. A better estimate of the uncertainty involved in ground motion

estimation could be made with this direct approach, rather than the approach applied

herein that required ratios of spectral ordinates.

60

Table 3.1 Geotechnical Site Categories (after Bray and Rodríguez-Marek 1997)

Site Categories

A Hard Rock

B Rock

C Weathered/Soft Rock or Shallow Stiff Soil

D Deep Stiff Soil

E Soft Clay

F Special, e.g., Liquefiable Sand

Site Description SitePeriod

Comments

A Hard Rock ≤ 0.1 s Hard, strong, intact rock (Vs ≥ 1500 m/s)B Rock ≤ 0.2 s Most "unweathered" California rock cases

(Vs ≥ 760 m/s or < 6 m. of soil)C-1 Weathered/Soft Rock ≤ 0.4 s Vs ≈ 360 m/s increasing to > 700 m/s,

weathered zone > 6 m and < 30 m -2 Shallow Stiff Soil ≤ 0.5 s Soil depth > 6 m and < 30 m -3 Intermediate Depth Stiff

Soil≤ 0.8 s Soil depth > 30 m and < 60 m

D-1 Deep Stiff Holocene Soil,either S (Sand) or C(Clay)

≤ 1.4 s Soil depth > 60 m and < 200 m. Sand haslow fines content (< 15%) or non-plasticfines (PI < 5). Clay has high fines content(> 15%) and plastic fines (PI > 5).

-2 Deep Stiff PleistoceneSoil, S (Sand) or C(Clay)

≤ 1.4 s Soil depth > 60 m and < 200 m. See D1for S or C sub-categorization.

-3 Very Deep Stiff Soil ≤ 2 s Soil depth > 200 mE-1 Medium Depth Soft Clay ≤ 0.7 s Thickness of soft clay layer 3 m to 12 m -2 Deep Soft Clay Layer ≤ 1.4 s Thickness of soft clay layer > 12 mF Special, e.g., Potentially

Liquefiable Sand or Peat≈ 1 s Holocene loose sand with high water table

(zw ≤ 6 m) or organic peat.

61

Table 3.2 Sites located on the footwall (FW) and hanging-wall (HW) in the NorthridgeEarthquake (adapted from Abrahamson and Somerville 1996).

Organization Station # Location ClassificationCDMG 24047 FW BCDMG 24207 FW BCDMG 24279 FW C3CDMG 24469 FW BCDMG 24514 FW D1CCDMG 24575 FW C2CDMG 24607 FW C1

USC 90057 FW D1SUSGS 127 FW C1CDMG 24396 HW C1DWP 75 HW D1SDWP 77 HW C2USC 90003 HW D1CUSC 90049 HW C2USC 90053 HW C3USC 90055 HW C2

USGS 637 HW D2CUSGS 655 HW FUSGS 5080 HW BUSGS 5081 HW C2USGS 5108 HW C1

62

Table 3.3a Regression coefficients and Standard Error for spectral acceleration values at5% damping for the Northridge Earthquake

B Sites C Sites D SitesT a b c σσ a b c σσ a b c σσ

PGA 2.3718 -1.2753 6.3883 0.3209 2.3718 -1.1538 6.3883 0.4686 2.6916 -1.2161 6.3883 0.35590.055 3.5192 -1.4829 10.2486 0.4343 3.5192 -1.3869 10.2486 0.4661 3.5126 -1.3703 10.2486 0.35600.06 3.7423 -1.5138 11.8103 0.4343 3.7423 -1.4266 11.8103 0.4655 3.7970 -1.4257 11.8103 0.36540.07 4.3982 -1.6291 14.5768 0.4310 4.3982 -1.5480 14.5768 0.4636 4.4475 -1.5472 14.5768 0.37050.08 4.8097 -1.7006 16.9734 0.4180 4.8097 -1.6152 16.9734 0.4619 4.9774 -1.6422 16.9734 0.37540.09 4.9993 -1.7175 18.0000 0.3935 4.9993 -1.6366 18.0000 0.4617 5.2637 -1.6826 18.0000 0.37790.1 4.9768 -1.6855 18.0000 0.3615 4.9768 -1.6089 18.0000 0.4642 5.3000 -1.6679 18.0000 0.3774

0.11 4.9365 -1.6614 18.0000 0.3457 4.9365 -1.5844 18.0000 0.4667 5.2529 -1.6439 18.0000 0.37660.12 4.8748 -1.6330 18.0000 0.3322 4.8748 -1.5530 18.0000 0.4703 5.1563 -1.6072 18.0000 0.37590.13 4.7753 -1.5991 18.0000 0.3226 4.7753 -1.5140 18.0000 0.4750 5.0044 -1.5586 18.0000 0.37580.14 4.6161 -1.5564 17.3303 0.3179 4.6161 -1.4646 17.3303 0.4808 4.7947 -1.4991 17.3303 0.37660.15 4.3937 -1.5041 16.0757 0.3182 4.3937 -1.4037 16.0757 0.4877 4.5454 -1.4330 16.0757 0.37860.16 4.1376 -1.4471 14.9021 0.3232 4.1376 -1.3364 14.9021 0.4952 4.2958 -1.3685 14.9021 0.38200.17 3.8807 -1.3907 13.7997 0.3315 3.8807 -1.2694 13.7997 0.5030 4.0778 -1.3133 13.7997 0.38650.18 3.7555 -1.3635 12.7603 0.3368 3.7555 -1.2373 12.7603 0.5069 3.9820 -1.2900 12.7603 0.38930.19 3.6370 -1.3378 11.7771 0.3418 3.6370 -1.2069 11.7771 0.5105 3.8913 -1.2680 11.7771 0.39180.2 3.4048 -1.2891 10.8444 0.3531 3.4048 -1.1508 10.8444 0.5174 3.7044 -1.2249 10.8444 0.3974

0.24 2.9146 -1.1904 7.5290 0.3759 2.9146 -1.0449 7.5290 0.5285 3.2196 -1.1160 7.5290 0.40710.28 2.6754 -1.1429 5.8000 0.3872 2.6754 -0.9965 5.8000 0.5330 2.9725 -1.0610 5.8000 0.41060.3 2.5178 -1.1149 4.9000 0.3983 2.5178 -0.9682 4.9000 0.5372 2.8087 -1.0250 4.9000 0.4129

0.34 2.4645 -1.1197 4.4254 0.4176 2.4645 -0.9768 4.4254 0.5463 2.7212 -1.0067 4.4254 0.41450.36 2.4594 -1.1242 4.3606 0.4242 2.4594 -0.9870 4.3606 0.5515 2.6916 -0.9999 4.3606 0.41420.4 2.4375 -1.1239 4.2415 0.4276 2.4375 -0.9935 4.2415 0.5570 2.6466 -0.9915 4.2415 0.4133

0.44 2.4279 -1.1279 4.1337 0.4277 2.4279 -1.0049 4.1337 0.5627 2.6269 -0.9946 4.1337 0.41190.5 2.4692 -1.1545 3.9890 0.4198 2.4692 -1.0526 3.9890 0.5739 2.7651 -1.0629 3.9890 0.4066

0.55 2.4447 -1.1582 3.8812 0.4140 2.4447 -1.0682 3.8812 0.5792 2.8613 -1.1091 3.8812 0.40230.6 2.3687 -1.1540 3.7828 0.4090 2.3687 -1.0710 3.7828 0.5843 2.9263 -1.1469 3.7828 0.3968

0.667 2.2699 -1.1513 3.6630 0.4060 2.2699 -1.0675 3.6630 0.5892 2.9650 -1.1752 3.6630 0.39010.7 2.1804 -1.1550 3.6084 0.4059 2.1804 -1.0660 3.6084 0.5937 2.9956 -1.1995 3.6084 0.3826

0.75 2.1276 -1.1664 3.5303 0.4090 2.1276 -1.0746 3.5303 0.5977 3.0096 -1.2199 3.5303 0.37500.8 2.1239 -1.1848 3.4573 0.4151 2.1239 -1.0966 3.4573 0.6009 2.9754 -1.2294 3.4573 0.3680

0.85 2.1516 -1.2064 3.3887 0.4235 2.1516 -1.1267 3.3887 0.6030 2.8866 -1.2261 3.3887 0.36210.9 2.1703 -1.2244 3.4413 0.4332 2.1703 -1.1539 3.4413 0.6041 2.7784 -1.2185 3.4413 0.3579

0.95 2.1451 -1.2353 3.2629 0.4435 2.1451 -1.1701 3.2629 0.6041 2.6965 -1.2187 3.2629 0.35561.0 2.0734 -1.2443 3.2048 0.4538 2.0734 -1.1775 3.2048 0.6033 2.6601 -1.2333 3.2048 0.35511.1 1.9888 -1.2635 3.0970 0.4637 1.9888 -1.1873 3.0970 0.6017 2.6461 -1.2583 3.0970 0.35631.2 1.9252 -1.2983 2.9986 0.4726 1.9252 -1.2071 2.9986 0.5995 2.6099 -1.2804 2.9986 0.35871.3 1.8811 -1.3390 2.9080 0.4799 1.8811 -1.2317 2.9080 0.5962 2.5295 -1.2884 2.9080 0.36181.4 1.8327 -1.3706 2.8242 0.4799 1.8327 -1.2510 2.8242 0.5909 2.4272 -1.2846 2.8242 0.36491.5 1.7582 -1.3853 2.7461 0.4799 1.7582 -1.2588 2.7461 0.5850 2.3331 -1.2785 2.7461 0.36641.7 1.5420 -1.3800 2.6045 0.4799 1.5420 -1.2565 2.6045 0.5800 2.1862 -1.2817 2.6045 0.38112.0 1.3896 -1.3970 2.4206 0.4799 1.3896 -1.2933 2.4206 0.5700 2.0500 -1.3154 2.4206 0.41302.2 1.2440 -1.3983 2.3128 0.4799 1.2440 -1.3004 2.3128 0.5600 1.8906 -1.3182 2.3128 0.42442.6 0.9829 -1.3739 2.1238 0.4799 0.9829 -1.2719 2.1238 0.5400 1.6293 -1.2941 2.1238 0.41453.0 0.6859 -1.3338 2.0000 0.4799 0.6859 -1.2207 2.0000 0.5200 1.3413 -1.2536 2.0000 0.3877

63

Table 3.3b Regression coefficients and Standard Error for spectral acceleration values at5% damping for the Loma Prieta Earthquake

B Sites C Sites D SitesT A b c σσ a b c σσ a b c σσ

PGA 0.7219 -0.7954 1.0000 0.4713 0.8212 -0.7502 1.0000 0.3111 0.5716 -0.6032 1.0000 0.38960.055 1.6308 -0.9794 1.0000 0.4566 1.4230 -0.8769 1.0000 0.3708 1.3201 -0.7767 1.0000 0.43340.06 1.8207 -1.0119 1.0000 0.4561 1.4804 -0.8841 1.0000 0.3747 1.2568 -0.7489 1.0000 0.43380.07 1.9001 -1.0181 1.0000 0.4554 1.4819 -0.8734 1.0000 0.3798 1.2413 -0.7315 1.0000 0.43400.08 2.0559 -1.0383 1.0000 0.4538 1.5348 -0.8701 1.0000 0.3886 1.3041 -0.7271 1.0000 0.43310.09 2.1619 -1.0489 1.0000 0.4518 1.5875 -0.8642 1.0000 0.3973 1.4037 -0.7300 1.0000 0.43030.1 2.2305 -1.0551 1.0000 0.4500 1.6419 -0.8595 1.0000 0.4027 1.5122 -0.7400 1.0000 0.4269

0.11 2.2946 -1.0607 1.0000 0.4481 1.7031 -0.8527 1.0000 0.4074 1.6341 -0.7524 1.0000 0.42200.12 2.3215 -1.0625 1.0000 0.4472 1.7361 -0.8492 1.0000 0.4091 1.6890 -0.7575 1.0000 0.41870.13 2.3462 -1.0642 1.0000 0.4464 1.7665 -0.8461 1.0000 0.4108 1.7395 -0.7622 1.0000 0.41570.14 2.3659 -1.0613 1.0000 0.4451 1.8339 -0.8450 1.0000 0.4124 1.7916 -0.7621 1.0000 0.40840.15 2.3410 -1.0484 1.0000 0.4448 1.9079 -0.8523 1.0000 0.4120 1.7724 -0.7458 1.0000 0.40040.16 2.2804 -1.0268 1.0000 0.4460 1.9696 -0.8621 1.0000 0.4095 1.7156 -0.7191 1.0000 0.39240.17 2.2370 -1.0125 1.0000 0.4476 1.9792 -0.8631 1.0000 0.4071 1.6926 -0.7066 1.0000 0.38880.18 2.1960 -0.9991 1.0000 0.4491 1.9882 -0.8640 1.0000 0.4049 1.6710 -0.6949 1.0000 0.38530.19 2.0939 -0.9675 1.0000 0.4545 1.9513 -0.8531 1.0000 0.3989 1.6405 -0.6754 1.0000 0.37970.2 1.9861 -0.9352 1.0000 0.4626 1.8633 -0.8291 1.0000 0.3923 1.5961 -0.6551 1.0000 0.3761

0.24 1.8523 -0.8933 1.0000 0.4797 1.6772 -0.7749 1.0000 0.3837 1.5154 -0.6295 1.0000 0.37540.28 1.8136 -0.8775 1.0000 0.5001 1.5268 -0.7272 1.0000 0.3796 1.5140 -0.6328 1.0000 0.37960.3 1.8860 -0.8959 1.0000 0.5149 1.4800 -0.7104 1.0000 0.3812 1.5933 -0.6598 1.0000 0.3859

0.34 1.9996 -0.9271 1.0000 0.5292 1.4613 -0.7033 1.0000 0.3868 1.7028 -0.6968 1.0000 0.39500.36 2.0373 -0.9373 1.0000 0.5358 1.4510 -0.7011 1.0000 0.3916 1.7453 -0.7128 1.0000 0.40120.4 2.0412 -0.9393 1.0000 0.5524 1.3972 -0.6925 1.0000 0.4092 1.7829 -0.7364 1.0000 0.4219

0.44 1.8966 -0.9057 1.0000 0.5600 1.3081 -0.6785 1.0000 0.4251 1.6708 -0.7148 1.0000 0.43960.5 1.5766 -0.8357 1.0000 0.5658 1.0905 -0.6402 1.0000 0.4486 1.3791 -0.6481 1.0000 0.4659

0.55 1.3683 -0.7909 1.0000 0.5678 0.9405 -0.6134 1.0000 0.4616 1.1859 -0.6031 1.0000 0.48080.6 1.2193 -0.7593 1.0000 0.5685 0.8299 -0.5944 1.0000 0.4699 1.0459 -0.5707 1.0000 0.4906

0.667 1.0380 -0.7209 1.0000 0.5694 0.6953 -0.5713 1.0000 0.4799 0.8757 -0.5314 1.0000 0.50250.7 0.9158 -0.6959 1.0000 0.5700 0.5954 -0.5543 1.0000 0.4867 0.7392 -0.4998 1.0000 0.5112

0.75 0.7412 -0.6602 1.0000 0.5708 0.4527 -0.5302 1.0000 0.4965 0.5444 -0.4547 1.0000 0.52350.8 0.6212 -0.6371 1.0000 0.5719 0.3418 -0.5106 1.0000 0.5038 0.3623 -0.4116 1.0000 0.5335

0.85 0.5083 -0.6155 1.0000 0.5728 0.2376 -0.4923 1.0000 0.5106 0.1913 -0.3712 1.0000 0.54280.9 0.2964 -0.5761 1.0000 0.5760 0.0693 -0.4630 1.0000 0.5215 -0.1385 -0.2932 1.0000 0.5598

0.95 0.0614 -0.5335 1.0000 0.5803 -0.0415 -0.4494 1.0000 0.5296 -0.3583 -0.2461 1.0000 0.57391.0 -0.1915 -0.4913 1.0000 0.5854 -0.0967 -0.4555 1.0000 0.5354 -0.4193 -0.2456 1.0000 0.58521.1 -0.4301 -0.4563 1.0000 0.5904 -0.1041 -0.4806 1.0000 0.5401 -0.3485 -0.2856 1.0000 0.59361.2 -0.6336 -0.4304 1.0000 0.5941 -0.0738 -0.5215 1.0000 0.5450 -0.2165 -0.3463 1.0000 0.59961.3 -0.8156 -0.4103 1.0000 0.5953 -0.0320 -0.5691 1.0000 0.5511 -0.0920 -0.4091 1.0000 0.60351.4 -1.0118 -0.3912 1.0000 0.5931 -0.0357 -0.6071 1.0000 0.5593 -0.0276 -0.4612 1.0000 0.60631.5 -1.2503 -0.3703 1.0000 0.5874 -0.1493 -0.6191 1.0000 0.5697 -0.0722 -0.4911 1.0000 0.60871.7 -1.5259 -0.3501 1.0000 0.5785 -0.3975 -0.6017 1.0000 0.5819 -0.2535 -0.4925 1.0000 0.61172.0 -1.7950 -0.3397 1.0000 0.5674 -0.7453 -0.5663 1.0000 0.5950 -0.5395 -0.4756 1.0000 0.61602.2 -1.9108 -0.3426 1.0000 0.5611 -0.9419 -0.5467 1.0000 0.6018 -0.7079 -0.4662 1.0000 0.61872.6 -2.0796 -0.3504 1.0000 0.5508 -1.2418 -0.5188 1.0000 0.6119 -0.9767 -0.4513 1.0000 0.62333.0 -2.1924 -0.3596 1.0000 0.5428 -1.4567 -0.5011 1.0000 0.6189 -1.1824 -0.4400 1.0000 0.62683.4 -2.3459 -0.3686 1.0000 0.5302 -1.7104 -0.4873 1.0000 0.6263 -1.5020 -0.4122 1.0000 0.63104.0 -2.4736 -0.3683 1.0000 0.5170 -1.8745 -0.4834 1.0000 0.6284 -1.7876 -0.3769 1.0000 0.6319

64

Table 3.4 Standard deviations for the Northridge Earthquake compared with standarddeviations from Somerville and Abrahamson (Somerville, personal comm.). Values of thestandard deviation of the sample standard deviation are given in parenthesis.

Period This StudySite B

This StudySite C

This StudySite D

Somerville &Abrahamson:

Rock

Somerville &Abrahamson:

SoilPGA .32 (.07) .47 (.04) .36 (.03) .53 .480.3 .40 (.08) .54 (.05) .41 (.04) .60 .511 .45 (.11) .60 (.05) .36 (.03) .62 .482 .48 (.12) .57 (.05) .41 (.04) .57 .60

65

Table 3.5 Subdivision of sites classified according to the presented classification systemby means of the 1997 UBC shear wave velocity-based classification system.

NORTHRIDGE

SiteClassification

(from this work)

sV basedClassification

Number ofsites

B UBC B 11UBC C 0

C UBC B 9UBC C 41UBC D 20

D UBC C 5UBC D 54

LOMA PRIETA

Site Classification(from this work)

sV basedClassification

Numberof sites

B UBC B 13UBC C 5

C UBC B 1UBC C 21UBC D 4

D UBC C 1UBC D 18

66

Table 3.6 Comparison of standard errors at selected periods for an analysis based on theclassification system presented herein and an analysis based on the 1997 UBC averageshear wave velocity-based classification system. Values in parenthesis are standarddeviations of the estimate of the standard error.

Northridge Loma PrietaT = 0.3 s T = 1.0 s T = 0.3 s T = 1.0 s

Site ThisStudy

UBC ThisStudy

UBC ThisStudy

UBC ThisStudy

UBC

B .40(.08) .46(.07) .45(.11) .52(.09) .51(.10) .52(.10) .58(.11) .61(.11)C .54(.05) .54(.06) .60(.05) .54(.06) .38(.05) .36(.05) .53(.08) .52(.07)D .41(.04) .42(.03) .36(.03) .41(.03) .39(.07) .39(.06) .59(.11) .64(.10)

67

Table 3.7a Spectral acceleration amplification factors with respect to Site B and standarddeviations for corresponding soil type. Geometric mean of the Loma Prieta andNorthridge earthquakes.

Site C Site DT PGA =

0.1 gPGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ PGA =0.1 g

PGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ

PGA 1.43 1.35 1.31 1.28 0.39 1.75 1.58 1.49 1.43 0.370.055 1.30 1.20 1.14 1.11 0.42 1.57 1.37 1.27 1.20 0.390.06 1.30 1.20 1.14 1.11 0.42 1.57 1.37 1.27 1.20 0.400.07 1.30 1.20 1.14 1.11 0.42 1.57 1.37 1.27 1.20 0.400.08 1.30 1.20 1.14 1.10 0.43 1.57 1.37 1.27 1.20 0.400.09 1.31 1.20 1.14 1.10 0.43 1.58 1.38 1.28 1.21 0.400.1 1.32 1.21 1.15 1.11 0.43 1.59 1.39 1.29 1.21 0.40

0.11 1.33 1.22 1.16 1.11 0.44 1.61 1.40 1.29 1.22 0.400.12 1.35 1.23 1.16 1.12 0.44 1.62 1.41 1.30 1.23 0.400.13 1.36 1.24 1.17 1.13 0.44 1.63 1.42 1.31 1.24 0.400.14 1.38 1.25 1.19 1.14 0.45 1.65 1.44 1.33 1.26 0.390.15 1.39 1.27 1.20 1.15 0.45 1.68 1.46 1.35 1.27 0.390.16 1.41 1.28 1.21 1.16 0.45 1.70 1.48 1.37 1.29 0.390.17 1.42 1.29 1.22 1.17 0.46 1.72 1.50 1.38 1.30 0.390.18 1.43 1.29 1.22 1.17 0.46 1.73 1.50 1.39 1.31 0.390.19 1.44 1.30 1.23 1.18 0.45 1.74 1.52 1.40 1.32 0.390.2 1.45 1.31 1.23 1.18 0.45 1.76 1.54 1.42 1.34 0.39

0.24 1.46 1.31 1.23 1.17 0.46 1.79 1.56 1.44 1.36 0.390.28 1.46 1.30 1.22 1.16 0.46 1.80 1.57 1.46 1.38 0.400.3 1.46 1.30 1.21 1.15 0.46 1.81 1.58 1.47 1.39 0.40

0.34 1.44 1.29 1.20 1.14 0.47 1.83 1.60 1.49 1.40 0.400.36 1.44 1.28 1.20 1.14 0.47 1.83 1.61 1.50 1.42 0.410.4 1.42 1.27 1.19 1.13 0.48 1.83 1.62 1.51 1.43 0.42

0.44 1.41 1.26 1.18 1.13 0.49 1.84 1.63 1.52 1.45 0.430.5 1.38 1.25 1.18 1.13 0.51 1.85 1.66 1.55 1.48 0.44

0.55 1.36 1.24 1.17 1.13 0.52 1.85 1.67 1.57 1.50 0.440.6 1.35 1.24 1.17 1.13 0.53 1.86 1.68 1.59 1.52 0.44

0.667 1.34 1.23 1.17 1.13 0.53 1.87 1.70 1.60 1.54 0.450.7 1.33 1.23 1.17 1.13 0.54 1.88 1.71 1.62 1.56 0.45

0.75 1.32 1.23 1.18 1.14 0.55 1.89 1.73 1.64 1.58 0.450.8 1.32 1.23 1.18 1.14 0.55 1.91 1.75 1.67 1.61 0.45

0.85 1.31 1.23 1.19 1.15 0.56 1.92 1.77 1.69 1.63 0.450.9 1.31 1.24 1.20 1.18 0.56 1.95 1.81 1.73 1.67 0.46

0.95 1.31 1.26 1.22 1.20 0.57 1.98 1.85 1.78 1.72 0.461.0 1.31 1.27 1.25 1.23 0.57 2.02 1.89 1.83 1.78 0.471.1 1.31 1.29 1.27 1.26 0.57 2.05 1.94 1.88 1.84 0.471.2 1.31 1.30 1.29 1.29 0.57 2.09 1.99 1.94 1.90 0.481.3 1.32 1.32 1.32 1.32 0.57 2.12 2.04 2.00 1.96 0.481.4 1.32 1.33 1.34 1.34 0.58 2.15 2.09 2.05 2.02 0.491.5 1.32 1.35 1.36 1.36 0.58 2.18 2.13 2.10 2.08 0.491.7 1.33 1.36 1.37 1.38 0.58 2.22 2.18 2.16 2.14 0.502.0 1.33 1.37 1.38 1.39 0.58 2.25 2.22 2.20 2.18 0.512.2 1.33 1.37 1.39 1.40 0.58 2.26 2.24 2.22 2.20 0.522.6 1.33 1.37 1.39 1.40 0.58 2.27 2.25 2.23 2.22 0.523.0 1.33 1.37 1.39 1.40 0.57 2.27 2.25 2.24 2.23 0.51

68

Table 3.7b Spectral acceleration amplification factors with respect to Site B and standarddeviations for corresponding soil type. Variance weighted geometric mean of the LomaPrieta and Northridge earthquakes.

Site C Site DT PGA =

0.1 gPGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ PGA =0.1 g

PGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ

PGA 1.45 1.37 1.33 1.30 0.37 1.74 1.60 1.53 1.48 0.360.055 1.29 1.19 1.13 1.09 0.42 1.56 1.38 1.28 1.22 0.370.06 1.29 1.19 1.13 1.09 0.42 1.56 1.38 1.28 1.22 0.380.07 1.29 1.19 1.13 1.10 0.43 1.56 1.38 1.28 1.22 0.380.08 1.30 1.19 1.14 1.10 0.43 1.57 1.38 1.29 1.23 0.380.09 1.31 1.20 1.15 1.11 0.44 1.57 1.39 1.30 1.24 0.390.1 1.33 1.22 1.17 1.13 0.44 1.58 1.41 1.32 1.26 0.39

0.11 1.35 1.24 1.18 1.14 0.44 1.59 1.42 1.34 1.28 0.380.12 1.36 1.25 1.19 1.16 0.45 1.60 1.44 1.35 1.29 0.380.13 1.38 1.27 1.21 1.17 0.45 1.61 1.45 1.36 1.31 0.380.14 1.40 1.28 1.22 1.18 0.45 1.63 1.47 1.38 1.32 0.380.15 1.41 1.30 1.23 1.19 0.46 1.66 1.49 1.40 1.34 0.380.16 1.43 1.31 1.24 1.20 0.46 1.68 1.50 1.41 1.35 0.380.17 1.44 1.31 1.25 1.20 0.46 1.70 1.52 1.42 1.36 0.390.18 1.45 1.32 1.25 1.20 0.46 1.72 1.53 1.43 1.36 0.390.19 1.45 1.32 1.25 1.21 0.46 1.73 1.54 1.44 1.37 0.390.2 1.46 1.33 1.25 1.21 0.45 1.75 1.56 1.45 1.39 0.39

0.24 1.47 1.32 1.24 1.19 0.45 1.79 1.58 1.47 1.40 0.400.28 1.47 1.32 1.23 1.18 0.45 1.80 1.60 1.49 1.41 0.400.3 1.46 1.31 1.23 1.17 0.45 1.81 1.61 1.50 1.42 0.41

0.34 1.45 1.30 1.21 1.16 0.46 1.83 1.63 1.52 1.44 0.410.36 1.44 1.29 1.21 1.15 0.46 1.84 1.64 1.53 1.45 0.410.4 1.43 1.28 1.20 1.15 0.48 1.84 1.65 1.55 1.48 0.42

0.44 1.42 1.28 1.20 1.15 0.49 1.85 1.67 1.57 1.50 0.420.5 1.39 1.26 1.20 1.15 0.51 1.86 1.70 1.61 1.55 0.42

0.55 1.38 1.26 1.20 1.15 0.53 1.87 1.72 1.63 1.57 0.410.6 1.36 1.25 1.19 1.15 0.53 1.87 1.73 1.65 1.60 0.41

0.667 1.35 1.25 1.19 1.15 0.54 1.88 1.75 1.68 1.63 0.400.7 1.34 1.25 1.19 1.15 0.55 1.89 1.77 1.70 1.65 0.40

0.75 1.33 1.24 1.19 1.16 0.56 1.91 1.79 1.72 1.68 0.390.8 1.33 1.24 1.20 1.16 0.56 1.92 1.81 1.75 1.70 0.38

0.85 1.32 1.24 1.20 1.17 0.57 1.94 1.83 1.77 1.72 0.380.9 1.32 1.25 1.22 1.19 0.57 1.97 1.86 1.80 1.76 0.37

0.95 1.32 1.26 1.23 1.21 0.58 2.00 1.90 1.84 1.80 0.371.0 1.32 1.28 1.25 1.24 0.58 2.04 1.94 1.89 1.85 0.371.1 1.32 1.29 1.28 1.26 0.58 2.08 1.99 1.94 1.90 0.371.2 1.32 1.31 1.30 1.29 0.58 2.12 2.03 1.99 1.95 0.371.3 1.33 1.32 1.32 1.32 0.58 2.15 2.08 2.04 2.01 0.371.4 1.33 1.34 1.34 1.34 0.58 2.19 2.13 2.09 2.06 0.381.5 1.33 1.35 1.36 1.36 0.58 2.23 2.17 2.14 2.11 0.381.7 1.34 1.36 1.37 1.38 0.58 2.28 2.23 2.19 2.17 0.392.0 1.35 1.37 1.38 1.39 0.58 2.31 2.27 2.24 2.21 0.432.2 1.35 1.37 1.39 1.39 0.57 2.32 2.28 2.25 2.23 0.442.6 1.35 1.37 1.39 1.40 0.56 2.33 2.29 2.26 2.24 0.433.0 1.34 1.37 1.39 1.41 0.54 2.32 2.29 2.27 2.25 0.41

69

Table 3.8a Spectral acceleration amplification factors with respect to Site C and standarddeviations for corresponding soil type. Geometric mean of the Loma Prieta andNorthridge earthquakes.

Site B Site DT PGA =

0.1 gPGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ PGA =0.1 g

PGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ

0.68 0.72 0.74 0.76 0.40 1.24 1.18 1.15 1.12 0.370.055 0.74 0.81 0.85 0.88 0.45 1.23 1.16 1.13 1.10 0.390.06 0.74 0.81 0.85 0.88 0.45 1.23 1.16 1.13 1.10 0.400.07 0.74 0.81 0.85 0.88 0.44 1.23 1.16 1.13 1.10 0.400.08 0.74 0.81 0.85 0.88 0.44 1.23 1.16 1.13 1.10 0.400.09 0.73 0.80 0.85 0.88 0.42 1.23 1.17 1.13 1.11 0.400.1 0.72 0.80 0.84 0.88 0.41 1.22 1.17 1.13 1.11 0.40

0.11 0.72 0.79 0.84 0.87 0.40 1.22 1.17 1.13 1.11 0.400.12 0.71 0.78 0.83 0.86 0.39 1.22 1.17 1.13 1.11 0.400.13 0.70 0.78 0.82 0.86 0.38 1.22 1.17 1.13 1.11 0.400.14 0.69 0.77 0.81 0.85 0.38 1.22 1.17 1.13 1.11 0.390.15 0.68 0.76 0.80 0.84 0.38 1.22 1.17 1.14 1.11 0.390.16 0.68 0.75 0.80 0.83 0.38 1.22 1.17 1.14 1.12 0.390.17 0.67 0.74 0.79 0.82 0.39 1.22 1.17 1.14 1.12 0.390.18 0.66 0.74 0.79 0.82 0.39 1.22 1.18 1.15 1.13 0.390.19 0.66 0.74 0.78 0.82 0.40 1.23 1.18 1.15 1.13 0.390.2 0.65 0.73 0.78 0.81 0.41 1.23 1.18 1.16 1.14 0.39

0.24 0.65 0.73 0.78 0.82 0.43 1.24 1.20 1.18 1.17 0.390.28 0.65 0.73 0.78 0.82 0.44 1.24 1.21 1.20 1.19 0.400.3 0.65 0.73 0.78 0.83 0.46 1.25 1.22 1.21 1.20 0.40

0.34 0.65 0.74 0.79 0.83 0.47 1.27 1.25 1.24 1.23 0.400.36 0.66 0.74 0.80 0.84 0.48 1.28 1.26 1.25 1.24 0.410.4 0.67 0.75 0.80 0.84 0.49 1.29 1.28 1.27 1.26 0.42

0.44 0.67 0.75 0.81 0.84 0.49 1.31 1.29 1.28 1.28 0.430.5 0.69 0.77 0.81 0.85 0.49 1.34 1.33 1.32 1.31 0.44

0.55 0.70 0.77 0.82 0.85 0.49 1.36 1.34 1.33 1.33 0.440.6 0.71 0.78 0.82 0.85 0.49 1.38 1.36 1.35 1.34 0.44

0.667 0.71 0.78 0.82 0.85 0.49 1.39 1.37 1.36 1.36 0.450.7 0.72 0.78 0.82 0.85 0.49 1.41 1.39 1.38 1.37 0.45

0.75 0.73 0.79 0.82 0.85 0.49 1.43 1.41 1.39 1.39 0.450.8 0.73 0.79 0.82 0.85 0.49 1.45 1.42 1.41 1.40 0.45

0.85 0.74 0.79 0.82 0.84 0.50 1.47 1.44 1.42 1.41 0.450.9 0.74 0.78 0.81 0.83 0.50 1.50 1.46 1.44 1.42 0.46

0.95 0.75 0.78 0.80 0.81 0.51 1.53 1.48 1.45 1.43 0.461.0 0.75 0.77 0.79 0.80 0.52 1.55 1.50 1.47 1.45 0.471.1 0.75 0.77 0.77 0.78 0.53 1.58 1.52 1.49 1.46 0.471.2 0.75 0.76 0.76 0.76 0.53 1.61 1.54 1.51 1.48 0.481.3 0.76 0.75 0.75 0.75 0.54 1.63 1.56 1.52 1.50 0.481.4 0.76 0.75 0.74 0.74 0.54 1.65 1.58 1.54 1.52 0.491.5 0.76 0.74 0.73 0.73 0.53 1.67 1.60 1.56 1.53 0.491.7 0.75 0.73 0.72 0.72 0.53 1.70 1.63 1.59 1.56 0.502.0 0.75 0.73 0.72 0.71 0.52 1.72 1.65 1.61 1.58 0.512.2 0.75 0.73 0.72 0.71 0.52 1.72 1.65 1.61 1.59 0.522.6 0.75 0.73 0.72 0.71 0.52 1.73 1.66 1.62 1.59 0.523.0 0.76 0.73 0.72 0.71 0.51 1.73 1.66 1.63 1.60 0.51

70

Table 3.8b Spectral acceleration amplification factors with respect to Site C and standarddeviations for corresponding soil type. Variance weighted geometric mean of the LomaPrieta and Northridge earthquakes.

Site B Site D

T PGA =0.1 g

PGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ PGA =0.1 g

PGA =0.2 g

PGA =0.3 g

PGA =0.4 g

σσ

PGA 0.67 0.71 0.73 0.75 0.36 1.17 1.15 1.14 1.14 0.360.055 0.74 0.81 0.86 0.89 0.45 1.14 1.12 1.10 1.09 0.370.06 0.74 0.81 0.86 0.89 0.45 1.14 1.12 1.10 1.09 0.380.07 0.74 0.81 0.86 0.89 0.44 1.14 1.12 1.10 1.09 0.380.08 0.74 0.81 0.85 0.89 0.44 1.14 1.12 1.11 1.10 0.380.09 0.73 0.80 0.85 0.88 0.42 1.14 1.12 1.11 1.10 0.390.1 0.72 0.79 0.83 0.86 0.40 1.14 1.12 1.11 1.10 0.39

0.11 0.71 0.78 0.82 0.85 0.39 1.13 1.12 1.11 1.10 0.380.12 0.70 0.77 0.81 0.84 0.37 1.13 1.12 1.11 1.10 0.380.13 0.70 0.76 0.80 0.83 0.36 1.13 1.12 1.11 1.10 0.380.14 0.69 0.75 0.79 0.82 0.36 1.13 1.12 1.11 1.10 0.380.15 0.68 0.74 0.79 0.81 0.36 1.13 1.12 1.11 1.10 0.380.16 0.67 0.74 0.78 0.81 0.36 1.14 1.12 1.11 1.10 0.380.17 0.66 0.73 0.77 0.81 0.37 1.14 1.13 1.12 1.11 0.390.18 0.66 0.73 0.77 0.81 0.38 1.15 1.13 1.12 1.11 0.390.19 0.66 0.73 0.77 0.80 0.38 1.15 1.14 1.13 1.12 0.390.2 0.65 0.72 0.77 0.80 0.39 1.16 1.15 1.14 1.13 0.39

0.24 0.65 0.72 0.77 0.81 0.42 1.18 1.17 1.16 1.16 0.400.28 0.65 0.72 0.78 0.81 0.43 1.19 1.18 1.18 1.18 0.400.3 0.65 0.73 0.78 0.82 0.44 1.20 1.20 1.19 1.19 0.41

0.34 0.65 0.73 0.79 0.83 0.46 1.23 1.22 1.22 1.22 0.410.36 0.66 0.74 0.79 0.83 0.47 1.24 1.24 1.24 1.24 0.410.4 0.66 0.74 0.79 0.83 0.48 1.25 1.26 1.26 1.26 0.42

0.44 0.67 0.75 0.80 0.84 0.48 1.27 1.28 1.28 1.28 0.420.5 0.69 0.76 0.80 0.84 0.47 1.30 1.31 1.32 1.33 0.42

0.55 0.69 0.76 0.81 0.84 0.47 1.32 1.34 1.34 1.35 0.410.6 0.70 0.77 0.81 0.84 0.46 1.34 1.36 1.37 1.38 0.41

0.667 0.71 0.77 0.81 0.84 0.46 1.36 1.38 1.39 1.40 0.400.7 0.72 0.78 0.81 0.84 0.46 1.37 1.40 1.41 1.42 0.40

0.75 0.72 0.78 0.81 0.84 0.46 1.39 1.42 1.43 1.45 0.390.8 0.73 0.78 0.81 0.84 0.47 1.41 1.44 1.45 1.47 0.38

0.85 0.73 0.78 0.81 0.83 0.48 1.43 1.45 1.47 1.48 0.380.9 0.74 0.78 0.80 0.82 0.49 1.45 1.48 1.49 1.50 0.37

0.95 0.74 0.77 0.79 0.81 0.50 1.48 1.50 1.51 1.52 0.371.0 0.74 0.77 0.78 0.80 0.51 1.50 1.52 1.53 1.54 0.371.1 0.75 0.76 0.77 0.78 0.51 1.53 1.54 1.55 1.56 0.371.2 0.75 0.76 0.76 0.77 0.52 1.55 1.56 1.57 1.58 0.371.3 0.75 0.75 0.75 0.75 0.53 1.58 1.59 1.59 1.60 0.371.4 0.75 0.74 0.74 0.74 0.53 1.60 1.61 1.61 1.62 0.381.5 0.75 0.74 0.73 0.73 0.53 1.63 1.63 1.63 1.64 0.381.7 0.74 0.73 0.72 0.72 0.52 1.67 1.66 1.66 1.67 0.392.0 0.74 0.73 0.72 0.72 0.52 1.70 1.69 1.69 1.69 0.432.2 0.74 0.73 0.72 0.71 0.52 1.71 1.70 1.70 1.70 0.442.6 0.74 0.73 0.72 0.71 0.51 1.71 1.71 1.71 1.71 0.433.0 0.75 0.73 0.72 0.71 0.51 1.72 1.71 1.70 1.70 0.41

71

Table 3.9a Short-period (Fa) and mid-period (Fv) spectral amplification factors from the1997 Uniform Building Code.

Fa PGA = .08 g PGA = .15 g PGA = .2 g PGA = .3 g PGA = .4 gB 1.0 1.0 1.0 1.0 1.0C 1.2 1.2 1.2 1.1 1.0D 1.6 1.5 1.4 1.2 1.1

Fv PGA = .08 g PGA = .15 g PGA = .2 g PGA = .3 g PGA = .4 gB 1.0 1.0 1.0 1.0 1.0C 1.7 1.7 1.6 1.5 1.4D 2.4 2.1 2.0 1.8 1.6

Table 3.9b Average spectral amplification periods over the short-period range (0.1 s – 0.5s) and the mid-period range (0.4 s – 2.0 s), denoted by Fa and Fv, respectively.

Fa PGA = .08 g PGA = .15 g PGA = .2 g PGA = .3 g PGA = .4 gB 1.0 1.0 1.0 1.0 1.0C 1.5 1.3 1.3 1.2 1.2D 1.8 1.6 1.6 1.5 1.4

Fv PGA = .08 g PGA = .15 g PGA = .2 g PGA = .3 g PGA = .4 gB 1.0 1.0 1.0 1.0 1.0C 1.4 1.3 1.3 1.3 1.2D 2.1 2.0 1.9 1.8 1.8

72

BLANK

73

Figure 3.1. Relationship between structural damage intensity and soil depth in the Caracas earthquake of 1967 (From Seed and Alonso 1974).

74

Figure 3.2a. Shear wave velocity versus depth for a generic stiff clay deposit. Shear wave velocity of underlying bedrock is 1220 m/s.c

Figure 3.2b. Spectral accelerations for the stiff soil deposit shown in Figure 3.2a, with PGA = 0.3 g for a Mw = 8.0 earthquake.

0

25

50

75

100

125

150

0 500 1000 1500 2000 2500

Shear Wave Velocity (m/s)

Dep

th (m

)

00.20.40.60.8

11.21.41.61.8

0 01 0.1 1 10Period (s)

Spec

tral A

ccel

erat

ion

(g)

30 m60 m150 m

Input

Depth to Bedrock

5% damping

75

Figure 3.2c. Spectral acceleration amplification ratio for the stiff soil deposit shown in Figure 3.2a with PGA = 0.3 g for a Mw = 8.0 earthquake. The predominant period of the site is indicated by a circle.

0

1

2

3

4

5

0.01 0.1 1 10Period (s)

Spec

tral A

mpl

ifica

tion

Rat

io

30 m

60 m

150 m

Depth to Bedrock

76

Figure 3.3a. Shear wave velocity profiles for generic sites. Shear wave velocity of underlying bedrock is 1220 m/s.

Figure 3.3b. Spectral acceleration amplification ratio for the soil profiles shown in Figure 3b, with PGA = 0.3 g for a Mw = 8.0 earthquake.

00.5

11.5

22.5

33.5

44.5

5

0.01 0.1 1 10Period (s)

Rat

io o

f Res

pons

e Sp

ectra

Stiff ClaySoft ClayLoose SandDense Sand

0

5

10

15

20

25

30

0 100 200 300 400 500 600 700 800 900

Shear Wave Velocity (m/s)

Dep

th (m

)

Stiff Clay\ Loose Sand Dense Sand Soft Clay

77

Figure 3.4a. Distribution of data by site type for the Northridge Earthquake.

Figure 3.4b. Distribution of data by site type for the Loma Prieta Earthquake.

58.9

1 10 100

Distance (km)

B sites

C sites

D sites

11

28

27

15

11

29

10

9

9

B

C1

C2

C3

D1C

D1S

D2C

D2S

F

1 10 100

Distance (km)

B sites

C sites

D sites

E sites

18

11

11

4

10

2

3

3

7

1

B

C1

C2

C3

D1C

D1S

D2C

D2S

E

F

78

Figure 3.5. Number of recordings as a function of period.

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5

Period (s)

Num

ber o

f Rec

ordi

ngs

B - Northridge

C - Northridge

D - Northridge

C - Loma Prieta

B - Loma Prieta

D - Loma Prieta

79

Figure 3.6. Regression coefficients for the Northridge Earthquake.

a

0

1

2

3

4

5

6

0.01 0.1 1 10Period (s)

a

B and CD

b

-2

-1.6

-1.2

-0.8

0.01 0.1 1 10Period (s)

b

BCD

c

0

4

8

12

16

20

0.01 0.1 1 10Period (s)

c

σσσσ

0.0

0.2

0.4

0.6

0.8

0.01 0.1 1 10Period (s)

σBCD

80

Figure 3.7. Regression coefficients for the Loma Prieta Earthquake. The coefficient "c" is equal to 1.0 for all periods.

a

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

0.01 0.1 1 10Period (s)

a

BCD

b

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.01 0.1 1 10Period (s)

b

BCD

σσσσ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.01 0.1 1 10Period (s)

σ BCD

81

Figure 3.8. Comparison of response spectra before smoothing and after smoothing regression coefficients. Corresponds to the Northridge Earthquake at R = 10 km.

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10

Period (s)

Spec

tral A

ccel

erat

ion

(g's

)

Site B

Site D

Site C

5% damping

82

Figure 3.9. Response spectra for the Northridge Earthquake. Thick lines represent median values, thin lines represent ± one standard deviation.

15 km

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.01 0.1 1 10Period (s)

Spec

tral A

ccel

erat

ion

(g's

)

Site BSite CSite D

5% damping

30 km

0

0.2

0.4

0.6

0.8

0.01 0.1 1 10Period (s)

Spec

tral A

ccel

erat

ion

(g's

)

Site BSite CSite D

5% damping

50 km

0

0.2

0.4

0.6

0.01 0.1 1 10Period (s)

Spec

tral A

ccel

erat

ion

(g's

)

Site BSite CSite D

5% damping

3

3

3

83

Figure 3.10. Response spectra for the Loma Prieta Earthquake. Thick lines represent median values, thin lines represent ± one standard deviation.

15 km

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.01 0.1 1 10Period (s)

Spec

tral A

ccel

erat

ion

(g's

)

Site BSite CSite D

5% damping

30 km

0

0.2

0.4

0.6

0.8

0.01 0.1 1 10Period (s)

Spec

tral A

ccel

erat

ion

(g's

)

Site BSite CSite D

5% damping

50 km

0

0.2

0.4

0.6

0.8

0.01 0.1 1 10Period (s)

Spec

tral A

ccel

erat

ion

(g's

)

Site BSite CSite D

5% damping

4

4

4

84

Figure 3.11. Median spectral values vs. distance for the Northridge Earthquake.

PGA

0.01

0.1

1

10

1 10 100Distance (km)

Spec

tral A

ccel

erat

ion

(g)

Site BSite CSite D

T = 0.3 s

0.01

0.1

1

10


Spec

tral A

ccel

erat

ion

(g)

Site BSite CSite D

T = 1.0 s

0.01

0.1

1

10


Spec

tral A

ccel

erat

ion

(g)

Site BSite CSite D

85

Figure 3.12. Median spectral values vs. distance for the Loma Prieta Earthquake.

PGA

0.01

0.1

1

10


Spec

tral A

ccel

erat

ion

(g)

Site BSite CSite D

T = 0.3 s

0.01

0.1

1

10


Spec

tral A

ccel

erat

ion

(g)

Site BSite CSite D

T = 1.0 s

0.01

0.1

1

10


Spec

tral A

ccel

erat

ion

(g)

Site BSite CSite D

86

Figure 3.13. Comparison of results with an earthquake specific attenuation relationship by Somerville and Abrahamson (1998). Response spectra at 5% damping for the Northridge Earthquake at R = 20 km.

0

0.2

0.4

0.6

0.8

0.01 0.1 1 10Period (s)

Spe

ctra

l Acc

eler

atio

n (g

)

Site B (11 sites)Site C (70 sites)Site D (59 sites)S&A: RockS&A: Soil

5% damping

87

PGA T=0.1 T=0.3 T=1 T=2 UBC B -0.08 -0.08 -0.14 -0.34 -0.44 UBC C -0.08 -0.07 -0.09 0.08 0.10 UBC D 0.14 0.13 0.19 0.19 0.18

Figure 3.14. Residuals with respect to regression analysis for Site C. All sites shown are classified as C sites in the classification system proposed in this study, but are differentiated with respect to their corresponding UBC classification based in the average shear wave in the upper 30 m.

PGA

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

UBC D UBC C UBC B

T = 0.3 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

UBC D UBC C UBC B

T = 1.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

UBC D UBC C UBC B

T = 2.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

UBC D UBC C UBC B

88

PGA T=0.3 T=1 T=3 C1 0.08 0.08 0.10 0.01 C2 -0.08 -0.10 0.01 0.05 C3 -0.10 -0.06 0.05 0.16

Figure 3.15a. Residuals for Site C, Northridge Earthquake. Table gives mean of residuals for each subgroup.

PGA

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

T = 2.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

T = 0.3 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

T = 1.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

89

PGA T=0.3 T=1 T=3 C1 -0.04 0.07 0.05 -0.04 C2 0.07 0.06 -0.01 0.01 C3 -0.08 -0.29 0.10 0.16

Figure 3.15b. Residuals for Site C, Loma Prieta Earthquake. Table gives mean of residuals for each subgroup.

PGA

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

T = 0.3 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

T = 1.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

T = 2.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

C1 C2 C3

90

PGA T=0.3 T=1 T=3 D1C 0.03 0.14 0.13 0.15 D1S 0.00 -0.02 0.01 -0.07 D2C 0.18 0.13 0.18 0.04 D2S -0.26 -0.29 -0.13 0.07 D1 0.01 0.02 0.04 -0.01 D2 -0.03 -0.07 0.04 0.05 DC 0.10 0.14 0.16 0.10 DS -0.06 -0.09 -0.02 -0.04

Figure 3.16a. Residuals for Site D, Northridge Earthquake. Table gives mean of residuals for each subgroup.

PGA

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

T = 0.3 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

T = 1.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

T = 2.0 s

-1.5

-1

-0.5

0

0.5

1

1.5


Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

91

PGA T=0.3 T=1 T=3 D1C 0.00 -0.02 0.21 0.24 D1S 0.15 0.23 0.04 -0.10 D2C 0.20 0.19 0.18 -0.32 D2S -0.46 -0.30 -0.78 -0.78 D1 0.03 0.02 0.18 0.19 D2 -0.13 -0.05 -0.30 -0.55 DC 0.05 0.03 0.20 0.11 DS -0.21 -0.09 -0.45 -0.51

Figure 3.16b. Residuals for Site D, Loma Prieta Earthquake. Table gives mean of residuals for each subgroup.

PGA

-1.5

-1

-0.5

0

0.5

1

1.5

10 100Distance (km)

Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

T = 0.3 s

-1.5

-1

-0.5

0

0.5

1

1.5

10 100Distance (km)

Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

T = 1.0 s

-1.5

-1

-0.5

0

0.5

1

1.5

10 100Distance (km)

Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

T = 2.0 s

-1.5

-1

-0.5

0

0.5

1

1.5

10 100Distance (km)

Res

idua

l (in

Ln

scal

e)

D1C D1S D2C D2S

92

Figure 3.17. Residuals for D sites within the Los Angeles Basin plotted as a function of depth to basement rock.

T = 1.0 s

-1

-0.5

0

0.5

1

0 500 1000 1500 2000 2500 3000

Depth to Basement Rock (m)

Res

idua

l (in

Ln

scal

e)

T = 2.0 s

-1

-0.5

0

0.5

1

0 500 1000 1500 2000 2500 3000

Depth to Basement Rock (m)

Res

idua

l (in

Ln

scal

e)

93

Figure 3.18a. Amplification factors with respect to Site B for the Northridge Earthquake.

C sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

0.01 0.1 1 10Period (s)

Fact

or

PGA = 0.1 gPGA = 0.2 gPGA = 0.3 gPGA = 0.4 g

D sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

0.01 0.1 1 10Period (s)

Fact

or PGA = 0.1 gPGA = 0.2 gPGA = 0.3 gPGA = 0.4 g

94

Figure 3.18b. Amplification factors with respect to Site C for the Northridge Earthquake.

B sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or PGA = 0.1 g

PGA = 0.2 g

PGA = 0.3 g

PGA = 0.4 g

D sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or PGA = 0.1 g

PGA = 0.2 g

PGA = 0.3 g

PGA = 0.4 g

95

Figure 3.18c. Amplification factors with respect to Site B for the Loma Prieta Earthquake.

C sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

0.01 0.1 1 10Period (s)

Fact

or


D sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

0.01 0.1 1 10Period (s)

Fact

or


96

Figure 3.18d. Amplification factors with respect to Site C for the Loma Prieta Earthquake.

B sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or


D sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or


97

Figure 3.19a. Amplification factors with respect to Site B. Geometric mean of the Northridge and Loma Prieta Earthquakes.

C sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

0.01 0.1 1 10Period (s)

Fact

or

D sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

0.01 0.1 1 10Period (s)

Fact

or



98

Figure 3.19b. Amplification factors with respect to Site C. Geometric mean of the Northridge and Loma Prieta Earthquakes.

B sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or

D sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or



99

Figure 3.20a. Amplification factors with respect to Site B. Weighted mean of the Northridge and Loma Prieta Earthquakes.

C sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

0.01 0.1 1 10Period (s)

Fact

or

D sites

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60

0.01 0.1 1 10Period (s)

Fact

or



100

Figure 3.20b. Amplification factors with respect to Site C. Weighted mean of the Northridge and Loma Prieta Earthquakes.

B sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or

D sites

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

0.01 0.1 1 10Period (s)

Fact

or



101

Figure 3.21. Earthquake weighting scheme used for calculating spectral amplification factors. Shown here is an average of the weights used for all periods. Weights are inversely proportional to the sample variance.

D/B ratios

Loma Prieta39% Northridge

61%

D/C ratios

Northridge74%

Loma Prieta26%

C/B Ratios

Northridge54%

Loma Prieta46%

102

Figure 3.22. Short-period (Fa) and intermediate-period (Fv) spectral amplification factors. Dotted lines are code values (UBC 1997), and continuous lines are values obtained from this study.

Fa

0.90

1.10

1.30

1.50

1.70

1.90

2.10

2.30

2.50

0 0.1 0.2 0.3 0.4 0.5PGA (g)

Am

plifi

catio

n Fa

ctor

, Fa

UBC CUBC DC (This Study)D (This Study)

Fv

0.90

1.10

1.30

1.50

1.70

1.90

2.10

2.30

2.50

0 0.1 0.2 0.3 0.4 0.5PGA (g)

Am

plifi

catio

n Fa

ctor

, Fv

UBC CUBC DC (This Study)D (This Study)

103

CHAPTER 4

EMPIRICAL CHARACTERIZATION OF NEAR-FAULT

GROUND MOTIONS

4.1 INTRODUCTION

The estimation of ground motions in close proximity to the ruptured fault for

medium to large magnitude earthquakes must account for the special characteristics of

near-fault ground motions. The near-fault zone is typically assumed to be restricted to

within 10 to 15 km of the causative fault. Of particular importance in the near-fault are

the effects of forward-directivity. Forward-directivity conditions produce ground motions

characterized by a strong pulse or series of pulses best observed in ground velocity-time

histories. The importance of forward-directivity records has prompted a number of

studies directed at addressing the prediction of this type of motions and their effect on

structural response (e.g. Somerville 1998, Krawinkler and Alavi 1998, Sasani and Bertero

2000). These studies have highlighted the importance of characterizing near-fault

records. However, more research is needed to account for the potential effects of local

soil conditions on these motions.

The effects of rupture directivity are generated because the velocity of fault

rupture is only slightly lower than the shear-wave propagation velocity. As the rupture

front propagates from the hypocenter, a shear wave front is formed by the accumulation

of the shear waves traveling ahead of the rupture front. When a site is located at one end

104

of the fault and rupture initiates at the other end of the fault and travels towards the site,

the arrival of the wave front is seen as a large pulse of motion occurring at the beginning

of the record (Somerville et al. 1997). This condition is known as forward-directivity,

and is illustrated in Figure 4.1. The radiation pattern of the shear dislocation on the fault

causes this large pulse of motion to be oriented in a direction perpendicular to the fault

plane (Somerville et al. 1997). The pulse is typically a long-period pulse that is best

observed in the velocity or displacement-time history. However, if a site is located at one

end of the fault and rupture propagates away from the site, the opposite effect is observed

and the motion is characterized by longer duration and lower amplitude ground motions.

This condition is termed backward-directivity.

Pulse-like motions can also be generated by a concentration of high slip in the

region of the fault near the recording site (Abrahamson, personal communication). Slip-

induced pulses, herein termed “fling-step”, have different characteristics than forward-

directivity pulses and are modeled by different terms in seismological fault modeling.

The fling-step normally generates one sided velocity pulses. It is also observed as a

discrete step in displacement-time histories that occurs parallel to strike of the fault with

strike-slip earthquakes, and in the dip direction for dip-slip events. Given their different

characteristics, it is desirable to treat the pulse motions originated from forward-

directivity and fling-step slip effects separately. In strike-slip events, forward-directivity

pulses can be identified by positive fault normal to fault parallel spectral ratios at long

periods. Whereas forward-directivity records have larger fault normal motions at long

periods, slip-induced pulses tend to have equal energy in both directions. In dip-slip

faults, permanent displacement can also occur in the fault normal direction. For these

105

motions, the fling-step should be removed before the effects of a forward-directivity pulse

can be evaluated.

Forward-directivity conditions can be present both for strike-slip and dip-slip

events. In strike-slip events, forward-directivity conditions are typically largest in sites

near the end of the fault when the rupture front is moving towards the site. In dip-slip

events, forward-directivity conditions occur in sites located in the up-dip projection of the

fault.

Currently, there is a lack of data with regards to the effects of site response on the

characteristic of pulse-type motions. This in itself constitutes an important motivating

factor for pursuing research on site effects for near-fault ground motions. Further

understanding of near-fault site effects is important for two reasons. First, pulse-type

motions have been identified as critical in the design of structures in the near-fault region

(e.g. Krawinkler and Alavi 1998, Sasani and Bertero 2000). Preliminary analysis of

elastic and inelastic multiple degree of freedom systems indicated that the amplitude and

period of the pulse are parameters that control the demand on the structure. Site effects

have the potential to significantly alter these parameters. The second important

motivating factor lies in the evaluation of existing near-fault records. Recent events, such

as the 1999 Kocaeli, Turkey, and Chi-Chi, Taiwan earthquakes, have increased

significantly the available data for large magnitude earthquakes in the near-fault. As

those data become available, improved empirical characterization of near-fault ground

motions is possible. A complete understanding of the effects of site conditions on these

106

motions will greatly aid in the development of predictive relationships for near-fault

events.

This chapter presents an evaluation of the currently available recordings of near-

fault, forward-directivity motions. Section 4.2 presents a review of the current

attenuation relationships used to predict near-fault effects in the frequency domain.

Section 4.3 studies the time domain representation of forward-directivity, near-fault

motions. Section 4.4 presents the development of base-line pulse-motions to be used as

input for site response analyses in Chapter 6. All the analyses in the remaining parts of

this work concentrate on pulses generated by forward-directivity effects.

4.2 FREQUENCY DOMAIN REPRESENTATION OF NEAR-

FAULT GROUND MOTIONS

In current practice, rupture directivity effects are generally taken into account by

modifications to the elastic acceleration response spectrum (at 5% damping). A detailed

model for the amplitude and duration effects of rupture directivity is presented by

Somerville et al. (1997). This model is widely used in conjunction with attenuation

relationships for the estimation of ground motions in the near-fault region. Besides from

its application on ground motion estimation, the model is important in its development of

a parameterization of the geometric conditions that lead to forward and backward

directivity conditions.

The ground motion parameters that are modified to account for the effects of

directivity are the average horizontal response spectra, the ratios of fault normal to fault

107

parallel response spectra, and the duration of the ground motion. The model parameters

used to define the geometric conditions for rupture directivity are illustrated in Figure 4.2.

The spatial variation of directivity effects depends on two parameters. First, the angle

between the direction of rupture propagation (θ for strike-slip faults, and φ for dip-slip

faults) and the direction of waves travelling from the fault to the site. Second, the

fraction of the fault rupture surface (X for strike-slip faults and Y for dip-slip faults) that

lies between the hypocenter and the site. The smaller the angle, the larger the directivity

conditions that are experienced at a site. Similarly, if a larger fraction of the fault lies

between the site and the hypocenter, the effects of directivity are larger. Somerville et al.

(1997) chose to model rupture directivity effects using the functions Xcosθ and Ycosφ for

strike-slip and dip-slip fault, respectively. The equations developed by Somerville et al.

(1997) are presented in Table 4.1 and their effect on spectral amplification ratios are

illustrated in Figure 4.3. The parameters that identify forward-directivity conditions in

the Somerville model can also be used to identify sites with a potential for experiencing

forward-directivity conditions. This is discussed further in Section 4.3.

The 1997 UBC accounts for near-fault effects by means of near-fault factors Na

and Nv applied to the low period (acceleration) and intermediate period (velocity) parts of

the acceleration response spectrum, respectively. The near source factors are specified

for distances less than 15 km and for three different fault types (Table 4.2). The near-

source factors in the UBC are compatible with the average of the fault normal and fault

parallel component in the Somerville et al. (1997) model. However, the code does not

specifically address the larger ground motions in the fault normal component (Somerville

1998).

108

4.3 TIME DOMAIN REPRESENTATION OF NEAR-FAULT

GROUND MOTIONS

4.3.1 General

Ground motion recordings at sites subject to forward-directivity are characterized

by the arrival of most of the seismic energy in a single large pulse of motion at the

beginning of the record (Somerville 1997). As described earlier, this large pulse of

motion is oriented in the direction perpendicular to the strike of the fault. Although some

design codes characterize near-fault ground motions by means of an amplified

acceleration response spectrum, there is a growing recognition that a time history

representation is better able to capture the effects of near-fault ground motions on

structures (e.g. Somerville 1998, Alavi and Krawinkler, 2000, Sasani and Bertero, 2000).

A time-domain representation is desired because the frequency domain characterization

of ground motion (i.e. through a response spectrum) implies a stochastic process having a

relatively uniform distribution of energy throughout the duration of the motion. When

the energy is concentrated in a single pulse of motion, the resonance phenomenon that the

response spectrum was conceived to represent has insufficient time to build up

(Somerville 1998). Somerville (1998) states:

… the inadequacy of the response spectrum as a sole design criterion becomes readily apparent when time histories are selected to represent a design response spectrum. It is well known that a suite of different ground motion time histories which all match the same response spectrum produce variations in the response of a structure subjected to non-linear time history analysis. However, when the input time history is a near-fault pulse this effect is accentuated to the point where small modifications of a near-fault

109

time history that have no significant effect on the response spectrum can have a major effect on the response of a structure when subjected to non-linear time history analysis. This demonstrates that the current standard of practice does not provide a reliable basis for providing near-fault ground motions time histories that are specified solely on the basis of a design response spectrum.

Studies by structural engineers, such as Krawinkler and Alavi (1998) and Sasani

and Bertero (2000), have shown that simplified representations of the velocity pulse are

capable of capturing the salient response features of structures subjected to near-fault

ground motions. The model by Krawinkler represents the observed pulses in a velocity

time history by means of the zero crossings and peak ground velocities of the dominant

pulses in a ground motion. The peaks can be joined by straight lines or can be fitted by

means of a spline to obtain a better fit (Somerville 1998). The period of the equivalent

pulse in Krawinkler's model is identified by a clear and global peak in the velocity

response spectrum of the ground motion. The equivalent pulse amplitude is obtained by

minimizing the differences between the maximum story ductility demand from the near-

fault record and the corresponding demand obtained from the equivalent pulse for a

certain range of ductility. Results indicated that in almost all cases, the equivalent pulse

velocity lies within 20% of the peak ground velocity of the record (Alavi and Krawinkler

2000). The simplified pulses developed by Krawinkler and Alavi (1998) and Sasani and

Bertero (2000) are shown in Figure 4.4.

These previously mentioned studies point to the importance of studying near-fault

motions in the time domain. In particular, structural response is sensitive to the long

period pulses that are best observed in a velocity time history. In the following sections,

110

the available empirical database of near-fault ground motions is evaluated with two main

objectives. First, to obtain a characterization of forward-directivity motions that includes

effects of site conditions, and second, to develop simplified velocity pulses that will be

used as input motions in site response studies. The results of the site response studies

will be validated with the trends observed in the data.

4.3.2 Parameterization of velocity pulses

Velocity time histories of near-fault ground motions can be satisfactorily

approximated by means of simplified pulse shapes. For the sake of simplicity, sine

functions will be used to represent the observed pulses in velocity time histories. Figure

4.5 illustrates the fit of sine pulses to selected near-fault recordings. The advantage of

using sine functions is that velocity pulses can be characterized by a reduced number of

parameters. This, in turn, will facilitate a systematic study of the effects of site response

on the characteristics of velocity pulses. Section 6.3 presents a detailed site response

analysis using the motions in Figure 4.5 as input motions. The site response analyses are

done using both the recorded and the simplified pulses as input motions. Results are

remarkably similar. This serves as further validation for the use of simplified pulses for

the study of site response in the near-fault region.

The larger fault-normal component of motion is considered critical and is

commonly used for structural response studies (Alavi and Krawinkler 2000). Ground

motion in the fault parallel direction, however, can also be significant and can induce

large strains in an affected soil column. Consequently, both directions of motion will be

considered in developing simplified velocity time histories. This is not to say that the

111

fault normal component is not the dominant component in near-fault recordings, but that

significant fault-parallel velocities are sometimes observed to occur in phase with fault-

normal pulses. Figure 4.6 illustrates two different recordings that are both oriented in the

fault normal direction, but have significantly different fault parallel components of

motion. The use of both components of motion will allow the evaluation of the effect of

fault parallel motions in site response. Figure 4.6 includes horizontal velocity-trace plots.

The horizontal velocity-trace plot is a plot of the fault parallel versus the fault normal

components of motion. It represents the magnitude and the orientation of the particle's

velocity during the seismic event. Horizontal velocity-trace plots illustrate well the

orientation of velocity pulses.

The simplified sine-pulse representations of velocity time histories are fully

defined by the number of equivalent half-cycles of motion, the period of each half-cycle,

and their corresponding amplitudes. In general, simplified velocity pulses can be defined

by their peak amplitude or peak ground velocity (PGV), approximate period of the

primary pulse, and the number of half-cycles of pulse motions. The time lag between the

initiation of the fault normal and fault parallel component must also be defined in

addition to the equivalent sine-pulse representation of the fault parallel component.

Figure 4.7 illustrates the parameters needed for a full characterization of the simplified

time histories. In the following section, the methodology to obtain the equivalent sine

pulse parameters is described.

112

Equivalent Pulse Representation of Ground Motions

Near-fault, forward-directivity records were selected from the ground motion

database described in Chapter Three (Silva, personal comm.). This ground motion

database was complemented with records from the 1999 Kocaeli, Turkey. Processing of

these recent records is described in Rathje et al. (2000). The parameters describing

directivity conditions for each station were compiled by Stewart (personal

communication). Forward-directivity conditions were determined by means of the

predicted ratio of fault normal to fault parallel spectral acceleration at a period of three

seconds. This ratio, termed RDI by Stewart (personal comm.) is predicted by an updated

version of the Somerville et al. (1997) model (Table 4.1). For strike-slip faults it is given

by (Stewart, personal comm.):

( ) ( ) ( )( ) ( )

<+−

=otherwise )395.0exp(

4.0cos if cos506.2605.0exp

wMR

wMR

MTRTXMTR TX

RDIθφ

(4.1a)

and for dip-slip faults by:

( ) ( ) ( )EndMTRTYRDI wMR cos559.0327.0exp φ+−= (4.1b)

where X, Y, θ, and φ are defined in Figure 4.2; R is closest distance to the fault plane

(Sadigh et al. 1993); Mw is moment magnitude; End is 0 if the station is on the ends of the

113

fault and 1 otherwise (Figure 4.2); and TR and TM are taper functions given by (Stewart,

personal comm.):

<≤<

=

otherwise 0

km 60 km 30 if 30

30 - -1km 30 if 1

)( RRR

RTR (4.2a)

≤<

>

=

otherwise 0

6.5 km 6.0 if 30

30 - -1

6.5 if 1

)( w

w

wwM MM

M

MT (4.2b)

Geometric conditions for forward-directivity exist for an RDI larger than 1.0. It is

important to note, however, that even when the geometric conditions for forward-

directivity are satisfied, the effects of forward-directivity may not exist. This could

happen if a station is at the end of a fault and rupture occurs towards the station (i.e.

forward-directivity conditions), but slip is concentrated near the end of the fault where the

station is located. Under these conditions, the forward-directivity pulse may not be

generated. Note, however, that forward-directivity conditions occur also from up-dip

rupture propagation. Hence, in a strike slip event where the hypocenter is at depth,

forward-directivity conditions might arise, especially if dip-slip movement along the fault

occurs also.

Earthquakes with moment magnitude equal or larger than 6.1 were considered.

Table 4.3 lists the earthquakes included in the study, along with the fault strike used to

determine fault normal orientation. Stations at a distance (closest distance to the fault

114

plane) less than or equal to 20 km for earthquakes with Mw ≥ 6.5 and 15 km for lower

magnitude events, and an RDI larger than 1.0 were considered. In addition, recordings

not possessing at least some features of forward-directivity characteristics were excluded

from the analysis. Forward-directivity characteristics are positive fault normal to fault

parallel response spectral ratios for long periods, and a reasonably well-defined velocity

pulse in the fault normal direction.

Simplified sine-pulses matching the dominant pulses of the selected ground

motions were developed using the characterization illustrated in Figure 4.7. In addition to

the time domain parameters of the pulse, the period corresponding to a clear and global

peak of the pseudo-velocity response spectra was also calculated for each recording (see

Figure 4.10). Table 4.4 details the methodology used to obtain the parameters in Figure

4.7. Subscripts N and P are used to describe the fault-normal and fault-parallel

components, respectively. While forward-directivity motions generally have well defined

velocity pulses in the fault-normal direction, the fault-parallel direction is not always

pulse-like. Nonetheless, the fault-parallel motion was also fit using the simplified sine-

pulse representation. The simplified representation of these motions have the effect of

filtering out the high frequency content of the pulses. The amplification of velocities in a

site response analysis is affected mainly by intermediate period energy. Consequently,

the simplified sine-pulse representation of the fault parallel motions constitutes a

reasonable scenario when these are used as input motions for a site response analysis. For

characterizing the fault parallel direction, only the time interval in which the pulses occur

for the fault normal component is considered. In most cases, this time interval contains

the peak velocity values in the fault parallel direction. Tables 4.5 and 4.6 lists the

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selected recordings with their corresponding simplified sine-pulse parameters. Table 4.7

lists the stations satisfying the geometrical conditions of forward-directivity that were

excluded from the analysis for not showing the characteristics of forward-directivity

motions.

Previous studies of structural response indicated that pulse period is an important

parameter to characterize the response of structures to pulse-type motions. In order to

simplify the analysis of the empirical database, a single pulse period associated with each

recording is desired. This simplification is justified in large part by the fact that the

energy of forward-directivity pulses is constrained to a relatively narrow band of

frequencies (Somerville 2000).

Three options were considered for the pulse period:

a) The period of the pulse with the maximum amplitude.

b) The weighted average period, which is calculated as the sum of the periods of each

half-cycle of motion weighted by the pulse amplitude and divided by the number of

half-cycles of motion.

c) The period corresponding to a clear and global peak (i.e. maximum value) in the

pseudo-velocity response spectra (Tv-p).

Figure 4.8 shows the relationship between the period of the pulse with maximum

amplitude and the weighted average period. The mean ratio of these two parameters is

0.99 with a standard deviation of 0.19. In almost all cases, the weighted average period is

within 20% of the period of the maximum pulse. Given the large uncertainty associated

with predictions of Tv, these two parameters can be used interchangeably. Note that the

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definition of pulse period in Table 4.4 uses either the zero crossing time or the time at

which velocity is equal to 10% of the peak velocity for this pulse. This is necessary for

pulses in which the pulse is preceded by a small drift in the velocity time history (Figure

4.9). A degree of subjectivity is involved in this definition and can lead to some

variations in the estimates of Tv. However, as will be shown later, the uncertainty

associated with predicting Tv is much larger than the possible errors in estimating Tv from

zero crossings. On the other hand, a parameter such as Tv-p is relatively unambiguous

(Figure 4.10). Table 4.8 shows the ratio of Tv-p to Tv for events for which there were

multiple near-fault recordings. For a velocity time-history defined by a one-cycle sine

pulse, Tv-p is approximately equal to Tv (i.e. Tv-p = 0.95Tv). For the recorded time

histories, the two measures of pulse period coincide for events characterized by one or

two dominant pulses of motion in the velocity time histories (i.e. Imperial Valley and

Northridge). For events with more complex velocity time histories (i.e. Loma Prieta and

Kobe) Tv-p is significantly lower than Tv. The overall average of the ratio between Tv and

Tv-p is 0.84 with a standard deviation of 0.28. A relationship between the two parameters

is plotted in Figure 4.11. Due to potential differences in the magnitude of Tv and Tv-p,

values of both parameters are obtained for the stations listed in Table 4.5. Given that the

emphasis of this work is the time-domain characterization of near-fault forward-

directivity motions, the parameter Tv will be emphasized. The choice also allows a better

match to recorded motions in the time domain.

The coincidence of Tv and Tv-p indicates that the velocity pulse in the ground

motion contains energy in a narrow period band. Figure 4.12 shows the normalized

power spectral density (PSD) of four ground motions. The El Centro records from the

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Imperial Valley earthquake have equal values of Tv-p and Tv. The PSD of these records

shows a clear peak at a frequency close to the frequency of the pulse. The fact that the

peak in the PSD and the pulse frequency (1/Tv) do not coincide is because the pulse is not

a perfect sine pulse. Also included in Figure 4.12 are the Los Gatos Presentation Center

(LGPC) record from the Loma Prieta earthquake and the Pacoima Dam Downstream

record from the Northridge earthquake. These records have significantly lower Tv-p than

Tv. Even tough the PSD of these records has a dominant peak, the motions also contain

energy at higher frequencies. This implies that the motion is not dominated by a single

pulse. The fact that the peak in the spectral velocity plot does not coincide with the

period of the largest pulse is not surprising for these types of records.

4.3.3 Statistical Evaluation of Equivalent Pulse parameters in the Fault Normal

Direction

Number of Significant Pulses

The number of significant pulses in the velocity time-history is an important

parameter for structural response. Multiple cycles of motion can dramatically increase

the damage potential of the ground motions (Alavi and Krawinkler 2000). In the present

study, the number of cycles of motion (referred to as the number of significant pulses) is

defined as the number of half-cycle velocity pulses that have an amplitude at least 50% of

the peak ground velocity of the ground motion (Table 4.4). For evaluating the number of

significant velocity pulses, only the fault-normal component of motion is considered.

Table 4.9 lists the number of significant pulses for the recordings in each of the

earthquakes included in this study. Figures 4.13 to 4.23 show the velocity-time histories

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for all the records used in the analysis. A tabulation of 15 recorded near-fault motions by

Somerville (1998) rendered a similar proportion of number of pulses, where the number

of pulses was defined using the equivalent motion model of Krawinkler (1998).

The 50% level chosen as a cut-off is arbitrary, however, the values listed in Table

4.9 are useful as a guideline for estimating the number of significant velocity pulses in an

earthquake. The number of pulses within the initial pulse sequence with a 33% cut-off is

also included in Table 4.9 to illustrate the sensitivity of number of significant pulses to

the cut-off level. It is important to note that, in general, each earthquake event has a well-

defined pulse sequence for nearly all of its near-fault motions. For example, most of the

recordings in the Imperial Valley earthquake consist of two significant velocity pulses or

a full velocity cycle of motion (Figure 4.15). With the exception of the OSAJ motion, all

the records in the Kobe earthquake consist of four significant velocity pulses or two full

velocity cycles (Figure 4.22). This might be expected for faults that have a relatively

uniform slip distribution or earthquakes where slip is concentrated over a single zone.

For these types of earthquakes, stations that are close to each other will be equidistant to

regions of high slip. Moreover, path effects are minimized for stations in the near-fault

region. For an earthquake with highly non-uniform slip, such as the Northridge

earthquake, the type of pulse sequence observed depends on the instrument distance

relative to the asperities. In fact, Somerville (1998) suggests that the number of half sine

pulses in the velocity time-history might be associated with the number of asperities in a

fault. From the point of view of ground motion prediction, this implies that the prediction

of number of significant velocity pulses in a given earthquake is associated with the

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determination of slip distribution in the causative fault. This of course is a difficult thing

to estimate a priori.

Pulse Period

The equivalent pulse period (Tv) of the velocity time-history and the period

corresponding to the peak of the pseudo-velocity response spectra (Tv-p) were evaluated

for all the sites listed in Table 4.5. In addition, each recording site was classified either as

rock (r) or soil (s). The rock or soil classification is along the lines of the simplified

classification schemes described in Chapter Three (Abrahamson and Silva 1997). In

terms of the geotechnically-based classification system introduced in Chapter Three, the

rock category includes competent rock sites (Site B) and shallow stiff soil/weathered rock

sites (Site C), while the soil category comprises deep stiff clay sites (Site D) and some

soft clay sites (Site E). Note that the grouping of competent rock sites and shallow stiff

soil/weathered rock sites into a single category increased the standard deviations in the

prediction of site response for motions at intermediate to long distances (see Chapter

Three). In light of this, the classification presented herein seems counter-intuitive.

However, results in Chapter Six indicate that for the study of long period velocity pulses,

shallow stiff soil/weathered rock sites do not modify significantly the input rock velocity

pulse. This observation holds only for long pulse periods. Given that the present analysis

concentrates on velocity pulses that typically have long periods, it was deemed

appropriate to follow this simplified rock vs. soil site classification scheme.

Somerville (1998) performed a regression analysis using data from 15 recorded

time histories augmented by 12 simulated time histories. The records correspond to a

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magnitude range of 6.2 to 7.5 and distances of 0 to 10 km. The pulse parameters modeled

were the period and amplitude of the largest cycle of motion in the velocity time-history.

The relationship obtained is

log10Tv = -2.5 + .425 Mw (4.3)

where Tv is the period of the largest cycle of motion (Somerville 1998). In a larger study

of slip distributions using slip models for 15 earthquakes, Somerville et al. (1999) provide

justifications to the use of self-similar scaling relationships to constrain fault parameters.

In a self-similar system, events of different sizes cannot be distinguished except by a scale

factor. Using this self-similar scaling model, the magnitude scaling in the above

relationship is 0.5 and the resulting equation is

log10Tv = -3.0 + .5 Mw (4.4)

The period of the velocity pulse is associated with the rise time of slip in the fault,

which measures the duration of slip at a single point on the fault. Somerville et al. (1999)

performed a regression analysis on the slip duration or rise time using slip models for 15

earthquakes. The relationship obtained is

log10TR = -3.34 + 0.5 Mw (4.5)

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where TR is the rise time. This relationship is also constrained by the self similar scaling

model. Joining Equations 4.4 and 4.5, the relationship between pulse period and rise time

is (Somerville 1998)

Tv = 2.2 TR (4.6)

The relationship between pulse duration and rise time can also be inferred from

the physics of the fault rupture phenomenon. If a fault is modeled as a point and

propagation effects are ignored, the duration of motion would be equal to the rise time

(Somerville 1998). Fault finiteness and propagation effects contribute to widening the

pulse. Rise time is then, in essence, a lower bound for pulse period.

A similar study relating pulse period to moment magnitude was presented by

Alavi and Krawinkler (2000) on the same data set used by Somerville (1998). Alavi and

Krawinkler defined the pulse period as the predominant period in a velocity response

spectrum plot (Tv-p). The relationship obtained is

log10Tv-p = -1.76 + 0.31 Mw (4.7)

Equations 4.6 and 4.7 do not account for the potential effect of local site

conditions on the pulse period. Moreover, a measure of the uncertainty associated with

the estimate of Tv and Tv-p is not provided.

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The records listed in Table 4.5 are used to develop a relationship between Tv and

Mw. A linear relationship between the logarithm of pulse period and moment magnitude

is assumed, consistently with the relationship by Somerville (1998). The number of

records for each earthquake varies from just one in some cases to 13 for the Imperial

Valley earthquake. To avoid a relationship that is controlled by the few events with a

large number of records, a random effects model is used (Abrahamson and Silva 1997).

The random effects model partitions the standard error associated with each data point

into an inter-event term (defines that part of the overall standard error resulting from

scatter between events) and an intra-event term (defines that part of the overall standard

deviation resulting from scatter within each event). Thus, the relationship for pulse

period becomes

ln(Tv)ij = a + bMw + ηi + εij (4.8)

where (Tv)ij is the pulse period of the jth recording from the ith event, a and b are the

model parameters, ηi is the inter-event term, and εij represent the intra-event variations.

The relationship is valid for moment magnitudes 6.1 to 7.4 and for distances less than 20

km. The inter-event and the intra-event error terms are assumed to be independent

normally distributed random variables with variances τ2 and σ2, respectively. The

standard error associated with the estimate of Tv is then

σtotal2 = τ2 + σ2

(4.9)

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The algorithm presented by Abrahamson and Youngs (1992) is used for the

regression analysis. The analysis is performed both on the whole data set, and on the

subsets of rock and soil motions separately. The values of the model parameters resulting

from the analyses are presented in Table 4.10 and illustrated in Figure 4.24. Note that

this relationship was developed with data from earthquakes with moment magnitudes

ranging from 6.1 to a maximum of 7.4. Equation 4.8, along with the parameters in Table

4.10, should be used only within this magnitude ranges. Moreover, the amount of data at

higher magnitudes is limited, so this predictive relationship should be applied with

caution. Finally, all the data are restricted to within 20 km from the fault plane, so the

resulting relationship should not be used for greater distances.

The random effect term associated with inter-event variations can be used to

evaluate the deviation from the median relationship for individual events. For given

model parameters, ηi is given by (Abrahamson and Youngs 1992)

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2

στ

µτη

+

−=

∑=

i

n

jijij

i n

yi

(4.10)

where ni is the number of records for event i, yij is the jth recording for the ith event, µij is

the pulse period predicted by the model (Equation 4.8), and the summation index j

represents a summation over all the recordings of event i. Equation 4.10 illustrates how

the random effect model partitions the error into inter and intra-event terms. For an event

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with a single recording (ni = 1), the percentage of the residuals that is assigned to the

inter-event term is given by the ratio

τ2/(τ2 + σ2) (4.11)

If the number of recordings for a single event is large (ni >> 1), the inter-event term

becomes the mean residual for event i. Table 4.11 gives the values of ni for each of the

11 earthquakes in the data set. Observe that the event term for the Imperial Valley

earthquake is much larger than those for the other events. This in particular illustrates the

need for an adequate way of partitioning the standard error between intra and inter-event

terms. If the inter-event error term is ignored and error is partitioned individually among

all recordings, the Imperial Valley records would control the regression data.

A comparison of the relationship developed herein with those of Somerville

(1998) and Alavi and Krawinkler (2000) is shown in Figure 4.25. The definition of pulse

period of Somerville is similar to that use in this study (Tv). On the other hand, the pulse

period of Alavi and Krawinkler (2000) is the period corresponding to a clear maximum in

the pseudo-velocity response spectra (Tv-p). For comparison purposes, Figure 4.25

includes a regression line for Tv-p from this study. The parameters for the regression are

included in Table 4.10. For lower magnitudes, the relationships developed in this study

render lower pulse periods than those of Somerville and Alavi and Krawinkler. For larger

magnitudes (Mw > 7.0), the regression line for Tv matches that of Somerville, while the

regression line for Tv-p matches the Alavi and Krawinkler line.

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The relationship illustrated in Figure 4.24 has important implications both for

design and for the further evaluation of the existing database of near-fault motions. The

regression analysis predicts longer periods at soil sites than at rock sites for lower

magnitude earthquake events. This difference diminishes as magnitude increases and

disappears for large magnitudes. From the point of view of structural design, the longer

periods predicted at soil sites for earthquakes of low to intermediate magnitudes might

lead to different design considerations when dealing with near-fault ground motions.

From the point of view of ground motion prediction, Figure 4.24 illustrates the

importance of considering local site effects in the evaluation of near-fault ground

motions. This becomes particularly relevant for the development of future attenuation

relationships that include predictions of near-fault effects.

The significant difference in pulse periods predicted for different magnitude

earthquakes also raises important questions for ground motion estimation. For many

short-period structures, the expected large pulse periods from large magnitude

earthquakes may not produce significant levels of damage for these structures. Lower

magnitude earthquake events may result in velocity pulses with periods closer to the

natural period of short-period structures, which are often more common in urban building

stocks. In this case, the lower magnitude earthquake may result in larger levels of

damage associated with the velocity pulse. Following the same reasoning, ground

motions recorded at near-fault sites for large magnitude earthquakes cannot be assumed

to be “worst-case scenarios” for the design of all structures. It is likely that in the future

development of attenuation relationships, accounting for near-fault effects could result in

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larger spectral accelerations for lower magnitude earthquakes at periods of 1 to 2 seconds,

especially at soil sites.

The standard deviations of the model in Equation 4.8 are relatively high. Figure

4.26 shows the predictive relationships for the median and the plus and minus one

standard deviation values for soil and rock. The standard deviation associated with the

prediction of Tv is larger than the predicted difference between soil and rock sites for the

high magnitude (long period) range. For these cases, however, the regression analysis

indicates a diminished difference between pulse period in soil and rock. However, the

difference in pulse period between soil and rock sites at low magnitudes is significant and

compares with the standard deviation of the estimates.

Additional evidence of period elongation due to site effects can be obtained by

looking at paired stations for individual events. However, in the database of near-fault

motions, the number of paired (rock and soil) stations that are relatively close to each

other and have similar pulse sequences are limited. For the present analysis, the Gilroy

motions from the 1989 Loma Prieta earthquake are selected. Parameters for these

stations are included in Table 4.5 and are repeated for this comparison in Table 4.12.

Figure 4.27 shows the time history and pseudo-velocity response spectral plots for

these motions. The observed fault-normal pulse sequence has the same characteristics, an

initial half-cycle pulse followed by two full cycles of motion. The corresponding pulse

periods, however, vary significantly depending on the soil type, as indicated in Table

4.12, with soil sites having larger velocity pulse periods than rock sites.

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Peak Ground Velocity

The random effects model (Abrahamson and Youngs 1992) was also used to

evaluate the PGVs listed in Table 4.5. PGV is affected by both magnitude and distance.

Attenuation relationships for PGV typically use elaborate functional forms to match the

data over all distance ranges (e.g. Campbell 1997). For near-fault ground motions,

however, the data, by definition, are restricted to relatively short distances, and the

functional forms can be simplified. Somerville (1998) proposed the use of a bilinear

relationship between the logarithm of PGV, magnitude, and the logarithm of distance. To

avoid unrealistic predictions of PGV at short distances, Somerville (1998) used a distance

cut-off at 3 km. A similar relationship was later proposed by Alavi and Krawinkler

(2000). For the study of near-fault data, however, it is desirable to obtain predictions of

PGV at close distances to the fault as well. For this reason, the following functional form

was used:

ln(PGV) = a + b Mw + c ln (R2 + d2) (4.12)

where PGV is in units of cm/s. This functional form results in a nearly zero slope at close

distances to the fault. The relationship becomes linear at larger distances. Other

functional forms, including a functional relationship on magnitude for the distance

scaling term c were attempted without significant improvements on the match of the data.

The functional form given by Equation 4.12 was chosen based on its simplicity.

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The robustness of the predictions of Equation 4.12 depends on the distribution of

the data both with distance and with magnitude (Figure 4.28). Observe that data are

scarce for larger magnitude earthquakes, thus magnitude scaling is controlled by data for

earthquakes with Mw ranging from 6.5 to 6.9. The parameters of the attenuation

relationships are listed in Table 4.13 and Table 4.14; and the resulting attenuation

relationships for soil and rock are illustrated in Figure 4.29. This attenuation relationship

is valid only within the distance and magnitude ranges of the data, that is, distances < 20

km and magnitudes of 6.1 to 7.4. Observe that the distance scaling term, c, is only

slightly larger for rock data set than for the soil data set, implying that the PGV is not

largely affected by intensity of motion. However, the magnitude scaling term in the rock

attenuation relationship is much larger than in the soil attenuation relationship, implying

that the ratio of PGV in soil to PGV in rock decreases as magnitude increases (Figure

4.30). Since magnitude is related to pulse period, this implies that for longer period

motions, there is lesser amplification of PGV in soils. However, due to the large scatter

in the data, the low amount of rock sites in the data set, and the poor distribution of data

across magnitude, caution must be exercised when interpreting these observations.

Numerical simulations of near-fault site effects are needed to complement the empirical

database due to the lack of data. Chapter Six deals in length with the effects of soil on

PGVs.

Figure 4.29b compares the relationship in Equation 4.12 with the relationships

developed by Somerville (1998) and Alavi and Krawinkler (2000). These researchers

augmented the empirical database with 12 simulated ground motions. The regression

analysis in Equation 4.12 differs from the other relationships mainly in the magnitude

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scaling term. Somerville (1998) and Alavi and Krawinkler (2000) propose a much

stronger variation of PGV with distance. The variation cannot be attributed to the

addition of the simulated time histories because Somerville (1998) indicates that the PGV

of the recorded time histories grows more rapidly with magnitude than for the simulated

time histories. The differences are likely due to the amount of data included. The current

work includes a larger database.

4.3.4 Characteristics of the fault parallel component of motion

The previous analysis dealt with the identification of the salient characteristics of

the fault normal component of near-fault ground motions. As previously indicated, this

component is typically larger than the fault-parallel component, and thus is commonly

used as input for structural response studies. The fault parallel component, however, can

also be significant as shown for example in Figure 4.6. The different types of pulses

resulting from varying intensities and frequency characteristics of fault parallel ground

motions are best observed by looking at a particle velocity-trace plot of the horizontal

component of motion. In this section, the different pulse shapes from near-fault ground

motions are evaluated.

No particular trend in the ratio of peak ground velocities in the fault parallel and

fault normal directions (PGVP/N) with respect to varying magnitude or distance is

observed. However, a trend of decreasing PGVP/N with increasing fault parallel velocity

is inferred from the available data (Figure 4.31). The trend is stronger for rock motions.

With the exception of one outlier, the PGVP/N ratio in rock is lower than 0.5 for fault

normal PGV larger than about 50 cm/s. Although the variability in the PGVP/N ratio is

130

large both for rock and soil sites across all values of fault normal PGV, it is important to

point out that for large values of fault normal PGV, this variability is larger for soil sites

than for rock sites. This variability might be due to the effects of site amplification. For

fault normal PGV values greater than 100 cm/s, the PGVP/N ratios in soil are generally

larger than in rock, while for lower values of fault normal PGV, no particular trend with

site condition is observed. Over all the records, the PGVP/N ratio is about the same

whether rock or soil records are taken. This average value is 0.64 with a standard

deviation of 0.26.

While the fault-normal component of velocity is generally well characterized by

simple pulses, the fault parallel pulse sometimes shows more irregular velocity time

histories. Nonetheless, a clearly defined pulse in the fault parallel direction exists in

many cases. Moreover, this pulse normally occurs during the same window of time as the

fault normal pulse occurs. Table 4.15 gives the ratio of pulse period in the fault parallel

direction to pulse period in the fault normal direction for different earthquakes. As can be

observed in these tables, the ratios have a mean value of 0.75 and a standard deviation of

0.33. In general, the fault parallel period is shorter than the period in the fault normal

direction.

4.3.5 Development of bi-directional simplified pulses for use as baseline input

motions

The preceding sections dealt with the development of simplified representations

of fault normal pulses (Section 4.3.3), and the description of general characteristics of

fault parallel pulses (Section 4.3.4). This section deals with the development of bi-

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directional pulses, that is, when both fault-normal and fault-parallel components of

motion are defined simultaneously. As previously indicated, typical structural analysis

procedures deal only with the fault normal component of motion, this one being

considered the controlling component for forward-directivity motions. The recordings on

Figure 4.6, however, illustrate that forward-directivity motions can also have significant

peak ground velocities in the fault parallel direction, particularly when lower input

velocities are involved. From the point of view of site response, the fault-parallel

component of motion can lead to larger levels of strain in the soil column, thus leading to

differences in site response. For this reason, a definition of bi-directional input pulses is

desired. Note that prescribing a fault-normal and fault-parallel velocity results in a given

pulse shape in the horizontal velocity-trace plot. For ease of notation, the term “pulse

shapes” will be used to define the shape of velocity pulses in a fault-normal versus fault-

parallel horizontal velocity plot.

The bi-directional horizontal velocity pulses developed in this section are used as

input motions for the site response analyses presented in Chapter Six. Ideally, input

pulses would be taken only from records on outcropping rock. The available number of

rock records, however, is relatively small and excluding soil sites proved to be too

restrictive. For this reason, the analysis of pulse shapes will be performed for all records

independently of their site classification. Note, however, that the pulse parameters used

in baseline input ground motions (e.g. PGV, PGVP/N, Tv) will be estimated from the

regression curves for rock motions.

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A selected number of the records listed in Table 4.5 were chosen for a more

detailed look at the pulse shape. A list of these motions is given in Table 4.16. The

selected motions are those that have all of the characteristics of forward-directivity

motions, that is:

- Most of the energy is concentrated on the initial pulse. This can be

determined by examining the plot of arias intensity versus time (Husid plot).

If most of the energy is concentrated in the velocity pulse, the slope of the

Husid plot is large for the time duration of the pulse. Quantitatively, the

condition stated above (most of the energy is concentrated on the initial pulse)

can be translated into the condition that at least 50% of the Arias Intensity

occurs before the end of the pulse.

- Positive fault normal to fault parallel spectral ratios for periods in the

neighborhood of the pulse period (in general, ratios are positive for periods

longer than one second).

Moreover, motions that have pulses not amenable to fitting with simple sine

pulses were not included in this list. Note that this creates an automatic bias towards

simpler pulses. This is inevitable if simplicity in the equivalent pulses is desired.

Motions that are not fit well by simple sine pulses (such as the Gilroy Gavilan College

record for the Loma Prieta earthquake) are also used without modification as input

motions for the analyses in Chapter Six. This is done in order to verify that the selection

of simple pulses does not adversely affect findings regarding site response.

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Seven motions with a dominant half-cycle of motion (with amplitude at least two

times larger than the amplitude of the remaining pulses) are shown in Figure 4.32. For

these eight motions, the ratio of fault parallel pulse period to fault-normal pulse period

ranges from 0.24 to 1.12, with an average value of 0.5. The ratio of peak velocities in the

fault-parallel to the fault-normal direction ranges from 0.21 to 0.7, with an average of 0.3.

Figure 4.33 presents motions with a pulse consisting of a full cycle of motion.

Fifteen such motions were studied in detail. In general, the positive and negative

amplitude of the half-cycle of motion are equal. Moreover, the periods of each half cycle

are within 10% of each other in most of the motions. The ratios of fault parallel

amplitude to fault normal amplitude, as well as the period ratios (Tv,FP/Tv,FN) fall within

the ranges of all the motions as presented in Section 4.3.4. Simplified representations for

the dominant velocity pulses of some of these motions are shown in Figure 4.34.

In light of the preceding discussion, the simplified pulses in Figure 4.35 were

developed. These pulses have fault parallel period ratios (Tv,FP/Tv,FN) consistent with the

records in Figure 4.32 and 4.33. Beyond the definition of the fault parallel and fault

normal ratios, the time delay (toff) between the initiation of the pulse in each direction is

also needed for the full definition of the pulse shape. The data do not suggest that there is

a particular trend for this parameter. The parameter was arbitrarily selected so that either

of the following three conditions occur: a) both motions start at the same time (toff = 0), b)

the fault parallel motion initiates when the fault normal is at a maximum (toff = TvN/4), or

c) the resulting maximum amplitude in the fault normal and fault parallel motion are

reached simultaneously. Preliminary site response analyses with these pulses indicated

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that Pulses 6, 7 and 8 cover the range of fault normal period elongation and amplification

of peak ground velocity. Consequently, these pulses were selected as the baseline input

motions for the analyses in Chapter Six.

Beyond the pulse shape, the pulse parameters must also be chosen for the input

motions. These were selected from the regression analyses for rock motions. In general,

the following criteria was used:

• Fault normal PGV ranges from 75 cm/s to 300 cm/s. The latter extreme value is

a judgmental choice to cover potential extreme values that can result from large

magnitude earthquakes. Records from the 1999 Chi-Chi earthquake in Taiwan

showed peak ground velocities in this range of values. These velocity peaks,

however, include slip effects (fling-step) and are not due solely to the effects of

forward-directivity (Abrahamson, personal comm.). Nonetheless, the value of

300 cm/s was chosen to a reasonable assumption for an upper limit on

velocities. Note that a PGV of 300 cm/s is also the median prediction of the

attenuation relationships for PGV in rock sites for a magnitude of 7.5 and a

distance of 0 km, and the plus one standard deviation prediction for a magnitude

of 7.0 and a distance of 1 km.

• The fault period in the fault normal direction was varied from 0.6 to 4.0

seconds. The lower value was also used by Somerville et al. (1997) in the

empirical analysis of directivity effects. Moreover, it corresponds to the

estimated period for an earthquake with a magnitude of 6.2. The upper limit

was taken as a reasonable period beyond which site amplification effects will

135

not be very significant for typical soil profiles (exception being very deep soft

soil deposits). Moreover, this period is beyond the range of interest for most

typical structures.

• The PGVP/N ratio was varied from 0.25 to 1.0. The lower value applies mainly

to large velocity input motions. A ratio higher than 0.5 is considered reasonable

only for motions lower than about 150 cm/s.

4.4 SUMMARY AND FINDINGS

Near-fault strong motion recordings that satisfied the geometric conditions for

forward directivity (Somerville et al. 1997) were selected and analyzed. The motions

were parameterized in the time domain by means of the number of significant pulses,

pulse amplitudes, and pulse periods of their velocity time-history. Several

characterizations of the pulse period were evaluated and compared. The representation of

pulse period by the period of the pulse with largest amplitude was selected. Some

important findings include:

• Near-fault, forward-directivity motions can be adequately represented by

simplified time histories consisting of one or a few sine-pulses. The number of

pulses is likely related to slip distributions in the causative fault, and

consequently is difficult to predict.

• Pulse period is a function of moment magnitude. For earthquakes with Mw =

6.1, the pulse period is about 0.6 s and increases to about 6.0 s for Mw = 7.5.

Most of the energy in forward-directivity ground motions is concentrated on the

136

narrow-period band centered on the pulse period. Consequently, lower

magnitude events might result in more damaging ground motions for typical low

period structures. Relatively high standard deviations are associated with the

prediction of pulse period (Figure 4.26).

• Local site conditions have an important effect on pulse period. Longer periods

occur at soil sites than at rock sites for events with magnitudes lower than about

Mw = 7.0. The difference diminishes as the magnitude increases. For events

with Mw ≈ 7.5, the pulse periods at rock and soil sites are approximately the

same (see Figure 4.24).

• Peak ground velocities in the near-fault region vary significantly with magnitude

and distance. The attenuation relationships developed (Equation 4.12 and Table

4.13) has relatively high standard deviation. Median peak ground velocities for

soils are larger than median peak ground velocities at rocks sites for low

magnitude events. The difference diminishes as the magnitude increases. This

observation, however, relies on a data-set with a small number of rock

recordings. Moreover, the difference between peak ground velocities in rock

and in soil is obscured by the large scatter in the data.

Simplified pulse representations based on the analysis of the database were

developed for use as input ground motions in site response analysis. Chapter Five

introduces the site response methodology that is used in Chapter Six for the site response

analyses.

137

BLANK

138

Table 4.1. Modification to ground motion parameters to account for directivity effects. Parameters X, Y, θ, and φ are defined in Figure 4.2, along with exclusion zones for strike slip faults. For coefficients and standard deviations see Somerville et al. (1997) (adapted from Somerville et al. 1997).

Ground Motion Parameter

Description Equation Range of Applicability

Amplitude Factor: Ratio of data/model

Bias in average horizontal response spectral acceleration (log) with respect to Abrahamson and Silva (1997)

Strike-Slip faults: y = C1+C2 Xcosθ Dip-Slip faults: y = C1+C2 Ycosφ

Moment Magnitude (Mw): 6.5 – 7.5 Distance (R): 0 – 50 km C1, C2 functions of period

Duration factor: Ratio of data/model (D0.05-0.75)

Bias in duration of acceleration with respect to Abrahamson and Silva (1997)

Strike-Slip faults: y = C1+C2 Xcosθ Dip-Slip faults: y = C1+C2 Ycosφ

6.5 ≤ Mw ≤ 7.5 0 ≤ R ≤ 20 km

Strike Normal/Average Amplitude

Natural logarithm of the ratio of strike normal to average horizontal spectral acceleration

y = cos2ξ [C1 + C2 ln(R + 1) + C3(Mw-6)]

6.0 ≤ Mw ≤ 7.5 0 ≤ R ≤ 50 km ξ = θ for strike-slip, φ for dip-slip. 0 < ξ < 90° C1, C2, C3 function of period. Given separately for cases in which dependence on ξ is included, and cases in which dependence on ξ is ignored.

139

Table 4.2. Near-source factors from the 1997 Uniform Building Code (UBC). (a) Short-period factor (Na)

Closest Distance to Known Seismic Source1 Seismic Source

Type ≤≤≤≤ 2 km 5 km ≥≥≥≥ 10 km

A 1.5 1.2 1.0

B 1.3 1.0 1.0

C 1.0 1.0 1.0

(b) Intermediate-period factor (Nv)

Closest Distance to Known Seismic Source1 Seismic Source

Type ≤≤≤≤ 2 km 5 km 10 km ≥≥≥≥ 15 km

A 2.0 1.6 1.2 1.0

B 1.6 1.2 1.0 1.0

C 1.0 1.0 1.0 1.0

(c) Description of seismic source types

Seismic Source Definition Seismic Source Type

Description Maximum Moment

Magnitude, Mw

Slip Rate, SR (mm/year)

A

Faults that are capable of producing large magnitude events

and that have a high rate of seismic activity

Mw ≥ 7.0 SR ≥ 5

B All faults other than Types A and C

Mw ≥ 7.0 Mw < 7.0 Mw ≥ 6.5

SR > 5 SR > 2 SR < 2

C

Faults that are not capable of producing large magnitude earthquakes and that have a

relatively low rate of seismic activity

Mw < 6.5 SR ≤ 2

1 The closest distance to seismic source shall be taken as the minimum distance between the site and the surface projection of the fault plane. The surface projection need not include portions of the source at depths of 10 km or greater.

140

Table 4.3. Earthquakes included in the study of near-fault ground motions. Fault parameters are obtained from Somerville et al. (1997).

Event # Earthquake Date Moment Magnitude

Mechanism1 Strike Dip

1 Parkfield 6/27/66 6.1 SS 317 90

2 San Fernando 2/9/71 6.6 TH 290 50

3 Imperial Valley 10/15/79 6.5 SS 143 90

4 Morgan Hill 4/24/84 6.2 SS 154 90

5 Superstition Hills (B) 11/24/87 6.6 SS 127 90

6 Loma Prieta 10/17/89 7.0 OB 128 70

7 Erzincan, Turkey 3/13/92 6.7 SS 300 86

8 Landers 6/28/92 7.3 SS 3512 90

9 Northridge 1/17/94 6.7 TH 122 40

10 Kobe 1/17/95 6.9 SS 50 85

11 Kocaeli 8/17/99 7.4 SS 90 90

1 SS: Strike-slip, OB: Oblique-slip, TH: Thrust 2 The Landers Earthquake occurred on a fault with multiple segments. The Lucerne record was rotated to the indicated strike corresponding to the orientation of highest fault normal peak ground velocity.

141

Table 4.4. Parameters used to define the simplified sine-pulse ground motions.

Parameter Abbreviation Methodology to obtain parameter

Number of significant pulses.

N Number of pulses (half-cycle pulses) in the velocity-time history with amplitudes at least 50% of the peak ground velocity of the record.

Pulse Period. Tv,i For each half sine pulse, Tv,i = 2 (t2 – t1), where t1 and t2 are either the zero-crossing time, or the time at which velocity is equal to 10% of the peak velocity for the pulse if this time is significantly different than the zero crossing time. Tv corresponding to the pulse with maximum amplitude is the overall representative velocity pulse period.

Predominant Period from Pseudo-velocity response spectra.

Tp-v Period corresponding to a clear and global peak in the pseudo-velocity response spectra at 5% damping.

Pulse Amplitude. Ai For each half sine pulse, the peak ground velocity in the time interval [t1,t2].

Peak ground velocity PGV Maximum velocity, defined by the maximum value of Ai. Note, however, that in very few exceptions, the maximum value of Ai in the fault parallel direction does not occur concurrently with the fault normal pulse.

Ratio of fault parallel to fault normal amplitude

PGVP/N Defined by the ratio of maximum AP divided by maximum AN, where the subscripts P and N denote fault-parallel and fault-normal motions respectively.

Time delay between fault normal and fault parallel pulse

toff Time of initiation of fault parallel pulse minus the time of initiation of fault normal pulse.

142

Tabl

e 4.

5. S

tatio

ns in

clud

ed in

the

anal

ysis

of n

ear-

faul

t gro

und

mot

ions

. St

atio

n

Age

ncy

Stat

ion

#Ev

ent 1

R2 (k

m)

RDI3

Site

C

ondi

tion4

PGA

(g)

PGV

(cm

/s)

PGV P

/N5

T v5

(s)

T v-p

5

(s)

Cho

lam

e #2

C

DM

G

1013

1

0.1

1.08

s

0.47

75

-6

0.67

0.

66

Tem

blor

C

DM

G

1438

1

9.9

1.08

r

0.29

17

.5

1.07

0.

44

0.4

Paco

ima

Dam

C

DM

G

279

2 2.

8 1.

04

r 1.

47

114

0.33

1.

44

1.15

B

raw

ley

Airp

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USG

S 50

60

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48

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21

36.1

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99

3.01

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5154

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7.6

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2.8

143

Stat

ion

A

genc

y St

atio

n #

Even

t 1 R2

(km

) RD

I3 Si

te

Con

ditio

n4 PG

A (g

) PG

V (c

m/s

)PG

V P/N

5 T v

5

(s)

T v-p

5

(s)

LA D

am

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9 2.

6 1.

14

r 0.

58

77

0.25

1.

24

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ll –

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#

CD

MG

24

279

9 7.

1 1.

16

s 0.

72

120

0.42

2.

04

0.69

N

ewha

ll –

W. P

ico

Can

yon

Rd.

U

SC

9005

6 9

7.1

1.15

s

0.43

87

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0.85

2.

19

2.0

Paco

ima

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(dow

nstr)

#

CD

MG

24

207

9 8

1.16

r

0.48

49

.9

0.46

0.

61

0.43

Pa

coim

a D

am (u

pper

left)

#

CD

MG

24

207

9 8

1.16

r

1.47

10

7 0.

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0.89

0.

72

Rin

aldi

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eivi

ng S

ta #

D

WP

77

9 7.

1 1.

13

s 0.

89

173

0.29

1.

31

1.05

Sy

lmar

– C

onve

rter S

ta #

D

WP

74

9 6.

2 1.

14

s 0.

8 13

0 0.

72

2.87

1.

05

Sylm

ar –

Con

verte

r Sta

Eas

t #

DW

P 75

9

6.1

1.14

s

0.84

11

6 0.

67

2.64

3.

0 Sy

lmar

– O

live

Vie

w M

ed F

F #

CD

MG

24

514

9 6.

4 1.

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123

0.44

1.

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2.35

KJM

A (K

obe)

- 10

0.

6 1.

14

r 0.

85

96

0.56

1.

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0.86

K

obe

Uni

vers

ity

CEO

R

- 10

0.

2 1.

48

r 0.

32

42.2

0.

92

1.59

1.

32

OSA

J

- 10

8.

5 1.

48

s 0.

08

19.9

0.

86

3.83

1.

18

Port

Isla

nd (0

m)

CEO

R

- 10

2.

5 1.

12

s 0.

38

84.3

0.

30

1.91

1.

32

Arc

elik

K

andi

lli

- 11

17

s 0.

21

42.3

0.

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5.

45

Duz

ce

ERD

-

11

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s 0.

37

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0.

82

1.92

1.

36

Geb

ze

ERD

-

11

17

r

0.26

40

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0.84

5.

04

4.5

Yar

imka

K

andi

lli

- 11

4.

4

s 0.

32

87.1

1

6.00

3.

80

1 Se

e Ta

ble

4.3

2 Clo

sest

dis

tanc

e to

the

faul

t pla

ne (S

adig

h et

al.

1993

) 3 E

quat

ion

4.1

4 Soi

l(s) o

r Roc

k ®

5 D

efin

ed in

Tab

le 4

.4

6 Th

e C

hola

me

#2 re

cord

in th

e Pa

rkfie

ld e

arth

quak

e tri

gger

ed o

nly

in o

ne d

irect

ion

(15o fr

om th

e fa

ult n

orm

al d

irect

ion)

144

Tabl

e 4.

6. P

ulse

par

amet

er fo

r sta

tions

incl

uded

in th

is st

udy.

St

atio

n

Even

t 1 N

2,

(50%

)N3

(33%

)AN

2

(cm

/s)

AP2

(cm

/s)

T vN

2 (s

) T v

P2 (s

) t off2

(s)

Cho

lam

e 1

2 2

-56.

9, 2

0, -2

0, 7

5

0.73

, 0.2

, 0.2

, 0.6

7

Tem

blor

1

4 5

12.2

, -5.

2, 8

.19,

-16.

3,

14.2

11

, -7.

2, 1

1.4,

-17.

5,

5.7

1.41

, 0.3

1, 0

.44,

0.3

7,

0.31

0.

44, 0

.29,

0.4

, 0.5

, 0.

25

0.62

Paco

ima

Dam

2

1 3

45.3

, -11

4, 4

6.2

15.3

, 38,

-17.

1 1.

25, 1

.44,

1.1

7 1.

21, 1

.47,

0.4

9 -0

.17

Bra

wle

y A

irpor

t 3

2 4

36.1

, -24

.4

15.6

, -35

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9.6,

-11

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1.3

2.56

, 3.8

9 0.

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.14,

0.6

8, 1

.55,

1.

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0.44

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ount

y C

ente

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4 35

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54.5

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1.5,

42.

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.78

1.21

, 1.7

6 1.

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EC M

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, -50

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Imp.

Co.

Cen

t 5

1 4

21.9

, -51

.9, 0

, -22

.1,

25.4

-1

0.5,

-8.9

, 36.

1, 6

, -25

.5

1.47

, 1.8

5, 0

.15,

0.5

6,

2.21

1.

82, 0

.47,

1.0

7, 0

.44,

2.

28

0.17

Para

chut

e Te

st si

te

5 2

2 -1

07, 8

0.9

45.8

, -45

.6

2.11

, 2.3

6 1.

47, 2

.41

0.13

G

ilroy

– G

avila

n C

oll.

6 2

6 -3

0.8,

0, -

14.9

15

.2, 1

0.5,

-26.

6 1.

16, 0

.05,

0.6

4 0.

74, 0

.78,

0.8

2 0.

08

Gilr

oy –

His

toric

Bld

g.

6 2

4 35

.8, -

36.8

, 0, 1

4.5

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, -24

.1, 1

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.9

1.62

, 1.3

3, 0

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1.0

9 1.

43, 0

.98,

1.2

6, 0

.67

0.13

Gilr

oy A

rray

#1

6 4

7 -3

8.6,

0, -

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16

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7 1.

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3 0.

02

Gilr

oy A

rray

#2

6 4

6 -4

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7 19

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7.6,

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3, -

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0.5

5, 0

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1.

81

1, 0

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.11,

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.6

0.05

Gilr

oy A

rray

#3

6 2

5 -4

9.3

34.1

1.

46

1.56

-0

.18

LGPC

6

6 10

-7

0.8,

102

-4

1.2,

51.

6 2.

92, 2

.14

2.74

, 1.3

3 0.

26

145

Stat

ion

Ev

ent 1

N2,

(50%

)N3

(33%

)AN

2

(cm

/s)

AP2

(cm

/s)

T vN

2 (s

) T v

P2 (s

) t off2

(s)

Sara

toga

– A

loha

Ave

6

3 5

-31.

5, 5

5.5,

43

16.3

, 0, -

43.3

3.

02, 2

.31,

1.8

1 4.

5, 1

.96,

2.0

1 -0

.08

Sara

toga

– W

Val

ley

Col

l. 6

2 3

71.3

-2

9.1,

60

1.71

1.

76, 2

.46

-0.1

5 Er

zinc

an

7 3

3 -5

9, 9

5.5,

-58

-17.

8, 4

5.9,

-36.

2 1.

43, 2

.27,

2.5

2 1.

59, 1

.71,

2.1

1 -0

.35

Luce

rne

# 8

1 2

-145

, 55.

7 29

.7, -

19.2

5.

39, 4

.43

4.68

, 1.7

6 -1

.59

Jens

en F

ilter

Pla

nt #

9

3 6

104,

-84.

7, 6

2.8,

-44

.9, 3

8.8

36.8

, -60

.7, 6

1, -9

4,

90.8

1.

99, 3

.2, 2

.7, 2

.2, 3

.61.

11, 1

.06,

2.6

7, 1

.64,

2.

23

0

LA D

am

9 1

2 77

, -35

.5

19, -

8 1.

24, 1

.55

1.39

, 1.4

3 -0

.25

New

hall

– Fi

re S

ta #

9

3 5

68.6

, 37.

7, 1

20

49.9

, -19

.3, 4

2.7

2.04

, 0.6

8, 0

.95

1.05

, 0.2

5, 0

.4

0.08

N

ewha

ll –

W. P

ico

Can

yon

Rd.

9

2 2

82.8

, -87

.7

-74.

7, 4

8.7

2.92

, 2.1

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71, 2

.51

0.92

Pa

coim

a D

am (d

owns

tr) #

9

1 3

49.9

, -19

, 24.

7 12

, -23

, 3

0.61

, 0.3

2, 0

.51

0.38

, 1.1

, 0.2

5 0.

18

Paco

ima

Dam

(upp

er le

ft) #

9

2 3

107,

-72.

7, 4

6 18

.9, -

27.2

, 25.

5, -

46.5

0.

89, 0

.94,

0.2

0.

52, 0

.48,

0.3

6, 0

.67

0.04

Rin

aldi

Rec

eivi

ng S

ta #

9

1 2

173,

-81.

1, 3

3 -1

7.3,

23,

-13,

-50

1.31

, 1.1

5, 0

.61

0.36

, 0.5

4, 0

.33,

2.0

6 -0

.04

Sylm

ar –

Con

verte

r Sta

#

9 3

6 13

0, 6

9, 3

3.2,

93.

8 75

.1, -

39.9

, 61.

5, -

93.3

2.

87, 2

.23,

1.2

5, 1

.51

1.7,

1.8

9, 2

.27,

1.4

3 -0

.32

Sylm

ar –

Con

verte

r Sta

Eas

t #

9 2

2 11

6, 3

7.3,

25.

4, 6

6.6

37.6

, -39

.5, 4

0.5,

-78

.3

2.64

, 2.5

3, 1

.32,

1.3

2 1.

04, 1

.73,

2.5

3, 1

.7

0

Sylm

ar –

Oliv

e V

iew

Med

FF

# 9

2 4

123,

-56.

3, 6

2.9

-22.

3, 5

4.4,

-40.

6 1.

76, 2

.72,

2.8

5 0.

83, 0

.78,

0.3

2 -0

.28

KJM

A (K

obe)

10

7

10

-73.

4, 6

7.6,

-88.

7,

95.7

-1

4.4,

13.

4, -3

4.2,

53

.4

1.03

, 0.8

7, 0

.85,

1.9

1 1.

32, 0

.37,

0.3

7, 0

.73

0.02

Kob

e U

nive

rsity

10

4

8 38

.3, -

20, 1

0.8,

-29.

7,

42.9

28

.8, -

39.6

, 16.

3, -

9.05

, 26.

4 1.

85, 1

.62,

0.4

9, 1

.55,

1.

59

1.02

, 1.2

1, 1

.74,

0.5

3,

1.62

-0

.02

OSA

J 10

2

3 18

.9-1

9.9,

5.7

-1

0.4,

16,

-17.

1 4.

16, 3

.83,

0.5

8 3.

7, 2

.64,

1.8

3 0.

32

Port

Isla

nd (0

m)

10

5 6

-48.

9, 5

8.2,

-33.

6,

84.3

, -76

.4

-9.6

, 16.

9, -1

9.4,

25.

6,

-18.

3 1.

86, 1

.16,

0.9

7, 1

.91,

2.

92

1.26

, 1.1

6, 1

.46,

2.1

7,

2.42

1.

13

Arc

elik

11

2

2 -4

2.2,

30.

6 12

.8, -

13.1

6.

82, 5

.16

11.2

, 4.2

4 -3

.9

Duz

ce

11

4 5

29.2

, -46

.5, 5

2.5,

42.

3-3

2.4,

37.

8, -1

0.3,

22

.4

1.61

, 1.4

6, 1

.92,

0.7

2.

47, 1

.02,

0.2

6, 0

.83

0.66

Geb

ze

11

2 2

13.1

, -40

.7, 3

1.6

-19.

2, 3

4.3,

-9.8

5.

76, 5

.04,

4.4

6 7.

2, 5

.5, 2

.5

-0.3

2 Y

arim

ka

11

2 4

87, -

42, 4

7.1

32.2

, -48

, 87

6, 3

.9, 3

.7

4.3,

2.9

, 5.9

-0

.4

1 See

Tabl

e 4.

3 2 D

efin

ed in

Tab

le 4

.4 (N

is th

e nu

mbe

r of h

alf-

cycl

e pu

lses

with

pea

k ve

loci

ties a

t lea

st 5

0% o

f the

PG

V o

f the

ent

ire re

cord

). 3 T

he 3

3% le

vel i

s add

ed to

illu

stra

te h

ow d

iffer

ent c

ut-o

ff v

alue

s res

ult i

n a

diff

eren

t num

ber o

f pul

ses

146

Table 4.7. Ground motion recordings satisfying geometric requirements for forward-directivity conditions not included in this study. Listed are only stations for earthquakes with Mw ≥ 6.5 and R < 20 km.

Earthquake Ground Motion Station Imperial Valley Parachute Test Site Imperial Valley El Centro Array #1 Imperial Valley El Centro Array #11 Imperial Valley El Centro Array #2 Imperial Valley El Centro Array #12 Kobe Fukushima Kobe Nishi-Akashi Kobe Shin-Osaka Kobe Amagasaki Kobe Kobc Kocaeli Izmit Loma Prieta Gilroy Array #6 Loma Prieta Lexington Dam Loma Prieta Watsonville Nahanni, Canada Site 3 - Battlement Creek Northridge Canyon Country - W Lost Canyon Northridge Dam toe Northridge Jensen Generator Building Northridge Sepulveda V.A. Superstition Hills(B) 11369 Westmorland Fire Sta Superstition Hills(B) 5060 Brawley

147

Table 4.8. Values of Tv-p/Tv for multiple-record events.

Earthquake Average Std. Dev. Imperial Valley .96 .11 Loma Prieta .64 .36 Northridge .90 .34 Kobe .57 .23

148

Table 4.9. Number of half-cycle pulses (N) by event for the recordings considered in this study. Value in parenthesis is the number of half-cycle pulses that corresponds to a cut-off value of 33% of the PGV (as opposed to 50% used to define N).

Number of Records with given number of half-cycle pulses (N) Earthquake Year Number of

Records 1 pulse 2 pulses 3 pulses > 3 pulses

Parkfield 66 2 0 (0) 1 (1) 0 (0) 1 (1) San Fernando 71 1 1 (0) 0 (0) 0 (1) 0 (0) Imperial Valley 79 13 1 (0) 10 (1) 1 (7) 1 (5) Morgan Hill 84 2 0 (0) 0 (0) 1 (0) 1 (2) Superstition Hills(B)

87 2 1 (0) 1 (1) 0 (0) 0 (1)

Loma Prieta 89 8 0 (0) 4 (0) 1 (1) 3 (7) Erzincan, Turkey 92 1 0 (0) 0 (0) 1 (1) 0 (0) Landers 92 1 1 (0) 0 (1) 0 (0) 0 (0) Northridge 94 10 3 (0) 4 (4) 3 (2) 0 (4) Kobe 95 4 0 (0) 1 (0) 0 (1) 3 (3) Kocaeli, Turkey 99 4 0 (0) 3 (2) 0 (0) 1 (2)

Totals 48 7 (0) 24 (10) 7 (13) 10 (25)

149

Table 4.10. Parameters from the regression analyses for the period of the pulse of maximum amplitude, Tv, and the period corresponding to the maximum pseudo-velocity response spectral value, Tv-p (Equation 4.8). a) Tv

Data Set a b σσσσ

ττττ

σσσσtotal Var(σσσσ2) Var(σσσσ2)

Cov(σσσσ2,ττττ2)

All Motions -8.33 1.33 0.36 0.40 0.54 0.0008 0.0078 -0.0003

Rock -11.10 1.70 0.31 0.41 0.51 0.0029 0.0140 -0.0018

Soil -5.81 0.97 0.32 0.40 0.51 0.0008 0.0100 -0.0003

b) Tv-p

Data Set a b σσσσ

ττττ

σσσσtotal Var(σσσσ2) Var(σσσσ2)


All Motions -6.92 1.08 0.48 0.45 0.66 0.0028 0.0154 -0.0009

Rock -9.53 1.42 0.37 0.61 0.71 0.0062 0.0555 -0.0041

Soil -5.66 0.91 0.41 0.45 0.61 0.0022 0.0181 -0.0008

150

Table 4.11. Inter-event error term from the random effects model for the attenuation relationship for pulse period.

All Motions Soil Rock Earthquake No. of

Records ηηηηi No. of


Records ηηηηi

Erzincan, Turkey 1 -0.03 1 -0.053 0 - Imperial Valley 13 0.92 13 0.76 0 - Kobe 4 -0.077 2 0.0662 2 -0.07 Kocaeli 4 -0.032 3 0.0498 1 0.076 Landers (B) 1 0.15 0 - 1 0.23 Loma Prieta 8 -0.40 5 -0.3879 3 -0.25 Morgan Hill 2 -0.066 0 - 2 0.30 Northridge 10 -0.197 7 -0.0923 3 -0.37 Parkfield 2 -0.294 1 -0.3203 1 -0.07 San Fernando 1 -0.057 0 - 1 0.146 Superstition Hills 2 0.057 2 -0.0238 0 -

151

Table 4.12. Comparison of pulse periods between rock and soil stations for recordings in the Gilroy area in the 1989 Loma Prieta earthquake.

Station R (km)

Site Condition

Tv (s)

Tv-p (s)

Gilroy#1 11.2 r 1.16 0.40 Gilroy Gavilan College 11.6 r 1.14 0.39 Gilroy#2 12.7 s 1.41 1.45 Gilroy#4 16.1 s 1.33 1.0

152

Table 4.13. Parameters from the regression analyses for peak ground velocity (Equation 4.12).

Data Set a b c d σσσσ

ττττ

σσσσtotal Var(σσσσ2) Var(ττττ2)


All Motions 2.44 0.50 -0.41 3.93 0.47 0.41 0.62 0.0026 0.011 8.2e-4

Rock 1.46 0.61 -0.38 3.93 0.53 0.25 0.59 0.023 0.019 -0.0120

Soil 3.86 0.30 -0.42 3.93 0.43 0.41 0.59 0.014 0.0026 8.6e-4

153

Table 4.14. Inter-event error term from the random effects model for the attenuation relationship for peak ground velocity.

All Motions Soil Rock Earthquake No. of



Records ηηηηi

Erzincan, Turkey 1 -0.045 1 -0.067 0 - Imperial Valley 13 -0.009 13 -0.113 0 - Kobe 4 -0.553 2 -0.422 2 -0.145 Kocaeli 4 -0.076 3 -0.031 1 -0.019 Landers (B) 1 0.020 0 - 1 0.026 Loma Prieta 8 0.086 5 0.142 3 -0.019 Morgan Hill 2 -0.012 0 - 2 0.047 Northridge 10 0.046 7 0.053 3 0.107 Parkfield 2 -0.214 1 -0.114 1 -0.099 San Fernando 1 0.127 0 - 1 0.085 Superstition Hills 2 0.116 2 *.079 0 -

154

Table 4.15. Ratios of pulse period in the fault parallel to pulse period in the fault normal direction.

Earthquake Tv,FP/Tv,FN Imperial Valley .60

Morgan Hill .88 Loma Prieta .86

Erzincan, Turkey .75 Landers .87

Northridge .89 Kobe .69

155

Table 4.16. Ground motions included in the determination of typical pulse shapes.

Earthquake Ground Motion Site Site Condition

Erzincan Erzincan Soil

Imperial Valley Brawley Airport Soil

Imperial Valley El Centro – Meloland Overpass Soil

Imperial Valley El Centro #10 Soil




Imperial Valley El Centro County Center Soil

Imperial Valley Holtville Post Office Soil

Kobe Port Island Soil

Landers Lucerne Rock

Loma Prieta Gilroy – Gavilan College Soil

Loma Prieta Gilroy – Historic Building Soil

Northridge LA Dam Rock

Northridge Newhall – West Pico Canyon Rd. Soil

Northridge Pacoima Dam Dwnstr. Rock

Northridge Pacoima Dam, Upper Left Ab. Rock

Northridge Rinaldi Receiving Station Soil

Northridge Sylmar – Olive View Hospital Soil

San Fernando Pacoima Dam Rock

Superstition Hills El Centro, Imperial County Center Soil

Superstition Hills Parachute Test Station Soil

156

Figure 4.1 Schematic diagram of rupture directivity effects for a vertical strike-slip fault. The rupture begins at the hypocenter and spreads at a speed that is about 80% of the shear wave velocity. The figure shows a snapshot of the rupture front at a given instant. Resulting time-histories close to and away from the hypocenter are represented by strike-normal velocity recordings of the 1993 Landers earthquake at Joshua Tree and Lucerne respectively (from Somerville et al. 1997).

DEP

TH (k

m)

abou

t to

slip

finis

hed

slip

ping

Healing front

Rupture Front

S wave front

hypocenter

0

5

10

S waves travelling right S waves travelling left

slip

ping

157

Figure 4.2. Definition of parameters used in defining rupture directivity conditions (adapted from Somerville et al. 1997).

158

(a) Average response spectra ratio, showing dependence on period and on

directivity function.

(b) Strike-normal to average horizontal response spectral ratio for maximum

forward-directivity conditions (Xcosθ = 1). Figure 4.3. Predictions from the Somerville et al. (1997) relationship for varying directivity conditions.

0

1

2

3

0.01 0.1 1 10Period (s)

Spec

tral

Acc

eler

atio

n Fa

ctor

0

1

2

3

0.01 0.1 1 10Period (s)Sp

ectr

al A

ccel

erat

ion

Fact

or

Xcosθ = 1.0

0.75

0.50

0.0

Ycosφ =1.0

0.0

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6Period (s)

FN/A

vera

ge R = 0 kmR = 10 kmR = 50 km

0.8

1

1.2

1.4

1.6

1.8

2

0.1 1 10 100Period (s)

FN/A

vera

ge M = 6.0M = 7.0M = 7.5

Mw = 7.5 T = 3.0 s

159

Figure 4.4. Simplified pulses used by other researchers.

(a) Krawinkler and Alavi (1998) (b) Sasani and Bertero (2000)

-1.5-1

-0.50

0.51

1.5

0 0.5 1 1.5 2Nor

mal

ized

Acc

eler

atio

n

-1.5-1

-0.50

0.51

1.5

0 0.5 1 1.5 2

Nor

mal

ized

Vel

ocity

-1.5-1

-0.50

0.51

1.5

0 0.5 1 1.5 2Nor

mal

ized

Dis

plac

emen

t

-1.5-1

-0.50

0.51

1.5

0 0.5 1 1.5 2Nor

mal

ized

Acc

eler

atio

n

-1.5-1

-0.50

0.51

1.5

0 0.5 1 1.5 2

Nor

mal

ized

Vel

ocity

-1.5-1

-0.50

0.51

1.5

0 0.5 1 1.5 2Nor

mal

ized

Dis

plac

emen

t

160

0 2 4 6 8 10

-100

-50

0

50

100

Pacoima Dam (San Fernando EQ.)

Time (s)

Velo

city

(cm

/s)

0 1 2 3 4 5-50

0

50Gilroy Gavilan College (Loma Prieta EQ.)

Time (s)

Velo

city

(cm

/s)

0 1 2 3 4 5

-60

-40

-20

0

20

40

60

Pacoima Dam Dwnst. (Northridge EQ.)

Time (s)

Velo

city

(cm

/s)

0 2 4 6 8 10-150

-100

-50

0

50

100

150KJMA (Kobe EQ.)

Time (s)

Velo

city

(cm

/s)

Figure 4.5. Recorded and simplification of the dominant pulses for selected fault-normal velocity time-histories. Thick lines correspond to motions representing a simplification of the dominant pulses.

161

0 5 10 15 20

-100

-50

0

50

100

Meloland Overpass, Imperial Valley EQ.Ve

loci

ty (c

m/s

)

-100 0 100

-100

-50

0

50

100

FP V

eloc

ity (c

m/s

)

0 5 10 15 20

-50

0

50

West Pico Canyon Road, Northridge EQ.

Velo

city

(cm

/s)

Time (s)

Thick: Fault NormalThin: Fault Parallel

-50 0 50

-50

0

50

FP V

eloc

ity (c

m/s

)

FN Velocity (cm/s)

Figure 4.6. Fault normal (FN) and fault parallel (FP) velocity time-histories and horizontal velocity-trace plots for two near-fault recordings. Both recordings have significant fault normal velocities, but the Meloland Overpass recording from the Imperial Valley earthquake has much lower fault parallel velocities than the West Pico Canyon Road record from the Northridge earthquake.

162

Figure 4.7. Parameters needed to define the fault parallel and fault normal components of simplified velocity pulses. Subscripts N and P indicate fault normal and fault parallel motions, respectively.

toff

½ TvN,1

½ TvN,2

½ TvN,3

AN,1

AN,2

AN,3

½TvP,1

½TvP,2

AP,1

AP,2

Faul

t Nor

mal

Vel

ocity

Fa

ult P

aral

lel V

eloc

ity

NN = 3

NP = 2

163

Average Period = 0.9647Tv

R2 = 0.91

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8Period of Maximum Pulse, Tv (s)

Wei

ghte

d Av

erag

e Pe

riod,

(s)

Regression Line+- 20%

1

1

Figure 4.8. Comparison of two different measures of pulse period: weighted average period and period of maximum pulse (Tv).

164

1 2 3 4 5 6 7 8 9 10-

-80

-60

-40

-20

0

20

40

60

80

100

Time (s)

FN V

eloc

ity (c

m/s

)

t1 = time for which Velocity = 0.1 PGV

t1 = zero crossing time

t2 = zero crossing time

Tv,1 = 2(t2 – t1)

0.1 PGV

-0.1 PGV

Figure 4.9. Determination of pulse period (Tv) for cases in which the pulse is preceded by a small drift in the velocity time-history.

165

10-2

10-1

100

101

0

20

40

60

80

100

120

140

160

180

200

Period (s)

Velo

city

(cm

/s)

Figure 4.10. Determination of period corresponding to the peak pseudo-velocity response spectral value. The pseudo-velocity response spectra shown is for the Erzincan record of the Erzincan, Turkey, earthquake.

Tv-p

5% damping

166

Tv-p = 0.8501Tv

R2 = 0.7469

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8Period of Maximum Pulse, Tv (s)

T v-p

(s)

+- 20% from 1:1 lineRegression Line

1

1

Figure 4.11. Comparison of the period of the maximum pseudo-velocity response spectral value (Tv-p) with the period of the maximum pulse, Tv.

167

Figure 4.12. Normalized power spectral densities of velocity time histories for selected ground motions. The two ground motions on the left have lower Tv-p than Tv, while those on the right have roughly coinciding values of Tv-p and Tv.

10-1

10 0

1010

0.2

0.4

0.6

0.8

1

LGPC, Loma Prieta EQ.

Period (s)

Nor

mal

ized

PSD

10-1

100

10 1 0

0.2

0.4

0.6

0.8

1

El Centro #4, Imp. Valley EQ.

Period (s)

Nor

mal

ized

PSD

10-1 100 10 1 0

0.2

0.4

0.6

0.8

1El Centro #8, Imp. Valley EQ.

Period (s)

Nor

mal

ized

PSD

10 -1 10

0 1010

0.2

0.4

0.6

0.8

1 Pacoima Dam Dwnstr., Northridge EQ.

Period (s)

Nor

mal

ized

PSD

Tv-p = 0.40 s Tv = 2.14 s Tv-p/Tv = 0.2

Tv-p = 0.43 s Tv = 0.61 s Tv-p/Tv = 0.7

Tv-p = 4.0 s Tv = 4.31 s Tv-p/Tv = 0.9

Tv-p = 4.0 s Tv = 3.98 s Tv-p/Tv = 1.0

168

0 2 4 6 8 10-20

-10

0

10

20Rock

Temblor

0 2 4 6 8 10

-50

0

50Soil

Cholame 02Ve

loci

ty (c

m/s

)

Velo

city

(cm

/s)

Time (s) Time (s)

Figure 4.13. Velocity time-histories for the 1966 Parkfield earthquake. The Temblor record corresponds to the fault normal direction. The seismograph at the Cholame 02 station recorded only acceleration in one direction that corresponds to 15° degrees from the fault normal direction. Dashed lines correspond to 50% and 33% of the PGV.

169

0 2 4 6 8 10

-100

-50

0

50

100 Rock

Pacoima DamVe

loci

ty (c

m/s

)

Figure 4.14. Fault normal velocity time-histories for the Pacoima Dam record in the 1971 San Fernando earthquake. Dashed lines correspond to 50% and 33% of the PGV.

Time (s)

170

0 5 10 15 20

-100

-50

0

50

100 Soil

El Centro - Meloland Overpass

0 5 10 15 20-100

-50

0

50

100 Soil

El Centro Array #7

0 5 10 15 20-100

-50

0

50

100Soil

El Centro Array #5

0 5 10 15 20

-100

-50

0

50

100 Soil

El Centro Array #6

0 5 10 15 20

-50

0

50 Soil

El Centro Array #8

0 5 10 15 20

-50

0

50Soil

El Centro Array #4

0 5 10 15 20

-50

0

50 Soil

El Centro Diff. Array

Time (s)

Velo

city

(cm

/s)

0 5 10 15 20

-50

0

50 Soil

Holtville Post Office

Time (s)

Velo

city

(cm

/s)

Figure 4.15a. Fault normal velocity time-histories for the 1979 Imperial Valley earthquake. Dashed lines correspond to 50% and 33% of the PGV.

171

0 5 10 15 20

-50

0

50 Soil El Centro - County Center

0 5 10 15 20 -40

-20

0

20

40Soil

Brawley Airport

0 5 10 15 20-50

0

50 Soil

El Centro Array #10

0 5 10 15 20 -50

0

50Soil

El Centro Array #3

Time (s)

0 5 10 15 20

-20

0

20 Soil

Westmorland Fire Station

Time (s)

Velo

city

(cm

/s)

Velo

city

(cm

/s)

Figure 4.15b. Fault normal velocity time-histories for the 1979 Imperial Valley earthquake. Dashed lines correspond to 50% and 33% of the PGV.

172

Figure 4.16. Fault normal velocity time-histories for the 1984 Morgan Hill earthquake. Dashed lines correspond to 50% and 33% of the PGV.

0 5 10 15-40

-20

0

20

40 Rock

Gilroy #6

Time (s) 0 5 10 15

-50

0

50Rock

Coyote Lake Dam

Time (s)

Velo

city

(cm

/s)

Velo

city

(cm

/s)

173

Figure 4.17. Fault normal velocity time-histories for the 1987 Superstition Hills earthquake. Dashed lines correspond to 50% and 33% of the PGV.

5 10 15 20 25-100

-50 0

50

100 Soil Parachute Test Site

Time (s) 5 10 15 20 25

-50

0

50 Soil

El Centro - Imp. Co. Cent.

Time (s)

Velo

city

(cm

/s)

Velo

city

(cm

/s)

174

0 5 10 15-100

-50

0

50

100 Rock

LGPC

0 5 10 15-40

-20

0

20

40Rock

Gilroy #1

0 5 10 15-40

-20

0

20

40Rock

Gilroy Gavilan College

0 5 10 15-50

0

50Soil

Gilroy #2

0 5 10 15-40

-20

0

20

40Soil

Gilroy - Historic Building

0 5 10 15

-50

0

50 Soil

Saratoga - Aloha Av.

0 5 10 15

-50

0

50Soil

Saratoga - West Valley Coll.

Time (s)

Vel

ocity

(cm

/s)

0 5 10 15-50

0

50Soil

Gilroy #3

Time (s)

Vel

ocity

(cm

/s)

Figure 4.18. Fault normal velocity time-histories for the 1989 Loma Prieta earthquake. Dashed lines correspond to 50% and 33% of the PGV.

175

Figure 4.19. Fault normal velocity time-histories for the Erzincan record in the 1992 Erzincan, Turkey, earthquake. Dashed lines correspond to 50% and 33% of the PGV.

0 2 4 6 8 10-100

-50

0

50

100 Soil

Erzincan

Time (s)

Velo

city

(cm

/s)

176

Figure 4.20. Fault normal velocity time-histories for the Lucerne record in the 1992 Landers earthquake. Dashed lines correspond to 50% and 33% of the PGV.

5 10 15 20

-100

0

100Rock

Lucerne

Time (s)

Velo

city

(cm

/s)

177

0 2 4 6 8 10

-50

0

50 Rock

LA Dam

0 2 4 6 8 10-50

0

50Rock

Pacoima Dam Dwnstr.

Time (s)

0 2 4 6 8 10-100

-50 0

50

100 Rock Pacoima Dam - Upper Left Ab.

Time (s)

Velo

city

(cm

/s)

Velocity

Figure 4.21a. Fault normal velocity time-histories for rock records from the 1994 Northridge earthquake. Dashed lines correspond to 50% and 33% of the PGV.

178

0 2 4 6 8 10

-100

-50

0

50

100 Soil

Sylmar Converter Station East

0 2 4 6 8 10

-100

-50

0

50

100Soil

Sylmar Converter Station

0 2 4 6 8 10-100

-50

0

50

100 Soil

Jensen Filtration Plant

0 2 4 6 8 10

-100

-50

0

50

100 Soil

Sylmar - Olive View Hospital

0 2 4 6 8 10

-100

0

100Soil

Rinaldi Receiving Station

0 2 4 6 8 10

-50

0

50Soil

Newhall - West Pico Canyon Road

Time (s)

0 2 4 6 8 10

-100

-50

0

50

100 Soil

Newhall Fire Station

Time (s)

Vel

ocity

(cm

/s)

Vel

ocity

(cm

/s)

Figure 4.21b. Fault normal velocity time-histories for soil records from the 1994 Northridge earthquake. Dashed lines correspond to 50% and 33% of the PGV.

179

Figure 4.22. Fault normal velocity time-histories for records from the 1995 Kobe, Japan, earthquake. Dashed lines correspond to 50% and 33% of the PGV.

0 5 10 15 20-50

0

50 Rock

Kobe University

0 5 10 15 20-100

-50

0

50

100Rock

KJMA

0 5 10 15 20

-50

0

50 Soil

Port Island

Time (s) 0 5 10 15 20

-20

-10

0

10

20Soil

OSAJ

Time (s)

Velo

city

(cm

/s)

Velo

city

(cm

/s)

180

0 5 10 15 20 25

-50

0

50 Soil

Yarimka

0 5 10 15 20 25

-50

0

50 Soil

Duzce

0 5 10 15 20 25-50

0

50 Soil

Arcelik

Time (s) 0 5 10 15 20 25

-50

0

50Rock

Gebze

Time (s)

Velo

city

(cm

/s)

Velo

city

(cm

/s)

Figure 4.23. Fault normal velocity time-histories for records from the 1999 Kocaeli, Turkey, earthquake. Dashed lines correspond to 50% and 33% of the PGV.

181

Figure 4.24. Attenuation relationship for pulse period (Tv).

0.1

1.0

10.0

6 6.5 7 7.5 8Moment Magnitude, Mw

Puls

e Pe

riod,

T v (s

) RockSoilAll DataRockSoil

182

Figure 4.25. Comparison of results form regression analysis with relationships proposed by other researchers. The definition of pulse period of Somerville (1998) is similar to that used in this study (Tv). On the other hand, the pulse period of Alavi and Krawinkler (2000) is the period of the maximum pseudo-velocity response spectral value (Tv-p).

0

1

2

3

4

5

6

6 7 8Moment Magnitude (Mw)

Puls

e Pe

riod

(s)

Alavi andKrawinkler (2000)Somerville (1998)

Tv, RegressionCurveTv-p, RegressionCurve

183

Figure 4.26. Attenuation relationship for pulse period (Tv) showing one standard deviation band.

0

2

4

6

8

10

6 7 8Moment Magnitude (Mw)

Puls

e Pe

riod

(s)

Soil

Rock

sd

184

0 2 4 6 8 10-40

-20

0

20

40Rock

Gilroy #1

-40 -20 0 20 40-40

-20

0

20

40

0 2 4 6 8 10-40

-20

0

20

40Rock


Velo

city

(cm

/s)

-40 -20 0 20 40-40

-20

0

20

40

Faul

t Par

alle

l Vel

ocity

(cm

/s)

0 2 4 6 8 10-50

0

50Soil

Gilroy #2

-50 0 50-50

0

50

0 2 4 6 8 10-40

-20

0

20

40Soil

Gilroy #4

Time (s)


-40 -20 0 20 40-40

-20

0

20

40

Fault Normal Velocity (cm/s)

Figure 4.27a. Velocity time-histories and velocity-trace plots for sites in the Gilroy area recorded in the 1989 Loma Prieta earthquake.

185

Figure 4.27b. Pseudo-velocity response spectra for the sites in Figure 2.27a. Observe the significant difference in frequency content at long periods between the soil and the rock sites.

10 -2

10-1

100

1010

20

40

60

80

100

120

140

160

180

200

Period (s)

Spec

tral V

eloc

ity (c

m/s

) Gilroy #1: Rock Gilroy Gavilan College: Rock Gilroy #2: Soil Gilroy #4: Soil

5% damping

186

10-1 100 101

6.5

7

7.5

Distance (km)

Mom

ent M

agni

tude

, Mw

SoilRock

Figure 4.28. Distribution of near-fault sites considered in this study.

187

Figure 4.29. Attenuation relationship for PGV in the near-fault region.

10-1

100

101

101

102

All D ataAll D ataAll D ataAll D ata

Soil; Mw = 6.0Soil; Mw = 7.5Rock; Mw = 6.0Rock; Mw = 7.5Model: Mw = 6.1 and 7.4

10-1

100

101

101

102

Peak

Gro

und

Velo

city

(cm

/s) Soi lSoi lSoi lSoi l

10-1

100

101

101

102

Distance (km)

RockRockRockRock

188

2 4 6 8 10 12 14 16 18 20 10

1

10 2

10 3

Rock

Peak

Gro

und

Velo

city

(cm

/s)

Distance (km)

Equation 4.12 Alavi and Krawinkler 2000) Somerville (1998)

Increasing Magnitude

Mw=7.4

Mw=6.1

Equation 4.12: ln(PGV) = 2.44 + 0.5 Mw

–0.41 ln(R2 + 3.932) Somerville (1998): ln(PGV) = -2.31 + 1.15 Mw

– 0.5 ln(R) Krawinkler and Alavi (2000): ln(PGV) = -5.11 + 1.59 Mw

– 0.58 ln(R)

Figure 4.29b. Comparison of results from regression analysis for PGV with relationships proposed by other researchers for a database of near-fault, forward-directivity motions.

189

Figure 4.30. Dependence of ratio of PGV from soil to rock on magnitude and distance.

6.5 7 7.5

0.9

1.0

1.1

1.2

1.3

Moment Magnitude, Mw

PGVs

oil/P

GVr

ock

PGVrock = 75 cm/s PGVrock = 100 cm/s

PGVrock increases

190

Figure 4.31. Relationship between the ratio of fault parallel to fault normal peak ground velocity (PGVP/N) to fault normal peak ground velocity (PGV). Regression line and equation are for Rock sites.

y = -0.3222Ln(x) + 1.9185R2 = 0.4449

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200

Fault Normal PGV (cm/s)

PGVP

/N Rock Soil

191

0 1 2 3 4 5

-100

-50

0

50

100 Rock

Pacoima Dam, San Fernando EQ.

-100 0 100

-100

-50

0

50

100

0 5 10 15

-100

0

100Rock

Lucerne, Landers EQ.

Velo

city

(cm

/s)

-100 0 100

-100

0

100

Faul

t Par

alle

l Vel

ocity

(cm

/s)

0 2 4 6-50

0

50Rock

Pacoima Dam Dwnstr., Northridge EQ.

-50 0 50-50

0

50

0 5 10 15 20

-50

0

50 Soil

El Centro, ICC, Superstition Hills. EQ.

Time (s)


-50 0 50

-50

0

50


Figure 4.32a. Selected motions with one dominant half-cycle pulse (N = 1).

192

0 2 4 6 8 10

-100

-50

0

50

100 Soil

El Centro - Meloland Overpass, Imp. Valley EQ.

-100 0 100

-100

-50

0

50

100

0 5 10 15

-50

0

50Rock

LA Dam, Northridge EQ.

Velo

city

(cm

/s)

-50 0 50

-50

0

50

Faul

t Par

alle

l Vel

ocity

(cm

/s)

0 2 4 6 8 10

-100

0

100Soil

Rinaldi Receiving Station, Northridge

Time (s)


-100 0 100

-100

0

100


Figure 4.32b. Selected motions with one dominant half-cycle pulse (N = 1).

193

0 2 4 6 8 10

-100

-50

0

50

100 Soil

El Centro #6, Imperial Valley EQ.

-100 0 100

-100

-50

0

50

100

0 5 10 15-40

-20

0

20

40Soil

Brawley Airport, Imperial Valley EQ.

Velo

city

(cm

/s)

-40 -20 0 20 40-40

-20

0

20

40

Faul

t Par

alle

l Vel

ocity

(cm

/s)

0 2 4 6 8 10-40

-20

0

20

40Soil

Gilroy - Historic Building, Loma Prieta EQ.

-40 -20 0 20 40-40

-20

0

20

40

0 5 10 15

-50

0

50Soil

Newhall, West Pico Canyon Rd., Northridge

Time (s)


-50 0 50

-50

0

50


Figure 4.33a. Selected motions with a dominant full cycle of motion (N = 2).

194

0 5 10 15 20-100

-50

0

50

100 Soil

Parachute Test Station, Superstition Hills EQ.

-100 0 100-100

-50

0

50

100

0 2 4 6 8 10-100

-50

0

50

100 Rock

Pacoima Dam, Upper Left Abb., Northridge EQ.

Velo

city

(cm

/s)

-100 0 100-100

-50

0

50

100

Faul

t Par

alle

l Vel

ocity

(cm

/s)

0 2 4 6 8 10

-50

0

50Soil


-50 0 50

-50

0

50

0 5 10 15-50

0

50Soil


Time (s)


-50 0 50-50

0

50


Figure 4.33b. Selected motions with a dominant full cycle of motion (N = 2).

195

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5

190056 Newhall - W. Pico Canyon Rd.

-1 0 1-1

-0.5

0

0.5

1

Nor

mal

ized

Fau

lt Pa

ralle

l Vel

ocity

0 1 2 3 4 5-1

-0.5

0

0.5

15060 Brawley Airport

Nor

mal

ized

Vel

ocity

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5

1942 El Centro Array #6

-1 0 1-1

-0.5

0

0.5

1

Normalized Fault Normal Velocity

0 1 2 3 4 5-1

-0.5

0

0.5

157476 Gilroy - Historic Bldg.

Time (s)


Figure 4.34a. Simplification of the dominant pulses in the recordings in Figure 4.33a.

196

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5

15051 Parachute Test site

-1 0 1-1

-0.5

0

0.5

1

Nor

mal

ized

Fau

lt Pa

ralle

l Vel

ocity

0 1 2 3 4 5-1

-0.5

0

0.5

124207 Pacoima Dam (upper left)

Nor

mal

ized

Vel

ocity

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5


-1 0 1-1

-0.5

0

0.5

1

Normalized Fault Normal Velocity0 1 2 3 4 5

-1

-0.5

0

0.5


Time (s)


Figure 4.34b. Simplification of the dominant pulses in the recordings in Figure 4.33b.

197

Figure 4.35a. Simplified sine-pulse representation of near-fault ground motions. The fault parallel PGV is set to 50% of the fault normal PGV.

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5

1Set 1

-1 0 1-1

-0.5

0

0.5

1

Nor

mal

ized

Fau

lt Pa

ralle

l Vel

ocity

0 1 2 3 4 5-1

-0.5

0

0.5

1Set 2

Nor

mal

ized

Vel

ocity

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5

1Set 3

-1 0 1-1

-0.5

0

0.5

1


-1

-0.5

0

0.5

1Set 4

Time (s)


198

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5

1Set 5

-1 0 1-1

-0.5

0

0.5

1

Nor

mal

ized

Fau

lt Pa

ralle

l Vel

ocity

0 1 2 3 4 5-1

-0.5

0

0.5

1Set 6

Nor

mal

ized

Vel

ocity

-1 0 1-1

-0.5

0

0.5

1

0 1 2 3 4 5-1

-0.5

0

0.5

1Set 7

-1 0 1-1

-0.5

0

0.5

1


-1

-0.5

0

0.5

1Set 8

Time (s)


Figure 4.35b. Simplified sine-pulse representation of near-fault ground motions. The fault parallel PGV is set to 50% of the fault normal PGV.

199

CHAPTER 5

SITE RESPONSE ANALYSIS METHODOLOGY

5.1 INTRODUCTION

Site response analysis has been an important element of seismic risk assessment

since the pioneering work of Seed and coworkers in the late 1960s. The advent of

computers and the increased experimental database on the cyclic behavior of soils has led

to the development of increasingly more sophisticated site response methods. The ability

of these methodologies to reproduce observed behavior has been well documented for a

number of earthquakes (e.g. Seed et al. 1991, Chang and Bray 1995, Borja et al. 1999).

This chapter presents the implementation of a site response analysis within the context of

the finite element method. The site response methodology is then used in Chapter 6 for

the analysis of site response to near-fault ground motions.

For a given soil profile, the site response problem consists in estimating the

ground motion at a specified depth at the site in response to a ground motion prescribed at

another depth at the site. Typically, the ground motion is assumed to be composed of

vertically propagating, horizontally polarized (SH) shear waves traveling through a

horizontally layered soil profile. This assumption reduces the site response analysis to the

solution of a one-dimensional wave propagation problem. The representation of the

nonlinear response of the soil is typically accounted for in some fashion. The effects of

soil non-linearity on seismic ground motions have long been recognized by the

200

engineering profession, and lately have also been accepted by leading seismologists (Chin

and Aki 1991, Silva et al. 1988, among others). The site response problem is generally

solved either by equivalent-linear analysis (e.g., Schnabel et al. 1972; Hudson et al.

1994), or using non-linear time domain solutions (e.g., Wang et al. 1990, Borja et al.

1999).

In the equivalent-linear analysis, both the secant shear modulus and equivalent

viscous damping values are calculated for an assumed strain level. The wave propagation

problem is solved assuming linear behavior for the assumed values of shear moduli and

damping for each layer. The procedure is iterated until the assumed strain level

corresponds to a specified percentage of the maximum strain calculated for each soil

layer. At each step, the linear 1-D wave propagation analysis is solved in the frequency

domain using a closed form solution to the wave equation (Schnabel et al. 1972). This

methodology involves important assumptions. The frequency domain approach together

with the assumption of linearity at each iteration step implies the assumption that the

ground motion is stationary. Moreover, a unique set of properties (e.g, stiffness and

damping) is used for the entire time history at each layer. The equivalent-linear

procedure has been validated for a number of case studies, primarily using the computer

code SHAKE (Schnabel et al. 1972), and its later edition SHAKE91 (Idriss and Sun

1992). However, it is important to point out that validations in general have been done

using "normal" ground motions, that is, ground motions at reasonable levels of intensity

that do not exhibit typical near-fault characteristics. Seed et al. (1991) indicate that the

difference between the equivalent-linear method and non-linear analyses could be

significant with higher levels of shaking. Moreover, peak strains in near-fault ground

201

motions occur during a single or a limited number of high intensity stress cycles. Strains

observed during these cycles are likely not representative of strain levels during the

remainder of the motion. Assuming soil properties at only a fraction of peak strain during

these cycles may render unreasonable estimates of soil stiffness and damping during near-

fault pulses.

Non linear computer programs differ in two main aspects: the tools used for the

solution of the equation of motion (i.e. the wave propagation equation) and the definition

of soil stress-strain behavior. Methods that have been applied to the solution of site

response problems include elastic wave propagation methods (e.g. SHAKE, Schnabel et

al. 1972), the method of characteristics (e.g. CHARSOIL, Streeter et al. 1974), finite

difference (e.g. Martin and Seed 1982), and finite element (e.g. DYNA1D, Prevost 1989;

QUAD4M, Hudson et al. 1994). Soil non-linearity is addressed by introducing

mathematical relationships that model the stress-strain behavior of the soil. A brief

summary of cyclic non-linear models was presented in Chapter 2. The most general

models able to represent soil behavior are based on plasticity theory. Plasticity-based

models permit the evaluation of soil response along a variety of stress paths and can be

easily expanded into three-dimensional stress space to model simultaneously the three

components of motion.

A site response method that is used for the analysis of near-fault, forward-

directivity ground motions must take into consideration their particular characteristics.

These motions are dominated by a small number of long-period pulses best observed in

velocity or displacement-time histories (see Chapter 4). These strong pulses of motion

202

likely generate large strain levels in soils, particularly for softer and deeper soils that have

site periods on the order of one to three seconds. Forward-directivity pulses also control

peak response values affected by long-period motions (i.e. PGV and peak ground

displacement, PGD). In addition, the strain levels generated by uni-directional shaking

are lower than those generated by multi-dimensional shaking. Hence, site response

analyses using all components of motion are more likely to provide reasonable results

than uni-directional analyses. This is important also for near-fault ground motions

because the analysis of the near-fault ground motion database indicated that, in some

cases, significant fault parallel velocities in phase with the fault normal velocity pulse are

observed. Thus, in summary, a site response analysis for near-fault ground motions

should be able to:

• Handle significant levels of strains

• Account for bi-directional input motions

• Deal with highly non-stationary input motions (i.e., time domain analysis)

These requirements are best met by finite element analysis in conjunction with an

appropriate constitutive model. This chapter presents the development of the site

response methodology used in Chapter 6 for the analysis of near-fault ground motions.

Section 5.2 summarizes the constitutive model used for the analysis. Section 5.3 presents

the implementation of this constitutive model into the finite element analysis program

GeoFEAP (Espinoza et al. 1995). Finally, a validation of the site response methodology

is presented in Section 5.4.

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5.2 CONSTITUTIVE MODEL

5.2.1 Selection of a constitutive model

A constitutive model is defined by a set of equations that relate the strains in a

material to the current state of stress and temperature through a set of internal variables.

The mechanical response of soil is generally very complex, and constitutive models that

aim at describing a comprehensive range of soil behavior (such as dilatancy, small strain

non-linearity, stress anisotropy, etc) are generally complicated and involve a large number

of input parameters. Conversely, relatively simple models might over-simplify important

aspects of soil response. In general, all constitutive models share the same basic

requirements (Luccioni 1999):

1. Assumptions inherent to the formulation of the constitutive laws should be

justified with respect to both the material and problem to be modeled. The

response predicted by constitutive models should match observed field and

laboratory response of soil. The latter requirement implies that particular

aspects of soil behavior relevant to the problem in question are addressed. For

example, a model that assumes a linear elastic region might be adequate when

dealing with the prediction of stresses in a slope, but is inadequate for dealing

with the intermediate strain cyclic behavior of soil.

2. Procedures to estimate model parameters should be provided along with the

formulation of the constitutive model.

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3. Numerical implementation of these constitutive laws into numerical codes

should ensure that when the converged state is achieved, governing equations

and energy and mass balance laws are satisfied.

A large number of constitutive models satisfying the above requirements have

been proposed. A brief summary of some of the models currently available for the cyclic

response of soils is presented in Chapter 2. In this section, a model capable of modeling

cyclic soil behavior is sought. The model is applied to site response problems involving

generalized site profiles (see Chapter 6). A number of the simplifying assumptions

involved in the generation of the site profiles have implications on the requirements for

the constitutive model. These assumptions and their implications on model development

include:

1. The generalized site profiles represent average site conditions for a given site.

Soil models that are used for these generalized profiles should capture general

aspects of soil behavior, but it is not required that the model be able to match

exactly each possible response of a particular type of soil.

2. Undrained soil behavior is assumed. Moreover, pore pressure increase during

shaking is assumed to be insignificant. These assumptions imply that the

constitutive model can be developed in a total stress framework. The use of an

effective stress framework in a constitutive model in general leads to wider

predictive capabilities at the cost of increased complexity in the definition of the

model. The use of a total stress soil model permits simplifying assumptions, such

as the uncoupling of the deviatoric and volumetric components soil response.

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This, in turns, permits the uncoupling of site response analysis to incoming shear

and compressive waves. The uncoupling of horizontal shear waves and

compressive waves has been verified with models that do account for the

interaction between volumetric and deviatoric behavior of soils (EPRI 1993). The

validity of the assumption that no significant pore pressure is generated can be

questioned for the case of near-fault motions, in particular due to the potentially

large pore pressures that may develop due to the large strains observed in a soil

profile subject to near-fault velocity pulses. However, with the exception of very

loose to medium dense, saturated sands and silts, and probably some special clays,

the small number of pulses involved in near-fault ground motions may preclude

the rapid development of sufficient pore pressures that significantly alter site

response. In any case, a first-look at the near-fault site response problem

considering soils not prone to the development of significant pore pressures

during loading provides insight for these types of soils. The uncoupling of

volumetric and deviatoric behavior permits the use of models using a constant

void ratio, essentially, the void ratio at small strains.

The soil constitutive model used should be able to accommodate the large strains

expected in near-fault ground motions. Moreover, the model should be able to account

for soil yielding for the cases in which the soil's shear strength becomes an issue (such as

for soft clays, or Site E in Table 3.1). For this study, the model presented by Borja and

Amies (1994) is selected from those available at the start of this research. This model

conforms to the requirements identified previously while balancing simplicity and

robustness in its implementation.

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The model by Borja and Amies (1994) is a multiaxial constitutive model

developed for clays. The model expands on the concept of bounding surface plasticity

with a vanishing elastic region (Dafalias and Popov 1977) by presenting general criteria

for loading and unloading that are applicable to general stress states. The concept of the

vanishing elastic region allows for the modeling of plastic strains developed at small

strains. This section presents a summary of the model by Borja and Amies (1994) that

includes the kinematic hardening of the boundary surface (Borja et al. 1999). The model

presented herein differs from the mentioned models (Borja and Amies 1994, Borja et al.

1999) solely in the introduction of Rayleigh damping to model small strain viscous

damping. The presentation, however, is slightly different from that in the aforementioned

papers.

5.2.2 Mathematical development

General Description

Assume the existence of a point Fo in three dimensional stress space defined by

the stress tensor σσσσo (Figure 5.1). This point defines the stress state where the soil

experienced the most recent unloading, where unloading is defined as the condition in

which the direction of the load step causes the instantaneous hardening modulus to

increase (Borja and Amies 1994). The unloading condition is presented with larger detail

later in this section. The point Fo is located within a bounding surface defined by the

function B. The soil is assumed to behave elastically at point Fo and to follow a linear

kinematic hardening plasticity law at the bounding surface B. In between the point Fo

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and the bounding surface B, the hardening modulus is interpolated between the values of

infinity at Fo (corresponding to elastic behavior) and Ho at the surface B (corresponding

to a linear, kinematic hardening plasticity law). A yield surface F is defined inside the

bounding surface passing through a point defined by the current stress tensor σσσσ. The

vanishing elastic region corresponds to the limit when the size of F is zero. The

interpolation of the hardening modulus H' is generated from well-accepted one-

dimensional models for soils (Borja and Amies 1994). A formal mathematical

development of the model is now introduced. For further details of the model, see Borja

and Amies (1994) and Borja et al. (1999).

Model Development

Assume that strains within the soil are infinitesimal. Furthermore, assume an

additive decomposition of the Cauchy stress tensor, σσσσ, into an inviscid component (σσσσinv)

and a viscous component (σσσσvis):

σσσσ = σσσσinv + σσσσvis (5.1)

Laboratory tests have shown that damping at large strain levels is essentially

frequency independent and can be attributed to the hysteretic behavior of soils. On the

other hand, recent experimental results with damping at very small strains indicated that

at these strain levels, soils experience damping levels that can not be attributed to

hysteretic mechanisms alone (Lanzo and Vucetic 1999). The mechanism that controls

energy dissipation at these strain levels has been attributed to rate-of-loading effects

(Lanzo and Vucetic 1999). Typically, damping at low strains is low and can be modeled

208

as equivalent viscous damping. Viscous damping is typically represented as a fraction of

critical damping, where critical damping corresponds to the smallest damping value that

inhibit oscillations completely. The viscous term in Equation 5.1 is assumed to follow

Rayleigh damping, that is (Cook et al. 1989),

Dεσ =vis (5.2)

where εεεε is the strain tensor and D is a rank-four damping tensor given by:

[ ] [ ]MCD 2e

1 αααααααα += (5.3)

where Ce is the rank four small-strain elasticity tensor and M is the mass matrix. The use

of Rayleigh damping results in frequency dependent damping. The constants α1 and α2 in

Equation 5.3 are calculated for a selected value of critical damping ratio, ξ, using two

reference frequencies, ω1 and ω2, at which the resultant damping ratio is matched to the

desired damping ratio ξ (Figure 5.2). The constants α1 and α2 are determined by

12121

212

21

2

2

ααααωωωωωωωωωωωωωωωωξξξξωωωωωωωωαααα

ωωωωωωωωξξξξαααα

=+

=

+=

(5.4)

The constants α1 and α2 control the values of stiffness and mass proportional

damping, respectively (Equation 5.3). Figure 5.2 illustrates the dependence of mass and

stiffness proportional damping on frequency. Mass-proportional damping is zero at high

frequencies. On the other hand, stiffness proportional damping results in over-damping at

high frequency. The use of both mass and stiffness proportional damping allows a fit for

209

the desired value of damping ratio within a selected range of frequencies (i.e., close to

and between ω1 and ω2). Between ω1 and ω2, the resulting damping is lower than the

desired critical damping ratio. Away from this range, the system is over-damped.

Hudson et al. (1994) proposed setting ω1 to the natural frequency of the soil deposit and

letting ω2 = n ω1, where n is an odd integer, such that n is the closest odd integer greater

than ωi/ ω1, where ωi is the predominant frequency of the input earthquake motion. The

choice of n as an odd integer is motivated by shear beam studies showing that the

frequencies of higher modes are odd multiples of the frequency of the fundamental mode

of the beam. In general, ω1 and ω2 can be chosen so as to cover a desired range of

frequencies.

The inviscid term in Equation 5.1 is represented in terms of the bounding surface

plasticity model outlined previously. The stress-strain relations are described using an

incremental formulation that relates infinitesimal increments of stress and strain. For rate

independent models, these equations can be written either in rate or differential format.

When these relations are written in rate format, time serves only to indicate precedence in

the history of events. Additive decomposition of elastic and plastic strains is assumed:

pe εεε DDD += (5.5)

where εεεε is the total strain tensor and the dot indicates differentiation with respect to time.

The generalized Hooke's law is used to relate elastic strains with the stress tensor, thus,

)(: pe εεCσ �� −= (5.6)

where Ce is the rank four elasticity tensor.

210

Plastic strains are associated with the yield surface F and the boundary surface B.

These surfaces are translated cylinders in stress space, and are given by:

0r:F 2FF =−= ζζζζζζζζ (5.7)

where ζζζζF = σσσσ' - α,α,α,α, αααα is a deviatoric back stress tensor representing the center of F, and r is

the radius of the cylinder; and

0R:B 2BB =−= ζζ (5.8)

where ζζζζB = σσσσ' - β,β,β,β, ββββ is a deviatoric back stress tensor representing the center of B, and R

is the radius of the bounding surface B.

The yield surface F is always contained in B. This implies that B = 0 if and only

if F = 0. For both the yield surface and the bounding surface, an associative flow rule is

used to define the direction of plastic strain increments, therefore

BBFF ˆˆ nnε λλ +=p� (5.9)

where λF and λB are the consistency parameters associated with the yield surface and the

bounding surface respectively, and Fn̂ and Bn̂ are unit normal tensors to each yield

surface ( Fn̂ = ∂F/∂σσσσ = ζζζζF/ Fζ and Bn̂ = ∂B/∂σσσσ = ζζζζB/ Bζ ). Equation 5.9 is similar to

generalized flow rules used for multiple yield surface plasticity models (Pestana 1994).

For a point in the surface of F, the unit normal Fn̂ is equal to ζζζζF/r. Note that Fn̂ is

ill defined in the limiting case of the vanishing elastic region when the radius of the yield

surface F tends to zero. Taking the limit as r goes to zero, Fn̂ if defined by:

211

σσζn�

�=

=→ r

ˆ lim0r

FF (5.10)

A restriction on the consistency parameters is placed such that

0FB =λλ (5.11)

which implies that at any given time, only one of the two surfaces (F and B) is the active

yield surface, in the sense that plastic flows occurs only in that surface. Loading

conditions in traditional rate-independent plasticity models are given by the Kuhn-Tucker

loading/unloading (complementarity) conditions (Simo and Hughes 1998). For any given

yield surface f, these conditions are given by:

λ > 0, f ≤ 0, λf = 0 (5.12)

where λ is the consistency. Equation 5.12 is used to define loading and unloading when

the bounding surface is the active yield surface (i.e. λB ≠ 0). These loading conditions,

however, are ill defined for the yield surface F due to the assumption of a vanishing

elastic region (Borja and Amies 1994). The loading/unloading condition for states of

stress within the bounding surface is presented later.

Associative flow is assumed for the hardening rules for parameters ββββ and αααα (note

that ∂F/∂αααα = - Fn̂ and ∂B/∂ββββ = - Bn̂ ). The hardening rules are

BBB

FFF

ˆ

ˆ

nβnα

dd

λλ

−=

−=C

C

(5.13)

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where dB and dF are generalized plastic moduli coefficients (Simo and Hughes 1998).

The plastic moduli coefficients are chosen to so that the uniaxial stress σ� = (3/2)1/2 σ� ′

is related to the uniaxial plastic strain pε� = (2/3)1/2 pε� by the hardening modulus H', so

that

pH εσ �� '= (5.14)

where H' is the uniaxial plastic modulus. For this effect, dB and dF are chosen to be equal

to 2/3 H'. The functional form of H' defines the stress-strain behavior of the soil.

The consistency conditions are applied in two mutually exclusive cases, (a) when

the stresses are inside the bounding surface, that is, when the active yield surface is F and

the model behaves as a bounding surface plasticity model (the necessary and sufficient

condition for this case is λB = 0); and (b) when the stresses are on the bounding surface

(i.e., B = 0). In the latter case, the bounding surface acts as a yield surface. The

application of the consistency conditions completes the required steps for the full

definition of the constitutive model and leads to the relationship between stresses and

strains. Both cases are presented separately below.

case i) Stresses are inside the bounding surface (λB = 0). The consistency condition is

defined by expanding F using Taylor series and neglecting terms higher than

linear,

0:2F FF == ξξ DD (5.15)

dividing by r and assuming r ≠ 0 we get

213

0)ˆ =−ασ(n DD:F (5.16)

Equation 5.6, relating the stress rate tensor to the strain rate tensor, can

now be rewritten using the associative flow rule (Equation 5.9) with λB = 0, and

observing that Fn̂ is a deviatoric tensor,

)ˆ(G2)K FFtr( nε1εσ λλλλ−′+= �� (5.17)

where K is the elastic bulk modulus, G is the elastic shear modulus, 1 is the rank-

two identity tensor, and tr(⋅) is the trace operator. Equation 5.17 assumes that the

elastic behavior of the soil is isotropic. By virtue of the vanishing elastic region,

the total deviatoric strain tensor ε′D is directed along the unit normal vector Fn̂ .

Using Equation 5.16 in conjunction with the hardening rule (Equation 5.13), the

consistency parameter λF can be expressed as

εεεελλλλ ′+

= �

G3'H1

1F (5.18)

From Equation 5.17 and 5.17, and the fact that the strain rate is oriented

parallel to Fn̂ , the rate constitutive equation is obtained:

ε1εσ ′

′++=

−

CCC

1

H31G2)K µµµµtr( (5.19)

Note that the assumption of a vanishing elastic region enters the derivation

solely through the limit in Equation 5.10.

214

case ii) The stress state is on the bounding surface (λF = 0, λB ≠ 0). At this point,

assume that the yield surface and the bounding surface coincide at the stress

point σσσσ'. This assumption must be assured in the implementation by making

sure that loading inside the bounding surface continues until the bounding

surface is encountered. This assumption ensures that the total strain is aligned

with the normal vector Bn̂ . The same condition was obtained inside the

bounding surface by using the vanishing elastic region assumption (Equation

5.10). The consistency parameter λB is then obtained in an identical fashion as

its counterpart inside the bounding surface ,λF (Equations 5.15 – 5.20), and is

given by

εD

G3H1

1′

+=Bλλλλ (5.20)

The resulting stress-strain difference equation is identical to Equation

5.19. The difference in both cases (i.e. when either F or B are the active yield

surface) lies on the definition of the hardening modulus H' that is presented later.

Note that the previous development could have been followed making use of a

single yield surface F, but with different hardening rules that specify separately when αααα

changes (in the present development, when λF ≠ 0) and when ββββ changes (i.e., when λB ≠

0).

215

Hardening Function

The hardening modulus H' can be defined in such a way as to fit accepted one-

dimensional cyclic stress-strain relationships. Along with the functional definition of H',

the unloading conditions must also be clearly defined, since plastic deformations grow

proportionally with distance from the point of last unloading (Borja and Amies 1994).

The hardening modulus is obtained from an interpolation between the elastic value (H' =

infinity) at the last unloading point (elastic nucleus), to a limiting value of Ho at the

bounding surface. Following the concept of bounding surface plasticity models (Dafalias

and Popov 1977), an image point σ̂ on B is defined such that (Borja and Amies 1994):

( )oBB σσζζ ′−′+= κκκκˆ (5.21)

where βσζ −′= ˆˆB , and σσσσo' is the stress tensor at Fo (Figure 5.1). The dimensionless scalar

quantity κ satisfies the condition.

( )oBB σσζζ ′−′+= κκκκˆ (5.22)

An exponential hardening modulus is chosen for this work. This model was

validated by Borja et al. (1999 and 2000) using events recorded at the Lotung and Gilroy

downhole arrays. The model is expressed by:

mhκH =′ (5.23)

where h and m are material parameters. For other interpolation functions for H', see

Borja and Amies (1994).

216

The loading and unloading conditions when the stress is on the bounding surface

(λB ≠ 0) are given by the Kuhn-Tucker conditions (Equation 5.12). Within the bounding

surface, a different condition must be applied because the singularity due to zero radius of

the elastic region renders the loading condition ill defined. Borja and Amies (1994)

define unloading as the condition in which the direction of the load step causes the

hardening modulus to increase. The loading condition is then postulated as (see Borja

and Amies 1994 for details)

( ) ( )( )( )

0ooB

oB >′′−′+′−′

′−′+++− ε:σσσσ:ζσσζ

2

11κκκκκκκκκκκκκκκκ (5.24)

Upon unloading, the position of the reference point Fo must be shifted to the

current position defined by the stress tensors σσσσ'. Equation 5.24 implies that the

loading/unloading condition depends solely on the direction of the strain tensor ε′D . This

is a highly desirable condition given that the finite element program GeoFEAP is strain

driven.

5.3 FINITE ELEMENT IMPLEMENTATION

5.3.1 General

The finite element method has been used extensively to solve numerous problems

in geotechnical engineering (e.g. Britto and Gunn 1987, Zienkiewicz et al. 1999). The

finite element method solves a boundary value problem by prescribing interpolation

functions between nodal values for the unknown variables. The solution is obtained by

217

minimizing an energy function or some error measure over the whole domain. The

present implementation is developed within the finite element program GeoFEAP

(Espinoza et al. 1995). GeoFEAP is a multi purpose, nonlinear finite element code

developed as a combined effort between the Computational Mechanics and Geotechnical

groups at the University of California at Berkeley.

For this implementation, the site is assumed to be composed of horizontal layers

of finite thickness and infinite lateral extent resting on an elastic half-space. The

condition of zero initial static shear stresses is assumed. Moreover, the seismic waves are

assumed to originate in the elastic half-space of the soil column and travel in the vertical

direction through the soil column. These assumptions imply that all the variables are

functions of depth and time only. The problem is thus reduced to one spatial dimension,

and the site can be represented by a vertical soil column. The boundary value problem

can then be defined as: Given a prescribed displacement function (of time) at the base of

the soil column, and given a constitutive relations that determines the stress-strain

behavior of the soil, solve for the time-displacement function at any other point in the

column for the given boundary conditions.

The boundary conditions are zero strain in the horizontal direction and zero stress

at the soil surface. The boundary condition for the base of the soil column must be

prescribed such that it acts as an elastic, semi-infinite half space (Section 5.3.3).

Although the problem is one-dimensional in space, the prescribed displacement time

function can have arbitrary spatial orientation, thus each node for each soil element is

allowed three degrees of freedom (rotational degrees of freedom are ignored).

218

The boundary value problem can also be stated in terms of the relative

displacements of the soil with respect to the base. In this case, the boundary condition at

the base is fixed and an inertial body force equal to ρ guDD , where guDD is the ground

acceleration, is applied to the soil column (Zienkiewicz and Taylor 1989). The numerical

computations for both cases are identical if the same initial conditions are assumed. The

advantage of the relative displacement formulation is that the input is prescribed in terms

of accelerations, which are often more accurately known (Zienkiewicz et al. 1999).

The governing equations of the dynamic boundary value problem are the balance

laws of linear momentum (i.e. Zienkiewicz and Taylor 1989). The dynamic problem

involves spatial and temporal independent variables. Typically, the time variable is

discretized and is solved using some type of finite difference approximation, while the

spatial problem is solved using the finite element method. Similarly, the solution can be

posed in a space-time domain in what is commonly referred to as space-time finite

element (Zienkiewicz and Taylor 1989). In this implementation, the former method is

used. The method chosen for advancing the solution in time was proposed by Hilber,

Hughes, and Taylor and is commonly known as the HHT method (Hilber et al. 1977).

This time integration scheme is second order accurate and unconditionally stable. The

HHT method is part of a larger family of single step algorithms presented by Katona and

Zienkiewicz (1985). The parameters in the HHT method can be used to control the

amount of numerical damping introduced by the time integration scheme.

The finite element approximation to the balance of linear momentum equations is

presented in detail in finite element textbooks (e.g., Zienkiewicz and Taylor 1989). In

219

this implementation, an iso-parametric formulation using three dimensional "brick"

elements with tri-linear shape functions is used. At any given time, the finite element

approximation together with the time stepping scheme yield a system of nonlinear

algebraic equations for the nodal displacements, F(u), where u is the vector of nodal

displacements. This system of equations represents the balance of internal and external

(applied) forces. In the site response problem, external forces correspond to the applied

inertial forces. Thus, from the global point of view, the finite element problem is load-

controlled. The system of equations F(u) is solved using a Newton-Raphson scheme

(Taylor 1998). The effectiveness of the solution depends on the "quality" of the tangent

used in the solution. Simo and Taylor (1985) have shown that the Newton-Raphson

method in combination with the algorithmic or 'consistent' tangent presents a very

efficient way of solving the nonlinear system of equations F(u). The tangent matrix

∂F/∂u is assembled by the finite element code from the element stiffness matrices. Both

element stiffness matrices and the internal forces are calculated at the element level for a

given value of the displacement node vector. Thus, at the element level, the problem

becomes strain (displacement) controlled. The integration of the constitutive equations at

the element level is presented in Section 5.3.2.

5.3.2 Local integration of the constitutive equations

The incremental solution of boundary value problem requires the integration of

the constitutive equations. In the context of the finite element method, displacements are

primary variables determined iteratively from the global balance laws. Hence, total strain

increments are considered as input data and the integration of the constitutive model

220

reduces to finding the stress state at time n+1 for the given strain increment and the

converged values at time n. The stresses found are used to determine the internal stress

vector used to solve the global set of equations. Moreover, the algorithm at the local

element level must provide the consistent tangent (Simo and Taylor 1985), the element

mass matrix, and the element damping matrix. The type of the mass matrix used in this

implementation is the consistent mass matrix (Zienkiewicz and Taylor 1989). The

element damping matrix is obtained using Equation 5.3. The consistent tangent arises

from the linearization of the stress increment with respect to the strain increment. The

presentation in this section is reduced to the integration of the constitutive equations and

the consistent tangent. The definition of the internal stress vector is presented in finite

element textbooks (e.g. Zienkiewicz and Taylor 1989 and Cook et al. 1989).

Let σσσσn, ββββn, and σσσσon denote the known stress tensor, the back-stress tensor, and the

last unloading point at time n. For a given strain increment ∆εεεε, the local integration

algorithm must return the updated values of the stress tensor, σσσσn+1, and the hardening

variable ββββn. The first step in the iteration is to check whether the current state of stress is

on the bounding surface, let

2nBnB R−= ζζ :B (5.25)

where B = 0 defines the bounding surface (Equation 5.8), and ζζζζBn is the algorithmic

counterpart of ζζζζ. In the case of B = 0, the bounding surface is the active yield surface (λB

≠ 0 and λF = 0) and the integration of the constitutive model is done through a radial

221

return mapping algorithm. The radial return mapping algorithm is presented in Box 5.1

(from Simo and Hughes 1998).

If B < 0, the current state of stresses is inside the bounding surface and the

integration of the constitutive model and the plastic strains are obtained from the

bounding surface plasticity model. The loading/unloading condition is determined using

the direction of the strain increment ∆εεεε. The condition for unloading is obtained by

taking the limit of Equation 5.24 as ∆t → 0 (Borja and Amies 1994),

( ) ( )( )( ) o

oonBn

oBn tol::

11nnn

nnnnn >′′

′−′+′−′′−′++−−

εε

σσσσζσσζ

n ∆∆∆∆∆∆∆∆

κκκκκκκκκκκκκκκκ

(5.26)

where tol0 is a numerical tolerance parameter. Upon each unloading, the stresses at the

last unload point, σσσσon are reset to σσσσn and the value of κ is set to infinity (i.e., an infinite

plastic modulus). In a strict sense, unloading occurs anytime the left hand side of

Equation 5.26 is larger than zero. Within a finite element implementation, however, the

iterative nature of the solution sometimes produces unloading at extremely small values

of strain. This unloading (termed 'numerical' unloading) can lead to instabilities in the

convergence to the solution. After a reload cycle, the instantaneous plastic modulus H' is

now interpolated between the new unload point at oσ′ and the bounding surface. When

the unloading step is small (i.e. with numerical unloading), the result is observed as a

jump in the stress-strain loop (Figure 3). For this reason, tol0 is set to a positive value <<

1. From the physical point of view, this is equivalent to establishing a small elastic

region around σσσσo where the material behaves elastically upon unloading. The value of

tol0 is determined solely from convergence requirements of the global finite element

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algorithm. Note, however, that tol0 can also be assigned a physical significance. That is,

tol0 is equivalent to the strain level that induces the soil to 'feel' unloading. This value

could be related to the elastic threshold strain (Pestana, personal comm.).

After the load/unload condition has been established, the integration of the

constitutive equation is performed following the algorithm of Borja and Amies (1994),

which is included in Box 5.2. If Equation 5.26 indicates a condition of numerical

unloading, the material unloads following the tangent stiffness at σ,σ,σ,σ, but the stress reversal

point Fo is not changed from its previous value.

Equation 1 in Box 5.2 is obtained by applying a generalized trapezoidal rule to

Equation 5.19. The parameter β is the trapezoidal integration parameter (i.e., explicit if β

= 0, implicit otherwise). The parameters ψ and κ in Equations 3 and 4 in Box 5.2 are

solved using a Newton iteration with initial estimates ψo = 2G and κn+1 = κn. If κ

converges to a negative value, it implies that the final stress state is outside the bounding

surface. In this case, the strain step is partitioned such that the first step takes place inside

the bounding surface, and the second step corresponds to a kinematically hardening

plasticity model with the bounding surface as the active yield surface. The strain tensor is

then given by:

FB : εεε ∆∆=∆ (5.27)

where ∆εεεεF = γ ∆εεεε is the strain within the bounding surface, and ∆εεεεB = (1 - γ) ∆εεεε is the

strain step corresponding to plastic yielding on the bounding surface. The parameter γ

corresponds to an intermediate time tn+γ between tn and tn+1 corresponding to the strain

223

step ∆εεεεF. It is assumed that the strain step is applied uniformly over the time step. The

value of γ is obtained following the algorithm in Box 5.3. After the partition, the stresses

σσσσn+ γ are obtained using the algorithm in Box 5.2 with ∆εεεε = ∆εεεεF = γ ∆εεεε. The strain step

∆εεεεB is then taken using the algorithm in Box 5.1 with σσσσn+γ as the initial condition. The

tangent modulus Cep is obtained by a linear interpolation of the tangent moduli returned

by Box 5.1 and Box 5.2:

( ) [ ]( ) ( )[ ] εεεεαααααααα

εεεεαααααααα∆∆∆∆

∆∆∆∆∆∆∆∆∆∆∆∆

:1)1(:::

epep

epep

BF

BF ep

CCCεCεC

−+=

−+= (5.28)

where epFC is the consistent tangent corresponding to the strain step inside the bounding

surface (Box 5.2), and epBC is the tangent for the kinematically hardening strain step (Box

5.1).

The algorithm in Box 5.2 is implicit for values of β larger than zero. Similarly,

the radial return mapping algorithm in Box 5.1 is obtained by applying the fully implicit

backward Euler integration scheme to the constitutive equation. Fully implicit algorithms

guarantee linear stability, also called A-Stability. Non-linear stability, or B-stability, can

be identified by combining the return mapping algorithm with a time discretization of the

algorithmic weak form of the finite element approximation (Simo and Hughes 1998).

Unfortunately, non-linear stability analyses become very cumbersome and have been

done only for relatively simple models (e.g., Simo and Govindjee 1991).

224

5.3.3 Radiation Boundary Conditions

General considerations

Figure 5.4 presents a schematic diagram of the site response problem. Note that

for simplicity, the column analyzed is called 'soil', and the half-space where the motion is

prescribed is called 'rock'. This terminology is used only for convenience. Clearly, the

input motion could be as easily be prescribed within the soil, as well as the site response

analyses could be performed for a rock column. An incoming stress wave is applied to

the soil column at point A. This stress wave travels through the soil and is reflected at the

free surface of the soil. The reflected wave hits the soil-rock interface and is both

transmitted across and reflected from the base rock. The actual energy of the reflected

and transmitted waves is a function of the impedance contrast (ρsVs/ρrVr) between the soil

and the rock layers. The boundary conditions prescribed in the finite element analysis

should account for the correct distribution of the reflected and transmitted energy at the

rock-soil boundary.

In Figure 5.4, both point A (underneath the soil column), and point B (at the free-

surface) are subject to the same incoming vertically propagating seismic shear waves.

Both points, however, are subject to different boundary conditions. Point A receives the

stress waves that are reflected from the soil surface. On the other hand, the boundary

condition at point B is that of a stress-free boundary. The incoming waves at the free

rock surface are fully reflected into the half space. The effect of the full reflection is that

the incoming and reflected wave have constructive interference and the displacements at

the surface are doubled. The process of modifying the free surface rock motion (typically

225

called the outcropping motion) to represent the motion underneath the soil column (called

the within motion) is called deconvolution. Deconvolution consists essentially in

eliminating the 'free surface' effect and in removing the energy corresponding to the wave

reflected from the surface. In the context of the finite element analysis, the deconvolution

process is accomplished by the appropriate definition of the boundary conditions at the

base of the soil column. In this section, the corresponding boundary conditions are

presented.

When the prescribed input motion in a site response analysis corresponds to an

outcropping motion, a number of conditions must be satisfied so that the deconvolution

process results in the actual motion at the base of the soil column. These conditions

apply in particular when site response analyses are performed to obtain a ground motion

at a particular location. In the present work, site response analyses are preformed for

generic site conditions (Chapter 6) and the considerations provided below are not

necessarily critical in all cases. However, these conditions are presented because of their

importance in the general context of site response analysis.

1) The materials underneath the soil column are assumed to behave as an elastic,

homogeneous half space. If this assumption does not hold in the field, a number

of phenomena which are not accounted for in the analysis can take place. Some of

the energy that is transmitted into the rock from downward traveling waves could

be reflected once more into the soil from larger impedance contrasts at depth.

This situation is likely to arise when the input motion is specified at a relatively

soft layer. At this soft layer, a large proportion of the downward traveling waves

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is transmitted into the underlying layer. This energy, however, will likely be

reflected back in part from basement rock at lower depths. This consideration

affects long period waves in particular, because these are able to travel larger

distances without scattering and they are attenuated less rapidly by material

damping. Conversely, when the material underlying the soil column is much

stiffer than the overlying material, most of the energy of the downward traveling

waves is reflected back into the soil and the effects of potential impedance

contrasts at greater depths are reduced.

2) The properties of the material underneath the soil column (point A in Figure 4) are

the same as the materials at the free surface (point B). This assumption is

commonly made in site response analyses. The stiffness of geo-materials,

including rock, is a function of confining stress. For this reason alone, it is

unlikely that this condition is satisfied in the field. In addition, rock that is

exposed to a free surface is more likely weathered and has larger joint spacing

than rock that is covered by soil. Both these effects imply softer conditions for

rock at B than at A.

3) The energy of the outcropping motion corresponds solely to vertically propagating

body waves. This condition is also not likely satisfied for most recordings at rock

sites. Silva (1988) found that about 75% of the power (87% of the energy) in a

free surface motion could be attributed to vertically propagating shear waves at

frequencies up to 15 Hz (Kramer 1996). Improvements in the deconvolution

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process can be realized by appropriate pre-filtering of the recorded ground motion

(Silva 1988).

Mathematical development

Let u = u(z,t) denote the relative horizontal displacement field in the soil column

with respect to the free field condition (input motion). Moreover, assume that only elastic

behavior exists. Under this condition, the laws of balance of momentum and the

constitutive relation σσσσxz = G ∂u/∂z yield the one dimensional wave equation

2

2

2s

2

2

tu

V1

zu

∂∂=

∂∂ (5.29)

where Vs is the shear wave velocity (Vs2 = G/ρ). The solution to this equation is given by

( ) ( )tVzutVzuu sOsI ++−= (5.30)

where and uI and uo are two waves traveling in the upward (incoming) and downward

direction (outgoing), respectively. At the rock-soil interface (point A), the boundary

condition should represent only an outgoing wave (recall that u is the relative

displacement with respect to the free field condition, and the displacement at the free

field is the displacement of the upward wave). Then we have (Zienkiewicz et al. 1999)

( )tVzuu sO += (5.31)

Using this condition, we observe that:

228

sVutu ′=∂∂ and u

zu ′=

∂∂ (5.32)

where ( )( )..

dduu =′ .

Therefore, on the boundary, we have:

tu

V1

zu

s ∂∂=

∂∂ (5.33)

The tangential shear stress at the boundary then becomes

tuV

zuG sxz ∂

∂=∂∂ ρ:σ (5.34)

This is equivalent to the requirement that a viscous dashpot acts on the boundary.

A similar result is obtained for compressive (P) waves. Representation or the boundary

conditions in the manner presented above was suggested independently by Lysmer and

Kuhlmeyer (1969) and Zienkiewicz and Newton (1969). Lysmer and Kuhlmeyer (1969)

show that this type of boundary condition results in appropriate energy transmitting

properties.

5.3.4 Model Parameters

The parameters of the constitutive model and the parameters used in its numerical

implementation are listed in Table 5.1. The methods used to obtain the appropriate value

of the parameters are also included in Table 5.1. The bounding surface plasticity model is

fully defined by two elastic constants, model parameters h and m, and the radius of the

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bounding surface R. In addition, the plastic modulus Ho defines the behavior of the

model after the bounding surface has been reached.

Parameter h and m are obtained by matching the shear modulus reduction (G/Gmax,

where Gmax is the small strain modulus) and material damping versus shear strain curves.

These curves are typically obtained from tests using radial loading, paths such as cyclic

triaxial, resonant column, and torsional shear tests (Borja et al. 1994). Using the stress

path defined by cyclic simple shear tests, the relationship between the moduli ratio and

the corresponding amplitude of shear strain-stress curves, χ, is given by (Borja et al.

1999):

ξξξξξξξξ

ξξξξχχχχθθθθ

χχχχγγγγθθθθ dH

G2

Rh

231

1

G2

0 o

m

maxmax

−

∫

+

−+= (5.35)

where θ = G/Gmax.

Equation 5.35 can be made to pass through two points in the measured shear

modulus reduction versus shear strain curve, and the parameters h and m can be defined

for a given value of R. Once the modulus reduction curve is matched, the predicted

damping curve should be checked to ensure it is within reasonable values for the soil in

question. More often than not, a perfect match of both shear modulus reduction and

damping is not possible, and a compromise between matching the damping curve or

matching the shear modulus curve must be accepted. Figure 5.5 and 5.6 shows the

influence of parameters h and m on shear modulus reduction curves. In general, an

increase in m causes the curvature of shear modulus reduction curves to increase, while

230

an increase in parameter h causes a shift to the right of both shear modulus reduction and

damping curves.

The parameter R, the radius of the bounding surface, is defined by the shear

strength of the soil. The value of R is given by R = 38 su, where su is the unconfined

compressive strength of the soil. Similarly, R can be defined by R = 2 τf, where τf is the

failure strength of soil in a simple shear tests. In terms of the shear modulus reduction

curve, R controls the abscissa of the stress-strain curve at large strains. The parameters R,

h, and m are inter-related. Ideally, the shear modulus reduction curves should be defined

independently of the strength of the soil.

The parameter Ho defines the plastic modulus after the soil has reached the

bounding surface. Ho can be determined from the slope of the stress-strain curve at large

strains. Soil typically do not exhibit work hardening, consequently, as they reach failure

the stress-strain curve becomes parallel to the strain axes. In this case, a very low value

of Ho can be assigned. With a low value of Ho, the radial loading component of the

constitutive model becomes nothing else than a numerical tool to deal with soil failure.

Figure 5.7 shows typical stress strain curves predicted by the model at different

amplitudes for regular cyclic loading. Figure 5.8 shows the stress-strain curves

experienced by one element subject to an earthquake loading.

5.3.5 General comments

This section presented the implementation of the bounding surface plasticity

model described in Section 5.2 into a finite element program. An evaluation of the ability

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of the model to represent observed soil behavior is presented in the following section.

One problem with the soil constitutive model presented here, which is common to many

other nonlinear models, is that variations in the model parameters that are used to shift

shear modulus reduction curves to the right also shift damping curves in the same

direction. Thus, a simultaneous match of recorded shear modulus reduction and damping

curves is not always possible. Pestana and Lok (2000) proposed a hysteretic model based

on the perfect hysteretic model formulated by Hueckel and Nova (1979). Pestana and

Lok used model parameters to describe separately small strain and large strain non-

linearity. In addition, a separate parameter is used to control the rebound modulus, which

describes the slope of the stress-strain curve at the point of strain reversal (Lok 1999). By

modifying this parameter, the shear modulus reduction and damping curves can be

adjusted in opposite directions allowing a better match of both curves (Figure 5.9). Some

soils exhibit strain softening behavior when approaching yield. The model implemented

herein cannot model this behavior. Consequently, problems controlled by strain softening

soil response should not be modeled using this implementation.

5.4 VALIDATION

5.4.1 General

The constitutive model presented in Section 5.2 has been previously validated

through its application to site response analyses for events recorded in the Lotung,

Taiwan and Gilroy, California, downhole arrays (Borja et al. 1999, 2000). Section 5.3

presented an implementation of the model into the finite element program GeoFEAP. In

232

this section, a validation of this implementation is presented with particular emphasis on

the ability of the implementation in capturing long period motions typically associated

with near-fault ground motions. Moreover, the case studies presented serve to illustrate

the stress-strain behavior predicted by the constitutive model. Four case studies are

analyzed. The first case study corresponds to the same one performed by the developers

of the constitutive model (Borja et al. 1999), where the constitutive model is validated for

a particular finite element implementation. The same case study is reconsidered here to

validate the implementation of the model into the finite element program GeoFEAP. The

remaining case studies correspond to an events recorded in the Chiba downhole array in

Japan (Katayama et al. 1990), and a series of shake-table analysis performed at the

University of California at Berkeley (Meymand 1998 ).

5.4.2 Lotung Array

General

The Lotung site is located approximately 30 miles south-east of Taipei, Taiwan.

This site was one of the reference sites used by the Electric Power Research Institute in an

extended study on ground motion estimation (EPRI 1993). The site is the location of the

Large-Scale-Soil-Structure Test (LSST) facility, which was constructed in 1985 jointly by

Taipower and EPRI. The site is instrumented with two downhole arrays and includes

1/12- and 1/4-scale models of nuclear containment structures (EPRI 1993). The two

downhole arrays are located at 3 and 49 m from the containment structures.

Accelerographs are placed at 0, 6, 11, 17 and 47 m, and have recorded a number of

earthquakes since their installment in 1986. The Lotung site has been used in numerous

233

site response studies (i.e. EPRI 1993, Chang et al. 1990, Borja et al. 1999, Borja et al.

2000, among others). The study of Borja et al. (1999, 2000) used the earthquake of May

20, 1986 (Mw = 6.5, R = 68 km) to validate the constitutive model presented in Section

5.2. In this section, the same analysis is repeated to validate the implementation of the

Borja soil constitutive model into the finite element program GeoFEAP. The array

furthest from the containment structures is used throughout validations to represent a

free-field soil response.

Site profile and model parameters

The LSST site is located within the SMART-1 strong motion array. It is situated

on a flat plain in a basin of triangular shape that is 15 km wide and 8 km long (EPRI

1993). The soils at the site generally consist of Holocene and Pleistocene deposits of

interlayered silt-sand and sandy-silt with some gravel overlying a thicker layer of silty-

clay. Bedrock is estimated at approximately 400 m below ground surface (EPRI 1993).

Groundwater is located within a few feet of the ground surface. The shear-wave velocity

profile at the site is presented in Figure 5.10 from the data presented in the

aforementioned EPRI report (1993). The shear-wave velocity profile was originally

reported by Anderson and Tang (1989). The shear wave velocity profile was validated

through elastic analysis of small amplitude earthquakes (EPRI 1993).

Zeghal et al. (1995) used data from recorded events in the LSST array to

backcalculate the shear modulus reduction and damping curves. The parameters h and m

were obtained by matching the shear modulus reduction curve at G/Gmax values of 0.9 and

0.5. The parameters R and Ho are obtained from the moduli ratio curve at 5% shear

234

strain. The resulting shear modulus reduction curves are shown in Figure 5.11 and the

model parameters are given in Table 5.2. Resonant column and cyclic torsional tests on

"undisturbed" soil from the Lotung site suggested that the shear modulus reduction ratio

curves agree well with the upper bound curve for sands proposed by Seed and Idriss

(1970), while the damping curve plots in between the upper and lower bound curves

(EPRI 1993). The EPRI curve as well as the Seed and Idriss curves are included in

Figure 5.11 for comparison. Further details on the properties of the soil at the site are

given in Borja et al. (1999), Zeghal et al. (1995), and EPRI (1993).

Results

The event of May 20, 1986, generally known in the literature as the LSST 7 event,

produced peak ground accelerations at the surface of 0.16 g and 0.21g in the east-west

and north-south directions, respectively. The motion recorded at a depth of 47 m is used

as the input motion in the analysis. A finite element mesh with 47 brick elements, 1 m

thickness, was used. Consistent with the presentation of the finite element approach to

site response, the analysis is performed with only one spatial dimension assuming body

waves propagating in the vertical direction. All nodes in the brick elements at a given

depth are constrained to have equal displacements. With this constraint, the brick

elements behave as one-dimensional stick elements with linear interpolation functions.

The finite element analysis is performed using relative displacements (with respect to the

free-field), and the input ground motion is entered in the form of inertial forces

throughout the finite element model. Since the input motion was recorded at the base of

235

the soil column, the prescribed boundary condition is zero relative displacements (with

respect to the input displacements used in the analysis).

The recorded and the predicted particle acceleration and velocity plots are shown

in Figure 5.12. Peak accelerations and velocities are predicted accurately by the site

response analysis. Time histories of velocity and acceleration at all depths are presented

in Figures 5.13 and 5.14. The matching of both accelerations and velocities at all depths

in the downhole array is excellent.

The same analysis was performed for an uni-directional input motion using

independently the north-south and east-west components of motion. The predicted peak

response values are similar for the uni-directional and bi-directional analyses. Figure

5.15 compares the velocity response spectra for both these analyses. Around the one

second period where maximum spectral velocities are concentrated, the bi-directional

analysis gives a marginally better fit to the recorded motion in the East-West and North-

South directions.

General Observations

The results shown in the previous section agree well with the recorded data.

Similar findings were presented by Borja et al. (1999 and 2000). The LSST event was

also used to perform a series of parametric studies on the influence of different model

parameters on the results. An extended parametric study is not presented here for reasons

of brevity, but a number of conclusions are listed. Observations by Borja and coworkers

in the above papers are also included to present a more complete picture of the

constitutive model used.

236

a) A small increase in the viscous ratio did not result in significantly different

motions at the surface. On the other hand, a decrease in ξ led to motions with

high frequency noise. Borja et al. (2000) indicate that viscous damping has the

effect of suppressing high frequency noise. Results indicated that variations in the

viscous damping affected mainly the initial portion of the ground motion. After

the first acceleration peak, hysteretic damping controls and viscous damping does

not have a significant effect on the results. The suppression of high frequency

noise, however, proved to be important beyond just the high frequency region of

the spectrum. High frequency noise causes a larger number of unloads in the soil.

Upon each unload, the stiffness in the soil changes thus affecting the solution.

b) Borja et al. (2000) repeated the analysis for shear modulus reductions curves

covering the upper and lower bounds of the data processed by Zeghal et al.

(1995). Peak accelerations are under-predicted when the lower bound curves are

used, but are unchanged when the upper bound curve is used.

c) The effectiveness of the program is a function of the convergence rate of the finite

element program for each time step. Ideally, the convergence of the global

Newton-Raphson iteration should be quadratic for starting points in the

neighborhood of the solution. Table 5.3 illustrates the convergence for an element

test for stress levels that remain within the bounding surface. The residual norm

is the norm of the internal minus the external forces. The one element test permits

the direct evaluation of the element consistent tangent. Note that convergence is

better than linear, but it is not quadratic. The cause for this is that the model is

237

slightly strain-step dependent due to the trapezoidal integration of the constitutive

equation and the exponential nature of the function defining H'. The lack of

quadratic convergence does not affect the accuracy of the implementation.

Moreover, for stress states on the bounding surface, the convergence is quadratic

as expected for the radial return algorithm applied to J2 type plasticity (Simo and

Hughes 1998).

d) The parameter tol0 controls the size of the elastic nucleus around an unloading

point. It was found that variations of tol0 within a certain range do not affect the

results. The range is from about 10-8 (in strain units) to the strain defined by the

'elastic threshold' of the model. For this study, the elastic threshold strain

corresponds to the strain level after which the shear modulus reduction curve is

less than 0.99.

e) Changes in the time step do not significantly affect the results. In general, smaller

time steps do increase the energy of high frequency motions. For these small time

steps, viscous damping becomes more important. The time step is in general

determined from the data sampling of the input motion. The value of ∆t also

establishes an upper limit to the frequencies that can be represented in a time

history (Nyquist frequency). The element size also provides a limit to the

frequencies that are affected by the finite element analysis. Frequencies higher

than those corresponding to a wavelength of twice the element length are not

captured by the analysis. The element size used (1 m) more than suffices for the

frequencies of typical earthquake motions (0.2 – 20 Hz) for typical values of soil

238

stiffness. The sensitivity of the results to mesh size were also tested by increasing

the number of elements in the mesh. Results were not affected by element size.

f) The model used herein uses total stresses; thus, it cannot predict pore pressure

generation. Although pore water pressure sensors were not installed during the

LSST 7 event, they were installed shortly thereafter. A later earthquake in

November 1986 of a slightly higher magnitude and similar distance (LSST16)

generated a pore pressure ratio of about 25%. These levels are unlikely to cause

significant degradation of soil shear strength and stiffness (Borja et al. 1999).

g) The model, as presented, can also deal with vertically propagating pressure (P)

waves that normally generate vertical motion. Borja et al. (1999) present an

analysis of the vertical motion on the LSST 7 event resulting in a good match with

observed motion. Those results were duplicated in this study. For the remainder

of the work, vertical motions are excluded. Although the vertical ground motions

can be potentially significant (EPRI 1993), the study of near-fault motions

presented herein is constrained to the more important horizontal motions.

A number of other site response analyses have been performed on the LSST 7 data

and are included on the EPRI study on ground motion estimation (1993). Borja et al.

(2000) compares the results of these studies with the predictions of the bounding surface

plasticity model. Borja's model performs satisfactorily when compared with other

nonlinear models such as DESRA (Lee and Finn 1991), SUMDES (Li et al. 1992), and

TESS (Pyke 1992). The equivalent linear model SHAKE91 (Idriss and Sun 1992) has

predictions that are as good as these predictions with the current model, although peak

239

values are slightly under-predicted by SHAKE91 (Borja et al. 2000). Further

comparisons with SHAKE91 are performed for the Chiba downhole array.

5.4.3 Chiba Downhole Array

Generalities

The Chiba downhole array is located approximately 30 km east of Tokyo on the

Chiba Plain. The Chiba array data are made available by the Institute of Industrial

Science of the University of Tokyo. The array consists of 44 three component

accelerographs located in 14 boreholes (Katayama et al. 1990). The accelerograms are

placed at different depths in each of the boreholes. The borehole at the center of the array

has accelerographs at depths of 1, 5, 10, 20, and 40 m and it is used in this study to

represent free-field soil response. Previous site response analyses at this site using

equivalent-linear analysis (e.g., Katayama et al. 1990) and a system identification

approach (Glaser and Gaskings 1998) resulted in a good match of recorded motions

indicating that typical site response analysis assumptions (i.e., vertically propagating

shear waves, no significant pore pressure generation) hold for this site.

Soil Profile and Model Parameters

The Chiba array is placed in a geologically and topographically simple location.

Fifteen boreholes were drilled and logged. The logs indicated spatially consistent

horizontal profiles. The site consists of approximately 3 to 5 m of loam, underlain by 2 to

4 m of sandy clay which is in turn underlain by a stiffer sand layer (Katayama et al. 1990).

The water table is located at a depth of 5 m. The sand layer contains thin clay layers and

240

has a stiffness that increases with depth. The inferred shear wave velocity profiled used

in the analysis is shown in Figure 5.16 (Katayama et al. 1990). The Vucetic and Dobry

(1991) damping and shear modulus reduction curves for clay with PI=30 are used to

match the shear modulus reduction curve for the clay layers. For the sand layers, the

confining pressure-dependent curves by Iwasaki et al (1978) are used. Unit weights are

assumed using reasonable values for the soil types in the profile.

The model and numerical parameters used in the analysis are listed in Table 5.4.

The shear modulus reduction and damping curves used are shown in Figure 5.17. In

matching the shear modulus reduction and damping curves, the parameter R was used as a

matching parameter and was not used matched to the soil strength. This allows a better

matching of both shear modulus reduction and damping curves for strains up to 1%. For

larger strains, the model over-predicts the strength of the soil. However, the largest

strains observed in the profile for these ground motions were only on the order of 0.1%.

Results

The Chibaken-Toho-Oki earthquake (MJMA = 6.7) was used for this study. The

recorded and predicted time histories of acceleration and velocity are shown in Figures

5.18 and 5.19. Peak ground accelerations at the surface in the direction of maximum

intensity (North-South) are under-predicted by about 15% (0.33 g compared to 0.27 g).

In the East-West direction, the peak ground acceleration is predicted to within 10% (0.2 g

for the recorded motion, 0. 18 for the predicted motion). The general shape of the

acceleration time history is predicted relatively well with the exception of the under-

prediction of some PGAs late in the motion. Velocities, on the other hand, are predicted

241

to within .5 % in the north south direction, and to within 5% in the east-west direction

(Figure 5.19). Prediction of velocities is important because near-fault ground motions

have high energies at periods controlled by velocities. Note, however, that the velocities

observed at Chiba are lower than velocities expected at near-fault sites. However,

downhole records of near-fault soil response not involving liquefaction (i.e. Port Island

Array in the 1995 Kobe earthquake) are not available. The Chiba records represents the

highest intensity records available at this time.

A site response analysis using the equivalent linear code SHAKE91 (Idriss and

Sun 1992) was also performed. Figure 5.20 shows the pseudo-velocity response spectra

in the North-South direction for both analysis (the GeoFEAP analysis is performed using

only the North-South directions to simulate the one-directional SHAKE91 analysis).

Both methods yield almost identical results at all periods except for the PGA. Figure

5.21 shows the maximum strain profile predicted by both methods. The profiles show

similar patterns and predict the same level of strains at all depths except near the surface,

where SHAKE91 predicts larger strains.

5.4.4 Analysis of Shaking Table Tests

General Description of Experiments

A series of scaled physical model experiments were performed at the U.C.

Berkeley/PEER Center Earthquake Simulator Laboratory to examine the seismic response

of soil-pile-superstructure interaction. The experiments are described in detail in

Meymand (1998) and Lok (1999). Each test included a borehole with accelerographs at

242

different depths located at a relatively large distance from the pile group. It is assumed

that this array reflects free-field response.

The soil used in the analyses is a model soil prepared to simulate the response of

San Francisco Bay Mud at the model scale. The model soil consisted of 72% kaolinite,

24% bentonite, and 4% type C fly ash by weight, and has a plasticity index of 75. The

soil has an undrained shear strength of 100 psf and a shear wave velocity of

approximately 32 m/s. The model container is 7.5 feet in diameter and 7 feet in height.

The container was filled with 6 feet of the model soil overlying 6 inches of sand. The

flexible-wall container was designed to provide pseudo-free-field conditions such that the

soil column can deform horizontally in pure shear mode (Meymand 1998, Lok 1999). An

evaluation of the response of the soil column, particularly as it pertains to representing

true free-field response, is included in Lok (1999). Lok was able to predict reasonably

well spectral accelerations at long periods using both an equivalent-linear model

(SHAKE91) and a fully nonlinear model (Lok 1999). This indicates that the free field

assumptions is likely acceptable, and influence of the boundary conditions generated by

the container, if any, is constrained to high frequencies.

The shaking table tests were performed using the Yerba Buena Island record from

the Loma Prieta earthquake and the Port Island record from the 1995 Kobe earthquake as

input motions. The motions were scaled both in the time domain (to comply with model

scaling laws) and in amplitude, to simulate earthquakes at varying intensities. In this

study, the tests with the Port Island motion are used. The Port Island record from the

243

1995 Kobe earthquake contains forward-directivity effects. Table 5.6 lists the input

motion PGA for the analyses performed in this study.

Soil Properties

The shear-wave velocity profile of the model soil in the container is shown in

Figure 5.22. The shear-wave velocity profile was obtained before the testing sequence

was initiated. The stiffness of the model soil column could be expected to decrease in

response to strong shaking due to shear modulus reduction and inelastic effects; and

increase with periods of rest due to thixotropy (Meymand 1998). Sine-sweep tests were

performed prior to each test to determine the natural period of the soil column prior to

each test. Meymand (1998) and Lok (1999) used the sine-sweep tests to calibrate the

shear wave velocity profiles for each individual site response analyses. The base-line

shear-wave velocity profile was multiplied by a constant to obtain the same first-mode

frequency measured in the sine-sweep tests. This, somewhat simplistically, implies that

the effects of stiffness degradation and thixotropy occurred uniformly across the entire

soil depth. This would not be expected due to the effects of boundary conditions and

shear strain concentrations during shaking. The shear wave velocity profile used by

Meymand (1998) and Lok (1999) in equivalent linear analyses was also increased by a

factor of 30%. This increase was justified on the grounds that the strains induced in the

soil by the sine sweep tests were high enough to cause a modulus degradation of this

magnitude, thus they did not capture the small-strain stiffness of the soil. This

justification relies both on the assumption that strains (and consequently moduli

degradation) are constant through the profile and also on the equivalent-linear assumption

244

of an 'equivalent strain' for the whole time history. In the present analysis, this 30%

increase was ignored and the baseline shear-wave velocity profiles modified to match sine

sweep tests were used.

The measured shear modulus reduction and damping curves are presented in

Figure 5.23 along with the curves predicted using model parameters calibrated to match

these curves. The model parameters used to match these curves are given in Table 5.5.

As in the analysis of the Chiba data, the parameter R was used to match shear modulus

reduction curves at strains lower than 10%. Thus, soil strength is overestimated for

strains larger than 10%. Nonetheless, maximum strains observed in the analysis are

lower than 4%. The properties of the bottom sand layer in the profile were not very well

defined by experiments. A subsequent parametric study, however, showed that results

were relatively insensitive to the shear-wave velocity of this layer. The shear-wave

velocity value estimated by Meymand (1998) as a 'best match' was used in the analyses.

Results

The predicted and observed velocity response spectra for all the runs in Table 5.6

are shown in Figures 5.24 to 5.27. Observe that in general, the shape of the velocity

response spectra is predicted well. However, at periods corresponding to peak response

spectral velocities, the model underpredicts peaks in the velocity response spectrum,

except for the test with the highest intensity (Figure 5.27). In the worst case, the under-

prediction is about 25 % (Test 2.53), however, in most cases the peak spectral velocities

are predicted to within 15 %. At high frequencies, the model fails to predict a peak in the

response spectra. Table 5.6 lists the recorded and calculated peak ground motion values.

245

Clearly, the match of velocity values is better than the acceleration values, reflecting the

better match at longer periods. The spectral velocities presented in Figure 5.24 to 5.27

are at the model scale. Velocities at field scale must be multiplied by the scaling factor

8 (Meymand 1998). The input velocities used in these analysis correspond to the

ranges of peak ground velocities expected at near-fault, forward directivity sites.

Discussion

Note that the high frequency peak in the recorded response spectra is missing for a

depth of 18 inches (Figure 5.24 to 5.26). At this depth, response spectral values have a

better match for the high frequency region of the spectrum. The better match is a result of

the fact that this depth lacks the high energy content at high frequencies that is observed

at other depths. Not coincidentally, this depth corresponds to a nodal point (i.e., a point

of zero displacements) for the second natural frequency of the system, that is, the

frequency corresponding to a wavelength equal to 4/3H, where H is the height of the soil

column. This observation implies that the finite element model does not capture well the

standing wave that develops at the second mode.

Another reason for the relatively poor match at high frequencies may be the

effects of viscous damping. Note that the time-scaling of the motion results in input

motions with uncharacteristically high energy at low frequencies. At these frequencies,

the effect of viscous damping is much higher. The high frequency energy present in the

motion might also be caused by the container and not a characteristic of soil-column

response.

246

Finally, observe that predictions are better for the fault normal direction than for

the fault parallel direction (Figures 5.25 – 5.28). This might occur because the pile

groups show much more intense motions in this direction (i.e., the direction of larger

intensity). Although the model was unable to predict high frequency response of the soil

in the shaking table experiments, velocities were predicted with relative accuracy. Since

near-fault ground motions are evaluated mainly on the long-period content, the results are

encouraging, especially because the input motion used has the characteristics of near-

fault, forward-directivity motions that are targeted in this work.

5.4.5 General Comments

In general, the case studies indicate that the finite element site response analysis

methodology can adequately model bi-directional site response. In particular, the

implementation showed ability to predict amplifications in the velocity-controlled region

of the spectra. The site response implementation is used in Chapter 6 to study site

response to near-fault ground motions.

247

BLANK

248

BOX 5.1. Integration of radial return algorithm (from Simo and Hughes 1998).

1. Assume the stress states is initially on the bounding surface B. Let:

1ntr1n G2 ++ ′= εσ ∆∆∆∆ (1)

ntr1n

tr βσζ −= +B (2)

2. Load/unload check

Rf tr1n

tr1n −= ++ ζ (3)

if trnf 1+ < 0, then set unloading point σσσσo to current stress. Proceed with Box 5.2.

3. Calculate

G3H1

′+

=ε�∆∆∆∆

Bλλλλ ; tr1n

tr1nˆ

+

+=ζζn (4)

4. Update stresses and hardening variables.

nββ ˆ32

Bo1 λHnn +=+ (5)

( ) ( )BBnε1εσσ ˆG2Ktrn1n λλλλ−++=+ ∆∆∆∆∆∆∆∆ (6)

5. Elasto-Plastic Tangent

BB1n1nep1n ˆˆG2

31G2K nn11I11C ⊗−

⊗−+⊗= +++ θθ (7)

where I is the rank-four symmetric identity tensor, and θn+1 and θ n+1 are given by

tr1n

B1n

G21+

+ −=ζ

λλλλθ and ( )1n1n 1

G3H1

1++ −−′

+θθ

249

BOX 5.2. Integration of bounding surface plasticity model (Borja and Amies 1994).

1. From Equation 5.19, applying a trapezoidal integration:

εσσ ′=′

′+

′−+′

+

∆∆∆∆∆∆∆∆∆∆∆∆ G2HH

1G31nn

ββββββββ (1)

where H' =h κm + Ho 2. Use the fact that ∆σσσσ' is directed along ∆εεεε', and obtain

G2HH

1G31nn

=

′+

′−+

+

ββββββββψψψψψψψψ (2)

were ψ is a positive scalar defined such that ∆σσσσ' = ψ ∆εεεε' 3. Apply Equation 5.22

( )onσεσζ ′−′+′+′+= + ∆∆∆∆∆∆∆∆ ψψψψκκκκεεεεψψψψ n1nnR (3) 4. Solve for ψ and κ from Equations 2 and 3 5. Apply the stress increment

( ) εεσσ ′∆+∆+=+ ψ11 Ktrnn (4)

250

BOX 5.3. Decomposition of strain step.

1. Let:

FB εεε ∆+∆=∆ (1) where ∆εεεεF is the strain that loads the soil to the bounding surface at time tn+γ, and ∆εεεεB is the strain the soil experiences after it has reached the bounding surface. 2. Solve for

+

′−+

=

oHH1G31

G2

n

ββββββββψψψψ γγγγ (2)

3. Apply Equation 5.22 at time tn+γ:

Rnnn =−+=+ ββββεεεεγγγγψψψψ γγγγγγγγ ∆∆∆∆σσ (3) where ∆εεεεF = γ∆εεεε. Now solve for α

( ) ( )

εεζζεεεζεζ

∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆

::R::: nn

22nn

γγγγψψψψγγγγ

−++−= (4)

251

Table 5.1. Parameters needed for the model implementation.

Parameter Function Procedure to obtain it

Elastic Parameters

Shear Modulus (G) Poisson Ratio (ν)

Any pair of elastic parameters. Describe the elastic behavior of the soil at unload points.

Shear modulus obtained from shear-wave velocity logs (G = ρVs

2) Poisson's ratio estimated or calculated from

−−

= 2s

2p

2s

2p

VV1VV2

21νννν ,

where Vs and Vp are the shear wave and compression wave velocities, respectively.

Material Density Density (ρ) Measured from boring log samples

h, m Interpolation function for the hardening modulus.

Matching modulus degradation curves (Equation 5.35). Exponential

Model Parameters Ho

Kinematic hardening parameter of the bounding surface.

Obtained from tangential shear modulus at large strains.

Strength Parameter R Radius of bounding

surface.

R = 38 su ; R = 2 τf where su is the soil strength obtained from unconfined compression tests, and τf is the soil strength from simple shear tests.

ξ

Determine the viscous component of stress-strain relationship.

ξ is damping ratio at very small strains, obtained from cyclic tests at small strain levels (e.g., Lanzo and Vucetic 1999). Small Strain

Damping

ω1, ω2

Frequency band where ξ is matched. Outside this band, the model results in damping higher than ξ.

ω1, ω2 typically chosen to encompass the natural period of the soil and the predominant period of input motion.

β

Trapezoidal integration parameter.

Choose β :> 0 for implicit integration (Box 5.2).

Numerical parameters tol0

Determines radius of region around unloading point where the soil remains elastic. Its function is to minimize 'numerical unloading', that is, unloading in the soil due to infinitesimal strain steps taken while the finite element algorithm searches for a solution.

Elastic threshold from laboratory modulus reduction curves (i.e. strain level at which G/Gmax < 1).

252

Table 5.2. Model and numerical parameters used for analysis of Lotung array. The shear-wave velocity profile is given in Figure 5.10.

Parameter Value

h/Gmax 0.63 m 0.97

R/Gmax 0.0015 Ho 0

ρ

1.95 v 0.48

ξ

2.0%

ω1 4

ω2 0

β

0.5 tol0 0.0001 %

Time step (∆t) 0.04 s

253

Table 5.3. An example of the rate of convergence of the Residual Norm in the finite element implementation.

Iteration Residual Norm

1 1 2 5e-2 3 1e-4 4 3e-7 5 8e-10 6 2e-12

7 convergence (< 1e-16)

254

Table 5.4. Model and numerical parameters used in the analysis of the Chiba downhole array. The shear-wave velocity profile is given in Figure 5.16. (a) Model parameters

Parameter Sand 1 ksc < σσσσv' < 3 ksc

Sand σσσσv' > 3 ksc

Clay

h/Gmax .088 0.366 0.126 m 1.0768 1.035 1.159

R/Gmax .005 .002 0.01 Ho 0 0 0 ξ

1% 1% 1% ρ

1.92 – 2.0 1.92-2.0 1.45-1.76 ν

0.49* 0.49* 0.49* * For incompressible (undrained) deformation, ν should be set to 0.5. The value 0.49 is used to avoid numerical errors associated with incompressible deformations. (b) Numerical Parameters

Parameter Value

ω1 11

ω2 33

β

0.5

tol0 0.0001 %

Time step (∆t) 0.01 s

255

Table 5.5. Model and numerical parameters used in the analysis of the Shaking Table tests. Shear-velocity profile is given in Figure 5.23.

Parameter Model Soil

h/Gmax 0.0613 m 0.859

R/Gmax .02 Ho 0

ρ

1.51

ν

0.49*

ξ

2%

ω1 20

ω2 0

β

0.5 tol0 0.0001 %

Time step (∆t) 0.01 s * For incompressible (undrained) deformation, ν should be set to 0.5. The value 0.49 is used to avoid numerical errors associated with incompressible deformations.

256

Table 5.6 Peak response values for the Shaking Table Runs.

Test Input PGA (g)

PGA recorded

(g)

PGA calculated

(g)

PGV recorded

(cm/s)

PGV calculated

(cm/s) 2.16 0.1 0.15 0.10 8.3 7.9 2.53 0.25 0.61 0.36 27.7 17.7 2.55 0.7 1.22 0.62 55.6 41.4 2.58 1.0 1.27 0.68 67.9 57.6

257

Figure 5.1. Schematic representation of bounding surface plasticity model. Fo is the unloading point, B is the bounding surface and, F is the yield (loading) surface. Contours of constant H' are centered about Fo, where H' is infinite, decreasing to zero on B (adapted from Borja and Amies 1994).

σ1

σ3

σ2

B

Fo

σσσσo

H' > Ho

Ho < H' < ∞

H'= ∞

F

σσσσ'σσσσ'

258

Figure 5.2. Fraction of critical damping versus frequency for Rayleigh damping. ξ is the target fraction of critical damping.

Frequency

Frac

tion

of C

ritic

al D

ampi

ng

Mass Proportional Damping

Stiffness Proportional Damping

Total Damping

ω1 ω2

targ

et ξ

259

Figure 5.3. Stress-strain loops for two different loading paths. The dotted line is for monotonic loading. The solid line is for monotonic loading with an unload cycle of magnitude ∆ε at 0.5% strain. a) Numerical unloading (∆ε ≈ 1e-6%). b) True unload (∆ε ≈ 0.2 %).

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shear Strain (%)

Nor

mal

ized

She

ar S

tress

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shear Strain (%)

Nor

mal

ized

She

ar S

tress

a) b)

260

Figure 5.4. Schematic representation of site response problem.

A B

Soil

Rock

Reflected Wave

Incoming Wave

x

z

261

Figure 5.5. Influence of hardening parameter h on modulus reduction and damping curves. R/Gmax = .02, m = 1, Ho = 0 (adapted from Borja and Amies 1994).

10 -4

10 -3

10-2

10-1

10 0

1010

5

10

15

20

25

30

Dam

ping

Rat

io

Strain

h = 0.01 Gmaxh = 0.1 Gmax h = 0.5 Gmax h = 1 Gmax h = 5 Gmax

10 -4

10 -3

10-2

10-1

10 0

1010

0.2

0.4

0.6

0.8

1

Mod

uli R

atio

Strain

Decreasing h

Decreasing h

262

Figure 5.6. Influence of hardening parameter m on modulus reduction and damping curves. R/ Gmax = .02, h/ Gmax = 1, Ho = 0 (adapted from Borja and Amies 1994).

10-4 10 -3 10-2 10-1 10

0 1010

5

10

15

20

25

30

Dam

ping

Rat

io

Strain (%)

m=0.5m=1 m=2 m=4 m=8

10-4 10 -3 10-2 10-1 10

0 1010

0.2

0.4

0.6

0.8

1

Mod

uli R

atio

Strain (%)

Decreasing m

Decreasing m

263

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Strain (%)

Nor

mal

ized

Stre

ss

Figure 5.7. Representative stress-strain loops at different amplitudes. R/ Gmax = .02, h/ Gmax = 1, m = 1, Ho = 0.

264

Figure 5.8. Finite element response when subjected to cyclic seismic loading R/Gmax = .02, m = 1, h/Gmax = 1, Ho = 001. The loading was the KJMA station in the Kobe earthquake.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Strain (%)

Nor

mal

ized

Stre

ss

265

Increasing c

Gmax

ττττ

γγγγ

(a)

10-4

10-2

100

0

0.5

1Modulus Degradation for Nonlinear Cyclic Model

G/G

max

10-4

10-2

100

0

5

10

15

20

25Damping for Nonlinear Cyclic Model

Shear Strain (%)

Dam

ping

Rat

io (%

)

c

c

(b) Figure 5.9. Influence of the slope of the rebound modulus in the model by Pestana and Lok. (a) Increase in parameter c reduces the slope of the stress-strain curve at the rebound point. (b) The resulting stress-strain curves can be adjusted in opposite directions (from Lok 1999).

266

0

10

20

30

40

50

0 100 200 300 400

Vs (m/s)

Dep

th(m

)

Figure 5.10. Shear wave velocity (Vs) profile for Lotung Site (Borja et al. 1999, EPRI 1993).

267

10 -4

10 -3

10 -2

10 -1

10 0

0

0.2

0.4

0.6

0.8

1

Strain (%)

G/G

max

10 -4

10 -3

10 -2

10 -1

10 0

0

10

20

30

40

Strain (%)

Dam

ping

Rat

io (

%)

Seed and Idriss (1970) Sand Curves EPRI Curves Model Prediction

Figure 5.11. Modulus degradation and damping curves for the Lotung site.

268

-0.2 0 0.2

-0.2

0

0.2

GeoFEAP

East-West Acceleration (g)

Nor

th-S

outh

Acc

eler

atio

n (g

)

-20 0 20

-20

0

20

East-West Velocity (cm/s)

Nor

th-S

outh

Vel

ocity

(cm

/s)

-0.2 0 0.2

-0.2

0

0.2

Recorded data

East-West Acceleration (g)

Nor

th-S

outh

Acc

eler

atio

n (g

)

-20 0 20

-20

0

20

East-West Velocity (cm/s)

Nor

th-S

outh

Vel

ocity

(cm

/s)

Figure 5.12. Comparison of recorded and computed ground motions at the surface for the Lotung array.

269

Figure 5.13. Recorded and computed acceleration time histories in the Lotung array. Thick lines are recorded accelerations. The input motion is at Elev. 47.

5 10 15 20-0 .2

0

0 .2Surface

Acc

. (g)

5 10 15 20-0 .2

0

0 .2Elev 6

Acc

. (g)

5 10 15 20-0 .2

0

0 .2Elev 11

Acc

. (g)

5 10 15 20-0 .2

0

0 .2Elev 17

Acc

. (g)

5 10 15 20-0 .2

0

0 .2Elev 47

Acc

. (g)

T ime(s)

Thick: RecordedThin: Calcula ted

5 10 15 20

-0.1

0

0.1

0.2

Sur face

Acc.

(g)

5 10 15 20

-0.1

0

0.1

0.2

Elev 6

Acc.

(g)

5 10 15 20

-0.1

0

0.1

0.2

Elev 11

Acc.

(g)

5 10 15 20

-0.1

0

0.1

0.2

Elev 17

Acc.

(g)

5 10 15 20

-0.1

0

0.1

0.2

Elev 47

Acc.

(g)

T ime(s)

(a) East-West direction (b) North-South direction

270

Figure 5.14. Recorded and computed velocity time histories in the Lotung array. Thick lines are recorded velocities. The input motion is at Elev. 47.

5 10 15 20

-20

0

20

Surface

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 6

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 11

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 17

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 47

Vel

.(cm

/s)

T ime(s)

Thick: Recor dedThin: Calcula ted

5 10 15 20

-20

0

20

Sur face

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 6

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 11

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 17

Vel

.(cm

/s)

5 10 15 20

-20

0

20

Elev 47

Vel

.(cm

/s)

time(s)

(a) East-West direction (b) North-South direction

271

East - West

0

10

20

30

40

50

60

70

0.01 0.1 1 10

Period (s)

Velo

city

Res

pons

e Sp

ectr

a (c

m/s

) RecordedBi-directionalUni-directional

North - South

0

10

20

30

40

50

60

70

0.01 0.1 1 10

Period (s)

Velo

city

Res

pons

e Sp

ectr

a (c

m/s

) Recorded Bi-directionalUni-directional

Figure 5.15. Comparison of uni-directional and bi-directional analyses for the Lotung array.

272

0

5

10

15

20

25

30

35

40

0 200 400 600

Vs (m/s)

Dep

th(m

)

Figure 5.16. Shear wave velocity (Vs) profile for the Chiba site (Katayama et al. 1990).

273

Figure 5.17. Modulus degradation and damping curves for soils in the Chiba site. Thin lines correspond the curves predicted by the model. Model parameters are matched to curves from the indicated references.

10-4

10-3

10-2

10-1

100

101

0

0.2

0.4

0.6

0.8

1

Strain (%)

G/G

max

10-4

10-3

10-2

10-1

100

101

0

5

10

15

20

25

30

35

40

Strain (%)

Dam

ping

Rat

io (%

)

PI =30 (Vucetic and Dobry 1991)Sands, CP = 1-3 ksc (Iwasaki 1978)Sands, CP > 3 ksc (Iwasaki 1978)Model: ClayModel: Sand, CP = 1-3 kscModel: Sand, CP > 3 ksc

274

a) East-West direction b) North-South direction

5 10 15

-0 .2

0

0.2

Surface

Acc

. (g)

5 10 15

-0 .2

0

0.2

Elev 10

Acc

. (g)

5 10 15

-0 .2

0

0.2

Elev 20

Acc

. (g)

5 10 15

-0 .2

0

0.2

Elev 40

Acc

. (g)

Time(s)

5 10 15

-0.2

0

0.2

Surface

Acc

. (g)

5 10 15

-0.2

0

0.2

Elev 10

Acc

. (g)

5 10 15

-0.2

0

0.2

Elev 20A

cc. (

g)

5 10 15

-0.2

0

0.2

Elev 40

Acc

. (g)

Time(s)

Thick: Recor dedThin: Calcula ted

Figure 5.18. Recorded and computed acceleration time histories in the Chiba array. Thick lines are recorded accelerations. The input motion is at Elev. 40. The accelerogram at 5 m depth did not trigger for this event.

275

Figure 5.19. Recorded and computed velocity time histories in the Chiba array. Input motion is at Elev. 40.

a) East-West direction b) North-South direction

5 10 15- 15

- 10

-5

0

5

10

15Surface

Vel.(

cm/s

)

5 10 15- 15

- 10

-5

0

5

10

15Elev 10

Vel.(

cm/s

)

5 10 15- 15

- 10

-5

0

5

10

15Elev 20

Vel.(

cm/s

)

5 10 15- 15

- 10

-5

0

5

10

15Elev 40

Vel.(

cm/s

)

Time(s)

Thick: RecordedThin: Ca lculated

5 10 15-15

-10

-5

0

5

10

15Surface

Vel

.(cm

/s)

5 10 15-15

-10

-5

0

5

10

15Elev 10

Vel

.(cm

/s)

5 10 15-15

-10

-5

0

5

10

15Elev 20

Vel

.(cm

/s)

5 10 15-15

-10

-5

0

5

10

15Elev 40

Vel

.(cm

/s)

Time(s)

276

Figure 5.20. Acceleration response spectra for the analyses of the North-South component of motion in the Chiba downhole array.

0

5

10

15

20

25

30

35

40

45

50

0.01 0.1 1 10

Period (s)

Velo

city

(cm

/s)

RecordedSHAKE91GeoFEAP

277

0

10

20

30

40

0 0.05 0.1

Strain (%)

Dep

th(m

)

SHAKE91GeoFEAP

Figure 5.21. Strains predicted by SHAKE91 (Idriss and Sun 1992) and the GeoFEAP analysis for the North-South component of the Chiba downhole array.

278

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 100 200 300

Vs (m/s)

Dep

th(m

)

Vs Profile

AccelerographLocations

-8

-18

-30

-48

-60

-66

-72

-75

0

Figure 5.22. Shear-wave velocity profile of the model clay soil used in the Shaking Table Test 2.53, including location of accelerograms.

279

10-4 10-3 10-2 10-1 100 1010

0.2

0.4

0.6

0.8

1

Strain (%)

G/G

max

10-4 10-3 10-2 10-1 100 1010

10

20

30

40

Strain (%)

Dam

ping

Rat

io (%

)

Young Bay Mud (Isenhower and Stokoe 1981)Laboratory Curve Predicted Curve

Figure 5.23. Modulus degradation and damping curves for model soil used in shaking table test.

280

0

10

20 Elev 0

0

10

20 Elev -8

0

10

20 Elev -18

0

10

20 Elev -30

0

10

20 Elev -48

0

10

20 Elev -60

Velo

city

Res

pons

e Sp

ectra

(cm

/s)

Thick: Recorded Thin: Calculated

0

10

20 Elev -66

0

10

20 Elev -72

0.01 0.1 1 100

10

20 Elev -75

Period (s)

0

10

20

0

10

20

0

10

20

0.01 0.1 1 10 0

10

20

Period (s)

a) Fault Normal Direction b) Fault Parallel Direction

Figure 5.24. Velocity response spectral for recorded and calculated motions in Test 2.16. The thick line corresponds to the recorded motions. The input motion is at Elev. –75.

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0

50Elev 0

0

50Elev - 8

0

50Elev - 18

0

50Elev - 30

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50Elev - 48

0

50Elev - 60

Vel

ocity

Res

pons

e S

pect

ra (c

m/s

)

Thick: RecordedThin: Calculated

0

50Elev - 66

0

50Elev - 72

0.01 0.1 1 100

50Elev - 75

Period (s)

0

50

0

50

0

50

0.01 0.1 1 100

50

Period (s)

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0

100Elev 0

0

100Elev - 8

0

100Elev - 18

0

100Elev - 30

0

100Elev - 48

0

100Elev - 60

Vel

ocity

Res

pons

e S

pect

ra (c

m/s

)


0

100Elev - 66

0

100Elev - 72

0.01 0.1 1 100

100Elev - 75

Period (s)

0

100

0

100

0

100

0.01 0.1 1 100

100

Period (s)

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0

100

200Elev 0

0

100

200Elev - 8

0

100

200Elev - 18

0

100

200Elev - 30

0

100

200Elev - 48

0

100

200Elev - 60

Vel

ocity

Res

pons

e S

pect

ra (c

m/s

)


0

100

200Elev - 66

0

100

200Elev - 72

0.01 0.1 1 100

100

200Elev - 75

Period (s)

0

100

200

0

100

200

0

100

200

0.01 0.1 1 100

100

200

Period (s)

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CHAPTER 6

SITE RESPONSE ANALYSIS OF NEAR-FAULT

GROUND MOTIONS

6.1 INTRODUCTION

The analysis of near-fault, forward-directivity ground motions presented in

Chapter Four indicated that, for some motions, velocity pulse-period and amplitude

depend on local site conditions. Only a limited amount of recorded data is available to

evaluate site effects in the near-fault region. However, the understanding of the possible

effects of site conditions on near-fault motions can be enhanced by the use of numerical

simulations. This chapter presents a numerical evaluation of site response to forward-

directivity motions. The trends in the recorded data of near-fault ground motions

presented in Chapter Four are used to validate the results of the analyses presented herein.

The Borja and Amies (1994) constitutive model was employed in this near-fault

seismic site response study. This model is implemented in the program GeoFEAP, as

described in Chapter Five. The advantages of using a bi-directional time-domain

formulation for the analysis of near-fault motions were discussed in Chapter Five. The

numerical parameters used in the validation of the model (Table 5.4b in Section 5.4) are

used for the baseline analyses presented in this chapter. Radiation boundary conditions

(Lysmer and Kuhlemeyer 1969) are used to model the elastic half-space underlying the

finite element model of the soil profile.

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A set of generalized site profiles for the primary site conditions presented in Table

3.1 is first developed. A set of recorded near-fault rock motions is then used as input for

seismic site response analyses using the generalized profiles. In order to validate the use

of simplified pulses for site response studies, the results of these studies are compared to

analyses using simplified representations of the same input motions. The simplified

pulses developed in Chapter Four are then used to perform an analysis of the effect of site

conditions on the characteristics of rock velocity pulses. Insights from these analyses

provide the basis for the findings presented at the end of this chapter.

6.2 GENERALIZED SITE PROFILES

6.2.1 General

The generalized site profiles were selected to match average shear wave velocity

profiles associated with the primary site categories developed in this study and listed in

Table 3.1. Namely, the generalized site profiles represent the Shallow, Very Stiff Soil

(Site C), Deep Stiff Soil (Site D), and Soft Clay (Site E) groups. To avoid the

complication of the influence of significant pore pressure generation from loose saturated

sand sites that might liquefy, loose to medium-dense saturated cohesionless soils (Site F)

were excluded from this analysis. The nonlinear stress-strain properties for each site

condition were selected to represent generic soils commonly found in each of these three

soil profiles (i.e. sites C, D, and E).

The shear wave velocity profiles selected are based on average shear wave

velocity profiles from a database compiled by Walter Silva (personal comm.). This

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database consists of shear wave velocity profiles from a large number of sites, which are

located largely within California. These sites are classified according to the 1997

Uniform Building Code shear wave velocity-based classification system. The 1997 UBC

site categories C, D, and E correspond approximately to this study's site categories C, D

and E, respectively (see Chapter Three). The mean and standard deviation of the

“average profile” for each site condition are obtained by averaging shear wave velocities

at each depth for all the profiles within the same group. Therefore, the standard

deviations represent the spatial variability of shear wave velocity across a horizontal

plane. When the number of profiles involved is relatively large, the resulting average

profiles vary smoothly with depth. The number of profiles for each site condition is given

in Table 6.1. Results with equivalent linear analyses have shown that the average profile

captures the median response of these site conditions to a wide range of earthquake

ground motions. The use of average profiles allows for the evaluation of trends in the

response, as well as to quantify the average response of a particular soil class. However,

the use of average profiles precludes an estimate of the standard deviations associated

with ground motion values in the soil deposits. The following paragraphs describe in

detail the characteristics of each of the generalized site profiles.

Deep Stiff Soil (Site D)

Figure 6.1 shows the shear wave velocity profile selected for the Deep Stiff Soil

site. The average and the one standard deviation band shear wave velocity profiles for the

Site D category (180 m/s < Vs ≤ 360 m/s) from the Silva database are also shown. The

shear wave velocity in the upper 10 m of the profile was selected to match the average

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profile of the Silva database. The variation of shear wave velocity with depth was

assumed to be proportional to σ'm(z)n, where σ'm is the mean effective stress at depth z.

The value of n was taken to be 0.25 (Hardin and Drnevich 1972). The shear wave

velocity profile obtained in this manner is approximately near or within one-half of the

standard deviation of the mean of the Silva profile. For larger depths, the value is lower

than the mean of the Silva profile because the latter includes also shear wave velocity

measurements of underlying rock deposits. The depth of the profile was varied from 30

to 200 m to study the effect of variations of depth on site response to velocity pulses. The

density of the soil is assumed to be 1.90 Mg/m3 for the upper 30 m and 1.95 Mg/m3 for

the deeper soil layers. The soil column is assumed to overlie an elastic half-space with a

shear wave velocity of 1200 m/s and a density of 2.4 Mg/m3. A linear transition, which

represent weathered rock, with a thickness of 3 m is placed between the soil and the

elastic half-space.

The selected nonlinear stress-strain properties of the soil in the profile correspond

to soils of low to medium plasticity. The soils were assumed to contain sufficient fines to

preclude significant pore pressure generation. The shear modulus reduction vs. strain and

equivalent viscous damping vs. strain curves were selected to lie within the range of both

the curves for sand from Seed and Idriss (1970), and the curves for clays with plasticity

indices (PI) of 15 and 30 from Vucetic and Dobry (1991). The shear modulus reduction

and damping curves within this range are appropriate for a relatively wide range of soil

types. Three sets of shear modulus reduction and damping curves were used. The sets

were selected to represent a generic low-plasticity soil, a clay with a PI of 15, and a clay

with a PI of 30. The parameters for the Borja and Amies model used in each of these sets

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are listed in Table 6.2. The resulting shear modulus reduction and material damping

curves are shown in Figure 6.2. A large value of model parameter R was used to give a

better match to both the shear modulus reduction and damping curves while preventing

yielding of the soil at low strains. The implicit assumption is that the soil will not fail,

therefore the results must be checked to verify that the resulting stresses in the profile

remain within acceptable ranges and that the assumption holds. The strength profile of

the clayey soil was developed assuming an su/σv' ratio of 0.8, where su is the undrained

shear strength as measured by an unconfined triaxial compression test and σv' is the

vertical effective stress. The water table was arbitrarily assumed to be 10 m deep. A

value of su = 150 kPa corresponding to an unconfined compressive strength (qu) of 300

kPa was used as a lower bound for the soil strength. A qu value of 300 kPa corresponds

to a Very Stiff Soil and is consistent with the generalized stiff soil profile developed

herein, assuming that the water table at 10 m produces desiccation and overconsolidation

in the upper clay layers. The same shear strength profile was assumed to be applicable to

the Generic Low-Plasticity Soil. The static shear strength was multiplied by a factor of

1.4 (Lefebvre and Leboeuf 1987) to account for rate effects during rapid earthquake

loading. The resulting dynamic shear strength as a function of depth is shown in Figure

6.3.

It is important to note that the assumption that the soil will not fail due to the

shear induced by the ground motion is justified for the purpose of this study. Soil

yielding during earthquake ground motion can result in significant ground deformation,

particularly if the soil is subject to large static stress levels. However, soil yielding also

results in a large level of damping in the soil due to plastic dissipation of energy, reducing

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the intensities of ground motion at the surface. Moreover, the input motions required to

develop yielding will vary largely depending on soil type and a generalization based on a

single value of soil shear strength would not be realistic.

Soft Clay Profile (Site E)

A large number of metropolitan areas in the world have clay deposits that are

characterized by high sensitivity, low stiffness, and low strengths, such as Boston Blue

Clay and San Francisco Bay Mud. Seismic site response issues associated with these

deposits could have a significant economic impact. The response of soft clay sites to

earthquake ground motions is highly site-dependent, which implies that the response of a

generic soft clay profile can only be used as an indication of possible trends in site

response for similar soil types.

Soil profiles from the San Francisco Bay with thick layers of Holocene Bay Mud

are used to develop a generic “soft clay” profile. These profiles typically consist of a

layer of artificial fill over Holocene Bay Mud overlying a thicker layer of Pleistocene Bay

Clay. A shear wave velocity of 110 m/s was selected for the Holocene Bay Mud. This

value is the average of a number of shear wave velocity measurements on normally

consolidated Holocene Bay Mud sites in the San Francisco Bay region. For the

Pleistocene Bay Clay, typical clay shear wave velocity values of 200 to 400 m/s were

used. The resulting generic shear wave velocity profile is shown in Figure 6.4.

Undrained shear strength values for the soft clay were obtained by using an su /σv'

value of 0.3. This value is typical of Bay Mud, which is a medium plasticity silty clay. A

minimum value of su = 25 kPa was used for soils near the surface of the profile. The

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strength profile for the Pleistocene Bay Mud was obtained using an su /σv' value of 0.8.

The water table was assumed to be at the surface. As with the Stiff Soil profile, a

dynamic loading multiplicative factor of 1.4 was used to account for the increased rate of

loading experienced during seismic events. The resulting dynamic shear strength profile

is shown in Figure 6.5. The nonlinear stress-strain properties for the Holocene Bay Mud

were obtained by matching the shear modulus reduction and material damping vs. strain

curves with the curves presented by Sun et al. (1988). Two different sets of model

parameters were developed to conform to different assumptions regarding the behavior of

the soft clay under seismic loading. The first set of parameters was developed under the

assumption that stresses in the soil will not reach the yield strength of the soil. An

artificially high strength value was used to better match the shear modulus reduction and

damping curves at low strains for this "non-yielding soil" case. However, preliminary site

response analyses indicated that the maximum shear stresses induced in the soil deposit

undergoing high intensity (near-fault, forward-directivity) motions exceeded the dynamic

strength of the soft clay. For this reason, a second set of curves was developed to match

the soil strength and the shear modulus reduction and damping curves over a wider range

of strains. A parametric study showed that for the cases in which the soil yields, the

parameter that controls soil behavior is the yield strength of the soil. Moreover, the peak

ground velocity (PGV) at the surface is not very sensitive to relatively large variations in

the nonlinear small strain behavior of the soil. The value of the Borja and Amies (1994)

soil model parameter R was varied with respect to depth to match the value of su at each

depth, according to the strength profile shown in Figure 6.5. The variation of R with

respect to depth results in different shear modulus reduction and damping curves at

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different depths. The resulting curves are shown in Figure 6.6. The model parameters are

listed in Table 6.3. For strain levels larger than 0.1 %, the equivalent viscous damping in

the model with variable strength is much higher than the damping expected in a soft clay

such as Holocene Bay Mud. However, when stresses in the soil reach the yield strength

of the material, energy dissipation is controlled by plastic yielding and large damping

levels can be expected.

Nonlinear model parameters for the Pleistocene Bay Clay were obtained by fitting

the shear modulus reduction and damping curves to the range determined by the Vucetic

and Dobry (1991) curves for clays with a PI between 15 and 30. The resulting curves are

shown in Figure 6.7 and the model parameters are listed in Table 6.3. The parameter R

was set to an artificially high value assuming that the soil would not reach failure stresses.

When this assumption did not hold, values of R consistent with the dynamic shear

strength profile shown in Figure 6.5 were used. Equivalent viscous damping values for

large strains are not realistic for values of R lower than 0.003. For these soils, however,

energy dissipation is controlled by plastic yielding. For low values of R, the shear

modulus reduction curve at low strains falls outside the range of the Vucetic and Dobry

curves for PI between 15 and 50. However, as indicated above, when the soil reaches

failure response is controlled by the yield strength of the soil and is relatively insensitive

to the shear modulus reduction curves at low strains.

Shallow, Very Stiff Soil Profile (Site C)

The average shear wave velocities obtained from the Silva database for profiles

classified by the 1997 UBC as Site C (360 m/s < sV ≤ 760 m/s) were used to develop a

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profile corresponding to a Shallow, Very Stiff Soil site category (Site C). The shear wave

velocity profile used is shown in Figure 6.8. The shear modulus reduction and material

damping curves selected are those for the Generic Low-Plasticity Soil profile used in the

Stiff Soil profile (Figure 6.2). The soil was assumed not to fail during earthquake

loading.

6.3 COMPARISON OF RESULTS FOR RECORDED AND

SIMPLIFIED NEAR-FAULT MOTIONS

6.3.1 General

The simplified velocity pulses developed in Chapter Four generally capture the

low frequency energy associated with the effects of forward-directivity conditions.

However, these simplified representations do not contain significant energy in the high

frequency range, which is also generally present in recorded ground motions. Although

local soil conditions can alter the underlying rock motions throughout a wide range of

frequencies, velocities tend not to be significantly affected by high frequencies. Hence, it

is appropriate for this examination of near-fault site effects to focus on the low frequency

energy represented by the simplified velocity pulses.

In the following section, the simplified velocity pulses are used to conduct a

parametric study of the influence of site response on this type of input motions. The

implicit assumption is that site response to the simplified pulses is similar to site response

to recorded near-fault ground motions. This assumption is verified in this section by

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comparing site response analyses to recorded near-fault ground motions to the site

response to the corresponding simplified velocity pulses.

Four ground motions recorded on rock sites were selected for this study. These

motions are the Pacoima Dam record from the San Fernando earthquake, the Gilroy

Gavilan College record from the Loma Prieta earthquake, the Pacoima Dam Downstream

record from the Northridge earthquake, and the KJMA record from the Kobe earthquake.

A stiff soil profile (Figure 6.1) with a depth of 45 m with the Generic Low-Plasticity Soil

parameters (Figure 6.2, Table 6.2) was used for these site response analyses. Three

different sets of input motions were developed for each of the ground motions:

a) the full ground motion record,

b) the time interval of the record containing the fault normal velocity pulse, and

c) the simplified representation of the velocity pulse.

The period and the amplitude of the half-sine pulses of the simplified ground

motions are given in Table 4.6. A comparison of the frequency content for the input

ground motions for the three cases considered above (i.e. the full recorded ground

motion, the time interval of the recorded ground motion containing the velocity pulse, and

the simplified velocity time-history) is shown in Figure 6.9. With the exception of the

Pacoima Dam record for the San Fernando earthquake, the power spectral density of the

full time-history is similar to that of the recorded pulse alone. This implies that for these

records, most of the energy is concentrated in the velocity pulse. The Pacoima Dam

record, however, has a secondary pulse that arrives late in the time-history. The energy in

the high frequency range corresponds to this secondary pulse. The effect of secondary

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pulses is addressed in a later section. A comparison of the frequency content of the

simplified pulse and the recorded pulse shows that the simplified pulse does not include

the higher frequency content of the motion

The input (rock) velocity time-histories and the resulting output (soil) velocity

time-histories at the top of the soil column are shown in Figures 6.10 to 6.13. A

comparison between the results for the recorded pulse and those for the simplified pulse

shows that, with the exception of the KJMA record, the output motion for the simplified

and the real pulses are similar, both in period and amplitude. This implies that the

exclusion of higher frequencies resulting from the use of simplified velocity time-

histories does not significantly affect output (soil) velocities. The same inference can be

made by observing the particle velocity-trace plots (Figure 6.14). The output particle

velocity-traces for the simplified velocity time-histories are remarkably similar to the

output for the recorded pulses. Results are also listed in Table 6.4. With the exception of

the KJMA record, the soil PGVs for the simplified input motion differ at most by 10%

from the soil PGVs calculated for the recorded time-histories.

The KJMA input motion is the only record for which the site response to the

simplified velocity pulse yielded results that were significantly different from the site

response to the recorded motion. The KJMA record indicates the potential problems

associated with determining pulse period from zero crossings. The simplified sine pulse

does not match well the recorded pulse (Figure 4.5); the recorded pulse could also have

been fit with a sine pulse with a lower period (Tv ≈ 0.9 s). As a result, the simplified

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pulse is affected more significantly by the soil column, both in the amplification of PGV

and in the elongation of the pulse period.

The results shown in Figures 6.10 to 6.14 and in Table 6.4 can also be used to

evaluate the effect of the soil column properties on input near-fault ground motions.

While the input PGV of the recorded ground motion is amplified for the 1971 Pacoima

Dam, 1989 Gilroy, and 1994 Pacoima Dam Downstream records, it is de-amplified for

the 1995 KJMA record. Also significant is the elongation of the pulse period due to site

effects. A similar trend is observed with the pulse period, which is longer on the top of

the soil column for all the input motions except the KJMA. As indicated before, the

KJMA record could as well be represented by a shorter period, thus the apparent

shortening of the pulse period is likely due to the poor measure of input pulse period.

The PGV de-amplification might be due to strains induced in the soil in the first cycles of

motions. These strains results in a reduction of the profile stiffness. The revised input

pulse period (Tv = 0.9) is likely shorter than the degraded natural period of the profile,

consequently PGV de-amplification is observed. For the same reason, the longer-period

simplified input motion results in a PGV amplification. The concept of degraded natural

period and its effect on site response discussed at length in Section 6.4

The simplified velocity pulses do not always provide realistic representations of

the accelerations observed in near-fault ground motions. This discrepancy occurs because

accelerations have high frequency content and, as indicated before, the simplified pulses

do not contain high frequencies. In some cases the peak ground acceleration develops

later in the time-history and does not coincide with the forward-directivity velocity pulse,

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as illustrated by the Pacoima Dam record for the San Fernando earthquake (Figure 6.15).

The simplified representation of the forward-directivity pulses should therefore not be

used to evaluate site effects on peak ground accelerations.

It is also important to point out that the sensitivity of the response to higher

frequencies is a function of the natural period of a site (i.e. of profile depth and stiffness).

Shallower, stiffer sites respond to higher frequencies and thus these sites might be more

affected by the high frequencies that are excluded from the simplified pulses. However,

forward-directivity velocity pulses occur at long periods; thus, the effect of shallow stiff

profiles on the pulse characteristics is minimal (see Section 6.4).

Figure 6.16 shows the maximum strains developed in the soil profile for the

recorded velocity pulses and the corresponding simplified velocity pulses. The maximum

strains observed in both cases are of the same order of magnitude. For some of the input

motions, the strains associated with the simplified pulses are slightly larger. One possible

explanation is that these larger strains are observed because the simplified pulses

primarily excite the lower frequency modes of the finite element mesh. Moreover,

recorded pulses also introduce energy into higher frequency modes. These higher modes

might lead to non-constructive interference resulting in lower strains. Induced strains are

important because they control the nonlinear response of the soil. Based on the observed

results, it is reasonable to conclude that the simplified pulses lead to reasonable estimates

of strains in the soil profile.

The results in this section can be summarized as follows:

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a) The majority of the energy of pulse-like forward-directivity motions is

generally concentrated within a narrow low-frequency band.

b) Simplified velocity pulses can be used to evaluate the effects of local site

conditions on the amplitude and period.

c) Simplified velocity pulses do not consistently lead to good approximations of

peak ground accelerations. Hence, these simplified pulses should not be used

to evaluate the effects of site response on accelerations.

6.4 SITE RESPONSE TO SIMPLIFIED PULSES

6.4.1 General

Results presented in Section 6.3 indicate that using simplified representations of

input rock velocity pulses can capture salient aspects of seismic site response to near-fault

ground motions containing forward-directivity. Simplified pulses that represent the

existing ground motion database of both rock and soil motions were presented in Chapter

Four (Section 4.3.5). The simplified velocity pulses identified as Sets 6, 7, and 8 (Figure

4.35) are used as input motions in the study of site response for the generalized profiles

presented in Section 6.2.

The use of simplified representations facilitates a systematic study of the influence

of pulse parameters on site response. The pulse parameters and the site conditions are

varied systematically to study the effect of site response on the characteristics of the

velocity pulse. Table 6.5 gives the different combinations of pulse parameters and site

conditions used in the site response analyses. The pulse parameters were chosen to

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represent typical values of rock pulses as described in Chapter Four. The results are

presented first for the Stiff Soil profiles (Sites C and D) and then for the Soft Soil profile

(Site E).

Before discussing the results of the nonlinear analyses, it is instructive to analyze

the response of a uniform, linear soil column to the simplified pulses. Input motion Set 6

is applied to an elastic soil deposit with a uniform shear wave velocity of 120 m/s and a

depth of 30 m (natural period: Tn = 4H/Vs = 1.0 s). A target damping ratio of 5%

(Rayleigh Damping) is used to simulate the hysteretic damping in soil. The site response

analysis is carried out using the finite element program GeoFEAP with linear-elastic brick

elements. Figure 6.17a presents the calculated PGV at the top of the soil column as a

function of input velocity pulse period. This type of plot is sometimes referred to as a

"shock spectrum" (Chopra 1995). The maximum amplification of PGV occurs at an input

period of 1.1 s. This period is only slightly longer than the undamped natural period of

the site. Figure 6.17b shows the ratio of the pulse period on the top of the soil column to

the period of the input motion. Observe that when the input pulse period is less than the

undamped natural period of the site, the pulse period at the top of the soil column is

longer than the input pulse period.

In frequency domain, equivalent-linear analysis, the equation of motion is solved

for an equivalent strain level that is constant over time for each soil layer. The constant

strain level allows the definition of a "degraded site period", which is the natural period

corresponding to the equivalent strain level used in the analysis. This degraded site

period corresponds to the first resonant frequency of the site. The amplification of input

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ground motions is a maximum at the first natural frequency and peaks at the subsequent

higher order frequencies of the site (Figure 6.18). In nonlinear analysis, the concept of a

constant site period over time is ill-defined, because the tangent shear modulus (and

hence the shear wave velocity) changes as strain levels in the soil change during seismic

loading. The concept of degraded site period, however, can be used as a conceptual tool

to help understand the response of a site.

An increase in the intensity of the input motion results in larger strains and hence

a decrease in soil stiffness with a consequent increase of the degraded site period. The

decrease in soil stiffness leads to a larger impedance contrast between the rock, which

behaves almost linearly, and the soil. This larger impedance contrast leads to larger

amplifications of the input motion (Figure 6.18a). An increase in strain levels in the soil,

however, also results in an increase in hysteretic damping levels. Larger damping levels

damping can lead to lower velocities (Figure 6.18b). The amplification or de-

amplification of motion results from a balance of the effects of a change in natural period,

the effects of increased damping, and the effects of the larger impedance contrast. For

non-forward-directivity motions, the effect of increased damping at large input motion

intensities is significant, particularly at high frequencies. On the other hand, damping is

not as significant when the input motion is pulse-like (Chopra 1995), such as is the case

with forward-directivity motions. Moreover, damping has less of an effect when the

input motion is a long period motion. Therefore, site response to pulse-type input

motions is most likely controlled by the larger impedance contrast and the shift in the

natural period of the site that has degraded dynamic stiffness due to its response to intense

near-fault input motions.

300

6.4.2 Stiff Soil Profiles (Sites C and D)

General

The shear wave velocity profile and the nonlinear properties of the Deep Stiff Soil

profile (Site D) and the Shallow, Very Stiff Soil profile (Site C) were presented in Section

6.2. Three different sets of model parameters corresponding to different shear modulus

reduction and damping curves were also presented in Section 6.2. This section presents

the results of site response analyses using the shear modulus reduction and damping

curves for the soil termed "Generic Low-Plasticity Soil." The influence of the input pulse

parameters is first evaluated. The effects of profile stiffness are then evaluated by

comparing results of the 30 m deep Stiff Soil profile to those of the Very Stiff profile

(Site C). The effect of varying depth to bedrock for the Stiff Soil profiles is then

evaluated. Finally, the influence of the nonlinear stress-strain properties of the soil is

evaluated by comparing the analysis results for the Generic Low-Plasticity Soil to those

for the clays with PI of 15 and 30.

Influence of input pulse period and input PGV

The particle velocity plots at the top of the soil column for the Deep Stiff Soil

profile with a depth of 60 m are given in Figures 6.19 to 6.21 for three different input

motion sets. The influence of both the input pulse period (Tv) and the intensity of the

input motion (PGV) on the resulting velocity plots is evident. It is apparent that the

effects of input pulse period and PGV are interdependent. For example, for an input

pulse period of 1.0 s, velocity amplification decreases as the input PGV increases.

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However, for an input pulse period of 4.0 s, velocity amplification increases with

increasing input PGV. Figure 6.22a presents the ratio of PGV in the soil to PGV in the

rock plotted as a function of input pulse period. PGV soil to rock ratios for this profile

range from 1.0 to 2.0 for the cases considered. By analogy with the results for a linear-

elastic soil profile (Figure 6.17), the degraded (strain-compatible) period of a site can be

identified with the period at which the largest PGV amplification occurs. The period at

which maximum PGV amplification is observed increases with increasing input motion

intensity for all three input motion sets. This is due to the larger strains induced by the

more intense motions. The shift in degraded site period with increasing intensity is well

illustrated by the PGV soil to rock ratios for the 30 m deep Stiff Soil profile shown in

Figure 6.23.

The input pulse period and input PGV also influence the period of the resultant

velocity pulse. The ratio of the resultant (soil) pulse period to the input (rock) pulse

period is plotted in Figure 6.21b as a function of input pulse period. The pulse period in

the soil is larger than the pulse period in the rock for low input pulse period values. The

same trends were observed in the empirical database of near-fault ground motions.

Influence of soil stiffness

The influence of soil stiffness on site response is evaluated by comparing the site

response results for the 30 m deep Stiff Soil profile (natural site period of 0.44 s) with the

results for the 30 m deep Very Stiff Soil profile (natural site period of 0.22 s). Both these

profiles have the same depth to bedrock and the same nonlinear stress-strain soil

properties, but they differ in shear-wave velocity profiles (Figures 6.1 and 6.8). Figure

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6.24a compares the amplification of PGV in the soil column for these two soil profiles.

For the Very Stiff Soil profile, maximum amplification of PGV for the cases considered

occurs at an input pulse period of 0.6 s, while essentially no PGV amplification is

observed for periods longer than 2.0 s. Conversely, the Stiff Soil profile displays

maximum PGV amplification for input pulse periods ranging from 1.5 to 2.5 s, reflecting

the longer natural site period. For periods in this range, PGV amplifications are larger

than the amplifications for the Stiff Soil profile. Amplification of PGV in the Very Stiff

Soil profile is nearly independent of the input PGV. For this profile, variations in the

input PGV affect the resulting PGV soil to rock ratio. This difference in response is due

to the different strain levels induced in the profile. At the larger strains in the Stiff Soil

profile, the shear modulus reduction curve is more sensitive to changes in strain.

The effect of these two site profiles on the period of the velocity pulse is

illustrated in Figure 6.24b. The stiffer profile (Very Stiff Soil profile) has very little

influence on the pulse period, except for the case of an input pulse period of 0.6 s.

Conversely, the softer profile (Stiff Soil profile) causes significant elongation of the input

pulse period for all periods lower than 2 seconds.

Influence of soil profile depth

Figure 6.25 shows the soil to rock PGV ratios for all the Stiff Soil Profiles for

input motion Set 6. The depth to bedrock of these profiles varies from 30 m to 200 m.

The ratio of peak ground velocity in the soil (output) to the rock (input) varies from 0.6 to

2.3 depending on the input pulse period and the profile depth. For the conditions

considered in this study, the input pulse period for which maximum amplification occurs

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increases with profile depth. This is a direct consequence of the larger site periods for

deeper profiles. Also, observe that in general, PGV amplification decreases with

increasing input motion intensity for input motions with short pulse periods. The more

intense motions result in a degraded site period that is much longer than the input pulse

period, hence resulting in lower amplifications. Conversely, for input motions with long

periods, the longer degraded site periods resulting from more intense ground motions are

closer to the input pulse period, thus resulting in PGV amplifications.

Figure 6.26 shows the ratio of pulse period on soil to pulse period on rock. As the

depth of the profile increases (and consequently the natural period of the sites increase),

the soil pulse period increases. In general, when the input motion is shorter than the

degraded period of the site, the soil has a tendency to oscillate at its degraded period,

resulting in an elongation of the pulse. For longer site periods (e.g., deeper profiles), the

resultant pulse period is consequently longer. However, if the input pulse period is much

longer than the degraded site period, the wavelength of the input motion is much larger

than the profile depth and the soil column moves more or less as a rigid body.

Influence of shear modulus reduction and damping curves

Three different soil types are used in the site response analyses to the same set of

input motions. These soils are the Generic Low-Plasticity Soil used in the previous

analyses and two soils whose nonlinear stress-strain behavior is "matched" to the Vucetic

and Dobry (1991) curves for PI of 15 and 30 (see Section 6.2). Figure 6.27 presents the

resulting ratios of PGV in the soil to PGV in the rock for the three soil types. For any

given input soil period and velocity, the PGV soil to rock ratios vary with soil type. For

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an input pulse period of 0.6 s, the PGV soil to rock ratios are larger for soils with larger PI

(i.e. PI = 30 soil > PI = 15 soil > low-plasticity soil). This reflects the fact that if two

soils with different PI have the same elastic shear modulus, for a given strain level the

soils with larger plasticity have higher stiffness. The higher stiffness implies that the soil

profile has a lower degraded site period and amplifies more shorter input pulse periods.

Conversely, at longer input pulse periods, the PGV of the low-plasticity soil is larger than

the PGV in the other two soils. These results highlight the importance of a good

characterization of shear modulus reduction and damping curves in site response

analyses. The effects of different shear modulus degradation curves are important

especially for low input motion intensities. For larger intensities, high strains are induced

and degraded site periods are closer for the three profiles.

The stress-strain properties of the soils also affect the pulse period of the soil

(Figure 6.28). Velocity pulse periods are larger for the Generic Low-Plasticity Soil. This

reflects the higher large-strain stiffness of the soils with higher plasticity (Figure 6.2).

The variation in pulse periods with site condition, however, is not very significant.

Influence of fault parallel motion and pulse shape

In the following paragraphs, the effects of the fault parallel input motion and the

pulse shape on site response are examined. The effects of varying the input fault parallel

to fault normal velocity ratio (PGVp/n) is first considered. The sensitivity to pulse shape,

as defined in Chapter Four, is then considered. Figure 6.29 shows the PGV soil to rock

ratios in the 45 m deep Stiff Soil for two input motion sets and for varying values of the

ratio of input fault parallel to fault normal velocity (PGVp/n). The amplification of PGV is

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consistently higher for input pulses with higher PGVp/n ratios. The larger fault parallel

input motion leads to increased strain levels in the soil. The larger strains induce a

reduction in the shear modulus which in turn increases the impedance contrast and shifts

the degraded site period towards the long period energy contained in these near-fault

forward-directivity input motions. These changes lead to larger amplifications of

velocities for most of the cases studied herein. The influence of the fault parallel motion

on the amplification of fault normal velocities is also a function of the frequency content

of the fault parallel motion. When this frequency content is such that it leads to larger

amplifications, the strains developed in the profile are consequently larger and the

influence on fault normal velocities increases. For a PGV of 75 cm/s, it is reasonable to

use a fault parallel PGV of up to 75% of the fault normal value. In this case, the largest

fault normal amplification is 1.8 for Set 7. For comparison, the largest amplification for

the same input motion set but a fault parallel PGV of 25% of the fault normal value is 1.5,

and if the fault parallel component is ignored in analysis, the largest calculated

amplification is only 1.4. This increase is important and justifies the use of both fault

normal and fault parallel input motions, even for cases in which the fault normal motion

is significantly larger than the fault parallel motion.

The PGVp/n ratios also affect period elongation when the input pulse period is

lower than the degraded site period. The larger strains for the higher PGVp/n ratios imply

larger degraded periods, thus the pulse periods on the top of the soil column are longer for

higher PGVp/n. This effect decreases as the period of the input motion approaches the site

period. This is illustrated for the site with profile depth of 45 m in Figure 6.30.

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Figure 6.31 compares the resulting PGV ratios for three different shapes of the

input pulses for all the Stiff Soil profiles. Recall that “pulse shape” was defined as the

shape of the particle velocity-trace plot of the pulse (Chapter Four). Maximum PGV

amplification occurs for Set 6. This shape is also the pulse shape that leads to largest

strains. In general, larger values of PGV are associated with larger shear strain levels in

the soil. This is not necessarily the case when the input motion is not pulse-like. For

non-pulse type motions, the damping associated with large strain levels has a more

significant effect in attenuating the ground motions because of the increased number of

loading cycles in the soil. This is accentuated by the fact that non-forward-directivity

motions often contain significant energy at higher frequencies, which are more affected

by damping.

Observations on resulting stresses and strains

The soil model used for the Site D profile (Section 6.2) included the assumption

that the soil does not reach the failure stresses defined by Figure 6.3. The soil shear

strength was never exceeded by the calculated stresses even for the input motion Set 6,

which yields the maximum stresses. Figure 6.32 shows the maximum strains calculated

for each depth in the profile for a number of runs on the 45 m deep Stiff Soil profile. The

strains shown are the vectorial sum of the fault normal and fault parallel strains. For an

input PGV of 75 cm/s, shear strains induced in the soil profile are largest when the input

period is 1 s to 2 s (Figure 6.32a). Note that for a given PGV, the lower the pulse period,

the larger the acceleration; thus a motion with a higher acceleration (i.e. the one with Tv =

0.6) develops a lower level of strains. On the other hand, for motions with equal

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accelerations, strains are larger for larger velocities (Figure 6.32b). In general, the strain

level calculated in the profile is generally associated with the amount of PGV

amplification and consequently with the site period.

Figure 6.33 shows the maximum shear strains developed in all the Stiff Soil

profiles for a given input motion. This figure also includes the Shallow, Very Stiff Soil

profile (Site C) in addition to the Stiff Soil profiles. For the shorter input pulse periods,

larger shear strains occur in the shallow profiles. As the period of the input motion

increases, larger strains are observed for progressively deeper profiles. This is consistent

with the observations presented throughout this chapter.

6.4.3 Soft clay profile

The particle velocity plots for input motion Set 6 for the Soft Clay profile (Site E)

are shown in Figures 6.34 to 6.36. The magnitude of the ground deformation at the top of

the soft clay profile is a function of both the period and the intensity of the input rock

velocity pulse. In general, the effects of input pulse period and input PGV on the

amplifications of velocities are interdependent. The effect of the intensity of the input

motion is more pronounced for the Soft Clay profile than for the Stiff Soil profiles. A

significant consideration is that the amplification of velocities can sometimes also result

in a change of the input pulse from a full cycle to a half cycle of motion for short input

pulse periods.

The shear stresses and strains induced in the soil profile are shown in Figure 6.37.

In the soft clay (elevation 3 to 27 m), the shear stresses are bounded by the strength of the

soft clay, i.e., in all the cases studied the soft clay reached failure. The occurrence of

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failure explains the large strains in the soft clay. For large intensity motions (input PGV

= 300 cm/s), failure is also reached in the underlying Pleistocene Clay deposit. As

indicated previously, a PGV of 300 cm/s represents an upper bound for predicted

velocities at rock sites. It is interesting to observe that in the soft clay layer, larger strains

are observed for input motions with longer input pulse periods; while in the Pleistocene

Clay layer, strains are larger for motions with intermediate input pulse periods. A

parametric study on the nonlinear stress-strain properties of the Soft Clay indicated that

the failure stress is the parameter that controls response in the Soft Clay deposit. In fact,

large variations in the shear modulus reduction and damping curves in the low strain

region did not affect either the predicted peak ground velocities at the top of the soil

column, or the resultant period of the velocity pulse. It is important to realize that

equivalent linear programs, such as SHAKE91, do not account for soil failure and thus

cannot be expected to correctly predict site response for the Soft Clay profiles for intense

near-fault ground motions.

A number of studies from paired stations in rock and soft clay have shown the

tendency of soft clays to attenuate high frequency motions and to amplify long period

motions. In particular, soft clays have a strong tendency to attenuate accelerations when

these are larger than about 0.4 g (Idriss 1991, Figure 2.1). Figure 6.38 shows the

maximum displacement, acceleration, and velocity values observed in the soft clay profile

for an input PGV of 160 cm/s. For the input pulse periods shorter than 4 s, there is a

decrease in maximum accelerations in the soft clay layer (3 – 20 m depth) with respect to

the underlying clay. Velocities are more sensitive to longer period energy. A decrease in

309

velocity with respect to the underlying clays is observed in the soft clay layer when the

input period is 0.6 s, but input periods of 2.0 s or more result in slightly larger velocities.

Figure 6.39 compares the PGV ratios (soil/rock) for the Soft Clay profile and a

Stiff Soil profile of equal depth (60 m). The level of amplification in both profiles is

approximately equal for a PGV of 75 cm/s. The soft clay profile would be expected to

have larger velocities because of the higher impedance contrast with the underlying

bedrock. However, yielding of the soft clay deposits results in a degraded site period that

is much larger than the period of the input motions, reducing the expected amplification.

Moreover, yielding in the soil introduces large levels of damping that attenuate the input

pulse. This effect is intensified for larger input motions (PGV = 160 cm/s and PGV = 300

cm/s). For the larger motions, the velocities in the soft clay are lower than in the stiff

soil. Figure 6.40 compares the amplification of PGV in the Stiff Soil with the

amplifications in the Soft Clay profile for all the input motion sets. For the Deep Stiff

Soil, the PGVs in the soil are generally larger than in the rock. Conversely, the Soft Clay

(Site E) de-amplifies input PGVs which are higher than a crossover value. This crossover

value is a function of the input period. The elongation of the input pulse period is slightly

larger in the soft clay site than in a stiff soil deposit of the same depth (Figure 6.41). This

occurs because the degraded period of the soft clay site is larger than that of the stiff soil

deposit.

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6.4.4 General Observations

Pulse Shape

The results presented so far have dealt primarily with the fault normal component

of motion. This component typically controls the response of structures to forward-

directivity motions. However, it is also important to consider the effects of site

amplification on the fault parallel component of motion. This section presents some

findings related to the change of pulse shape due to site effects.

If the frequency content of the two perpendicular directions of motion is different,

site response will affect each direction in different proportions, altering the ratio of fault

parallel to fault normal velocity. Typical values of fault parallel to fault normal PGVs in

the entire database of forward-directivity motions are shown in Figure 4.26. The input

motions used in the analyses are defined by input PGVp/n ratios ranging from 0.0 to 0.75.

The variation in PGVp/n ratios due to site effects is shown in Figure 6.42. For the

Shallow, Very Stiff Soil (Site C), the PGVp/n ratios of the output motions are independent

of input pulse period and its values are equal to the PGVp/n ratios of the input motions.

For the Deep Stiff Soil with 60 m depth (Site D) and the Soft Clay (Site E) sites, PGVp/n

ratios tend to increase with increasing input period for the range of values considered in

this study. The changes in PGVp/n ratio reflect the varying frequency content of the fault

parallel motions. The fault parallel pulses in the simplified input pulse Set 7 have shorter

pulse periods. Therefore, when the fault normal period is longer than the degraded site

period the fault parallel pulse period matches the degraded site period, resulting in a

larger amplification of PGV in the fault parallel direction. Note that in nonlinear analysis,

311

the fault normal and fault parallel components cannot be separated. In fact, large

amplifications of fault parallel motions (resulting in a decrease of PGVp/n ratios) are not

observed when site response analyses are performed with only the fault parallel

component of motion. The amplifications of fault parallel motion are associated with the

shear modulus reduction induced by large fault normal velocities. In recorded near-fault

ground motions, the period of the fault parallel pulse is generally shorter than the period

of the fault normal pulse (see Chapter Four); thus, the results obtained herein lie within

the bounds of the empirical data. If the initial PGVp/n ratio is high, as some of the low

intensity motions in the database indicate, site amplification can lead to PGVp/n ratios

larger than one. This is an important observation with respect to seismic demand

estimation for structural analysis in the near-fault region. High PGVp/n ratios are also

observed in some of the recordings in the strong motion database.

Number of Cycles of Motion

In the dynamic analysis of nonlinear structures, the number of cycles of motion is

an important input parameter. Site response can affect the number of cycles of motion in

two ways:

a) Energy is trapped in the softer soil layers. This case applies in particular to sites with

a large impedance contrast. At a large scale, this phenomenon causes the basin effects

observed in a number of earthquakes (e.g., Aki and Larner 1970, Bard and Bouchon

1985). At individual sites, the trapping of energy in softer soil layers is responsible

for the longer duration of strong shaking in soil sites than in rock sites (Abrahamson

and Silva 1997). The results presented herein do not indicate that there are velocities

312

of significant magnitude beyond the initial velocity pulses. With pulse-type motions,

the initial pulse of motion typically has the largest amplitude. The amplitude of the

ground motion resulting from the trapping of vertically-propagating waves in the soil

profile is generally lower than the amplitude of the initial pulse.

b) When the period of the input pulse is significantly shorter than the natural period of

the site, input motions with a full cycle of motion tend to be one-sided pulses. This

occurs because the soil elongates as a result of the initial pulse and does not capture

the second half of the pulse (Figures 6.18 to 6.20 for Tv = 0.6, and Figures 6.34 to

6.36 for Tv ≤ 1 s). This effect has to be considered when evaluating a database for

estimating the number of cycles of near-fault ground motions.

Consecutive Pulses

The analyses presented in this chapter were conducted using input pulses with a

full cycle of motion in the fault normal direction. Although this condition characterizes a

large number of records in the empirical database, some near-fault recordings, such as the

Gilroy records for the Loma Prieta earthquake and the KJMA record for the Kobe

earthquake, have two consecutive full cycles of pulse-like motion. The number of cycles

is generally related to the slip distribution in the causative fault. On occasions in which

the slip distribution is uneven, the pulses of motion for near-fault, forward-directivity

motion may arrive at separate times. To account for both of these factors, additional

analyses were performed with input motions consisting of two consecutive cycles of

motion at different offsets. Peak amplification normally occurs in the first full cycle of

motion. This implies that the results presented in this chapter in terms of the effects on

313

peak values of velocity are also valid for ground motions with more than two consecutive

cycles, as long as the largest velocity corresponds to the first cycle of motion. However,

near-fault motions containing consecutive pulses (i.e. extended durations) will

undoubtedly affect the nonlinear response of structures that have yielded.

A further look at pulse shape

The shape of the pulse in a particle-velocity plot is also affected by the delay

between the fault normal and fault parallel pulses. The effects of changing the delay

between the fault normal and fault parallel pulses on site response were explored. A fault

normal pulse with an amplitude of 160 cm/s and a full cycle of motion with a period of

2.0 s was used as input motion for site response analyses of the 60 m deep Stiff Soil

deposit. The amplitude of the fault parallel motion was assumed to have a value of 66%

of the fault normal amplitude, with a full cycle of motion and a period of 75% of the fault

normal period. The offset between the fault normal and fault parallel pulses was varied

between a range of two pulse periods. The maximum difference in fault normal PGVs in

the soil is about 13%. Maximum amplifications occur when the fault parallel pulse leads

the fault normal pulse. Considering the existing level of uncertainty in the prediction of

PGVs, the effect of the offset is not considered significant at this time. Similar

conclusions apply to the velocity pulse period.

314

6.5 FINDINGS

General findings

This chapter presented the results of ground motion analyses to near-fault ground

motions using simplified velocity pulses. The use of simplified representations of pulse

motions was validated by comparing the results of site response analyses to recorded

ground motions with the results of site response analyses to simplifications of the

recorded ground motions.

The surface PGV values obtained from the site response analyses of the Stiff Soil

deposits using simplified velocity pulses are plotted against the input PGVs in Figure

6.43. Ratios of soil to rock PGV generally lie between 1.0 and 2.0. The findings related

to PGV amplification in Stiff Soil can be summarized as follows:

• PGV amplification is a function of both the period and the intensity of the input

pulse (Figure 6.44). The period for which maximum amplification of PGV

occurs can be associated with the degraded natural period of the site (Figure

6.23).

• The degraded natural period of a site increases with increasing profile depth, it

decreases with increasing profile stiffness, and it increases with increasing input

motion intensity. Consequently, sites with long degraded site periods (i.e. Stiff

Soil sites deeper than 60 m, Soft Clay sites, or Shallow Stiff Clay sites subject to

intense shaking) tend to amplify input pulses with periods longer than 2.0 s.

Conversely, sites with short degraded site periods (i.e. Shallow, Very Stiff Soil

315

sites, or sites subject to low intensity input motions) tend to amplify input pulses

with short periods (≤ 1 s) and are not affected by long pulse periods.

• The nonlinear stress-strain properties of the soil have an important effect on the

PGV amplification. For a given strain level, soils that exhibit lesser shear

modulus reduction tend to amplify motions with shorter input pulse periods

more than soils with larger shear modulus reduction.

• Amplification of input pulses with long periods is observed for input PGVs as

large as 300 cm/s when the input pulse period is longer than 1 s.

• The mid-period amplification factor for Site D in the 1997 UBC (180 m/s < sV

≤ 360 m/s) in Zone 4 is 1.6. This factor agrees with the range of Stiff Soil PGV

amplification obtained in the site response analyses for input pulse amplitudes

lower than 300 cm/s (Figure 6.43). Mid-period amplification factors can be

compared to velocity amplification factors, because mid-period spectral values

are generally controlled by velocities.

• PGV amplification for recorded ground motions (Section 6.3) is comparable to

PGV amplifications obtained from the simplified pulses (Figure 6.43).

In addition, site response also affects the period of the velocity pulse. Figure 6.45

illustrates the ratios of the calculated velocity pulse period to the input pulse periods.

Figure 6.45 also includes the ratio of the median regression curves for pulse period in soil

and rock obtained from the database of near-fault ground motions (Chapter Four). The

trends in the results from the numerical simulations agree well with the regression line

316

developed using empirical data. Some observations relating to the influence of site

response on the velocity pulse period include:

• When the input pulse period is shorter than the degraded natural period of the

site, the soil column tends to elongate the velocity pulse, because of its tendency

to oscillate at its natural site period. This results in a significant velocity pulse

period increase for short input pulse periods.

• Input pulse periods that are significantly longer than the degraded site period

have much longer wavelengths than the profile depth. Consequently, the soil

column moves more or less as a rigid body, and the period (i.e. and amplitude)

of the input pulse is not affected significantly by the response of the soil

column.

• For the recorded near-fault ground motions presented in Section 6.3, the

magnitudes of the calculated pulse periods lie in the same range as the

calculated pulse periods for simplified velocity pulses with comparable

amplitudes (Figure 6.46).

Figure 6.40 showed a summary of the results of the analyses for the Soft Clay

profile to a number of the simplified velocity pulses. Contrary to the results obtained for

the Stiff Soil profiles, PGV amplifications in the Soft Clay profile diminish as the

intensity of the input motion increases. Some important findings relating to the response

of soft clay deposits to the simplified velocity pulses are:

• The Soft clay profile used in this study (static su = 25 kPa) reaches its yield

strength even for input velocity pulse amplitudes as small as 75 cm/s. This

317

value is well within the expected range of velocities for near-fault ground

motions (Chapter Four).

• The response of the Soft Clay profile is controlled by the value of the yield

strength of the soil. The analyses indicated that soft soils are expected to yield

at input velocities typical of near-fault ground motions. Thus, the evaluation of

the dynamic shear strength of soft clays is important for the proper evaluation of

seismic site response to near-fault motions. The mid-period amplification factor

for Site E ( sV ≤ 180 m/s) in the 1997 UBC in Zone 4 is 2.4, a value larger than

the estimated PGV amplification factors from the site response analyses. The

large code factor is likely due to extrapolation from equivalent linear analyses

that do not account for soil failure.

Limitations

As indicated in Chapter Five, the finite element implementation of site response

used in this study assumes that horizontal shear waves propagate vertically. This

assumption is commonly made in most site response analyses, and it applies to sites away

from the fault. For these sites, non-vertically travelling waves are refracted as they

propagate from the source to progressively softer soils, resulting in the observed

vertically-propagating waves. However, the short travel paths to near-fault sites could

prevent a complete reorientation of the wave front causing some of the energy to be

originated from non-vertically-propagating waves. In addition, constructive interference

with long-period surface waves originating from basin effects might contribute to a large

portion of the long-period energy of recorded motions, particularly near basin edges.

318

The evaluation of site response effects on velocity pulses was achieved by

comparing the recorded soil motions to the simplified velocity pulses used as input

motions in the site response analyses. These simplified velocity pulses are assumed to be

outcropping rock motions. This assumption implies that the simplified pulses already

capture the effects of the geological structures underlying the soil profile. In general,

site response affects the input ground motions only at periods shorter than about 1.5 to 2

times the degraded site period. Input motions with longer periods have wavelengths that

are much larger than the profile depth and thus are unaffected by the profile. However,

these long period motions are affected by the deeper velocity structure underlying the site.

As mentioned, it is assumed that the outcropping velocity pulses already capture these

effects.

319

Table 6.1. Number of profiles for each Site Type (based on UBC classification scheme,Table A3) in the database of shear wave velocity profiles of Silva (personal comm.).

Site Type Number of ProfilesB 35C 227D 343E 42

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Table 6.2. Model parameters used for the soils considered in the Stiff Soil profiles. Theshear-wave velocity profile is given in Figure 6.1.

Parameter Generic Low-Plasticity Soil Clay, PI = 15 Clay, PI = 30

h/Gmax 0.032 0.126 0.126

m 1.4 1.16 1.16

R/Gmax .005 .005 .01

Ho 1e-6 1e-6 1e-6

ξ 1% 1% 1%

ρ=(Mg/m3) 1.9 – 1.95 1.9 - 1.95 1.9 - 1.95

ν 0.49* 0.49* 0.49*

* For incompressible (undrained) deformation, ν should be set to 0.5. The value 0.49 is used to avoidnumerical errors associated with incompressible deformations.

321

Table 6.3. Model parameters used for the soils considered in the Soft Clay profiles. Theshear-wave velocity and density profile is given in Figure 6.4.

Parameter Holocene Bay MudLow Strain curve

Holocene Bay MudVariable Strength

CurveClay, PI = 30

Variable Strength Curve

h/Gmax 0.218 1.7 0.8

m 0.97 1.1 0.8

R/Gmax 0.0.014 us38 /Gmax**

us38 /Gmax**

Ho 1e-6 1e-6 1e-6

ξ 1% 1% 1%

ν 0.49 0.49 0.49*

* For incompressible (undrained) deformation, ν should be set to 0.5. The value 0.49 is used to avoidnumerical errors associated with incompressible deformations.

** su profile given in Figure 6.5.

322

Table 6.4. Input motions and results of site response analyses on recorded near-faultground motions (PCD = Pacoima Dam , San Fernando earthquake; GIL = Gilroy GavilanCollege, Loma Prieta earthquake; PAC= Pacoima Dam Downstream, Northridgeearthquake; KJM = KJMA, Kobe earthquake).

(a) Input MotionsFault Normal Fault Parallel

PGA (g) PGV (cm/s) Tv (s) PGA (g) PGV (cm/s) Tv (s)Full Recorded

Ground Motion 1.47 114 1.44 0.78 38.0 1.47

RecordedVelocity Pulse 0.78 113 1.44 0.36 39.7 1.47PCD

SimplifiedVelocity Pulse 0.51 114 1.44 0.22 36.2 1.47

Full RecordedGround Motion 0.29 30.8 1.16 0.41 26.6 0.82

RecordedVelocity Pulse 0.29 31.6 1.16 0.41 26.3 0.82GIL

SimplifiedVelocity Pulse 0.25 31.1 1.16 0.33 26.7 0.82


RecordedVelocity Pulse 0.48 52.7 0.61 0.31 23.5 1.1PAC



RecordedVelocity Pulse 0.85 96.8 1.91 0.55 53.5 0.73KJM


(b) Calculated soil responseFault Normal Fault Parallel

PGA (g) PGV (cm/s) Tv (s) PGA (g) PGV (cm/s) Tv (s)Full Recorded

Ground Motion 0.47 159 1.78 0.36 65.8 1.60

RecordedVelocity Pulse 0.48 166 1.80 0.34 63.5 1.66PCD



RecordedVelocity Pulse 0.33 41.4 1.52 0.39 38.0 1.70GIL



RecordedVelocity Pulse 0.31 55.3 1.08 0.32 26.0 1.00PAC



RecordedVelocity Pulse 0.50 77.7 1.40 0.54 77.6 1.24KJM


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Table 6.5. Values of parameters used in the parametric study of site response tosimplified pulse motions.

Parameter Values

Tv,N 0.6, 1.0, 2.0, 4.0

Fault Normal PGV (cm/s) 75, 160, 300

PGVp/n 0, 0.25, 0.5, 0.75

Input Motions Sets (PulseShapes) Set 6, Set 7, and Set 8 (Figure 4.35)

Vs profile Shallow Very Stiff Soil, Deep Stiff Soil (varying profile depth),and Soft Clay (Section 6.2)

Depth to bedrock For Stiff Soil profile: 30 m, 45 m, 60 m, 100 m, 200 m.

324

Figure 6.1. Selected shear wave velocity profile for the Stiff Soil profile. Included in thegraph are the median and ± one standard deviation for a database of 343 profiles (Silva,personal comm.) classified as Site D by the Uniform Building Code (e.g. 180 m/s < Vs ≤360 m/s).

0

20

40

60

80

100

120

140

160

180

200

0 200 400 600 800 1000

Vs (m/s)

Dep

th(m

)

Selected ProfileMedian and +_ one standard deviation from 343 profiles

325

Figure 6.2. Shear modulus degradation and equivalent viscous damping curves for thesoils used in the Stiff Soil profile.

10-4 10-3 10-2 10-1 100 1010

0.2

0.4

0.6

0.8

1

Strain (%)

Mod

ulus

Red

uctio

n

10-4 10-3 10-2 10-1 100 1010

10

20

30

Strain (%)

Dam

ping

(%)

Sand Curves (Seed and Idriss 1970)PI = 15 (Vucetic and Dobry 1991) PI = 30 (Vucetic and Dobry 1991) Generic Low-Plasticity model curvePI = 15 model curve PI = 30 model curve

326

Figure 6.3. Dynamic undrained shear strength for the Stiff Soil profile.

0

20

40

60

80

100

120

140

160

180

200

0 500 1000 1500 2000 2500

Undrained Strength, su (kPa)D

epth

(m)

su/σv' = 1.4 (0.8)

327

Figure 6.4. Generic Soft Clay shear wave velocity and density profiles. The profilerepresents typical Bay Mud sites from the San Francisco Bay region.

0

10

20

30

40

50

60

70

0 500 1000 1500

Vs (m/s)

Dep

th (m

)Fill

Soft

Cla

ySt

iff C

lay

Rock

0

10

20

30

40

50

60

70

1.5 2 2.5

Density (Mg/cm3)

328

Figure 6.5. Dynamic shear strength profile for the generic Soft Clay site.

0

10

20

30

40

50

60

0 100 200 300 400 500 600

Undrained Strength, su (kPa)D

epth

(m)

329

Figure 6.6. Shear modulus degradation and equivalent viscous damping curves for theSoft Clay soil. The Large-Strain model curve is given for the average strength value ofthe clay. The curves shift to the right for an increase in strength, and to the left for adecrease in strength.

10-4 10-3 10-2 10-1 100 1010

0.2

0.4

0.6

0.8

1

Strain (%)

Mod

ulus

Red

uctio

n

10-4 10-3 10-2 10-1 100 1010

10

20

30

40

Strain (%)

Dam

ping

(%)

Young Bay Mud (Isenhower and Stokoe 1981)Low-strain model Large-strain model

330

Figure 6.7. Shear modulus degradation and equivalent viscous damping curves for thePleistocene clay used in the generic Soft Clay profile. The upper- and lower-boundmodel curve corresponds to the upper bound and lower bounds, respectively, for thestrength parameter R. The upper bound curve is used when soil yielding is not ofconcern.

10-4 10-3 10-2 10-1 1000

0.2

0.4

0.6

0.8

1

Strain (%)

Mod

ulus

Red

uctio

n

10-4 10-3 10-2 10-1 1000

10

20

30

40

Strain (%)

Dam

ping

(%)

PI = 15 (Vucetic and Dobry 1991)PI = 30 PI = 50 Upper bound model curve Lower bound model curve

331

Figure 6.8. Selected shear wave velocity profile for the Very Stiff Soil profile (Site C).Included in the graph are the median and ± one standard deviation for a databaseincluding 227 profiles (Silva, personal comm.) classified as Site C by the UniformBuilding Code (e.g. 360 m/s < Vs ≤ 760 m/s).

0

5

10

15

20

25

30

0 500 1000 1500

Vs (m/s)

Dep

th (m

)

Selected ProfileAverage and +_ one standard deviation for 227 profiles

332

Figure 6.9. Power spectral densities, PSD (normalized by the maximum value of thePSD) for the ground motions used in the site response analyses. PSD are given for thefull recorded ground motion, for the time interval of the ground motion containing thevelocity pulse, and for the simplified representation of the velocity pulse.

10-2 100 1020

0.5

1

Frequency (Hz)

Nor

mal

ized

PS

D

Pacoima Dam, SF Eq.

10-2 100 1020

0.5

1

Frequency (Hz)

Nor

mal

ized

PS

D


10-2 100 1020

0.5

1

Frequency (Hz)

Nor

mal

ized

PS

D

Pacoima Dam, North. Eq.

10-2 100 1020

0.5

1

Frequency (Hz)

Nor

mal

ized

PS

D

KJMA

Recorded Ground Motion Recorded Velocity Pulse Simplified Velocity Pulse

333

Figure 6.10. Site response analyses results. Pacoima Dam record for the 1971 SanFernando Earthquake.

0 5 10-200

-100

0

100

200

Vel

ocity

(cm

/s)

Recorded Velocity Time History

0 5 10-200

-100

0

100

200

Vel

ocity

(cm

/s)

Recorded Velocity Pulse

0 5 10-200

-100

0

100

200

Vel

ocity

(cm

/s)

Time (s)

Simplified Velocity Pulse

0 5 10-200

-100

0

100

200

0 5 10-200

-100

0

100

200

0 5 10-200

-100

0

100

200

Time (s)

Output(Soil)Input (Rock)

Fault Normal Fault Parallel

334

Figure 6.11. Site response analyses results. Gilroy Gavilan College record for the 1989Loma Prieta Earthquake.


0 1 2 3 4 5-50

0

50V

eloc

ity (c

m/s

)


0 1 2 3 4 5-50

0

50

Vel

ocity

(cm

/s)


0 1 2 3 4 5-50

0

50

Vel

ocity

(cm

/s)

Time (s)


0 1 2 3 4 5-50

0

50

0 1 2 3 4 5-50

0

50

0 1 2 3 4 5-50

0

50

Time (s)


335

Figure 6.12. Site response analyses results. Pacoima Dam downstream record for the1994 Northridge Earthquake.

2 3 4 5 6 7

-50

0

50

Vel

ocity

(cm

/s)


0 1 2 3 4 5

-50

0

50

Vel

ocity

(cm

/s)


0 1 2 3 4 5

-50

0

50

Vel

ocity

(cm

/s)

Time (s)


2 3 4 5 6 7

-50

0

50

0 1 2 3 4 5

-50

0

50

0 1 2 3 4 5

-50

0

50

Time (s)



336

Figure 6.13. Site response analyses results. KJMA record for the 1995 KobeEarthquake.


0 5 10

-100

0

100

Vel

ocity

(cm

/s)


0 5 10

-100

0

100

Vel

ocity

(cm

/s)


0 5 10

-100

0

100

Vel

ocity

(cm

/s)

Time (s)


0 5 10

-100

0

100

0 5 10

-100

0

100

0 5 10

-100

0

100

Time (s)


337

Figure 6.14. Site response analyses results. Input (rock) and output (soil) particlevelocity-trace plots.

-200 0 200-200

-100

0

100

200

-200 0 200-200

-100

0

100

200

Pacoima Dam, San Fernando EQ

-50 0 50-50

0

50

-50 0 50-50

0

50Gilroy Gavilan College, Loma Prieta EQ.

-50 0 50

-50

0

50

-50 0 50

-50

0

50

Pacoima Dam Downstream, Northridghe EQ

-100 0 100

-100

-50

0

50

100


Faul

t Par

alle

l Vel

ocity

(cm

/s)

-100 0 100

-100

-50

0

50

100


KJMA, Kobe EQ

Input Output

Recorded Simplified

338

Figure 6.15. Fault-normal acceleration and velocity time-histories for the Pacoima Damrecord of the 1971 San Fernando earthquake. The dominant velocity pulse is highlightedto show that the maximum accelerations do not coincide with the fault-normal velocitypulse.

0 5 10 15-1

-0.5

0

0.5

1

Time (s)

Acc

eler

atio

n (g

)

0 5 10 15

-100

-50

0

50

100

Time (s)

Vel

ocity

(cm

/s)

339

Figure 6.16. Calculated maximum strains using the recorded velocity pulse and thesimplified velocity pulse as input motions.

0 0.1 0.2 0.3 0.4-50

-40

-30

-20

-10

0Gilroy Gavilan College, Loma Prieta EQ

Shear Strain (%)

Dep

th (m

)

0 0.5 1 1.5-50

-40

-30

-20

-10

0Pacoima Dam, San Fernando EQ

Shear Strain (%)

Dep

th (m

)

0 0.1 0.2 0.3 0.4-50

-40

-30

-20

-10

0Pacoima Dam Dwnstr., Northridge EQ

Shear Strain (%)

Dep

th (m

)

0 0.5 1 1.5-50

-40

-30

-20

-10

0KJMA, Kobe EQ

Shear Strain (%)

Dep

th (m

)

Recorded Pulse Simplified Pulse

340

Figure 6.17. Response of a linear-elastic soil column subject to the simplified velocitypulses. The natural period of the soil deposit is 1.0 s.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5Linear Elastic Soil Deposit, Tsoil = 1 s

Input Pulse Period, Tv (s)

PG

Vso

il/P

GV

rock

Amplitude of first half cycle Amplitude of second half cycleMaximum Amplitude

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

Input Pulse Period, Tv (s)

Tv,s

oil/T

v,ro

ck

Period of first half cycle Period of second half cycle Period of pulse with Maximum Amplitude

(a) Amplification of PGV.

(b) Ratio of soil pulse period to rock pulse period.

341

Figure 6.18. Amplification of input motions by a linear-elastic profile (ω is thefrequency of the input motion, Vs is the shear wave velocity of the elastic soil, and H isthe soil depth (from Kramer 1996).

0

2

4

6

8

0 1 2 3 4 5 6 7

kH

Am

plifi

catio

n Fa

ctor

Impedance ratio = 0

0.3

0.5

0

2

4

6

8

10

12

14

0 5 10 15 20

ωωωωH/Vs

Am

plifi

catio

n Fa

ctor

ξ = 5%

10%20%

(a) Amplification factors for a soil with zero damping for varyingimpedance contrast.

(b) Amplification factors for an elastic soil on rigid bedrock forvarying levels of soil damping.

ωωωωH/Vs

342

Figure 6.19. Results from the site response analyses for the 60 m deep Stiff Soil profile.Particle velocity-trace plots for the input (rock) and output (soil) motions. Input motionset is Set 6. PGVp/n of input motion is 0.5.

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 0.6 s

PG

Vro

ck =

75

cm/s

FP V

el. (

cm/s

)

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 1.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 2.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 4.0 s

-200 0 200

-200

0

200

FN Vel. (cm/s)

PG

Vro

ck =

160

cm

/sFP

Vel

. (cm

/s)

-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)

-500 0 500-500

0

500

FN Vel. (cm/s)

FP V

el. (

cm/s

)

Input (rock) Output (soil)

-500 0 500-500

0

500

FN Vel. (cm/s)-500 0 500

-500

0

500

FN Vel. (cm/s)

PG

Vro

ck =

300

cm

/s

343


-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 0.6 sP

GV

rock

= 7

5 cm

/sFP

Vel

. (cm

/s)

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 1.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 2.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 4.0 s

-200 0 200

-200

0

200

FN Vel. (cm/s)

PG

Vro

ck =

160

cm

/sFP

Vel

. (cm

/s)

-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)

-500 0 500-500

0

500

FN Vel. (cm/s)

FP V

el. (

cm/s

)


-500 0 500-500

0

500

FN Vel. (cm/s)-500 0 500

-500

0

500

FN Vel. (cm/s)

PG

Vro

ck =

300

cm

/s

344


-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 0.6 sP

GV

rock

= 7

5 cm

/sFP

Vel

. (cm

/s)

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 1.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 2.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 4.0 s

-200 0 200

-200

0

200

FN Vel. (cm/s)

PG

Vro

ck =

160

cm

/sFP

Vel

. (cm

/s)

-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)

-500 0 500-500

0

500

FN Vel. (cm/s)

FP V

el. (

cm/s

)


-500 0 500-500

0

500

FN Vel. (cm/s)-500 0 500

-500

0

500

FN Vel. (cm/s)

PG

Vro

ck =

300

cm

/s

345

(a) PGV soil to rock ratios.

(b) Ratio of pulse period in soil to pulse period in rock.

Figure 6.22. Results of site response analyses for the 60 m deep Stiff Soil site for threedifferent input motion sets. Results are plotted with fault normal input pulse period in theabscissa.

0 2 40.5

1

1.5

2

2.5

3Set 6

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

3Set 7

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

3Set 8

PGVs

oil/P

GVr

ock

Tv,FN (s)

PGVrock = 75 cm/s PGVrock = 160 cm/sPGVrock = 300 cm/s

0 2 40.5

1

1.5

2

2.5

3Set 6

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

3Set 7

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

3Set 8

Tv,s

oil/T

v,ro

ck

Tv,FN (s)


346

Figure 6.23. Site response analyses results for the 30 m deep Stiff Soil profile. The inputmotion is Set 6, with a PGVp/n ratio of 0.5. Observe how the period at which maximumamplification of PGV occurs shifts to the right with increasing input motion intensity.Also, note that the level of PGV amplification decreases slightly with increasing inputmotion intensity.

0 0.5 1 1.5 2 2.5 3 3.5 40.5

1

1.5

2

2.5

3

Tv,FN

PG

Vso

il/P

GV

rock

PGVrock = 75 cm/s PGVrock =160 cm/s PGVrock = 300 cm/s

347

Figure 6.24. Comparison of site response studies for two profiles with equal depth tobedrock (30 m) and varying profile stiffness. Input motion is Set 6 with a PGVp/n ratio of0.5.

0 1 2 3 40.5

1

1.5

2Very Stiff Soil Site

PGVs

oil/P

GVr

ock

Tv,FN (s)0 1 2 3 4

0.5

1

1.5

2Stiff Soil Site

PGVs

oil/P

GVr

ock

Tv,FN (s)


0 1 2 3 40.5

1

1.5

2Very Stiff Soil Site

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 1 2 3 4

0.5

1

1.5

2Stiff Soil Site

Tv,s

oil/T

v,ro

ck

Tv,FN (s)


(a) PGV soil to rock ratios

(b) Ratio of pulse period in soil to pulse period in rock.

348

Figure 6.25. Comparison of site response analyses results for the Stiff Soil profile. Thedepth to bedrock is varied from 30 m to 200 m. The input motion is Set 6 with a PGVp/nratio of 0.5.

0 2 40.5

1

1.5

2

2.5

330 m

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

345 m

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

360 m

PGVs

oil/P

GVr

ock

Tv,FN (s)

0 2 40.5

1

1.5

2

2.5

3100 m

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

3200 m

PGVs

oil/P

GVr

ock

Tv,FN (s)


349

Figure 6.26. Ratio of pulse period in soil to pulse period in rock for site responseanalyses on Stiff Soil profiles with varying depth to bedrock. The input motion is Set 6with a PGVp/n ratio of 0.5.

0 2 4

1

2

3

4

530 m

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 2 4

1

2

3

4

545 m

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 2 4

1

2

3

4

560 m

Tv,s

oil/T

v,ro

ck

Tv,FN (s)

0 2 4

1

2

3

4

5100 m

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 2 4

1

2

3

4

5200 m

Tv,s

oil/T

v,ro

ck

Tv,FN (s)


350

Figure 6.27. Comparison of site response analyses for soils with different shear modulusreduction and damping curves. Results for input motion Set 6 with a PGVp/n ratio of 0.5.

0 1 2 3 40.5

1

1.5

2

2.5

3PGV = 75 cm/s

PGVs

oil/P

GVr

ock

Tv,FN (s)0 1 2 3 4

0.5

1

1.5

2

2.5

3PGV = 160 cm/s

PGVs

oil/P

GVr

ock

Tv,FN (s)Generic Low PI soilPI = 15 Clay PI = 30 Clay

351

Figure 6.28. Comparison of calculated ratio of soil pulse period to rock pulse period.Results for site response analyses for soils with different shear modulus reduction anddamping curves. Input motion is Set 6, with PGVp/n ratios of 0.5.

0 2 40.5

1

1.5

2

2.5

3Generic Low-plasticity soil

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

3PI = 15 Clay

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5

3PI = 30 Clay

Tv,s

oil/T

v,ro

ck

Tv,FN (s)


352

Figure 6.29. Comparison of site response analyses results to input motions with varyinglevels of fault parallel velocity. Results for a 45 m deep Stiff Soil for PGVp/n ratios of0, 0.25 and 0.75. Input fault-normal PGV is 75 cm/s.

0 1 2 3 40.5

1

1.5

2Input Motion Set 7

PGVs

oil/P

GVr

ock

Tv,FN (s)0 1 2 3 4

0.5

1

1.5

2Input Motion Set 8

PGVs

oil/P

GVr

ock

Tv,FN (s)PGVp/n = 0 PGVp/n = 0.25PGVp/n = 0.75

353

Figure 6.30. Comparison of site response analyses to input motions with different fault-parallel velocities. Results for the 45 m deep Stiff Soil profile, Input motion Set 6.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.5

1

1.5

2

2.5

3Tv

,soi

l/Tv,

rock

Tv,FN (s)

PGVp/n = 0 PGVp/n = .25PGVp/n = .75

354

Figure 6.31. Comparison of site response analyses results for the different input motionsets. The analyses are for the Stiff Soil profiles with varying depth. The input PGVp/nratio is 0.25 for Sets 7 and 8, and 0.5 for Set 6.

0 2 40.5

1

1.5

2

2.530 m

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.545 m

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.560 m

PGVs

oil/P

GVr

ock

Tv,FN (s)

0 2 40.5

1

1.5

2

2.5100 m

PGVs

oil/P

GVr

ock

Tv,FN (s)0 2 4

0.5

1

1.5

2

2.5200 m

PGVs

oil/P

GVr

ock

Tv,FN (s)

Set 6Set 7Set 8

355

Figure 6.32. Comparison of strain levels for runs with (a) equal PGV, and (b) equalPGA. Stiff Soil profile with a depth of 45 m. Input motion is Set 6 with PGVp/n = 0.5.

0 1 2-50

-40

-30

-20

-10

0

Strain (%)

Dep

th (m

)

Tv = .6 s; PGA = 0.8 g Tv = 1.0 s; PGA = 0.48 gTv = 2.0 s; PGA = 0.24 gTv = 4.0 s; PGA = 0.12 g

0 5 10-50

-40

-30

-20

-10

0

Strain (%)

Dep

th (m

)

Tv = 1 s; PGV = 75 cm/s Tv = 2.0 s; PGV = 160 cm/sTv = 4.0 s; PGV = 300 cm/s

(a) PGV = 75 cm/s

(b) PGA = 0.5 g

356

Figure 6.33. Calculated strains for various Stiff Soil profiles. Input motion is Set 6 withPGVp/n ratios of 0.5, and input PGV of 160 cm/s.

0 2 4 6-200

-150

-100

-50

0

Max. Strain (%)

Dep

th (m

)

0 2 4 6-200

-150

-100

-50

0

Max. Strain (%)D

epth

(m)

0 2 4 6-200

-150

-100

-50

0

Max. Strain (%)

Dep

th (m

)

0 2 4 6-200

-150

-100

-50

0

Max. Strain (%)

Dep

th (m

)

Very Stiff Soil (30 m)Stiff Soil (30 m) Stiff Soil (60 m) Stiff Soil (100 m) Stiff Soil (200 m)

Tv = 0.6 s

Tv

Tv

Tv

357

Figure 6.34. Results of site response analyses. Particle velocity-trace plots of input andoutput motions. Soft Clay site, input motion is Set 6 with PGVp/n ratios of 0.5.

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 0.6 s

PG

Vro

ck =

75

cm/s

FP V

el. (

cm/s

)

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 1.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 2.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 4.0 s

-200 0 200-200

0

200

FN Vel. (cm/s)

PG

Vro

ck =

160

cm

/sFP

Vel

. (cm

/s)

-200 0 200-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)

-400-200 0 200 400-400

-200

0

200

400

FN Vel. (cm/s)

FP V

el. (

cm/s

)


-400-200 0 200 400-400

-200

0

200

400

FN Vel. (cm/s)-400-200 0 200 400

-400

-200

0

200

400

FN Vel. (cm/s)

PG

Vro

ck =

300

cm

/s

358


-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 0.6 s

PG

Vro

ck =

75

cm/s

FP V

el. (

cm/s

)

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 1.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 2.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 4.0 s

-200 0 200-200

0

200

FN Vel. (cm/s)

PG

Vro

ck =

160

cm

/sFP

Vel

. (cm

/s)

-200 0 200-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)

-400-200 0 200 400-400

-200

0

200

400

FN Vel. (cm/s)

FP V

el. (

cm/s

)


-400-200 0 200 400-400

-200

0

200

400

FN Vel. (cm/s)-400-200 0 200 400

-400

-200

0

200

400

FN Vel. (cm/s)

PG

Vro

ck =

300

cm

/s

359


-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 0.6 sP

GV

rock

= 7

5 cm

/sFP

Vel

. (cm

/s)

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 1.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 2.0 s

-100 0 100

-100

0

100

FN Vel. (cm/s)

Tv,rock = 4.0 s

-200 0 200-200

0

200

FN Vel. (cm/s)

PG

Vro

ck =

160

cm

/sFP

Vel

. (cm

/s)

-200 0 200-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)-200 0 200

-200

0

200

FN Vel. (cm/s)

-400-200 0 200 400-400

-200

0

200

400

FN Vel. (cm/s)

FP V

el. (

cm/s

)


-400-200 0 200 400-400

-200

0

200

400

FN Vel. (cm/s)-400-200 0 200 400

-400

-200

0

200

400

FN Vel. (cm/s)

PG

Vro

ck =

300

cm

/s

360

Figure 6.37. Calculated stresses and strains for the Soft Clay profile. Input motion is Set6 with PGVp/n = 0.5.

0 10 20 30-70

-60

-50

-40

-30

-20

-10

0

Max. Strain (%)

Dep

th (m

)

Tv = 0.75 sTv = 1.0 s Tv = 4.0 s

0 500 1000-70

-60

-50

-40

-30

-20

-10

0

Max. Stress (kPa)D

epth

(m)

Tv = 0.75 s Tv = 1.0 s Tv = 4.0 s Strength Profile

0 10 20 30-70

-60

-50

-40

-30

-20

-10

0

Max. Strain (%)

Dep

th (m

)

Tv = 1.0 sTv = 2.0 sTv = 4.0 s

0 500 1000-70

-60

-50

-40

-30

-20

-10

0

Max. Stress (kPa)

Dep

th (m

)

Tv = 1.0 s Tv = 2.0 s Tv = 4.0 s Strength Profile

(a) PGV of input motion = 160 cm/s

(b) PGV of input motion = 300 cm/s

361

0 200 400-80

-60

-40

-20

0

Displacement (cm)

Dep

th (m

)

0 200 400-80

-60

-40

-20

0

Velocity (cm/s)0 1 2

-80

-60

-40

-20

0

Acceleration (g)

Tv = .6 Tv = 2.0Tv = 4.0

Figure 6.38. Calculated profiles of displacement, velocity, and acceleration for the SoftClay site. Input motion is Set 6 with PGVp/n of 0.5 and input PGV of 160 cm/s.

362

Figure 6.39. Comparison of results of site response analyses for two profiles with equaldepth to bedrock but different soil profiles. Ratios of PGV in soil to PGV in rock. Inputmotion is Set 6 with PGVp/n of 0.5.

0 1 2 3 40.5

1

1.5

2Stiff Soil Site, 60 m depth

PGVs

oil/P

GVr

ock

Tv,FN (s)0 1 2 3 4

0.5

1

1.5

2Soft Clay Site

PGVs

oil/P

GVr

ock

Tv,FN (s)PGVrock = 75 cm/sPGVrock = 160 cm/sPGVrock = 300 cm/s

363

Figure 6.40. Comparison of site response analyses for the Soft Clay profile and the StiffSoil profile with equal depth (60 m). Results shown are for all input motion sets withPGVp/n values ranging from 0.25 to 0.5.

0 100 200 300 400 5000

100

200

300

400

500Stiff Soil

PGVs

oil (

cm/s

)

PGVrock (cm/s)

0 100 200 300 400 5000

100

200

300

400

500

Soft Clay Site

PGVs

oil (

cm/s

)

PGVrock(cm/s)

Tv = 0.6Tv = 1.0Tv = 2.0Tv = 4.0

1

1

1

1

2

1

1

2

364

Figure 6.41. Comparison of results of site response analyses for two profiles with equaldepth to bedrock but different soil profiles. Ratios of pulse period in soil to pulse periodin rock. Input motion is Set 6 with PGVp/n of 0.5.

0 1 2 3 4

1

2

3

4Stiff Soil Site, 60 m depth

Tv,s

oil/T

v,ro

ck

Tv,FN (s)0 1 2 3 4

1

2

3

4Soft Clay Site

Tv,s

oil/T

v,ro

ck

Tv,FN (s)PGVrock = 75 cm/s PGVrock = 160 cm/sPGVrock = 300 cm/s

365

Figure 6.42. Effect of site response on the PGVp/n ratio. Results for three differentprofiles and for input motion sets 7 and 8. PGV of input motion is 75 cm/s.

0 2 40

0.5

1

1.5PG

Vp/n

Very Stif f Soil

0 2 40

0.5

1

1.5Stiff Soil (45 m depth)

0 2 40

0.5

1

1.5Soft Soil

0 2 40

0.5

1

1.5

PGVp

/n

Tv,FN (s)0 2 4

0

0.5

1

1.5

Tv,FN (s)0 2 4

0

0.5

1

1.5

Tv,FN (s)

Input PGVp/n = 0.25Input PGVp/n = 0.75

Se

t 7

Set 8

366

Figure 6.43. Calculated amplification of peak ground velocity. Results for site responseanalyses for all the Stiff Clay profiles and all the simplified velocity poulse input motionsets. Large symbols are the predicted PGV amplification for the Stiff Soil site with adepth of 45 m using the indicated records as input motions.

0 100 200 300 400 500 6000

100

200

300

400

500

600PG

Vsoi

l (cm

/s)

PGVrock (cm/s)

Tv = 0.6 sTv = 1.0 s

Tv = 2.0 sTv = 4.0 s

11

21

×××× Pacoima Dam, SF EQ.

○ Gilroy Gav. Coll. LP EQ.

□ Pacoima Dam Dwnst.Northridge EQ.

∆ KJMA, Kobe EQ.

Actual Records

Pulse Motions

367

Figure 6.44. Relationship between PGV in soil and PGV in rock. Results for the 45 mdeep Stiff Soil deposit. Input motion is Set 6 with PGVp/n ratio of 0.5. Observe thedependence of PGV soil to rock ratio on pulse period (Tvp) and intensity of motion(PGVrock).

0

100

200

300

400

500

0 100 200 300 400 500

PGVrock (cm/s)

PGVs

oil (

cm/s

)

Tvp = 0.6 sTvp = 1.0 sTvp = 2.0 sTvp = 4.0 s

1

11

1.5

368

Figure 6.45. Calculated ratios of pulse period in the soil over pulse period in rock.Results from site response analyses on Stiff Clay profiles using all simplified velocitypulse input motions sets and PGVp/n ratios ranging from 0.25 to 0.75. The heavy linerepresents the ratio of the average pulse period in soil over the average pulse period inrock from the regression analysis of the near-fault ground motion database.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5Tv

,soi

l/Tv,

rock

Tv,FN (s)


Regression line forrecorded near-faultground motions

369

Figure 6.46. Calculated ratio of pulse period in soil to pulse period in rock for the 45 mdeep Stiff Soil profile. Results are shown for all the input motion sets with input PGVlower than 300 m/s. Results for the indicated ground motions are included forcomparison with the results of the simplified velocity pulses.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5Tv

,soi

l/Tv,

rock

Tv,FN (s)

Stiff Clay Sites

PGVrock = 75 cm/sPGVrock = 160 cm/s ×××× Pacoima Dam, SF EQ.

(PGVrock = 114 cm/s )

○ Gilroy Gav. Coll. LP EQ.(PGVrock = 31 cm/s)

□ Pac. Dam D.. North.EQ. (PGVrock = 50cm/s )

∆ KJMA, Kobe EQ.(PGV 96 / )

Actual RecordsPulse

370

CHAPTER 7

CONCLUSION

7.1 SUMMARY

This dissertation presented an evaluation of site effects on seismic ground

motions, with a particular emphasis on near-fault, forward-directivity ground motions.

The contributions of this work include the following:

1) The database of ground motions from the 1989 Loma Prieta and the 1994

Northridge earthquakes was used to evaluate a proposed new site

classification system to account for site amplification. The proposed site

classification system is based on a general characterization of the site that

includes depth to bedrock. This site classification system was used to evaluate

the influence of depth to bedrock and stiffness on site amplification. Site

amplification factors for different site conditions were proposed.

2) The database of near-fault ground motion records containing forward-

directivity effects was evaluated. Recorded near-fault forward-directivity

motions were represented by simplified velocity time-histories consisting of a

sequence of half-cycle sine pulses. The database was used to develop

attenuation relationships for the period and amplitude of the pulse. The

effects of local site conditions on these parameters were evaluated and

371

relationships between pulse period on rock and pulse period on soil were

proposed.

3) A nonlinear site response methodology using a bounding surface plasticity

model developed by Borja and Amies (1994) was implemented in the finite

element program GeoFEAP. The numerical simulations can represent the

two-dimensional response of a soil column to bi-directional motions (i.e.

fault-normal and fault-parallel components). The implementation was

evaluated with an emphasis on its ability to predict long-period site response.

The long-period site response was emphasized because near-fault ground

motions are characterized by long-period pulses of motion.

4) Simplified velocity time-histories representative of near-fault, forward-

directivity ground motions were developed to be used as input motions in site

response analyses. The time histories are developed both for the fault-normal

and fault-parallel components of motion.

5) Site response analyses using pulse-like simplified velocity time-histories as

input motions were performed. The ability of the simplified time-histories to

capture amplification of velocities and site effects on the input pulse motions

was validated by comparing results to site response analyses of recorded

ground motions. Site response analyses were performed on generalized site

profiles representing average site conditions for different site categories.

Conclusions regarding the influence of site conditions on the input velocity

pulses were presented.

372

7.2 FINDINGS

7.2.1 General

The primary findings in this dissertation are presented in this section.

Observations regarding the characterization of site response using the proposed

classification system are first presented. The results from the empirical analyses of near-

fault ground motions are then discussed. Finally, conclusions regarding site response to

near-fault ground motions are summarized.

7.2.2 Characterization of site response

A site classification scheme based on a general geotechnical characterization of

the site and the depth to bedrock was introduced. This site classification scheme was

evaluated with data from the 1989 Loma Prieta and 1994 Northridge earthquakes. The

main findings can be summarized as follows:

• The use of the proposed site classification scheme in ground motion estimation

results in a reduction in the standard deviation associated with median

estimates of ground motion when compared with a simpler "rock vs. soil"

classification system.

• Simplified classification systems commonly used in attenuation relationships

divide sites into "rock" and "soil" categories. The "rock" category groups

competent rock sites and weathered soft rock/shallow stiff soil sites. Results

showed that the response of these two site conditions is significantly different.

For an improved site categorization system for defining site-dependent ground

373

motions, the "rock" category should be divided along the indicated sub-

categories.

• Current attenuation relationships use the generic "rock" category as the baseline

site condition. The "rock" sites in the current database of strong ground

motions are dominated by weathered soft rock/shallow stiff soil sites; hence,

rock attenuation relationships likely reflect the response of these types of soils.

This, in turn, emphasizes the need to review the database of ground motions to

redefine the baseline site condition. Competent rock motions have spectral

acceleration values (at 5% damping) that are on average 30% less than those

predicted by widely used "rock" attenuation relationships.

• The standard deviations resulting from the proposed classification system are

comparable with the standard deviations obtained using a more burdensome

average shear wave velocity classification system. This illustrates that an

estimate of profile depth can compensate for a less exhaustive estimate of soil

stiffness. Profile depth is an important parameter for the estimation of seismic

site response.

7.2.3 Empirical study of near-fault, forward-directivity ground motions

Near-fault ground recordings that satisfied geometric conditions for forward-

directivity were selected. These recordings are characterized by long-period pulses of

motion that can be best observed in velocity or displacement time-histories. The

recordings in the database were characterized by a pulse period and their peak ground

velocity (PGV). The database was used for a statistical analysis of these parameters.

374

Regression analyses for pulse period as a function of magnitude and pulse amplitude as a

function of magnitude and distance were presented. The findings of the empirical study

can be summarized as follows:

• Near-fault, forward-directivity motions can be adequately represented by

simplified time-histories consisting of one or a few sine-pulses. The number of

pulses is likely related to slip distributions in the causative fault, and

consequently is difficult to predict.

• Pulse period is a function of moment magnitude (Mw). For earthquakes with a

magnitude of 6.1, the pulse period is about 0.6 s; and increases to about 6.0 s

for a magnitude of 7.5. Relatively high standard deviations are associated with

the median prediction of pulse period.

• Most of the energy in forward-directivity ground motions is concentrated in the

narrow-period band centered on the pulse period. Consequently, low

magnitude events might result in more damaging ground motions for typical

low period structures. Hence, care must be exercised when seismic demand is

estimated using deterministic analyses for Maximum Credible Earthquakes.

• Local site conditions have an important effect on pulse period. Longer periods

occur at soil sites than at rock sites for events with magnitudes lower than

about Mw = 7.0. The difference diminishes as the magnitude increases. For

events with Mw ≈ 7.5, the pulse periods at rock and soil sites are approximately

the same.

375

• PGVs in the near-fault region vary significantly with magnitude and distance.

Estimates of these velocities have large standard deviations. Median PGVs for

soils are larger than median PGVs for rock sites for low magnitude events. The

difference diminishes as the magnitude increases.

7.2.4 Site response analyses to near-fault ground motions

The finite element implementation of the bounding surface plasticity model by

Borja and Amies (1994) was evaluated with recordings at the Lotung array in Taiwan,

and at the Chiba array in Japan. The implementation reproduced accurately the surface

accelerations and velocities recorded in these arrays. In addition, a series of shaking table

tests performed at U.C. Berkeley (Meymand 1998, Lok 1999) using a near-fault recording

as the input motion were also evaluated. The implementation was able to represent well

the long-period site response and the PGVs for these experiments for PGVs as large as

190 cm/s in field scale (model scale values of PGV of up to 70 cm/s).

Site response analyses were performed on recorded near-fault ground motions and

on a series of simplified bi-directional velocity pulses developed from a statistical

analysis of the database of near-fault, forward-directivity ground motions. Generic soil

profiles used to represent different site categories were used in the analyses. The main

findings can be summarized as follows:

• In most cases, site response to simplified representation of recorded near-fault,

forward-directivity ground motions rendered results equivalent to those from

site response analyses to the corresponding recorded ground motions.

376

However, not all near-fault motions can be represented by simplified sine-pulse

velocity time-histories.

• PGV amplification is a function of both the period and the intensity of the input

pulse. The period for which maximum amplification of PGV occurs can be

associated with the degraded natural period of the site.

• The degraded natural period of a site increases with increasing profile depth, it

decreases with increasing profile stiffness, and it increases with increasing

input motion intensity. Consequently, sites with long degraded site periods

(i.e. Stiff Soil sites deeper than 60 m, Soft Clay sites, or Shallow Stiff Clay

sites subject to intense shaking) tend to amplify input pulses with periods

longer than 2.0 s. Conversely, sites with short degraded site periods (i.e.

Shallow, Very Stiff Soil sites, or sites subject to low intensity input motions)

tend to amplify input pulses with short periods (≤ 1 s) and are not significantly

affected by long pulse periods.

• The nonlinear stress-strain properties of the soil have an important effect on

PGV amplification. For a given strain level, soils that exhibit lesser shear

modulus reduction tend to amplify motions with shorter input pulse periods

more than soils with relatively more shear modulus reduction at the given strain

level.

• For near-fault ground motions, the response of soft clays is controlled by its

dynamic shear strength (i.e. soil model yield stress). For the cases studied in

this work, the soft clay reached its dynamic shear strength for input velocity

377

pulse amplitudes as low as 75 cm/s. This value is well within the expected

range of velocities for near-fault ground motions. Hence, the evaluation of the

shear strength of soft clays is important for the proper evaluation of site

response to near-fault ground motions.

• Results indicated that PGVs in stiff soils are amplified by a factor ranging from

one to two. The amplification is observed for input PGVs as large as 300 cm/s

when the input pulse period is longer than 1 s. For the cases considered in this

study, soft soils amplify PGVs when the input pulse period has amplitudes

lower than about 150 cm/s. For higher intensities of the input motion, PGVs at

the surface of the Soft Soil profile used in this study decrease with respect to

the input PGV.

• The mid-period amplification factor for Site D in the 1997 UBC (180 m/s < sV

≤ 360 m/s) in Zone 4 is 1.6. This factor agrees with the range of Stiff Soil

PGV amplification obtained in the site response analyses for input pulse

amplitudes lower than 300 m/s. Mid-period amplification factors can be

compared to velocity amplification factors because mid-period spectral values

are generally controlled by velocities. The amplification for Site E in the UBC

( sV ≤ 180 m/s) in Zone 4 is 2.4. This factor is larger than the amplification

factors obtained from the site response analyses for the Soft Clay site (Figure

6.40). The large code factors are likely due to extrapolations from equivalent

linear analyses that ignore the effect of soil failure.

378

• Site effects change the period of the velocity pulse. Typically, the calculated

pulse period is longer than the period of the input velocity pulse. An increase

in pulse period occurs when the input pulse period is shorter than the degraded

natural period of the site. In these cases, the soil column tends to oscillate at

the natural site period; hence, a significant increase in velocity pulse period is

observed.

• Input pulse periods that are significantly longer than the degraded site period

have much longer wavelengths than the profile depth. Consequently, a soil

column subject to these input pulses moves more or less as a rigid body and the

period and amplitude of the input pulse is not affected by the soil column.

• The median value of the calculated ratios of soil to rock pulse period for all the

cases considered in this study agree with the ratios estimated from the

regression analysis of the ground motion database of near-fault, forward

directivity ground motions.

7.3 RECOMENDATIONS FOR FURTHER RESEARCH

While the work in this thesis has increased the understanding of seismic site

response, it also identified a number of issues that warrant further investigation:

• Inclusion of additional earthquakes to the database of ground motions used in

this study for the evaluation of site response. The extended database should be

used in the development of updated attenuation relationships that use the

geotechnical site categories presented in this study.

379

• The data from the recent 1999 Chi-Chi, Taiwan, earthquake contains a large

number of near-fault recordings. Properly processed records from this and

other recent events should be included in the database of near-fault ground

motions for further analyses.

• Seismological fault modeling is gaining acceptance as an alternative method

for the prediction of ground motions, particularly for the large-and short

distance range where there are not a large number of recorded ground motions.

The important conclusions regarding site effects on near-fault ground motions

presented in this study should be included in the seismological modeling of

these types of motions.

• Soil constitutive models able to match simultaneously shear modulus

degradation and equivalent viscous damping curves are necessary for the

proper evaluation of site response. For near-fault ground motions, it is

important that these models represent the large strain behavior of soils while

also satisfying the extended Masing rules (Pyke 1979) for irregular cyclic

loadings. Models such as the model by Pestana and Lok (Lok 1999) should be

implemented and used for the evaluation of near-fault site response.

• The work in this dissertation did not include response of liquefiable soils. Soil

liquefaction, however, is a considerable problem and issues relating to the

particular characteristics of near-fault ground motions, such as the influence of

the reduced number of cycles on liquefaction potential, are not well

understood. Both empirical and analytical studies using effective stress soil

380

models and coupled mechanical-pore-water dissipation analysis should be

performed to address these issues.

• Near-fault, forward-directivity ground motions typically have long-period

energy. Basin effects affect ground motions in the same period-range. Three-

dimensional models including soil non-linearity should be used to evaluate the

effect of basin topography on long-period pulses.

• The relationship between earthquake magnitude and pulse period, together with

the narrow-banded nature of near-fault ground motions, imply that for certain

period bands, larger intensities could be expected from earthquakes of lower

magnitudes. The effects of this observation on building codes and on the use

of Maximum Credible Earthquakes for design should be evaluated.

381

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Taylor, P. W. and Larkin, T. J. (1978). "Seismic response of nonlinear soil media."Journal of the Geotechnical Engineering Division, ASCE, 104(GT3) 369-183.

Taylor, R. L. (1998). FEAP: A Finite Element Analysis Program. Department of Civiland Environmental Engineering, University of California at Berkeley, Berkeley,CA.

Taylor, R.L. (1998) “FEAP – A Finite Element Analysis Program,” Department of CivilEngineering, University of California, Berkeley.

Thiel, C. C., and Schneider, J. F. (1993). "Investigation of Thirty-three Loma PrietaEarthquake Strong Motion Recording Sites." California Universities for Researchin Strong Motion Seismograph Sites in California.

396

Thomson, W. T. (1950). "Transmission of elastic waves through a stratified solidmedium." Journal of Applied Physics.. 21.

Toro, G. R., Abrahamson, N. A., and Schneider, J. F. (1997). "Model of strong groundmotions from earthquakes in Central and Eastern North America: Best Estimatesand Uncertainties." Seismological Research Letters, Vol. 68(1), 41-57.

Trifunac, M. D., and Todorovska, M. I. (1996). Nonlinear Soil Response - 1994Northridge, California, Earthquake." Journal of Geotechnical Engineering, ASCE,122 (9), 725-735.

Vucetic, M. and Doroudian, M. (1995). "The Characteristics of Earthquake GroundMotions for Seismic Design. Task H-5: Geotechnical Site Database for SouthernCalifornia Digitized Boring Logs." Southern California Earthquake Center,University of Southern California, Los Angeles, California.

Vucetic, M., and Dobry, R. (1991). "Effect of Soil Plasticity on Cyclic Response."Journal of Geotechnical Engineering, ASCE, Vol. 117(1).

Wang, Z. L., Dafalias, Y. F. and Shen, C. K. (1990). "Bounding surface hypo-plasticitymodel for sands." Journal of Engineering Mechanics, ASCE, 116(5) 993-1001.

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Youngs, R. R. (1993). "Soil amplification and vertical and horizontal ratios for analysisof strong motion data from active tectonic regions, Appendix 2C in Guidelinesfor Determining Design Basis Ground Motions (see EPRI 1993).

Youngs, R. R., Chiou, S. –J., Silva, W. J., Humphrey, J. R. (1997). "Strong groundmotion attenuation relationships for subduction zone earthquakes." SeismologicalResearch Letters, Vol. 68(1), 58-73.

Zeghal, M., Elgamal, A. W., Tang, H. T., and Stepp, J. C. (1995). "Lotung downholearray. II: Evaluation of soil nonlinear properties." Journal of GeotechnicalEngineering, ASCE, 121(4), 363-378.

397

Zienkiewicz, O. C. and Taylor, R. L. (1989). The Finite Element Method. Volumes I andII. McGraw-Hill Book Company, London.

Zienkiewicz, O. C., and R. E. Newton. "Coupled vibrations of a structure submerged in acompressible fluid" Proceedings International Symposium on Finite ElementTechniques, Stuttgart 1-15, May 1969.

Zienkiewicz, O. C., Chan, A. H. C., Pastor, M., Schrefler, B. A., Shiomi, T. (1999).Computational Geomechanics with Special Reference to Earthquake Engineering.John Wiley and Sons, New York, NY.

Zienkiewicz, O. C., Chang, C. T., and Hinton, E. (1978). "Nonlinear seismic responseand liquefaction." International Journal of Numerical Analysis Methods inGeomechanics, 2, 381-404.

398

APPENDIX A

List of Ground Motion Sites with Corresponding Site Classification

399

Table A-1. Ground motion stations showing site classification, Northridge Earthquake.

Station Name Agency Sta. # SurfaceGeology

Seed &Dickenson(Table A-4)

UBC 97(Table A-3)

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(m)IR(2) Conrey's

depth(3)

(ft)

Source (5)

Alhambra - Fremont School CDMG 24461 Holocene B1? C C3? Fair >9 0? SCEC

Anacapa Island # CDMG 25169 Miocene AB2? D? C2? Poor - - - Geol.

Anaheim - W Ball Rd USC 90088 Holocene C3? D D1S? Fair >23.5 - 7000 SCEC

Anaverde Valley - City R # CDMG 24576 Holocene AB2? C? C?2 Poor - - - Geol.

Antelope Buttes # CDMG 24310 Jurassic-Cretaceous

AB1 B C?1 Poor - - - Geol. ,T.

Arcadia - Arcadia Av USC 90099 Holocene C3? D D?1S? Fair >40 - - SCEC

Arcadia - Campus Dr. USC 90093 Holocene C3? D D?1S? Fair 19 1.8 - SCEC

Arleta - Nordhoff Fire Sta # CDMG 24087 Holocene C3 D D1S Good >150 - - SCEC,ROSRINE

Baldwin Park - N. Holly Ave USC 90069 Holocene AB2 C C2 Good 21.5 5.5 - SCEC

Bell Gardens - Jaboneria USC 90094 Holocene C3 D D1S? Fair >29 - 6000 SCEC

Beverly Hills - 12520 Mulhol USC 90014 Miocene AB1 C C1 Fair 22(w?) 3.3 - SCEC

Beverly Hills - 14145 Mulhol USC 90013 Miocene AB? D C1 Fair 24(w?) 4.0 - SCEC

Big Tujunga, Angeles Nat F USC 90061 Mesozoic AB2 C C1 Fair 21.5(w) 3.3 - SCEC

Brea - S. Flower Ave. USC 90087 Pleistocene B2? C2? D? D?2C Fair >35 - - SCEC

Brentwood V.A. Hospital USGS 638 Pleistocene C2? D? D?2S? Poor >30? - - SCEC, Geol.

400



UBC 97(Table A-3)

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(m)IR(2) Conrey's

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(ft)

Source (5)



UBC 97(Table A-3)

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(m)IR(2) Conrey's

depth(3)

(ft)

Source (5)

Buena Park - La Palma USC 90086 Holocene C3? D D1S (F?) Poor >30 - 7000 SCEC

Burbank - Howard Rd. USC 90059 Cretaceous+

A1 B C?1 Good 6 - 9 (w) 3.0 - SCEC

Camarillo CDMG 25282 Holocene C2 D D1C? Fair >30 - - SCEC

Canoga Park - Topanga Can USC 90053 Holocene C2 D C3 Good 50 (to softrock)

2.2 - SCEC, G.e.a.

Canyon Country - W LostCany

USC 90057 Holocene B1? C3?AB1?

D? D?1S Poor >24 - - SCEC

Carson - Catskill Ave USC 90040 Pleistocene C3 D D2S Fair >22 - 3000 SCEC

Carson - Water St. USC 90081 Holocene F F F Fair 220 4.0 3000 SCEC,ROSRINE

Castaic - Old Ridge Route # CDMG 24278 Miocene AB2 C C?1 Fair 8 (w) - - USGS, D.e.a.,CSMIP

Compton - Castlegate St USC 90078 Holocene F F F Poor - - 5000 SCEC

Covina - S. Grand Ave. USC 90068 Pleistocene B D C3 Fair 25.5 3.0 - SCEC

Covina - W. Badillo USC 90070 Holocene B D C3 Fair 23 2.2 - SCEC

Downey - Birchdale USC 90079 Holocene C3 D D1S Good 120 3.0 6000 SCEC

Downey - Co Maint Bldg # CDMG 14368 Holocene B1? C2?C3?

D D1S? Fair >120? - 9000 SCEC, CSMIP,G.

Duarte - Mel Canyon Rd. USC 90067 Holocene? AB2 (F?) C (F?) C2? Good 17 8.0 - SCEC

401



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(m)IR(2) Conrey's

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(ft)

Source (5)

El Monte - Fairview Av USC 90066 Holocene F F F Fair 31 2.8 - SCEC

Elizabeth Lake # CDMG 24575 Holocene AB2? D? C? C?2 Poor - - - Geol.

Featherly Park - Pk MaintBldg #

CDMG 13122 Holocene B1? C3? D? D?1S? Poor - - - G.

Garden Grove - Santa Rita USC 90085 Holocene C2 D D?1C Fair >30 - 8000 SCEC

Glendale - Las Palmas USC 90063 Pleistocene AB2 D C3?? Fair 33 2.6 - SCEC

Glendora - N. Oakbank USC 90065 Holocene B C C3? Fair 41 2.0 - SCEC

Hacienda Hts - Colima Rd USC 90073 Holocene B2 D C3?? Fair 34? 2.0 - SCEC

Hemet - Ryan Airfield # CDMG 13660 Holocene C2? C3? D? D1C? Poor - - - Geol.

Hollywood - Willoughby Ave USC 90018 Holocene C2 D D2C Good 100 - 1000 SCEC

Huntington Bch - Waikiki USC 90083 Holocene C2 D D1C? Check - - 4000 Geol.

Huntington Beach - Lake St#

CDMG 13197 Holocene-Pleistocene

C2? D? D2S? Poor - - 3000 G.

Inglewood - Union Oil # CDMG 14196 Pleistocene C3 D? D2S? Poor - - 4000 G.

Jensen Filter Plant # USGS 655 Holocene F F F Good >93 - - G.e.a., Stew.

LA - 116th St School # CDMG 14403 Pleistocene C2 C? D2C Poor 152 - - SCEC, D&L,G

LA - Baldwin Hills # CDMG 24157 Pleistocene C2 D D2C? Good >85 - 2000 SCEC,G,D&L,ROSRINE

LA - Centinela St USC 90054 Pleistocene AB2 D C3 Fair 22 3.6 5000 SCEC

402



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LA - Century City CC North#

CDMG 24389 Pleistocene C2 D? D2C? Fair >120 2.6 - S&W,G

LA - Chalon Rd USC 90015 Jurassic? AB1 C C1 Good 15 (w) >2.0 - SCEC

LA - City Terrace # CDMG 24592 Pliocene AB1? A1? B? C? C?1 Fair <10? - 0 CSMIP,SCEC,S&S

LA - Cypress Ave USC 90033 Holocene AB2 C C2 Fair 21 3.5 - SCEC

LA - E Vernon Ave USC 90025 Holocene C2?C3? D D1C? Fair >30 - 3000 SCEC

LA - Fletcher Dr USC 90034 Holocene C3?B1? D D?1S Fair - - - SCEC

LA - Hollywood Stor FF # CDMG 24303 Holocene C2 D D1C Good 103.6 2.8 1000 USGS, Chang

LA - N Faring Rd USC 90016 Jurassic? AB1 D C1 Good 20 (w)

LA - N Westmoreland USC 90021 Holocene AB2 C C2 Good 22 3.8 - SCEC

LA - N. Figueroa St. USC 90032 Holocene AB2 C C2 Fair 24 2.0 - SCEC

LA - Obregon Park # CDMG 24400 Holocene F? F? F? Poor - <1000 Geol., SCEC

LA - Pico & Sentous # CDMG 24612 Holocene B1? D D1S Fair 120 - <1000 SCEC, D&L

LA - S Grand Ave USC 90022 Holocene C3 D D?1S Good >24.5? - 3000 SCEC

LA - S. Vermont Ave USC 90096 Holocene C2 D D1C Good >150 - 2000 SCEC, S&S

LA - Saturn St USC 90091 Holocene F F F Good - - 2000 SCEC

LA - Temple & Hope # CDMG 24611 Miocene AB1 D? C1 Good 64 (w) 2.4 0 CSMIP,D&L,S&S

403



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depth(3)

(ft)

Source (5)

LA - UCLA Grounds CDMG 24688 Pleistocene AB2 C C2 Good 20 (w) 2.0 - S&S

LA - Univ. Hospital # CDMG 24605 Miocene AB1 D? C1 Good 20? - - S&S

LA - W 15th St USC 90020 Pleistocene C3 D D2S Fair - - 1000 SCEC

LA - Wonderland Ave USC 90017 Cretaceous A1 B B Good 4 (w) 2.7 - SCEC

La Crescenta - New York USC 90060 Pleistocene AB2 D C3 Good 30 2.0 - SCEC

LA Dam USGS 0 Pliocene AB1 C C1 Good 0 - - G.e.a.

La Habra - Briarcliff USC 90074 Pleistocene C2?C3? D D2C? Fair - - 2000 SCEC

La Puente - 504 RimgroveAve

USC 90072 Holocene B D D?1S? Fair - - - SCEC

Lake Hughes #1 # CDMG 24271 Holocene C2 C D2C Good 260 - - G., USGS, S&W

Lake Hughes #12A # CDMG 24607 Paleocene A1 C C?1 Good 10 5 - USGS, S&W, G.

Lake Hughes #4 - CampMend #

CDMG 24469 Mesozoic A1 B B Good 5 (w) - - G., S&W

Lake Hughes #4B - CampMend #

CDMG 24523 Mesozoic A2 B B Good 6 (w) - - G., S&W

Lake Hughes #9 # USGS 127 Precambrian

A1 B C?1 Good 8 (w) - - USGS, S&W, G.

Lakewood - Del Amo Blvd USC 90084 Holocene C3? D D1S? Fair >25 - 8000 SCEC

Lancaster - Fox AirfieldGrnds

CDMG 24475 Holocene C3 D D1S Good - - - S&S

Lawndale - Osage Ave USC 90045 Pleistocene C2? D D2C? Poor - - 3--- SCEC

404



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depth(3)

(ft)

Source (5)

LB - City Hall # CDMG 14560 Pleistocene C3 D? D2S? Fair - - 3000 SCEC, Geol.

LB - Rancho Los Cerritos # CDMG 14242 Pleistocene C3? D? D2S? Poor - - 5000 SCEC

Leona Valley #1 # CDMG 24305 Pleistocene A1? B? B Fair - - - Geol., CSMIP

Leona Valley #2 # CDMG 24306 Holocene AB2? C? C2?? Poor - - - Geol.

Leona Valley #3 # CDMG 24307 Holocene? A1? B? B? Poor - - - CSMIP

Leona Valley #4 # CDMG 24308 Pleistocene AB2? C? C1? Poor - - - CSMIP, Geol.

Leona Valley #5 - Ritter # CDMG 24055 Holocene AB2? C? C2? Poor - - - Geol.

Leona Valley #6 # CDMG 24309 Holocene AB2? C? C?1 Poor - - - Geol.

Littlerock - Brainard Can # CDMG 23595 Mesozoic A1? B? B? Poor - - - Geol., CSMIP

Malibu - Point Dume Sch # CDMG 24396 Pleis?Miocene?

AB1? C? C1? Poor - - - Geol., SCEC

Manhattan Beach -Manhattan

USC 90046 Holocene C3? D D1S Fair >22 - 3000 Geol.

Mojave - Hwys 14 & 58 # CDMG 34093 D? D?1?S? Poor

Mojave - Oak Creek Canyon#

CDMG 34237 D? D?1?S? Poor

Montebello - Bluff Rd. USC 90011 Holocene F F F Good >21.5 - 1000? SCEC

Moorpark - Fire Sta # CDMG 24283 Holocene C3-2? C?D? D1S Poor 150 - - Geol.

Mt Baldy - Elementary Sch # CDMG 23572 Holo. nearCret.

AB?A? B?C? C?1 Fair - - - Geol., CSMIP

405



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(m)IR(2) Conrey's

depth(3)

(ft)

Source (5)

Mt Wilson - CIT Seis Sta # CDMG 24399 Mesozoic AB2 B C1 Fair 23 (w) 11.5 - Geol., CSMIP,D&L,G.,S&S

N. Hollywood - ColdwaterCan

USC 90009 Holocene AB2 C C2 Good 17 4.0 - SCEC

Neenach - Sacatara Ck # CDMG 24586 Holocene C? C? D1S? Poor

Newhall - Fire Sta # CDMG 24279 Holocene B1 D C3 Good 55 2.8 - ROSRINE

Newhall - W. Pico CanyonRd.

USC 90056 Holocene AB2? D? C2? Check - - - Geol.

Newport Bch - Irvine Ave.F.S. #

CDMG 13160 Pleistocene B1? C?D? D?2S? Poor - - - Geol.

Newport Bch - Newp &Coast #

CDMG 13610 Miocene AB1 C? C1 Fair - - - Geol.,CSMIP,S&S

Northridge - 17645 SaticoySt

USC 90003 Holocene C3 D D1C Good 81 2.5 - SCEC

Pacific Palisades - SunsetBlvd

USC 90049 Holocene AB2 D C2 Good 21 4.3 - SCEC

Pacoima Dam (downstr) # CDMG 24207 Mesozoic A1 B? B Fair 0 3 - CSMIP, Geol.

Pacoima Dam (upper left) # CDMG 24207 Mesozoic A1 B B Fair 0 3 - D.e.a., G.

Pacoima Kagel Canyon # CDMG 24088 Miocene AB1 C C1 Good 20 (w) 2 - G., ROSRINE

Palmdale - Hwy 14 &Palmdale #

CDMG 24521 Holocene-Pleistocene

B1? C C3? Fair >30,<60?

High?

- S&S, SCEC

Pardee - SCE 0

Pasadena - N Sierra Madre USC 90095 Holocene -Pleistocene

AB2 D C3? Good 34 2.0 SCEC

Phelan - Wilson Ranch # CDMG 23597 Holocene C2?C3? D? D1S? Poor - - - Geol.

406



UBC 97(Table A-3)

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Quality Depth(1)

(m)IR(2) Conrey's

depth(3)

(ft)

Source (5)

Playa Del Rey - Saran USC 90047 Holocene F F F Fair - - 3000 SCEC

Point Mugu - Laguna Peak # CDMG 25148 Miocene AB1? C? C?1 Poor - - - Geol.

Port Hueneme - Naval Lab.#

CDMG 24281 Holocene C3 D D1S Fair >30 - - Geol., SCEC

Rancho Cucamonga - DeerCan #

CDMG 23598 Mesozoic A1? B? B? Poor - - - Geol., CSMIP

Rancho Palos Verdes -Hawth #

CDMG 14404 Miocene A1? B? C1 Fair - - 0 Geol.

Rancho Palos Verdes -Luconia

USC 90044 Miocene AB1 D C1? Poor 10.5 2.0 - Geol. ,SCEC

Rinaldi Receiving Sta # DWP 77 Holocene(thin)

AB2 C C2 Fair 40-Pico250

2.03.0

- G.e.a.

Riverside - Airport # CDMG 13123 Pleistocene AB2? C C2 Poor - - - Geol.

Rolling Hills Est-RanchoVista

CDMG 14405 Pliocene AB1? C? C1 Poor - - - Geol.

Rosamond - Airport # CDMG 24092 Holocene B? D? D?1S? Poor - - - Geol.

San Bernardino - CSUSB Gr#

CDMG 23672 Holocene B1?C3? D? D?1S? Poor - - - Geol. ,S&S

San Bernardino - E & Hosp#

CDMG 23542 Holocene C3? D D?1S Poor - - - CSMIP, SCEC

San Gabriel - E. Grand Ave. USC 90019 Holocene A1 C C?2 Good 6.5 >10 0 SCEC

San Jacinto - CDF Fire Sta # CDMG 12673 Holocene C2?C3? D? D1C? Poor - - - Geol.

San Marino, SW Academy # CDMG 24401 Holocene C3? C?D? C3?? Poor >30 - - D&L,USGS,D&L,G

San Pedro - Palos Verdes # CDMG 14159 Miocene AB1? C? C?1 Poor - - 0 Geol.

407



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(m)IR(2) Conrey's

depth(3)

(ft)

Source (5)

Sandberg - Bald Mtn # CDMG 24644 Mesozoic AB1? B?C? C?1 Poor - - - Geol.

Santa Barbara - UCSBGoleta #

CDMG 25091 Holocene AB1 C? C1 Good >5 - - SCEC

Santa Fe Spr - E. Joslin USC 90077 Holocene C? D C3?? Check 25.5 3.0 - SCEC

Santa Monica City Hall # CDMG 24538 Pleistocene B1 D D2C Good 183+ 6.0 2000 Chang, SCEC

Santa Susana Ground # USGS 5108 Cretaceous A1?AB1? B?C? C1 Fair 2 - - S&S

Seal Beach - Office Bldg # CDMG 14578 Pleistocene C3? D? D2S Fair - - 3000 Geol., S&S

Sepulveda VA # USGS 637 Pleistocene C2 C D2C Good >76 1.8 - G.e.a.

Simi Valley - Katherine Rd USC 90055 Holocene A1 C C2 Good 12.5 4.0 - G.e.a.

Stone Canyon # MWD 78

Sun Valley - Roscoe Blvd USC 90006 Holocene AB2 C C2 Fair - - - SCEC

Sunland - Mt Gleason Ave USC 90058 Holocene AB2 C C3 Good 24.5 2.7 - SCEC

Sylmar - Converter Sta # DWP 74 Holocene C3 D D1S Fair >92 2.9 - G.e.a.

Sylmar - Converter Sta East#

DWP 75 Holocene C3? D? D1S? Poor - - - G.e.a.

Sylmar - Olive View Med FF#

CDMG 24514 Holocene C2 C D1C? Good 79 2.5 - G.e.a.

Tarzana, Cedar Hill # CDMG 24436 Pliocene AB2 D C2? Fair - - - Geol., CSMIP,ROSRINE

Terminal Island - S Seaside USC 90082 Holocene C2 D D1C Fair > 17 - 1000 SCEC

408



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(m)IR(2) Conrey's

depth(3)

(ft)

Source (5)

Topanga - Fire Sta # USGS 5081 Holo. nearMioc.

AB? C? C2? Poor - - - Geol.

Tustin - E. Sycamore USC 90089 Holocene F F F Fair - - 1000 SCEC

Vasquez Rocks Park # CDMG 24047 Pleistocene A?AB? C?B? B? Poor - - - Geol., CSMIP

Ventura - Harbor &California

CDMG 25340 Holocene C2 D D1C Fair >24 - - S&S

Villa Park - Serrano Ave USC 90090 Pleistocene(Shallow)

A1 C C?2 Good 7 - - SCEC

West Covina - S. OrangeAve

USC 90071 Holocene AB2 D C2 Good 16 - - SCEC

Whittier - S. Alta Dr USC 90075 Pleistocene A1 C C2 Good 7 - - SCEC

Wrightwood - Jackson Flat # CDMG 23590 Mesozoic A1? B? B? Poor - - - Geol.

Wrightwood - Nielson Ranch#

CDMG 23573 Holocene AB2? C?D? C2?? Poor - - - Geol.

Wrightwood - Swarthout # CDMG 23574 Holocene AB2? C?D? C2?? Poor - - - Geol.

Malibu Canyon, Monte NidoFire (4)

USGS 5080 Miocene A?AB? B? B? Poor - - - Geol.

Point Mugu - Naval AirStation (4)

CDMG 25147 Holocene C? D? D1C? Poor - - - Geol.

Malibu, W. Pacific CoastHwy. (4)

USC 90051 Pleistocene AB2 C C2 Good 14 4.0 - SCEC

Rancho Cucamonga - L&J(4)

CDMG 23497 Holocene C3 D D1S Good 210 - - S&S, G.

(1) Depth to bedrock obtained from boring log. If nothing is specified it refers to depth of soil cover over weathered rock.(2) Estimated Impedance Ratio.(3) Depth to base of Pliocene deposits (Conrey 1967).

409

(4) Ground motion sites added to the Walter Silva Database.(5) Abbreviated source for geotechnical and geological data. Corresponds to the following references:

Chang - Chang et. al. (1997).CSMIP - CSMIP (1992).D.e.a. - Duke et al. (1971).D&L - Duke and Leeds (1962).G - Geomatrix Consultants (1993).G e.a. - Gibbs et al. (1996).Geol. - Local geological maps (CDMG and Dibblee).ROSRINE - Internet Site (ROSRINE 1998).S&S - Stewart and Stewart (1997).S&W - Shannon and Wilson (1980).SCEC - Vucetic and Dourodian (1995).Stew - Stewart et. al. (1994)T. - Trifunac and Todorovska (1996).USGS - Fumal, Gibbs, and Roth (1981, 1982, and 1984).

411

Table A-2. Ground motion stations showing site classification, Loma Prieta Earthquake.



UBC 97(Table A-3)

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Quality Depth(1)

(m)IR(2) Source(4)

Agnews State Hospital CDMG 57066 Holocene C2 D D1C Good 300? - T&S,G

Alameda NAS Navy Holocene C4-D1-E2 E E1 Good 141 - T&S

Anderson Dam(Downstream)

USGS 1652 Pleistocene B1? C C3? Good > 50 ? - G, 94-552

APEEL 10 - Skyline CDMG 58373 Eocene AB1 C C1 Good 17(w) 1.7 79-1619,G

APEEL 2 - Redwood City USGS 1002 Holocene C4-D1-E2 E E1 Good 84.7 8.6 93-376,G,79-1619

APEEL 2E Hayward MuirSchool

CDMG 58393 Pleistocene C2 D D2C Good 152 - 79-1619

APEEL 3E HaywardCSUH

CDMG 58219 Pliocene AB2 C C?1? Good 12 3.6 79-1619,T&S

APEEL 7 - Pulgas CDMG 58378 Eocene AB1 C C1 Good 16 (w) 1.3 G

APEEL 9 - Crystal SpringResidence

USGS 1161 Pleistocene B1 C C3?? Good >30 1.3 79-1619,G

Bear Valley Sta 10, WebbResidence(3)

USGS 1479 Holocene -Pleistocene

B2 D C2 Good 21 1.8 94-552

Bear Valley Sta 12,Williams(3)

USGS 1481 Holocene C3 D D1C Good >60 94-552

Bear Valley Sta 5, CallensRanch(3)

USGS 1474 Holocene B2 C C2 Good 30 2.6 94-552

Bear Valley Sta. 7,Pinnacles(3)

USGS PNM Miocene(Rhyolite)

A1 B B Fair 91-311

Belmont - Envirotech CDMG 58262 Franciscan AB1 C C1 Good 13(w) 1.2 T&S,G

Berkeley LBL CDMG 58471 Cretaceous AB1 C C1 Fair shallow soil?

1.3

BRAN UCSC 13 AB1? B? B? Poor

412



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(m)IR(2) Source(4)



UBC 97(Table A-3)

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Quality Depth(1)

(m)IR(2) Source(4)

Capitola CDMG 47125 Holocene B1 D C2?? Check >183? ,16? 2.6 T&S

Corralitos CDMG 57007 Holocene AB2 C C1 Good 5(soft),32(Hard)

1.8 G

Coyote Lake Dam(Downst)

CDMG 57504 Holocene AB2 C C3? Fair <50? 1.5? G

Foster City - 355Menhaden

USGS 1515 Holocene C4-D1-E2 E E1 Good 115 High G

Foster City -RedwoodShores(3)

CDMG 58375 Holocene C4-D1-E2 E E2 Good 188 93-376

Fremont - Emerson Court USGS 1686 Pleistocene B1?C3? D D?2S Good >46 - 94-222, G

Fremont - Mission SanJose

CDMG 57064 Pleistocene B1 D D?2S Fair 5? 78? - G

Gilroy - Gavilan Coll. CDMG 47006 Pleistocene AB2 C C2 Good 13 6 S&W, G

Gilroy - Historic Bldg. CDMG 57476 Holocene AB2 C C2?? Poor - - G

Gilroy Array #1 CDMG 47379 Franciscan A1 B B Good 1 2.85 G

Gilroy Array #2 CDMG 47380 Holocene C2 D D1C Good 165 - G,92-387,T&S

Gilroy Array #3 CDMG 47381 Holocene C2 D D1C Good >60, 480? - 92-387,T&S,G

Gilroy Array #4 CDMG 57382 Holocene C2 D D1C Good >30, 800? - G

Gilroy Array #6 CDMG 57383 Eocene-Paleocene

AB1 C C1 Good 1 - G

Gilroy Array #7 CDMG 57425 Pleistocene AB2 D C2 Good 17m 3.5 G, 92-376

Golden Gate Bridge USGS 1678 Franciscan AB1 C? C?1 Poor 5? 1.3

413



UBC 97(Table A-3)

ThisStudy

(Table 1)

Quality Depth(1)

(m)IR(2) Source(4)

Halls Valley CDMG 57191 Holocene AB2 D C2?? Fair 17 Soft, 43Hard

5.2 T&S,G

Hayward - BART Sta CDMG 58498 Pleistocene B2? D? D?2C Poor - - Stewart

Hayward City Hall, groundsite(3)

USGS 1129 Pliocene(Rhyolite)

A1 C B Good 10 94-222

Hollister - SAGO Vault USGS 1032 Mesozoic A1 B B Fair 0 2.5 G

Hollister - South & Pine CDMG 47524 Holocene C2 D D1C? Fair > 105? G

Hollister City Hall USGS 1028 Holocene C2? D D1C Good >105 79-1619,G

Hollister Diff. Array USGS 1656 Holocene C2? D D1C? Good >106 79-1619,G

Larkspur Ferry Terminal(3) USGS 1590 Holocene C4-D1-E2 E E2 Good 27.5 5 94-222

LGPC UCSC 16 AB1 C? C?1 Poor

Monterey City Hall CDMG 47377 Mesozoic AB1 C C1? Good 6s, >8(w) - T&S

Oakland - Title & Trust CDMG 58224 Pleistocene C2 D D2C Good 137 - 93-376,T&S

Oakland Outer Harbor (3) CDMG 58472 Holocene C2 D D1C Good 150 3.8 Chang

Palo Alto - 1900 Embarc. CDMG 58264 Holocene C4-D1-E2 E E1 Good >55 - 93-376

Palo Alto - SLAC Lab USGS 1601 Pleistocene AB2 C C2 Good 12 - 94-222

Piedmont Jr High CDMG 58338 Franciscan A1 B B Good 4 9 T&S

Point Bonita CDMG 58043 Cretaceous A1 B B Good 2 (w) 2 T&S

Richmond City Hall CDMG 58505 Pliocene-Pleist.

C2 D C3?? Good 58 2.5 T&S

SAGO South - Surface CDMG 47189 Mesozoic A1 B B Good 4.5(w) 2.5 T&S

414



UBC 97(Table A-3)

ThisStudy

(Table 1)

Quality Depth(1)

(m)IR(2) Source(4)

Salinas - John & Work CDMG 47179 Holocene B?C? D D?1C? Poor - - G

San Francisco, 1295Shafter, Fire Station(3)

USGS 1675 Franciscan A1 B B Fair 7 91-311

San Jose - Sta. TeresaHills(3)

CDMG 57563 ? A1 B? B? Poor G.

Saratoga - Aloha Ave CDMG 58065 Pliocene AB1? C C2? Poor - - G

Saratoga - W Valley Coll CDMG 58235 Pleistocene AB2? C C2? Poor 4-10? - G

SF - Cliff House CDMG 58132 Franciscan A1?AB1? C?B? B? Poor 0? -

SF - Diamond Heights CDMG 58130 Franciscan AB1 C C1 Good 4 s, 11 (w) 4.7 T&S

SF - Pacific Heights CDMG 58131 Cretaceous A1 B B Good 6-9(w) 1.5 T&S

SF - Presidio CDMG 58222 Franciscan?

AB1 C C1 Good 17.5 1.9 93-376

SF - Rincon Hill CDMG 58151 Franciscan A1?AB1? C? B? Poor 5 (w) T&S (may bewrong boring)

SF - Telegraph Hill CDMG 58133 Franciscan A1?AB1? C? B? Fair 6? - T&S

SF Intern. Airport CDMG 58223 Holocene C4-D1-E2 E E1 Good 134 - T&S,G,92-287

So. San Francisco, SierraPt.

CDMG 58539 Franciscan A1 B B Good 4-5(w) 1.3 T&S

Sunnyvale - Colton Ave. USGS 1695 Holocene C2 D D1C? Good >60 - 94-222,G

Sunol Fire St (CalaverasArray) (3)

USGS SNF Pleistocene B1 C D?2S Good 35(>48) 2 94-552

Treasure Island CDMG 58117 Holocene F F F Check 88 - 92-287

UCSC UCSC 15 AB2 C? C2? Poor

UCSC Lick Observatory CDMG 58135 Paleozoic AB2 C C1 Fair >13 (w) - T&S

415



UBC 97(Table A-3)

ThisStudy

(Table 1)

Quality Depth(1)

(m)IR(2) Source(4)

Woodside CDMG 58127 Eocene AB2 C B Good 6 1.6 93-376

Yerba Buena Island CDMG 58163 Franciscan A1 B C1 Good 0? 2.5 92-287

(1) Depth to bedrock obtained from boring log. If no modifier is specified next to the depth value it refers to depth of soil cover over weatheredrock. If (w) is indicated, indicates depth of weathering (depth of weathered rock to slightly weathered or unweathered rock)

(2) Estimated Impedance Ratio.(3) Ground motion sites added to the Walter Silva Database.(4) Unless specifically omitted, information for all sites was also obtained from Fumal (1991). All other abbreviated source for geological data

correspond to the following references:79-1619 (Silverstein 1979).92-287 - (Gibbs et al. 1992).93-376 - (Gibbs et al. 1993).94-222 - (Gibbs et al. 1994).94-552 - (Gibbs and Fumal (1993).Chang - Chang and Bray (1995).D&L - Duke and Leeds (1962).G - Geomatrix Consultants (1993).Geol. - Local geological maps (CDMG).S&S - Stewart and Stewart (1997).S&W - Shannon and Wilson (1980).Stewart - Stewart, J. P. (personal comm.)T&S - Thiel and Schneider (1993).

416

Table A-3. Site categories in the 1997 UBC. sV is the average shear wave velocitymeasured over the upper 100 feet.

SA Hard rock with measured shear wave velocity, sV > 5000 ft/s.SB Rock with 2500 ft/s < sV ≤ 5000 ft/sSC Very dense soil and soft rock with 1200 ft/s < sV ≤ 2500 ft/s or with either N >

50 or us ≥ 2000 psf., where N is the average Standard Penetration blowcountover the upper 100 ft, and us is the average undrained shear strength over theupper 100 feet.

SD Stiff soil with 600 ft/s ≤ sV ≤ 1200 ft/s or with 15 ≤ N ≤ 50 or 1000 psf ≤ us ≤2000 psf.

SE A soil profile with sV < 600 ft/s or any profile with more than 10 ft. of soft claydefined as soil with PI > 20, wmc ≥ 40 percent and su < 500 psf.

SF Soils requiring site-specific evaluation.1. Soils vulnerable to potential failure or collapse under seismic loading such as

liquefiable soils, quick and highly sensitive clays, collapsible weaklycemented soils.

2. Peats and/or highly organic clays (H > 10 ft of peat and/or highly organic claywhere H = thickness of soil.

3. Very high plasticity clays (H > 25 ft with PI > 75).4. Very thick soft/medium stiff clays (H > 120 ft).

417

Table A4. Seed et al. (1991) site classification system (from Dickenson 1994).

SiteClass

SiteCondition 4.1 General Description Site Characteristics1,2

(Ao) Ao Very Hard Rock. Vs (avg.) > 5,000 ft/sec in top 50 ft.A A1 Competent rock with little or no soil

and/or weathered rock veneer.2,500 ft/sec ≤ Vs (rock) ≤ 5,000 ft/sec,and Hsoil + weathered rock < 40 ft with Vs > 800ft/sec (in all but the top few feet3).

AB AB1 Soft, fractured and/or weathered rock. For both AB1 and AB2:AB2 Stiff, very shallow soil over rock and/or

weathered rock.40 ft ≤ Hsoil + weathered rock ≤ 150 ft, and Vs ≥800 ft/sec (in all but the top few feet3).

B1 Deep, primarily cohesionless4 soils.(Hsoil ≤ 300 ft.)

No "Soft Clay" (See Note 5), andHcohesive soil < 0.2 Hcohesionless soil.

B B2 Medium depth, stiff cohesive soilsand/or mix of cohesionless with stiffcohesive soils; no "Soft Clay."

Hall soils≤ 200 ft., and Vs(cohesive soils)> 500 ft/sec.(See Note 5).

C1 Medium depth, stiff cohesive soilsand/or mix of cohesionless with stiffcohesive soils; thin layer(s) of "SoftClay."

Same as B2 above, except0 < Hsoft clay ≤ 10 ft.(See Note 5)

C2 Deep, stiff cohesive soils and/or mix ofcohesionless with stiff cohesive soils; no"Soft Clay."

Hsoil > 200 ft, andVs (cohesive soils) > 500 ft/sec.

C3 Very deep, primarily cohesionless soils. Same as B1 above, exceptHsoil > 300 ft.

C4 Soft, cohesive soil at small to moderatelevels of shaking.

10 ft. ≤ Hsoft clay ≤ 100 ft, andAmax,rock ≤ 0.25 g

D D1 Soft, cohesive soil at medium to stronglevels of shaking.

10 ft. ≤ Hsoft clay ≤ 100 ft, and0.25 g ≤ Amax,rock ≤ 0.45 g, or[0.25 g ≤ Amax,rock ≤ 0.45 g and M ≤7.25]

E1 Very deep, soft cohesive soil. Hsoft clay ≥ 100 ft. (See Note 5)

(E)4E2 Soft, cohesive soil and very strong

shakingHsoft clay ≥ 10 ft., and either:Amax,rock ≥ 0.55 g, orAmax,rock ≥ 0.45 g and M > 7.25

E3 Very high plasticity clays Hclay > 30 ft with PI > 75% andVs < 800 ft/sec.

F1 Highly organic and/or peaty soils. H > 20 ft. of peat and/or highlyorganic soils

(F)7 F2 Sites likely to suffer ground failure dueeither to significant soil liquefaction ofother potential modes of groundinstability.

Liquefaction and/or other types ofground failure analysis required.

(See next page for "notes")

418

Notes for Table A-4.

1. H = total (vertical) depth of soils of the type or types referred to.

2. Vs = seismic shear wave velocity (ft/sec) at small shear strains (shear strain ≈ 10-4 %).

3. If surface soils are cohesionless, Vs may be less than 800 ft/sec in top 10 feet.

4. "Cohesionless soils" = soils with less than 30% "fines" by dry weigth;"Cohesive soils" = soils with more than 30% "fines" by dry weight, and 15% ≤ PI (fines) ≤ 90%.Soils with more than 30% fines, and PI (fines) < 15% are considered "silty" soils herein, and theseshould be (conservatively) treated as "cohesive" soils for site clasification purposes in this Table.(Evaluation of approximate Vs for these "silty" soils should be based either on penetration resistance ordirect field Vs measurement; see Note 8 below).

5. "Soft Clay" is defined herein as cohesive soil with: (a) Fines content ≥ 30%, (b) PI (fines) ≥ 20%, and(c) Vs ≤ 500 ft/sec.

6. Site-specific geotechnical investigations and dynamic site response analyses are stronglyrecommended for these conditions. Variability of response characteristics within this class (E) of sitestends to be more highly variable than for classes Ao through D, and the very approximate responseprojections herein should be applied conservatively in the absence of (strongly recommended) site-specific studies.

7. Site-specific geotechnical investigations and dynamic site response analyses are recquired for theseconditions. Potentially significant ground failure must be mitigated, and/or it must be demonstratedthat the proposed structure/facility can be engineered to satisfactorily withstand such ground failure.

8. The following approaches are recommended for evaluation of Vs:

(a) For all site conditions, direct (in-situ) measurement of Vs is recommended.(b) In lieu of direct measurement, the following empirical approaches can be used:

(i) For sandy cohesionless soils: either SPT-based or CPT-based empirical correlations may beused.

(ii) For clayey soils: empirical correlations based on undrained shear strength and/or somecombination of one or more of the following can be used (void ratio, water content, plasticityindex, etc.) Such correlations tend to be somewhat approximate, and should be interpretedaccordingly.

(iii) Silty soils of low plasticity (PI ≤ 15%) should be treated as "largely cohesionless" soils here;SPT-based or CPT-based empirical correlations may be used (ideally with some "fines"correction relative to "clean sand" correlations.) Silty soils of medium to high plasticityshould be treated more like "clayey" soils as in (iii) above.

(iv) "Other" soil types (e.g. gravelly soils, rockfill, peaty and organic soils, etc.) requireconsiderable judgement, and must be evaluated on an individual basis; no simplified"guidance" can appropriately be offered herein.

419

APPENDIX B

Site Visits to Selected Ground Motion Sites

420

CONTENTS OF APPENDIX B

CDMG 24461- Alhambra - Fremont School................................................................... 421USC 99 - Arcadia - 855 Arcadia Av. .............................................................................. 422USC 93- Arcadia - Campus Dr. ...................................................................................... 423USC 14 - Beverly Hills - 12520 Mulholland .................................................................. 424USC 13 - Fire Station # 99, 14145 Mulholland .............................................................. 425USC 57: Canyon country - W. Lost Canyon ................................................................... 426USC 65 - Glendora - N. Oakbank ................................................................................... 427USC 34 - LA Fletcher Dr. ............................................................................................... 428CDMG 24271 - Lake Hughes #1 - Fire Station #78 ....................................................... 429Leona Valley sites (1-6) .................................................................................................. 430

CDMG 24305............................................................................................................... 430CDMG 24306............................................................................................................... 430CDMG 24307............................................................................................................... 430CDMG 24308............................................................................................................... 430CDMG 24055............................................................................................................... 430CDMG 24309............................................................................................................... 430

CDMG 24396 - Malibu, Point Dume School.................................................................. 431CDMG 14404 - Rancho Palos Verdes - Hawthorne Blvd............................................... 432USC 44 - Rancho Palos Verdes - 30511 Lucania Dr. ..................................................... 433CDMG 14405 - Rolling Hills Estates - Rancho Vista School ........................................ 434CDMG 14159 - San Pedro - Palos Verdes ...................................................................... 435CDMG 24644 - Sandberg - Bald Mountain .................................................................... 436USC 77 - Santa Fe Spring - 11500 E. Joslin ................................................................... 437USGS 5081...................................................................................................................... 438CDMG 24047 - Vasquez Rock Park ............................................................................... 439USC 51 - Malibu - St. Adams Episcopal Church............................................................ 440

421

CDMG 24461- Alhambra - Fremont SchoolDate: June 2, 1998Time: 2:00 p.m.Instrument: Could not access to school.

Description: The school is located in a flat area north of the Los Angeles basin. Somehills are seen in the SW and NE (about 200 m to the SW and 300 m to the NE) thatcorrespond to units of Miocene Shale in geological maps (Dibblee).

422

USC 99 - Arcadia - 855 Arcadia Av.Date: June 3, 1998Time: 3:00 p.m.Instrument: The instrument is located in private property. Could not access it nor locateit.Description: Building is located in a very flat region. Closest relief are the beginning ofthe San Gabriel Mountains about 1 mile north.

423

USC 93- Arcadia - Campus Dr.Date: June 3, 1998Time: 3:30 p.m.Instrument: Inside the school, but not located.

Description: The school is located in a very flat region. Closest relief are some hills tothe north (about 1 mile, the beginning of the San Gabriel Mountains). The site is farenough from any topographic feature that, from observation alone, it would be assumedthat it has thick sedimentary deposits. The C2 classification may be reviewed.

424

USC 14 - Beverly Hills - 12520 MulhollandDate: June 3, 1998Time: 11:00 p.m..Instrument: Located in the equipment room of Fire Station #108, about 50 m from thesouthern end of an esplanade along a cliff on Mulholland Dr.

Description: The building is located next to Mulholland Dr., along a cliff in the SantaMonica Mountains. There are canyons on both sides of the fire station, which is build onan esplanade facing lower ranges of the Sta. Monica Mountains in the south side. On theeast and west side the slopes are relatively steep (~ 45º). There is some exposed rock inthe south of the station, a well-cemented, medium-size grained sandstone (about 100 m.south of the station, in the slope of the south side of the station). There is a rock cut onthe north side of Mulholland Dr. showing what could be basalt as indicated in the Dibbleemap. A cut in the parking lot (a small 1 m3 manhole opening) shows what could be shalein the south edge of the cut, and sandstone in the north side. Probably is unit Tt inDibblee.

From this we can infer that the site is in between Tt and Ttls (see Dibblee map forBeverly Hills Quadrangle).

Geological Units mentioned:Tt Middle Topanga formation—mostly interbedded gray to tan semi-friablesandstone and gray micaceous claystone, locally includes lenses of pebbly sandstoneand pebble-cobble conglomerateTtls Lower Topanga formation—mostly tan, semi-friable to hard arkosic sandstone;locally includes gray micaceous shale.

Recommended Classification from Site Visit: C1

425

USC 13 - Fire Station # 99, 14145 MulhollandDate: June 3, 1998Time: 10:00 a.m.Instrument: At the time of the visit, the instrument was not installed because of workbeing done in the floor of the building. Normally the instrument is located underneath astairwell in the office (2 story building).

Description: The fire station is located on Mulholland Drive, which is placed along aridge on the Santa Monica Mountains. To the north, you can overlook the San FernandoValley, and the Sta. Monica Mountains continue to the south. The location is surroundedby vegetation and heavy brush. There are some recent landslides west of the station,about 1 mile away (on some steep cliffs). Exposed surface beneath the fire station lookslike soil, likely residual soil.

An outcrop on 13030 Mulholland, shows highly jointed, yellow-brown, poorlycemented sandstone. Major joint sets dips north (about 70º). In the same road cut there isalso a medium to coarse grained sandstone, bedded. Sedimentation plains dip ≈ 70º N,crumbles easily with hand pressure. Another exposure further west shows a deeper soilhorizon, also outcrop of sandstone, less weathered and more cemented than the previousone.


426

USC 57: Canyon country – W. Lost CanyonDate: June 2, 1998Time: 2:00 p.m.Instrument: Bolted to concrete floor on a one floor building (Main office, closet next tothe lounge)Description: The instrument is located in a valley next to the Santa Clara River.

427

USC 65 - Glendora - N. OakbankDate: June 3, 1998Time: 4:00 p.m.Instrument: Could not access the church. The instrument is located in a church in N.Oakbank.

Description:

Very close to the foothills of the San Gabriel Mountains. Site is located in the pointwhere the slope starts to increase, probably in the edge of recent alluvial fans. Noinferences can be made with respect to the soil depth.

428

USC 34 - LA Fletcher Dr.Date: June 3, 1998Time: 1:00 p.m.Instrument: Located inside the fire station (LA Fire Station 52). Could not accessinstrument in site visit, building was closed.

Description: The site is located in a very broad valley, possibly a deep site.

429

CDMG 24271 - Lake Hughes #1 - Fire Station #78Date: June 2, 1998Time: 10:00 a.m.Instrument: Located in the fire station building (1 story garage), bolted on concrete.

Description: On edge of hills overlooking the rift valley formed by the San Andreasfault, in the north side of the fault. The fire station is located in the mouth of a smallcanyon; this may explain why the soil cover is so thick in this station.

430

Leona Valley sites (1-6)CDMG 24305CDMG 24306CDMG 24307CDMG 24308CDMG 24055CDMG 24309

Date: June 2, 1998Time: 12:00 noonInstrument: All of the instruments located directly on soil. All but #5 have metalinstrument shelters with an antenna. #5 has an older instrument shelter.

Description:

#1 Located in a large hill in the middle of a (cherry?) orchard. There is a rock outcropnext to the stations (moderately weathered crystalline rock). According to the geologicmap, the rock is likely to be Pelona Schist. From this observation, I would classify thesite as B.# 2 Located in a small, broad valley, low relief. To the north there is a hill where #3 islocated at about 50 m., to the south is the hill where #1 is located. The bottom of the hillis at about 100 m. from the station. There is a small creek next to the station, exposesabout 1 m of gravel and clay. It is impossible to infer the depth of soil from just a cursoryobservation, but given proximity of hills and older deposits, I would classify this site asC2 or C3.#3 Located on the top of a hill with moderate slope (slope angle ≈ 20º). Definitely notmetamorphic bedrock as indicated in CSMIP. Most likely a C1 site, probably a B site.#4 Located on the slope of a small hill (about 5 -10 m. high). On the hill there is highlyweathered sedimentary rock exposed; crumbles with slight pressure from the hand. Youcan find quartz pebbles around the station. Inferring from rock behind the instrument, thesite is on highly weathered sedimentary rock, likely a C1 site.#5 On a flat region near a small hill with mild slope. This site is located near a hill, butcloser to the larger valley where the SAF is located. This site may be a C2 or a C3 site.#6 Located on a hill on the north side of the road (N2). The road has been recentlymoved to allow for the construction of a spillway in the creek running parallel to the SanAndreas fault (SAF) (Amargosa creek). The station is located apparently to the north ofthe main trace of the SAF. to the north of the station there is a small valley and on theother side of the valley there are larger mountains dividing the rift valley from theAntelope valley. Soil around the station is gravelly, angular. A canyon on a creek about300 m. east of the station exposes sedimentary rock. Due to this observation and thegeologic map, I would classify this as a C1 site.

431

CDMG 24396 - Malibu, Point Dume SchoolDate: June 4, 1998Time: 10:00 a.m.Instrument: Located in the Point Dume community center office (copy room)

Description: Instrument located at the foot of a mild sloping hill (~70 m. from the baseof the hill). A shallow soil cover may exist, probably formed from materials transportedfrom the hill. According to the map, the site is in Tertiary Marine deposits.I would classify the site C1 because in general the tertiary deposits in this zone areweathered. Note that USC 51 is close by and on the same formation; however it is on ahill at a very different elevation than this site.

432

CDMG 14404 - Rancho Palos Verdes - Hawthorne Blvd.Date: June 3, 1998Time: 6:30 p.m.Instrument: Located in a small maintenance building (1 story). It could not be accessed.The site, formerly a Loyola Marymount University building, now belongs to the SalvationArmy.

Description:Very weathered sandstone, light tan (looks the same formation as for CDMG 14159 Site).Building located against a hill on what appears to be a cut section.

433

USC 44 - Rancho Palos Verdes - 30511 Lucania Dr.Date: June 3, 1998Time: 8:00 p.m.Instrument: Located in Mira Catalina School.

Description: Site located in the side of a hill facing south. No rock exposures wherefound close to the site, but the geology is likely to be similar to the CDMG 14159 Site.

434

CDMG 14405 - Rolling Hills Estates - Rancho Vista SchoolDate: June 3, 1998Time: 7:00 p.m.Instrument: Located in Rancho Vista School, the school was closed at the time of thevisit and access to the exact location of the instrument was not possible.

Description: Site is on an esplanade in a hillside facing towards the LA Valley (N).

435

CDMG 14159 - San Pedro - Palos VerdesDate: June 3, 1998Time: 6:00 p.m.Instrument: Located in a fire station on 25th Street in front of Whites Park. Theinstruments in a deposit in the east side of the building (1 story building). The instrumentis not the same one that recorded the Northridge earthquake, the previous one was locatedin the other side of the wall from its current location.

Description: The fire station is located in top of a hill overlooking the ocean (at least 80m from the top of the slope). In Whites Park, south of the site there are some impressiverock exposures:

Sandstone, varies in grain size and color across the exposure, weathered to veryweathered at spots. The cut in the slope where rock is exposed is about 40 - 50 m. high.If station were located here, it would be a C1 site, but probably bordering on a B site.Given that the cut has been exposed, weathering in the cut may be due to its exposure,and may not reflect true depth of weathering; therefore the site may also be a B site.

Small slide W of the station (about 500 m) on the side of the road that goes downto Whites Park exposes sandstone, horizontal bedding, lightly tan, fine grained, veryweathered. Also layers of hard shale, dark tan, very weathered.

Recommended Classification from Site Visit: C?1

436

CDMG 24644 - Sandberg - Bald MountainDate: June 2, 1998Time: 8:00 a.m.Instrument: Instrument Shelter, placed on ground surface.

Description: Site located on the top of a mountain. Low vegetation cover. Thesurrounding peaks all are dome-shaped; no abrupt peaks are seen. A recently scrapedroad next to the instrument exposes soil (clay?) with considerable amount of coarse sand-size particles of granitic origin; some larger (1in. - 2 in.) granitic pebbles also found. Nooutcrops of intact rock visible in the near vicinity.

A small slide (10 m. wide) caused by road work about 0.5 miles east of the stationshows what seems to be residual soil at least 3 m deep. Grain size of mother rock(granite, pink) increases downward. Some highly weathered granite blocks seen also atabout 1m. deep.


437

USC 77 - Santa Fe Spring - 11500 E. JoslinDate: June 3, 1998Time: 5:00 p.m.Instrument: Located in Lake View School

Description: The site was visited because it had the particularity that it was the only C2site in the LA basin, however, nothing particular in the topography surrounding the sitewas noticed. From the site visit alone, it is recommended to change the classification to aD site, given that the C2 classification is inferred only from the SCEC shear wavevelocity database (see Boore and Brown 1998).

438

USGS 5081Date: June 3, 1998Time: 2:00 p.m.Instrument: Instrument located in the NW side of the station, in a small wash room (1story building) in a fire station. (FS is in the intersection of Topanga Canyon Rd. withFernwood-Pacific Dr.)

Description:

Station located in the wall of a steep canyon (W side of canyon). There is a landslide inthe road in front of the station on Topanga Canyon Rd. A driller on the site indicated tome that they found superficial colluvium over sandstone (soft, not well cemented).Strangely, they found a 10 ft. layer of river sand interbedded with the Sandstone.

Recommended Classification from Site Visit: C2, although it may be also C1. Stationis likely located in landslide deposits.

439

CDMG 24047 - Vasquez Rock ParkDate: June 2, 1998Time: 3:00 p.m.Instrument: Not located

Description: The instrument could not be exactly located due to the high vegetation.The site where the instrument should be located is near a creek bed just north of VasquezRock Park.

440

USC 51 - Malibu - St. Adams Episcopal ChurchDate: June 3, 1998Time: 11:00 a.m.Instrument: Located in a small deposit adjacent to the men's room in the church.

Description:Instrument is located in a hill on the N side of Hwy. 1. The hill has relatively steep gradedslopes. The instrument is about 30 m from the edge of the slope.

441

APPENDIX C

Equations to Obtain Combined Spectral Acceleration Ratios for theNorthridge and Loma Prieta Earthquakes

442

Formula for Spectral Amplification Factors from Site j to Site i

The amplification factors from Site j to Site i (Fi/j) for a given reference peak

ground acceleration (PGAref) and a given spectral period are given by the following

formula:

( ) ( )[ ] ( )[ ]ln ln ln/ / , / , / , / , / , / ,F w a b R c w a b R ci j i j N i j N i j N N N i j L i j L i j L L L= + + + + + (C1)

where the subscripts N and L denote coefficients for the Northridge and Loma Prieta

Earthquakes respectively, ai/j and bi/j are given by Equation 4 (rewritten here for

convenience), and are listed in Table C1;

ai/j = a(Site i) - a(Site j) (4a)

bi/j = b(Site i) - b(Site j) (4b)

a, b, and c are period-dependent coefficients given in Table 3 (rewritten as Table C4 for

convenience) for each earthquake; R is the distance corresponding to the reference PGA

(PGAref), and is given by:

ceR baPGAref

−=�

��

� −)ln(

(C2)

443

where a, b, and c are the coefficients given in Table 3 (Table C4) corresponding to the

peak ground acceleration (PGA) and wi/j are weights for each earthquakes. If a simple

geometric mean of both earthquakes is used, the weights are equal to 0.5 for each

earthquake. If the variance-weighted geometric mean is used, the weights are obtained

by:

( )( ) ( ) 1

L,j/i1

N,j/i

1N,j/i

N,j/i VARVAR

VARw −−

−

+= (C3)

for the Northridge Earthquake and:

( )( ) ( ) 1

L,j/i1

N,j/i

1L,j/i

L,j/i VARVAR

VARw −−

−

+= (C4a)

for the Loma Prieta Earthquake. The variance of the sample mean, VARi/j, is obtained for

each earthquake using the following formula:

j

2j

i

2i

j/i NNVAR

σ+σ= (C4b)

where σ and N are the standard deviation and number of sites corresponding to each site

condition, spectral period, and earthquake. The standard deviations are given in Table 3

444

(Table C4) and the number of sites is a function of period and is given in Table C2. The

resulting weights for the variance weighted scheme (wi/j) are given in Table C3.

Standard Deviations

The standard deviations associated with each site condition are obtained using a similar

weighting scheme as the amplification factors. For Site i, the standard deviations are

given by:

( )( ) ( )( )2L,iL,i

2N,iN,i

2i ww σ′+σ′=σ (C5)

where σi is the standard deviation and w'i the weight for site condition i for each

earthquake. The weights w'i are equal to 0.5 if a simple geometric mean of both

earthquakes is used. If the variance-weighted geometric mean is used, the weights are

obtained by:

( )( ) ( ) 1

L),i(VAR1

N),i(VAR

1N),i(VAR

N,i VARVAR

VARw −−

−

+= (C6a)

for the Northridge Earthquake and:

( )( ) ( ) 1

L),i(VAR1

N),i(VAR

1L),i(VAR

L,j/i VARVAR

VARw −−

−

+= (C6b)

445

for the Loma Prieta Earthquake. The variance of the sample variance, VARVAR(i) is

estimated for each earthquake by:

( ) 4i2

i

i)i(VAR N

1N2VAR σ

−= (C7)

where σ and N are the standard deviation and number of sites corresponding to each site

condition, spectral period, and earthquake. The resulting weights, w'i, are given in Table

C5.

446

Table C1. Coefficients for determining spectral amplification ratios (Equation C1).Northridge Loma Prieta

T ac/b bc/b ad/b bd/b ad/c bd/c ac/b bc/b ad/b bd/b ad/c bd/cPGA 0.0000 0.1215 0.3198 0.0592 0.3198 -0.0623 0.0993 0.0452 -0.1503 0.1922 -0.2496 0.14700.055 0.0000 0.0922 0.1404 0.0728 0.1404 -0.0195 -0.3276 0.1334 -0.5213 0.2640 -0.1937 0.13060.06 0.0000 0.0914 0.1449 0.0709 0.1449 -0.0205 -0.3414 0.1374 -0.5322 0.2673 -0.1907 0.12990.07 0.0000 0.0893 0.1597 0.0653 0.1597 -0.0240 -0.3589 0.1426 -0.5456 0.2717 -0.1867 0.12900.08 0.0000 0.0875 0.1801 0.0585 0.1801 -0.0290 -0.3859 0.1515 -0.5655 0.2790 -0.1796 0.12750.09 0.0000 0.0873 0.2017 0.0527 0.2017 -0.0346 -0.4068 0.1595 -0.5793 0.2856 -0.1724 0.12610.1 0.0000 0.0896 0.2193 0.0497 0.2193 -0.0398 -0.4141 0.1637 -0.5821 0.2891 -0.1680 0.12540.11 0.0000 0.0917 0.2257 0.0497 0.2257 -0.0421 -0.4150 0.1665 -0.5788 0.2913 -0.1638 0.12480.12 0.0000 0.0946 0.2306 0.0507 0.2306 -0.0440 -0.4124 0.1673 -0.5740 0.2918 -0.1616 0.12450.13 0.0000 0.0982 0.2343 0.0525 0.2343 -0.0456 -0.4100 0.1680 -0.5695 0.2922 -0.1595 0.12420.14 0.0000 0.1022 0.2373 0.0552 0.2373 -0.0470 -0.4002 0.1682 -0.5545 0.2917 -0.1543 0.12350.15 0.0000 0.1066 0.2401 0.0584 0.2401 -0.0482 -0.3874 0.1676 -0.5346 0.2898 -0.1472 0.12220.16 0.0000 0.1112 0.2430 0.0621 0.2430 -0.0491 -0.3740 0.1665 -0.5110 0.2866 -0.1370 0.12010.17 0.0000 0.1158 0.2463 0.0661 0.2463 -0.0497 -0.3679 0.1659 -0.4975 0.2843 -0.1297 0.11840.18 0.0000 0.1179 0.2482 0.0682 0.2482 -0.0498 -0.3621 0.1654 -0.4848 0.2822 -0.1227 0.11680.19 0.0000 0.1200 0.2501 0.0701 0.2501 -0.0498 -0.3537 0.1648 -0.4572 0.2768 -0.1035 0.11190.2 0.0000 0.1237 0.2544 0.0742 0.2544 -0.0496 -0.3500 0.1650 -0.4293 0.2705 -0.0793 0.10550.22 0.0000 0.1267 0.2593 0.0780 0.2593 -0.0487 -0.3512 0.1661 -0.4018 0.2637 -0.0506 0.09760.24 0.0000 0.1288 0.2649 0.0815 0.2649 -0.0474 -0.3541 0.1671 -0.3881 0.2600 -0.0340 0.09290.26 0.0000 0.1294 0.2683 0.0829 0.2683 -0.0465 -0.3568 0.1679 -0.3755 0.2566 -0.0187 0.08870.28 0.0000 0.1300 0.2714 0.0843 0.2714 -0.0457 -0.3650 0.1700 -0.3508 0.2494 0.0142 0.07930.3 0.0000 0.1301 0.2796 0.0862 0.2796 -0.0438 -0.3736 0.1719 -0.3279 0.2423 0.0457 0.07040.32 0.0000 0.1291 0.2901 0.0869 0.2901 -0.0422 -0.3769 0.1723 -0.3170 0.2389 0.0599 0.06650.34 0.0000 0.1272 0.3039 0.0861 0.3039 -0.0410 -0.3799 0.1728 -0.3067 0.2357 0.0732 0.06290.36 0.0000 0.1244 0.3218 0.0836 0.3218 -0.0408 -0.3804 0.1724 -0.2964 0.2325 0.0840 0.06010.4 0.0000 0.1209 0.3444 0.0792 0.3444 -0.0417 -0.3737 0.1688 -0.2669 0.2238 0.1068 0.05500.44 0.0000 0.1169 0.3718 0.0730 0.3718 -0.0439 -0.3561 0.1626 -0.2461 0.2183 0.1101 0.05570.5 0.0000 0.1081 0.4385 0.0562 0.4385 -0.0519 -0.3135 0.1491 -0.2137 0.2110 0.0998 0.06190.55 0.0000 0.1038 0.4753 0.0467 0.4753 -0.0571 -0.2834 0.1399 -0.1935 0.2067 0.0899 0.06680.6 0.0000 0.0998 0.5120 0.0373 0.5120 -0.0625 -0.2578 0.1324 -0.1770 0.2034 0.0808 0.07100.667 0.0000 0.0962 0.5466 0.0288 0.5466 -0.0674 -0.2267 0.1232 -0.1571 0.1994 0.0697 0.07620.7 0.0000 0.0933 0.5774 0.0217 0.5774 -0.0715 -0.1980 0.1149 -0.1380 0.1955 0.0599 0.08060.75 0.0000 0.0910 0.6030 0.0167 0.6030 -0.0743 -0.1569 0.1030 -0.1109 0.1900 0.0461 0.08700.8 0.0000 0.0895 0.6227 0.0139 0.6227 -0.0756 -0.1148 0.0910 -0.0811 0.1839 0.0338 0.09280.85 0.0000 0.0888 0.6365 0.0135 0.6365 -0.0753 -0.0753 0.0798 -0.0531 0.1781 0.0222 0.09830.9 0.0000 0.0889 0.6448 0.0154 0.6448 -0.0735 0.0161 0.0540 0.0173 0.1630 0.0011 0.10900.95 0.0000 0.0898 0.6486 0.0195 0.6486 -0.0704 0.1146 0.0266 0.0999 0.1448 -0.0146 0.11821.0 0.0000 0.0914 0.6491 0.0251 0.6491 -0.0663 0.2165 -0.0017 0.1932 0.1237 -0.0233 0.12531.1 0.0000 0.0936 0.6475 0.0320 0.6475 -0.0616 0.3180 -0.0297 0.2941 0.1002 -0.0239 0.12981.2 0.0000 0.0961 0.6449 0.0395 0.6449 -0.0565 0.4150 -0.0564 0.3984 0.0753 -0.0167 0.13171.3 0.0000 0.0987 0.6423 0.0472 0.6423 -0.0515 0.5039 -0.0809 0.5014 0.0503 -0.0025 0.13111.4 0.0000 0.1013 0.6402 0.0547 0.6402 -0.0466 0.5815 -0.1022 0.5984 0.0263 0.0169 0.12851.5 0.0000 0.1036 0.6389 0.0614 0.6389 -0.0422 0.6458 -0.1199 0.6853 0.0044 0.0395 0.12431.7 0.0000 0.1072 0.6389 0.0721 0.6389 -0.0350 0.6961 -0.1338 0.7592 -0.0145 0.0631 0.11932.0 0.0000 0.1090 0.6411 0.0787 0.6411 -0.0302 0.7329 -0.1441 0.8187 -0.0301 0.0859 0.11402.2 0.0000 0.1094 0.6424 0.0806 0.6424 -0.0287 0.7459 -0.1477 0.8423 -0.0363 0.0964 0.11142.6 0.0000 0.1096 0.6434 0.0818 0.6434 -0.0277 0.7634 -0.1526 0.8753 -0.0451 0.1119 0.10753.0 0.0000 0.1096 0.6441 0.0825 0.6441 -0.0272 0.7733 -0.1554 0.8958 -0.0506 0.1224 0.1048

447

Table C2. Number of sites for each earthquake as a function of site condition andspectral period. The number of sites for periods lower than one second is equal to thenumber of sites for peak ground acceleration.

Northridge Loma PrietaT B C D B C D

PGA 11 70 59 18 26 181.0 11 70 59 18 26 181.1 11 70 58 18 26 181.2 11 70 58 18 26 181.3 11 70 58 18 26 181.4 11 69 58 18 26 181.5 11 69 58 18 26 181.6 11 69 58 18 26 181.7 11 67 58 18 26 181.8 11 67 58 18 26 181.9 11 67 58 18 26 182.0 11 67 58 18 26 182.2 11 65 57 18 26 182.4 11 65 57 18 26 182.6 11 65 57 18 26 182.8 10 48 41 18 26 183.0 10 47 41 18 26 18

448

Table C3. Weights used to combine spectral amplification ratios for the Northridge andLoma Prieta Earthquakes. Weights are inversely proportional to the variance of thesample mean.

Northridge Loma PrietaT b/c b/d c/d b/c b/d c/d

PGA 0.56 0.64 0.70 0.44 0.36 0.300.055 0.45 0.53 0.75 0.55 0.47 0.250.06 0.46 0.53 0.75 0.54 0.47 0.250.07 0.46 0.53 0.75 0.54 0.47 0.250.08 0.48 0.54 0.75 0.52 0.46 0.250.09 0.50 0.57 0.75 0.50 0.43 0.250.1 0.54 0.60 0.75 0.46 0.40 0.250.11 0.56 0.61 0.75 0.44 0.39 0.250.12 0.57 0.63 0.74 0.43 0.37 0.260.13 0.58 0.64 0.74 0.42 0.36 0.260.14 0.58 0.64 0.73 0.42 0.36 0.270.15 0.58 0.63 0.73 0.42 0.37 0.270.16 0.57 0.62 0.72 0.43 0.38 0.280.17 0.56 0.61 0.71 0.44 0.39 0.290.18 0.56 0.60 0.70 0.44 0.40 0.300.19 0.55 0.60 0.69 0.45 0.40 0.310.2 0.54 0.58 0.68 0.46 0.42 0.320.22 0.53 0.58 0.67 0.47 0.42 0.330.24 0.52 0.57 0.66 0.48 0.43 0.340.26 0.52 0.57 0.66 0.48 0.43 0.340.28 0.52 0.57 0.66 0.48 0.43 0.340.3 0.52 0.57 0.66 0.48 0.43 0.340.32 0.52 0.57 0.67 0.48 0.43 0.330.34 0.51 0.56 0.67 0.49 0.44 0.330.36 0.51 0.56 0.67 0.49 0.44 0.330.4 0.53 0.58 0.69 0.47 0.42 0.310.44 0.54 0.59 0.71 0.46 0.41 0.290.5 0.55 0.61 0.73 0.45 0.39 0.270.55 0.56 0.63 0.74 0.44 0.37 0.260.6 0.57 0.64 0.74 0.43 0.36 0.260.667 0.57 0.65 0.75 0.43 0.35 0.250.7 0.58 0.65 0.76 0.42 0.35 0.240.75 0.58 0.65 0.77 0.42 0.35 0.230.8 0.57 0.65 0.77 0.43 0.35 0.230.85 0.57 0.65 0.78 0.43 0.35 0.220.9 0.56 0.65 0.79 0.44 0.35 0.210.95 0.56 0.65 0.80 0.44 0.35 0.201.0 0.56 0.65 0.80 0.44 0.35 0.201.1 0.55 0.64 0.81 0.45 0.36 0.191.2 0.55 0.64 0.81 0.45 0.36 0.191.3 0.55 0.63 0.81 0.45 0.37 0.191.4 0.55 0.63 0.82 0.45 0.37 0.181.5 0.55 0.63 0.82 0.45 0.37 0.181.7 0.55 0.63 0.82 0.45 0.37 0.182.0 0.55 0.62 0.82 0.45 0.38 0.182.2 0.55 0.62 0.82 0.45 0.38 0.182.6 0.55 0.62 0.83 0.45 0.38 0.173.0 0.52 0.59 0.80 0.48 0.41 0.20

449

Table C4a. Regression coefficients and Standard Error for spectral acceleration values at5% damping for the Northridge Earthquake.

B Sites C Sites D SitesT a b c σσσσ a b c σσσσ a b c σσσσ

PGA 2.3718 -1.2753 6.3883 0.3209 2.3718 -1.1538 6.3883 0.4686 2.6916 -1.2161 6.3883 0.35590.055 3.5192 -1.4829 10.2486 0.4343 3.5192 -1.3869 10.2486 0.4661 3.5126 -1.3703 10.2486 0.35600.06 3.7423 -1.5138 11.8103 0.4343 3.7423 -1.4266 11.8103 0.4655 3.7970 -1.4257 11.8103 0.36540.07 4.3982 -1.6291 14.5768 0.4310 4.3982 -1.5480 14.5768 0.4636 4.4475 -1.5472 14.5768 0.37050.08 4.8097 -1.7006 16.9734 0.4180 4.8097 -1.6152 16.9734 0.4619 4.9774 -1.6422 16.9734 0.37540.09 4.9993 -1.7175 18.0000 0.3935 4.9993 -1.6366 18.0000 0.4617 5.2637 -1.6826 18.0000 0.37790.1 4.9768 -1.6855 18.0000 0.3615 4.9768 -1.6089 18.0000 0.4642 5.3000 -1.6679 18.0000 0.37740.11 4.9365 -1.6614 18.0000 0.3457 4.9365 -1.5844 18.0000 0.4667 5.2529 -1.6439 18.0000 0.37660.12 4.8748 -1.6330 18.0000 0.3322 4.8748 -1.5530 18.0000 0.4703 5.1563 -1.6072 18.0000 0.37590.13 4.7753 -1.5991 18.0000 0.3226 4.7753 -1.5140 18.0000 0.4750 5.0044 -1.5586 18.0000 0.37580.14 4.6161 -1.5564 17.3303 0.3179 4.6161 -1.4646 17.3303 0.4808 4.7947 -1.4991 17.3303 0.37660.15 4.3937 -1.5041 16.0757 0.3182 4.3937 -1.4037 16.0757 0.4877 4.5454 -1.4330 16.0757 0.37860.16 4.1376 -1.4471 14.9021 0.3232 4.1376 -1.3364 14.9021 0.4952 4.2958 -1.3685 14.9021 0.38200.17 3.8807 -1.3907 13.7997 0.3315 3.8807 -1.2694 13.7997 0.5030 4.0778 -1.3133 13.7997 0.38650.18 3.7555 -1.3635 12.7603 0.3368 3.7555 -1.2373 12.7603 0.5069 3.9820 -1.2900 12.7603 0.38930.19 3.6370 -1.3378 11.7771 0.3418 3.6370 -1.2069 11.7771 0.5105 3.8913 -1.2680 11.7771 0.39180.2 3.4048 -1.2891 10.8444 0.3531 3.4048 -1.1508 10.8444 0.5174 3.7044 -1.2249 10.8444 0.39740.22 3.1681 -1.2413 9.1112 0.3646 3.1681 -1.0982 9.1112 0.5234 3.4809 -1.1745 9.1112 0.40260.24 2.9146 -1.1904 7.5290 0.3759 2.9146 -1.0449 7.5290 0.5285 3.2196 -1.1160 7.5290 0.40710.26 2.7904 -1.1657 6.6312 0.3818 2.7904 -1.0198 6.6312 0.5308 3.0913 -1.0874 6.6312 0.40890.28 2.6754 -1.1429 5.8000 0.3872 2.6754 -0.9965 5.8000 0.5330 2.9725 -1.0610 5.8000 0.41060.3 2.5178 -1.1149 4.9000 0.3983 2.5178 -0.9682 4.9000 0.5372 2.8087 -1.0250 4.9000 0.41290.32 2.4644 -1.1117 4.4939 0.4087 2.4644 -0.9657 4.4939 0.5415 2.7420 -1.0110 4.4939 0.41410.34 2.4645 -1.1197 4.4254 0.4176 2.4645 -0.9768 4.4254 0.5463 2.7212 -1.0067 4.4254 0.41450.36 2.4594 -1.1242 4.3606 0.4242 2.4594 -0.9870 4.3606 0.5515 2.6916 -0.9999 4.3606 0.41420.4 2.4375 -1.1239 4.2415 0.4276 2.4375 -0.9935 4.2415 0.5570 2.6466 -0.9915 4.2415 0.41330.44 2.4279 -1.1279 4.1337 0.4277 2.4279 -1.0049 4.1337 0.5627 2.6269 -0.9946 4.1337 0.41190.5 2.4692 -1.1545 3.9890 0.4198 2.4692 -1.0526 3.9890 0.5739 2.7651 -1.0629 3.9890 0.40660.55 2.4447 -1.1582 3.8812 0.4140 2.4447 -1.0682 3.8812 0.5792 2.8613 -1.1091 3.8812 0.40230.6 2.3687 -1.1540 3.7828 0.4090 2.3687 -1.0710 3.7828 0.5843 2.9263 -1.1469 3.7828 0.39680.667 2.2699 -1.1513 3.6630 0.4060 2.2699 -1.0675 3.6630 0.5892 2.9650 -1.1752 3.6630 0.39010.7 2.1804 -1.1550 3.6084 0.4059 2.1804 -1.0660 3.6084 0.5937 2.9956 -1.1995 3.6084 0.38260.75 2.1276 -1.1664 3.5303 0.4090 2.1276 -1.0746 3.5303 0.5977 3.0096 -1.2199 3.5303 0.37500.8 2.1239 -1.1848 3.4573 0.4151 2.1239 -1.0966 3.4573 0.6009 2.9754 -1.2294 3.4573 0.36800.85 2.1516 -1.2064 3.3887 0.4235 2.1516 -1.1267 3.3887 0.6030 2.8866 -1.2261 3.3887 0.36210.9 2.1703 -1.2244 3.4413 0.4332 2.1703 -1.1539 3.4413 0.6041 2.7784 -1.2185 3.4413 0.35790.95 2.1451 -1.2353 3.2629 0.4435 2.1451 -1.1701 3.2629 0.6041 2.6965 -1.2187 3.2629 0.35561.0 2.0734 -1.2443 3.2048 0.4538 2.0734 -1.1775 3.2048 0.6033 2.6601 -1.2333 3.2048 0.35511.1 1.9888 -1.2635 3.0970 0.4637 1.9888 -1.1873 3.0970 0.6017 2.6461 -1.2583 3.0970 0.35631.2 1.9252 -1.2983 2.9986 0.4726 1.9252 -1.2071 2.9986 0.5995 2.6099 -1.2804 2.9986 0.35871.3 1.8811 -1.3390 2.9080 0.4799 1.8811 -1.2317 2.9080 0.5962 2.5295 -1.2884 2.9080 0.36181.4 1.8327 -1.3706 2.8242 0.4799 1.8327 -1.2510 2.8242 0.5909 2.4272 -1.2846 2.8242 0.36491.5 1.7582 -1.3853 2.7461 0.4799 1.7582 -1.2588 2.7461 0.5850 2.3331 -1.2785 2.7461 0.36641.7 1.5420 -1.3800 2.6045 0.4799 1.5420 -1.2565 2.6045 0.5800 2.1862 -1.2817 2.6045 0.38112.0 1.3896 -1.3970 2.4206 0.4799 1.3896 -1.2933 2.4206 0.5700 2.0500 -1.3154 2.4206 0.41302.2 1.2440 -1.3983 2.3128 0.4799 1.2440 -1.3004 2.3128 0.5600 1.8906 -1.3182 2.3128 0.42442.6 0.9829 -1.3739 2.1238 0.4799 0.9829 -1.2719 2.1238 0.5400 1.6293 -1.2941 2.1238 0.41453.0 0.6859 -1.3338 2.0000 0.4799 0.6859 -1.2207 2.0000 0.5200 1.3413 -1.2536 2.0000 0.3877

450

Table C4b. Regression coefficients and Standard Error for spectral acceleration values at5% damping for the Loma Prieta Earthquake.

B Sites C Sites D SitesT a b c σσσσ a b c σσσσ a b c σσσσ

PGA 0.7219 -0.7954 1.0000 0.4713 0.8212 -0.7502 1.0000 0.3111 0.5716 -0.6032 1.0000 0.38960.055 1.6308 -0.9794 1.0000 0.4566 1.4230 -0.8769 1.0000 0.3708 1.3201 -0.7767 1.0000 0.43340.06 1.8207 -1.0119 1.0000 0.4561 1.4804 -0.8841 1.0000 0.3747 1.2568 -0.7489 1.0000 0.43380.07 1.9001 -1.0181 1.0000 0.4554 1.4819 -0.8734 1.0000 0.3798 1.2413 -0.7315 1.0000 0.43400.08 2.0559 -1.0383 1.0000 0.4538 1.5348 -0.8701 1.0000 0.3886 1.3041 -0.7271 1.0000 0.43310.09 2.1619 -1.0489 1.0000 0.4518 1.5875 -0.8642 1.0000 0.3973 1.4037 -0.7300 1.0000 0.43030.1 2.2305 -1.0551 1.0000 0.4500 1.6419 -0.8595 1.0000 0.4027 1.5122 -0.7400 1.0000 0.42690.11 2.2946 -1.0607 1.0000 0.4481 1.7031 -0.8527 1.0000 0.4074 1.6341 -0.7524 1.0000 0.42200.12 2.3215 -1.0625 1.0000 0.4472 1.7361 -0.8492 1.0000 0.4091 1.6890 -0.7575 1.0000 0.41870.13 2.3462 -1.0642 1.0000 0.4464 1.7665 -0.8461 1.0000 0.4108 1.7395 -0.7622 1.0000 0.41570.14 2.3659 -1.0613 1.0000 0.4451 1.8339 -0.8450 1.0000 0.4124 1.7916 -0.7621 1.0000 0.40840.15 2.3410 -1.0484 1.0000 0.4448 1.9079 -0.8523 1.0000 0.4120 1.7724 -0.7458 1.0000 0.40040.16 2.2804 -1.0268 1.0000 0.4460 1.9696 -0.8621 1.0000 0.4095 1.7156 -0.7191 1.0000 0.39240.17 2.2370 -1.0125 1.0000 0.4476 1.9792 -0.8631 1.0000 0.4071 1.6926 -0.7066 1.0000 0.38880.18 2.1960 -0.9991 1.0000 0.4491 1.9882 -0.8640 1.0000 0.4049 1.6710 -0.6949 1.0000 0.38530.19 2.0939 -0.9675 1.0000 0.4545 1.9513 -0.8531 1.0000 0.3989 1.6405 -0.6754 1.0000 0.37970.2 1.9861 -0.9352 1.0000 0.4626 1.8633 -0.8291 1.0000 0.3923 1.5961 -0.6551 1.0000 0.37610.22 1.8879 -0.9051 1.0000 0.4731 1.7414 -0.7944 1.0000 0.3861 1.5361 -0.6348 1.0000 0.37480.24 1.8523 -0.8933 1.0000 0.4797 1.6772 -0.7749 1.0000 0.3837 1.5154 -0.6295 1.0000 0.37540.26 1.8196 -0.8825 1.0000 0.4858 1.6182 -0.7570 1.0000 0.3815 1.4963 -0.6245 1.0000 0.37590.28 1.8136 -0.8775 1.0000 0.5001 1.5268 -0.7272 1.0000 0.3796 1.5140 -0.6328 1.0000 0.37960.3 1.8860 -0.8959 1.0000 0.5149 1.4800 -0.7104 1.0000 0.3812 1.5933 -0.6598 1.0000 0.38590.32 1.9446 -0.9120 1.0000 0.5223 1.4703 -0.7068 1.0000 0.3841 1.6498 -0.6788 1.0000 0.39060.34 1.9996 -0.9271 1.0000 0.5292 1.4613 -0.7033 1.0000 0.3868 1.7028 -0.6968 1.0000 0.39500.36 2.0373 -0.9373 1.0000 0.5358 1.4510 -0.7011 1.0000 0.3916 1.7453 -0.7128 1.0000 0.40120.4 2.0412 -0.9393 1.0000 0.5524 1.3972 -0.6925 1.0000 0.4092 1.7829 -0.7364 1.0000 0.42190.44 1.8966 -0.9057 1.0000 0.5600 1.3081 -0.6785 1.0000 0.4251 1.6708 -0.7148 1.0000 0.43960.5 1.5766 -0.8357 1.0000 0.5658 1.0905 -0.6402 1.0000 0.4486 1.3791 -0.6481 1.0000 0.46590.55 1.3683 -0.7909 1.0000 0.5678 0.9405 -0.6134 1.0000 0.4616 1.1859 -0.6031 1.0000 0.48080.6 1.2193 -0.7593 1.0000 0.5685 0.8299 -0.5944 1.0000 0.4699 1.0459 -0.5707 1.0000 0.49060.667 1.0380 -0.7209 1.0000 0.5694 0.6953 -0.5713 1.0000 0.4799 0.8757 -0.5314 1.0000 0.50250.7 0.9158 -0.6959 1.0000 0.5700 0.5954 -0.5543 1.0000 0.4867 0.7392 -0.4998 1.0000 0.51120.75 0.7412 -0.6602 1.0000 0.5708 0.4527 -0.5302 1.0000 0.4965 0.5444 -0.4547 1.0000 0.52350.8 0.6212 -0.6371 1.0000 0.5719 0.3418 -0.5106 1.0000 0.5038 0.3623 -0.4116 1.0000 0.53350.85 0.5083 -0.6155 1.0000 0.5728 0.2376 -0.4923 1.0000 0.5106 0.1913 -0.3712 1.0000 0.54280.9 0.2964 -0.5761 1.0000 0.5760 0.0693 -0.4630 1.0000 0.5215 -0.1385 -0.2932 1.0000 0.55980.95 0.0614 -0.5335 1.0000 0.5803 -0.0415 -0.4494 1.0000 0.5296 -0.3583 -0.2461 1.0000 0.57391.0 -0.1915 -0.4913 1.0000 0.5854 -0.0967 -0.4555 1.0000 0.5354 -0.4193 -0.2456 1.0000 0.58521.1 -0.4301 -0.4563 1.0000 0.5904 -0.1041 -0.4806 1.0000 0.5401 -0.3485 -0.2856 1.0000 0.59361.2 -0.6336 -0.4304 1.0000 0.5941 -0.0738 -0.5215 1.0000 0.5450 -0.2165 -0.3463 1.0000 0.59961.3 -0.8156 -0.4103 1.0000 0.5953 -0.0320 -0.5691 1.0000 0.5511 -0.0920 -0.4091 1.0000 0.60351.4 -1.0118 -0.3912 1.0000 0.5931 -0.0357 -0.6071 1.0000 0.5593 -0.0276 -0.4612 1.0000 0.60631.5 -1.2503 -0.3703 1.0000 0.5874 -0.1493 -0.6191 1.0000 0.5697 -0.0722 -0.4911 1.0000 0.60871.7 -1.5259 -0.3501 1.0000 0.5785 -0.3975 -0.6017 1.0000 0.5819 -0.2535 -0.4925 1.0000 0.61172.0 -1.7950 -0.3397 1.0000 0.5674 -0.7453 -0.5663 1.0000 0.5950 -0.5395 -0.4756 1.0000 0.61602.2 -1.9108 -0.3426 1.0000 0.5611 -0.9419 -0.5467 1.0000 0.6018 -0.7079 -0.4662 1.0000 0.61872.6 -2.0796 -0.3504 1.0000 0.5508 -1.2418 -0.5188 1.0000 0.6119 -0.9767 -0.4513 1.0000 0.62333.0 -2.1924 -0.3596 1.0000 0.5428 -1.4567 -0.5011 1.0000 0.6189 -1.1824 -0.4400 1.0000 0.62683.4 -2.3459 -0.3686 1.0000 0.5302 -1.7104 -0.4873 1.0000 0.6263 -1.5020 -0.4122 1.0000 0.63104.0 -2.4736 -0.3683 1.0000 0.5170 -1.8745 -0.4834 1.0000 0.6284 -1.7876 -0.3769 1.0000 0.6319

451

Table C5. Weights used to combine standard deviations for the Northridge and LomaPrieta Earthquakes. Weights are inversely proportional to the variance of the samplevariance.

Northridge Loma PrietaT B C D B C D

PGA 0.75 0.34 0.82 0.25 0.66 0.180.055 0.44 0.51 0.87 0.56 0.49 0.130.06 0.44 0.52 0.86 0.56 0.48 0.140.07 0.44 0.54 0.86 0.56 0.46 0.140.08 0.47 0.57 0.85 0.53 0.43 0.150.09 0.52 0.59 0.84 0.48 0.41 0.160.1 0.60 0.60 0.84 0.40 0.40 0.160.11 0.64 0.60 0.83 0.36 0.40 0.170.12 0.68 0.60 0.83 0.32 0.40 0.170.13 0.70 0.59 0.83 0.30 0.41 0.170.14 0.71 0.59 0.81 0.29 0.41 0.190.15 0.71 0.57 0.80 0.29 0.43 0.200.16 0.70 0.55 0.78 0.30 0.45 0.220.17 0.68 0.53 0.76 0.32 0.47 0.240.18 0.67 0.52 0.75 0.33 0.48 0.250.19 0.66 0.49 0.74 0.34 0.51 0.260.2 0.65 0.46 0.72 0.35 0.54 0.280.22 0.64 0.44 0.70 0.36 0.56 0.300.24 0.63 0.42 0.69 0.37 0.58 0.310.26 0.62 0.41 0.69 0.38 0.59 0.310.28 0.64 0.40 0.70 0.36 0.60 0.300.3 0.64 0.40 0.71 0.36 0.60 0.290.32 0.63 0.40 0.71 0.37 0.60 0.290.34 0.62 0.40 0.72 0.38 0.60 0.280.36 0.62 0.40 0.73 0.38 0.60 0.270.4 0.64 0.43 0.77 0.36 0.57 0.230.44 0.65 0.46 0.80 0.35 0.54 0.200.5 0.68 0.50 0.84 0.32 0.50 0.160.55 0.69 0.51 0.87 0.31 0.49 0.130.6 0.70 0.52 0.88 0.30 0.48 0.120.667 0.71 0.54 0.90 0.29 0.46 0.100.7 0.71 0.54 0.91 0.29 0.46 0.090.75 0.71 0.56 0.92 0.29 0.44 0.080.8 0.70 0.56 0.93 0.30 0.44 0.070.85 0.68 0.57 0.94 0.32 0.43 0.060.9 0.66 0.59 0.95 0.34 0.41 0.050.95 0.65 0.61 0.96 0.35 0.39 0.041.0 0.64 0.62 0.96 0.36 0.38 0.041.1 0.63 0.63 0.96 0.37 0.37 0.041.2 0.61 0.64 0.96 0.39 0.36 0.041.3 0.60 0.66 0.96 0.40 0.34 0.041.4 0.60 0.68 0.96 0.40 0.32 0.041.5 0.59 0.70 0.96 0.41 0.30 0.041.7 0.57 0.72 0.95 0.43 0.28 0.052.0 0.55 0.75 0.94 0.45 0.25 0.062.2 0.54 0.77 0.93 0.46 0.23 0.072.6 0.52 0.80 0.94 0.48 0.20 0.063.0 0.49 0.78 0.94 0.51 0.22 0.06

Documents

Near-Fault Seismic Site Response